Author / David D. Nolte

Posts by David D. Nolte

E. M. Purcell Distinguished Professor of Physics and Astronomy at Purdue University
by David D. Nolte

Climate Change Physics 101

When our son was ten years old, he came home from a town fair in Battleground, Indiana, with an unwanted pet—a goldfish in a plastic bag.  The family rushed out to buy a fish bowl and food and plopped the golden-red animal into it.  In three days, it was dead!

It turns out that you can’t just put a gold fish in a fish bowl.  When it metabolizes its food and expels its waste, it builds up toxic levels of ammonia unless you add filters or plants or treat the water with chemicals.  In the end, the goldfish died because it was asphyxiated by its own pee.

It’s a basic rule—don’t pee in your own fish bowl.

The same can be said for humans living on the surface of our planet.  Polluting the atmosphere with our wastes cannot be a good idea.  In the end it will kill us.  The atmosphere may look vast—the fish bowl was a big one—but it is shocking how thin it is.

Turn on your Apple TV, click on the screen saver, and you are skimming over our planet on the dark side of the Earth. Then you see a thin blue line extending over the limb of the dark disc.  Hold!  That thin blue line!  That is our atmosphere! Is it really so thin?

When you look upwards on a clear sunny day, the atmosphere seems like it goes on forever.  It doesn’t.  It is a thin veneer on the surface of the Earth barely one percent of the Earth’s radius.  The Earth’s atmosphere is frighteningly thin. 

Fig. 1  A thin veneer of atmosphere paints the surface of the Earth.  The radius of the Earth is 6360 km, and the thickness of the atmosphere is 100 km, which is a bit above 1 percent of the radius.

Consider Mars.  It’s half the size of Earth, yet it cannot hold on to an atmosphere even 1/100th the thickness of ours.  When Mars first formed, it had an atmosphere not unlike our own, but through the eons its atmosphere has wafted away irretrievably into space.

An atmosphere is a precious fragile thing for a planet.  It gives life and it gives protection.  It separates us from the deathly cold of space, holding heat like a blanket.  That heat has served us well over the eons, allowing water to stay liquid and allowing life to arise on Earth.  But too much of a good thing is not a good thing.

Common Sense

If the fluid you are bathed in gives you life, then don’t mess with it.  Don’t run your car in the garage while you are working in it.  Don’t use a charcoal stove in an enclosed space.  Don’t dump carbon dioxide into the atmosphere because it also is an enclosed space.

At the end of winter, as the warm spring days get warmer, you take the winter blanket off your bed because blankets hold in heat.  The thicker the blanket, the more heat it holds in.  Common sense tells you to reduce the thickness of the blanket if you don’t want to get too warm.  Carbon dioxide in the atmosphere acts like a blanket.  If we don’t want the Earth to get too warm, then we need to limit the thickness of the blanket.

Without getting into the details of any climate change model, common sense already tells us what we should do.  Keep the atmosphere clean and stable (Don’t’ pee in our fishbowl) and limit the amount of carbon dioxide we put into it (Don’t let the blanket get too thick).

Some Atmospheric Facts

Here are some facts about the atmosphere, about the effect humans have on it, and about the climate:

Fact 1.  Humans have increased the amount of carbon dioxide in the atmosphere by 45% since 1850 (the beginning of the industrial age) and by 30% since just 1960.

Fact 2.  Carbon dioxide in the atmosphere prevents some of the heat absorbed from the Sun to re-radiate out to space.  More carbon dioxide stores more heat.

Fact 3.  Heat added to the Earth’s atmosphere increases its temperature.  This is a law of physics.

Fact 4.  The Earth’s average temperature has risen by 1.2 degrees Celsius since 1850 and 0.8 degrees of that has been just since 1960, so the effect is accelerating.

These facts are indisputable.  They hold true regardless of whether there is a Republican or a Democrat in the White House or in control of Congress.

There is another interesting observation which is not so direct, but may hold a harbinger for the distant future: The last time the Earth was 3 degrees Celsius warmer than it is today was during the Pliocene when the sea level was tens of meters higher.  If that sea level were to occur today, all of Delaware, most of Florida, half of Louisiana and the entire east coast of the US would be under water, including Houston, Miami, New Orleans, Philadelphia and New York City.  There are many reasons why this may not be an immediate worry. The distribution of water and ice now is different than in the Pliocene, and the effect of warming on the ice sheets and water levels could take centuries. Within this century, the amount of sea level rise is likely to be only about 1 meter, but accelerating after that.

Fig. 2  The east coast of the USA for a sea level 30 meters higher than today.  All of Delaware, half of Louisiana, and most of Florida are under water. Reasonable projections show only a 1 meter sea level rise by 2100, but accelerating after that. From https://www.youtube.com/watch?v=G2x1bonLJFA

Balance and Feedback

It is relatively easy to create a “rule-of-thumb” model for the Earth’s climate (see Ref. [2]).  This model is not accurate, but it qualitatively captures the basic effects of climate change and is a good way to get an intuitive feeling for how the Earth responds to changes, like changes in CO2 or to the amount of ice cover.  It can also provide semi-quantitative results, so that relative importance of various processes or perturbations can be understood.

The model is a simple energy balance statement:  In equilibrium, as much energy flows into the Earth system as out.

This statement is both simple and immediately understandable.  But then the work starts as we need to pin down how much energy is flowing in and how much is flowing out.  The energy flowing in comes from the sun, and the energy flowing out comes from thermal radiation into space. 

We also need to separate the Earth system into two components: the surface and the atmosphere.  These are two very different things that have two different average temperatures.  In addition, the atmosphere transmits sunlight to the surface, unless clouds reflect it back into space.  And the Earth radiates thermally into space, unless clouds or carbon dioxide layers reflect it back to the surface.

The energy fluxes are shown in the diagram in Fig. 3 for the 4-component system: Sun, Surface, Atmosphere, and Space. The light from the sun, mostly in the visible range of the spectrum, is partially absorbed by the atmosphere and partially transmitted and reflected. The transmitted portion is partially absorbed and partially reflected by the surface. The heat of the Earth is radiated at long wavelengths to the atmosphere, where it is partially transmitted out into space, but also partially reflected by the fraction a’a which is the blanket effect. In addition, the atmosphere itself radiates in equal parts to the surface and into outer space. On top of all of these radiative processes, there is also non-radiative convective interaction between the atmosphere and the surface.

Fig. 3 Energy flux model for a simple climate model with four interacting systems: the Sun, the Atmosphere, the Earth and Outer Space.

These processes are captured by two energy flux equations, one for the atmosphere and one for the surface, in Fig. 4. The individual contributions from Fig. 3 are annotated in each case. In equilibrium, each flux equals zero, which can then be used to solve for the two unknowns: Ts0 and Ta0: the surface and atmosphere temperatures.

Fig. 4 Energy-balance model of the Earth’s atmosphere for a simple climate approximation.

After the equilibrium temperatures Ts0 and Ta0 are found, they go into a set of dynamic response equations that governs how deviations in the temperatures relax back to the equilibrium values. These relaxation equations are

where ks and ka are the relaxation rates for the surface and atmosphere. These can be quite slow, in the range of a century. For illustration, we can take ks = 1/75 years and ka = 1/25 years. The equilibrium temperatures for the surface and atmosphere differ by about 50 degrees Celsius, with Ts = 289 K and Ta = 248 K. These are rough averages over the entire planet. The solar constant is S = 1.36×103 W/m2, the Stefan-Boltzman constant is σ = 5.67×10-8 W/m2/K4, and the convective interaction constant is c = 2.5 W m-2 K-1. Other parameters are given in Table I.

Short WavelengthLong Wavelength
as = 0.11
ts = 0.53t’a = 0.06
aa = 0.30a’a = 0.31

The relaxation equations are in the standard form of a mathematical “flow” (see Ref. [1]) and the solutions are plotted as a phase-space portrait in Fig. 5 as a video of the flow as the parameters in Table I shift because of the addition of greenhouse gases to the atmosphere. The video runs from the year 1850 (the dawn of the industrial age) through to the year 2060 about 40 years from now.

Fig. 5 Video of the phase space flow of the Surface-Atmosphere system for increasing year. The flow vectors and flow lines are the relaxation to equilibrium for temperature deviations. The change in equilibrium over the years is from increasing blanket effects in the atmosphere caused by greenhouse gases.

The scariest part of the video is how fast it accelerates. From 1850 to 1950 there is almost no change, but then it accelerates, faster and faster, reflecting the time-lag in temperature rise in response to increased greenhouse gases.

What if the Models are Wrong?  Russian Roulette

Now come the caveats.

This model is just for teaching purposes, not for any realistic modeling of climate change. It captures the basic physics, and it provides a semi-quantitative set of parameters that leads to roughly accurate current temperatures. But of course, the biggest elephant in the room is that it averages over the entire planet, which is a very crude approximation.

It does get the basic facts correct, though, showing an alarming trend in the rise in average temperatures with the temperature rising by 3 degrees by 2060.

The professionals in this business have computer models that are orders of magnitude more more accurate than this one. To understand the details of the real climate models, one needs to go to appropriate resources, like this NOAA link, this NASA link, this national climate assessment link, and this government portal link, among many others.

One of the frequent questions that is asked is: What if these models are wrong? What if global warming isn’t as bad as these models say? The answer is simple: If they are wrong, then the worst case is that life goes on. If they are right, then in the worst case life on this planet may end.

It’s like playing Russian Roulette. If just one of the cylinders on the revolver has a live bullet, do you want to pull the trigger?

Matlab Code

function flowatmos.m

mov_flag = 1;
if mov_flag == 1
    moviename = 'atmostmp';
    aviobj = VideoWriter(moviename,'MPEG-4');
    aviobj.FrameRate = 12;
    open(aviobj);
end

Solar = 1.36e3;		% Solar constant outside atmosphere [J/sec/m2]
sig = 5.67e-8;		% Stefan-Boltzman constant [W/m2/K4]

% 1st-order model of Earth + Atmosphere

ta = 0.53;			% (0.53)transmissivity of air
tpa0 = 0.06;			% (0.06)primes are for thermal radiation
as0 = 0.11;			% (0.11)
aa0 = 0.30;			% (0.30)
apa0 = 0.31;        % (0.31)
c = 2.5;               % W/m2/K

xrange = [287 293];
yrange = [247 251];

rngx = xrange(2) - xrange(1);
rngy = yrange(2) - yrange(1);

[X,Y] = meshgrid(xrange(1):0.05:xrange(2), yrange(1):0.05:yrange(2));

smallarrow = 1;
Delta0 = 0.0000009;
for tloop =1:80
    
    Delta = Delta0*(exp((tloop-1)/8)-1);   % This Delta is exponential, but should become more linear over time
    date = floor(1850 + (tloop-1)*(2060-1850)/79);
    
    [x,y] = f5(X,Y);
    
    clf
    hold off
    eps = 0.002;
    for xloop = 1:11
        xs = xrange(1) +(xloop-1)*rngx/10 + eps;
        for yloop = 1:11
            ys = yrange(1) +(yloop-1)*rngy/10 + eps;
            
            streamline(X,Y,x,y,xs,ys)
            
        end
    end
    hold on
    [XQ,YQ] = meshgrid(xrange(1):1:xrange(2),yrange(1):1:yrange(2));
    smallarrow = 1;
    [xq,yq] = f5(XQ,YQ);
    quiver(XQ,YQ,xq,yq,.2,'r','filled')
    hold off
    
    axis([xrange(1) xrange(2) yrange(1) yrange(2)])
    set(gcf,'Color','White')
    
    fun = @root2d;
    x0 = [0 -40];
    x = fsolve(fun,x0);
    
    Ts = x(1) + 288
    Ta = x(2) + 288
    
    hold on
    rectangle('Position',[Ts-0.05 Ta-0.05 0.1 0.1],'Curvature',[1 1],'FaceColor',[1 0 0],'EdgeColor','k','LineWidth',2)
    
    posTs(tloop) = Ts;
    posTa(tloop) = Ta;
    
    plot(posTs,posTa,'k','LineWidth',2);
    hold off
    
    text(287.5,250.5,strcat('Date = ',num2str(date)),'FontSize',24)
    box on
    xlabel('Surface Temperature (oC)','FontSize',24)
    ylabel('Atmosphere Temperature (oC)','FontSize',24)
    
    hh = figure(1);
    pause(0.01)
    if mov_flag == 1
        frame = getframe(hh);
        writeVideo(aviobj,frame);
    end
    
end     % end tloop

if mov_flag == 1
    close(aviobj);
end

    function F = root2d(xp)   % Energy fluxes 
        
        x = xp + 288;
        feedfac = 0.001;      % feedback parameter 
        
        apa = apa0 + feedfac*(x(2)-248) + Delta;  % Changes in the atmospheric blanket
        tpa = tpa0 - feedfac*(x(2)-248) - Delta;
        as = as0 - feedfac*(x(1)-289);
        
        F(1) = c*(x(1)-x(2)) + sig*(1-apa)*x(1).^4 - sig*x(2).^4 - ta*(1-as)*Solar/4;
        F(2) = c*(x(1)-x(2)) + sig*(1-tpa - apa)*x(1).^4 - 2*sig*x(2).^4 + (1-aa0-ta+as*ta)*Solar/4;
        
    end

    function [x,y] = f5(X,Y)   % Dynamical flow equations
        
        k1 = 1/75;   % 75 year time constant for the Earth
        k2 = 1/25;   % 25 year time constant for the Atmosphere
        
        fun = @root2d;
        x0 = [0 0];
        x = fsolve(fun,x0);   % Solve for the temperatures that set the energy fluxes to zero
        
        Ts0 = x(1) + 288;   % Surface temperature in Kelvin
        Ta0 = x(2) + 288;   % Atmosphere temperature in Kelvin
        
        xtmp = -k1*(X - Ts0);   % Dynamical equations
        ytmp = -k2*(Y - Ta0);
        
        nrm = sqrt(xtmp.^2 + ytmp.^2);
        
        if smallarrow == 1
            x = xtmp./nrm;
            y = ytmp./nrm;
        else
            x = xtmp;
            y = ytmp;
        end
        
    end     % end f5

end       % end flowatmos

This model has a lot of parameters that can be tweaked. In addition to the parameters in the Table, the time dependence on the blanket properties of the atmosphere are governed by Delta0 and by feedfac for feedback of temperature on the atmosphere, such as increasing cloud cover and decrease ice cover. As an exercise, and using only small changes in the given parameters, find the following cases: 1) An increasing surface temperature is moderated by a falling atmosphere temperature; 2) The Earth goes into thermal run-away and ends like Venus; 3) The Earth initially warms then plummets into an ice age.

By David D. Nolte Oct. 16, 2022

References

[1] D. D. Nolte, Introduction to Modern Dynamics: Chaos, Networks, Space and Time, 2nd Ed. (Oxford University Press, 2019)

[2] E. Boeker and R. van Grondelle, Environmental Physics (Wiley, 1995)

[3] Recent lecture at the National Academy of Engineering by John Holdren.

by David D. Nolte

Is There a Quantum Trajectory? The Phase-Space Perspective

At the dawn of quantum theory, Heisenberg, Schrödinger, Bohr and Pauli were embroiled in a dispute over whether trajectories of particles, defined by their positions over time, could exist. The argument against trajectories was based on an apparent paradox: To draw a “line” depicting a trajectory of a particle along a path implies that there is a momentum vector that carries the particle along that path. But a line is a one-dimensional curve through space, and since at any point in time the particle’s position is perfectly localized, then by Heisenberg’s uncertainty principle, it can have no definable momentum to carry it along.

My previous blog shows the way out of this paradox, by assembling wavepackets that are spread in both space and momentum, explicitly obeying the uncertainty principle. This is nothing new to anyone who has taken a quantum course. But the surprising thing is that in some potentials, like a harmonic potential, the wavepacket travels without broadening, just like classical particles on a trajectory. A dramatic demonstration of this can be seen in this YouTube video. But other potentials “break up” the wavepacket, especially potentials that display classical chaos. Because phase space is one of the best tools for studying classical chaos, especially Hamiltonian chaos, it can be enlisted to dig deeper into the question of the quantum trajectory—not just about the existence of a quantum trajectory, but why quantum systems retain a shadow of their classical counterparts.

Phase Space

Phase space is the state space of Hamiltonian systems. Concepts of phase space were first developed by Boltzmann as he worked on the problem of statistical mechanics. Phase space was later codified by Gibbs for statistical mechanics and by Poincare for orbital mechanics, and it was finally given its name by Paul and Tatiana Ehrenfest (a husband-wife team) in correspondence with the German physicist Paul Hertz (See Chapter 6, “The Tangled Tale of Phase Space”, in Galileo Unbound by D. D. Nolte (Oxford, 2018)).

The stretched-out phase-space functions … are very similar to the stochastic layer that forms in separatrix chaos in classical systems.

The idea of phase space is very simple for classical systems: it is just a plot of the momentum of a particle as a function of its position. For a given initial condition, the trajectory of a particle through its natural configuration space (for instance our 3D world) is traced out as a path through phase space. Because there is one momentum variable per degree of freedom, then the dimensionality of phase space for a particle in 3D is 6D, which is difficult to visualize. But for a one-dimensional dynamical system, like a simple harmonic oscillator (SHO) oscillating in a line, the phase space is just two-dimensional, which is easy to see. The phase-space trajectories of an SHO are simply ellipses, and if the momentum axis is scaled appropriately, the trajectories are circles. The particle trajectory in phase space can be animated just like a trajectory through configuration space as the position and momentum change in time p(x(t)). For the SHO, the point follows the path of a circle going clockwise.

Fig. 1 Phase space of the simple harmonic oscillator. The “orbits” have constant energy.

A more interesting phase space is for the simple pendulum, shown in Fig. 2. There are two types of orbits: open and closed. The closed orbits near the origin are like those of a SHO. The open orbits are when the pendulum is spinning around. The dividing line between the open and closed orbits is called a separatrix. Where the separatrix intersects itself is a saddle point. This saddle point is the most important part of the phase space portrait: it is where chaos emerges when perturbations are added.

Fig. 2 Phase space for a simple pendulum. For small amplitudes the orbits are closed like those of a SHO. For large amplitudes the orbits become open as the pendulum spins about its axis. (Reproduced from Introduction to Modern Dynamics, 2nd Ed., pg. )

One route to classical chaos is through what is known as “separatrix chaos”. It is easy to see why saddle points (also known as hyperbolic points) are the source of chaos: as the system trajectory approaches the saddle, it has two options of which directions to go. Any additional degree of freedom in the system (like a harmonic drive) can make the system go one way on one approach, and the other way on another approach, mixing up the trajectories. An example of the stochastic layer of separatrix chaos is shown in Fig. 3 for a damped driven pendulum. The chaotic behavior that originates at the saddle point extends out along the entire separatrix.

Fig. 3 The stochastic layer of separatrix chaos for a damped driven pendulum. (Reproduced from Introduction to Modern Dynamics, 2nd Ed., pg. )

The main question about whether or not there is a quantum trajectory depends on how quantum packets behave as they approach a saddle point in phase space. Since packets are spread out, it would be reasonable to assume that parts of the packet will go one way, and parts of the packet will go another. But first, one has to ask: Is a phase-space description of quantum systems even possible?

Quantum Phase Space: The Wigner Distribution Function

Phase-space portraits are arguably the most powerful tool in the toolbox of classical dynamics, and one would like to retain its uses for quantum systems. However, there is that pesky paradox about quantum trajectories that cannot admit the existence of one-dimensional curves through such a phase space. Furthermore, there is no direct way of taking a wavefunction and simply “finding” its position or momentum to plot points on such a quantum phase space.

The answer was found in 1932 by Eugene Wigner (1902 – 1905), an Hungarian physicist working at Princeton. He realized that it was impossible to construct a quantum probability distribution in phase space that had positive values everywhere. This is a problem, because negative probabilities have no direct interpretation. But Wigner showed that if one relaxed the requirements a bit, so that expectation values computed over some distribution function (that had positive and negative values) gave correct answers that matched experiments, then this distribution function would “stand in” for an actual probability distribution.

The distribution function that Wigner found is called the Wigner distribution function. Given a wavefunction ψ(x), the Wigner distribution is defined as

Fig. 4 Wigner distribution function in (x, p) phase space.

The Wigner distribution function is the Fourier transform of the convolution of the wavefunction. The pure position dependence of the wavefunction is converted into a spread-out position-momentum function in phase space. For a Gaussian wavefunction ψ(x) with a finite width in space, the W-function in phase space is a two-dimensional Gaussian with finite widths in both space and momentum. In fact, the Δx-Δp product of the W-function is precisely the uncertainty production of the Heisenberg uncertainty relation.

The question of the quantum trajectory from the phase-space perspective becomes whether a Wigner function behaves like a localized “packet” that evolves in phase space in a way analogous to a classical particle, and whether classical chaos is reflected in the behavior of quantum systems.

The Harmonic Oscillator

The quantum harmonic oscillator is a rare and special case among quantum potentials, because the energy spacings between all successive states are all the same. This makes it possible for a Gaussian wavefunction, which is a superposition of the eigenstates of the harmonic oscillator, to propagate through the potential without broadening. To see an example of this, watch the first example in this YouTube video for a Schrödinger cat state in a two-dimensional harmonic potential. For this very special potential, the Wigner distribution behaves just like a (broadened) particle on an orbit in phase space, executing nice circular orbits.

A comparison of the classical phase-space portrait versus the quantum phase-space portrait is shown in Fig. 5. Where the classical particle is a point on an orbit, the quantum particle is spread out, obeying the Δx-Δp Heisenberg product, but following the same orbit as the classical particle.

Fig. 5 Classical versus quantum phase-space portraits for a harmonic oscillator. For a classical particle, the trajectory is a point executing an orbit. For a quantum particle, the trajectory is a Wigner distribution that follows the same orbit as the classical particle.

However, a significant new feature appears in the Wigner representation in phase space when there is a coherent superposition of two states, known as a “cat” state, after Schrödinger’s cat. This new feature has no classical analog. It is the coherent interference pattern that appears at the zero-point of the harmonic oscillator for the Schrödinger cat state. There is no such thing as “classical” coherence, so this feature is absent in classical phase space portraits.

Two examples of Wigner distributions are shown in Fig. 6 for a statistical (incoherent) mixture of packets and a coherent superposition of packets. The quantum coherence signature is present in the coherent case but not the statistical mixture case. The coherence in the Wigner distribution represents “off-diagonal” terms in the density matrix that leads to interference effects in quantum systems. Quantum computing algorithms depend critically on such coherences that tend to decay rapidly in real-world physical systems, known as decoherence, and it is possible to make statements about decoherence by watching the zero-point interference.

Fig. 6 Quantum phase-space portraits of double wave packets. On the left, the wave packets have no coherence, being a statistical mixture. On the right is the case for a coherent superposition, or “cat state” for two wave packets in a one-dimensional harmonic oscillator.

Whereas Gaussian wave packets in the quantum harmonic potential behave nearly like classical systems, and their phase-space portraits are almost identical to the classical phase-space view (except for the quantum coherence), most quantum potentials cause wave packets to disperse. And when saddle points are present in the classical case, then we are back to the question about how quantum packets behave as they approach a saddle point in phase space.

Quantum Pendulum and Separatrix Chaos

One of the simplest anharmonic oscillators is the simple pendulum. In the classical case, the period diverges if the pendulum gets very close to going vertical. A similar thing happens in the quantum case, but because the motion has strong anharmonicity, an initial wave packet tends to spread dramatically as parts of the wavefunction less vertical stretch away from the part of the wave function that is more nearly vertical. Fig. 7 is a snap-shot about a eighth of a period after the wave packet was launched. The packet has already stretched out along the separatrix. A double-cat-state was used, so there is a second packet that has coherent interference with the first. To see a movie of the time evolution of the wave packet and the orbit in quantum phase space, see the YouTube video.

Fig. 7 Wavefunction of a quantum pendulum released near vertical. The phase-space portrait is very similar to the classical case, except that the phase-space distribution is stretched out along the separatrix. The initial state for the phase-space portrait was a cat state.

The simple pendulum does have a saddle point, but it is degenerate because the angle is modulo -2-pi. A simple potential that has a non-degenerate saddle point is a double-well potential.

Quantum Double-Well and Separatrix Chaos

The symmetric double-well potential has a saddle point at the mid-point between the two well minima. A wave packet approaching the saddle will split into to packets that will follow the individual separatrixes that emerge from the saddle point (the unstable manifolds). This effect is seen most dramatically in the middle pane of Fig. 8. For the full video of the quantum phase-space evolution, see this YouTube video. The stretched-out distribution in phase space is highly analogous to the separatrix chaos seen for the classical system.

Fig. 8 Phase-space portraits of the Wigner distribution for a wavepacket in a double-well potential. The packet approaches the central saddle point, where the probability density splits along the unstable manifolds.

Conclusion

A common statement often made about quantum chaos is that quantum systems tend to suppress chaos, only exhibiting chaos for special types of orbits that produce quantum scars. However, from the phase-space perspective, the opposite may be true. The stretched-out Wigner distribution functions, for critical wave packets that interact with a saddle point, are very similar to the stochastic layer that forms in separatrix chaos in classical systems. In this sense, the phase-space description brings out the similarity between classical chaos and quantum chaos.

By David D. Nolte Sept. 25, 2022

YouTube Video

YouTube Video of Dynamics in Quantum Phase Space

For more on the history of quantum trajectories, see Galileo Unbound from Oxford Press:

Order from Amazon

References

1. T. Curtright, D. Fairlie, C. Zachos, A Concise Treatise on Quantum Mechanics in Phase Space.  (World Scientific, New Jersey, 2014).

2. J. R. Nagel, A Review and Application of the Finite-Difference Time-Domain Algorithm Applied to the Schrödinger Equation, ACES Journal, Vol. 24, NO. 1, pp. 1-8 (2009)

by David D. Nolte

Is There a Quantum Trajectory?

Heisenberg’s uncertainty principle is a law of physics – it cannot be violated under any circumstances, no matter how much we may want it to yield or how hard we try to bend it.  Heisenberg, as he developed his ideas after his lone epiphany like a monk on the isolated island of Helgoland off the north coast of Germany in 1925, became a bit of a zealot, like a religious convert, convinced that all we can say about reality is a measurement outcome.  In his view, there was no independent existence of an electron other than what emerged from a measuring apparatus.  Reality, to Heisenberg, was just a list of numbers in a spread sheet—matrix elements.  He took this line of reasoning so far that he stated without exception that there could be no such thing as a trajectory in a quantum system.  When the great battle commenced between Heisenberg’s matrix mechanics against Schrödinger’s wave mechanics, Heisenberg was relentless, denying any reality to Schrödinger’s wavefunction other than as a calculation tool.  He was so strident that even Bohr, who was on Heisenberg’s side in the argument, advised Heisenberg to relent [1].  Eventually a compromise was struck, as Heisenberg’s uncertainty principle allowed Schrödinger’s wave functions to exist within limits—his uncertainty limits.

Disaster in the Poconos

Yet the idea of an actual trajectory of a quantum particle remained a type of heresy within the close quantum circles.  Years later in 1948, when a young Richard Feynman took the stage at a conference in the Poconos, he almost sabotaged his career in front of Bohr and Dirac—two of the giants who had invented quantum mechanics—by having the audacity to talk about particle trajectories in spacetime diagrams.

Feynman was making his first presentation of a new approach to quantum mechanics that he had developed based on path integrals. The challenge was that his method relied on space-time graphs in which “unphysical” things were allowed to occur.  In fact, unphysical things were required to occur, as part of the sum over many histories of his path integrals.  For instance, a key element in the approach was allowing electrons to travel backwards in time as positrons, or a process in which the electron and positron annihilate into a single photon, and then the photon decays back into an electron-positron pair—a process that is not allowed by mass and energy conservation.  But this is a possible history that must be added to Feynman’s sum.

It all looked like nonsense to the audience, and the talk quickly derailed.  Dirac pestered him with questions that he tried to deflect, but Dirac persisted like a raven.  A question was raised about the Pauli exclusion principle, about whether an orbital could have three electrons instead of the required two, and Feynman said that it could—all histories were possible and had to be summed over—an answer that dismayed the audience.  Finally, as Feynman was drawing another of his space-time graphs showing electrons as lines, Bohr rose to his feet and asked derisively whether Feynman had forgotten Heisenberg’s uncertainty principle that made it impossible to even talk about an electron trajectory.

It was hopeless.  The audience gave up and so did Feynman as the talk just fizzled out.  It was a disaster.  What had been meant to be Feynman’s crowning achievement and his entry to the highest levels of theoretical physics, had been a terrible embarrassment.  He slunk home to Cornell where he sank into one of his depressions.  At the close of the Pocono conference, Oppenheimer, the reigning king of physics, former head of the successful Manhattan Project and newly selected to head the prestigious Institute for Advanced Study at Princeton, had been thoroughly disappointed by Feynman.

But what Bohr and Dirac and Oppenheimer had failed to understand was that as long as the duration of unphysical processes was shorter than the energy differences involved, then it was literally obeying Heisenberg’s uncertainty principle.  Furthermore, Feynman’s trajectories—what became his famous “Feynman Diagrams”—were meant to be merely cartoons—a shorthand way to keep track of lots of different contributions to a scattering process.  The quantum processes certainly took place in space and time, conceptually like a trajectory, but only so far as time durations, and energy differences and locations and momentum changes were all within the bounds of the uncertainty principle.  Feynman had invented a bold new tool for quantum field theory, able to supply deep results quickly.  But no one at the Poconos could see it.

Fig. 1 The first Feynman diagram.

Coherent States

When Feynman had failed so miserably at the Pocono conference, he had taken the stage after Julian Schwinger, who had dazzled everyone with his perfectly scripted presentation of quantum field theory—the competing theory to Feynman’s.  Schwinger emerged the clear winner of the contest.  At that time, Roy Glauber (1925 – 2018) was a young physicist just taking his PhD from Schwinger at Harvard, and he later received a post-doc position at Princeton’s Institute for Advanced Study where he became part of a miniature revolution in quantum field theory that revolved around—not Schwinger’s difficult mathematics—but Feynman’s diagrammatic method.  So Feynman won in the end.  Glauber then went on to Caltech, where he filled in for Feynman’s lectures when Feynman was off in Brazil playing the bongos.  Glauber eventually returned to Harvard where he was already thinking about the quantum aspects of photons in 1956 when news of the photon correlations in the Hanbury-Brown Twiss (HBT) experiment were published.  Three years later, when the laser was invented, he began developing a theory of photon correlations in laser light that he suspected would be fundamentally different than in natural chaotic light. 

Because of his background in quantum field theory, and especially quantum electrodynamics, it was fairly easy to couch the quantum optical properties of coherent light in terms of Dirac’s creation and annihilation operators of the electromagnetic field. Glauber developed a “coherent state” operator that was a minimum uncertainty state of the quantized electromagnetic field, related to the minimum-uncertainty wave functions derived initially by Schrödinger in the late 1920’s. The coherent state represents a laser operating well above the lasing threshold and behaved as “the most classical” wavepacket that can be constructed.  Glauber was awarded the Nobel Prize in Physics in 2005 for his work on such “Glauber states” in quantum optics.

Fig. 2 Roy Glauber

Quantum Trajectories

Glauber’s coherent states are built up from the natural modes of a harmonic oscillator.  Therefore, it should come as no surprise that these coherent-state wavefunctions in a harmonic potential behave just like classical particles with well-defined trajectories. The quadratic potential matches the quadratic argument of the the Gaussian wavepacket, and the pulses propagate within the potential without broadening, as in Fig. 3, showing a snapshot of two wavepackets propagating in a two-dimensional harmonic potential. This is a somewhat radical situation, because most wavepackets in most potentials (or even in free space) broaden as they propagate. The quadratic potential is a special case that is generally not representative of how quantum systems behave.

Fig. 3 Harmonic potential in 2D and two examples of pairs of pulses propagating without broadening. The wavepackets in the center are oscillating in line, and the wavepackets on the right are orbiting the center of the potential in opposite directions. (Movies of the quantum trajectories can be viewed at Physics Unbound.)

To illustrate this special status for the quadratic potential, the wavepackets can be launched in a potential with a quartic perturbation. The quartic potential is anharmonic—the frequency of oscillation depends on the amplitude of oscillation unlike for the harmonic oscillator, where amplitude and frequency are independent. The quartic potential is integrable, like the harmonic oscillator, and there is no avenue for chaos in the classical analog. Nonetheless, wavepackets broaden as they propagate in the quartic potential, eventually spread out into a ring in the configuration space, as in Fig. 4.

Fig. 4 Potential with a quartic corrections. The initial gaussian pulses spread into a “ring” orbiting the center of the potential.

A potential with integrability has as many conserved quantities to the motion as there are degrees of freedom. Because the quartic potential is integrable, the quantum wavefunction may spread, but it remains highly regular, as in the “ring” that eventually forms over time. However, integrable potentials are the exception rather than the rule. Most potentials lead to nonintegrable motion that opens the door to chaos.

A classic (and classical) potential that exhibits chaos in a two-dimensional configuration space is the famous Henon-Heiles potential. This has a four-dimensional phase space which admits classical chaos. The potential has a three-fold symmetry which is one reason it is non-integral, since a particle must “decide” which way to go when it approaches a saddle point. In the quantum regime, wavepackets face the same decision, leading to a breakup of the wavepacket on top of a general broadening. This allows the wavefunction eventually to distribute across the entire configuration space, as in Fig. 5.

Fig. 5 The Henon-Heiles two-dimensional potential supports Hamiltonian chaos in the classical regime. In the quantum regime, the wavefunction spreads to eventually fill the accessible configuration space (for constant energy).

Youtube Video

Movies of quantum trajectories can be viewed at my Youtube Channel, Physics Unbound. The answer to the question “Is there a quantum trajectory?” can be seen visually as the movies run—they do exist in a very clear sense under special conditions, especially coherent states in a harmonic oscillator. And the concept of a quantum trajectory also carries over from a classical trajectory in cases when the classical motion is integrable, even in cases when the wavefunction spreads over time. However, for classical systems that display chaotic motion, wavefunctions that begin as coherent states break up into chaotic wavefunctions that fill the accessible configuration space for a given energy. The character of quantum evolution of coherent states—the most classical of quantum wavefunctions—in these cases reflects the underlying character of chaotic motion in the classical analogs. This process can be seen directly watching the movies as a wavepacket approaches a saddle point in the potential and is split. Successive splits of the multiple wavepackets as they interact with the saddle points is what eventually distributes the full wavefunction into its chaotic form.

Therefore, the idea of a “quantum trajectory”, so thoroughly dismissed by Heisenberg, remains a phenomenological guide that can help give insight into the behavior of quantum systems—both integrable and chaotic.

As a side note, the laws of quantum physics obey time-reversal symmetry just as the classical equations do. In the third movie of “A Quantum Ballet“, wavefunctions in a double-well potential are tracked in time as they start from coherent states that break up into chaotic wavefunctions. It is like watching entropy in action as an ordered state devolves into a disordered state. But at the half-way point of the movie, the imaginary part of the wavefunction has its sign flipped, and the dynamics continue. But now the wavefunctions move from disorder into an ordered state, seemingly going against the second law of thermodynamics. Flipping the sign of the imaginary part of the wavefunction at just one instant in time plays the role of a time-reversal operation, and there is no violation of the second law.

By David D. Nolte, Sept. 4, 2022

YouTube Video

YouTube Video of Quantum Trajectories

For more on the history of quantum trajectories, see Galileo Unbound from Oxford Press:

Order from Amazon

References

[1] See Chapter 8 , On the Quantum Footpath, in Galileo Unbound, D. D. Nolte (Oxford University Press, 2018)

[2] J. R. Nagel, A Review and Application of the Finite-Difference Time-Domain Algorithm Applied to the Schrödinger Equation, ACES Journal, Vol. 24, NO. 1, pp. 1-8 (2009)

by David D. Nolte

Quantum Chaos and the Cheshire Cat

Alice’s disturbing adventures in Wonderland tumbled upon her like a string of accidents as she wandered a world of chaos.  Rules were never what they seemed and shifted whenever they wanted.  She even met a cat who grinned ear-to-ear and could disappear entirely, or almost entirely, leaving only its grin.

The vanishing Cheshire Cat reminds us of another famous cat—Arnold’s Cat—that introduced the ideas of stretching and folding of phase-space volumes in non-integrable Hamiltonian systems.  But when Arnold’s Cat becomes a Quantum Cat, a central question remains: What happens to the chaotic behavior of the classical system … does it survive the transition to quantum mechanics?  The answer is surprisingly like the grin of the Cheshire Cat—the cat vanishes, but the grin remains.  In the quantum world of the Cheshire Cat, the grin of the classical cat remains even after the rest of the cat vanished. 

The Cheshire Cat fades away, leaving only its grin, like a fine filament, as classical chaos fades into quantum, leaving behind a quantum scar.

The Quantum Mechanics of Classically Chaotic Systems

The simplest Hamiltonian systems are integrable—they have as many constants of the motion as degrees of freedom.  This holds for quantum systems as well as for classical.  There is also a strong correspondence between classical and quantum systems for the integrable cases—literally the Correspondence Principle—that states that quantum systems at high quantum number approach classical behavior.  Even at low quantum numbers, classical resonances are mirrored by quantum eigenfrequencies that can show highly regular spectra.

But integrable systems are rare—surprisingly rare.  Almost no real-world Hamiltonian system is integrable, because the real world warps the ideal.  No spring can displace indefinitely, and no potential is perfectly quadratic.  There are always real-world non-idealities that destroy one constant of the motion or another, opening the door to chaos.

When classical Hamiltonian systems become chaotic, they don’t do it suddenly.  Almost all transitions to chaos in Hamiltonian systems are gradual.  One of the best examples of this is the KAM theory that starts with invariant action integrals that generate invariant tori in phase space.  As nonintegrable perturbations increase, the tori break up slowly into island chains of stability as chaos infiltrates the separatrixes—first as thin filaments of chaos surrounding the islands—then growing in width to take up more and more of phase space.  Even when chaos is fully developed, small islands of stability can remain—the remnants of stable orbits of the unperturbed system.

When the classical becomes quantum, chaos softens.  Quantum wave functions don’t like to be confined—they spread and they tunnel.  The separatrix of classical chaos—that barrier between regions of phase space—cannot constrain the exponential tails of wave functions.  And the origin of chaos itself—the homoclinic point of the separatrix—gets washed out.  Then the regular orbits of the classical system reassert themselves, and they appear, like the vestige of the Cheshire Cat, as a grin.

The Quantum Circus

The empty stadium is a surprisingly rich dynamical system that has unexpected structure in both the classical and the quantum domain.  Its importance in classical dynamics comes from the fact that its periodic orbits are unstable and its non-periodic orbits are ergodic (filling all available space if given long enough).  The stadium itself is empty so that particles (classical or quantum) are free to propagate between reflections from the perfectly-reflecting walls of the stadium.  The ergodicity comes from the fact that the stadium—like a classic Roman chariot-race stadium, also known as a circus—is not a circle, but has a straight stretch between two half circles.  This simple modification takes the stable orbits of the circle into the unstable orbits of the stadium.

A single classical orbit in a stadium is shown in Fig 1. This is an ergodic orbit that is non-periodic and eventually would fill the entire stadium space. There are other orbits that are nearly periodic, such as one that bounces back and forth vertically between the linear portions, but even this orbit will eventually wander into the circular part of the stadium and then become ergodic. The big quantum-classical question is what happens to these classical orbits when the stadium is shrunk to the nanoscale?

Fig. 1 A classical trajectory in a stadium. It will eventually visit every point, a property known as ergodicity.

Simulating an evolving quantum wavefunction in free space is surprisingly simple. Given a beginning quantum wavefunction A(x,y,t0), the discrete update equation is

Perfect reflection from the boundaries of the stadium are incorporated through imposing a boundary condition that sends the wavefunction to zero. Simple!

A snap-shot of a wavefunction evolving in the stadium is shown in Fig. 2. To see a movie of the time evolution, see my YouTube episode.

Fig. 2 Snapshot of a quantum wavefunction in the stadium. (From YouTube)

The time average of the wavefunction after a long time has passed is shown in Fig. 3. Other than the horizontal nodal line down the center of the stadium, there is little discernible structure or symmetry. This is also true for the mean squared wavefunction shown in Fig. 4, although there is some structure that may be emerging in the semi-circular regions.

Fig. 3 Time-average wavefunction after a long time.
Fig. 4 Time-average of the squared wavefunction after a long time.

On the other hand, for special initial conditions that have a lot of symmetry, something remarkable happens. Fig. 5 shows several mean-squared results for special initial conditions. There is definite structure in these cases that were given the somewhat ugly name “quantum scars” in the 1980’s by Eric Heller who was one of the first to study this phenomenon [1].

Quantum Scars
Quantum Scars
Quantum Scars
Fig. 5 Quantum scars reflect periodic (but unstable) orbits of the classical system. Quantum effects tend to quench chaos and favor regular motion.

One can superpose highly-symmetric classical trajectories onto the figures, as shown in the bottom row. All of these classical orbits go through a high-symmetry point, such as the center of the stadium (on the left image) and through the focal point of the circular mirrors (in the other two images). The astonishing conclusion of this exercise is that the highly-symmetric periodic classical orbits remain behind as quantum scars—like the Cheshire Cat’s grin—when the system is in the quantum realm. The classical orbits that produce quantum scars have the important property of being periodic but unstable. A slight perturbation from the symmetric trajectory causes it to eventually become ergodic (chaotic). These scars are regions with enhanced probability density, what might be termed “quantum trajectories”, but do not show strong interference patterns.

It is important to make the distinction that it is also possible to construct special wavefunctions that are strictly periodic, such as a wave bouncing perfectly vertically between the straight portions. This leads to large-scale interference patterns that are not the same as the quantum scars.

Quantum Chaos versus Laser Speckle

In addition to the bouncing-wave cases that do not strictly produce quantum scars, there is another “neutral” phenomenon that produces interference patterns that look a lot like scars, but are simply the random addition of lots of plane waves with the same wavelength [2]. A snapshot in time of one of these superpositions is shown in Fig. 6. To see how the waves add together, see the YouTube channel episode.

Fig. 6 The sum of 100 randomly oriented plane waves of constant wavelength. (A snapshot from YouTube.)

By David D. Nolte, Aug. 14, 2022

YouTube Video

YouTube Video of Quantum Chaos

For more on the history of chaos theory, buy Galileo Unbound from Oxford Press:

Order from Amazon

References

[1] Heller E J, Bound-state eigenfunctions of classically chaotic hamiltonian-systems – scars of periodic-orbits, Physical Review Letters 53 ,1515 (1984)

[2] Gutzwiller M C, Chaos in classical and quantum mechanics (New York: New York : Springer-Verlag, 1990)

by David D. Nolte

George Cantor meets Machine Learning: Deep Discrete Encoders

Machine learning is characterized, more than by any other aspect, by the high dimensionality of the data spaces it seeks to find patterns in.  Hence, one of the principle functions of machine learning is to reduce the dimensionality of the data to lower dimensions—a process known as dimensionality reduction.

There are two driving reasons to reduce the dimensionality of data: 

First, typical dimensionalities faced by machine learning problems can be in the hundreds or thousands.  Trying to visualize such high dimensions may sound mind expanding, but it is really just saying that a data problem may have hundreds or thousands of different data entries for a single event.  And many, or even most, of those entries may not be independent.  While many others may be pure noise—or at least not relevant to the pattern.  Deep learning dimensionality reduction seeks to find the dependences—many of them nonlinear and non-single-valued (non-invertible)—and to reject the noise channels.

Second, the geometry of high dimension is highly unintuitive.  Many of the things we take for granted in our pitifully low dimension of 3 (or 4 if you include time) just don’t hold in high dimensions.  For instance, in very high dimensions almost all random vectors in a hyperspace are orthogonal, and almost all random unit vectors in the hyperspace are equidistant.  Even the topology of landscapes in high dimensions is unintuitive—there are far more mountain ridges than mountain peaks—with profound consequences for dynamical processes such as random walks (see my Blog on a Random Walk in 10 Dimensions).  In fact, we owe our evolutionary existence to this effect!  Therefore, deep dimensionality reduction is a way to bring complex data down to a dimensionality where our intuition can be applied to “explain” the data.

But what is a dimension?  And can you find the “right” dimensionality when performing dimensionality reduction?  Once again, our intuition struggles with these questions, as first discovered by a nineteenth-century German mathematician whose mind-expanding explorations of the essence of different types of infinity shattered the very concept of dimension.

George Cantor

Georg Cantor (1845 – 1918) was born in Russia, and the family moved to Germany while Cantor was still young.  In 1863, he enrolled at the University of Berlin where he sat on lectures by Weierstrass and Kronecker.  He received his doctorate in 1867 and his Habilitation in 1869, moving into a faculty position at the University of Halle and remaining there for the rest of his career.  Cantor published a paper early in 1872 on the question of whether the representation of an arbitrary function by a Fourier series is unique.  He had found that even though the series might converge to a function almost everywhere, there surprisingly could still be an infinite number of points where the convergence failed.  Originally, Cantor was interested in the behavior of functions at these points, but his interest soon shifted to the properties of the points themselves, which became his life’s work as he developed set theory and transfinite mathematics.

Georg Cantor (1845 – 1918)

In 1878, in a letter to his friend Richard Dedekind, Cantor showed that there was a one-to-one correspondence between the real numbers and the points in any n-dimensional space.  He was so surprised by his own result that he wrote to Dedekind “I see it, but I don’t believe it.”  Previously, the ideas of a dimension, moving in a succession from one (a line) to two (a plane) to three (a volume) had been absolute.  However, with his newly-discovered mapping, the solid concepts of dimension and dimensionality began to dissolve.  This was just the beginning of a long history of altered concepts of dimension (see my Blog on the History of Fractals).

Mapping Two Dimensions to One

Cantor devised a simple mapping that is at once obvious and subtle. To take a specific example of mapping a plane to a line, one can consider the two coordinate values (x,y) both with a closed domain on [0,1]. Each can be expressed as a decimal fraction given by

These two values can be mapped to a single value through a simple “ping-pong” between the decimal digits as

If x and y are irrational, then this presents a simple mapping of a pair of numbers (two-dimensional coordinates) to a single number. In this way, the special significance of two dimensions relative to one dimension is lost. In terms of set theory nomenclature, the cardinality of the two-dimensional R2 is the same as for the one-dimensions R.

Nonetheless, intuition about dimension still has it merits, and a subtle aspect of this mapping is that it contains discontinuities. These discontinuities are countably infinite, but they do disrupt any smooth transformation from 2D to 1D, which is where the concepts of intrinsic dimension are hidden. The topological dimension of the plane is clearly equal to 2, and that of the line is clearly equal to 1, determined by the dimensionality D+1 of cuts that are required to separate the sets. The area is separated by a D = 1 line, while the line is separated by a D= 0 point.

Fig. 1 A mapping of a 2D plane to a 1D line. Every point in the plane has an associated point on the line (except for a countably infinite number of special points … see the next figure.)

While the discontinuities help preserve the notions of dimension, and they are countably infinite (with the cardinality of the rational numbers), they pertain merely to the representation of number by decimals. As an example, in decimal notation for a number a = 14159/105, one has two equivalent representations

trailing either infinitely many 0’s or infinitely many 9’s. Despite the fact that these representations point to the same number, when it is used as one of the pairs in the bijection of Fig. 1, it produces two distinctly different numbers of t in R. Fortunately, there is a way to literally sweep this under the rug. Any time one retrieves a number that has repeating 0’s or 9’s, simply sweep it to the right it by dividing by a power of 10 to a region of trailing zeros for a different number, call it b, as in Fig. 2. The shifted version of a will not overlap with the number b, because b also ends in repeating 0’s or 9’s and so is swept farther to the right, and so on to infinity, so none of these numbers overlap, each is distinct, and the mapping becomes what is known as a bijection with one-to-one correspondence.

Fig. 3 The “measure-zero” fix. Numbers that have two equivalent representations can be shifted to the right to replace other numbers that are shifted further to the right, and so on to infinity. There is infinite room within the reals to accommodate the countably infinite number of repeating-decimal numbers.

Space-Filling Curves

When the peripatetic mathematician Guiseppe Peano learned of Cantor’s result for the mapping of 2D to 1D, he sought to demonstrate the correspondence geometrically, and he constructed a continuous curve that filled space, publishing the method in Sur une courbe, qui remplit toute une aire plane [1] in 1890.  The construction of Peano’s curve proceeds by taking a square and dividing it into 9 equal sub squares.  Lines connect the centers of each of the sub squares.  Then each sub square is divided again into 9 sub squares whose centers are all connected by lines.  At this stage, the original pattern, repeated 9 times, is connected together by 8 links, forming a single curve.  This process is repeated infinitely many times, resulting in a curve that passes through every point of the original plane square.  In this way, a line is made to fill a plane.  Where Cantor had proven abstractly that the cardinality of the real numbers was the same as the points in n-dimensional space, Peano created a specific example. 

This was followed quickly by another construction, invented by David Hilbert in 1891, that divided the square into four instead of nine, simplifying the construction, but also showing that such constructions were easily generated [2].  The space-filling curves of Peano and Hilbert have the extreme properties that a one-dimensional curve approaches every point in a two-dimensional space.  These curves have topological dimensionality of 1D and a fractal dimensionality of 2D. 

Fig. 3 Peano’s (1890) and Hilbert’s (1891) plane-filling curves.  When the iterations are taken to infinity, the curves approach every point of two-dimensional space arbitrarily closely, giving them a dimension D = 2.

A One-Dimensional Deep Discrete Encoder for MNIST

When performing dimensionality reduction in deep learning it is tempting to think that there is an underlying geometry to the data—as if they reside on some submanifold of the original high-dimensional space.  While this can be the case (for which dimensionality reduction is called deep metric learning) more often there is no such submanifold.  For instance, when there is a highly conditional nature to the different data channels, in which some measurements are conditional on the values of other measurements, then there is no simple contiguous space that supports the data.

It is also tempting to think that a deep learning problem has some intrinsic number of degrees of freedom which should be matched by the selected latent dimensionality for the dimensionality reduction.  For instance, if there are roughly five degrees of freedom buried within a complicated data set, then it is tempting to believe that the appropriate latent dimensionality also should be chosen to be five.  But this also is missing the mark. 

Take, for example, the famous MNIST data set of hand-written digits from 0 to 9.  Each digit example is contained in a 28-by28 two-dimensional array that is typically flattened to a 784-element linear vector that locates that single digit example within a 784-dimensional space.  The goal is to reduce the dimensionality down to a manageable number—but what should the resulting latent dimensionality be?  How many degrees of freedom are involved in writing digits?  Furthermore, given that there are ten classes of digits, should the chosen dimensionality of the latent space be related to the number of classes?

Fig. 4 A sampling of MNIST digits.

To illustrate that the dimensionality of the latent space has little to do with degrees of freedom or the number of classes, let’s build a simple discrete encoder that encodes the MNIST data set to the one-dimensional line—following Cantor.

The deep network of the encoder can have a simple structure that terminates with a single neuron that has a piece-wise linear output.  The objective function (or loss function) measures the squared distances of the outputs of the single neuron, after transformation by the network, to the associated values 0 to 9.  And that is it!  Train the network by minimizing the loss function.

Fig. 5 A deep encoder that encodes discrete classes to a one-dimensional latent variable.

The results of the linear encoder are shown in Fig. 6 (the transverse directions are only for ease of visualization…the classification is along the long axis). The different dots are specific digits, and the colors are the digit value. There is a clear trend from 1 through 10, although with this linear encoder there is substantial overlap among the point clouds.

Fig. 6 Latent space for a one-dimensional (line) encoder. The transverse dimensions are only for visualization.

Despite appearances, this one-dimensional discrete line encoder is NOT a form of regression. There is no such thing as 1.5 as the average of a 1-digit and a 2-digit. And 5 is not the average of a 4-digit and a 6-digit. The digits are images that have no intrinsic numerical value. Therefore, this one-dimensional encoder is highly discontinuous, and intuitive ideas about intrinsic dimension for continuous data remain secure.

Summary

The purpose of this Blog (apart from introducing Cantor and his ideas on dimension theory) was to highlight a key question of dimensionality reduction in representation theory: What is the intrinsic dimensionality of a dataset? The answer, in the case of discrete classes, is that there is no intrinsic dimensionality. Having 1 degree of freedom or 10 degrees of freedom, i.e. latent dimensionalities of 1 or 10, is mostly irrelevant. In ideal cases, one is just as good as the other.

On the other hand, for real-world data with its inevitable variability and finite variance, there can be reasons to choose one latent dimensionality over another. In fact, the “best” choice of dimensionality is one less than the number of classes. For instance, in the case of MNIST with its 10 classes, that is a 9-dimensional latent space. The reason this is “best” has to do with the geometry of high-dimensional simplexes, which will be the topic of a future Blog.

By David D. Nolte, June 20, 2022

[1] G.Peano: Sur une courbe, qui remplit toute une aire plane. Mathematische Annalen 36 (1890), 157–160.

[2] D. Hilbert, “Uber die stetige Abbildung einer Linie auf ein Fllichenstilck,” Mathemutische Anna/en, vol. 38, pp. 459-460, (1891)

by David D. Nolte

The Anharmonic Harmonic Oscillator

Harmonic oscillators are one of the fundamental elements of physical theory.  They arise so often in so many different contexts that they can be viewed as a central paradigm that spans all aspects of physics.  Some famous physicists have been quoted to say that the entire universe is composed of simple harmonic oscillators (SHO).

Despite the physicist’s love affair with it, the SHO is pathological! First, it has infinite frequency degeneracy which makes it prone to the slightest perturbation that can tip it into chaos, in contrast to non-harmonic cyclic dynamics that actually protects us from the chaos of the cosmos (see my Blog on Chaos in the Solar System). Second, the SHO is nowhere to be found in the classical world.  Linear oscillators are purely harmonic, with a frequency that is independent of amplitude—but no such thing exists!  All oscillators must be limited, or they could take on infinite amplitude and infinite speed, which is nonsense.  Even the simplest of simple harmonic oscillators would be limited by nothing other than the speed of light.  Relativistic effects would modify the linearity, especially through time dilation effects, rendering the harmonic oscillator anharmonic.

Despite the physicist’s love affair with it, the SHO is pathological!

Therefore, for students of physics as well as practitioners, it is important to break the shackles imposed by the SHO and embrace the anharmonic harmonic oscillator as the foundation of physics. Here is a brief survey of several famous anharmonic oscillators in the history of physics, followed by the mathematical analysis of the relativistic anharmonic linear-spring oscillator.

Anharmonic Oscillators

Anharmonic oscillators have a long venerable history with many varieties.  Many of these have become central models in systems as varied as neural networks, synchronization, grandfather clocks, mechanical vibrations, business cycles, ecosystem populations and more.

Christiaan Huygens

Already by the mid 1600’s Christiaan Huygens (1629 – 1695) knew that the pendulum becomes slower when it has larger amplitudes.  The pendulum was one of the best candidates for constructing an accurate clock needed for astronomical observations and for the determination of longitude at sea.  Galileo (1564 – 1642) had devised the plans for a rudimentary pendulum clock that his son attempted to construct, but the first practical pendulum clock was invented and patented by Huygens in 1657.  However, Huygens’ modified verge escapement required his pendulum to swing with large amplitudes, which brought it into the regime of anharmonicity. The equations of the simple pendulum are truly simple, but the presence of the sinθ makes it the simplest anharmonic oscillator.

Therefore, Huygens searched for the mathematical form of a tautochrone curve for the pendulum (a curve that is traversed with equal times independently of amplitude) and in the process he invented the involutes and evolutes of a curve—precursors of the calculus.  The answer to the tautochrone question is a cycloid (see my Blog on Huygen’s Tautochrone Curve).

Hermann von Helmholtz

Hermann von Helmholtz (1821 – 1894) was possibly the greatest German physicist of his generation—an Einstein before Einstein—although he began as a medical doctor.  His study of muscle metabolism, drawing on the early thermodynamic work of Carnot, Clapeyron and Joule, led him to explore and to express the conservation of energy in its clearest form.  Because he postulated that all forms of physical processes—electricity, magnetism, heat, light and mechanics—contributed to the interconversion of energy, he sought to explore them all, bringing his research into the mainstream of physics.  His laboratory in Berlin became world famous, attracting to his laboratory the early American physicists Henry Rowland (founder and first president of the American Physical Society) and Albert Michelson (first American Nobel prize winner).

Even the simplest of simple harmonic oscillators would be limited by nothing other than the speed of light.  

Helmholtz also pursued a deep interest in the physics of sensory perception such as sound.  This research led to his invention of the Helmholtz oscillator which is a highly anharmonic relaxation oscillator in which a tuning fork was placed near an electromagnet that was powered by a mercury switch attached to the fork. As the tuning fork vibrated, the mercury came in and out of contact with it, turning on and off the magnet, which fed back on the tuning fork, and so on, enabling the device, once started, to continue oscillating without interruption. This device is called a tuning-fork resonator, and it became the first door-bell buzzers.  (These are not to be confused with Helmholtz resonances that are formed when blowing across the open neck of a beer bottle.)

Lord Rayleigh

Baron John Strutt, the Lord Rayleigh (1842 – 1919) like Helmholtz also was a generalist and had a strong interest in the physics of sound.  He was inspired by Helmholtz’ oscillator to consider general nonlinear anharmonic oscillators mathematically.  He was led to consider the effects of anharmonic terms added to the harmonic oscillator equation.  in a paper published in the Philosophical Magazine issue of 1883 with the title On Maintained Vibrations, he introduced an equation to describe the self-oscillation by adding an extra term to a simple harmonic oscillator. The extra term depended on the cube of the velocity, representing a balance between the gain of energy from a steady force and natural dissipation by friction.  Rayleigh suggested that this equation applied to a wide range of self-oscillating systems, such as violin strings, clarinet reeds, finger glasses, flutes, organ pipes, among others (see my Blog on Rayleigh’s Harp.)

Georg Duffing

The first systematic study of quadratic and cubic deviations from the harmonic potential was performed by the German engineer George Duffing (1861 – 1944) under the conditions of a harmonic drive. The Duffing equation incorporates inertia, damping, the linear spring and nonlinear deviations.

Fig. 1 The Duffing equation adds a nonlinear term to the spring force when alpha is positive, stiffening or weakening it for larger excursions when beta is positive or negative, respectively. And by making alpha negative and beta positive, it describes a damped driven double-well potential.

Duffing confirmed his theoretical predictions with careful experiments and established the lowest-order corrections to ideal masses on springs. His work was rediscovered in the 1960’s after Lorenz helped launch numerical chaos studies. Duffing’s driven potential becomes especially interesting when α is negative and β is positive, creating a double-well potential. The driven double-well is a classic chaotic system (see my blog on Duffing’s Oscillator).

Balthasar van der Pol

Autonomous oscillators are one of the building blocks of complex systems, providing the fundamental elements for biological oscillators, neural networks, business cycles, population dynamics, viral epidemics, and even the rings of Saturn.  The most famous autonomous oscillator (after the pendulum clock) is named for a Dutch physicist, Balthasar van der Pol (1889 – 1959), who discovered the laws that govern how electrons oscillate in vacuum tubes, but the dynamical system that he developed has expanded to become the new paradigm of cyclic dynamical systems to replace the SHO (see my Blog on GrandFather Clocks.)

Fig. 2 The van der Pol equation is the standard simple harmonic oscillator with a gain term that saturates for large excursions leading to a limit cycle oscillator.

Turning from this general survey, let’s find out what happens when special relativity is added to the simplest SHO [1].

Relativistic Linear-Spring Oscillator

The theory of the relativistic one-dimensional linear-spring oscillator starts from a relativistic Lagrangian of a free particle (with no potential) yielding the generalized relativistic momentum

The Lagrangian that accomplishes this is [2]

where the invariant 4-velocity is

When the particle is in a potential, the Lagrangian becomes

The action integral that is minimized is

and the Lagrangian for integration of the action integral over proper time is

The relativistic modification in the potential energy term of the Lagrangian is not in the spring constant, but rather is purely a time dilation effect.  This is captured by the relativistic Lagrangian

where the dot is with respect to proper time τ. The classical potential energy term in the Lagrangian is multiplied by the relativistic factor γ, which is position dependent because of the non-constant speed of the oscillator mass.  The Euler-Lagrange equations are

where the subscripts in the variables are a = 0, 1 for the time and space dimensions, respectively.  The derivative of the time component of the 4-vector is

From the derivative of the Lagrangian with respect to speed, the following result is derived

where E is the constant total relativistic energy.  Therefore,

which provides an expression for the derivative of the coordinate time with respect to the proper time where

The position-dependent γ(x) factor is then

The Euler-Lagrange equation with a = 1 is

which gives

providing the flow equations for the (an)harmonic oscillator with respect to proper time

This flow represents a harmonic oscillator modified by the γ(x) factor, due to time dilation, multiplying the spring force term.  Therefore, at relativistic speeds, the oscillator is no longer harmonic even though the spring constant remains truly a constant.  The term in parentheses effectively softens the spring for larger displacement, and hence the frequency of oscillation becomes smaller. 

The state-space diagram of the anharmonic oscillator is shown in Fig. 3 with respect to proper time (the time read on a clock co-moving with the oscillator mass).  At low energy, the oscillator is harmonic with a natural period of the SHO.  As the maximum speed exceeds β = 0.8, the period becomes longer and the trajectory less sinusoidal.  The position and speed for β = 0.9999 is shown in Fig. 4.  The mass travels near the speed of light as it passes the origin, producing significant time dilation at that instant.  The average time dilation through a single cycle is about a factor of three, despite the large instantaneous γ = 70 when the mass passes the origin.

Fig. 3 State-space diagram in relativistic units relative to proper time of a relativistic (an)harmonic oscillator with a constant spring constant for several relative speeds β. The anharmonicity becomes pronounced above β = 0.8.
Fig. 4 Position and speed in relativistic units relative to proper time of a relativistic (an)harmonic oscillator with a constant spring constant for β = 0.9999.  The period of oscillation in this simulation is nearly three times longer than the natural frequency at small amplitudes.

By David D. Nolte, May 29, 2022

[1] W. Moreau, R. Easther, and R. Neutze, “RELATIVISTIC (AN)HARMONIC OSCILLATOR,” American Journal of Physics, Article vol. 62, no. 6, pp. 531-535, Jun (1994)

[2] D. D. Nolte, Introduction to Modern Dynamics: Chaos, Networks, Space and Time, 2nd. ed. (Oxford University Press, 2019)

by David D. Nolte

The Many Worlds of the Quantum Beam Splitter

In one interpretation of quantum physics, when you snap your fingers, the trajectory you are riding through reality fragments into a cascade of alternative universes—one for each possible quantum outcome among all the different quantum states composing the molecules of your fingers. 

This is the Many-Worlds Interpretation (MWI) of quantum physics first proposed rigorously by Hugh Everett in his doctoral thesis in 1957 under the supervision of John Wheeler at Princeton University.  Everett had been drawn to this interpretation when he found inconsistencies between quantum physics and gravitation—topics which were supposed to have been his actual thesis topic.  But his side-trip into quantum philosophy turned out to be a one-way trip.  The reception of his theory was so hostile, no less than from Copenhagen and Bohr himself, that Everett left physics and spent a career at the Pentagon.

Resurrecting MWI in the Name of Quantum Information

Fast forward by 20 years, after Wheeler had left Princeton for the University of Texas at Austin, and once again a young physicist was struggling to reconcile quantum physics with gravity.  Once again the many worlds interpretation of quantum physics seemed the only sane way out of the dilemma, and once again a side-trip became a life-long obsession.

David Deutsch, visiting Wheeler in the early 1980’s, became convinced that the many worlds interpretation of quantum physics held the key to paradoxes in the theory of quantum information (For the full story of Wheeler, Everett and Deutsch, see Ref [1]).  He was so convinced, that he began a quest to find a physical system that operated on more information than could be present in one universe at a time.  If such a physical system existed, it would be because streams of information from more than one universe were coming together and combining in a way that allowed one of the universes to “borrow” the information from the other.

It took only a year or two before Deutsch found what he was looking for—a simple quantum algorithm that yielded twice as much information as would be possible if there were no parallel universes.  This is the now-famous Deutsch algorithm—the first quantum algorithm [2].  At the heart of the Deutsch algorithm is a simple quantum interference.  The algorithm did nothing useful—but it convinced Deutsch that two universes were interfering coherently in the measurement process, giving that extra bit of information that should not have been there otherwise.  A few years later, the Deutsch-Josza algorithm [2] expanded the argument to interfere an exponentially larger amount of information streams from an exponentially larger number of universes to create a result that was exponentially larger than any classical computer could produce.  This marked the beginning of the quest for the quantum computer that is running red-hot today.

Deutsch’s “proof” of the many-worlds interpretation of quantum mechanics is not a mathematical proof but is rather a philosophical proof.  It holds no sway over how physicists do the math to make their predictions.  The Copenhagen interpretation, with its “spooky” instantaneous wavefunction collapse, works just fine predicting the outcome of quantum algorithms and the exponential quantum advantage of quantum computing.  Therefore, the story of David Deutsch and the MWI may seem like a chimera—except for one fact—it inspired him to generate the first quantum algorithm that launched what may be the next revolution in the information revolution of modern society.  Inspiration is important in science, because it lets scientists create things that had been impossible before. 

But if quantum interference is the heart of quantum computing, then there is one physical system that has the ultimate simplicity that may yet inspire future generations of physicists to invent future impossible things—the quantum beam splitter.  Nothing in the study of quantum interference can be simpler than a sliver of dielectric material sending single photons one way or another.  Yet the outcome of this simple system challenges the mind and reminds us of why Everett and Deutsch embraced the MWI in the first place.

The Classical Beam Splitter

The so-called “beam splitter” is actually a misnomer.  Its name implies that it takes a light beam and splits it into two, as if there is only one input.  But every “beam splitter” has two inputs, which is clear by looking at the classical 50/50 beam splitter.  The actual action of the optical element is the combination of beams into superpositions in each of the outputs. It is only when one of the input fields is zero, a special case, that the optical element acts as a beam splitter.  In general, it is a beam combiner.

Given two input fields, the output fields are superpositions of the inputs

The square-root of two factor ensures that energy is conserved, because optical fluence is the square of the fields.  This relation is expressed more succinctly as a matrix input-output relation

The phase factors in these equations ensure that the matrix is unitary

reflecting energy conservation.

The Quantum Beam Splitter

A quantum beam splitter is just a classical beam splitter operating at the level of individual photons.  Rather than describing single photons entering or leaving the beam splitter, it is more practical to describe the properties of the fields through single-photon quantum operators

where the unitary matrix is the same as the classical case, but with fields replaced by the famous “a” operators.  The photon operators operate on single photon modes.  For instance, the two one-photon input cases are

where the creation operators operate on the vacuum state in each of the input modes.

The fundamental combinational properties of the beam splitter are even more evident in the quantum case, because there is no such thing as a single input to a quantum beam splitter.  Even if no photons are directed into one of the input ports, that port still receives a “vacuum” input, and this vacuum input contributes to the fluctuations observed in the outputs.

The input-output relations for the quantum beam splitter are

The beam splitter operating on a one-photon input converts the input-mode creation operator into a superposition of out-mode creation operators that generates

The resulting output is entangled: either the single photon exits one port, or it exits the other.  In the many worlds interpretation, the photon exits from one port in one universe, and it exits from the other port in a different universe.  On the other hand, in the Copenhagen interpretation, the two output ports of the beam splitter are perfectly anti-correlated.

Fig. 1  Quantum Operations of a Beam Splitter.  A beam splitter creates a quantum superposition of the input modes.  The a-symbols are quantum number operators that create and annihilate photons.  A single-photon input produces an entangled output that is a quantum superposition of the photon coming out of one output or the other.

The Hong-Ou-Mandel (HOM) Interferometer

When more than one photon is incident on a beam splitter, the fascinating effects of quantum interference come into play, creating unexpected outputs for simple inputs.  For instance, the simplest example is a two photon input where a single photon is present in each input port of the beam splitter.  The input state is represented with single creation operators operating on each vacuum state of each input port

creating a single photon in each of the input ports. The beam splitter operates on this input state by converting the input-mode creation operators into out-put mode creation operators to give

The important step in this process is the middle line of the equations: There is perfect destructive interference between the two single-photon operations.  Therefore, both photons always exit the beam splitter from the same port—never split.  Furthermore, the output is an entangled two-photon state, once more splitting universes.

Fig. 2  The HOM interferometer.  A two-photon input on a beam splitter generates an entangled superposition of the two photons exiting the beam splitter always together.

The two-photon interference experiment was performed in 1987 by Chung Ki Hong and Jeff Ou, students of Leonard Mandel at the Optics Institute at the University of Rochester [4], and this two-photon operation of the beam splitter is now called the HOM interferometer. The HOM interferometer has become a center-piece for optical and photonic implementations of quantum information processing and quantum computers.

N-Photons on a Beam Splitter

Of course, any number of photons can be input into a beam splitter.  For example, take the N-photon input state

The beam splitter acting on this state produces

The quantity on the right hand side can be re-expressed using the binomial theorem

where the permutations are defined by the binomial coefficient

The output state is given by

which is a “super” entangled state composed of N multi-photon states, involving N different universes.

Coherent States on a Quantum Beam Splitter

Surprisingly, there is a multi-photon input state that generates a non-entangled output—as if the input states were simply classical fields.  These are the so-called coherent states, introduced by Glauber and Sudarshan [5, 6].  Coherent states can be described as superpositions of multi-photon states, but when a beam splitter operates on these superpositions, the outputs are simply 50/50 mixtures of the states.  For instance, if the input scoherent tates are denoted by α and β, then the output states after the beam splitter are

This output is factorized and hence is NOT entangled.  This is one of the many reasons why coherent states in quantum optics are considered the “most classical” of quantum states.  In this case, a quantum beam splitter operates on the inputs just as if they were classical fields.

By David D. Nolte, May 8, 2022

References

[1] David D. Nolte, Interference: The History of Optical Interferometry and the Scientists who Tamed Light, (Oxford, July 2023)

[2] D. Deutsch, “Quantum-theory, the church-turing principle and the universal quantum computer,” Proceedings of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences, vol. 400, no. 1818, pp. 97-117, (1985)

[3] D. Deutsch and R. Jozsa, “Rapid solution of problems by quantum computation,” Proceedings of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences, vol. 439, no. 1907, pp. 553-558, Dec (1992)

[4] C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between 2 photons by interference,” Physical Review Letters, vol. 59, no. 18, pp. 2044-2046, Nov (1987)

[5] Glauber, R. J. (1963). “Photon Correlations.” Physical Review Letters 10(3): 84.

[6] Sudarshan, E. C. G. (1963). “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams.” Physical Review Letters 10(7): 277-&.; Mehta, C. L. and E. C. Sudarshan (1965). “Relation between quantum and semiclassical description of optical coherence.” Physical Review 138(1B): B274.

New from Oxford Press: The History of Optical Interferometry (Late Summer 2023)

Pre-order Today

by David D. Nolte

Post-Modern Machine Learning: The Deep Revolution

The mysteries of human intelligence are among the final frontiers of science. Despite our pride in what science has achieved across the past century, we have stalled when it comes to understanding intelligence or emulating it. The best we have done so far is through machine learning — harnessing the computational power of computers to begin to mimic the mind, attempting to answer the essential question:

How do we get machines to Know what we Know?

In modern machine learning, the answer is algorithmic.

In post-modern machine learning, the answer is manifestation.

The algorithms of modern machine learning are cause and effect, rules to follow, producing only what the programmer can imagine. But post-modern machine learning has blown past explicit algorithms to embrace deep networks. Deep networks today are defined by neural networks with thousands, or tens of thousands, or even hundreds of thousands, of neurons arrayed in multiple layers of dense connections. The interactivity of so many crossing streams of information defies direct deconstruction of what the networks are doing — they are post-modern. Their outputs manifest themselves, self-assembling into simplified structures and patterns and dependencies that are otherwise buried unseen in complicated data.

Fig. 1 A representation of a deep network with three fully-connected hidden layers. Deep networks are typically three or more layers deep, but each layer can have thousands of neurons. (Figure from the TowardsDataScience blog.)

Deep learning emerged as recently as 2006 and has opened wide new avenues of artificial intelligence that move beyond human capabilities for some tasks.  But deep learning also has pitfalls, some of which are inherited from the legacy approaches of traditional machine learning, and some of which are inherent in the massively high-dimensional spaces in which deep learning takes place.  Nonetheless, deep learning has revolutionized many aspects of science, and there is reason for optimism that the revolution will continue. Fifty years from now, looking back, we may recognize this as the fifth derivative of the industrial revolution (Phase I: Steam. Phase II: Electricity. Phase III: Automation. Phase IV: Information. Phase V: Intelligence).

From Multivariate Analysis to Deep Learning

Conventional machine learning, as we know it today, has had many names.  It began with Multivariate Analysis of mathematical population dynamics around the turn of the last century, pioneered by Francis Galton (1874), Karl Pearson (1901), Charles Spearman (1904) and Ronald Fisher (1922) among others.

The first on-line computers during World War II were developed to quickly calculate the trajectories of enemy aircraft for gunnery control, introducing the idea of feedback control of machines. This was named Cybernetics by Norbert Wiener, who had participated in the development of automated control of antiaircraft guns.

Table I. Evolution of Names for Machine Learning

A decade later, during the Cold War, it became necessary to find hidden objects in large numbers of photographs.  The embryonic electronic digital computers of the day were far too slow with far too little memory to do the task, so the Navy contracted with the Cornell Aeronautical Laboratory in Cheektowaga, New York, a suburb of Buffalo, to create an analog computer capable of real-time image analysis.  This led to the invention of the Perceptron by Frank Rosenblatt as the first neural network-inspired computer [1], building on ideas of neural logic developed by Warren McColloch and Walter Pitts. 

Fig. 2 Frank Rosenblatt working on the Perceptron. (From the Cornell Chronicle)
Fig. 3 Rosenblatt’s conceptual design of the connectionism of the perceptron (1958).

Several decades passed with fits and starts as neural networks remained too simple to accomplish anything profound.  Then in 1986, David Rumelhart and Ronald Williams at UC San Diego with Geoff Hinton at Carnegie-Mellon discovered a way to train multiple layers of neurons, in a process called error back propagation [2].  This publication opened the floodgates of Connectionism — also known as Parallel Distributed Processing.  The late 80’s and much of the 90’s saw an expansion of interest in neural networks, until the increasing size of the networks ran into limits caused by the processing speed and capacity of conventional computers towards the end of the decade.  During this time it had become clear that the most interesting computations required many layers of many neurons, and the number of neurons expanded into the thousands, but it was difficult to train such systems that had tens of thousands of adjustable parameters, and research in neural networks once again went into a lull.

The beginnings of deep learning started with two breakthroughs.  The first was by Yann Lecun at Bell Labs in 1998 who developed, with Leon Bottou, Yoshua Bengio and Patrick Haffner, a Convolutional Neural Network that had seven layers of neurons that classified hand-written digits [3]. The second was from Geoff Hinton in 2006, by then at the University of Toronto, who discovered a fast learning algorithm for training deep layers of neurons [4].  By the mid 2010’s, research on neural networks was hotter than ever, propelled in part by several very public successes, such as Deep Mind’s machine that beat the best player in the world at Go in 2017, self-driving cars, personal assistants like Siri and Alexa, and YouTube recommendations.

The Challenges of Deep Learning

Deep learning today is characterized by neural network architectures composed of many layers of many neurons.  The nature of deep learning brings with it two main challenges:  1) efficient training of the neural weights, and 2) generalization of trained networks to perform accurately on previously unseen data inputs.

Solutions to the first challenge, efficient training, are what allowed the deep revolution in the first place—the result of a combination of increasing computer power with improvements in numerical optimization. This included faster personal computers that allowed nonspecialists to work with deep network programming environments like Matlab’s Deep Learning toolbox and Python’s TensorFlow.

Solutions to the second challenge, generalization, rely heavily on a process known as “regularization”. The term “regularization” has a slippery definition, an obscure history, and an awkward sound to it. Regularization is the noun form of the verb “to regularize” or “to make regular”. Originally, regularization was used to keep certain inverse algorithms from blowing up, like inverse convolutions, also known as deconvolution. Direct convolution is a simple mathematical procedure that “blurs” ideal data based on the natural response of a measuring system. However, if one has experimental data, one might want to deconvolve the system response from the data to recover the ideal data. But this procedure is numerically unstable and can “blow up”, often because of the divide-by-zero problem. Regularization was a simple technique that kept denominators from going to zero.

Regularization became a common method for inverse problems that are notorious to solve because of the many-to-one mapping that can occur in measurement systems. There can be a number of possible causes that produce a single response. Regularization was a way of winnowing out “unphysical” solutions so that the physical inverse solution remained.

During the same time, regularization became a common tool used by quantum field theorists to prevent certain calculated quantities from diverging to infinity. The solution was again to keep denominators from going to zero by setting physical cut-off lengths on the calculations. These cut-offs were initially ad hoc, but the development of renormalization group theory by Kenneth Wilson at Cornell (Nobel Prize in 1982) provided a systematic approach to solving the infinities of quantum field theory.

With the advent of neural networks, having hundreds to thousands to millions of adjustable parameters, regularization became the catch-all term for fighting the problem of over-fitting. Over-fitting occurs when there are so many adjustable parameters that any training data can be fit, and the neural network becomes just a look-up table. Look-up tables are the ultimate hash code, but they have no predictive capability. If a slightly different set of data are fed into the network, the output can be anything. In over-fitting, there is no generalization, the network simply learns the idiosyncrasies of the training data without “learning” the deeper trends or patterns that would allow it to generalize to handle different inputs.

Over the past decades a wide collection of techniques have been developed to reduce over-fitting of neural networks. These techniques include early stopping, k-fold holdout, drop-out, L1 and L2 weight-constraint regularization, as well as physics-based constraints. The goal of all of these techniques is to keep neural nets from creating look-up tables and instead learning the deep codependencies that exist within complicated data.

Table II. Regularization Techniques in Machine Learning

By judicious application of these techniques, combined with appropriate choices of network design, amazingly complex problems can be solved by deep networks and they can generalized too (to some degree). As the field moves forward, we may expect additional regularization tricks to improve generalization, and design principles will emerge so that the networks no longer need to be constructed by trial and error.

The Potential of Deep Learning

In conventional machine learning, one of the most critical first steps performed on a dataset has been feature extraction. This step is complicated and difficult, especially when the signal is buried either in noise or in confounding factors (also known as distractors). The analysis is often highly sensitive to the choice of features, and the selected features may not even be the right ones, leading to bad generalization. In deep learning, feature extraction disappears into the net itself. Optimizing the neural weights subject to appropriate constraints forces the network to find where the relevant information lies and what to do with it.

The key to finding the right information was not just having many neurons, but having many layers, which is where the potential of deep learning emerges. It is as if each successive layer is learning a more abstract or more generalized form of the information than the last. This hierarchical layering is most evident in the construction of convolutional deep networks, where the layers are receiving a telescoping succession of information fields from lower levels. Geoff Hinton‘s Deep Belief Network, which launched the deep revolution in 2006, worked explicitly with this hierarchy in mind through direct design of the network architecture. Since then, network architecture has become more generalized, with less up-front design while relying on the increasingly sophisticated optimization techniques of training to set the neural weights. For instance, a simplified instance of a deep network is shown in Fig. 4 with three hidden layers of neurons.

Fig. 4 General structure of a deep network with three hidden layers. Layers will typically have hundreds or thousands of neurons. Each gray line represents a weight value, and each circle is a neural activation function.

The mathematical structure of a deep network is surprisingly simple. The equations for the network in Fig. 4, that convert an input xa to an output ye, are

These equations use index notation to denote vectors (single superscript) and matrices (double indexes). The repeated index (one up and one down) denotes an implicit “Einstein” summation. The function φ(.) is known as the activation function, which is nonlinear. One of the simplest activation functions to use and analyze, and the current favorite, is known as the ReLU (for rectifying linear unit). Note that these equations represent a simple neural cascade, as the output of one layer becomes the input for the next.

The training of all the matrix elements assumes a surprisingly simply optimization function, known as an objective function or a loss function, that can look like

where the first term is the mean squared error of the network output ye relative to the desired output y0 for the training set, and the second term, known as a regularization term (see section above) is a quadratic form that keeps the weights from blowing up. This loss function is minimized over the set of adjustable matrix weights.

The network in Fig. 4 is just a toy, with only 5 inputs and 5 outputs and only 23 neurons. But it has 30+36+36+30+23 = 155 adjustable weights. If this seems like overkill, but it is nothing compared to neural networks with thousands of neurons per layer and tens of layers. That massive overkill is exactly the power of deep learning — as well as its pitfall.

The Pitfalls of Deep Learning

Despite the impressive advances in deep learning, serious pitfalls remain for practitioners. One of the most challenging problems in deep learning is the saddle-point problem. A saddle-point in an objective function is like a mountain pass through the mountains: at the top of the pass it slopes downward in two opposite directions into the valleys but slopes upward in the two orthogonal directions to mountain peaks. A saddle point is an unstable equilibrium where a slight push this way or that can lead the traveller to two very different valleys separated by high mountain ridges. In our familiar three-dimensional space, saddle points are relatively rare and landscapes are dominated by valleys and mountaintops. But this intuition about landscapes fails in high dimensions.

Landscapes in high dimensions are dominated by neutral ridges that span the domain of the landscape. This key understanding about high-dimensional space actually came from the theory of evolutionary dynamics for the evolution of species. In the early days of evolutionary dynamics, there was a serious challenge to understand how genetic mutations could allow such diverse speciation to occur. If the fitness of a species were viewed as a fitness landscape, and if a highly-adapted species were viewed as a mountain peak in this landscape, then genetic mutations would need to drive the population state point into “valleys of death” that would need to be crossed to arrive at a neighboring fitness peak. It would seem that genetic mutations would likely kill off the species in the valleys before they could rise to the next equilibrium.

However, the geometry of high dimensions does not follow this simple low-dimensional intuition. As more dimensions come available, landscapes have more and more ridges of relatively constant height that span the full space (See my recent blog on random walks in 10-dimensions and my short YouTube video). For a species to move from one fitness peak to another fitness peak in a fitness landscape (in ultra-high-dimensional mutation space), all that is needed is for a genetic mutation to step the species off of the fitness peak onto a neutral ridge where many mutations can keep the species on that ridge as it moves ever farther away from the last fitness peak. Eventually, the neutral ridge brings the species near a new fitness peak where it can climb to the top, creating a new stable species. The point is that most genetic mutations are neutral — they do not impact the survivability of an individual. This is known as the neutral network theory of evolution proposed by Motoo Kimura (1924 – 1994) [5]. As these mutation accumulate, the offspring can get genetically far from the progenitor. And when a new fitness peak comes near, many of the previously neutral mutations can come together and become a positive contribution to fitness as the species climbs the new fitness peak.

The neutral network of genetic mutation was a paradigm shift in the field of evolutionary dynamics, and it also taught everyone how different random walks in high-dimensional spaces are from random walks in 3D. But although neutral networks solved the evolution problem, they become a two-edged sword in machine learning. On the positive side, fitness peaks are just like the minima of objective functions, and the ability for partial solutions to perform random walks along neutral ridges in the objective-function space allows optimal solutions to be found across a broad range of the configuration space of the neural weights. However, on the negative side, ridges are loci of unstable equilibrium. Hence there are always multiple directions that a solution state can go to minimize the objective function. Each successive run of a deep-network neural weight optimizer can find equivalent optimal solutions — but they each can be radically different. There is no hope of averaging the weights of an ensemble of networks to arrive at an “average” deep network. The averaging would simply drive all weights to zero. Certainly, the predictions of an ensemble of equivalently trained networks can be averaged—but this does not illuminate what is happening “under the hood” of the machine, which is where our own “understanding” of what the network is doing would come from.

Post-Modern Machine Learning

Post-modernism is admittedly kind of a joke — it works so hard to pull down every aspect of human endeavor that it falls victim to its own methods. The convoluted arguments made by its proponents sound like ultimate slacker talk — circuitous logic circling itself in an endless loop of denial.

But post-modernism does have its merits. It surfs on the moving crest of what passes as modernism, as modernism passes onward to its next phase. The philosophy of post-modernism moves beyond rationality in favor of a subjectivism in which cause and effect are blurred.  For instance, in post-modern semiotic theory, a text or a picture is no longer an objective element of reality, but fragments into multiple semiotic versions, each one different for each different reader or watcher — a spectrum of collaborative efforts between each consumer and the artist. The reader brings with them a unique set of life experiences that interact with the text to create an entirely new experience in each reader’s mind.

Deep learning is post-modern in the sense that deterministic algorithms have disappeared. Instead of a traceable path of sequential operations, neural nets scramble information into massively-parallel strings of partial information that cross and interact nonlinearly with other massively-parallel strings. It is difficult to impossible to trace any definable part of the information from input to output. The output simply manifests some aspect of the data that was hidden from human view.

But the Age of Artificial Intelligence is not here yet. The vast multiplicity of saddle ridges in high dimensions is one of the drivers for one of the biggest pitfalls of deep learning — the need for massive amounts of training data. Because there are so many adjustable parameters in a neural network, and hence so many dimensions, a tremendous amount of training data is required to train a network to convergence. This aspect of deep learning stands in strong contrast to human children who can be shown a single picture of a bicycle next to a tricycle, and then they can achieve almost perfect classification accuracy when shown any number of photographs of different bicycles and tricycles. Humans can generalize with an amazingly small amount of data, while deep networks often need thousands of examples. This example alone points to the marked difference between human intelligence and the current state of deep learning. There is still a long road ahead.

By David D. Nolte, April 18, 2022

[1] F. Rosenblatt, “THE PERCEPTRON – A PROBABILISTIC MODEL FOR INFORMATION-STORAGE AND ORGANIZATION IN THE BRAIN,” Psychological Review, vol. 65, no. 6, pp. 386-408, (1958)

[2] D. E. Rumelhart, G. E. Hinton, and R. J. Williams, “LEARNING REPRESENTATIONS BY BACK-PROPAGATING ERRORS,” Nature, vol. 323, no. 6088, pp. 533-536, Oct (1986)

[3] LeCun, Yann; Léon Bottou; Yoshua Bengio; Patrick Haffner (1998). “Gradient-based learning applied to document recognition”. Proceedings of the IEEE. 86 (11): 2278–2324.

[4] G. E. Hinton, S. Osindero, and Y. W. Teh, “A fast learning algorithm for deep belief nets,” Neural Computation, vol. 18, no. 7, pp. 1527-1554, Jul (2006)

[5] M. Kimura, The Neutral Theory of Molecular Evolution. Cambridge University Press, 1968.

by David D. Nolte

The Physics of Starflight: Proxima Centauri b or Bust!

The ability to travel to the stars has been one of mankind’s deepest desires. Ever since we learned that we are just one world in a vast universe of limitless worlds, we have yearned to visit some of those others. Yet nature has thrown up an almost insurmountable barrier to that desire–the speed of light. Only by traveling at or near the speed of light may we venture to far-off worlds, and even then, decades or centuries will pass during the voyage. The vast distances of space keep all the worlds isolated–possibly for the better.

Yet the closest worlds are not so far away that they will always remain out of reach. The very limit of the speed of light provides ways of getting there within human lifetimes. The non-intuitive effects of special relativity come to our rescue, and we may yet travel to the closest exoplanet we know of.

Proxima Centauri b

The closest habitable Earth-like exoplanet is Proxima Centauri b, orbiting the red dwarf star Proxima Centauri that is about 4.2 lightyears away from Earth. The planet has a short orbital period of only about 11 Earth days, but the dimness of the red dwarf puts the planet in what may be a habitable zone where water is in liquid form. Its official discovery date was August 24, 2016 by the European Southern Observatory in the Atacama Desert of Chile using the Doppler method. The Alpha Centauri system is a three-star system, and even before the discovery of the planet, this nearest star system to Earth was the inspiration for the Hugo-Award winning sci-fi trilogy The Three Body Problem by Chinese author Liu Cixin, originally published in 2008.

It may seem like a coincidence that the closest Earth-like planet to Earth is in the closest star system to Earth, but it says something about how common such exoplanets may be in our galaxy.

Artist’s rendition of Proxima Centauri b. From WikiCommons.

Breakthrough Starshot

There are already plans to send centimeter-sized spacecraft to Alpha Centauri. One such project that has received a lot of press is Breakthrough Starshot, a project of the Breakthrough Initiatives. Breakthrough Starshot would send around 1000 centimeter-sized camera-carrying laser-fitted spacecraft with 5-meter-diameter solar sails propelled by a large array of high-power lasers. The reason there are so many of these tine spacecraft is because of the collisions that are expected to take place with interstellar dust during the voyage. It is possible that only a few dozen of the craft will finally make it to Alpha Centauri intact.

Relative locations of the stars of the Alpha Centauri system. From ScienceNews.

As these spacecraft fly by the Alpha Centauri system, possibly within one hundred million miles of Proxima Centauri b, their tiny HR digital cameras will take pictures of the planet’s surface with enough resolution to see surface features. The on-board lasers will then transmit the pictures back to Earth. The travel time to the planet is expected to be 20 or 30 years, plus the four years for the laser information to make it back to Earth. Therefore, it would take a quarter century after launch to find out if Proxima Centauri b is habitable or not. The biggest question is whether it has an atmosphere. The red dwarf it orbits sends out catastrophic electromagnetic bursts that could strip the planet of its atmosphere thus preventing any chance for life to evolve or even to be sustained there if introduced.

There are multiple projects under consideration for travel to the Alpha Centauri systems. Even NASA has a tentative mission plan called the 2069 Mission (100 year anniversary of the Moon landing). This would entail a single spacecraft with a much larger solar sail than the small starshot units. Some of the mission plans proposed star-drive technology, such as nuclear propulsion systems, rather than light sails. Some of these designs could sustain a 1-g acceleration throughout the entire mission. It is intriguing to do the math on what such a mission could look like, in terms of travel time. Could we get an unmanned probe to Alpha Centauri in a matter of years? Let’s find out.

Special Relativity of Acceleration

The most surprising aspect of deriving the properties of relativistic acceleration using special relativity is that it works at all. We were all taught as young physicists that special relativity deals with inertial frames in constant motion. So the idea of frames that are accelerating might first seem to be outside the scope of special relativity. But one of Einstein’s key insights, as he sought to extend special relativity towards a more general theory, was that one can define a series of instantaneously inertial co-moving frames relative to an accelerating body. In other words, at any instant in time, the accelerating frame has an inertial co-moving frame. Once this is defined, one can construct invariants, just as in usual special relativity. And these invariants unlock the full mathematical structure of accelerating objects within the scope of special relativity.

For instance, the four-velocity and the four-acceleration in a co-moving frame for an object accelerating at g are given by

The object is momentarily stationary in the co-moving frame, which is why the four-velocity has only the zeroth component, and the four-acceleration has simply g for its first component.

Armed with these four-vectors, one constructs the invariants

and

This last equation is solved for the specific co-moving frame as

But the invariant is more general, allowing the expression

which yields

From these, putting them all together, one obtains the general differential equations for the change in velocity as a set of coupled equations

The solution to these equations is

where the unprimed frame is the lab frame (or Earth frame), and the primed frame is the frame of the accelerating object, for instance a starship heading towards Alpha Centauri. These equations allow one to calculate distances, times and speeds as seen in the Earth frame as well as the distances, times and speeds as seen in the starship frame. If the starship is accelerating at some acceleration g’ other than g, then the results are obtained simply by replacing g by g’ in the equations.

Relativistic Flight

It turns out that the acceleration due to gravity on our home planet provides a very convenient (but purely coincidental) correspondence

With a similarly convenient expression

These considerably simplify the math for a starship accelerating at g.

Let’s now consider a starship accelerating by g for the first half of the flight to Alpha Centauri, turning around and decelerating at g for the second half of the flight, so that the starship comes to a stop at its destination. The equations for the times to the half-way point are

This means at the midpoint that 1.83 years have elapsed on the starship, and about 3 years have elapsed on Earth. The total time to get to Alpha Centauri (and come to a stop) is then simply

It is interesting to look at the speed at the midpoint. This is obtained by

which is solved to give

This amazing result shows that the starship is traveling at 95% of the speed of light at the midpoint when accelerating at the modest value of g for about 3 years. Of course, the engineering challenges for providing such an acceleration for such a long time are currently prohibitive … but who knows? There is a lot of time ahead of us for technology to advance to such a point in the next century or so.

Figure. Time lapsed inside the spacecraft and on Earth for the probe to reach Alpha Centauri as a function of the acceleration of the craft. At 10 g’s, the time elapsed on Earth is a little less than 5 years. However, the signal sent back will take an additional 4.37 years to arrive for a total time of about 9 years.

Matlab alphacentaur.m

% alphacentaur.m
clear
format compact

g0 = 1;
L = 4.37;

for loop = 1:100
    
    g = 0.1*loop*g0;
    
    taup = (1/g)*acosh(g*L/2 + 1);
    tearth = (1/g)*sinh(g*taup);
    
    tauspacecraft(loop) = 2*taup;
    tlab(loop) = 2*tearth;
    
    acc(loop) = g;
    
end

figure(1)
loglog(acc,tauspacecraft,acc,tlab,'LineWidth',2)
legend('Space Craft','Earth Frame','FontSize',18)
xlabel('Acceleration (g)','FontSize',18)
ylabel('Time (years)','FontSize',18)
dum = set(gcf,'Color','White');
H = gca;
H.LineWidth = 2;
H.FontSize = 18;

To Centauri and Beyond

Once we get unmanned probes to Alpha Centauri, it opens the door to star systems beyond. The next closest are Barnards star at 6 Ly away, Luhman 16 at 6.5 Ly, Wise at 7.4 Ly, and Wolf 359 at 7.9 Ly. Several of these are known to have orbiting exoplanets. Ross 128 at 11 Ly and Lyuten at 12.2 Ly have known earth-like planets. There are about 40 known earth-like planets within 40 lightyears from Earth, and likely there are more we haven’t found yet. It is almost inconceivable that none of these would have some kind of life. Finding life beyond our solar system would be a monumental milestone in the history of science. Perhaps that day will come within this century.

By David D. Nolte, March 23, 2022

Further Reading

R. A. Mould, Basic Relativity. Springer (1994)

D. D. Nolte, Introduction to Modern Dynamics : Chaos, Networks, Space and Time, 2nd ed.: Oxford University Press (2019)

by David D. Nolte

Democracy against Authoritarians: The Physics of Public Opinion

An old joke goes that Democracy is a terrible form of government … except it’s better than all the others!

Our world today is faced with conflict between democracy and dictatorship. On the one side is the free world, where leaders are chosen by some form of representation of large numbers of citizens and sometimes even a majority. On the other side is authoritarianism where a select few are selected by a select few to govern everyone else.

[I]t has been said that democracy is the worst form of Government except all those other forms that have been tried from time to time; but there is the broad feeling in our country that the people should rule, and that public opinion expressed by all constitutional means, should shape, guide, and control the actions of Ministers who are their servants and not their masters.

Winston Churchill (1947)

An argument in favor of democracy is freedom of choice for the largest segment of the population, plus the ability to remove leaders who fail to provide for the perceived welfare of the most citizens. This makes democracy adaptive, shifting with the times. It also makes leaders accountable for their actions and crimes. An argument in favor of authoritarianism is the myth of the benevolent dictator–someone who knows what’s best for the people even if the people don’t know it themselves.

But dictators are rarely benevolent, and as they become saturated with power, they are corrupted. The criminal massacres, perpetrated by Putin, of Ukrainian civilians is one of the strongest recent arguments against authoritarianism. A single man decides, on a whim, the life and death of thousands or maybe more. The invasion of Ukraine is so egregious and unwarranted, that we wonder how the Russian people can put up with their isolated and manic leader. Yet by some measure more than 60% of the people in Russia approve of the war.

How can the free world see the invasion as the atrocity it is, while Russia’s majority sees it as a just war? The answer is a surprising result of population dynamics known as the replicator-mutator equation. The challenge for us here in the free world is to learn how to game the replicator-mutator equation to break up the monopoly of popular opinion and make Putin pay for his arrogance. This blog explains how “mass hysteria” can arise from forces within a complex environment, and how to construct a possible antidote.

Replicator-Mutator Equation

There are several simple models of population dynamics that try to explain the rise and fall of the number of individuals that belong to varying cohorts within the population. These models incorporate aspects of relative benefit of one group over another, plus the chance to change sides–defection. The dynamics under these conditions can be highly nonlinear and highly non-intuitive. One of the simplest of these models is known as the replicator-mutator model where replication follows the fitness of the cohort, and where individuals can defect to a “more fit” cohort.

The basic dynamics of the model are

where xa is the fraction of the population that is in cohort a, Wab is a transition probability, and φ is the average fitness of the full population. The transition matrix is given by

where fb is the fitness of cohort b and Qba is a stochastic matrix that allows for defection of an individual from one cohort to another. The fitness of a cohort is given by

where pbc is the pay-off matrix for the relative benefit of one cohort at the expense of another. Finally the average fitness is

The Einstein implicit summation convention is assumed in all of these equations, and the metric space in which the dynamics are embedded is “flat” so that there is no essential difference between superscripts and subscripts. There is also a conservation law that the sum over all population fractions equals unity.

In the language of population dynamics, this model has frequency-dependent fitness, with defection and pay-off, in a zero-sum game.

One of the simplest questions to answer with this model is how so many people can come to believe one thing. This is known as “opinion uniformity”.

Uniformity versus Diversity

This replicator-mutator model explains the property of opinion uniformity, as well as the opposite extreme of opinion diversity. The starting point for both is the pay-off matrix pbc which is assumed to be unity on the diagonal for b = c and to a constant factor a for b ~= c. This pay-off is symmetric so that all opinions are equally “believable”. The stochastic defection matrix is close to unity on the diagonal, and has random terms on the off-diagonal that are proportional to a constant ε. The defection matrix allows a person from one cohort to defect to the belief system of another cohort if they believe that the new cohort has more merit. Cohorts with greater merit (fitness) gain more members over time, while cohorts with lower merit have fewer members over time.

Note that the fitness increases with the number of members in the cohort. This is the bandwagon effect. A belief system is perceived to have more merit if there are more people who believe it. This clearly creates a positive feedback that would cause this cohort to grow. Even though all the cohorts have this same positive feedback, the zero-sum rule only allows one of the cohorts to grow to its maximum extent, taking members away from all the other cohorts. This is illustrated in Fig. 1. One belief system wins, taking almost the full population with it.

Fig. 1 Population fractions evolving as a function of time for a = 0.5 and a small defection rate ε = 0.02. One winner takes almost all the population. These are two views of the same data on semilog and log-log.

What allows the winner to take all is the positive feedback where the fitness of the cohort increases with the number of members, combined with the ability for that cohort to take members from other cohorts through the defection matrix.

However, all of the cohorts are trying the same thing, and the pay-off matrix is fully symmetric and equal for all cohorts, so no cohort is intrinsically “better” than another. This property opens the door to a strong alternative to opinion uniformity. In fact, as more members are allowed to defect, it creates a trend counter to winner-take-all, helping to equalize the cohorts. Suddenly, a bifurcation is passed when the winner-take-all converts discontinuously to a super-symmetric situation when all opinions are held by equal numbers of people. This is illustrated in Fig. 2 for a slightly higher defection rate ε = 0.03. The parameters are identical to those in Fig. 1, but the higher defection rate stabilizes the super-symmetric state of maximum diversity.

Fig. 2 Population fractions for higher defection rate of 0.03. In super-symmetric state, all opinions are held at the same rate with maximum diversity.

These two extreme results of the replicator-mutator equation, that switch suddenly from one to the other dependent on the defection rate, may seem to produce solutions neither of which are ideal for a healthy democracy. One the one hand, in the uniform case where the wining opinion is monolithic, everyone is a carbon-copy of everyone else, which is a form of cultural death (lack of diversity). But, on the other hand, one might argue that maximum opinion diversity is just as concerning, because no-one can agree on anything. If all opinions are equivalent, then everyone in the society believes something different and there is no common ground. But in the diversity case, at least there is no state-level control of the population. In the case of opinion uniformity, the wining opinion can be manipulated by propaganda.

The Propaganda Machine

A government can “seed” the belief networks with propaganda that favors the fitness of what they want their citizens to hear. Because of the positive feedback, any slight advantage of one opinion over others can allow that opinion to gain large numbers through the bandwagon effect. Of course, even stronger control that stifles dissent, for instance by shutting down the free press, makes it that much more likely that the state-controlled story is believed. This may be one reason for the 60% (as of the writing of this blog) support Putin’s war, despite the obvious lies that are being told. This is illustrated in Fig. 3 by boosting the payoff between two similar lies that the government wants its people to believe. These rise to take about 60% of the population. Members of the cohort are brain-washed, not by the government alone, but by all their neighbors who are parroting the same thing.

Fig. 3 Government propaganda acts as a “seed” that makes the propaganda grow faster than other beliefs, even for a defection rate of 0.03 which is above the threshold of Fig. 2.

Breaking the Monopoly of Thought

How do we fight back? Not just against the Kremlin’s propaganda, but also against QANON and Trump’s Big Lie and the pernicious fallacy of nationalism? The answer is simple: diversity of thought! The sliver bullet in the replicator-mutator model is the defection matrix. The existence of a bifurcation means that a relatively small increase in the amount of diverse opinion, and the freedom to swap opinions, can lead to a major qualitative collapse of the monolithic thought, even when supported by government propaganda, as shown in Fig. 4. More people may still believe in the state-supported propaganda than the others, but it is no longer a majority.

Fig. 4 Increasing the defection rate can help equalize free opinions against the state-supported propaganda

The above models were all very homogeneous. It is more realistic that people are connected through small-world networks. In this case, there is much more diversity, as shown in Fig. 5, although the defection rate needs to be much higher to prevent a monolithic opinion from dominating. The state-supported propaganda is buried in the resulting mix of diverse ideas. Therefore, to counteract state control, people must feel free to hop about in their choice of beliefs and have access to other beliefs.

Fig. 5 The defection matrix is multiplied by the adjacency matrix of a small-world network. There is significant diversity of thought, but a relatively high defection rate is needed. The state-supported propaganda is buried in this mix.

This is a bit paradoxical. On the one hand, the connectivity of the internet has fostered the rise of conspiracy theories and other odd-ball ideas. But sustained access to multiple sources of information is the best defense against all that crazy stuff winning out. In other words, not only do we have to put up with the lunatic fringe if we are to have full diversity of thought, but we need to encourage everyone to feel free to “shop around” for different ideas, even if some of them are crazy. Our free society shouldn’t be cancelling people who have divergent opinions, because that sets us down the path to authoritarianism. As a recent add said in the New York Times, “Cancel culture cancels culture.” Unfortunately, authoritarianism is on the rise around the world, and the US almost suffered that fate on Jan. 6, 2021. Furthermore, with Xi aligning with Putin and giving him the green light on Ukraine–cynically on the eve of the Olympic Games (of peace)–the new world order will revolve around that axis for decades to come, if the world survives that long. Diversity and freedom may be the only antidote.

By David D. Nolte, March 24, 2022

Matlab Program: Repmut.m

function repmut
% https://github.itap.purdue.edu/nolte/Matlab-Programs-for-Nonlinear-Dynamics

clear
format compact

N = 63;     
p = 0.5;

mutype = 1;     % 0 = Hamming   1 = rand
pay = 1;        % 0 = Hamming   1 = 1/sqrt(N) 
ep = 0.5;      % average mutation rate: 0.1 to 0.01 typical  (0.4835)

%%%%% Set original population
x0temp = rand(1,N);    % Initial population
sx = sum(x0temp);
y0 = x0temp/sx;
Pop0 = sum(y0);


%%%%% Set Adjacency

%node = makeglobal(N);
%node = makeER(N,0.25);       % 0.5     0.25 
%node = makeSF(N,6);       % 12         6
node = makeSW(N,7,0.125);   % 15,0.5    7,0.5
[Adj,degree,Lap] = adjacency(node);

%%%%%% Set Hamming distance
for yloop = 1:N
    for xloop = 1:N
        H(yloop,xloop) = hamming(yloop-1,xloop-1);
    end
end

%%%%%%% Set Mutation matrix
if mutype == 0
    Qtemp = 1./(1+H/ep);    %Mutation matrix on Hamming
    Qsum = sum(Qtemp,2);
    mnQsum = mean(Qsum);
    
    % Normalize mutation among species
    for yloop = 1:N
        for xloop = 1:N
            Q(yloop,xloop) = Qtemp(yloop,xloop)/Qsum(xloop);
        end
    end
    
elseif mutype == 1  
    S = stochasticmatrix(N);
    Stemp = S - diag(diag(S));
    Qtemp = ep*Stemp;
    sm = sum(Qtemp,2)';
    Q = Qtemp + diag(ones(1,N) - sm);
end

figure(1)
imagesc(Q)
title('Mutation Matrix')
colormap(jet)

%%%%%%% Set payoff matrix
if pay == 1
    payoff = zeros(N,N);
    for yloop = 1:N
        payoff(yloop,yloop) = 1;
        for xloop = yloop + 1:N
            payoff(yloop,xloop) = p;
            payoff(xloop,yloop) = p;
            payoff(1,N) = 1;    % Propaganda
            payoff(N,1) = 1;
        end
    end
elseif pay == 0
    payoff = zerodiag(exp(-1*H));
end

figure(2)
imagesc(payoff)
title('Payoff Matrix')
colormap(jet)

% Run time evolution
tspan = [0 4000];
[t,x] = ode45(@quasispec,tspan,y0);

Pop0
[sz,dum] = size(t);
Popend = sum(x(sz,:))

for loop = 1:N
    fit(loop) = sum(payoff(:,loop)'.*x(sz,:));
end

phistar = sum(fit.*x(sz,:))       % final average fitness

xend = x(sz,:)
sortxend = sort(xend,'descend');
coher = sum(sortxend(1:2))

figure(3)
clf
h = colormap(lines);
for loop = 1:N
    plot(t,x(:,loop),'Color',h(round(loop*64/N),:),'LineWidth',1.25)
    hold on
end
hold off

figure(4)
clf
for loop = 1:N
    semilogx(t,x(:,loop),'Color',h(round(loop*64/N),:),'LineWidth',1.25)
    hold on
end
hold off

figure(5)
clf
for loop = 1:N
    semilogy(t,x(:,loop),'Color',h(round(loop*64/N),:),'LineWidth',1.25)
    hold on
end
hold off

figure(6)
clf
for loop = 1:N
    loglog(t,x(:,loop),'Color',h(round(loop*64/N),:),'LineWidth',1.25)
    hold on
end
hold off

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    function yd = quasispec(~,y)
        
        for floop = 1:N
            f(floop) = sum(payoff(:,floop).*y);
        end
        
        Connect = Adj + eye(N);
        
        % Transition matrix
        for yyloop = 1:N
            for xxloop = 1:N
                W(yyloop,xxloop) = f(yyloop)*(Connect(yyloop,xxloop)*Q(yyloop,xxloop));
            end
        end
        
        phi = sum(f'.*y);   % Average fitness of population
        
        yd = W*y - phi*y;
        
    end     % end quasispec
end

Further Reading

M. A. Nowak, Evolutionary Dynamics: Exploring the Equations of Life. Cambridge, Mass.: Harvard University Press, 2006.