I had submitted a report for evaluation of the term project, which I have, similar to my previous blog, converted from LaTex to Markdown using the All-Mighty pandoc. So I invite you to brew a cup of masala chai, put your glasses on, and start reading this terribly written blogpost of mine.

History

In light of the new MANIAC(Metropolis and von Neumann Install Awful Computers, acronym courtesy Gamow) computer, Fermi, Pasta, Ulam, and Tsingou[13] set out to study some physical problem using numerical simulations. They chose to study a 1D chain of particles connected by nonlinear springs. Their goal was to understand how energy would distribute among the normal modes of the system over time, expecting that the nonlinearity would lead to thermalization and equipartition of energy among the modes.

For the more interested reader with higher levels of motivation and reading prowess than the nincompoop author of this blog, can also refer to the extensive review by Giovanni Gallavotti[6]

Problem Setup

The FPUT problem considers a one-dimensional chain of \(N\) particles connected by springs with linear and nonlinear restoring forces. The Hamiltonian is as follows:

where \(p_i\) is the momentum of the \(i^{th}\) particle, \(x_i\) is its position, and \(\alpha, \beta\) are the coefficients for the nonlinear terms. The boundary conditions are fixed at both ends: \(x_0 = x_{N+1} = 0\). The equations of motion derived from the Hamiltonian are given by:

for \(i = 1, 2, \ldots, N\), with appropriate boundary conditions.

Normal Modes

For the linear Hamiltonian \((H_0)\) , we can express the displacements in terms of the normal mode coordinates \(Q_k\):

where \(k = 1, 2, \ldots, N\). Each normal mode oscillates with its own frequency \(\omega_k\):

The normal mode coordinates are easily found using the Fourier expansion and plugging in the appropriate boundary conditions. Using the normal mode coordinates, the equations of motion for the linear system can be written as:

with the \(k^{th}\) mode energy given by:

Thus, we can decouple the oscillators in the linear case. No energy exchange between modes. The original Hamiltonian with linear springs \((H_0)\) would lead to normal modes of vibration, each with a specific frequency. The expectation was that the addition of nonlinear terms\((H_1)\) with perturbation, either \(\alpha\) or \(\beta\), would cause the energy to spread out among these modes over time, leading to thermalization and equipartition of energy.

For the \(\alpha\) model, the equations of motion in terms of normal modes become:

where \(C_{j l k}\) are coupling coefficients that depend on the mode indices. Similarly, for the \(\beta\) model, we have:

where \(D_{j l m k}\) are the corresponding coupling coefficients.

A Classical Mechanics Refresher

A constant of motion is a function \(F(q_i, p_i, t)\) that remains constant along the trajectories of the system in phase space for a given Hamiltonian \(H(q_i, p_i, t)\). Mathematically, \(F\) is a constant of motion if its total time derivative vanishes:

For example, in a system with a time-independent Lagrangian, the Hamiltonian itself is a constant of motion, representing the total energy of the system.

Integrability

A Hamiltonian system with \(N\) degrees of freedom is said to be integrable if it possesses \(N\) independent constants of motion. In such systems, the equations of motion simplify significantly, and the dynamics can be described as linear motion on an invariant tori in phase space.

More formally, a Hamiltonian \(H(q_i, p_i)\) is integrable if there exists a single-valued analytical canonical transformation to action-angle variables \((J_i, \theta_i)\) such that the Hamiltonian depends only on the action variables:

In these variables, the equations of motion become:

where \(\omega_i(J) = \frac{\partial H}{J_i}\) are the frequencies of the system. This motion can be mapped to an invariant torus in the appropriate phase space. The action represents radii of the tori, while the angle variable describes the evolution on the surface of the tori.

A result a la Poincaré

Consider perturbed Hamiltonians of the form:

where \(\epsilon\) is a small parameter, \(H_0(J)\) is integrable, and the frequencies of the unperturbed hamiltonian \(\omega_i = \frac{\partial H_0}{J_i}\) are functionally independent.

A nice sketch of this proof can be seen at [5]. Basically, when Poincaré starts with the Hamiltonian \(H = H_0 + \mu H_1\). Then he looks for constants of the motion of the form:

where \(Q,P\) are the position and momentum variables and where all \(\Phi_k\) are all analytic functions of \(Q,P\). Specifically, since \(\Phi\) is a constant of the motion, he inserts the above expression in the Poisson Bracket equation \(\{H,\Phi\} = 0\) and then insists that for each coefficient of each \(\mu^k\) equal to zero. This yields \(\{H_0,\Phi_0\} = 0\), which means \(\{H_0,\Phi_k\} = - \{H_0, \Phi_{k-1}\} ~ \forall ~ k>0\). We can solve these equations iteratively for all \(\Phi_k\) once \(\Phi_0\), which is a constant of motion for the integrble Hamiltonian, is specified. Poincaré wants to analytically continue \(\Phi_0\). Then he shows that this is not possible due to the presence of denominators which become zero for a dense set of hypersurfaces. As a result, the only constant of motion that can be analytically continued is the Hamiltonian itself.

Fermi’s Proof

Fermi started out his career trying to prove the ergodic hypothesis. He proved that the class of Poincaré-type Hamiltonians are ergodic. The proof sketch is quite straightforward. He proceeds via contradiction. Suppose, the system is not ergodic. In consequence, at least two distinct regions of phase space exist that are invariant under Hamiltonian flow. Fermi now asserts that there must a surface separating these two regions. This surface must be a constant of motion, since the Hamiltonian flow cannot cross it. However, by Poincaré’s result, no such constant of motion exists for Poincaré-type Hamiltonians. Therefore, the system must be ergodic. The problem with this proof is that Fermi assumed that the separating surface is analytic, which is not necessarily true. The KAM theorem shows that these surfaces are disjoint sets that fill most of the phase space, but are not analytic.

Under a canonical transformation, the FPUT Hamiltonian can be expressed as a Poincaré-type Hamiltonian. Therefore, by Fermi’s proof, the FPUT system is ergodic. We cannot measure ergodicity directly. However, since Fermi’s Proof dictates that the FPUT system is ergodic, and therefore should follow the principle of equal a priori probabilities. This would imply that over long times, the law of equipartition of energy should hold. This is how the original authors wanted to verify Fermi’s proof.

A more detailed explanation of this connection will be provided later in the report.

The Problem with The Problem

However, the numerical simulations by Fermi, Pasta, Ulam and Tsingou showed that the system did not thermalize as expected. Instead of energy spreading out evenly among all modes, it exhibited a phenomenon known as "recurrence," where the energy returned to the initially excited mode after some time.

Numerical Simulation

To verify the results of Fermi, Pasta, Ulam and Tsingou, we perform our own numerical simulations of the FPUT system. We used both Euler and Velocity-Verlet integration methods to solve the equations of motion.

• The Euler method is a simple first-order method, but it can be less accurate and does not conserve energy well over long simulations.

• The Velocity-Verlet method is a second-order symplectic integrator that is more accurate and better at conserving energy in Hamiltonian systems.

  A nice handout on these simulations can be found at [7]

Euler Method

Velocity Verlet

We have seen that the numerical simulations do not match the theoretical predictions by the original creators of this model. There are some questions that need to answered, in light of this discrepancy between the theoretical and numerical predictions.

Fermi’s Folly

The proof by Fermi has been shown to be incorrect. To see this, a landmark theorem by Kolmogorov, Arnold and Moser is required.

The KAM theorem provides a resolution to why the FPUT system does not thermalize for small perturbations. The surviving invariant tori still show signatures of integrable behavior, preventing ergodicity. Fermi in his proof assumed that the surface dividing the regions of phase space invariant under Hamiltonian flow, is analytic. The work by Kolmogorov, shows that these surfaces are not analytic, and are in fact "pathological monstrosities".

Inquisitions

We try to answer some questions that can be asked and people have studied.

• Is the FPUT system integrable or non-integrable? Note Poincaré’s result states that we can’t analytically continue the constants of motion from the unperturbed system to the perturbed system. It does not say anything about introducing new constants of motion.

• Does the FPUT system not thermalize at all, or does it thermalize over very long timescales?

• Does the FPUT system thermalize for certain initial conditions and not for others?

• Is the FPUT system ergodic after all?

Let’s explore these questions one by one.

Is the FPUT system integrable or non-integrable?

People have thought for some time that the FPUT system might be integrable after all. However, the simple answer to our question is NO. Let us consider the \(\alpha\) model with \(N=3\) masses with periodic boundary conditions. The Hamiltonian is given by:

Under a simple canonical transformation, we obtain the Hénon-Heiles Hamiltonian, which is a well known non-integrable system, that exhibits chaotic behavior above a certain energy threshold.

Hénon Heiles Hamiltonian

The Hénon-Heiles Hamiltonian is given by:

This system was originally proposed to model the motion of a star around a galactic center. For energies above a certain threshold, the system exhibits chaotic behavior, with trajectories that are highly sensitive to initial conditions.

As a result, we get a clue that the FPUT system might be chaotic for higher perturbations or energies.

Does the FPUT system not thermalize at all, or does it thermalize over very long timescales?

After some time of evolution, we see a dip in the energy of the first mode. People had conjectured that over longer timescales, the system might thermalize.

Super Period

Even though there an energy dip after some time, after observing a bit longer, we see some sort of super recurrence. Almost all of the energy( over 99%) flows back into the first mode. The super period was first observed by Tuck and Tsingou(then Menzel) in 1972.

We simulated the same using both Euler and Velocity Verlet algorithms. We observe super periods in both methods, however for the euler method, quite less than 99% of energy returns to the first mode. This problem has been eliminated when the velocity verlet method is used.

There have been some studies that suggest that the FPUT system might thermalize over extremely long timescales. Similar to glassy behavior in condensed matter systems, the FPUT system might be stuck in a metastable state for long times before eventually reaching thermal equilibrium. KAM theorem hints that for small perturbations, there are invariant tori that survive. Therefore, for initial conditions lying on these tori, the system will not thermalize. Building on this Idea F.M. Izrailev and B.V. Chirikov proposed the stochasticity threshold, also called the Chirikov criterion[4]. According to this criterion, when the perturbation strength exceeds a certain threshold, the invariant tori break down, leading to widespread chaos in phase space and subsequent thermalization.The original FPUT paper considers two models, the \(\alpha\) and \(\beta\) models, with certain initial conditions. Israilev and Chirikov showed for the \(\beta\) model,the initial conditions lie below the stochasticity threshold for the respective models, explaining the lack of thermalization.

There is region of conditions where due to the KAM theorem, the system does not thermalize. This criterion is a property of a lot of other systems as well. The system here exhibits Kolmogorov Stability. The criterion shows that very high nonlinear couplings are required for thermalization when low frequency modes are excited initially. For higher modes and large \(N\), the threshold is much lower, and thermalization can occur for smaller nonlinear couplings.

Resolutions

Which neighbour is Integrable?

This energy recurrence phenomenon is explained by the fact that there is an integrable Hamiltonian in the neighbourhood of the FPUT system.(We can map a Hamiltonian to a space where this sentence makes perfect sense). That cannot be the linear model, since the Poincare surfaces of the FPU are quite different from that of the linear one. In the quest for finding a suitable candidate, two of them have been the leading candidates,

1. The KdV equation.

2. The toda Lattice

The KdV Equation

We shall look at the connection to the KdV equation in more detail. The FPUT system can be regarded a discrete approximation to the integrable Korteweg-de Vries (KdV) equation. Let’s see how. We shall consider the \(\alpha\) model for simplicity. The connection to the KdV equation is a tiny bit technical, so bear with me. We will show that the naive approach to the continuum limit does not work, and leads to unphysical results. We follow the approach by [12].

The FPUT equations of motion for the \(\alpha\) model with arbitrary \(m\) and \(k\) are given by:

We approximate the spring mass system as a continuous string of length L. Let the equilibrium positions of the masses be given by \(x_i^0 = ih\), where \(h = L/(N+1)\) is the spacing between masses. Denote \(\rho\) is the density of the string, then \(m = \rho h\). Let \(\kappa\) denotes the Young’s modulus for the string (i.e., the spring constant for a piece of unit length) Then k = \(\kappa/h\) will be the spring constant for a piece of length h.

Defining \(c = \sqrt{\kappa/\rho}\) we obtain,

Let \(u(x,t)\) be the function measuring the displacement of the string from equilibrium at position \(x\) and time \(t\). Let us analyse for one particular \(x=x_i\) Then,

• \(x_i(t) = u(x, t)\)

• \(x_{i+1}(t) = u(x + h, t)\)

• \(x_{i-1}(t) = u(x - h, t)\)

We can see that \(\ddot{x}_i = u_{tt} (x, t)\). Using Taylor series expansion about \(x\), we have,

Similarly,

We arrive at,

where \(\epsilon = 2 \alpha h\). We obtain the PDE,

One can draw parallels between this equation and the inviscid Burgers equation, which is known to develop shocks in finite time, for generic initial conditions. For wave like solutions, the rising part of the wave with goes faster with \(u_x > 0\) than the fallling part with \(u_x < 0\), leading to wave steepening and shock formation. One can see how following the inviscid Burgers equation develops shocks over a finite time. This happens over a characteristic time scale \(t_s\) which is found to be much smaller than the recurrence time observed in the FPUT simulations.

This is unphysical, since the FPUT system with small nonlinearity does not exhibit such shock formation. To resolve this issue, we follow the approach by Zabusky and Kruskal. The correct approach is to keep higher order terms in the Taylor series expansion. Keeping terms upto order \(h^2\), we obtain the PDE,

The additional fourth order derivative term acts as a dispersive term, preventing shock formation. We now differentiate w.r.t. \(x\) and define \(w = u_x\), to obtain,

This is known as the Boussinesq equation. This admits wave like solutions that do not form shocks, these are periodic water waves.

Note that for small values \(\alpha\) and \(h\), the wave like solutions should qualitatively behave like solutions to the linear wave equation. In general, the solutions will be superpositions of right and left moving waves. Here, these two cases are treated differently. To be specific, let us consider only right moving waves. We would like to look for solutions, such that behave more and more like right moving waves for longer and longer times as \(\alpha, h \to 0\).

To interpret this physically, the function \(u(x,t) = y(x - ct, \alpha h c t)\) uniformly approximates the right moving wave \(u^0(x,t) = y(x - ct, \alpha h c t_0)\) over the time interval \(|t - t_0| < T\). To restate this, \(u(x,t) = y(x - ct, \alpha h c t)\) is approximately a right moving wave whose shape gradually changes over time.

We now substitute \(w(x,t) = y(x - ct, \alpha h c t)\) into the (KZ) equation, and divide by \(- 2 \alpha h\). Neglecting terms of order \(O(h^4)\), we obtain the equation,

We begin by making the substitutions, \(\xi = x - ct\) and \(\tau = \alpha h c t\). Now, we can pass it into the continuum limit \(\alpha, h \to 0\). We assume that \(h\) and \(\alpha\) are related such that \(\alpha, h \to 0\) and \(h/\alpha\) tend to a positive limit. We then define \(\delta = \lim_{h \rightarrow 0} \sqrt{\frac{h}{24 \alpha}}\). This also means that \(\alpha h = O(h^2)\).

Our wave operator reduces to

In this limit, we obtain the celebrated Korteweg-de Vries Equation by taking \(v(\xi, \tau) = y_\xi(\xi, \tau)\):

In their seminal 1965 paper, Zabusky and Kruskal performed numerical simulations of the KdV equation and discovered solitons, which are stable, localized wave packets that maintain their shape while traveling at constant speed. They then observed how solitons interfere with each other, and found that they pass through each other without changing shape. Using this connection to solitons, they gave a phenomenological explanation for the recurrence phenomenon observed in the FPUT system. We now see how these solitons arise in the KdV equation, from a small section of the report [8].

Initially, the first two terms of the KdV equation dominate and the classical overtaking phenomenon occurs; that is, \(u\) steepens in regions where it has a negative slope. Second, after \(u\) has steepened sufficiently, the third term becomes important and serves to prevent the formation of a discontinuity. Instead, oscillations of small wavelength (of order \(\delta\)) develop on the left of the front. The amplitudes of the oscillations grow and finally each oscillation achieves an almost steady amplitude (which increases linearly from left to right) and has a shape almost identical to that of an individual solitary-wave solution of the KdV equation. Finally, each such "solitary-wave pulse"or "soliton" begins to move uniformly at a rate (relative to the background value of u from which the pulse rises) which is linearly proportional to its amplitude. Thus, the solitons spread apart. Because of the periodicity, two or more solitons eventually overlap spatially and interact nonlinearly. Shortly after the interaction, they reappear virtually unaffected in size or shape. In other words, solitons "pass through" one another without losing their identity.

Here we have a nonlinear physical process in which interacting localized pulses do not scatter irreversibly.

They explained the recurrence phenomenon on an FPUT system with periodic boundary conditions. These solitons arise and interfere periodically, leading to the recurrence of the initial state after some time. Note that these derivation also agrees with the Chirikov criterion. The KdV approximation is valid for small nonlinearity, which is precisely the regime where the stochasticity threshold is high, preventing thermalization.

Toda Lattice

The N-particle Toda lattice also serves as a leading candidate as an integrable Hamiltonian in the neighbourhood of the FPUT lattice.

where the index \(k\) runs from \(1\) to \(N\) and we impose periodic boundary consitions \(Q_{N+1} = Q_1\). The Toda lattice can be in a sense, the discretized form of the KdV equation. One can show that the Toda Lattice is completely integrable using LAX pairs.

Connection to Statistical Mechanics

Statistical mechanics has been one the most successful theories in physics. It provides a microscopic explanation for macroscopic thermodynamic phenomena, for widly different systems. Even after more than a century after Boltzmann’s pioneering work, the foundations of statistical mechanics are still not quite rigourously justified. Some of them include what the thermodynamic limit really means, whether the free energy functionals are well defined at that limit, why do these systems thermalize, and most importantly, the ergodic hypothesis. One might think it would be too naive to apriori expect equipartition of energy in a slightly perturbed system. But if we establish a connection between ergodicity and the basic building block of statistical mechanics, the principle of equal a priori probabilities, we can see why the FPUT problem is so rich in its physical implications.

Ergodicity

This is a restatement of the hypothesis that Boltzmann used to derive the microcanonical ensemble and the law of equipartition of energy. Let’s see a heuristic proof of this statement.

Consider a system with Hamiltonian \(H(q_i, p_i)\) and total energy \(E\). Let us discretize the phase space into small cells of volume \(\Delta V_\Gamma\). The total time spent by the system in a cell \(\Delta V_\Gamma\) over a long time \(T\) is given by:

Therefore, the probability of finding the system in that cell, in our long time observation is given by:

Taking the limit \(\Delta V_\Gamma \to 0\), we obtain the probability density function:

This principle is the cornerstone of statistical mechanics. It allows us to derive the microcanonical ensemble and subsequently other ensembles. The previous definition has a certain nuance that is often overlooked. The word accessible is very important here. The microcanonical ensemble is,generally, defined for an isolated system with fixed total energy \(E\). Suppose for a system which has an additional constant of motion \(\Phi(p_i, q_i)\) . Then for a certain initial condition, both \(H(p_i, q_i)\) and \(\Phi(p_i, q_i)\) will be conserved. Therefore, our energy hypersurface will be further constrained to a submanifold defined by \(\Phi(p_i, q_i) = \phi_0\). So the probability density function will be given by:

We need to ensure that there are no additional constants of motion other than the Hamiltonian itself, to obtain the microcanonical ensemble in its widely used form. Recall that Poincaré’s result states that for perturbed Hamiltonians of that form, there exists no constant of motion \(\Phi(Q_k, P_k, t)\) that is analytic in \(Q_k, P_k\) and \(\epsilon\), other than the Hamiltonian itself. This motivated Fermi’s proof to show that those Hamiltonians are ergodic. The Ergodic hypothesis also helps us understand systems and their validity in the real world. This relates time averages to ensemble averages.

For real systems, we can only measure time averages. The ergodic hypothesis allows us to equate these to ensemble averages, which are easier to compute theoretically. A proof of this hypothesis still eludes us. There have been some system specific proofs, by von Neumann and Birkhoff, but a general proof is still unknown.

Energy Equipartition

The principle of equal a priori probabilities leads to the law of equipartition of energy. This is a fairly standard derivation. One can look at any statistical mechanics textbook for details, for example [9]. Since equipartition of energy is a direct consequence of the principle of equal a priori probabilities, FPUT decided to test this as a signature of ergodicity in their system.

Remarks

The foundations of statistical mechanics are still not completely rigourously justified. Khinchin [10] and Ruelle [11] have a lot of opinions and criticisms about these foundations. The FPUT problem is a classic example that highlights the subtleties involved in these justifications. It launched the fields of studying solitons and chaos theory.

Moreover, FPUT forever pioneered the usage of computers in physics research.

That’s all folks

Initially the experiment was widely referred to as the FPU problem, since Mary Tsingou was only recognised in a footnote for all the programming done for this experiment. A very illuminating read about the problem focusing on giving the deserving credits to Mrs. Mary Tsingou Menzel is here. Unlike my previous blogpost, I shall end with a relevant quote.

                       We thank Miss Mary Tsingou

Danke Schön

Feel free to direct your feedback and curses to my email: iamsabarno@gmail.com

References

[1] G. P. Berman and F. M. Izrailev, The Fermi-Pasta-Ulam problem: Fifty years of progress, Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 15, no. 1, 2005. DOI: 10.1063/1.1855036.

[2] Wolfram Research, Inc., Henon–Heiles System, 2002. Available at: http://mathworld.wolfram.com/Henon-HeilesSystem.html

[3] J. L. Tuck and M. T. Menzel, The superperiod of the nonlinear weighted string (FPU) problem, Advances in Mathematics, vol. 9, no. 3, pp. 399–407, 1972. DOI: 10.1016/0001-8708(72)90024-2.

[4] F. M. Izrailev and B. V. Chirikov, Statistical properties of a nonlinear string, Doklady Akademii Nauk SSSR, vol. 166, no. 1, pp. 57–59, 1966.

[5] J. Ford, The Fermi-Pasta-Ulam problem: Paradox turns discovery, Physics Reports, vol. 213, no. 5, pp. 271–310, 1992. DOI: 10.1016/0370-1573(92)90116-H.

[6] G. Gallavotti, The Fermi-Pasta-Ulam Problem, Lecture Notes in Physics, Springer, 2008. DOI: 10.1007/978-3-540-72995-2.

[7] T. Dauxois, M. Peyrard, and S. Ruffo, The Fermi–Pasta–Ulam “numerical experiment”: history and pedagogical perspectives, European Journal of Physics, vol. 26, no. 5, pp. S3–S11, 2005. DOI: 10.1088/0143-0807/26/5/S01.

[8] N. J. Zabusky and M. D. Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial states, Physical Review Letters, vol. 15, no. 6, pp. 240–243, 1965. DOI: 10.1103/PhysRevLett.15.240.

[9] K. Huang, Statistical Mechanics, 2nd ed., John Wiley & Sons, 1987.

[10] A. Ya. Khinchin, Mathematical Foundations of Statistical Mechanics, Dover Publications, 2013.

[11] D. Ruelle, Statistical Mechanics: Rigorous Results, World Scientific Publishing, 1999.

[12] R. S. Palais, The Symmetries of Solitons, arXiv:dg-ga/9708004, 1997.

[13] E. Fermi, J. Pasta, and S. Ulam, Studies of Nonlinear Problems, Los Alamos National Laboratory, Technical Report LA-1940, 1955.

This is technically my first blog post containing actual material here. I gave a presentation for the Maths Club at my University. The slides for which were made in LaTeX using beamer and can be found here, on my GitHub. I just converted the latex document to markdown using Pandoc and after some minor tweaks, I have posted it here. The blog aims to demonstrate how to deal with statistically large systems(systems with a large number of degrees of freedom) and the conditions under which they might attain equilibrium. Enjoy :)

A Stochastic Description

I should probably start by giving an introduction to this stochastic stuff. Thermodynamics is a topic a lot of us are familiar with. What happens in thermodynamics? We have something we want to study(a system) and everything else(the surroundings). In thermodynamics, we take large systems, large I mean systems with a large number of particles. A system with a small number of particles can’t help us glean information on macroscopic properties. We can’t measure the velocities of say even 100 particles for a large amout of time, and conclude anything useful. Rather, we observe large systems and infer their macroscopic properties, like Temperature and pressure. Our first stepping stone into understanding how we get from the microscopic description to the phenomenologically modeled thermodynamics, is statistical mechanics. For inferring stuff using statistical mechanics, we take some number of particles and then scale the system to reach thermodynamics.

Here, we are interested in a length scale between microscopic and macroscopic scales, the mesoscopic regime. Here, again the number of particles are quite large enough for particle by particle descriptions to be completely useless and the number of particles aren’t large enough for a thermodynamic description to make sense.

So here, we initially lay out how we model such systems. We take particles, but instead of following the particles trajectory and accounting for all the forces exerted, say from the collisions, the particle to particle interactions, temperature fluctuations and all other phenomenon a sane person can describe. So what we do here is chalk every force that can be to some random force following some rules. So we describe these systems, using Newton’s third alongwith a random force, often fancifully called the Langevin Equation. For further information you can refer to [1][3].

Stochastic Process

An example would be Brownian Motion, which is described by the Wiener process.

\[P(\hat{W}(t+\Delta t)=x | \hat{W}(t)=x’) = \frac{1}{\sqrt{2 \pi \Delta t}}\exp(-\frac{(x-x’)^2}{2 \Delta t})\]

Let’s see what this equation says. Suppose we have a particle moving in 1D, at a position \(x = x'\) at \(t = t_0\). Then the probability of finding the particle at some position \(x\) after some time \(\Delta t\), is described by a gaussian distribution \(N(x',\Delta t)\). This equation makes sense intuitively. The gaussian probability peaks at the mean \(x'\) and a measure of its spread is given by \(\Delta t\). Here is a small illustration,

We can see that the width increases with increase in the standard deivation. This just means if we take a larger time step \(\Delta t\), the particle has more time to move to further and further distances, and the distribution flattens. Let’s see what the Wiener Process looks like, before we move onto the topic of this blog.

Markov Chains

Most of the physical processes that we study in classical statistical physics is modelled as Markov Chains.

Essentially the future of the system is independent of the past given the present state.

Continuous Time Markov Chains(CTMC)

These are markov chains with the index as time \(\{X(t)\}_t\). Let us define a state space as the set of all values \(S\) a random variable can assume. Then we use the notation

We can rewrite the markov property as

where \(s,s_i \in S ~ \forall i\).

Now using the theorem of total probability we can write the Chapman Kolmogorov equation as

We now choose a time interval \(dt\) and rewrite the Chapman Kolmogorov equation, dropping the explicit dependence on \(x_0,t_0\)

Since this is an infinitesimal time interval, we can write the above equation as

where \(Q\) is the transition rate matrix.

Transition Rate Matrix

The transition rate matrix should satisfy the following conditions

1. \(Q_{ss'} \geq 0\) for \(s \neq s'\)

2. \(Q_{ss} = -\sum_{s' \neq s} Q_{ss'}\)

The first condition ensures that the transition rate is non-negative and the second condition essentially ensures that the probability of transitioning from a state \(s\) to any state is 1.

Thus we can write

where \(\mathbb{I}\) is the identity matrix.

We define \(k_{xx'}\) as the jump rate from state \(x'\) to \(x\). We thus define the elements of the transition rate matrix as

And the diagonal elements are thus given as

We also assume that the elements \(k_{xx'}\) do not change with time.

The change in probability of being in state \(x\) at time \(t\) is the inflow of probability from all other states which is \(k_{xx'}p(x';t)\) minus the outflow of probability \(k_{x'x}p(x;t)\). We can now write the master equation as

This becomes clearer when we define the probability current.

Master Equations

These equations describe the time evolution of the system that are modelled as being in the a probabilistic combination of a set of states at any given time. These are phenomenologically modelled first order differential equations.

These are used in a variety of fields in physics, and other related fields. These are used for studying birth and death processes, non-equilibrium statistical mechanics, chemical reactions, and a whole lot more.
The most common one that we see is the one used here is of the form

where \(\mathbb{A}\) is the matrix of connections and \(\bold{p}\) is the probability distribution column vector.

Probability Current

The master equation can now be written as

A bit of physics coming up ahead \(:)\)

Jump Networks

These systems are quite often represented as jump networks, where the nodes of the graph represent the states \(x\) and the arrows(edges) \(x' \rightarrow x\) represent the allowed jumps i.e. the jumps with non-zero transition rates(\(k_{xx'}>0\)).

This property has some physical justification. Coupled together with microscopic reversibility, this property ensures that the graphs aren’t disconnected. Disconnected graphs often lead to systems with non-interacting components, which can be studied independently reducing to the strongly connected graph.

Strongly connected graphs also posses a property called Irreducibility. Irreducibility is a necessary condition for the Perron Frobenius theorem to hold.

Here are is an example of a jump network :

Steady State & Equilibrium Distribution

Conditions

Steady state and equilibrium distribution are often used interchangeably. However, they are defined quite differently. The condition for a system to be in equilibrium is a bit more constrained than that of a steady state.

The necessary condition for both to hold is that the probability of being in a state \(x\) at time \(t\) is independent of time. Thus, essentially our master equation equates to \(0\).

We will show that under certain conditions, the system relaxes to a stationary state.

Detailed Balance Condition

The independence of \(p(x)\) on time can be ensured by the following conditions:

1. \(\sum_{x' \neq x} J_{xx'} = 0\)

2. \(J_{xx'} = 0\) for all states \(x,x'\)

Condition 1 is the condition for a steady state, while condition 2 is the condition for equilibrium.

When the jump networks satisfy the detailed balance condition, the system is said to be in equilibrium.

Perron Frobenius Theorem

Irreducibility

We can easily see that if all the elements of \(\mathbb{A}\) are positive, then the matrix is irreducible. This is a necessary condition for the Perron Frobenius theorem to hold.

We can reframe the Irreducibility problem in terms of graphs.

We have already assumed that the jump network is strongly connected.

Strongly Connected Graphs

We can also think of Irreducibility from the point of view of the markov chains. Two states are said to be communicating if there is a non-zero probability of transitioning from one state to the other in a finite number of steps. This is an equivalence relation and thus partitions the state space into equivalence classes.
The markov chain is said to be irreducible if there is only one equivalence class, which is the state space of the Markov chain.

Here in our jump networks any state can be reached from any other state in a finite number of steps. Thus our markov chain is irreducible.

Primitive Matrices

The Perron Frobenius Theorem

Stationary State Distribution

Proof
All stochastic matrices \(\bold{P}\) have a spectral radius \(\rho(\bold{P}) =1\). All column sums equal to 1. Thus, \(e\) is an eigenvector of \(\bold{P}^T\) with eigenvalue \(1\), where \(e\) is the column vector with all ones in its entries. We also have \(\lVert{\bold{P}}\rVert_{1} = 1\). Using the fact that \(\rho(\star) \leq \lVert\star \rVert\) for every matrix norm(See [2]), we get that

We know that \(\bold{P}\) is primitive with \(\rho(\bold{P}) = 1\), we know that

where \(p\) and \(q\) are the perron vectors for \(\bold{P}\) and \(\bold{P}^T\) respectively. The perron vector for \(\bold{P}^T\) is \(e\) and let the perron vector for \(\bold{P}\) be \(\pi\).

We thus have,

From the conservation of probability we have that \(\sum_i \pi_i =1\), we have that \(e^T\pi = \sum_i \pi_i = 1\).Thus we have

Now we have

Again \(e^Tp(0) = \sum_i p_i(0) = 1\). This leads to

Note that the stationary limit is independent of the initial distribution \(p(0)\) here.

We now actually verify that \(\pi\) is indeed the stationary distribution. Let us recast the master equation into matrix form.

where \(Q\) is the transition rate matrix. Now we have

Thus the infinite time limit of the probability distribution is indeed the stationary distribution.

Now the stationary distribution must be unique since the Perron root of any irreducible matrix has a one dimensional eigenspace. Thus the eigenvector must be unique due to the conservation of probability.

SIDE NOTE: After completing the presentation, I found a proof that doesn’t use Perron Frobenius theorem. However, that only proves it for a transition rate matrix with all positive entries.[1]

We use Cesaro sums to define the stationary state probability distribution. However, we won’t deal with it here.

Graph Theory

Preliminaries

• A graph is a collection of vertices and edges. The edges connect the vertices.

• The degree of a vertex is the number of edges connected to it.

• A walk is a sequence of vertices connected by edges.

• A trail is a walk with no repeated edges.

• A cycle is a non-empty trail that starts and ends at the same vertex.

• A connected graph is a graph where there is a path between every pair of vertices.

Trees

Handshaking Lemma

An "obvious" Theorem

Equilibrium Distribution

Trees

Proof
We know that there exists at least \(2\) vertices with degree \(1\). Let us label one of them as \(x\). Thus, for this vertex \(\sum_{x' \neq x} J_{xx'} = 0\) for all states \(x'\) connected to it, under the stationary distribution. Since this is only connected to another vertex say \(a\), We have \(J_{xa} = 0\). We can then remove the vertex \(x\) and the edge \(\{x,a\}\). This is still a tree, so we can do this iteratively. We can reach a point where we have only one vertex left. For the lone vertex, the detailed balance condition is trivially satisfied. Thus, the equilibrium distribution is the stationary distribution.

Cyclic Networks

We can see that non-vanishing stationary currents can only survive in loops. If we allow for all cycles the forward and back jump rates to be equal, we can then allow for the balance of probability currents.

Note

Note that this also includes the acyclic cases as well. For all sequences \((x_0,...,x_n)\) that is not a cycle with \(x_0\) we get \(k_{x_0 x_n} = k_{x_n x_0} = 0\). Thus, this condition is trivially satisfied for acyclic networks.

That’s all folks

Somehow in all of this quite convoluted and somewhat fun maths is the physical result that we can attain equilibrium under certain conditions. This has been the quite abrupt end of my first blog. I never know how to end, so I’ll end on an unoriginal quote that makes absolutely zero sense in this context.

            He who has the biscuits gets to tell the story.

Danke Schön

Feel free to direct your feedback and curses to my email: iamsabarno@gmail.com

References

[1] : Luca Peliti and Simone Pigolotti, Stochastic Thermodynamics, Princeton University Press, Princeton, NJ, 2023.

[2]: Carl D. Meyer, Applied Linear Algebra and Matrix Analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000.

[3]: Naoto Shiraishi, An Introduction to Stochastic Thermodynamics: From Basic to Advanced, Springer Nature, Fundamental Theories of Physics, vol. 212, 2023. ISBN: 978-981-19-8186-9. DOI: 10.1007/978-981-19-8186-9

[4]: Nico G. van Kampen, Stochastic Processes in Physics and Chemistry, 3rd ed., North-Holland, 2007. ISBN: 978-0444529657.

[5]: William Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed., Wiley, 1968. ISBN: 978-0471257080.

[6]: Reinhard Diestel, Graph Theory, 5th ed., Springer, 2017. ISBN: 978-3662536216. Link to Online Version

[7]: Richard J. Trudeau, Introduction to Graph Theory, Dover Publications, 1993. ISBN: 978-0486678702.

[8]: Christopher Jarzynski, Equalities and Inequalities: Irreversibility and the Second Law of Thermodynamics at the Nanoscale, Annual Review of Condensed Matter Physics, vol. 2, pp. 329–351, 2011. DOI: 10.1146/annurev-conmatphys-062910-140506.

I made this website because I wanted a website. After doing that, the website lay dormant for a year because I was too lazy to even write anything for it.So now in an effort to actually do something about this website, I’ve been thinking about posting blogs about random stuff I find cool. Also, I need to improve my writing skills because it is currently abysmal and resembles how Luke Dunphy might talk after a concussion. I might also write some album reviews in a different page, but I am lazy. Finally, welcome to the Octopus Garden and it’s horrendously written blogs. This is a list of the stuff I have written till now.(I will hyperlink them later).

• Systems attaining Equilibrium.