Nonlinear Partial Differential Equations pp 299360  Cite as
(Ir)reversibility and Entropy
Abstract
In the 1860’s emerges a revolutionary idea: many properties of the world around us can be explained by combining the atomistic hypothesis with the statistical theory. Some of the great scientific conquests from this time are the Boltzmann equation, which triggers one of the first qualitative studies of a complicated nonlinear partial differential equation; the notion of statistical entropy, which would later be fundamental in other areas of physics and mathematics, including information theory; and the notion of macroscopic irreversibility emerging from microscopically reversible laws. Thus the basic rules of statistical physics were set until Boltzmann’s irreversibility paradigm was shaken by Landau’s discovery of the Landau damping effect, about 80 years later, which opened the idea that equilibration is compatible with preservation of information, and led to a number of problems concerning the statistical theory of matter.
Keywords
Boltzmann Equation Entropy Production Collision Operator Vlasov Equation Microscopic DynamicTime’s arrow is part of our daily life and we experience it every day: broken mirrors do not come back together, human beings do not rejuvenate and rings grow unceasingly in tree trunks. In sum, time always flows in the same direction! Nonetheless, the fundamental laws of classical physics do not favor any time direction and conform to a rigorous symmetry between past and future. It is possible, as discussed in the article by T. Damour in this same volume, that irreversibility is inscribed in other physical laws, for example on the side of general relativity or quantum mechanics. Since Boltzmann, statistical physics has advanced another explanation: time’s arrow translates a constant flow of less likely events toward more likely events. Before continuing with this interpretation, which constitutes the guiding principle of the whole exposition, I note that the flow of time is not necessarily based on a single explanation.
At first glance, Boltzmann’s suggestion seems preposterous: it is not because an event is probable that it is actually achieved, but time’s arrow seems inexorable and seems not to tolerate any exception. The answer to this objection lies in a catchphrase: separation of scales. If the fundamental laws of physics are exercised on the microscopic, particulate (atoms, molecules,…) level, phenomena that we can sense or measure involve a considerable number of particles. The effect of this number is even greater when it enters combinatoric computations: if N, the number of atoms participating in an experiment, is of order 10^{10}, this is already considerable, but N! or 2^{ N } are supernaturally large, invincible numbers.
The innumerable debates between physicists that have been pursued for more than a century, and that are still pursued today, give witness to the subtlety and depth of Maxwell’s and Boltzmann’s arguments, banners of a small scientific revolution that was accomplished in the 1860’s and 1870’s, and which saw the birth of the fundamentals of the modern kinetic theory of gases, the universal concept of statistical entropy and the notion of macroscopic irreversibility. In truth, the arguments are so subtle that Maxwell and Boltzmann themselves sometimes went astray, hesitating on certain interpretations, alternating naive errors with profound concepts; the greatest scientists at the end of the nineteenth century, e.g. Poincaré and Lord Kelvin, were not to be left behind. We find an overview of these delays in the book by Damour already mentioned; for my part, I am content to present a “decanted” version of Boltzmann’s theory. At the end of the text I shall evoke the way in which Landau shattered Boltzmann’s paradigm, discovering an apparent irreversibility where there seemed not to be any and opening up a new mine of mathematical problems.
In retracing the history of the statistical interpretation of time’s arrow, I shall have occasion to make a voyage to the heart of profound problems that have agitated mathematicians and physicists for more than a century.
The notation used in this exposition are generally classical; I denote ℕ={1,2,3,…} and log=natural logarithm.
1 Newton’s Inaccessible Realm
I shall adopt here a purely classical description of our physical universe, in accordance with the laws enacted by Newton: the ambient space is Euclidean, time is absolute and acceleration is equal to the product of the mass by the resultant of the forces.
In the case of the description of a gas, these hypotheses are questionable: according to E.G.D. Cohen, the quantum fluctuations are not negligible on the mesoscopic level. The probabilistic nature of quantum mechanics is still debated; we nevertheless accept that the resulting increased uncertainty due to taking these uncertainties into account can but arrange our affairs, at least qualitatively, and we thus concentrate on the classical and deterministic models, “à la Newton”.
1.1 The Solid Sphere Model

We suppose that initially the particles are well separated (\( i\neq j\implies X_{i}  X_{j} > 2r \)) and separated from the walls (d(X _{ i },∂Λ)>r for all i).

While these separation conditions are satisfied, the movement is uniformly rectilinear: \(\ddot{X_{i}}(t) = 0 \) for each i, where we denote \( \ddot{X} = d^{2}X/dt^{2} \), the acceleration of X.
 When two particles meet, their velocities change abruptly according to Descartes’ laws: if X _{ i }(t)−X _{ j }(t)=2r, thenwhere n _{ ij }=(X _{ i }−X _{ j })/X _{ i }−X _{ j } denotes the unit vector joining the centers of the colliding balls.$$ \begin{cases}\dot{X_i}(t^+) = \dot{X_i}(t^)  2\langle \dot{X}_i(t^) \dot{X}_j(t^),n_{ij}\rangle n_{ij},\\[4pt]\dot{X_j}(t^+) = \dot{X_j}(t^)  2\langle \dot{X}_j(t^) \dot{X}_i(t^),n_{ji}\rangle n_{ji},\end{cases}$$
 When a particle encounters the boundary, its velocity also changes: if X _{ i }−x=r with x∈∂Λ, thenwhere n(x) is the exterior normal to Λ at x, supposed well defined.$$\dot{X}_i(t^+) = \dot{X}_i(t^)  2\langle \dot{X}_i(t),n(x)\rangle n(x),$$
These rules are not sufficient for completely determining the dynamics: we cannot exclude a priori the possibility of triple collisions, simultaneous collisions between particles and the boundary, or again an infinity of collisions occurring in a finite time. However, such events are of probability zero if the initial conditions are drawn at random with respect to Lebesgue measure (or Liouville measure) in phase space [40, Appendix 4.A]; we thus neglect these eventualities. The dynamic thus defined, as simple as it may be, can then be considered as a caricature of our complex universe if the number N of particles is very large. Studied for more than a century, this caricature has still not yielded all its secrets; far from that.
1.2 Other Newtonian Models

replace dimension 3 by an arbitrary dimension d≥2 (dimension 1 is likely pathological);

replace the boundary condition (elastic rebound) by a more complex law [40, Chap. 8];

or, instead, eliminate the boundaries, always delicate, by setting the system in the whole space ℝ^{ d } (but we may then add that the number of particles must then be infinite so as keep a nonzero global mean density) or in a torus of side L, Open image in new window , which will be my choice of preference in the sequel;

replace the contact interaction of hard spheres by another interaction between point particles, e.g. associated with an interaction potential between two bodies: ϕ(x−y)=potential exerted at point x by a material point situated at y.

the Coulomb potential: ϕ(x−y)=1/x−y;

the Newtonian potential: ϕ(x−y)=−1/x−y;

the Maxwellian potential: ϕ(x−y)=1/x−y^{4}.
The Maxwellian interaction was artificially introduced by Maxwell and Boltzmann in the context of the statistical study of gases; it leads to important simplifications in certain formulas. There exists a taxonomy of other potentials (LennardJones, Manev…). The hard spheres correspond to the limiting case of a potential that equals 0 for x−y>r and +∞ for x−y<2r.
1.3 Distribution Functions
To resume: our knowledge of the particle system is achieved only through the behavior of the empirical measure in a weak topology that models the macroscopic limitation of our experiments—laboratory experiments as well as sensory perceptions.
Evidently, by continuing up until k=N, we find a measure \(\hat{\mu}^{N;N}(dx_{1}\,\ldots \,dv_{N})\) concentrated at the vector of particle positions and velocities (the mean over all permutations of the particles). But in practice we never go to k=N: k remains very small (going to 3 would already be a feat), whereas N is huge.
1.4 Microscopic Randomness
If the sense of the empirical measure is transparent (it’s the “true” particle density), that of the microscopic probability measure is less evident. Let us assume that the initial state has been prepared by a great combination of circumstances about which we know little: we can only make suppositions and guesses. Thus \( \mu_{0}^{N} \) is a probability measure on the set of possible initial configurations. A physical statement involving \( \mu_{0}^{N} \) will, however, scarcely make sense if we use the precise form of this distribution (we cannot verify it, since we do not observe \( \mu_{0}^{N} \)); but it will make good sense if a \( \mu_{0}^{N} \)almost certain property is stated, or indeed with \( \mu_{0}^{N} \)probability of 0.99 or more.
1.5 Micromegas
In this section I shall introduce two very different statistical descriptions: the macroscopic description f(t,x,v) dx dv and the microscopic probabilities \( \mu_{t}^{N}(dx^{N}\,dv^{N}) \). Of course, the quantity of information contained in μ ^{ N } is considerably more important than that contained in the macroscopic distribution: the latter informs us about the state of a typical particle, whereas a draw following the distribution \( \mu_{t}^{N} \) informs us about the state of all particles. Think that if we have 10^{20} degrees of freedom, we will have to integrate 99999999999999999999 of them. For handling such vertiginous dimensions, we will require a fundamental concept: entropy.
2 The Entropic World
The concept and the name entropy were introduced by Clausius in 1865 as part of the theory—then under construction—of thermodynamics. A few years later Boltzmann (certainly influenced by the statistical ideas put forward by Laplace, Quetelet and others) revolutionized the concept by giving it a statistical interpretation based on atomic theory. In addition to this section, the reader can consult e.g. Balian [9, 10] about the notion of entropy in physical statistics.
2.1 Boltzmann’s Formula
Let a physical system be given, which we suppose is completely described by its microscopic state \( Z\in \mathcal{Z}\). Experimentally we only gain access to a partial description of that state, say \( \pi (Z)\,\in \,\mathcal{Y}\), where \( \mathcal{Y}\) is a space of macroscopic states. I will not give precise hypotheses on the spaces \( \mathcal{Z}\) and \( \mathcal{Y}\), but with the introduction of measure theory we will implicitly assume that these are “Polish” (separable complete metric) spaces.
How can we estimate the amount of information that is lost when we summarize the microscopic information by the macroscopic? Assuming that \( \mathcal{Y}\) and \( \mathcal{Z}\) are denumerable, it is natural so suppose that the uncertainty associated with a state \( y\in \mathcal{Y}\) is a function of the cardinality of the preimage, i.e. #π ^{−1}(y).
In numerous cases, the space \( \mathcal{Z}\) of microscopic configurations is continuous, and in applying Boltzmann’s formula it is customary to replace the counting measure by a privileged measure: for example by Liouville measure if we are interested in a Hamiltonian system. Thus W in (4) can be the volume of microscopic states that are compatible with the macroscopic state y.
If this last point is not at all evident, the universality is nonetheless verified in the particular case that interests us where the microscopic state \( \mathcal{Z}\) is the space of configurations of N particles, i.e. \( {{\mathcal{Y}}^{N}}\), and where we begin by taking the limit N→∞. In this limit, as we will see, the mean entropy per molecule tends to a finite value and we can subsequently take the limit ε→0, which corresponds to an arbitrarily precise macroscopic measure. The result is, within a sign, nothing other than Boltzmann’s famous H function.
2.2 The Entropy Function H
If ν is a probability measure, or more generally a measure of finite mass, it is easy to extend this formula to all probabilities μ by setting H _{ ν }(μ)=+∞ if μ is not absolutely continuous with respect to ν. If ν is a measure of infinite mass, more precautions must be taken; we could require at the very least the finiteness of ∫f(logf)_{−} dν.
We then note that if the macroscopic space \( \mathcal{Y}\) bears a measure ν, then the microscopic space \( \mathcal{Z}\in {{\mathcal{Y}}^{N}}\) bears a natural measure ν ^{⊗N }.
If ν is a probability measure, statement (6) is known as Sanov’s theorem [43] and is a leading result in the theory of large deviations. Before giving the interpretation of (6) in this theory, note that once we know how to treat the case where ν is a probability measure we easily deduce the case where ν is a measure of finite mass; however, I have no knowledge of any rigorous discussion in the case where ν is of infinite mass, even though we may expect that the result remains true.
2.3 Large Deviations
Let ν be a probability measure and suppose that we independently draw random variables y _{ i } according to ν. The empirical measure \( \hat{\mu} = N^{1}\sum \delta_{y_{j}} \) is then a random measure, almost certainly convergent to ν as N→∞ (it’s Varadarajan’s theorem, also called the fundamental law of statistics [49]). Of course it’s possible that appearances deceive and that we think we are observing a measure μ distinct from ν. This probability decreases exponentially with N and is roughly proportional to exp(−N H _{ ν }(μ)); in other words, the Boltzmann entropy dictates the rarity of conditions that lead to the “unexpected” observation μ.
2.4 Information
Information theory was born in 1948 with the remarkable treatise of Shannon and Weaver [94] on the “theory of communication” which is now a pillar for the whole industry of information transmission.
In Shannon’s theory, somewhat disembodied for its reproduction and impassionate discussion, the quantity of information carried by the decoding of a random signal is defined as a function of the reciprocal of the probability of the signal (which is rare and precious). Using the logarithm allows having the additivity property, and Shannon’s formula for the mean quantity gained in the course of decoding is obtained: \({\mathbb{E}}\log (1/p(Y))\), where p is the law of Y. This of course gives Boltzmann’s formula again!
2.5 Entropies on All Floors
Entropy is not an intrinsic concept; it depends on the observer and the degree of knowledge that can be acquired through experiments and measures. The notion of entropy will consequently vary with the degree of precision of the description.
Boltzmann’s entropy, as has been seen, informs us of the rarity of the kinetic distribution function f(x,v) and the quantity of microscopic information remaining to be discovered once f is known.
2.6 The Universality of Entropy
Initially introduced within the context of the kinetic theory of gases, entropy is an abstract and evolving mathematical concept, which plays an important role in numerous areas of physics, but also in branches of mathematics having nothing to do with physics, such as information theory and other sciences.
Some mathematical implications of the concept are reviewed in my survey HTheorem and beyond: Boltzmann’s entropy in today’s mathematics [106].
3 Order and Chaos
Intuitively, a microscopic system is ordered if all its particles are arranged in a coordinated, correlated way. On the other hand, it is chaotic if the particles, doing just as they please, act entirely independently from one another. Let us reformulate this idea: a distribution of particles is chaotic if each of the particles is oblivious to all the others, in the sense that a gain of information obtained for a given particle brings no gain in information about any other particle. This simple notion, key to Boltzmann’s equation, presents some important subtleties that we will briefly mention.
3.1 Microscopic Chaos
To say that random particles that are oblivious to each other is equivalent to saying that their joint law is tensorial. Of course, even if the particles are unaware of each other initially, they will enter into interaction right away and the independence property will be destroyed. In the case of hard spheres, the situation is still worse: the particles are obliged to consider each another since the spheres cannot interpenetrate. Their independence is thus to be understood asymptotically when the number of particles becomes very large; and experiments seeking to measure the degree of independence will involve but a finite number of particles. This leads naturally to the definition that follows.
Of course, the definition can be quantified by introducing an adequate notion of distance, permitting us to measure the gap between μ ^{ N } and μ ^{⊗N }. We can then say that a distribution μ ^{ N } is more or less chaotic. We again emphasize: what matters is the independence of a small number k of particles taken from among a large number N.
It can be shown (see the argument in [99]) that it is equivalent to impose property (7) for all k∈ℕ, or simply for k=2. Thus chaos means precisely that 2 particles drawn randomly from among N are asymptotically independent when N→∞. The proof proceeds by observing the connections between chaos and empirical measure.
3.2 Chaos and Empirical Measure
By the law of large numbers, chaos automatically implies an asymptotic determinism: with very high probability, the empirical measure approaches the statistical distribution of an arbitrary particle when the total number of particles becomes gigantic.
It turns out that, conversely, correlations accommodate very badly a macroscopic prescription of density. Before giving a precise statement, we will illustrate this concept in a simple context. Consider a box with two compartments, in which we distribute a very large number N of indistinguishable balls. A highly correlated state would be a one in which all the particles occupy the same compartment: if I draw two balls at random, the state of first ball informs me completely about the state of the second. But of course, from the moment when the respective numbers of balls in the compartments are fixed and both nonzero, such a state of correlation is impossible. In fact, if the particles are indistinguishable, when two are drawn at random, the only information gotten is obtained by exploiting the fact that they are distinct, so that knowledge of the state of the first particle reduces slightly the number of possibilities for the state of the second. Thus, if the first particle occupies state 1, then the chances of finding the second particle in state 1 or 2 respectively are not f _{1}=N _{1}/N and f _{2}=N _{2}/N, but \(f_{1}^{\prime} = (N_{1}  1)/(N  1)\) and f _{2}=N _{2}/(N−1). The joint distribution of a pair of particles is thus very close to the product law.
 (i)
(μ ^{ N }) is μchaotic;
 (ii)
the empirical measure \( \widehat{\mu}^{N} \) associated with μ ^{ N } converges in law toward the deterministic measure μ.
The notion of chaos thus presented is weak and susceptible to numerous variants; the general idea being that μ ^{ N } must be close to μ ^{⊗N }. The stronger concept of entropic chaos was introduced by Ben Arous and Zeitouni [13]: there \( H_{\mu^{\otimes N}}(\mu^{N}) = o(N) \) is imposed. A related notion was developed by Carlen, Carvalho, Le Roux, Loss and Villani [32] in the case where the microscopic space is not a tensor product, but rather a sphere of large dimension; the measure μ ^{⊗N } is replaced by the restriction of the product measure to the sphere. Numerous other variants remain to be discovered.
3.3 The Reign of Chaos

if we can act on the macroscopic configuration, we will not have access to the microscopic structure and it is very difficult to impose correlations;

the laws that underlie the microscopic variations are unknown to us and we may suppose that they involve a large number of factors destructive to correlations;

the macroscopic measure observed in practice seems always well determined and not random;

if we fix the macroscopic distribution, the entropy of a chaotic microscopic distribution is larger than the entropy of a nonchaotic microscopic distribution.
Let us explain the last argument. If we are given a probability μ on \( \mathcal{Y}\), then the product probability μ ^{⊗N } is the maximum entropy among all the symmetric probabilities μ ^{ N } on \( {{\mathcal{Y}}^{N}}\) having μ as marginal. In view of the large numbers N in play, this represents a phenomenally larger number of possibilities.
The microscopic measure \( \mu_{0}^{N} \) can be considered as an object of Bayesian nature, an a priori probability on the space of possible observations. This choice, in general arbitrary, is made here in a canonical manner by maximization of the entropy: in some way we choose the distribution that leaves the most possibilities open and makes the observation the most likely. We thus join the scientific approach of maximum likelihood, which has proved its robustness and effectiveness—while skipping the traditional quarrel between frequentists and Bayesians!
The problem of the propagation of chaos consists of showing that our chaos hypothesis, made on the initial data (it’s not entirely clear how), is propagated by the microscopic dynamic (which is well defined). The propagation of chaos is essential for two reasons: first, it shows that independence is asymptotically preserved, providing statistical information about the microscopic dynamic; secondly, it guarantees that the statistical measure remains deterministic, which allows hope for the possibility of a macroscopic equation governing the evolution of this empirical measure or its approximation.
3.4 Evolution of Entropy
A recurrent theme in the study of dynamical systems, at least since Poincaré, is the search for invariant measures; the best known example is Liouville measure for Hamiltonian systems. This measure possesses the remarkable additional property of tensorizing itself.
Suppose that we have a microscopic dynamic on \( {{\mathcal{Y}}^{N}}\) and a measure ν on the space \( \mathcal{Y}\) such that ν ^{⊗N } is an invariant measure for the microscopic dynamic; or more generally that there exists a νchaotic invariant measure on \( {{\mathcal{Y}}^{N}}\). What happens with the preservation of microscopic volume in the limit N→∞?
A simple consequence of preservation of volume is conservation of macroscopic information \( H_{\nu^{\otimes N}}(\mu_{t}^{N}) \), where \( \mu_{t}^{N} \) is the image measure of \( \mu_{0}^{N} \) through the microscopic evolution. In fact, since \( \mu_{t}^{N} \) is preserved by the flow (by definition) and ν ^{⊗N } likewise, the density f ^{ N }(t,y _{1},…,y _{ N }) is constant along the trajectories of the system, and it follows that ∫f ^{ N }logf ^{ N } dν ^{⊗N } is likewise constant.
Matters are more subtle for macroscopic information. Of course, if the various particles evolve independently from one another, the measure \( \mu_{t}^{N}\) remains factored for all time, and we easily deduce that the macroscopic entropy remains constant. In general, the particles interact with one another, which destroys independence; however if there is propagation of chaos in a sufficiently strong sense, the independence is restored as N→∞, and we consequently have determinism for the empirical measure. So all the typical configurations for the microscopic initial measure \(\mu_{0}^{N}\) give way, after a time t, to an empirical measure \(\hat{\mu}_{t}^{N}\simeq \mu_{t}\), where μ _{ t } is well determined. But it is possible that other microscopic configurations are compatible with the state μ _{ t }, configurations that have not been obtained by evolution from typical initial configurations.
But then why not reverse the argument and say that chaos at time t implies chaos at time 0, by reversibility of the microscopic dynamic? This argument is in general inadmissible unless an exact notion of the chaos propagated is specified. The initial data prepared “at random” with just one kinetic distribution constraint, is supposed chaotic in a less strong sense; this depends on the microscopic evolution.
The notion of scale of interaction plays an important role here. Certain interactions take place on a macroscopic scale, other on a microscopic scale, which is to say that all or part of the interaction law is coded in parameters that are invisible on the macroscopic level. In this last case, the notion of chaos conducive to the propagation of the dynamic risks not being visible on the macroscopic scale and we can expect a degradation of the notion of chaos.
From there, the discussion must involve the details of the dynamic, and our worst troubles begin.
4 Chaotic Equations
After the introduction of entropy and chaos, we can return to the Newtonian systems of Sect. 1, for which the phase space is composed of positions and velocities. A kinetic equation is an evolution equation bearing on the distribution f(t,x,v); the important role of the velocity variable v justifies the terminology kinetic. By extension, in the case where there are external degrees of freedom (orientation of molecules for example), by extension we still speak of kinetic equations.
As descendents of Boltzmann, we pose the problem of deducing the macroscopic evolution starting from the underling microscopic model. This problem is in general of considerable difficulty. The fundamental equations are those of Vlasov, Boltzmann, Landau and Balescu–Lenard, published respectively in 1938, 1867, 1936 and 1960 (the more or less logical order of presentation of these equations does not entirely follow the order in which they were discovered…).
4.1 Vlasov’s Equation
We note well that \(\hat{\mu}_{t}^{N}\) in (8) is a weak solution of Vlasov’s equation, so that the passage to the limit is conceptually very simple: it is simply a stability result for the Vlasov equation.
Quite clearly I have gone a bit far, for this equation is nonlinear. If \(\hat{\mu}\simeq f\) in the sense of the weak topology of measures, then \(F\ast \hat{\mu}\) converges to F∗∫f dv in a topology determined by the regularity of F, and if this topology is weaker than uniform convergence, nothing guarantees that \((F\ast \hat{\mu})\hat{\mu}\simeq (F\ast f)f\).
This act of faith is very effective since the Vlasov–Poisson model, in which F=−∇W, where W a fundamental solution of ±Δ, is the universally accepted classic model in plasma physics [42, 71] as well as in astronomy [15]. In the first instance a particle is an electron, in the second a star! The only difference lies in the sign: repulsive interaction for electrons, attractive for stars. We should not be astonished to see stars considered in this way as microscopic objects: they are effectively so on the scale of a galaxy (which can tally 10^{12} stars…).
The theory of the Vlasov–Poisson equation itself remains incomplete. We can distinguish presently two principal theories, both developed in the entire space. That of Pfaffelmoser, simplified by Schaeffer and exposited for example in [51], supposes that the initial data f _{ i } is C ^{1} with compact support; later this unsatisfactory compactness assumption was removed by Horst [114] by an improvement of the Pfaffelmoser–Schaeffer method. The concurrent theory is that Lions–Perthame, reviewed in [23]. Pfaffelmoser’s theory has been adapted in spatially periodic context (see [12] or modify [114]), which is not the case for the Lions–Perthame theory.
4.2 Boltzmann’s Equation
Vlasov’s equation loses its relevance when the interactions have a short range. A typical example is that of rarefied gas, for which the dominant interactions are binary and are uniquely produced in the course of “collisions” between particles.
Boltzmann’s equation was established by Maxwell [80] and Boltzmann [21, 22]; it describes a situation where the interactions are of short range and where each particle undergoes O(1) impacts per unit of time. Much more subtle than the situation of Vlasov’s mean field, the Boltzmann situation is nonetheless simpler than the hydrodynamic one where the particles undergo a very large number of collisions per unit of time.
We start by establishing the equation informally. The movement of a particle occurs with alternation of rectilinear trajectories and collisions, during the course of which its velocity changes so abruptly that we can consider the event as instantaneous and localized in space. We first consider the emblematic case of hard spheres of radius r: a collision occurs when two particles, with respective positions x and y and with respective velocities v and w, are found in a configuration where x−y=2r and (w−v)⋅(y−x)<0. We then speak of a precollisional configuration. We let ω=(y−x)/x−y.
To do this, in the 1960s Grad proposed a precise mathematical limit: have r tend toward 0 and at the same time N toward infinity, so that Nr ^{2}→1, which is to say that the total effective section remains constant. Thus a given particle, moving among all the others, will typically encounter a finite number of them in a unit of time. One next starts with a microscopic probability density \(f_{0}^{N}(x^{N},v^{N})\,dx^{N}\,dv^{N}\), which is allowed to evolve by the Newtonian flow \( {{\mathcal{N}}_{t}}\), and one attempts to show that the first marginal f ^{1;N }(t,x,v) (obtained by integrating all the variables except the first position variable and the first velocity variable) converges in the limit to a solution of the Boltzmann equation.
The Boltzmann–Grad limit is also often called the low density limit [40, p. 60]: in fact, if we start from the Newtonian dynamic and fix the particle size, we will dilate the spatial scale by a factor \(1/\sqrt{N}\) and the density will be of the order N/N ^{3/2}=N ^{−1/2}.
At the beginning of the 1970’s, Cercignani [37] showed that Grad’s program could be completed if one proved a number of plausible estimates; shortly thereafter, independently, Lanford [69] sketched the proof of the desired result.
Lanford’s theorem is perhaps the single most important mathematical result in kinetic theory. In this theorem, we are given microscopic densities \(f_{0}^{N}\) such that for each k the densities \(f_{0}^{k;N}\) of the k particle marginals are continuous, satisfy Gaussian bounds at large velocities and converge uniformly outside the collisional configurations (those where the positions of two distinct particles coincide) to their limit \(f_{0}^{\otimes k}\). The conclusion is that there exists a time t _{∗}>0 such that \(f_{t}^{k;N}\) converges almost everywhere to \(f_{t}^{\otimes k}\), where f _{ t } is a solutions of Boltzmann’s equation, for all time t∈[0,t _{∗}].
Lanford’s estimates were later rewritten by Spohn [95] and by Illner and Pulvirenti [61, 62] who replaced the hypothesis of small time by a smallness hypothesis on the initial data, permitting Boltzmann’s equation to be treated as a perturbation of free transport. These results are reviewed in [40, 90, 95].
The technique used by Lanford and his successors goes through the BBGKY hierarchy (Bogoliubov–Born–Green–Kirkwood–Yvon), the method by which the evolution of the marginal for a particle f ^{1;N } is expressed as a function of the marginal for two particles f ^{2;N }; the evolution of a twoparticle marginal f ^{2;N } as a function of a threeparticle marginal f ^{3;N }, and so forth. This procedure is especially uneconomical (in the preceding heuristic argument, we only use f ^{1;N } and f ^{2;N }, but there is no known alternative).
Each of the equations of the hierarchy is then solved via Duhamel’s formula, applying successively the free transport and collision operators, and by summing over all the possible collisional history. The solution at time t is thus formally expressed, as with an exponential operator, as a function of the initial data and we can apply the chaos hypothesis on \((f_{0}^{N})\).
We then pass to the limit N→∞ in each of the equations, after having verified that we can neglect pathological “recollisions”, where a particle again encounters a particle that it had already encountered beforehand, and which is thus not unknown to it. This point is subtle: in [40, Appendix 4.C] a dynamic that is a priori simpler than that of solid spheres, due to Uchiyama, is discussed, with only four velocities in the plane, for which the recollision configurations cannot be neglected, and the kinetic limit does not exist.
It remains to identify the result with the series of tensor products of the solution to Boltzmann’s equation and conclude by using a uniqueness result.
Spohn [95, Sect. 4.6] shows that one can give more precise information on the microscopic distribution of the particles: on the small scale, this follows a homogeneous Poisson law in phase space. This is consistent with the intuitive idea of molecular chaos.
Lanford’s theorem settled a controversy that had lasted since Boltzmann himself; but it leaves numerous questions in suspense. In the first place, it is limited to a small time interval (on which only about 1/6 of the particles have had time to collide… but the conceptual impact of the theorem is nonetheless important). The variant of Illner and Pulvirenti lifts this restriction of small time, but the proof does not lend itself to a bounded geometry. As for lifting the smallness restriction, at the moment it is but a distant dream.
Next, to this day the theorem has only been proved for a system of solid spheres; longrange interactions are not covered. Cercignani [36] notes that the limit of Boltzmann–Grad for such interactions poses subtle problems, even from the formal viewpoint.
Finally, the most frustrating thing is that Lanford avoided discussion of precollisional chaos, the notion that particles that are about to collide are not correlated. This notion is very subtle, because just after the collision, correlations have inevitably taken place. In other words, we have precollisional chaos, but not postcollisional.
In the particular case s=5, the collision kernel depends no longer on the relative velocity, but only on the deviation angle: we speak of Maxwellian molecules. By extension, we say that B(v−v _{∗},σ) is a Maxwellian collision kernel if it depends only on the angle between v−v _{∗} and σ. The Maxwellian molecules are above all a phenomenological model, even if the interaction between a charged ion and a neutral particle in a plasma is regulated by such a law [42, Vol. 1, p. 149]. The potentials in 1/r ^{ s−1} for s>5 are called hard potentials, for s<5 soft potentials. Often the angular singularity b(cosθ) is truncated to small values of θ.
The Boltzmann equation is important in modeling rarefied gases, as explained in [39]. Nonetheless, because of its eventful history and its conceptual content, as well as the impact of Boltzmann’s treatise [22], this equation has exerted a fascination that goes far beyond its usefulness. The first mathematical works dedicated to it are those of Carleman^{2} [26, 27], followed by Grad [57]. Besides the article by Lanford [69] already mentioned, a result that has had a great impact is the weak stability theorem of DiPerna–Lions [47]. The equation is well understood in the spatially homogeneous setting for hard potentials with angular truncation, see e.g. [84]; and in the setting close to equilibrium, see e.g. [60]. We refer to the reference treatises [38, 40, 103] for a number of other results.
4.3 Landau’s Equation
Boltzmann’s collisional integral loses its meaning for Coulomb interactions because of the extremely slow decrease of the Coulomb potential. The grazing collisions, with large impact parameter, then become dominant.
The Landau approximation is now well understood mathematically in the context of a limit called grazing collision asymptotics; [3] can be consulted for a detailed discussion of this problem.
The Landau operator, both diffusive and integral, presents a remarkable structure. It is easily generalized to arbitrary dimensions d≥2, and the coefficient L/z can be changed to L z^{ γ+2}, where γ is the exponent appearing in (17). The models of hard potential type with γ>0 have been completely studied in the spatially homogeneous case [45]; but it is definitely the case γ=−3 in dimension 3 that is physically interesting. In this case we only know how to prove the existence of weak solutions in the spatially homogeneous case (by adapting [1, Sect. 7] and the existence of strong solutions for perturbations of equilibrium [59]). This situation is entirely unsatisfactory.
4.4 The Balescu–Lenard Equation
The mathematical theory of the Balescu–Lenard equation is wide open, both with regard to establishing it and to studying its qualitative properties; one of the rare rigorous papers on the subject is the linearized study of R. Strain [96]. Even though little used, the Balescu–Lenard equation is nonetheless the most respected of the collisional models in plasmas and it’s an intermediary that allows justification for using the Landau collision operator to represent long duration fluctuations in particle systems; its theory represents a formidable challenge.
5 Boltzmann’s Theorem H
In this section we will start with Boltzmann’s equation and examine several of its most striking properties. Much more information can be found in my long review article [103].
5.1 Modification of Observables by Collisions
5.2 Theorem H
The impact of this result is crucial. First, the heuristic microscopic reasoning of Sect. 3.4 has been replaced by a simple argument that leads directly to the limit equation. Next, even if it is a manifestation of the second law of thermodynamics, the increase in the entropy in Boltzmann’s model is deduced by logical reasoning and not by a postulate (a law) which one accepts or not. Finally, of course, in doing so, Boltzmann displayed an arrow of time associated with his equation.
Not only is this macroscopic irreversibility not contradictory with microscopic reversibility, but it is in fact intimately linked to it: as has already been explained, it’s the conservation of microscopic volume in phase space that guarantees the nondecrease of entropy. For the rest, as L. Carleson was already surprised to discover in 1979 while examining simplified models of Boltzmann’s equation [35], it is precisely when the parameters of the dynamics are adjusted in such a way to achieve microscopic reversibility, that the H theorem holds. The phenomenon is well known in the context of the physics of granular media [105]: there the microscopic dynamic is dissipative (nonreversible), including a loss of energy due to friction, and the macroscopic dynamic does not satisfy Theorem H!
From the informational point of view, the increase in entropy means that the system always runs toward macroscopic states that are more and more probable. This probabilistic idea is exacerbated by the formidable power of the combinatorics: we suppose for example that we are considering a gas with N≃10^{16} particles (which is roughly what we find in 1 mm^{3} of gas under ordinary conditions!), and between time t=t _{1} and time t=t _{2} the entropy increases only by 10^{−5}. The volume of microscopic possibilities is then multiplied by \( e^{N\,[S(t_{2})S(t_{1})]}=e^{10^{11}}\gg 10^{10^{10}} \). This phenomenal factor far exceeds the number of protons in the universe (10^{100}?) or the number of 1000page books that could be written by combining all the alphabetic characters of all the languages in the world….
The intuitive interpretation of Theorem H is thus rather eloquent: the high entropy states occupy, at the microscopic level, a place so monstrously larger than the states of low entropy, that the microscopic system goes to them automatically. As we have seen, the logical reasoning justifying this scenario is complex and indirect, involving the propagation of chaos and macroscopic determinism—and to this day only a small portion of the program has been rigorously achieved.
5.3 Vanishing of Entropy Production
The increase in entropy admits a complement that is no less profound, frequently stated as a second part of Theorem H: the characterization of cases of equality, i.e. states for which the production of entropy vanishes.
Maxwell already noticed that (34) makes Boltzmann’s collision operator vanish: Q(f,f)=0. Such a distribution is called Maxwellian in his honor. However, it is Boltzmann who first gave a convincing argument that the distributions (34) are the only solutions of the equation PE(f)=0, and consequently the only solutions of Q(f,f)=0. Let’s honor him by sketching a variant of his original proof.
They are called local Maxwellians or else hydrodynamic states. In accordance with the kinetic description, these states are characterized by a considerable reduction in complexity, since they depend on but three fields: the density field ρ, the field of macroscopic velocities u and the temperature field T. These are the fields that enter into the hydrodynamic equations, whence the above terminology.
This discovery establishes a bridge between the kinetic and hydrodynamic descriptions: in a process where collisions are very numerous (weak Knudsen number), the finiteness of entropy production forces the dynamic to be concentrated near distributions that makes the entropy production vanish. This remark makes way for a vast program of hydrodynamic approximation of Boltzmann’s equation, to which Hilbert alludes in his Sixth Problem. Readers can consult [54, 55, 93]. If the Boltzmann equation can be approached both by compressible and incompressible equations, we should note that it does not lead to the whole range of hydrodynamic equations, but only to those for perfect gases, i.e. those that conform to a law where pressure is proportional to ρT.
6 Entropic Convergence: Forced March to Oblivion
If Maxwell discovered the importance of Gaussian velocity profiles, he did not, as Boltzmann remarks, prove that these profiles are actually induced by the dynamic. Boltzmann wanted to complete this program, and for this recover the Maxwellian profiles not only as equilibrium distributions, but also as limits of the kinetic equation asymptotically as time becomes large (t→∞). This conceptual leap aimed at basing equilibrium statistical mechanics on its nonequilibrium counterpart—usually much more delicate—is still topical in innumerable contexts.
I have written a good bit on this topic and readers can consult the survey article [103, Chap. 3], the course [108], the research article [46] or the research memoir [109]. In the sequel, in order to fix the ideas, I will suppose that the position variable lives in the torus \(\mathbb{T}^{d}\).
6.1 Global Maxwellian
This distribution is thus an equilibrium for Boltzmann’s equation. Moreover, it is easy to verify that it is the distribution that maximizes entropy under the constraints of fixed mass, linear momentum and total energy. This selection criterion foreshadows the classical theory of equilibrium statistical mechanics and Gibbs’ famous canonical ensembles (Gibbs measure).
6.2 The Entropic Argument

entropy increases strictly unless it is in a hydrodynamic state,

the global Maxwellian, stationary, is the only hydrodynamic solution of Boltzmann’s equation.
The image that emerges is that the entropy will continue to increase as much as possible since the distribution never remains “stuck” on a hydrodynamic solution; the entropy will end up approaching the maximal entropy of the global Maxwellian, and convergence results.
Boltzmann’s reasoning is essentially correct and it’s not difficult to transform it into a rigorous argument by showing that sufficiently regular solutions of the Boltzmann equation approach Maxwellian equilibrium. In the context of spatially homogeneous solutions, T. Carleman formalized this reasoning in 1932 [26].
However, Boltzmann did not have the means for making his argument qualitative; it would be necessary to wait almost a century before anyone dared to pose the problem of the speed of convergence toward Gaussian equilibrium, especially pertinent since the range of validity of Boltzmann’s equation is not eternal and is limited in time by phenomena such as the Poincaré recurrence theorem.
6.3 The Probabilistic Approach of Mark Kac
At the beginning of the 1950s, Kac [66] attempted to understand convergence toward equilibrium for the Boltzmann equation and began by simplifying the model. Kac ignores positions, simplifies the collision geometry extravagantly and invents a stochastic model where randomness is present in the interaction: whenever two particles interact, one draws at random the parameters describing the collision. Positions being absent, the particles all interact, each with all the others, and thus a “mean field” model is produced. This simplified probabilistic model is for Kac an opportunity for formalizing mathematically the notion of propagation of chaos in the mean field equations, which would prove so fertile and would be taken up later by Sznitman [98] and many others.
Suspicious of Boltzmann’s equation, Kac wants to explain the convergence by microscopic probabilistic reasoning on the level of an Nparticle system; he attempts to obtain spectral gap estimates that are uniform in N. His approach seems naive nowadays in that it underestimates the difficulty of treating dimension N; nonetheless, the problem of determining the optimal spectral gap, resolved a halfcentury later, has proved to be very interesting [31, 65, 79]. For this subject readers can equally well consult [104, Sect. 6] and [32], where there is interest in the entropic version of this “microscopic” program.^{3}
In 1966 McKean [81] resumed Kac’s work and drew a parallel with the problems of the central limit theorem. He introduced the tools of information theory to the subject, in particular Fisher information [41], which measures the difficulty in estimating a parameter such as the velocity of the particles. The program would be completed by Tanaka [100], who discovered new contracting distances, and would culminate with the work of Carlen, Carvalho, Gebetta, Lu, Toscani on “the central limit theorem for the Boltzmann equation” [29, 30, 33, 34]. This theory encompasses basic convergence theorems based on the combinatorics of interactions between particles and tools from the study of central limit theorem (weak distances…), as well as counterexamples demonstrating extremely slow convergence to equilibrium.
This stochastic program allows us to dispense with Theorem H; in fact it has also permitted the updating of several Lyapunov functionals: Tanaka’s contracting distance (of optimal transport), Fisher’s information. Nonetheless, from a technical point of view, the whole theory remains essentially confined to Maxwellian interactions (in which \(\tilde{B}(vv_{\ast},\omega)\) depends only on the angle between v−v _{∗} and ω) and to spatially homogeneous gases. Chapter 4 of [103] is dedicated to particular properties, highly elegant moreover, of these interactions.
For gaining generality and for studying inhomogeneous situations or nonMaxwellian interactions, the only robust approach known to this day is based on the H Theorem.
6.4 Cercignani’s Conjecture
Boltzmann’s H Theorem is general and relevant, so that it is natural to look for its quantitative refinements. At the beginning of the 1980s, C. Cercignani asked if one could estimate from below the production of local entropy as a function of the “nonGaussianity” of the kinetic distribution, ideally by a multiple of the information H _{ M }(f). It was not until a decade later that Carlen and Carvalho [28], Desvillettes [44], without answering Cercignani’s question, could nonetheless present quantitative lower bounds for the production of entropy.
6.5 Conditional Convergence
Inequality (38) concerns a function f=f(v) but does not include spatial dependence; this is inevitable since the variable x does not enter into the study of the global production of entropy. Of course, (38) immediately implies (modulo the regularity bounds) convergence in O(t ^{−∞}) for the spatially homogeneous equation, i.e. the distance between the distribution and equilibrium tends to 0 faster than t ^{−k } for each k; yet this inequality does not resolve the inhomogeneous problem. The obstacle to be overcome is the degeneracy of entropy production on hydrodynamic states. A key to the study over long time of Boltzmann’s equation thus consists in showing that not too much time is spent in a hydrodynamic, or approximately hydrodynamic, state. To avoid this trap, we can only depend on the transport, represented by the operator v⋅∇_{ x }. Grad [56] had understood in 1965, in a moreover rather obscure paper: “the question is whether the deviation from a local Maxwellian, which is fed by molecular streaming in the presence of spatial inhomogeneity, is sufficiently strong to ultimately wipe out the inhomogeneity” (…) “a valid proof of the approach to equilibrium in a spatially varying problem requires just the opposite of the procedure that is followed in a proof of the HTheorem, viz., to show that the distribution function does not approach too closely to a local Maxwellian.”
We end this section with several commentaries on the hypotheses. The regularity theory of the Boltzmann equation allows reduction of the general bounds to very particular bounds, e.g. it is known that the kinetic distribution is automatically minorized by a multiple of \(e^{v^{q}}\) if, for example, the equation is set on the torus and the solution is regular. It is also known that bounds on the moments of low order allow having bounds on arbitrarily high moments, etc. But regularity in the general context remains a celebrated open problem. The conditional convergence result shows that it’s the final obstacle separating us from quantitative estimates of convergence to equilibrium; it likewise unifies the already known results on convergence: both the case of spatially homogeneous distributions and that of distributions close to equilibrium are situations in which we have an almost complete regularity theory.
In studying convergence toward equilibrium for the Boltzmann equation, we observe a subtle interaction between the collision operator (nonlinear, degeneratively dissipative) and the transport operator (linear, conservative). Neither of the two, taken separately, would be sufficient for inducing convergence, but the combination of the two succeeds. This situation arises rather frequently and recalls the hypoellipticity problem in the theory of partial differential equations. By analogy, the hypocoercivity problem is the study of the convergence properties for potentially degenerate equations.
A somewhat systematic study of these situations, both for linear and nonlinear equations, was made in my memoir [107]. The general strategy consists of constructing Lyapunov functionals adapted to the dynamic, while adjoining by a natural functional (like entropy) a well chosen term of lesser order. A case study is the “A ^{∗} A+B theorem”, inspired by Hörmander’s sum of squares theorem, which gives sufficient conditions on the commutators between operators A and B, with B antisymmetric, for the evolution \(e^{t(A^{\ast}A + B)}\) to be hypocoercive. In the simpler variant, one of these conditions reminiscent of Hörmander’s Lie algebra condition, is the coercivity of A ^{∗} A+[A,B]^{∗}[A,B].
Hypercoercivity theory has now taken on a life of its own and there are already a number of striking results; it continues to expand, especially in the nonlinear context. This is true as well both for kinetic theory, as in the paper [58] that will be mentioned in the next section, and outside of kinetic theory, as in the paper by Liverani and Olla on the hydrodynamic limits of certain particle systems [73].
In a nonlinear context, the principal result remains [109, Theorem 51]; this general statement allows for simplification of the proof of the conditional convergence theorem for the Boltzmann equation, and includes new interactions and limit conditions. See [109, Part III] for more details.
6.6 Linearized System
The rate of convergence to equilibrium can be determined by a linearized study. We begin by flushing out a classical logical mistake: the linearized study can in no case be a substitute for the nonlinear study, since linearization is only valid beginning from where the distribution is very close to equilibrium.

quantitatively estimating the spectral gap for the linearized collision operator;

performing a spectral study of the linearization in a space appropriate for the nonlinear problem, so as to achieve a “connection” between the nonlinear study and the linearized study;

take into account the hydrodynamic degeneracy from a hypercoercive perspective: in fact, the linearized equation is just as degenerate as the nonlinear equation.
All of these difficulties have been resolved in the last decade by C. Mouhot and his collaborators Baranger, Gualdani and Mischler [11, 58, 82], at least in the emblematic case of solid spheres. Thus the recent article [58] establishes a conditional convergence result with exponential rate O(e ^{−λt }) instead of O(t ^{−∞}), and the rate λ is estimated in a constructive manner.
Exponential convergence is not a universal characteristic of Boltzmann’s equation: we do not expect it for hard potentials or even for the moderately soft. To fix our ideas, let us suppose that the collision kernel behaves like v−v _{∗}^{ γ } b(cosθ). In the case where b(cosθ)sin^{ d−2} θ is integrable (often by angular truncation for the grazing collisions), the linearized collision operator only admits a spectral gap for γ≥0. An abundance of grazing collisions permits extending this condition, as Mouhot and Strain [83] showed: if b(cosθ)sin^{ d−2} θ≃θ ^{−(1+ν)} for θ→0 (important grazing collisions), then the linearized collision operator only allows a spectral gap for γ+ν≥0. The regularity theory is presently under development for such equations (work of Gressman–Strain, Alexandre–Morimoto–Ukai–Xu–Yang), and we can wager that within a few years the linearized theory will cover all these cases.
For too soft potentials (or for the Landau’s model of Coulomb collisions), there is no spectral gap and the best result we can expect is fractional exponential convergence \(O(e^{\lambda t^{\beta }})\), 0<β<1. Such estimates can be found in the paper of Guo and Strain [60].
6.7 Qualitative Evolution of Entropy
Here the Boltzmann equation nicely reveals its double nature, relevant for both transfer of information via collisions and fluid mechanics via the transport operator. It is often the marriage between the two aspects that proves delicate.
The relative importance of transport and collisions can be modulated by the boundary conditions; in the periodic context it comes down to the size of the box. A large box will permit important spatial variations, giving the hydrodynamic effects free rein, as in the above simulation. Nonetheless, we clearly see that even in this case, and contrary to an idea well ingrained even with specialists, the asymptotic regime is not hydrodynamic, in the sense that the ratio between hydrodynamic entropy and total kinetic entropy does not increase significantly as time passes, oscillating rather between minimum and maximum values.
The conclusion that we can draw from this figure is precisely opposite to our intuition, according to which the hydrodynamic effects dominate in the long run: quite the contrary, starting from a hydrodynamic situation, we quickly arrive at a situation that is almost homogeneous (visually we have the impression that at time ≃0.7 the hydrodynamic information represents scarcely more that 1% of the total information!). The inhomogeneous effects then resume their rights (at time t=1 the information is divided into parts of the same order), after which it becomes resolutely homogeneous. In this example, the homogenization has proceeded faster than convergence to equilibrium. We’ll return to this figure, which has cause some perplexity, in Sect. 8.
6.8 Two Nonconventional Problems
I end this section by mentioning two peculiar problems linked to time’s arrow in Boltzmann’s equation that are perhaps just curiosities. The first is the classification of eternal solutions of Boltzmann’s equation: I tried to show in my doctoral thesis that, at least for the spatially homogeneous Boltzmann equation with Maxwellian molecules, there do not exist eternal solutions with finite energy. The second would be to instead look for selfsimilar solutions with finite energy that do not converge to Maxwellian equilibrium. For the first problem, [111] can be consulted for partial results; the conjecture is still viable, and Bobylev and Cercignani [18] have been able to show that there does not exist any eternal solution having finite moments of all orders. As to the second problem, it has been resolved by the same authors [17] using Fourier transform techniques.
7 Isentropic Relaxation: Living with Ones Memories
Additionally, whereas Boltzmann’s equation allows but a very small number of equilibria (the Maxwellians determined by the conservation laws), Vlasov’s equation allows a considerable number of them. For example, all the homogeneous distributions f ^{0}=f ^{0}(v) are stationary. There exist yet many other stationary distributions, for example the family of Bernstein–Greene–Kruskal waves [14]. For all these reasons, there is nothing a priori that would lead us to conjecture a well determined behavior over the long term and there is no indication at all of time’s arrow. However, in 1946, L. Landau, released several years earlier from the soviet communist prisons where his frank speech had led him, suggested a very specific long term behavior for Vlasov’s equation [68]. It is based on an analysis of the linearized equation near a homogeneous equilibrium. Landau’s prediction provoked a shock and a conceptual change which still today raises lively discussions [92]; in its sequel it began to be suspected that convergence toward equilibrium is not necessarily tied to an increase in entropy. This section is devoted to a survey of the question of isentropic convergence, while emphasizing the perturbation regime, which is the only one for which there are sound elements. More details can be found in my course [110].
7.1 Linearized Analysis
The stability of Volterra equations is a classical problem. If u satisfies Open image in new window , then the rate of decrease of u is dictated by the worse of two rates: the rate of decrease of S of course, and on the other hand the width of the largest band {0≤ℜξ≤Λ} that does not intersect any solution of the equation K ^{ L }=1, where K ^{ L } is the Laplace transform of K. If Λ>0, we thus have exponential stability for the linearization.
In summary, the linearized Vlasov equation about a stable homogeneous equilibrium (in the sense of Penrose) predicts an exponential dampening of the force, in an apparently irreversible manner. This discovery brought back the problematic of time’s arrow in the theory of Vlasov’s equation.
The study of the linearized Vlasov equation can be found in advanced treatises on plasma physics, like [71]; however, the treatment there is systematically obscured by the use of contour integrals in the complex plane, which arise from the inversion of the Laplace transform. This has been avoided in the presentation of [85, Sect. 3], based on the simple Fourier transform; or in the short version [111].
7.2 Nonlinear Landau Dampening
There are other reasons for distrusting the linearization. First, the eliminated term, F[h]⋅∇_{ v } h, is of higher degree in terms of derivatives of h with respect to velocity. Next, the linearization eliminates conservation of entropy, and favors the particular state f ^{0}, which voids the discussion of reversibility.
In 2009, Mouhot and I established such a result [85]. If the interaction potential W is not too singular, in the sense that \(\hat{W}(k)=O(1/k^{2})\) (this hypothesis allows just Coulomb and Newton interactions!), and if f ^{0} is an analytic homogeneous equilibrium satisfying Penrose’s stability condition, then there is nonlinear dynamic stability: starting with initial data f _{ i }, analytic and such that ∥f _{ i }−f ^{0}∥=O(δ) when δ is very small, we have decrease of the force in O(e ^{−2πλt}) for all λ<min(λ _{0},λ _{ i },λ _{ L }), where λ _{0} is the width of the band of complex analyticity of f ^{0} about \(\mathbb{R}_{v}^{d}\), λ _{ i } is the width of the band of complex analyticity of f _{ i } in the variable v, and λ _{ L } is the rate of the Landau convergence. In brief, linear damping implies nonlinear damping, with an arbitrarily small loss in rate of convergence.
The theorem also establishes the weak convergence of f(t,⋅) to an asymptotically homogeneous state f _{∞}(v). More precisely, the equation being invariant under time reversal, there is an asymptotic profile f _{+∞} for t→+∞, and another profile f _{−∞} for t→−∞. If Vlasov’s equation is viewed as a dynamical system, there is then a remarkable behavior: the homoclinic/heteroclinic trajectories are so numerous that they fill an entire neighborhood of f ^{0} in analytic topology.

a reinterpretation of the problem in terms of regularity: instead of showing that there is damping, it is shown that f(t,x,v) is “as regular” as the free transport solution, uniformly in time;

“deflection” estimates: a particle placed in an exponentially decreasing force field follows a free transport asymptotic trajectory in a sense that can be quantified precisely;

the stabilizing role of retarded response, in echoes, of the plasma: when one of the modes of the plasma is perturbed, the reaction of the other modes is not instantaneous, but follows with a slight retardation, because the effect of the modes is compensated outside of some instants of resonance;

a Newton scheme that takes advantage of the fact that the linearized Vlasov equation is in some way completely integrable; the speed of convergence of this scheme compensates for the loss of decay that accompanies the solution of the linearization.
All these ingredients are described in more detail in [110]. The special place of the Newton scheme and of the complete integrability form an unexpected bridge with KAM (Kolmogorov–Arnold–Moser) theory. In some way the Vlasov nonlinear Vlasov equation, in the perturbative process, inherits some of the good properties of the completely integrable linearized Vlasov equation.
From the physical point of view, information goes toward the small kinetic scales: the oscillations of the distribution function are amplified when time becomes large, and become invisible. LyndenBell [75, 76] clearly understood this and used a striking formula for explaining: “a [galactic] system whose density has achieved a steady state will have information about its birth still stored in the peculiar velocities of its stars.”
7.3 Gliding Regularity
The nonlinear damping theorem is based on a recent reinterpretation, in terms of regularity, that deserves some comments. We begin by talking about the cascade associated with free transport, represented on the diagram below:
This image, which is derived from formula (43), shows that the frequencies that matter vary over time: there is an overall movement toward high kinetic frequencies, and this movement is all the faster than the frequency is high. More precisely, the spatial mode of frequency k oscillates in the velocity variable with period of order O(1/kt). The challenge of Landau damping is to show that this cascade, although distorted, is globally preserved by the effect of the interactions that couple the different modes.
These strong oscillations preclude any hope of obtaining bounds that are uniform in time, e.g. analytically regular in the usual sense. A key idea in [85] consists of concentrating on the Fourier modes that matter for the free transport solution, and thus to follow the cascade over the course of time. This concept is called gliding regularity and comes with a degradation of the regularity bounds in velocity, but simultaneously with an improvement of regularity in position, once velocity averages have been formed. Our interpretation of Landau damping is thus a transfer of regularity away from the variable v and toward the variable x, the regularity of the force improving, which implies that its amplitude dies off.
7.4 Nonlinear Echoes and Critical Regularity
These estimates can be adapted to the original Vlasov–Poisson equation; we then find, in the solution of the linearized equation about a nonstationary solution, a loss of regularity/decay that is fractional exponential. Under good assumption (as strong as the Penrose condition in the gravitational case, even stronger in the Coulomb case) we find essentially exp((kt)^{1/3}); in the more general case the growth remains like a fractional exponential in kt. As it remains subexponential, this loss of regularity can be compensated by the exponential decay coming from the linear problem.
The loss of regularity depends essentially on the interaction, whereas the linear gain depends foremost on the regularity of the data: exponential for analytic data, polynomial for C ^{ r } data, fractional exponential for Gevrey data. The preceding discussion thus suggests that it is possible to extend the nonlinear damping theorem to Gevrey data. E.g., in the gravitational case, the critical exponent 1/3 corresponds to a critical regularity Gevrey3. We recall that a function is called Gevreyν if its successive derivatives do not grow faster than O(n!^{ ν }). Even if losing arbitrary little over ν, it is equivalent to requiring that its Fourier transform decay as a fractional exponential exp(c ξ^{1/ν }).
7.5 Speculations
The nonlinear Landau damping theorem opens a large number of questions. First, its extensions to geometries other than \(\mathbb{T}^{d}\) is a real challenge, because then we lose the magical Fourier transform. The extension of inhomogeneous equilibria is still a distant dream; in fact, the linear stability of Bernstein–Greene–Kruskal waves is still not known!
Next, we have seen that it is known how to deal with damping under Gevrey regularity; but that the extension to lower regularities such as C ^{ r } regularity is an open problem. We have already emphasized the parallel with KAM theory, in which we know how to treat problems of class C ^{ r }; but in KAM the loss of regularity is only polynomial, and here it is much more severe. Certain variants of the KAM problem lead to a fractional exponential loss of regularity, and then it’s likewise an open problem to work with regularity lower than Gevrey. In the immediate future, the only progress in C ^{ r } regularity suggested by [85] is the possibility of proving damping on time scales much larger than the nonlinear scale (O(1/δ) instead of \(O(1/\sqrt{\delta })\), see [85, Sect. 13]); this development seems to depend on an original conjecture concerning the optimal constant occurring in certain interpolation inequalities. In Sect. 8 we shall again discuss a strategy permitting us to conceptually bypass this limitation of very high regularity.
Whatever the optimal regularity, it is not possible to obtain a Landau damping in the space with natural energy associated with the physical conservation laws. In fact, Lin and Zeng [72] show that nonlinear Landau damping is false if there are strictly less than two derivatives, in an appropriate sense.
Finally, even if Landau damping is but a perturbative phenomenon, it should be noted that its conceptual importance remains considerable because, at the present moment, it’s the only little island that we are succeeding in exploring in the ocean of open problems related to isentropic relaxation. By its discovery, Landau raised awareness that physical systems may relax without there being any irreversibility and increase in entropy. In the 1960s, LyndenBell [75, 76] invoked this conceptual advance for solving the galactic relaxation problem, which appears in an approximately quasistationary state, whereas the relaxation times associated with the galactic Vlasov equation are vastly greater than the age of the universe. Subsequently, the violent relaxation principle—relaxation of the force field over certain times characteristic of the dynamic—has been generally accepted by astrophysicists, without there being any theoretical explanation to promote it. We have here a major scientific challenge.
8 Weak Dissipation

it’s a more realistic physical model than the “pure” Vlasov equation without collisions;

it permits quantification, as a function of the small parameter ε, of the relative velocities of the homogenization (Landau damping) and entropic convergence phenomena;

it permits bypassing the obstacle of Gevrey regularity that confronts the study of the noncollisional model.
Everything remains to be done here and I will merely sketch a longterm program.
8.1 A Plausible Scenario
Starting with a perturbation of homogeneous equilibrium with very rapid velocity decay, we should, in the course of temporal evolution per (46), remain close to a homogeneous regime; this is in the spirit of results of Arkeryd, Esposito and Pulvirenti [5] on the weakly inhomogeneous Boltzmann equation. In the homogeneous context, the operator of the right hand side undoubtedly has the same regularization properties as a Laplacian in velocity, at least locally (the regularization properties become very weak at large velocities, but a very strong velocity decay is imposed). Assuming that this remains true in a weakly inhomogeneous context, we are left with a hypoelliptic equation that will regularize in all the variables, surely more quickly in velocity than in the position variable.
The hypoelliptic regularization in the Gevrey class has been but little studied, but using dimensional arguments we might think that in this context there is regularization in the Gevrey1/α class, with velocity O(exp((εt)^{−α/(2−α)})) in v, and O(exp((εt)^{−3α/(2−3α)})) in x.
From another direction, in the Gevrey1/α class, for α>1/3 we must have decay toward the homogeneous regime like O(exp−t ^{ α }).
By combining the two effects we obtain homogenization on a O(ε ^{−ζ}) time scale, with ζ<1, which is a more rapid rate than the rate of increase of the entropy in O(ε ^{−1}).
Balancing accounts, the coefficient ζ we might hope for is disappointing, of order 8/9. Among the steps used, the weakest link seems to be Gevrey regularization in x, which is extremely costly and perhaps not optimal since this regularity is not necessary in linear analysis. This motivates the development of a version of the nonlinear damping theorem in low regularity in x. If this regularity occurs, the coefficient becomes much better, of order 1/6….
8.2 Reexamining Simulations
9 Metastatistics
Here I use the word “metastatistics” to talk about statistics on the distribution function, which itself has a statistical content. This section will be short because we have scarcely more than speculations on the matter.
Now is there a natural probability measure Π on the space of admissible profiles? Ideally, such a measure will be invariant under the dynamic. In the context of the Boltzmann equation, the question really is not posed: only trivial measures, borne by Maxwellian equilibria, remain in contention. However, in the context of Vlasov’s equation, the construction of nontrivial invariant measures is a fascinating problem. Such measures reflect the Hamiltonian nature of Vlasov’s equation, studied for simplified interactions by Ambrosio and Gangbo [4].
A rather serious candidate for the status of invariant measure is Sturm’s entropic measure [97], issuing from optimal transport theory, formally of the form \(P=e^{\beta H_{\nu}}\). Its complex definition has until now impeded success in proving its invariance. It should not be very difficult to modify the construction by appending an energy term. Sturm’s measure is defined on a compact space, and there are perhaps subtleties in extending it to a kinetic context where the velocity space is not bounded. But the worst difficulty comes no doubt from the singularity of the typical measures: it is expected that ℙalmost every measure is totally foreign to Lebesgue measure, and is supported by a set of codimension 1. This seems to close the door to every statistical study of damping based on regularity, and increases the mystery.^{4}
Robert [91] and others have attempted to build a statistical theory of Vlasov’s equation, starting from the notion of entropy, trying to predict the likely asymptotic state of dynamic evolution. The theory has gained some success, but it remains controversial. Furthermore, since the asymptotic state is obtained by a weak limit, the question arises of knowing whether an equality or inequality should be imposed on the constraints involving nonlinear functional density. For this topic readers can consult [102].
Then, this theory does not take into account the underlying evolution equation, postulating a certain universality with respect to the interaction. Isichenko [63] has remarked that the longterm asymptotic state, if it exists, must depend on fine details of the initial distribution and of the interaction, whereas the measures constructed by statistical theory only depend on invariants: energy, entropy, or other functionals of the form ∬A(f) dx dv. This objection has found substance with the counterexamples constructed in [85, Sect. 14], which show that the transformation f(x,v)→f(x,−v) can modify the final asymptotic state, while it preserves all the known invariants of the dynamic. The objection is perhaps surmountable, because these counterexamples are constructed in analytic regularity, i.e. in a class that must be invisible to a statistical treatment; but these counterexamples show the subtlety of the problem, and reinforce the feeling of difficulty in the construction of invariant measures.
10 Paradoxes Lost
In this last section I will review a series of more or less famous paradoxes involving time’s arrow and kinetic equations, and present their commonly accepted resolutions. A certain number of them involve infinity, a classic source of paradoxes such as “Hilbert’s hotel” with an infinite number of rooms, where it is always possible to find a place for a new arrival even if the hotel is already full. On our scale, this paradox reflects our incapacity to account for the appearance or disappearance of a particle in relation to the gigantic number that make up our universe. The limit N→∞ (or the asymptotic N≫1, if like Boltzmann one prefers to avoid manipulating infinities) being the basis for statistical mechanics, it’s not surprising that this paradox should arise.
In all the sequel, when I mention positive or negative time, or precollisional or postcollisional configuration, I am referring to the absolute microscopic time of Newton’s equations.
10.1 The Poincaré–Zermelo Paradox
In 1895 Poincaré [89] cast doubt on Boltzmann’s theory because it seemed to contradict the fundamental properties of dynamical systems. A little later Zermelo [113] developed this point and noted that the inexorable increase in entropy prohibited the return of the system to the initial state, which is however predicted by the recurrence theorem (within an arbitrarily small error).
The same objection can be applied to the Landau damping problem: if the distribution tends to a homogeneous equilibrium, it will never return close to its initial state.
From the mathematical point of view, this reasoning clearly does not apply, since the Boltzmann equation involves an infinite number of degrees of freedom; it is only for a fixed number of particles that the recurrence theorem applies. From the physical point of view, the answer is a bit more subtle. On the one hand, the recurrence time diverges when the number N of particles tends to infinity, and this divergence is likely monstrously rapid! For a system of macroscopic size, albeit small, the recurrence theorem simply never applies, for it involves times well greater than the age of the universe. On the other hand, the validity of the Boltzmann equation is not eternal: for N fixed, the quality of the approximation will degrade with time, because chaos (simple or precollisional) is preserved only approximately. By the time that Poincaré recurrence takes place, the Boltzmann equation will have long ceased to be valid!!^{5}
10.2 Microscopic Conservation of the Volume
Poincaré’s recurrence theorem is based on conservation of the volume in microscopic phase space (preservation of Liouville measure). The entropy is a direct function of the volume of microscopic admissible states; how can it increase if the volume of microscopic states is constant?
The answer to this question may seem surprising: it can be argued that the increase of entropy does not occur despite the conservation of microscopic volume, but because of this conservation; more precisely, it’s what keeps entropy from decreasing. In fact, let us start at the initial time from all the typical configurations associated with a distribution f _{ i }. After a time t, these typical configurations have evolved and are now associated with a distribution f _{ t }, the transition from f _{ i } to f _{ t } being governed by the Boltzmann equation. The typical configurations associated with f _{ t } are thus at least as numerous as the typical configurations associated with f _{ i }, which clearly means that entropy cannot decrease.
In a microscopic irreversible model, we will typically have a contraction of microscopic phase space, linked to a dissipative phenomenon. The preceding argument does not apply then, and one can imagine that the entropy decreases, at least for certain initial data. It is indeed what happens, for example, in models of granular gases undergoing inelastic collisions.
10.3 Spontaneous Appearance of Time’s Arrow
How, starting from a microscopic equation that does not favor any time direction, can the Boltzmann equation predict an inexorable evolution toward positive times?
The answer is simple: there is not any inexorable evolution toward positive times, and the double direction of time is preserved. There is simply a particular choice of the initial data (instant of preparation of the experiment), that has fixed a particular time, say t=0. Starting from there, one has a double arrow of time; entropy increases for positive times, and decreases for negative times.
10.4 Loschmidt’s Paradox
Loschmidt’s paradox [74] formalizes the apparent contradiction that exists of a reversible microscopic dynamic and an irreversible evolution of the entropy. Let us suppose that we start from a given initial configuration and that at time t we stop the gas and reverse the velocities of all the particles. This operation does not change the entropy, and starting from this new initial data we can let the dynamic act anew. By microscopic reversibility, at the end of time 2t we will have returned to the point of departure; but the entropy will not have ceased to increase, whence the contradiction.
This paradox can be resolved in several ways, all of which come down to the same finding: the degradation of the notion of chaos between the initial time and time t>0. On the mathematical level it is only known how to prove the weak convergence of \(\mu _{t}^{1;N}\) to f(t,⋅) as N→∞, whereas the convergence is supposed uniform at the initial time. In fact, it is conjectured that the data \((\mu _{t}^{N})\) satisfies the property (still to be defined…) of precollisional chaos, whereas the initial data is supposed to satisfy a complete chaos property. When the velocities are reversed, the hypothesis of precollisional chaos is transformed into postcollisional chaos, and the relevant equation is no longer the Boltzmann equation, but the “reverse” Boltzmann equation, in which a negative sign is placed before the collision operator. Entropy then increases toward negative times and no longer toward positive times, and all contradiction disappears.
To state matters in a more informal way: at the initial time the particles are all strangers to one another. After a time t, the particles that have just collided know each other still, while those which are about to collide do not know each other: the particles have a memory of the past and not of the future. When the velocities are reversed, the particles have a memory of the future and not of the past, and time begins to flow backwards!
Legend has it that Boltzmann, confronted with the velocity reversal paradox, responded: “Go ahead, reverse them!” Behind the jest is hidden a profound observation: reversal of velocities is an operation that is inaccessible to us because it requires microscopic knowledge of the system; and the notion of entropy emerges precisely from what we can only act upon macroscopically. Beginning in the 1950s, spin echo experiments allowed us to see the paradox from another angle [10].
10.5 Nonuniversal Validity of Boltzmann’s Equation
This paradox is a variant of the preceding. Having understood that Boltzmann’s equation does not apply after reversal of velocities, we will exploit this fact to put Herr Boltzmann in default. We redo the preceding experiment and choose as initial data the distribution obtained after reversal of velocities at time t. We let time act, and the relevant equation certainly is not Boltzmann’s equation.
This paradox effectively shows that there are microscopic configurations that do not lead to Boltzmann’s equation. Nevertheless, and it’s thus that Boltzmann argued, these configurations are rare: precisely, they cause the appearance of correlations between precollisional velocities. This is not rarer than correlations between postcollisional velocities, but it’s rarer than not having correlations at all! The Boltzmann equation is approximately true if we depart from a typical configuration, which is to say drawn according to a “strongly chaotic” law, but it does not hold for all initial configurations. Once these grand principles are stated, the quantitative work remains to be done.
10.6 Boltzmann’s Arbitrary Procedure
To establish the Boltzmann equation, the encounter probabilities of particles are expressed in terms of the precollisional probabilities, which are arbitrary. If instead postconditional probabilities had been used, a different equation would have been obtained, with a negative sign before the collision operator! Why then have confidence in Boltzmann?
The answer is still the same, of course, and depends on which side of the origin one is placed: for negative times, these are precollisional probabilities that are almost factored, whereas for positive times, these are postcollisional probabilities.
10.7 Maxwell’s Demon
Maxwell conceived a thought experiment in which a malicious demon positions himself in a box with two compartments and adroitly manipulates a small valve so that there is a flow of balls going from the right compartment toward the left but not the other way. The system thus evolves toward increased order, and the entropy decreases.
Of course, this cannot be considered an objection to the law of increasing entropy, and the experiment is intended to make us think: first, the demon should be part of the model and himself subjected to reversible mechanical laws, taking into account the energy needed for recognizing that a particle is approaching and for evaluating its velocity, for the mental work done, etc. If a complete account is made, we will find again, for sure, that the second law of thermodynamics is not violated.^{6}
We note in this regard that recently experiments with Maxwell’s demon have been realized with granular gases: as I myself saw with stupefaction in a film made of an experiment, initially there is a receptacle with two vertically separated compartments and an opening above that allows communication, the two compartments are filled with inelastic particles in approximately equal number, the whole thing is agitated automatically, and little by little one of the compartments is emptied in favor of the other. An underlying principle is that in the fuller compartment the abundance of collisions results in cooling by dissipation of energy; and the particles jump less high, rendering it more and more difficult for them to leave the full compartment. We find again on this occasion the principle—already mentioned—according to which a dissipative (irreversible) dynamic does not necessarily lead to an increase of entropy, but to the opposite.
10.8 Convergence and Reversibility
This paradox is a variant of the Loschmidt paradox; it applies both to the theme of Boltzmann entropic relaxation and to nonlinear Landau damping: how can there be convergence when t→+∞ if there is reversibility of the dynamic? The answer is of childish simplicity: there is also convergence when t→−∞. For Vlasov, this was accomplished with the same equation, and we thus have a phenomenon of generalized homoclinic/heteroclinic. For Boltzmann, the equation changes according as to whether times are considered which are prior or subsequent to the chaotic data.
10.9 Stability and Reversibility
This paradox is more subtle and applies to nonlinear Landau damping: asymptotic stability and reversibility of the dynamic automatically imply an instability, which seems contradictory.
We detail the argument. If we have stability in time t→+∞, let f _{∞}(v) be an asymptotically stable profile, which we assume to be even. We take a solution \(\bar{f}(t,x,v)\), inhomogeneous, which converges toward f _{∞}(v). We then choose as initial data f(T,x,−v) with T very large; we thus have data arbitrarily close to f _{∞}(−v)=f _{∞}(v), and which brings us back after time T to the data \(\bar{f}(0,x,v)\), rather removed from f _{∞}(v). In other words, the distribution f _{∞} is unstable. How is this compatible with stability?
The answer, as explained e.g. in [25], lies in the topology: in the theorem of asymptotic stability (nonlinear Landau damping), the convergence over large time occurs in the sense of the weak topology, with frenetic oscillations in the velocity distribution, which is compensated locally. When we say that a distribution f ^{0} is stable, that means that if we depart close to f ^{0} in the sense of the strong topology (e.g. analytic or Gevrey), then we remain close to f ^{0} in the sense of the weak topology. The asymptotic stability combined with the reversibility thus imply instability in the sense of the weak topology, which is perfectly compatible with stability in the sense of the strong topology.
10.10 Conservative Relaxation
This problem is of a rather general nature. Vlasov’s equation comes with a preservation of the amount of microscopic uncertainty (conservation of entropy). Moreover the distribution at time t>0 allows reconstructing exactly the distribution at time t=0: it suffices to solve Vlasov’s equation after reversal of the velocities. We can say that Vlasov’s equation forgets nothing; but convergence consists precisely in forgetting the episodes of the dynamic evolution!
The answer again lies in the weak convergence and the oscillations. Information will be lodged in these oscillations, information which is invisible because in practice we never measure the complete distribution function, but averages of this distribution function (recall the quote of LyndenBell reproduced at the end of Sect. 7.2). Every observable will converge toward its limit value, and there will be a “forgetfulness”. The force field, obtained as mean of the kinetic distribution, converges toward 0 without this being contradictory to preservation of information: the information leaves the spatial variables so as to go into the kinetic variables. In particular, the spatial entropy ∫ρlogρ (where ρ=∫f dv) tends toward 0, whereas the total kinetic entropy ∫flogf is conserved (but does not converge! information is conserved for all time, but because of the weak convergence there is a loss of information in the passage to the limit t→∞).
Similarly, in nonlinear Landau damping, the energy of interaction—which is ∫W(x−y)ρ(x)ρ(y) dx dy—tends toward 0, and it is converted into kinetic energy (which can increase or decrease as a function of the interaction).
10.11 The Echo Experiment
In this famous experiment [77, 78] a plasma, prepared in a state of equilibrium, is excited at the initial time by a spatial frequency impulse k. At the end of a time τ, after relaxation of the plasma, it is excited anew by a spatial frequency ℓ, collinear and in the direction opposite to k, and of greater amplitude. We then wait and observe spontaneous response from the electric field of the plasma, called echo, which is produced with spatial frequency k+ℓ and around the time t _{ e }=(ℓ/k+ℓ)τ.
This experiment shows that the kinetic distribution of the plasma has kept track of past impulses: even if the force field has died off to the point of becoming negligible, the kinetic oscillations of the distribution remain present and evolve over the course of time. The first impulse subsists in the form of very rapid oscillations of period (kt)^{−1}, the second in the form of oscillations of period (ℓ(t−τ))^{−1}. A calculation, found e.g. in [110, Sect. 7.3] shows that the distribution continues to oscillate rapidly in velocity, and the associated force remains negligible, up until the two trains of oscillations compensate almost exactly, which is manifested by an echo.
Footnotes
 1.
Even if this formula accurately reflects Boltzmann’s thoughts, it was Planck who first wrote it in this particular form, around 1900.
 2.
The monography [27] was incomplete at the time of Carleman’s passing away, and was completed by Carleson.
 3.
This program culminates in a recent manuscript by Mischler and Mouhot.
 4.
According to a personal communication by Mouhot, there are clues that Sturm’s measure may be too singular to do the job.
 5.
In real life, I think it likely that the validity of the Boltzmann equation is longer, because of slight nonNewtonian randomness, like quantum perturbations, which “renew” the equation; but this does not invalidate the reasoning.
 6.
Maxwell’s Demon has been the object of many discussions, in particular by Smoluchowski, Szilard, Gabor, Brillouin, Landauer and Bradbury; it has also inspired novelists like Pynchon. A recent paper by Binder and Danchin suggests to look for such concepts in the heart of living mechanisms.
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