# Lanford’s Theorem and the Emergence of Irreversibility

## Abstract

It has been a longstanding problem to show how the irreversible behaviour of macroscopic systems can be reconciled with the time-reversal invariance of these same systems when considered from a microscopic point of view. A result by Lanford (Dynamical systems, theory and applications, 1975, Asterisque 40:117–137, 1976, Physica 106A:70–76, 1981) shows that, under certain conditions, the famous Boltzmann equation, describing the irreversible behaviour of a dilute gas, can be obtained from the time-reversal invariant Hamiltonian equations of motion for the hard spheres model. Here, we examine how and in what sense Lanford’s theorem succeeds in deriving this remarkable result. Many authors have expressed different views on the question which of the ingredients in Lanford’s theorem is responsible for the emergence of irreversibility. We claim that these interpretations miss the target. In fact, we argue that there is no time-asymmetric ingredient at all.

### Keywords

Statistical mechanics Irreversibility Time-reversal invariance Lanford## 1 Introduction

The Boltzmann equation is one of the most important tools of statistical physics. It describes the evolution of a dilute gas towards equilibrium, and serves as the key to the derivation of further hydrodynamical equations. A striking aspect of this equation is that it is irreversible, i.e., it is not invariant under time reversal. Indeed, when Boltzmann [2] presented this equation, he immediately derived from it a celebrated theorem, now commonly known as the \(H\)-theorem, which shows that a certain quantity \(H\) of the gas can only change monotonically in time, so that the gas displays an evolution towards equilibrium.

Despite its long-standing legacy, the status of the \(H\)-theorem has remained controversial. The reversibility objection by Loschmidt [19] questioned the validity of the \(H\)-theorem by constructing a counterexample. Essentially, this objection raised the problem of how an irreversible macro-evolution equation can be obtained from the time-reversal invariant micro-evolution equations governing molecular motion. More than twenty years later, Culverwell [10] posed the same problem and inaugurated a famous debate in *Nature* with a provocative question: “Will anyone say exactly what the \(H\)-theorem proves?”.

In his responses to the reversibility objection, [4, 5] suggested an alternative approach and reading of the \(H\)-theorem, which the [12] called the “modified formulation of the \(H\)-theorem”, and which we will refer to as the statistical \(H\)-theorem. Yet, the problem of providing a rigorous statistical counterpart of the Boltzmann equation and the \(H\)-theorem was left unsolved. It is widely believed that a theorem by Oscar [15, 16, 17] provides the best available candidate for a rigorous derivation of the Boltzmann equation and the \(H\)-theorem from statistical mechanics, in the limiting case of an infinitely diluted gas system described by the hard spheres model, at least for a very brief time.

The proof of Lanford’s result is cast in the formalism developed by Bogolyubov, Born, Green, Kirkwood and Yvon (BBGKY). This formalism provides, departing from the Hamiltonian formulation of statistical mechanics, a hierarchy of equations for the time-evolution of macroscopic systems, called the BBGKY hierarchy. On the other hand, the Boltzmann equation itself can also be reformulated in the form of a hierarchy (the Boltzmann hierarchy). Lanford’s theorem then shows how the Boltzmann hierarchy can be obtained from the BBGKY hierarchy for the hard spheres model in the Boltzmann–Grad limit under specific conditions. To be sure, the technical assumptions needed in this rigorous derivation present on several points severe limitations. In particular, the convergence obtained in this Boltzmann–Grad limit holds for a very brief time only, and the Boltzmann–Grad limit itself implies that the density of the gas-model goes to zero, which is quite incompatible with the hydrodynamic limit where the Boltzmann equation is actually supposed to work. These clauses of course imply that the theorem will hardly apply to realistic circumstances.

Still, Lanford’s theorem has recently been called “maybe the most important mathematical result of kinetic theory” [28]. The importance of this theorem is that it claims to show how the conceptual gap between macroscopic irreversibility and microscopic reversibility can in principle be overcome, at least in simple cases. However, Lanford’s papers suggested various answers to the question exactly how the irreversibility embodied in the Boltzmann equation or the ensuing \(H\)-theorem arises in this rigorous statistical mechanical setting. Later authors on Lanford’s theorem (e.g.: [8, 9, 18, 22, 23, 24, 26]) have also expressed mutually incompatible views on this particular issue. So, one may well ask: “Will anyone say exactly what Lanford’s theorem proves?”.

The present paper addresses this question. We analyse the problem of how Lanford’s theorem gives rise to the irreversible behaviour of the Boltzmann equation and show that most previous interpretations of the emergence of irreversibility in this theorem miss the target. In fact, we argue that there is no genuine irreversibility in Lanford’s theorem. We begin by reviewing the Boltzmann equation and the \(H\)-theorem in the kinetic theory of gases for the hard spheres model, along with the quest for a statistical \(H\)-theorem (Sect. 2). Section 3 discusses the connection between the BBGKY hierarchy for the hard spheres model and the Boltzmann hierarchy. Lanford’s theorem is then stated in Sect. 4. We take up the issue of irreversibility in Sect. 5, and present our conclusions in Sect. 6. To summarize, we conclude that Lanford’s theorem does not contain any time asymmetric ingredient. In fact, we argue that all the assumptions of the theorem are time-reversal invariant. We also show that, contrary to claims by Lanford [15], Cercignani et al. [9] and Cercignani [8] the technical procedure of rewriting all collision integrals in terms of an incoming representation, which is used in the proof, does not introduce time asymmetry. In particular, while the initial conditions allowed by the theorem allow one to derive the Boltzmann equation for positive times, they also allow one to derive the so-called ’anti-Boltzmann equation’ (the time-reversal transform of the Boltzmann equation) for negative times. Whereas the solutions of the Boltzmann equation lead to an increase of entropy, solutions of the anti-Boltzmann equation lead to a decrease of entropy. The upshot of our analysis is that Lanford’s result is time-reversal invariant, and thus it is neutral with respect to the arrow of time. As a consequence, there cannot be any source of irreversibility in the theorem. These conclusions mirror observations that have been made many times concerning Boltzmann’s statistical \(H\)-theorem. Thus, although Lanford’s theorem does not give rise to irreversibility, it does nevertheless provide a mathematically rigorous underpinning of the statistical \(H\)-theorem.

## 2 Boltzmann’s Derivation of the Boltzmann Equation and the \(H\)-Theorem

In the kinetic theory of gases, one considers a gas as a system consisting of a very large number \(N\) of molecules, moving in accordance with the laws of classical mechanics, enclosed in a container \(\Lambda \) with perfectly elastic reflecting and smooth walls. In the hard spheres model, these molecules are further idealized as rigid and impenetrable spheres of diameter \(a\) interacting only by collisions. The instantaneous state of the gas system at time \(t\) is represented by a *distribution function*\(f_t(\vec {q}, \vec {p})\), such that \(f_t(\vec {q}, \vec {p}) d \vec {q} d \vec {p}\) is supposed to give the relative number of molecules in the gas with positions between \(\vec {q}\) and \(\vec {q} +d \vec {q}\) inside the container \(\Lambda \) and momenta between \(\vec {p}\) and \(\vec {p} + d \vec {p}\).

*Stoßzahlansatz*, or “assumption about the number of collisions”, also often referred to as the Hypothesis of Molecular Chaos, which provides a constraint on the way in which collisions between the particles take place. There are (at least) two distinct versions of this assumption in the literature, which we would like to distinguish:

- Factorization The relative number of pairs of particles, with positions within \(d\vec {q}_{1}\) and momenta within \(d\vec {p}_{1}\), and within \(d\vec {q}_{2}\) and \(d\vec {p}_{2}\), respectively, is given by$$\begin{aligned} f_{t}^{(2)}(\vec {q}_{1}, \vec {p}_{1}; \vec {q}_{2}, \vec {p}_{2}) d\vec {p}_{1} \, d\vec {q}_{1}d\vec {p}_{2}d\vec {q}_{2} = f_{t}(\vec {q}_{1}, \vec {p}_{1}) \, f_{t}(\vec {q}_{2}, \vec {p}_{2})\, d\vec {p}_{1}d\vec {q}_{1}d\vec {p}_{2} d \vec {q}_2. \end{aligned}$$(2)
- Pre-collision The relative number \(N( \vec {q}, \vec {p}_{1}; \vec {q},\vec {p}_{2})\) of pairs of particles which are
*about to collide*in a region \(d\vec {q}\) and within a time span \(dt\) is proportional to the product \(f_{t}(\vec {q}, \vec {p}_{1}) f_t( \vec {q}, \vec {p}_{2})\) and the volume \(dV\) of the “collision cylinder”, i.e. the spatial region around the position \(q\) at which the particles are located when colliding, i.e.where$$\begin{aligned} N( \vec {q}, \vec {p}_{1}; \vec {q},\vec {p}_{2}) =N f_{t}(\vec {q}, \vec {p}_{1}) f_t(\vec {q}, \vec {p}_{2}) \, dV d\vec {p}_1 d\vec {p}_2, \end{aligned}$$(3)$$\begin{aligned} dV = a^2 \pi \, \vec {\omega }_{12} \cdot \left( \frac{\vec {p}_1 - \vec {p}_2}{m}\right) d t d\vec {\omega }_{12}. \end{aligned}$$(4)

*Stoßzahlansatz*have an importantly different interpretation. The factorization condition (2) may be interpreted as saying that particle pairs are uncorrelated, i.e. finding a first particle at a particular position \(\vec {q}_1\) and moving with momentum \(\vec {p}_1\) gives no information about whether we find the second particle at position \(\vec {q}_2\) and with momentum \(\vec {p}_2\). Note that the condition (2) cannot be literally true for all \(( \vec {q}_1, \vec {p}_{1}; \vec {q}_2,\vec {p}_{2})\): due to their finite diameter, no two particles can have positions such that \(\Vert \vec {q}_1 - \vec {q}_2 \Vert < a\). It is plausible, therefore that Boltzmann implicitly assumed some limit in which \(a \longrightarrow 0\).

The pre-collision condition, on the other hand, says that, when we focus on those pairs of particles that are just about to undergo a collision, they are to be regarded as uncorrelated. It is this condition, and not the factorization condition that is actually used in deriving the Boltzmann equation and the \(H\)-theorem. Of course, one could note that this condition, again, cannot be literally true for the hard-spheres model for the same reason as mentioned before: as long as the particles have a finite diameter, no two particles can be found at the same position \(\vec {q}\) (which would indeed raise questions about the definition of \(\omega _{12}\)). But again, Boltzmann’s derivation of the Boltzmann equation should be, at best, regarded as a heuristic argument, rather than a rigorous proof.

We also note that the literature is somewhat confusing in the terminology here. Many authors use the name “Molecular Chaos” for the factorization condition (2) alone, rather than the pre-collision condition.

*S*tosszahlansatz the Boltzmann equation, which describes the change of the distribution function in the course of time:

However, the validity of the \(H\)-theorem was called into question soon after its formulation. Loschmidt’s reversibility objection, as rephrased by Boltzmann [4] goes basically as follows. Take a non-equilibrium initial distribution of state \(f_0\) for which the \(H\)-theorem holds and let it evolve for a certain amount of time \(t\), so that \(H[f_t] < H[f_0]\). Then, suddenly reverse the velocities of all particles. The particles will now simply retrace all their previous motions back to their original spatial configuration at time \(2t\). If at that point we reverse their velocities again the distribution of state at time \(2t\) will be identical to \(f_0\). But since \(H\), as defined by (9) is invariant under a velocity reversal, this means that under the evolution from \(t\) to \(2t\), \(H\) must have been increasing. In other words, for every dynamically allowed evolution for the particles during which \(H\) decreases, one can construct another for which \(H\) increases, but also allowed by the dynamics.

This argument relies on the tension between the time-reversal invariance of the dynamics governing the motion of the particles and the explicit time-reversal non-invariance of the \(H\)-theorem. In fact, one can trace this time-reversal non-invariance back to the Boltzmann equation from which the \(H\)-theorem has been derived. To be explicit, the behaviour under time reversal of the Boltzmann equation may be checked by implementing the usual transformations: (i) replace \( \partial /\partial t\) by \( - \partial /\partial t\); (ii) reverse the direction of all momenta. It is easy to verify that under these transformations, the left-hand side of (8) changes sign, but the collision integral does not, so that the equation is indeed not time-reversal invariant. This is shown explicitly in *Proposition*1 in the Appendix. The equation one obtains by applying a time-reversal transformation to the Boltzmann equation is commonly called the *anti-Boltzmann equation*, i.e. the version of the Boltzmann equation with an additional minus sign in front of the collision term.

The upshot of the reversibility objection is that the irreversible time-evolution of macroscopic systems cannot be a consequence of the laws of Hamiltonian mechanics alone. There must be some additional non-dynamical ingredient in the \(H\)-theorem, or indeed in the Boltzmann equation from which it follows, that picks out a preferred direction in time. As we now know, the Stoßzahlansatz is the culprit. The pre-collision condition introduces a time-asymmetric element, since it is assumed to hold only for particle pairs immediately *before* collisions, but not for pairs immediately *after* they collided. This is responsible for the failure of the Boltzmann equation to be time-reversal invariant. Indeed, if we had supposed, instead of the pre-collision condition, a similar condition for the momenta immediately after collision, we would, by the same argument, have obtained the anti-Boltzmann equation, and accordingly, we would have derived an anti-\(H\) theorem, i.e. \(dH[f_t]/dt \ge 0\). Hence the irreversible behaviour in the macro-evolution of non-equilibrium distributions towards equilibrium is due to the preference of this pre-collision rather than a corresponding post-collision condition. But this preference cannot be grounded in the dynamics.

Boltzmann [4] response to Loschmidt already argued that one cannot prove that every initial distribution function should always evolve towards the equilibrium distribution function, but rather that there are infinitely many more initial states that do evolve, in a given time, towards equilibrium than do evolve away from equilibrium, and that even these latter states will evolve towards equilibrium after an even longer time. However, Boltzmann did not provide proofs for these claims. A more detailed argument can be found in [5, 6]. To any microstate one can associate a curve (the \(H\)-curve), representing the behavior of \(H[f_t]\) in the course of time. Boltzmann claimed that, with the exception of certain ‘regular’ microstates, the curve exhibits the following properties: (i) for most of the time, \(H[f_t] \) is very close to its minimal value \(H_{m in}\); (ii) occasionally the \(H\)-curve rises to a peak well above the minimum value; (iii) higher peaks are extremely less probable than lower ones. If at time \(t = 0\) the curve takes on a value \(H_[f_0]\) much greater than \(H_{min}\), the function may evolve only in two alternative ways. Either \(H[f_{0}]\) lies in the neighborhood of a peak, and hence \(H[f_t]\) decreases in both directions of time; or it lies on an ascending or descending slope of the curve, and hence \(H[f_t]\) would correspondingly decrease or increase. However, statement (iii) entails that the first case is much more probable than the second. One would thus conclude that there is a very high probability that at time \(t = 0\) the entropy of the system, associated with the negative of the \(H\)-function, would increase for positive time; likewise there is a very high probability that the entropy would increase for negative time. It is this conclusion that is sometimes called the *statistical*\(H\)*-theorem*.

Nevertheless, Boltzmann gave no proof of these claims, nor did he indicate whether or how they might still depend on the Boltzmann equation, or the *Stoßahlansatz*. In fact, the statistical \(H\)-theorem is hardly a theorem at all. The problem of finding an analogue of Boltzmann’s \(H\)-theorem in statistical mechanics thus remained unsettled. In order to make progress upon this problem, many authors have called upon Lanford’s theorem. Indeed, this theorem is often presented as providing a rigorous derivation of the Boltzmann equation and the associated \(H\)-theorem from statistical mechanics.

## 3 Lanford’s Derivation of the Boltzmann Equation from Hamiltonian Statistical Mechanics

### 3.1 From the Hamiltonian Framework to the BBGKY Hierarchy

Strictly speaking, the Hamiltonian should also contain a term corresponding to the elastic wall potential, describing the interaction when individual particles collide with the boundary \(\partial \Lambda \) of the vessel. However, there are ways to suppress this complication. The easiest way is to suppose that each particle \(i\) undergoes specular reflection when it hits the wall and identify the values \((\vec {q}_i, \vec {p}_i)\) just before such a collision and the values \((\vec {q}_i, \vec {p}^{\prime }_{i})\) immediately after. In this move, the phase-space \(\Gamma _N\) is endowed with the topology of a torus, and the dynamics under wall collisions becomes smooth. Indeed, a collision with the wall becomes indistinguishable from free motion, and consideration of the wall potential becomes redundant.

*how many*particles have certain molecular properties, or how many pairs have certain relations to each other, etc., but not on their particle labels. It thus becomes attractive to study the dynamics in terms of reduced probability densities obtained by conveniently integrating out most of the variables. For this purpose, one defines a hierarchy of reduced or marginal probability densities:

These dynamical equations for the rescaled reduced densities of the statistical state \(\mu \) constitute the BBGKY hierarchy. Note that, taken together, they are strictly equivalent to the Hamiltonian evolution, i.e. nothing else has been assumed yet, except for the rather harmless permutation invariance of \(\rho \) and the specific form of the Hamiltonian (11). As one might expect, therefore, solving these equations is just as hard as for the original Hamiltonian equations. Indeed, to find the time-evolution of \(\rho _{1}\) from 21, we need to know \(\rho _{2,t}\). But to solve the dynamical equation for \(\rho _{2}\), we need to know \(\rho _{3,t}\) etc. Moreover, the equations of the BBGKY hierarchy are still perfectly time-reversal invariant.

Nevertheless, the above might already make one hopeful that a counterpart of the Boltzmann equation can be obtained from the exact Hamiltonian dynamics. Indeed, if we tentatively identify Boltzmann’s \(f\) function with \(\rho _{1}\), (21) looks somewhat similar to the Boltzmann equation (8). Of course, much work still remains to be done: first of all, the Boltzmann equation pertains to the hard-sphere model, whereas the Eq. 21 assumes a smooth pair-potential \( \phi (\vec {q}_i - \vec {q}_j)\). More importantly, we would have to justify the tentative relationship between \(\rho _{1}\) and \(f\). These tasks will be addressed in the following subsections.

### 3.2 From Smooth Potentials to the Hard-spheres Gas Model

While the BBGKY hierarchy provides a generally useful format for studying the evolution of a statistical state for a system of indistinguishable particles interacting by smooth pair potentials, it is our purpose here to apply it to the hard-spheres potential 12.

That is to say, we can either delete this unwanted set \(\Delta \) of measure zero from our phase space \(\Gamma ^{(a)}_{N, \ne }\), and in doing so guarantee that there is a Hamiltonian flow \(\{ T_t, t\in {\mathbb {R}}\}\) defined on the smaller phase space \(\Gamma _{N, \ne } \setminus \Delta \), or continue with the original space, with the provision that its Hamiltionian flow is defined only almost everywhere, i.e. outside of the above set \(\Delta \).

*Proposition*2 in the Appendix.

However, one more step is needed in order to obtain Lanford’s theorem. Let us split the integral over the unit sphere \(S^2\) into two parts: the hemisphere \(\vec {\omega }_{i,k+1} \cdot (\vec {p}_{i} - \vec {p}_{k+1}) \ge 0 \), and the hemisphere \(\vec {\omega }_{i,k+1} \cdot (\vec {p}_{i} -\vec {p}_{k+1}) \le 0 \). In the first hemisphere, the collision configuration \(( \vec {q}_i, \vec {p}_i; \vec {q}_i + a \vec {\omega }_{i,k+1}, \vec {p}_{k+1})\) represents a collision between particles \(i\) and \(k+1\) with incoming momenta \(\vec {p}_i, \vec {p}_{k+1}\), and we leave the integrand as it is.

### 3.3 From the Boltzmann Equation to the Boltzmann Hierarchy

In this section, we start from the other side of the bridge that we aim to cross. That is, we take the Boltzmann equation as given, and reformulate it in a mathematically equivalent hierarchy of distribution functions. This idea is captured by the lemma below, which is spelled out by Lanford [15], p. 88.

**Lemma 3.1**

*Boltzmann hierarchy*, for the functions \((f_{1}, f_{2}, \ldots )\) under the assumption of a factorization condition (33). One can write this hierarchy more compactly by regarding the \(f_{k}\) as components of a vector: \(\varvec{f}=(f_{1}, f_{2}, \ldots )\). Then we can write (34) schematically as:

The second virtue of the lemma is that it brings the Boltzmann equation in a form which is more similar to the results from the BBGKY formalism discussed above, which likewise take the form of a hierarchy, and this alleviates the effort to build a rigorous bridge between them.^{1}

As we have remarked above, the factorization condition (33), taken as a generalization of Boltzmann’s condition (2), is sometimes called ‘molecular chaos’. This is an unfortunate habit because (33) does not contain an accompanying condition to single out the pre-collision coordinates, as a generalization of (3). Nevertheless, it is worth noting that if the initial data of the Boltzmann hierarchy at time \(t = 0\) take the form (33), then this factorization is preserved through time, i.e. it holds for the solution of (34) for all time \(t\), with \(f_{t}\) being a solution of the Boltzmann equation. This important property of the Boltzmann hierarchy is commonly known as ‘propagation of chaos’.^{2} We emphasize, however, that this factorization, and its preservation in time, has nothing to do with the pre-collision condition (3) mentioned in Sect. 2 as a crucial ingredient of the molecular chaos hypothesis Boltzmann used to obtain the Boltzmann equation.

Finally, we stress that the Boltzmann hierarchy, being an equivalent way of expressing the Boltzmann equation, is just as time-reversal non-invariant as the original Boltzmann equation. In fact, by applying a time-reversal transformation to it, one obtains a hierarchy of evolution equations which has the same form as (34) except for a minus sign in front of the collision term \(C_{k, k+1}\). We refer to the latter as the *anti-Boltzmann hierarchy*. The ‘propagation of chaos’ property which we just noted for the Boltzmann hierarchy is also valid for this anti-Boltzmann hierarchy. Also, notice that both the collision operators and the distribution functions in (36) resemble those involved in (32), except that they do not depend on the diameter \(a\) of the particles. The crucial point in Lanford’s theorem is to demonstrate that all relevant terms in the BBGKY hierarchy tend to their counterparts in the Boltzmann hierarchy in the Boltzmann–Grad limit, whereby \(a \longrightarrow 0\). That would establish that the Boltzmann equation can be obtained from Hamiltonian mechanics.

## 4 Lanford’s Theorem

So far, we have seen how the Hamiltonian dynamics for the hard-spheres model leads, under a relatively harmless assumption of permutation invariance, to a hierarchy of BBGKY equations describing the evolution of reduced density functions of a statistical state. And we have also seen how the Boltzmann equation can be reformulated as a hierarchy of equations in close resemblance to the BBGKY hierarchy. The question still remains how to bridge the gap between these two descriptions. Lanford’s theorem establishes the convergence of the BBGKY hierarchy to the Boltzmann hierarchy in the so-called Boltzmann–Grad limit.

This Boltzmann–Grad limit defines a particular limiting regime within the hard spheres model. In this limit, one not only lets the number of particles grow to infinity, i.e. \(N \rightarrow \infty \), but also requires that their diameter goes to zero, i.e. \(a \rightarrow 0\), while keeping the volume \(|\Lambda |\) of the container fixed. The limit is taken in such a way that the quantity \(N a^{2}\) remains constant, or at least approaches a finite non-zero value. This guarantees that the collision term in the Boltzmann equation or Boltzmann hierarchy, which is proportional to this quantity, does not vanish. Accordingly, the ‘mean free path’ \(\lambda := \frac{|\Lambda |}{2 \pi N a^{2}}\), which is the typical scale-distance traveled by any particle between two subsequent collisions in an equilibrium state, also remains of order one. The same holds for the ‘mean free time’, i.e. the typical duration between collisions in equilibrium, which is of the order \( \sqrt{(\beta /3)} \pi a^2N/|\Lambda |\), where \(\beta \) is the inverse temperature.

We are now ready to state the precise version of Lanford’s theorem, as given by Spohn [23], Theorem 4.5. Here, when we write \(\lim _{a \longrightarrow 0}\), the Boltzmann–Grad limit is understood, i.e. it is assumed that \(N\longrightarrow \infty \) simultaneously, while keeping \(Na^2\) a fixed non-zero constant.

LANFORD’S THEOREM

With the notation introduced in Sect. 3, take \(0<a<a_0\) and let \(\rho ^{(a)}_{k,t}\) be a family of functions defined on \(\Gamma ^{(a)}_{k,\ne }\), and assume that for all such \(a\), the following conditions hold at time \(t = 0\).

- (i)There exists positive real constants \( z, \beta ,M\), independent of \(a\), such thatfor any \(k= 1,2,\ldots \), where \(h_{\beta }(\vec {p}_{i})\) denotes the normalized Maxwellian distribution over momenta: \( h_{\beta }(\vec {p}_{i}) = (\frac{\beta }{2 \pi })^{\frac{3}{2}} \cdot e^{-\frac{\beta \vec {p}^{2}_{i}}{2}}\) at inverse temperature \(\beta \), and the spatial distribution is constant inside the vessel \(\Lambda \) with density \(z\).$$\begin{aligned} \rho _{k, 0}^{(a)}(x_{1}, \ldots , x_{k}) \le M z^k \prod _{i=1}^{k} h_{\beta }(\vec {p}_{i}), \end{aligned}$$(43)
- (ii)There exist continuous functions \(f_{k,0}\) on \(\Gamma _k\), for \(k=1,2\ldots \) such thatfor all compact subsets \(K \subset \Gamma _{k ,\ne }(s)\) for some \(s \ge 0\).$$\begin{aligned} \lim _{a \longrightarrow 0} \underset{(x_1,\ldots x_k) \in K}{\text {ess sup}}|\rho _{k, 0}^{(a)}(x_{1}, \ldots , x_{k}) - f_{k, 0}(x_{1}, \ldots x_{k})| =0, \end{aligned}$$(44)

Then, there exists a strictly positive time \(\tau \), such that for all times \(0\le t \le \tau \)

for any \(k= 1,2, \ldots \) and compact subset \(K \subset \Gamma _{k, \ne }(s+t)\).

Here, \(\rho ^{(a)}_{k,t}\) are solutions of the BBGKY hierarchy with initial conditions \(\rho ^{(a)}_{k,0} \) and \(f_{k,t}\) solutions of the Boltzmann hierarchy with initial conditions \(f_{k,0}\).

By assumption (ii), the sequence of initial conditions for the BBGKY hierarchy converge to functions \(f_{k,0}\) that serve as initial conditions of the Boltzmann hierarchy. The theorem then states that this convergence is maintained through time, at least for \(t\in [0,\tau ]\), so that solutions \(\rho _{k, t}^{(a)}\) of the BBGKY hierarchy converge to solutions \(f_{k, t}\) of the Boltzmann hierarchy as \(a \rightarrow 0\) for all \(k\), except for the phase-points comprised in the set \( \Gamma _{k ,=}(s+t)\). The size of such exceptional sets increases in time, i.e. \(\Gamma _{k, =}(s + t) \subset \Gamma _{k, =}(s + t')\) if \( 0<t < t'\). It follows that the type of convergence obtained for \(\rho ^{(a)}_{k,t}\) in (45) is weaker than the convergence assumed for the initial conditions \(\rho ^{(a)}_{k,0}\) in (55). This is actually to be expected due to the fact that the BBGKY hierarchy is time-reversal invariant, while the Boltzmann hierarchy is not. Note, however, that, being hypersurfaces of codimension one, the exceptional sets \( \Gamma _{k ,=}(s+t)\) all have Lebesgue measure zero for any time \(t\), and hence they also have probability zero for any statistical state which is absolutely continuous with respect to the Lebesgue measure.

*almost*imply factorization of the limiting densities. Moreover, since the Boltzmann hierarchy is linear and propagates chaos, it will preserve this convex combination for later times, i.e., if the initial conditions for the Boltzmann hierarchy take the form of the right-hand side of Eq. (46), they will evolve in time to

Another issue, that is emphasized by Lanford himself and nearly all subsequent commentators, concerns the limited time of validity of the theorem. In fact, the time bound \(\tau \), for which the sought-after convergence of solutions of the BBGKY hierarchy to solutions of the Boltzmann hierarchy is assured, proves extremely short. An explicit estimate given by Spohn [23], p. 62 shows that \(\tau \le 0.2 \sqrt{(\beta /3)} \pi a^2z \), and hence the result holds only during one-fifth of the mean free time between collisions for the given Maxwellian. Since such a time-scale, for realistic gas systems under ordinary circumstances, will be of the order of milliseconds, the theorem will hardly be enough ammunition to provide a justification of the Boltzmann equation through macroscopic time scales, or even the time scale in which equilibration sets in.^{3} Yet, this is not too short to make irreversibility unobservable: in a duration of \({1}/{5}^{th}\) of the mean collision time, one expects that about 20 % of the particles will have collided, and this can be sufficient for a significant increase of entropy of the gas. As Lanford put it,

Although these results apply only to small positive times, the times involved are large enough for Boltzmann’s \(H\)-function to decrease a strictly positive amount. Thus our results show unambiguously that there is no contradiction between the reversibility of molecular dynamics and irreversibility implied by the \(H\)-theorem. [15], p. 99.

But is there really irreversibility in Lanford’s theorem? And, if so, where does it come from? This question seems particularly pressing since this theorem does not explicitly rely on the *Stosszahlansatz* or an analogous statement. We take up these questions in the next section.

## 5 Irreversibility in Lanford’s Theorem

Lanford’s theorem shows how one can derive the Boltzmann equation from the Hamiltonian equations of motion under precise assumptions. As a statistical version of Boltzmann’s \(H\)-theorem, it seems to account for the approach to equilibrium for a general class of non-equilibrium initial conditions characterized by the regularity condition (i), at least during the time-interval \([0, \tau ]\). The most important question is then how the implied irreversibility of this macro-evolution arises. On this point Lanford and other authors on his theorem made remarks that are not quite univocal. We first survey and criticize these different views and then present our own argument on the emergence of irreversibility.

### 5.1 Views on the Emergence of Irreversibility in the Literature

In the final pages of his first paper, Lanford offers a diagnosis for the emergence of irreversibility. There, he stresses that the factorization condition cannot be the time-asymmetric ingredient needed to derive the Boltzmann equation.

The Boltzmann hierarchy, like the Boltzmann equation is not invariant under time reversal. That is, irreversibility appears in passing to the limit \(a\longrightarrow 0\), not in the assumption that the rescaled correlation functions factorize. [15], p. 110

Indeed, Lanford’s result does not require a factorization condition to get convergence towards a solution of the Boltzmann hierarchy, but only to guarantee that the latter becomes equivalent to the Boltzmann equation. In fact, the factorization condition, at least in the version adopted by Lanford, (i.e. (47), as distinguished from the Stoßzahlansatz) is itself time-reversal invariant. Therefore, it surely does not yield an explanation for irreversibility. This point is fortified by the fact that, as we saw in section 4.1, factorization is also invoked in the version of theorem holding for negative times to derive the anti-Boltzmann equation. Instead, since irreversible behaviour already appears at the level of the Boltzmann hierarchy (or the anti-Boltzmann hierarchy), Lanford puts the blame on the procedure to take the limit from the BBGKY hierarchy to the Boltzmann hierarchy. (In the above quote, the notation \(a\longrightarrow 0\) ought to be understood as equivalent to \(N \longrightarrow \infty \), as the Boltzmann–Grad limit prescribes.) We obviously agree here with [15] that any irreversibility is not due to the factorization assumption, but should we thereby conclude that it is due to the Bolzmann-Grad limiting procedure?

^{4}The lesson Lanford draws from this is the following:

None of this, however, really implies that irreversible behavior

mustoccur in the limiting regime; it merely makes this behavior plausible. For a really compelling argument in favor of irreversibility, it seems to be necessary to rely on some version of Boltzmann’s original proof of the \(H\)-theorem [17], p. 75.

Unfortunately, Lanford did not specify how appealing to (some version of) Boltzmann’s derivation of the \(H\)-theorem would provide a compelling argument in favor of irreversibility. But it seems reasonable to suppose that he meant to go back to the Stoßzahlansatz, and its distinction between pre- and post-collision configurations.

An explanation of what Lanford intended in the last quote may perhaps be traced back to the analysis he develops in his 1975 work. When presenting his own re-examination of the derivation of the Boltzmann equation from the BBGKY hierarchy, i.e. (32) specialized to the case \(k=1\), Lanford comments:

We obtained [the BBGKY hierarchy for the hard spheres model with the collision term expressed by Eq. (32) in the present paper] by systematically writing collision phase points in their incoming representations. We could have equally well have written them in their outgoing representations; if we then assumed factorization we would have obtained the Boltzmann collision term with its sign reversed. It is thus essential, in order to get the Boltzmann equation, to assume

for

incomingcollision points \((x_1, x_2)\) and not for outgoing ones. [15], p. 88

*vice versa*as we did in (32). Let us make this point explicit. In analogy to what we did in Sect. 3.2, going back to Eqn. (29), we again split the integral over \(S^{2}\) into two hemispheres, but on the hemisphere with \( \vec {\omega }_{i, k+1} \cdot ( \vec {p}_{k+1} - \vec {p}_i) \ge 0\) we would leave the configuration for the pair \((\vec {q}_i, \vec {p}_i, \vec {q}_i+ a \vec {\omega }_{i, k+1} , \vec {p}_{k+1})\) as it is (i.e. outgoing). On the hemisphere characterized by \( \vec {\omega }_{i, k+1} \cdot ( \vec {p}_{k+1} - \vec {p}_i) \le 0\) we replace the coordinates \((\vec {q}_i, \vec {p}_i, \vec {q}_i+ a \vec {\omega }_{i, k+1} , \vec {p}_{k+1})\) by \((\vec {q}_i, \vec {p} ^{\ \prime }_i, \vec {q}_i+ a \vec {\omega }_{i, k+1} , \vec {p}^{\ \prime }_{k+1})\) as well as change the sign of the integration variable \(\vec {\omega }_{i, k+1}\) so that the two hemisphere integrals have a common domain. The result is that Eq. (29) now goes over in

What is puzzling about this view, though, is that in an earlier passage [15], p. 86 had argued for the *identification* of phase points which differ only by having an incoming collision configuration replaced by the corresponding outgoing collision configuration. This suggests that the origin of irreversibility in the Boltzmann equation would now lie in a conventional choice of representation of the *same* phase point. An obvious objection to this view is that it is not clear at all how physical irreversibility can be due to a conventional choice of representation. Since the question of whether we derive \(dH/dt \le 0\) or \( dH/dt \ge 0\) is a substantive issue, such a difference cannot be a matter of mere convention.

Cercignani et al. [9] and Cercignani [8] have argued that there actually is a non-conventional justification for adopting the incoming representation, and that the choice for this representation indeed has a dynamical underpinning:

We are compelled to ask whether the representation in terms of ingoing configurations is the right one, i.e. physically meaningful. As we shall later see, in a more careful analysis of the validity problem, the representation in terms of ingoing configurations follows automatically from hard-spheres dynamics and is, indeed, not a matter of an a priori choice [9], p. 74.

So, for Cercignani et al., the preference for the incoming configuration over the ongoing one is an automatic consequence of the hard-spheres dynamics. This view might actually seem to settle the issue in a non-conventional way. However, in Sect. 5.2 below, we will argue in detail against the claim that irreversibility is due to the adoption of one representation rather than another.

An entirely different argument for the emergence of irreversibility is presented by Spohn [22, 23] and Lebowitz [18]. A similar argument is also sketched by Lanford [16]. Spohn [22], p. 596 devotes a paragraph to the question “how Lanford’s theorem escapes the conflict between the reversible character and the irreversible character of the Boltzmann hierarchy see also [23], p. 66. He points out how Lanford’s theorem will not sustain the construction of a counterexample as in the original reversibility objection to the \(H\)-theorem. Recall that in that construction we assumed an initial distribution function \(f_0(x)\) that evolves in accordance with the Boltzmann equation from the initial time \(0\) to some positive time \(t\), when the distribution function is \(f_t(x)\), and then suddenly reverse all the velocities of all particles. Due to the time-reversal invariance of the microdynamics, \(H[f]\) would have to increase during the interval \([t,2t]\). Spohn discusses what happens if we try to run this same argument on the basis of Lanford’s theorem (a more elaborate version of his reasoning is given by Lebowitz [18]). The crucial point in his analysis is that the set \(\Gamma _{k ,=}(s + t)\) of phase points for which the convergence need not hold increases with time. Hence, if we consider the rescaled densities \(\rho ^{(a)}_{k,t}\) at time \(t\), with \(0 < t < \tau \), and then reverse the velocities, the ensuing evolution of these functions will no longer be guaranteed by Lanford’s result: in fact, for the theorem to be applicable at the new initial time \(t\) after the velocity-reversal, the convergence of solutions of the BBGKY hierarchy to solutions of the Boltzmann hierarchy would have be assured again over the domain of convergence \(\Gamma _{k,\ne }(s)\) as demanded by the convergence condition (ii). Yet, from the result of Lanford’s theorem from the evolution during \([0,t]\), we would only have convergence on \(\Gamma _{k,\ne }(s + t)\). Given that \(\Gamma _{k,\ne }(s) \supset \Gamma _{k,\ne }(s + t)\), we cannot apply Lanford’s theorem (after velocity-reversal) to obtain an evolution during the time \([t,2t]\) convergent to a solution of the Boltzmann equation with increasing \(H\) on the same domain of rescaled density functions as we started out with. An argument along the lines of the reversibility objection against Lanford’s theorem is thereby blocked.

This suggests a very different view on the emergence of irreversibility than the previous ones. The time-asymmetric ingredient in the theorem is now identified in its assumptions, specifically in the convergence condition (ii). Indeed, the prescribed domain of convergence is not invariant under time reversal in the sense that \(\Gamma _{k,\ne }(s) \ne \Gamma _{k,\ne }(-s)\). Therefore, according to such an interpretation, the source of irreversibility would lie in time-asymmetric assumptions of Lanford’s theorem.

To summarize the different views on the emergence of irreversibility in Lanford’s theorem available in the literature: Lanford first identified the Boltzmann–Grad limit as the source of irreversibility, but later he mitigated this claim; a next argument of Lanford is that irreversibility arises by adopting the incoming representation for collision points, instead of the outgoing representation. We have argued against this view that it would make the appearance of irreversibility seem to be a matter of mere conventional choice of representation; Cercignani et al. claimed that a privileged role of the incoming representation follows from the hard-spheres dynamics; Spohn and Lebowitz instead pointed out a time-asymmetric ingredient in the domain of assumption (ii) of the theorem. Below, we argue that all these views fail to provide a satisfactory account of the status of irreversibility in Lanford’s result. Moreover, we will argue that, in our opinion, there is no genuine irreversibility in the theorem.

### 5.2 Is there Really Irreversibility Embodied in Lanford’s Theorem?

Let us begin by discussing the notion of the incoming representation, as opposed to the outgoing representation, of a collision point. As [15], p. 87 puts it “[T]hese two are really just different representations of the same phase point.” Even [18], p. 8 argues similarly when he writes about the incoming and outgoing momenta as being “just two different representations of the same phase point.” However, this very identification of the two configurations as representations of the same phase point seems to make the distinction between incoming and outgoing collisions, which is allegedly at the heart of the issue of irreversibility—at least according to some of the quotes we have just discussed—quite hard to maintain.

Yet, taking this way of speaking too literally may expose one to the risk of a misleading interpretation of the theorem. Indeed, we are free in adopting any topology we like on the boundary of \(\Gamma ^{(a)}_{\ne }\) (as long as it extends the Euclidean topology on its interior), and in particular we can choose a topology to make an instantaneous transition from \(x_{\text{ in }}\) to \(x_{\text{ out }}\) appear as a smooth trajectory. Such a choice of topology entails that every metric, or distance function, \(d\) on \(\Gamma ^{(a)}_{N \ne }\), compatible with it would have the property that \(d(x_{\text{ in }} , x_{\text{ out }}) =0\), and hence (by the usual definition of a metric) it would follow that \(x_{\text{ in }} = x_{\text{ out }}\), i.e. those points are identified. But, when choosing a topology, we are not forced to introduce a metric. Moreover, the topological identification of phase-space coordinates ought not to be granted physical significance. Even if we identify the incoming and outgoing points \(x_{\text{ in }}\) and \(x_{\text{ out }}\) for the purpose of topological or metrical considerations, it does not follow that they thereby are *physically* identical. Indeed, that would overlook the distinctive and physically relevant fact that the momenta are quite different in these two points. Furthermore, if the diameter \(a\) of the particles is not null, as it happens in real physical situations described by the hard spheres model, their positions are also necessarily distinct, even if they are colliding. In other words, all that this choice of topology enforces is that a trajectory connecting points like \(x_{\text{ in }}\) and \(x_{\text{ out }}\) becomes smooth, but not that these points are physically one and the same. In fact, the very use of the terminology “representation” appears quite inappropriate in this context. We shall therefore use the term “configuration” instead.

*continuity across collisions*: Denote the phase point in which two particles, say \(i\) and \(k + 1\), are touching each other, i.e., \((x_{1}, \ldots , x_{i-1},\vec {q}_{i} ,\vec {p}_{i}, x_{i+1}, \ldots x_k, \vec {q}_{i} + a \vec {\omega }_{} , \vec {p}_{k+1})\) by the abbreviation \(( \vec {q}_i, \vec {p}_i; \vec {q}_i + a \vec {\omega }_{i,k+1}, \vec {p}_{k+1})\). Continuity across collisions then requires that, if \(\vec {\omega }_{i,k+1} \cdot (\vec {p}_{i} - \vec {p}_{k+1}) \ge 0 \), then

The claim we wish to argue against here is that the choice of the incoming configuration over the outgoing configuration could be the source of irreversibility in Lanford’s theorem. The basis for our argument rests on two main points. On the one hand, as we have just seen, these two configurations are in fact different from each other in a physically relevant sense. On the other hand, to counter the claim of Cercigiani *et al.*, we argue that the hard-spheres dynamics will not provide a preference for writing the collision term of the BBGKY hierarchy in the incoming configuration rather than the outgoing configuration. Indeed, *Proposition*3 in the Appendix shows that, if (53) holds, then the BBGKY hierarchy with the collision term expressed by (32) and the BBGKY hierarchy with the collision term expressed by (52) are perfectly equivalent. Thus, the choice of either one of the two collision configurations does not make any difference at the level of the BBGKY hierarchy. In particular, one can derive the Boltzmann hierarchy, as well as the anti-Boltzmann hierarchy, from the BBGKY hierarchy rewritten in terms of either the incoming or the outgoing configurations without having to choose the “right” one.

Furthermore, *Proposition*3 proves that continuity across collisions is sufficient to guarantee the time-reversal invariance of the BBGKY hierarchy with the collision term expressed by (32), or equivalently by (52). In other words, this choice is neutral with respect to the direction of time. This indicates that the source of irreversibility in Lanford’s theorem does not lie in the adoption of either one between the incoming and the outgoing configurations of collision points.

^{5}An obvious lesson one should draw from this argument is that, for an irreversible approach to equilibrium to obtain, the continuity at collisions condition, which maintains time-reversal invariance of the BBGKY hierarchy written in any collision configuration, has no analogue in the Boltzmann–Grad limit.

We now turn to the view offered by Spohn and Lebowitz. In our opinion, they convincingly showed that the reversibility objection cannot be run against Lanford’s theorem in the same way as is it was used by Loschmidt and Culverwell against Boltzmann’s original presentation of the \(H\)-theorem. However, we believe that it is one thing to show how the reversibility objection is evaded, but it is quite another thing to explain the emergence of irreversibility in Lanford’s theorem. And although the Spohn-Lebowitz argument is successful in the first objective, we feel it does little to offer the sought-after explanation. In particular, the suggestion that the source of irreversibility is to be traced back to time-asymmetric initial conditions employed in the theorem seems unconvincing. First of all, Lanford’s theorem holds also if one sets \(s = 0\) in assumption (ii). That is actually how [15] first formulated his theorem. The domain of convergence \(\Gamma _{k, \ne } (0)\) in condition (ii) then corresponds to the largest set of initial configurations one can admit. The key point is to recognize that this domain is clearly invariant under velocity-reversal, and thus condition (ii) is now time-reversal invariant too.^{6} Yet, the theorem still implies the irreversible Boltzmann hierarchy for positive times, as well as the irreversible anti-Boltzmann hierarchy for negative times. The problem concerning the emergence of irreversibility thus still presents itself, even when the time-reversal non-invariance of assumption (ii) is avoided by choosing \(s=0\).

Secondly, even if we choose \(s>0\) in conditions (ii) and (ii’), recall that the set of exceptional states \(\Gamma _{k,=}(s + t)\cap \Gamma _{k, \ne } (s) \), comprising those microstates in the initial domain of convergence \(\Gamma _{k, \ne } (s)\) for which the solution \(\rho _{k, t}^{(a)}\) of the BBGKY hierarchy for hard spheres may not converge to a solution \(f_{k, t}\) of the Boltzmann hierarchy, has Lebesgue measure zero for all times \(t\). Now, in the spirit in which Lanford’s theorem has been formulated, Lebesgue mejasure zero sets in phase-space are not held to be physically significant, and we have already neglected several such measure-zero sets from the outset. Thus, it would seem that the sets of exceptional states, even if they are not invariant under time-reversal, ought to be neglected as physically irrelevant too. However, in order to obtain an emergence of irreversibility, one would like to see that the overwhelming majority of initial phase space points will evolve in the course of time in such a way to obtain the Boltzmann hierarchy, rather than the anti-Boltzmann hierarchy. Considerations of the time-reversal non-invariance of measure zero sets will not be helpful in this regard.

So far, we have criticized the available views on the emergence of irreversibility in Lanford’s theorem. None of them, in our opinion, really succeeds in identifying an ingredient responsible for the irreversible behaviour of the Boltzmann equation. The claim we wish to make now is that there is no such a time-asymmetric ingredient at all.

In order to substantiate this claim, let us stress that Lanford’s theorem can be proven also for negative times. In fact, while the statement of the theorem in Sect. (4) is formulated for positive times, one can derive an analogous result for negative times \( -\tau <t<0\), as shown by Lanford [15], p. 109–110, and more explicitly by Lebowitz [18], p. 9–10. For this purpose, one ought to take the time-reversal of the assumptions of the theorem and verify that one obtains the time-reversal of the conclusion. The regularity assumption (i) is time-independent, and as such it is time-reversal invariant. Therefore, one leaves it in the same form. Assumption (ii), instead, makes an explicit reference to time in the domain of convergence \(\Gamma _{k, \ne }(s)\). Hence, if one keeps \(s \ge 0\), one can rewrite it as

One can then prove^{7} that there exists a strictly positive time \(\tau \), such that equation (45) holds for any \(k= 1,2, \ldots \) and compact subset \(K \subset \Gamma _{k, \ne }(- s - t)\) during the time interval \(t \in [\tau , 0]\), where now the solutions \(\rho ^{(a)}_{k,t}\) of the BBGKY hierarchy with initial conditions \(\rho ^{(a)}_{k,0} \) are taken to converge to solutions \(f_{k,t}\) of the anti-Boltzmann hierarchy with initial conditions \(f_{k,0}\). Since the domain of convergence \(\Gamma _{k, \ne }(- s - t)\) is the time-reversal of \(\Gamma _{k, \ne }(s + t)\) and the anti-Boltzmann hierarchy is the time-reversal of the Boltzmann hierarchy, this conclusion is just the time-reversal of the conclusion of the theorem as stated for positive times. Further, if one additionally assumes the factorization condition (46), one would derive a solution \(f_{t}\) of the anti-Boltzmann equation.

As a consequence, if we consider a time \(t\) such that \( -\tau \le t \le \tau \), Lanford’s theorem is clearly neutral with respect to time reversal: that is, for positive times we obtain convergence to solutions of the Boltzmann equation, and hence a decrease of \(H\), just as the \(H\)-theorem requires, but for negative times we obtain convergence towards a solution of the anti-Boltzmann equation, and hence an increase of \(H\). This is analogous to Boltzmann’s 1897 argument based on the \(H\)-curve. Indeed, in this understanding, the theorem proves that for most initial microstates the \(H\)-function lies at a local peak of the \(H\)-function. So, at the initial time instant \(t = 0\), \(H[f_t]\) is expected to decrease in both directions of time. This offers a mathematical formalization of Boltzmann’s claim that, apart from equilibrium, the most probable case is that the \(H\)-function is at a maximum of the curve. As such, Lanford’s result provides a rigorous version of the statistical \(H\)-theorem sketched by Boltzmann. More importantly, with respect to the issue of emergence of irreversibility, the thus-established time-reversal invariance of the theorem does supply evidence that there is no time-asymmetric ingredient in the theorem.

On the other hand, though, one should notice that the behavior of the rescaled probability densities \(\rho ^{(a)}_{k,t}\) implied by the theorem for negative times has a serious drawback. In fact, this behavior conflicts with the expectation from thermodynamics that entropy of an isolated gas system should increase rather than decrease even during the interval \([-\tau , 0]\). This issue has already been discussed many times in the literature cf. [21], Feynman *et al.* 1964, [24], [11] and [27].

## 6 Conclusion

We discussed the problem of the emergence of irreversibility in Lanford’s theorem. We argued that all the different views on the issue presented in the literature miss the target, in that they fail to identify a time-asymmetric ingredient that, added to the Hamiltonian equations of motion, would obtain the Boltzmann equation. More to the point, we argued that there is no such an ingredient at all, as one can infer from the fact that the theorem is indeed time-reversal invariant.

## Footnotes

- 1.
The similarity is not complete, of course. For example, the BBGKY hierarchy is finite while the Boltzmann hierarchy is not.

- 2.
- 3.
It is true that Illner and Pulvirenti [14], have derived a longer validity but only under much more stringent conditions, i.e. for a gas cloud expanding into a vacuum. As a matter of fact, this repeated attention to time scale has deluded attention from more serious problems. Indeed, Lanford already pointed out that there is a simple, if merely technical, “fix” to the above problem: one would only need to require that assumption (i) of the theorem holds for arbitrary times, and not just at \(t = 0\), and Lanford’s result may be extended to all times (see [27] for a discussion of this issue).

- 4.
This can actually be seen as a counterexample to [20], who claimed that the irreversible approach to equilibrium would follow from taking the limit for \(N \longrightarrow \infty \).

- 5.
This argument is really an adaptation to the present context of a discussion of the notion of pre-collisional chaos contained in [28], p.35.

- 6.
In fact, in this case, that is when \(s = 0\), the condition coincides with its time-reversal.

- 7.
The proof of this result proceeds in the same way as that for the theorem for positive times, except that here one explicitly appeals to the outgoing configuration instead of the incoming one. However, in light of

*Proposition*3 establishing the equivalence of the BBGKY hierarchy written in terms of each configuration, the latter step is not an independent ingredient.

## Notes

### Acknowledgments

The authors would like to thank Herbert Spohn for extensive comments on the technical and conceptual subtleties of Lanford’s theorem, as well as two anonymous referees for helpful remarks.

### References

- 1.Alexander, R.K.: The infinite hard-sphere system, Ph.D. Thesis, University of California at Berkeley (1975)Google Scholar
- 2.Boltzmann, L.: Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen, Wien. Ber. 66, 275–370 (1872); in Boltzmann (1909) vol.I, pp. 316–402Google Scholar
- 3.Boltzmann, L.: Über das Wärmegleichgewicht Von Gasen, auf welche äußere Kräfte wirken, Wien. Ber. 72, 427–457 (1872); in Boltzmann (1909) vol.II, pp. 1–30Google Scholar
- 4.Boltzmann, L.: Bemerkungen über einige probleme der mechanische Wärmetheorie, Wien. Ber. 66, 275–370 (1877); in Boltzmann (1909) vol.II, pp. 112–148Google Scholar
- 5.Boltzmann, L.: On certain questions of the theory of gases. Nature, 51, 413–415 (1895); in Boltzmann (1909) vol.III, pp. 535–544Google Scholar
- 6.Boltzmann L.: Zu Hrn Zermelos Abhandlung Über die mechanische Erklärung irreversibler Vorgänge, Wied. Ann. 60, 392–398 (1897); in Boltzmann (1909), Vol III, paper 119Google Scholar
- 7.Boltzmann, L.: Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. In: Hasenöhrl, F. (ed.) Wissenschaftliche Abhandlungen. Chelsea, Leipzig (1909)Google Scholar
- 8.Cercignani, C.: 134 years of Boltzmann equation. In: Gallavotti, G., Gallavotti, W.L., Yngvason, J. (eds.) Boltzmann’s Legacy, pp. 107–128. Zürich: European Mathematical Society, Zürich (2008)CrossRefGoogle Scholar
- 9.Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Diluted Gases. Springer, New York (1994)CrossRefGoogle Scholar
- 10.Culverwell, E.P.: Dr. Watson’s proof of Boltzmann’s theorem on permanence of distributions. Nature
**50**, 617 (1894)CrossRefADSMATHGoogle Scholar - 11.Drory, A.: Is there a reversibility paradox? Recentering the debate on the thermodynamic time arrow. Stud. Hist. Philos. Mod. Phys.
**39**, 889–913 (2008)CrossRefMATHMathSciNetGoogle Scholar - 12.Ehrenfest, P., Ehrenfest-Afanassjewa, T.: The Conceptual Foundations of the Statistical Approach in Mechanics. Cornell University Press, New York (1912)Google Scholar
- 13.Feynman, R., Leighton, R., Sands, M.: The Feynman Lectures on Physics. Addison Wesley, Reading (1964)Google Scholar
- 14.Illner, R., Pulvirenti, M.: Global validity of the Boltzmann equation for a two-dimensional rare gas in a vacuum. Commun. Math. Phys. 105, 189–203 (1986); erratum ibidem 121 143–146 (1989)Google Scholar
- 15.Lanford, O.E.: Time evolution of large classical systems. In: Moser, J. (ed.) Dynamical Systems, Theory and Applications. Lecture Notes in Theoretical Physics, vol. 38, pp. 1–111. Springer, Berlin (1975)CrossRefGoogle Scholar
- 16.Lanford, O.E.: On the derivation of the Boltzmann equation. Asterisque
**40**, 117–137 (1976)MathSciNetGoogle Scholar - 17.Lanford, O.E.: The hard sphere gas in the Boltzmann-Grad limit. Physica
**106A**, 70–76 (1981)CrossRefADSGoogle Scholar - 18.Lebowitz, J.L.: Microscopic dynamics and macroscopic laws. In: Horton Jr, C.W., Reichl, L.E., Szebehely, V.G. (eds.) Long-Time Prediction in Dynamics, pp. 220–233. Wiley, New York (1983)Google Scholar
- 19.Loschmidt, J.: Über die Zustand des Wärmegleichgewichtes eines Systems von Körpern mit Rücksicht auf die Schwerkraft. Wien. Ber.
**73**, 139 (1877)Google Scholar - 20.Goldstein, S.: Boltzmann’s approach to statistical mechanics. In: Bricmon, J., et al. (eds.) Chance in Physics: Foundations and Perspectives. Lecture Notes in Physics, pp. 39–54. Springer, Berlin (2001)CrossRefGoogle Scholar
- 21.Schrödinger, E.: Irreversibility. Proc. R. Ir. Acad.
**53**, 189–195 (1950)Google Scholar - 22.Spohn, H.: Kinetic equations from Hamiltonian dynamics: markovian limits. Rev. Mod. Phys.
**53**, 569–615 (1980)CrossRefADSMathSciNetGoogle Scholar - 23.Spohn, H.: Large Scale Dynamics of Interacting Particles. Springer, Berlin (1991)CrossRefMATHGoogle Scholar
- 24.Spohn, H.: Loschmidt’s reversibility argument and the H-theorem. In: Fleischhacker, W., Schlonfeld, T. (eds.) Pioneering Ideas for the Physical and Chemical Sciences, Loschmidt’s Contributions and Modern Developments in Structural Organic Chemistry, Atomistics and Statistical Mechanics, pp. 153–158. Plenum Press, New York (1997)Google Scholar
- 25.Spohn, H.: On the integrated form of the BBGKY hierarchy for hard spheres. http://arxiv.org/pdf/math-ph/0605068 (2006). Accessed 25 May 2006
- 26.Uffink, J.: Compendium of the foundations of classical statistical physics. In: Butterfield, J., Earman, J. (eds.) Handbook for the Philosophy of Physics, pp. 923–1074. Elsevier, Amsterdam (2008)Google Scholar
- 27.Valente, G.: The approach towards equilibrium in Lanford’s theorem. Eur. J. Philos. Sci.
**4**(3), 309–335 (2014)CrossRefMathSciNetGoogle Scholar - 28.Villani, C.: (Ir)reversibilité et entropie. Séminaire Poincaré XV, Le Temps, pp. 17–75. Institut Henri Poincaré, Paris (2010)Google Scholar