The approach towards equilibrium in Lanford’s theorem

Abstract

This paper develops a philosophical investigation of the merits and faults of a theorem by Lanford (1975), Lanford (Asterisque 40, 117–137, 1976), Lanford (Physica 106A, 70–76, 1981) for the problem of the approach towards equilibrium in statistical mechanics. Lanford’s result shows that, under precise initial conditions, the Boltzmann equation can be rigorously derived from the Hamiltonian equations of motion for a hard spheres gas in the Boltzmann-Grad limit, thereby proving the existence of a unique solution of the Boltzmann equation, at least for a very short amount of time. We argue that, by establishing a statistical H-theorem, it offers a prospect to complete Boltzmann’s combinatorial argument, without running against the objections which plug other typicality-based approaches. However, we submit that, while recovering the irreversible approach towards equilibrium for positive times, it fails to predict a monotonic increase of entropy for negative times, and hence it yields the wrong retrodictions about the past evolution of a gas.

Appendix

We now offer a more technical analysis of Lanford’s rigorous derivation of the Boltzmann equation. Although the original proof of the result is spelled out by Lanford (1975) within the BBGKY approach, here we just refer to the formulation of the theorem by Lanford (1976). We then conclude by making a few remarks on the limitation in time of the theorem and its possible extension to arbitrary times.

1.1 A rigorous derivation of the Boltzmann equation

Recall that Lanford’s theorem is espressed in terms of sequences of probability measures. In order to guarantee that the probability measures in the sequence $\mu ^{(a)} := \{ \mu _{k}^{(a_{k})} \}_{k= 1, ..., N}$ remain finite in the B-G limit, they must be suitably renormalized. To this extent, Lanford introduced the rescaled correlation functions

$$ f_{k}^{(a)}(x_{1}, ..., x_{k}) = \frac{N!}{(N - k)!} \frac{1}{N^{k}} \int x_{k+1}, ..., x_{N} \mu^{(a)}(x_{1}, ..., x_{N}) $$

(8)

As the number k of particles increases, such functions are defined on increasingly larger phase-spaces Γ _k := (Λ × R ³)^k. For N → ∞, the renormalization factor $\frac {N!}{(N - k)!} \frac {1}{N^{k}}$ in front of the integral tends to 1, which assures that $f_{N}^{(a)}(x_{1}, ..., x_{N}) = \mu ^{(a)}(x_{1}, ..., x_{N})$. It can then be shown that μ ^(a) is an approximating sequence for f just in case for all k = 1,..., N

$$ \lim\limits_{a \rightarrow 0} f_{k}^{(a)}(x_{1}, ..., x_{k})dx_{1} \cdot \cdot \cdot dx_{k} = \Pi_{i=1}^{k}f(x_{i}) dx_{i} $$

(9)

in the sense of weak*-convergence of measures. This means that, for any fixed k, the correlation functions $f_{k}^{(a)}$ converge almost everywhere to a limiting function $\lim\limits _{a \rightarrow 0} f_{k}^{(a)}$ on Γ _k , which in turn factorizes into the product of the values of f computed on the different mechanical states x _i of the particles. The factorization condition embedded in the above formula 9 thus connects the multi-particle description of the system with the description in terms of the one-particle phase space on which solutions of B.E. are defined. The time-evolved correlation functions $f_{k, t}^{(a)}$ are obtained from the time-evolved sequence μ ^(a)∘T _t by means of formula 8.

The assumptions of Lanford’s theorem are formulated as precise conditions on the correlation functions at the initial time t = 0. The first assumption is a regularity condition, that rules out those initial correlations functions $f_{k}^{(a)}$ not bounded by the product of the gas density z with the equilibrium Maxwellian distribution $h_{\beta }(\vec {p}_{k}) = (\frac {\beta }{2 \pi m})^{\frac {3}{2}} \cdot e^{-\frac {\beta \vec {p}^{2}_{k}}{2m}}$.

Assumption (1): There is a positive real constant M such that for any k = 1,..., N

$$ f_{k}^{(a)}(x_{1}, ..., x_{k}) \leq M \cdot z^{k} \mbox{ } \Pi_{i=1}^{k} h_{\beta}(\vec{p}_{k}) $$

(10)

Accordingly, one only admits correlations functions which drop off exponentially as the Maxwellian distribution. From a physical point of view, it prohibits the presence of particles with very high energy. This restricts one to a special class of well-behaving probability measures.

The second assumption requires that, for any fixed k, the correlations functions $f_{k}^{(a)}$ converge in the B-G limit to a well-defined limiting function $\lim\limits _{a \rightarrow 0} f_{k}^{(a)}$ uniformly on all compact subsets of the set $\Gamma _{k, \neq }(0)= \{ (x_{1}, ..., x_{k}) \in \Gamma _{k} \mbox { } | \mbox { } \vec {q_{i}} \neq \vec {q_{j}} \mbox { for all } i \neq j \}$. This domain of convergence comprises all the points for which no pair of particles would occupy the same position, and hence it excludes those configurations for which a collision between particles takes place. Furthermore, one requires that $\lim\limits _{a \rightarrow 0} f_{k}^{(a)}$ factorizes into the product of the values that the continuous function f ₀ takes on at each mechanical state of the individual particles.

Assumption (2): $\lim\limits _{a \rightarrow 0} f_{k}^{(a)}(x_{1}, ..., x_{k}) = \Pi _{i=1}^{k}f_{0}(x_{i})$ uniformly on all compact subsets of Γ _{k, ≠}(0).

Crucially, the set Γ _k /Γ _{k, ≠} (0) for which uniform convergence is not guaranteed has measure zero with respect to the Lebesgue measure, and thus the exceptional points in such a set can be neglected. This assumption plays a similar role as the factorization condition 9 in Boltzmann’s Stoßzahlansatz for k = 2. However, it makes no explicit reference to pre-collision or post-collision particles, and in fact, contrary to the Stoßzahlansatz, it is time-reversal invariant. Furthermore, the factorization condition in Lanford’s assumption has the property that, once it is posited at t = 0, it will hold continuously in the course of time.

Within this framework, we can now see how one obtains the statement of the theorem we presented in Section 3. Indeed, Lanford proved that, under assumptions (1) and (2), for all k = 1,..., N the time-evolved correlation functions converge in the B-G limit to a well-defined limiting function uniformly on all compact subsets of the set $\Gamma _{k, \neq }(t) = \{ (x_{1}, ..., x_{k}) \in \Gamma _{k} \mbox { } | \mbox { } \vec {q_{i}} - s\vec {p_{i}} \neq \vec {q_{j}} - s\vec {p_{j}} \mbox { for all } i \neq j \mbox { and } s \in [0, t] \}$at all positive t up to the time-bound τ. By factorization, such a limiting function $\lim\limits _{a \rightarrow 0} f_{k, t}^{(a)}(x_{1}, ..., x_{k})$ is equal to $\Pi _{i=1}^{k}f_{t}(x_{i})$, with f _t being a solution of the B.E. with initial value f ₀. The domain of convergence Γ _{k, ≠} (t) is strictly smaller than the initial domain of convergence Γ _{k, ≠} (0). In fact, uniform convergence fails for some initial points due to the fact that the correlation functions evolve in accordance with the time-reversal invariant Hamiltonian flow. Although Γ _{k, ≠} (t) becomes smaller and smaller in the course of time, the set Γ _k,/Γ _{k, ≠} (t) of exceptional points remains of Lebesgue measure zero. As a consequence, convergence almost everywhere on Γ _k is guaranteed for all k = 1,..., N in the B-G limit, which implies that condition 9 is satisfied. One can thus conclude that μ ^(a)∘T _−t is an approximating sequence for f _t for all t ∈ [0, τ].

1.2 Can the theorem be extended in time?

Let us begin by explaining how the time-bound τ arises. Recall that assumptions (1) and (2) in the theorem imply that the continuous function f ₀ taken as the initial data at time t = 0 satisfies the bound expressed by formula (6). Lanford proved his result by working with a formal series expansion for solutions of the Boltzmann equation, where the quadratic term is treated as a perturbation on the linear term, and he proved that, if f ₀ satisfies a bound of the same form as (6), such a series converges up to the positive time τ. His proof proceeds by making estimations on the amount of time for which the required convergence is guaranteed. Unfortunately, one is not able to control the convergence of the expansion series for longer times than τ when the bound given by inequality (6) fails. Notice that the time-interval of validity of the result being a fraction of the mean free time t̄ depends on the density z and the inverse temperature β of the gas. Such parameters are introduced in the theorem by the regularity assumption (1). In fact, it is the failure of the latter to guarantee the existence of bounds of the form (6) that prevents Lanford’s proof from carrying on after time τ.

Yet, there are indications that the result can hold for longer times. To this effect, Lanford himself sketched the following argument:

While the technique of the proof definitely does not extend to larger times, examination of the proof suggests (at least to me) that the result should remain true. One can show, for example, that if the bound (1) holds not only for the initial correlation functions but also for the time-dependent correlation functions up to time T, and if (2) holds at time zero, then μ ^(a)∘T _−t is an approximating sequence for f _t up to time T + τ. [Lanford (1976), p.14, , where the notation has been suitably modified]

So, the suggestion is that the result could be extended in time by strengthening the assumptions of the original theorem. Of course, one cannot require that assumption (2) holds after t = 0, at the price of begging the question: in fact, that would be tantamount to assuming that, for all k, the correlation functions $f^{(a)}_{k, t}$ converge uniformly on all compact subsets of Γ _{k, ≠} (t) to a limiting function which factorizes into the product of the values of the solution f _t of the Boltzmann equation at t > 0; however, by condition (9), this would entail that μ ^(a)∘T _t is an approximating sequence for f _t , which is actually the conclusion itself of the theorem. Instead, one could demand that the regularity assumption (1) holds continuously for some arbitrary positive time T. Let us try to develop Lanford’s suggestion.

A stronger version of the regularity assumption stated by condition (10) would require that there is a positive real constant M such that for all k = 1,..., N

$$f_{k, t}^{(a)}(x_{1}, ..., x_{k}) \leq M \cdot z^{k} \mbox{ } \Pi_{i=1}^{k} h_{\beta}(\vec{p}_{k}) $$

for any t ∈ [0, T]. This, together with assumption (2), now implies that the time-evolution of f ₀ satisfies the bound (6) up to the positive time T. Thus, by applying the technique of Lanford’s proof, one can show that there exists a unique solution f _t of the Boltzmann equation for all t ∈ [0, T + τ]. In principle, since T is arbitrary, by this procedure one could extend Lanford’s result at all times. This clearly seems cold comfort in that one would not obtain a derivation of the Boltzmann equation solely from initial conditions: the strong regularity assumption, as it is stated above, is a condition which ought to hold continuously for some time t > 0. Perhaps, one may try and interpret such an assumption as a condition on the class of well-behaving probability distributions which one admits at the initial time. That is, among the initial correlation functions $f^{(a)}_{k}$ s which satisfy the bound (10) at time t = 0 according to the regularity assumption, one admits only those which do not develop singularity in the course of time, at least for some fixed time T. So, under this interpretation, appealing to the stronger version of the regularity assumption would further restrict the class of well-behaving initial probability distributions allowed by the original Lanford’s theorem. However, this class would become smaller and smaller as one fixes higher and higher times T, and hence the strong regularity assumption appears as rather restraining.

Be it as it may, the main lesson of this argument is that, in order to obtain a time-extension of Lanford’s result from his original assumptions (1) and (2), one needs to adopt a different technique of the proof. That is an outstanding challenge in contemporary mathematical physics.