Boltzmannian Equilibrium in Stochastic Systems

Abstract

Equilibrium is a central concept of statistical mechanics. In previous work we introduced the notions of a Boltzmannian α-ɛ-equilibrium and a Boltzmannian γ-ɛ-equilibrium (Werndl and Frigg, Stud Hist Philos Mod Phys 44:470–479, 2015a; Philos Sci 82:1224–1235, 2015b). This was done in a deterministic context. We now consider systems with a stochastic micro-dynamics and transfer these notions from the deterministic to the stochastic context. We then prove stochastic equivalents of the Dominance Theorem and the Prevalence Theorem. This establishes that also in stochastic systems equilibrium macro-regions are large in requisite sense.

Appendices

Appendix

A.1 Proof of the Stochastic Dominance Theorem

First of all, let us show that a stationary stochastic process {Z _t } can be represented by a measure-preserving deterministic system $(X,\Sigma _{X},\mu _{X},T_{t})$. Let X be the set of all possible realisations, i.e., functions x(τ) from $\mathbb{R}$ or $\mathbb{Z}$ to $\bar{X}$. Let $\Sigma _{X}$ be the σ-algebra generated by the cylinder-sets

$$\displaystyle{ C_{i_{1}\ldots i_{n}}^{A_{1}\ldots A_{n} }\!\! =\!\!\{ x\! \in \! X\,\vert \,x(i_{1})\!\! \in \!\! A_{1},\ldots,x(i_{n})\!\! \in \!\! A_{n},A_{j}\! \in \! \Sigma _{\bar{X}},i_{j}\! \in \! \mathbb{R}\,\,\text{or}\,\,\mathbb{Z},\,i_{1}\!\! <\ldots <\!\! i_{n},1\! \leq j\! \leq n\}. }$$

(20.11)

Let μ _X be the unique probability measure arising by assigning to each $C_{i_{1}\ldots i_{n}}^{A_{1}\ldots A_{n}}$ the probability $P\{Z_{i_{1}} \in A_{1},\ldots,Z_{i_{n}} \in A_{n}\}$. The evolution functions shift a realisation t times to the left, i.e., T _t (x(τ)) = x(τ + t). The T _t are invariant under the dynamics because {Z _t } is stationary. $(X,\Sigma _{X},\mu _{X},T_{t})$ is a measure-preserving deterministic system called the deterministic representation (cf. Doob 1953, 621–622; Werndl 2009, 2011).

Let W = { x(τ) ∈ X  |  x(τ) = Z _τ (ω) for all τ for a $\omega \in \Omega ^{{\ast}}\}$. Note that μ _X (W) ≥ 1 −ɛ. Consider first the case of an α-ɛ-equilibrium M _α-ɛ-eq. Then it follows that for all x ∈ W:

$$\displaystyle{ LF_{X_{Q_{ M_{\alpha \text{-}\varepsilon \text{-eq}}}}}(x) \geq \alpha, }$$

(20.12)

where $Q_{M_{\alpha \text{-}\varepsilon \text{-eq}}} =\{ x \in X\,\,\vert \,\,x(0) \in \bar{ X}_{M_{\alpha \text{-}\varepsilon \text{-eq}}}\}$.

Hence $Q_{M_{\alpha \text{-}\varepsilon \text{-eq}}}$ is an α-ɛ-equilibrium of $(X,\Sigma _{X},\mu _{X},T_{t})$. It follows from the (deterministic) Dominance Theorem (Frigg and Werndl 2015a) that $\mu _{X}(Q_{M_{\alpha \text{-}\varepsilon \text{-eq}}}) >\alpha (1-\varepsilon )$, which immediately implies that P{M _α-ɛ-eq} > α(1 −ɛ).

A.2 Proof of the Stochastic Prevalence Theorem

The proof proceeds in the same fashion as the previous one. That is, consider again the measure-preserving deterministic system $(X,\Sigma _{X},\mu _{X},T_{t})$ that represents the stationary stochastic process {Z _t }. Suppose that M _γ-ɛ-eq is an γ-ɛ-equilibrium.

As before, let W = { x(τ) ∈ X  |  x(τ) = Z _τ (ω) for all τ for a $\omega \in \Omega ^{{\ast}}\}$. Note that μ _X (W) ≥ 1 −ɛ.

Then for all x ∈ W and all M ≠ M _γ-ɛ-eq it holds that

$$\displaystyle{ LF_{X_{Q_{ M_{\gamma \text{-}\varepsilon \text{-eq}}}}}(x) \geq LF_{X_{Q_{M}}} +\gamma -\varepsilon, }$$

(20.13)

where $Q_{M_{\gamma \text{-}\varepsilon \text{-eq}}} =\{ x \in X\,\,\vert \,\,x(0) \in \bar{ X}_{M_{\gamma \text{-}\varepsilon \text{-eq}}}\}$ and $Q_{M} =\{ x \in X\,\,\vert \,\,x(0) \in \bar{ X}_{M}\}$. Hence $Q_{M_{\gamma \text{-}\varepsilon \text{-eq}}}$ is an γ-ɛ-equilibrium of $(X,\Sigma _{X},\mu _{X},T_{t})$.

It follows from the (deterministic) Prevalence Theorem (cf. Werndl and Frigg 2015a) that $\mu _{X}(Q_{M_{\gamma \text{-}\varepsilon \text{-eq}}}) \geq \mu _{X}(Q_{M}) +\gamma -\varepsilon$ for all M ≠ M _γ-ɛ-eq. This immediately implies that P{M _γ-ɛ-eq} ≥ P{M} +γ −ɛ for all M ≠ M _γ-ɛ-eq.