Abstract
This chapter will discuss statistical mechanics in the manner of Ludwig Boltzmann (1844–1906). I will focus, in particular, on Boltzmann’s account of thermodynamic irreversibility and the second law of thermodynamics—the most profound legacy of the great Austrian physicist, in which the role of typicality comes out particularly clearly. In contemporary philosophical literature, this account is sometimes referred to as “neo-Boltzmannian,” a term that strikes me as overly flattering to both Boltzmann’s critics and his contemporary defenders.
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Notes
- 1.
- 2.
Poincaré recurrence theorem: Let \(A\subset \Gamma \) be a set of positive Liouville measure. Then, for almost all \(X \in A\), there exists a sequence of times \(t_i \to \infty \) such that \(X(t_i)=\Phi _{t_i,0}(X)\in A, \, \forall i \in \mathbb {N}\).
- 3.
The confusion has other sources that I can’t go into here, from misunderstandings of Maxwell’s demon to formal analogies between the Gibbs entropy and the information-theoretic Shannon entropy that lead to confounding the two concepts.
- 4.
Glenn Shafer has pointed out to me that it doesn’t even make sense to say that a fluctuating quantity “increases” or “decreases” without a specification of the time scales to which such a statement refers.
- 5.
A rigorous estimate:
$$\displaystyle \begin{aligned} \begin{aligned} \lvert \langle f \rangle - (1-\epsilon)\xi\rvert &\leq \int_{\Gamma_{\mathrm{eq}}} \lvert f(x) - \xi \rvert\, \rho(x) \, \mathrm{d}x + \int_{\Gamma\setminus\Gamma_{\mathrm{eq}}} \lvert f(x) \rvert\rho(x) \, \mathrm{d}x \\ &\leq (1-\epsilon) \Delta\xi + \epsilon \sup_{x \in \Gamma\setminus\Gamma_{\mathrm{eq}}} \lvert f(x)\rvert, \end{aligned}\end{aligned}$$and thus \( \langle f \rangle \approx \xi \) assuming \(\epsilon \sup _{x \in \Gamma \setminus \Gamma _{\mathrm {eq}}} \lvert f(x)\rvert \ll \lvert \xi \rvert \).
- 6.
Together with some appropriate bound on the variation of the macro-variable.
- 7.
Unless that average just happens to correspond itself to one of the Boltzmann equilibrium values.
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Lazarovici, D. (2023). Boltzmann’s Statistical Mechanics. In: Typicality Reasoning in Probability, Physics, and Metaphysics. New Directions in the Philosophy of Science. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-031-33448-1_8
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