Abstract
This paper analyzes the ergodic hypothesis in the context of Boltzmann’s late work in statistical mechanics, where Boltzmann lays the foundations for what is today known as the typicality account. I argue that, based on the concepts of stationarity (of the measure) and typicality (of the equilibrium state), the ergodic hypothesis, as an idealization, is a consequence rather than an assumption of Boltzmann’s account. More precisely, it can be shown that every system with a stationary measure and an equilibrium state (be it a typical state with respect to the phase space or the time average) behaves essentially as if it were ergodic. I claim that Boltzmann was aware of this fact as it grounds both his notion of equilibrium, relating it to the thermodynamic notion of equilibrium, and his estimate of the fluctuation rates.
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Notes
- 1.
Consider, for instance, an isolated system. Within that system, total energy E is conserved. Hence, trajectories are restricted to the constant-energy hypersurface \(\Gamma _E = \{(q,p)\in \Gamma | H(q,p)=E\}\), from which it follows that the microcanonical measure
$$\begin{aligned} d\mu _E = \prod _{i=1}^{3N} dq_i dp_i\ \delta (H(q,p)-E) \end{aligned}$$is the appropriate stationary measure of the dynamics in that case.
- 2.
- 3.
Reference [10] gives a definition of ergodicity in terms of invariant sets (where a set \(A\in \mathcal {B}(\Gamma )\) is called invariant if and only if \(T^{-1}A=A\)). If, for all sets \(A\in \mathcal {B}(\Gamma )\) with \(T^{-1}A=A\),
$$\begin{aligned} \mu (A)=0\qquad \textrm{or} \qquad\mu (A) =1, \end{aligned}$$then the system is called ‘ergodic’. Thus a system is called ‘ergodic’ if and only if all invariant sets are of full or zero measure. In other words, there exist no two (or more) disjoint invariant sets of non-zero measure. The two definitions of ergodicity relate to one another via Birkhoff’s theorem.
- 4.
There is a little caveat to this statement. While it is definitely true whenever phase space is finite and the measure is normalizable, one has to be careful with infinite phase spaces and non-normalizable measures. For problems related to the latter, see [25] or [19]. The distinction between the notions of probability and typicality has been drawn and discussed elsewhere (see, e.g., [26, 27] or [28]).
- 5.
[29] proves the existence of a region of overwhelming phase space measure for a large class of realistic physical systems.
- 6.
- 7.
Based on the apparently missing connection between the time and the phase space average of equilibrium, Frigg and Werndl assert that Boltzmann’s account of thermodynamic behaviour, which has later become known as the ‘typicality account’, is simply ‘mysterious’ [42, p. 918]. In follow-up papers (cf. [36, 37]) they even claim that the typicality account doesn’t relate to thermodynamics at all because it doesn’t draw the connection between Boltzmann’s definition of equilibrium (in terms of the phase space average) and the thermodynamic definition of equilibrium (in terms of the time average). Here essential ergodicity counters the critique and closes the explanatory gap as it connects the time and phase space averages of the equilibrium state in a mathematically precise way.
- 8.
- 9.
Goldstein makes a similar point when he asserts that, even without ergodicity, the value of any thermodynamic variable is constant ‘to all intents and purposes’ [34, p. 46].
- 10.
This agrees with the time estimate Boltzmann presents in his letter to Zermelo [15, p. 577].
- 11.
This quote was one of the first quotes (and essays) that were given to me by Detlef Dürr, to whom this memorial volume is dedicated. It is the style of writing that Detlef liked and that he himself employed on similar occasions.
- 12.
Known to the author from private conversation. The original version is about a person’s approach from non-equilibrium (here: an oasis) to equilibrium (here: the remainder of the desert), where it is the atypical initial condition, the special fact of ‘being in an oasis’ in the very beginning, which is in need of explanation. The fact that a person, walking around in an unspecific and maybe even random way, walks out of the oasis into the desert is merely typical (we call it typical within atypicality; see [46] for this phrasing). According to [34], it is the explanation of the atypical initial condition which constitutes the hard part of any explanation of thermodynamic irreversibility.
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Reichert, P. (2024). The Ergodic Hypothesis: A Typicality Statement. In: Bassi, A., Goldstein, S., Tumulka, R., Zanghì, N. (eds) Physics and the Nature of Reality. Fundamental Theories of Physics, vol 215. Springer, Cham. https://doi.org/10.1007/978-3-031-45434-9_20
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