Source
Frigg & Werndl (2023), LIT-114, the Stanford Encyclopedia of Philosophy entry “Philosophy of Statistical Mechanics”, read in full in NOTE-103.
What was actually shown
The obstacles. A system is a dynamical system whose flow is deterministic and measure-preserving, as Liouville’s theorem gives for Hamiltonian systems (§3). Loschmidt’s reversibility objection (1876) and Zermelo’s recurrence objection (1896), the latter from Poincaré recurrence in bounded measure-preserving systems, block a strict approach to equilibrium (§4.3). In the Gibbsian framework the fine-grained entropy S_G = −k∫ρ log ρ dx is a constant of motion, and a distribution that is not stationary can never become stationary (§6.3). These are stated as mathematical fact.
The Boltzmannian accounts (§§4.4–4.7) weaken the target. Ergodicity gives time-fractions equal to measure-fractions for almost all initial conditions, but many paradigm systems are not ergodic, the ideal gas among them. Typicality makes the approach overwhelmingly likely, and needs a dynamical assumption to do it. The Mentaculus conditionalizes on the Past-Hypothesis, a low-entropy initial condition. The authors’ long-run residence account defines equilibrium as the macro-state the system occupies most of the time.
The Boltzmann equation (§5). Lanford’s theorem recovers a statistical H-theorem only up to about two-fifths of the mean free time, “in the order of microseconds” for air. Where its irreversibility comes from is disputed.
The Gibbsian repairs (§§6.4–6.6) add something. Coarse-graining adds a grid, needs mixing and reaches quasi-equilibrium only as t → ∞. Interventionism adds an environment, has no theorem that an arbitrary environment drives equilibrium, and faces a regress up to the universe. The epistemic account adds the observer’s knowledge, and its most fundamental problem is that kettles do not boil because of what we know.
What could have come out otherwise. An account could have derived strict, monotonic irreversibility from the isolated dynamics. The formal facts rule that out, and no account in the survey claims it.
What this does not say
- Which account is right. Each repair is reported with its standing objections. Several verdicts are reported rather than adjudicated (NOTE-103, Limitations).
- That the classification is proved. That every Boltzmannian account drops strict irreversibility is stated account by account, not argued as a general result (NOTE-103, C4). An account outside the survey would need its own check.
- That the authors’ own account is established. The long-run residence definition and its existence theorem are described sympathetically, without a formal statement, and the theorem is cited as forthcoming (C5).
- That statistical mechanics reduces thermodynamics. The entry declines to frame its aim as reduction (§1, §7.5), and the phase-transition debate is left open.
- Anything about quantum statistical mechanics or the direction of time. The first is flagged as understudied (§4.8); the second has its own entry.
- That coarse-graining or maximum entropy are refuted. Coarse-graining’s indistinguishability premise is contested by the spin-echo experiment, and there is “no consensus” on the significance or “even cogency” of the maximum-entropy account. Contested is not refuted.
Connections
- THEORY-030 (Landauer’s principle) already cites this entry’s report of Norton’s verdict that the Landauer literature is “too fragile” (§7.2).
- The entry’s open problem, how Boltzmannian and Gibbsian statistical mechanics relate (§6.7), is where a THEORY on entropy or coarse-graining in the record should look first.
- LIT-082 (observational entropy with maximum-entropy principles) is the constructive side of the coarse-graining and maximum-entropy threads.