[Paper Review] Is ergodicity a reasonable hypothesis?

Bernard Gaveau, L. S. Schulman|arXiv (Cornell University)|Jan 25, 2014

Advanced Thermodynamics and Statistical Mechanics9 references3 citations

TL;DR

This paper argues that ergodicity—whereby a macroscopic system explores all accessible microscopic states over time—is physically impossible due to the astronomically vast phase space of even a single mole of gas. Using quantum mechanical state counting and time-scale estimates, the authors show that the fraction of states visited in the age of the universe is effectively zero, challenging the foundational assumption of statistical mechanics that all microstates are equally probable.

ABSTRACT

In the physics literature "ergodicity" is taken to mean that a system, including a macroscopic one, visits all microscopic states in a relatively short time. We show that this is an impossibility even if that time is billions of years. We also suggest that this feature does not contradict most physical considerations since those considerations deal with correlations of only a few particles.

Motivation & Objective

• To demonstrate that the assumption of ergodicity—visiting all accessible microstates—is physically untenable for macroscopic systems.
• To quantify the impossibility of ergodicity by estimating the fraction of phase space visited by a mole of nitrogen gas over the age of the universe.
• To reconcile the failure of ergodicity with the empirical success of statistical mechanics by showing that low-order correlations remain unaffected.
• To argue that the success of statistical mechanics does not rely on full ergodicity, but on typicality and local relaxation of observables.

Proposed method

• Counting the number of quantum mechanical microstates in a cubic meter of N₂ gas at 300 K using Avogadro's number and spatial discretization at 10 nm resolution.
• Estimating the time required for a molecule to traverse a 10 nm box, yielding a passage time of ~1.93 × 10⁻¹¹ s.
• Calculating the total number of state transitions possible since the Big Bang (~13.8 billion years), resulting in ~10²⁸.⁴ transitions.
• Comparing this to the total number of microstates (~10^(2.9193 × 10²³)), showing the fraction visited is effectively zero.
• Using quantum mechanical state counting with de Broglie wavelengths and phase space volume to bound the number of distinct states.
• Analyzing model systems like the stadium billiard and cat-map dynamics to show that even without ergodicity, local observables (e.g., density, entropy) equilibrate rapidly.

Experimental results

Research questions

• RQ1 Is it physically possible for a macroscopic system, such as a mole of gas, to visit all accessible microstates within the age of the universe?
• RQ2 Does the failure of full ergodicity invalidate the foundations of statistical mechanics, particularly the equal a priori probability postulate?
• RQ3 Why do standard statistical mechanics predictions remain accurate despite the impossibility of visiting all microstates?
• RQ4 Can relaxation of low-order correlation functions (e.g., one-particle distribution) occur without full ergodicity?
• RQ5 To what extent is the success of statistical mechanics based on typicality rather than exhaustive exploration of phase space?

Key findings

• The fraction of microstates visited by a mole of N₂ gas in the age of the universe is approximately 10^(-(2.9193 - ε) × 10²³), effectively zero.
• Even with idealized assumptions (non-interacting particles, 10 nm spatial resolution), the number of states visited is negligible compared to the total number of accessible states.
• The time required to explore all states is vastly longer than the age of the universe, making ergodicity an unphysical hypothesis for macroscopic systems.
• Despite the failure of ergodicity, low-order observables such as particle density and pressure fluctuations remain well-described by statistical mechanics.
• Model systems like the stadium billiard and cat-map dynamics show that local relaxation and uniform density emerge quickly, even when full phase space exploration is impossible.
• The success of statistical mechanics is not due to ergodicity per se, but due to the typicality of the states that are actually visited, which yield correct macroscopic predictions.