Gibbs Distribution From Sequentially Predictive Form of the Second Law

TL;DR

This paper introduces a sequential predictive framework for work extraction in thermodynamics, demonstrating that the initial state distribution converges to the Gibbs distribution under the second law, with implications for control and randomness testing.

Contribution

It establishes a prequential formulation of the second law, linking work extraction, Gibbs distribution convergence, and game-theoretic interpretation without assuming initial probability distributions.

Findings

Empirical distribution of initial states converges to Gibbs distribution.

Prequential second law implies strong law of large numbers for conjugate variables.

Work extraction can test the randomness of Gibbs-distributed generators.

Abstract

We propose a prequential or sequentially predictive formulation of the work extraction where an external agent repeats the extraction of work from a heat engine by cyclic operations based on his predictive strategy. We show that if we impose the second law of thermodynamics in this situation, the empirical distribution of the initial microscopic states of the engine must converge to the Gibbs distribution of the initial Hamiltonian under some strategy, even though no probability distribution are assumed. We also propose a protocol where the agent can change only a small number of control parameters linearly coupled to the conjugate variables. We find that in the restricted situation the prequential form of the second law of thermodynamics implies the strong law of large numbers of the conjugate variables with respect to the control parameters. Finally, we provide a game-theoretic interpretation of our formulation and find that the prequential work extraction can be interpreted as a testing procedure for random number generator of the Gibbs distribution.