[Paper Review] Laws of thermodynamics and game theory

Lev Sakhnovich|arXiv (Cornell University)|May 23, 2011

Advanced Thermodynamics and Statistical Mechanics3 citations

TL;DR

This paper applies game theory to derive the Gibbs distribution in classical and quantum statistical mechanics via a variational principle, offering a novel, unified derivation of the second law of thermodynamics. It rigorously proves the third law of thermodynamics, showing that residual entropy arises when the ground state is degenerate (m > 1), and conjectures a logarithmic asymptotic form for classical entropy at low temperatures.

ABSTRACT

Using a game theory approach and a new extremal problem, Gibbs formula is proved in a most simple and general way for the classical mechanics case. A corresponding conjecture on the asymptotics of the classical entropy is formulated. For the ordinary quantum mechanics case, the third law of thermodynamics is derived. Some results on the number of ground states and residual entropy are obtained rigorously.

Motivation & Objective

• To provide a new, game-theoretic derivation of the Gibbs distribution in statistical mechanics.
• To establish a rigorous connection between the third law of thermodynamics and the degeneracy of the ground state in quantum systems.
• To formulate and support a conjecture on the asymptotic behavior of classical entropy at zero temperature.
• To unify the treatment of classical and quantum entropy using extremal principles grounded in game theory.

Proposed method

• Formulates a compromise function F = λE + S, where λ = -1/(kT), representing a strategic trade-off between energy and entropy.
• Applies calculus of variations to the functional F_c = λE_c + S_c under normalization constraint ∫P dpdq = 1.
• Derives the canonical distribution P(p,q) = exp(λH(p,q))/Z_c via Euler-Lagrange equation for the classical case.
• Uses Lagrange multipliers and functional derivatives to solve the conditional extremum problem for both classical and quantum systems.
• Applies asymptotic analysis to the partition function Z_q(β) as β → ∞ (T → 0) to study low-temperature limits.
• Derives the entropy S_q(β) = βE_q(β) + log Z_q(β) and analyzes its behavior in the ground state regime.

Experimental results

Research questions

• RQ1 How can the Gibbs distribution in classical statistical mechanics be derived using game theory principles?
• RQ2 What is the relationship between the degeneracy of the ground state and the residual entropy in quantum systems?
• RQ3 Does classical entropy exhibit a logarithmic divergence at zero temperature, as conjectured in this work?
• RQ4 Can the second and third laws of thermodynamics be derived from a single variational principle framed in game-theoretic terms?

Key findings

• The classical Gibbs distribution P(p,q) = exp(λH(p,q))/Z_c is derived via a game-theoretic extremal principle using calculus of variations.
• The second law of thermodynamics is reinterpreted as a compromise between energy minimization and entropy maximization through the functional F = λE + S.
• For quantum systems, the third law of thermodynamics holds if and only if the ground state is non-degenerate (m = 1); otherwise, residual entropy log(m) remains.
• In the quantum case, the entropy S_q(β) → log(m) as β → ∞, confirming the existence of residual entropy for degenerate ground states.
• The classical entropy S_c(β) is conjectured to behave asymptotically as c₁ + c₂ log β + o(1) as β → ∞, indicating logarithmic divergence at zero temperature.
• The derivation confirms that the canonical ensemble emerges naturally from a game-theoretic optimization framework, unifying classical and quantum statistical mechanics.