Derivation of Boltzmann Principle

Abstract

We present a derivation of Boltzmann principle $S_B=k_B\ln \mathcal{W}$ based on classical mechanical models of thermodynamics. The argument is based on the heat theorem and can be traced back to the second half of the nineteenth century with the works of Helmholtz and Boltzmann. Despite its simplicity, this argument has remained almost unknown. We present it in a modern, self-contained and accessible form. The approach constitutes an important link between classical mechanics and statistical mechanics.

References in corpus (4)

• Microcanonical quantum fluctuation theorems
• Finite Bath Fluctuation Theorem
• On a new definition of quantum entropy
• Mechanical Proof of the Second Law of Thermodynamics Based on Volume Entropy

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• Thermostated Hamiltonian dynamics with log-oscillators
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• On the foundations of statistical mechanics
• Increase of quantum volume entropy in presence of degenerate eigenenergies
• Resolving the Debate Between Boltzmann and Gibbs Entropy: Relative Energy Window Eliminates Thermodynamic Inconsistencies and Allows Negative Absolute Temperatures
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• Verification of finite bath fluctuation theorem for a non-ergodic system
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