[Paper Review] A short derivation of Boltzmann distribution and Gibbs entropy formula from the fundamental postulate

D. Lairez|arXiv (Cornell University)|Nov 4, 2022

Advanced Thermodynamics and Statistical Mechanics4 citations

TL;DR

This paper presents a concise, undergraduate-accessible derivation of the Boltzmann distribution and Gibbs entropy formula directly from the fundamental postulate of statistical mechanics—equal a priori probability of microstates in an isolated system. By analyzing energy exchange between subparts and exploiting symmetry and normalization, it derives the exponential form of the probability distribution and identifies the entropy as $ S = -\sum_i p_i \ln p_i $, linking statistical mechanics directly to thermodynamics without Lagrange multipliers or Stirling's approximation.

ABSTRACT

Introducing the Boltzmann distribution very early in a statistical thermodynamics course (in the spirit of Feynmann) has many didactic advantages, in particular that of easily deriving the Gibbs entropy formula. In this note, a short derivation is proposed from the fundamental postulate of statistical mechanics and basics calculations accessible to undergraduate students.

Motivation & Objective

• To provide a pedagogically accessible derivation of the Boltzmann distribution and Gibbs entropy formula for undergraduate-level statistical thermodynamics courses.
• To eliminate reliance on Lagrange multipliers and Stirling's approximation, which are often challenging for beginners.
• To ground the Gibbs entropy formula in the fundamental postulate of statistical mechanics, rather than introducing it arbitrarily or via information theory.
• To demonstrate that the connection between thermodynamic entropy and statistical entropy is not arbitrary, but follows directly from symmetry and probability.
• To support early introduction of the Boltzmann distribution in teaching, following the historical and conceptual path of pioneers like Boltzmann and Feynman.

Proposed method

• Starts from the fundamental postulate: all microstates of an isolated system are equally probable.
• Considers a system composed of N subparts with discrete energy levels $ \epsilon_i = i\epsilon $, and tracks energy exchange via elementary transitions.
• Uses the independence of distant subparts to derive that the probability distribution must be exponential: $ p_i \propto e^{-\beta \epsilon_i} $.
• Applies normalization and symmetry arguments to show that the same functional form applies to all subparts, leading to the canonical distribution.
• Uses the partition function $ Z = \sum_i e^{-\beta \epsilon_i} $ and derives the free energy $ F = -T \ln Z $.
• Connects the derivative of $ \ln Z $ with respect to $ 1/T $ to internal energy, and identifies $ \beta = 1/T $, leading to the Gibbs entropy formula.

Experimental results

Research questions

• RQ1 Can the Boltzmann distribution be derived without using Lagrange multipliers or Stirling's approximation?
• RQ2 How can the Gibbs entropy formula $ S = -\sum_i p_i \ln p_i $ be derived directly from the fundamental postulate of statistical mechanics?
• RQ3 What is the physical meaning of the parameter $ \beta $, and how is it related to temperature?
• RQ4 Why is the exponential form of the probability distribution necessary for consistency across independent subsystems?
• RQ5 How can the link between thermodynamic entropy and statistical entropy be made rigorous and intuitive for undergraduate students?

Key findings

• The Boltzmann distribution $ p_i = \frac{1}{Z} e^{-\beta \epsilon_i} $ is derived from symmetry and normalization of independent subparts, without Lagrange multipliers or Stirling's approximation.
• The parameter $ \beta $ is identified as $ \beta = 1/T $ by matching the thermodynamic relation $ U = F + TS $ with the statistical expression for internal energy.
• The Gibbs entropy formula $ S = -\sum_i p_i \ln p_i $ is derived as a direct consequence of the probability distribution and the free energy relation.
• The derivation establishes that Clausius entropy and Gibbs entropy are the same physical quantity, resolving a common conceptual gap in textbooks.
• The method is accessible to undergraduates, relying only on basic probability and algebra, making it suitable for early introduction in statistical mechanics courses.
• The approach provides a logically consistent and pedagogically sound foundation for statistical mechanics, grounded in the fundamental postulate and physical intuition.