Distribution of microstates at equilibrium

Let $𝑀 : 𝜉 \to {𝑀 𝑖}$ be a discrete random variable representing the microstate of a thermodynamic system and let $𝑃 𝑖 = ℙ (𝑀 = 𝑀 𝑖)$ . Maximum entropy thermodynamics or MaxEnt thermodynamics derives all results from the general Principle of maximum entropy: The distribution ${𝑃 𝑖}$ of $𝑀$ at equilibrium is that which maximizes its Shannon entropy

˜ 𝑆 ({𝑃 𝑖}) = 𝑘 𝐵 𝐻 [𝑀] = - 𝑘 𝐵 \sum 𝑖 𝑃 𝑖 l n 𝑃 𝑖

and that this coïncides with the thermodynamic entropy of the system. Thus determination of the distribution ${𝑃 𝑖}$ is reduced to an optimization problem, and thus the method of Lagrange multipliers may be used. See Ensembles for the distributions in different scenarios.

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Distribution of microstates at equilibrium

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