Microcanonical ensemble

The microcanonical ensemble represents a system in thermal isolation, i.e. whose energy is exactly specified. As such, its natural variables are

$𝑁$ — Number of particles in the system
$𝑉$ — System volume
$𝐸$ — Total energy of the system

and all $Ω (𝑁, 𝑉, 𝐸)$ accessible microstates are equally likely, where

𝑃 𝑖 = 1 Ω (𝑁, 𝑉, 𝐸) 𝑆 (𝑁, 𝑉, 𝐸) = 𝑘 𝐵 l n Ω (𝑁, 𝑉, 𝐸)

Alternative approaches to statistical thermodynamics take this as their axiomatic starting point, but these notes opt for MaxEnt thermodynamics.

Derivation

By the symmetry of the situation, it should be obvious that if there exists a unique distribution which maximizes Shannon entropy, it must be that described above. derivation

It follows

An Irreversible process is one which increases $Ω$
A Reversible process is one which maintains $Ω$

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Microcanonical ensemble

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