# 4.7: The Grand Canonical Ensemble

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Grand as in the French “big” rather than as in the English “magnificent”. This ensemble is “bigger” than the canonical ensemble in that there are more possible microstates.

## Summary

The grand canonical ensemble for a pure classical monatomic fluid:

The probability that the system has N particles and is in the microstate ΓN is proportional to

$e^{-\beta\left(H\left(\Gamma_{N}\right)-\mu N\right)},$

where

$\beta=\frac{1}{k_{B} T}.$

Writing out all the normalizations correctly gives: the probability that the system has N particles and is in some microstate within the phase space volume element dΓN about ΓN is

$\frac{e^{-\beta\left(H\left(\Gamma_{N}\right)-\mu N\right)}}{N ! h_{0}^{3 N} \Xi(\beta, V, \mu)} d \Gamma_{N},$

where the "grand canonical partition function" is

$\Xi(\beta, V, \mu)=\sum_{N=0}^{\infty} \frac{1}{N ! h_{0}^{3 N}} \int e^{-\beta\left(H\left(\Gamma_{N}\right)-\mu N\right)} d \Gamma_{N}$

$=\sum_{N=0}^{\infty} e^{\beta \mu N} Z(\beta, V, N).$

This sum is expected to converge when µ is negative.

The connection to thermodynamics is that

$\Pi(T, V, \mu)=-p(T, \mu) V=-k_{B} T \ln \Xi(T, V, \mu).$

4.7: The Grand Canonical Ensemble is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.