Equivalence of Canonical and Grand Canonical Ensemble

1. Introduction

In statistical mechanics, different ensembles are used to describe systems in thermal equilibrium.

Although these ensembles have different constraints, they become equivalent in the thermodynamic limit.

The canonical partition function is:

\[ Z(N,V,T) = \sum_i e^{-\beta E_i} \]

Helmholtz free energy:

\[ F = -k_B T \ln Z \]

In the grand canonical ensemble, particle number can fluctuate.

The grand partition function is:

\[ \Xi = \sum_{N=0}^{\infty} e^{\beta \mu N} Z(N,V,T) \]

Grand potential:

\[ \Omega = -k_B T \ln \Xi \]

The grand potential is related to Helmholtz free energy by:

\[ \Omega = F - \mu N \]

Also,

\[ \Omega = - P V \]

In the grand canonical ensemble, the particle number fluctuates.

\[ (\Delta N)^2 = k_B T \left( \frac{\partial N}{\partial \mu} \right)_{T,V} \]

Relative fluctuation:

\[ \frac{\Delta N}{N} \propto \frac{1}{\sqrt{N}} \]

As \(N \to \infty\), relative particle number fluctuation → 0.

The thermodynamic limit is defined as:

\[ N \to \infty, \quad V \to \infty, \quad \frac{N}{V} = \text{constant} \]

In this limit:

In the thermodynamic limit, canonical and grand canonical ensembles give identical thermodynamic results.

Thus,

lead to the same equation of state and thermodynamic quantities.

For macroscopic systems, all ensembles (microcanonical, canonical, and grand canonical) are equivalent.

However, for small systems (nanoscopic systems), fluctuations become important and ensembles may not be strictly equivalent.