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Unit 2
Ensembles
Equivalence of Canonical and Grand Canonical Ensemble
1. Introduction
In statistical mechanics, different ensembles are used to describe systems
in thermal equilibrium.
- Canonical Ensemble → Fixed \(N, V, T\)
- Grand Canonical Ensemble → Fixed \(\mu, V, T\)
Although these ensembles have different constraints, they become
equivalent in the thermodynamic limit.
2. Canonical Ensemble
The canonical partition function is:
\[
Z(N,V,T) = \sum_i e^{-\beta E_i}
\]
Helmholtz free energy:
\[
F = -k_B T \ln Z
\]
3. Grand Canonical Ensemble
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:
\[
\Omega = -k_B T \ln \Xi
\]
4. Relation Between Thermodynamic Potentials
The grand potential is related to Helmholtz free energy by:
\[
\Omega = F - \mu N
\]
Also,
\[
\Omega = - P V
\]
5. Particle Number Fluctuation
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.
6. Thermodynamic Limit
The thermodynamic limit is defined as:
\[
N \to \infty, \quad V \to \infty, \quad \frac{N}{V} = \text{constant}
\]
In this limit:
- Energy fluctuation becomes negligible
- Particle number fluctuation becomes negligible
- Macroscopic observables become sharply defined
7. Equivalence of Ensembles
In the thermodynamic limit, canonical and grand canonical ensembles
give identical thermodynamic results.
Thus,
- \(F(N,V,T)\) from canonical ensemble
- \(\Omega(\mu,V,T)\) from grand canonical ensemble
lead to the same equation of state and thermodynamic quantities.
8. Important Result
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.
