Classical statistical mechanics
Statistical ensembles: microcanonical and canonical
A statistical ensemble is a conceptual collection of system replicas used to describe microstate probabilities under specified macroscopic constraints.
A statistical ensemble is a conceptual collection of system replicas used to describe microstate probabilities under specified macroscopic constraints.
Definition: Quantities and meaning
The microcanonical ensemble fixes E,V,N and assigns equal probability within an allowed energy window. The canonical ensemble fixes T,V,N, exchanges energy with a bath, and uses Boltzmann weights; β=1/(kBT), Z=Σᵣe⁻ᵝᴱʳ. An ensemble is a probability model for macroscopic averages, not literal simultaneous copies.
Quantitative relation
Example: Worked example
Two states have E₀=0,E₁=2kBT in a canonical ensemble. Find P₁/P₀.
Solution
P₁/P₀=e^{−(E₁−E₀)/(kBT)}=e⁻²≈0.135.
Example: Example: energy probabilities in the canonical ensemble
Consider energy levels 0 and ε with degeneracies g₀=1 and g₁=3. In the canonical ensemble, the total probability of level ε includes its degeneracy: P₁/P₀=3e^(−βε). For ε=2kBT, this ratio is 3e⁻²≈0.406; normalization gives P₁≈0.289 and P₀≈0.711. Several microstates at higher energy can therefore compete with a lower level. Ludwig Boltzmann pioneered the probabilistic interpretation of thermodynamics.
Macroscopic observables follow from ensemble averages; for example, ⟨E⟩=−∂ln Z/∂β and the energy variance is Var(E)=∂²ln Z/∂β²=kBT²C_V. Thus the partition function encodes both equilibrium thermodynamics and the response to temperature changes. For a large system with short-range interactions, canonical and microcanonical ensembles usually agree for local observables. Differences can matter in small systems, with long-range interactions, or near a first-order transition, where energy fluctuations and phase coexistence require care. A large canonical ensemble can be understood by treating a small subsystem as exchanging energy with the rest, which acts as a heat reservoir. In the thermodynamic limit, relative fluctuations shrink and ensemble averages become representative of measurements. The partition function also gives the Helmholtz free energy, F=−kBT ln Z, linking probabilities to thermodynamics. Agreement of ensembles does not mean that individual microstates have identical probabilities. Choose the ensemble to match the experimental constraints; ensembles are models, not physical copies.
Quick check
In the canonical ensemble, what is fixed and what may the system exchange?
In the microcanonical ensemble, what probabilities are assigned to accessible states in a narrow energy window?
References
- Charles Kittel and Herbert Kroemer (1980). Thermal Physics
- L. D. Landau and E. M. Lifshitz (1980). Statistical Physics