The canonical ensemble System  Heat Reservoir R T=const. adiabatic wall Consider system at constant temperature and volume We have shown in thermodynamics that system with (T,V)=const. in equilibrium is at a minimum of the Helmholtz free energy, F (T=const, V=const.) Q = -Q R

We use a similar approach now in deriving density function and partition function System  can exchange energy with the heat reservoir: Find maximum of S under the constraint that average (internal) energy is given found by maximizing under constraints Using again Lagrange multiplier technique

with Partition function of the canonical ensemble Next we show From the V,N constant With the equilibrium distribution back into the entropy expression

V,N constant Usingwe find With Gives meaning to the Lagrange parameter