1. Using Pj = e^-BE/Z in the Gibb's form for

the entropy, show that

F = kb Tln (Z), where F = U - TS is

the Helmholtz free energy. Here Z=

e^-BE is the canonical partition

function ...

Solution. The Gibbs formula for the entropy is

S = - KB∑pi ㏑ (pi).

Using the Boltzmann probability in the canonical ensemble pi = exp (-βEi) / Z, we have,

S = - kB∑ pi[ - (Ei : KBT) - ㏑ (Z) ] = (U:T) + kBln (Z) so U - T S = - kBTln (Z),

where we used∑ P i = 1, U = ∑i piEi