grand canonical partition function

From Physics wiki

Jump to: navigation, search

The grand canonical partition function of a system is a sum over all microstates weighted by their Gibbs factors:

\mathcal{Z} = \sum_{N=0}^\infty \sum_{s(N)} e^{-\frac{(E_{s(N)} - N \mu)}{k_B T}}\,,

and is the generalization of the canonical partition function to systems where the number of particles N\, is allowed to vary; i.e., systems that can exchange particles with a reservoir.

Contents

Probability

In analogy to the canonical partition function, the probability of the system being in a microstate s_i\, with energy E_i\, and particle number N_i\, at temperature T\, and chemical potential \mu\, is given by

P(E_i, N_i) = \frac{ e^{-\beta (E_i-N_i \mu) }}{ \mathcal{Z} }\,.

Number of particles

The average number of particles N \equiv \langle N_i \rangle\, is given by the given by the weighted sum N = \sum_i N_i P(E_i, N_i)\,, so that

N = \frac{1}{\beta} \left(\frac{\partial \ln \mathcal{Z}}{\partial \mu}\right)_{\beta, V}\,

Energy

The average internal energy U = \left\langle E \right\rangle\, is given by the weighted sum U = \sum_i E_i P(E_i, N_i)\,, so that

U = -\left(\frac{\partial \ln \mathcal{Z}}{\partial \beta} \right)_{\mu, V}+ \mu N\,

Grand canonical potential

Define the grand canonical potential to be

\Omega_G \stackrel{def}{=} - \frac{1}{\beta} \ln \mathcal{Z} \,.

Then, at equilibrium,

\Omega_G = U - TS - \mu N\,.
Retrieved from "http://www.physics.thetangentbundle.net/wiki/Statistical_mechanics/grand_canonical_partition_function"
Personal tools