The Cluster's Partition Function
A page about the partition function — the central object of statistical mechanics.
The partition function
The partition function Z is the sum over all microstates of the Boltzmann factor: Z = sum_i exp(-beta E_i), where beta = 1/(k_B T). All thermodynamic quantities can be derived from Z. The free energy is F = -k_B T log Z, the entropy is S = -dF/dT, the energy is U = -d log Z / dbeta. In the cluster, the partition function is the sum over all edit configurations of the Boltzmann factor. All edit thermodynamic quantities are derived from Z.
The canonical ensemble
The canonical ensemble describes a system in thermal equilibrium with a heat bath at temperature T. The probability of state i is P_i = exp(-beta E_i) / Z. In the cluster, the canonical ensemble describes a cluster section in thermal equilibrium with an edit bath at temperature T. The probability of edit configuration i is P_i = exp(-beta E_i) / Z.
The grand canonical ensemble
The grand canonical ensemble describes a system that can exchange both energy and particles with a reservoir. The partition function is Xi = sum_N sum_i exp(-beta (E_i - mu N)), where mu is the chemical potential. In the cluster, the grand canonical ensemble describes a cluster section that can exchange both edit energy and edit particles with a reservoir. The partition function includes both energy and edit number.
The path integral representation
The partition function can be written as a path integral: Z = integral D phi exp(-S_E[phi]), where S_E is the Euclidean action. In the cluster, the partition function is written as a path integral over edit configurations: Z = integral D[edit] exp(-S_E[edit]). The Euclidean action encodes the edit energy.
This partition
This page is about the partition function. Z = sum_i exp(-beta E_i). F = -k_B T log Z. S = -dF/dT. U = -d log Z / dbeta. The canonical ensemble gives P_i = exp(-beta E_i) / Z. The grand canonical ensemble includes chemical potential. The path integral represents Z as a functional integral. The partition function is real.