This quantity can be obtained via a path integral of the Euclidean action, which can be shown by Wick rotating the path integral used to obtain the propagator.
The 6N-dimensional integral used to calculate this quantity can be rewritten with the assumption that the potential only includes pairwise interactions - that technique is called Mayer cluster expansion. A common technique used to solve the Ising model is to express this quantity as the trace of a product of transfer matrices. This quantity can be obtained via a path integral of the Euclidean action, which can be shown by Wick rotating the path integral used to obtain the propagator. For a system of N indistinguishable particles, the total value for this quantity equals one individual particle's value for this quantity raised to the N power divided by N factorial. The expectation value of total energy equals the negative of the derivative of the natural log of this quantity with respect to thermodynamic beta. Helmholtz free energy equals negative kT times the natural log of this quantity, which is defined as the sum over all energy states of e raised to the energy of the state divided by kT. For 10 points, name this normalization constant for the Boltzmann distribution, which is symbolized Z.
The 6N-dimensional integral used to calculate this quantity can be rewritten with the assumption that the potential only includes pairwise interactions - that technique is called Mayer cluster expansion.
grand partition function [or Z until it is read; prompt on "Q", "xi", and "Y"]