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We present the path-integral solutions to the distributions in classical(Gibbs) and quantum (Wigner) statistical mechanics. The kernel of thedistributions are derived in two ways - one by time slicing and defining theappropriate short-time interval phase space matrix element and second by makinguse of the kernel in the path-integral approach to quantum mechanics. We showthat the two approaches are perturbatively identical. We also present anothercomputation for the Wigner kernel, which is also the Liouville kernel, for theharmonic oscillator and free particle. These kernels may be used as thestarting point in the perturbative expansion of the Wigner kernel for anypotential. With the kernel solved, we essentially solve also the distributionsin classical and quantum statistical mechanics.

Path Integral Solutions to the Distributions of Statistical Mechanics