Matinyan, SG; Müller, B, *The partition function in the Wigner–Kirkwood expansion*,
Journal of Physics A: Mathematical and General, vol. 39 no. 18
(May, 2006),
pp. L285-L292 [pdf], [doi] .
**Abstract:**

*We study the semiclassical Wigner-Kirkwood (WK) expansion of the partition function Z(t) for arbitrary even homogeneous potentials, starting from the Bloch equation. As is well known, the phase-space kernel of Z satisfies the so-called Uhlenbeck-Beth equation, which depends on the gradients of the potential. We perform a chain of transformations to obtain novel forms of this equation that invite analogies with various physical phenomena and formalisms, such as diffusion processes, the Fokker-Planck equation and supersymmetric quantum mechanics. © 2006 IOP Publishing Ltd.*