Brownian motion in en external field revisited

Abstract

In many interesting physical examples, the partition function is divergent, as first pointed out in 1924 by Fermi (for the hydrogen-atom case). Thus, the usualtoolbox of statistical mechanics becomes unavailable, notwithstanding the well known fact that the pertinent system may appear to be in a thermal steady state.We tackle and overcome these difficulties here by appeal to firmly established but not too well known mathematical recipes and obtain finite values for a typical divergent partitionfunction, that of a Brownian particle in an external field. This allows not only for calculating thermodynamic observables of interest, but for also instantiating other kind of statistical mechanics´ novelties.