Improved classical path approximation for the Boltzmann density matrix

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Abstract
A classical path approximation for diagonal matrix elements of the Boltzmann density operator, i.e., the equilibrium particle density, is derived and its properties analyzed, which is not only more accurate than an earlier result, but considerably simpler to apply to systems with many degrees of freedom. The most important simplifying feature is that it is not necessary to deal with classical trajectories with double‐ended boundary conditions. The partition function, for example, is given by a phase space average over initial conditions of an exponential function of a classical action integral along the trajectory with these initial conditions.