Phase space considerations for interacting particles obeying the Fokker–Planck equation

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Abstract
The N-particle Fokker–Planck equation has been cast into a form which is invariant to coordinate transformations. For a pair of particles interacting through a radial potential V(R), the equation of motion may be projected onto a one-dimensional problem in R when the conjugate momenta of the center of mass and internal coordinates have symmetric distributions, as in Boltzmann distributions. However, the R2 dependence of the density of configuration space demands that V(R) be supplemented with an effective repulsive potential −2kT ln R, a result which has consequences for Langevin simulations of chemical processes. For N interacting particles a subset of coordinates can often be identified as having such symmetric distributions, and if these are not of interest, they can be eliminated with the result that the potential containing the remaining coordinates is supplemented by −kT ln(‖gkl‖)1/2, where ‖gkl‖ is the determinant of that portion of the covariant metric tensor pertaining to unwanted coordinates k and l.