TheNhyphen;particle Fokkerndash;Planck equation has been cast into a form which is invariant to coordinate transformations. For a pair of particles interacting through a radial potentialV(R), the equation of motion may be projected onto a onehyphen;dimensional problem inRwhen the conjugate momenta of the center of mass and internal coordinates have symmetric distributions, as in Boltzmann distributions. However, theR2dependence of the density of configuration space demands thatV(R) be supplemented with an effective repulsive potential minus;2kTthinsp;lnthinsp;R, a result which has consequences for Langevin simulations of chemical processes. ForNinteracting 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 minus;kTthinsp;ln(Verbar;gklVerbar;)1/2, where Verbar;gklVerbar; is the determinant of that portion of the covariant metric tensor pertaining to unwanted coordinateskandl.
展开▼
