A PDE Formulation of Non-Equilibrium Statistical Mechanics for Ionic Permeation

When there is a steady net flux in a system of interacting particles, the microscopic structure of the system can no longer be determined from the Boltzmann equilibrium distribution (partition function). Nonetheless, the microscopic structure of a finite system of diffusing interacting particles can be described by Poisson-Nernst-Planck-type partial differential equations. These equations, defined in a finite domain, are the non-equilibrium generalization of the BBGKY hierarchy of equilibrium statistical mechanics. Indeed, when no-flux conditions are imposed on the domain boundaries, equilibrium results are recovered. When non-homogeneous boundary conditions are given for these equations, the solutions describe densities and electrostatic potentials of particle systems not in equilibrium. The construction of a pair correlation function under these conditions will be a new result in statistical physics. As in the equilibrium case, a closure relation between a higher and a lower order correlation function has to be assumed. However, since we are considering a finite system, boundary conditions for the higher order correlation functions must also be derived. In applications to the permeation of ions through protein channels of biological membranes the computation of the pair correlation function will lead to a prediction of current through an open channel, given the spatial structure and fixed charge distribution. The pair correlation function contains finite size effects that lead to blocking in a narrow channel and possibly to selectivity.