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 h...