Some New Features of Linear Theory for Describing Non-linear Diffusion

Abstract

The non-linear flux equation, the non-linear Fokker–Planck equation (or Smoluchowski equation), and the non-linear Langiven equation are the basic equations for describing particle diffusion in non-ideal system subjected to time-dependent external fields. Nevertheless, the exact solution of those equations is still a challenge because of their inherent complexity of the non-linear mathematics. Li et al. found that, based on the defined apparent variables, the nonlinear Fokker–Planck equation and the non-linear flux equation could be transformed to linear forms under the condition of strong friction limit or local equilibrium assumption. In this paper, some new features of the theory were found: (i) The linear flux equation for describing non-linear diffusion can be obtained from the irreversible thermodynamic theory; (ii) The linear non-steady state diffusion equation for describing non-linear diffusion of the non-steady state, which was described by the non-linear Fokker–Planck equation, can be derived more consistently from the microscopic molecular statistical theory; (iii) In the theory, the non-linear Langiven equation also bears a linear form; (iv) For some special cases, e.g. diffusion in a periodic total potential system, the local equilibrium assumption or the strong friction limit is not required in establishing the linear theory for describing non-linear diffusion, so the linear theory may be important in the study of Brown motor.