Nonlinear equations generalizing the continuous-time random walk model to the case of finite concentrations have been derived. The equations take into account two factors responsible for the emergence of anomalous diffusion: nonlinearity and local equilibrium breaking. In locally equilibrium conditions, these equations are reduced to the nonlinear Fokker-Plank equation that can be interpreted as the transport equation for fermions with multiple energy levels. As a consequence of the nonlinear equations, two linear non-Markov equations with concentration-dependent memory functions have been obtained. One of these equations describes diffusion of a small deviation from the equilibrium state, while the other describes diffusion of tagged particles in the equilibrium system. It is shown that the emergence of anomalous diffusion is favored by low concentrations.
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