Anomalous diffusion and transport processes in the presence of spatial and energy disorder, strong correlation and memory effects, and nonlinear friction

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The proposal is devoted to an interplay of nonlinearity, disorder and strong memory effects in anomalous diffusion and transport of Brownian particles in some generic physical models of complex disordered soft glass-like media like complex polymeric solutions and cytosol of living cells, both at or near to the thermal equilibrium, and very far away from it. In particular, we will investigate various models of spatially correlated and uncorrelated Gaussian potentials for one- and two-dimensional fractional type subdiffusion and transport in such potentials governed by generalized Langevin equation with an algebraically decaying memory. One of the purposes is to clarify the physical origin of Sinai type subdiffusion and transport in Gaussian potentials with bounded energy fluctuations. The widespread occurrence of Gaussian disorder and noise in nature makes this problem of fundamental significance in statistical physics, as well as of broad applied interest for material sciences, hard and soft matter physics, and biophysics. Furthermore, nonlinear models of viscoelastic friction with memory will also be studied, especially in the context of diffusion in nonlinear multistable potentials, also in the presence of time-dependent driving fields. Here, we will develop a generalization of linear viscoelastic Brownian dynamics with long-lasting memory towards nonlinear viscoelasticity. It will be done within a nonlinear Maxwell-Langevin model of viscoelasticity based on a multi-dimensional Markovian embedding. This proposed generalization takes into account the dependence of memory friction on the Brownian particle velocity. We base it on the physical idea that the elastic couplings to the viscoelastic modes of environment depend on the velocity of Brownian particle which disturbs this environment when it moves under the influence of an external field. The role of such nonlinear friction effects for viscoelastic subdiffusion and transport will be clarified. Besides, we will also address the role of hydrodynamic memory effects in this context, especially for transport driven far away from thermal equilibrium. In particular, the possibility of a transient superdiffusion regime hugely enhanced by driving will be clarified.