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

本文致力于研究布朗粒子异常扩散和输运中的非线性、无序和强记忆效应的相互作用，这些模型是复杂无序的软玻璃状介质的一般物理模型，如处于或接近热平衡的复杂聚合物溶液和活细胞的胞浆，并且远离热平衡。特别是，我们将研究空间相关和不相关高斯势的各种模型，这些模型适用于一维和二维分数型次扩散和输运，这些势由具有代数衰减记忆的广义朗之万方程支配。其中一个目的是阐明能量涨落有界的高斯势中Sinai型次扩散和输运的物理起源。自然界中广泛存在的高斯无序和噪声使得这一问题在统计物理中具有基础意义，并在材料科学、硬物质和软物质物理以及生物物理中引起了广泛的应用兴趣。此外，还将研究具有记忆的粘弹性摩擦的非线性模型，特别是在非线性多稳势扩散的背景下，以及在随时间变化的驱动场存在的情况下。在这里，我们将把具有长记忆的线性粘弹性布朗动力学推广到非线性粘弹性。它将在基于多维马尔科夫嵌入的非线性麦克斯韦-朗之万粘弹性模型中进行。这一推广考虑了记忆摩擦对布朗粒子速度的依赖。我们基于这样的物理思想，即与环境的粘弹性模式的弹性耦合取决于布朗粒子的速度，当布朗粒子在外场的影响下运动时，布朗粒子的速度会对环境产生扰动。这种非线性摩擦效应对粘弹性次扩散和输运的作用将被阐明。此外，我们还将讨论流体动力记忆效应在这方面的作用，特别是对于远离热平衡的运输。特别是，将澄清驾驶极大地增强了瞬时超扩散制度的可能性。

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.