The transport properties of glass-forming systems have many features that are not found in normal liquids. Among them is a very strong sensitivity of the relaxation times upon a change in temperature and the presence of so-called dynamical heterogeneities. In this review we discuss these unusual properties and present the results of a simple lattice gas model that helps to understand the origin of these heterogeneities from the microscopic level. Furthermore we discuss a simple analytical model, the continuous time random walk, and show that it allows to describe some aspects of the relaxation dynamics of glass-forming systems on a semi-quantitative level.

The glassy wormlike chain model is a highly successful phenomenological model recently introduced to describe anomalously slow subdiffusive dynamics in biopolymer networks and living cells. Here we extend this model by proposing a generic scheme how to include nonlinear plastic effects by introducing the possibility of force-dependent opening and closing of internal bonds. Further, we discuss physiological implications of this bond kinetics. Stability arguments lead us to the postulation of a “physiological sheet” in the parameter space. This sheet defines the set of parameters characterizing cells which are flexible enough to perform biological tasks while still being able to bear external perturbations characteristic of their surroundings and their internally generated prestress without damage. At the end of this contribution, we speculate about the connection between prestress and cell stiffness and about the mechanism by which the cell adapts to its mechanical environment.

Using the Fluctuation Theorem (FT), we give a first-principles derivation of Boltzmann’s postulate of equal a priori probability in phase space for the microcanonical ensemble. Using a corollary of the Fluctuation Theorem, namely the Second Law Inequality, we show that if the initial distribution differs from the uniform distribution over the energy hypersurface, then under very wide and commonly satisfied conditions, the initial distribution will relax to that uniform distribution. This result is somewhat analogous to the Boltzmann H-theorem but unlike that theorem, applies to dense fluids as well as dilute gases and also permits a nonmonotonic relaxation to equilibrium. We also prove that in ergodic systems the uniform (microcanonical) distribution is the only stationary, dissipationless distribution for the constant energy ensemble.

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For a continuum percolation model, it has been shown recentlythat the crossover from pure subdiffusion to normal diffusion extends over five decades in time [1, 2]; in addition, the asymptotic behavior is slowly approached and the large corrections cannot simply be ignored. Thus, it is of general interest to develop a systematic description of universal corrections to scaling in percolating systems. For percolating systems, we propose a universal exponent relation connecting the leading corrections to scaling of the cluster size distribution with the dynamic corrections to the asymptotic transport behavior at criticality. Our derivation is based on a cluster-resolved scaling theory unifying the scaling of both the cluster size distribution and the dynamics of a random walker. We corroborate our theoretical approach by extensive simulations for a site percolating square lattice and numerically determine both the static and dynamic correction exponents [3].

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We investigate the diffusion of particles on heterogeneous lattices with two kinds of nonequivalent sites. General analytical expressions for the chemical and jump diffusion coefficients have been derived in the case of strong inhomogeneity. We have calculated coverage dependencies of the diffusion coefficients and other necessary thermodynamic quantities for some representative values of the lateral pairwise interaction between the particles. The analytical data have been compared with the numerical data obtained by the kinetic Monte Carlo simulations. Almost perfect agreement between the respective results has been found.