Nonlinear response of the trap model in the aging regime: exact results in the strong-disorder limit

Disorder-Induced Anomalous Mobility Enhancement in Confined Geometries

Strong, scale-free disorder disrupts typical transport properties like the Stokes-Einstein relation and linear response, leading to anomalous diffusion observed in amorphous materials, glasses, living cells, and other systems. Our study reveals that the combination of scale-free quenched disorder and geometrical constraints induces unconventional single-particle mobility behavior. Specifically, in a two-dimensional channel with width w, under external drive, tighter geometrical constraints (smaller w) enhance mobility. We derive an explicit form of the response to an external force by utilizing the double-subordination approach for the quenched trap model. The observed mobility enhancement occurs in the low-temperature regime where the distribution of localization times is scale-free.

From quenched disorder to continuous time random walk

This work focuses on quantitative representation of transport in systems with quenched disorder. Explicit mapping of the quenched trap model to continuous time random walk is presented. Linear temporal transformation, t→t/Λ^{1/α}, for a transient process in the subdiffusive regime is sufficient for asymptotic mapping. An exact form of the constant Λ^{1/α} is established. A disorder averaged position probability density function for a quenched trap model is obtained, and analytic expressions for the diffusion coefficient and drift are provided.

Random walk in two-dimensional self-affine random potentials: strong-disorder renormalization approach

We consider the continuous-time random walk of a particle in a two-dimensional self-affine quenched random potential of Hurst exponent H>0 . The corresponding master equation is studied via the strong disorder renormalization procedure introduced in Monthus and Garel [J. Phys. A: Math. Theor. 41, 255002 (2008)]. We present numerical results on the statistics of the equilibrium time t(eq) over the disordered samples of a given size L x L for 10 < or = L< or = 80. We find an "infinite disorder fixed point," where the equilibrium barrier Gamma(eq) identical with t(eq) scales as Gamma(eq)=L(H)u where u is a random variable of order O(1). This corresponds to a logarithmically slow diffusion |r(t)-r(0)| approximately (ln t)(1/H) for the position r(t) of the particle.

Einstein relation and effective temperature for systems with quenched disorder

According to the Einstein relation the ratio between independent measurements of fluctuations (e.g., diffusivity) and the response to a weak external field (e.g., mobility) is equal to the thermal temperature when the system is kept close to thermal equilibrium. For strongly disordered systems, which are not self-averaging this ratio is a random variable and hence in this sense the Einstein relation is not valid. Thus effective temperatures found using the fluctuation dissipation ratio are at least in some cases stochastic variables. This scenario is tested with the quenched trap model. An average over an ensemble of systems yields an averaged effective temperature, which is compared with results obtained from the mean-field continuous-time random-walk model.

Driven interfaces in random media at finite temperature: existence of an anomalous zero-velocity phase at small external force

The motion of driven interfaces in random media at finite temperature T and small external force F is usually described by a linear displacement h{G}(t) approximately V(F,T)t at large times, where the velocity vanishes according to the creep formula as V(F,T) approximately e;{-K(T)F;{mu}} for F-->0 . In this paper, we question this picture on the specific example of the directed polymer in a two-dimensional random medium. We have recently shown [C. Monthus and T. Garel, J. Phys. A 41, 255002 (2008)] that its dynamics for F=0 can be analyzed in terms of a strong disorder renormalization procedure, where the distribution of renormalized barriers flows towards some "infinite disorder fixed point." In the present paper, we obtain that for small F , this "infinite disorder fixed point" becomes a "strong disorder fixed point" with an exponential distribution of renormalized barriers. The corresponding distribution of trapping times then only decays as a power law P(tau) approximately 1tau;{1+alpha} , where the exponent alpha(F,T) vanishes as alpha(F,T) proportional, variant F micro as F-->0 . Our conclusion is that in the small force region alpha(F,T)<1 , the divergence of the averaged trapping time tau[over ]=+infinity induces strong non-self-averaging effects that invalidate the usual creep formula obtained by replacing all trapping times by the typical value. We find instead that the motion is only sublinearly in time h{G}(t) approximately t;{alpha(F,T)} , i.e., the asymptotic velocity vanishes V=0 . This analysis is confirmed by numerical simulations of a directed polymer with a metric constraint driven in a traps landscape. We moreover obtain that the roughness exponent, which is governed by the equilibrium value zeta{eq}=23 up to some large scale, becomes equal to zeta=1 at the largest scales.

Random Walk Delayed on Percolation Clusters

We study a continuous-time random walk on the d-dimensional lattice, subject to a drift and an attraction to large clusters of a subcritical Bernoulli site percolation. We find two distinct regimes: a ballistic one, and a subballistic one taking place when the attraction is strong enough. We identify the speed in the former case, and the algebraic rate of escape in the latter case. Finally, we discuss the diffusive behavior in the case of zero drift and weak attraction.

Random Walk Delayed on Percolation Clusters

We study a continuous-time random walk on thed-dimensional lattice, subject to a drift and an attraction to large clusters of a subcritical Bernoulli site percolation. We find two distinct regimes: a ballistic one, and a subballistic one taking place when the attraction is strong enough. We identify the speed in the former case, and the algebraic rate of escape in the latter case. Finally, we discuss the diffusive behavior in the case of zero drift and weak attraction.

Field-induced dispersion in subdiffusion

We discuss the response of continuous-time random walks to an oscillating external field within the generalized master equation approach. We concentrate on the time dependence of the two first moments of the walker's displacement. We show that for power-law waiting-time distributions with 0<alpha<1 corresponding to a semi-Markovian situation showing nonstationarity, the mean particle position tends to a constant; namely, the response to the external perturbation dies out. On the other hand, the oscillating field leads to a new additional contribution to the dispersion of the particle position, proportional to the square of its amplitude and growing with time. These new effects, amenable to experimental observation, result directly from the nonstationary property of the system.

Finite-size effects and intermittency in a simple aging system

We study the intermittent dynamics and the fluctuations of the dynamic correlation function of a simple aging system. Given its size L and its coherence length xi , the system can be divided into N independent subsystems, where N= (L/xi)(d), and d is the dimension of space. Each of them is considered as an aging subsystem which evolves according to an activated dynamics between energy levels. We compute analytically the distribution of trapping times for the global system, which can take power-law, stretched-exponential or exponential forms according to the values of N and the regime of times considered. An effective number of subsystems at age t(w), N(eff) (t(w)), can be defined, which decreases as t(w) increases, as well as an effective coherence length, xi (t(w)) approximately tw ((1-mu)/d), where mu<1 characterizes the trapping times distribution of a single subsystem. We also compute the probability distribution functions of the time intervals between large decorrelations, which exhibit different power-law behaviors as t(w) increases (or N decreases), and which should be accessible experimentally. Finally, we calculate the probability distribution function of the two-time correlator. We show that in a phenomenological approach, where N is replaced by the effective number of subsystems N(eff) (t(w)), the same qualitative behaviour as in experiments and simulations of several glassy systems can be obtained.