Diffusion in a random catalytic environment, polymers in random media, and stochastically growing interfaces

The influences of correlated spatially random perturbations on first passage time in a linear-cubic potential

The influences of correlated spatially random perturbations (SRPs) on the first passage problem are studied in a linear-cubic potential with a time-changing external force driven by a Gaussian white noise. First, the escape rate in the absence of SRPs is obtained by Kramers' theory. For the random potential case, we simplify the escape rate by multiplying the escape rate of smooth potentials with a specific coefficient, which is to evaluate the influences of randomness. Based on this assumption, the escape rates are derived in two scenarios, i.e., small/large correlation lengths. Consequently, the first passage time distributions (FPTDs) are generated for both smooth and random potential cases. We find that the position of the maximal FPTD has a very good agreement with that of numerical results, which verifies the validity of the proposed approximations. Besides, with increasing the correlation length, the FPTD shifts to the left gradually and tends to the smooth potential case. Second, we investigate the most probable passage time (MPPT) and mean first passage time (MFPT), which decrease with increasing the correlation length. We also find that the variation ranges of both MPPT and MFPT increase nonlinearly with increasing the intensity. Besides, we briefly give constraint conditions to guarantee the validity of our approximations. This work enables us to approximately evaluate the influences of the correlation length of SRPs in detail, which was always ignored previously.

Faceted patterns and anomalous surface roughening driven by long-range temporally correlated noise

We investigate Kardar-Parisi-Zhang (KPZ) surface growth in the presence of power-law temporally correlated noise. By means of extensive numerical simulations of models in the KPZ universality class we find that, as the noise correlator index increases above some threshold value, the surface exhibits anomalous kinetic roughening of the type described by the generic scaling theory of Ramasco et al. [Phys. Rev. Lett. 84, 2199 (2000)PRLTAO0031-900710.1103/PhysRevLett.84.2199]. Remarkably, as the driving noise temporal correlations increase, the surface develops a characteristic pattern of macroscopic facets that completely dominates the dynamics in the long time limit. We argue that standard scaling fails to capture the behavior of KPZ subject to long-range temporally correlated noise. These phenomena are not not described by the existing theoretical approaches, including renormalization group and self-consistent approaches.

Simulating flexible polymers in a potential of randomly distributed hard disks

We perform equilibrium computer simulations of a two-dimensional pinned flexible polymer exposed to a quenched disorder potential consisting of hard disks. We are especially interested in the high-density regime of the disorder, where subtle structures such as cavities and channels play a central role. We apply an off-lattice growth algorithm proposed by Garel and Orland [J. Phys. A 23, L621 (1990)], where a distribution of polymers is constructed in parallel by growing each of them monomer by monomer. In addition we use a multicanonical Monte Carlo method in order to cross-check the results of the growth algorithm. We measure the end-to-end distribution and the tangent-tangent correlations. We also investigate the scaling behavior of the mean square end-to-end distance in dependence on the monomer number. While the influence of the potential in the low-density case is merely marginal, it dominates the configurational properties of the polymer for high densities.

Dynamics of perturbations in disordered chaotic systems

We study the time evolution of perturbations in spatially extended chaotic systems in the presence of quenched disorder. We find that initially random perturbations tend to exponentially localize in space around static pinning centers that are selected by the particular configuration of disorder. The spatiotemporal behavior of typical perturbations deltau(x,t) is analyzed in terms of the Hopf-Cole transform h(x,t) identical withlnmid R:deltau(x,t)mid R: . Our analysis shows that the associated surface h(x,t) self-organizes into a faceted structure with scale-invariant correlations. Scaling analysis of critical roughening exponents reveals that there are three different universality classes for error propagation in disordered chaotic systems that correspond to different symmetries of the underlying disorder. Our conclusions are based on numerical simulations of disordered lattices of coupled chaotic elements and equations for diffusion in random potentials. We propose a phenomenological stochastic field theory that gives some insights on the path for a generalization of these results for a broad class of disordered extended systems exhibiting space-time chaos.

Localization in disordered media, anomalous roughening, and coarsening dynamics of faceted surfaces

We study a surface growth model related to the Kardar-Parisi-Zhang equation for nonequilibrium kinetic roughening, but where the thermal noise is replaced by a static columnar disorder eta(x) . This model is one of the many representations of the problem of particle diffusion in trapping or amplifying disordered media. We find that probability localization in the latter translates into facet formation in the equivalent surface growth problem. Coarsening of the pattern can therefore be identified with the diffusion of the localization center. The emergent faceted structure gives rise to nontrivial scaling properties, including anomalous surface roughening in excellent agreement with an existing conjecture for kinetic roughening of faceted surfaces. In a wider context, our study sheds light onto the scaling properties in other systems displaying this kind of patterned surface.

Large-time dynamics and aging of a polymer chain in a random potential

We study the out-of-equilibrium large-time dynamics of a Gaussian polymer chain in a quenched random potential. The dynamics studied is a simple Langevin dynamics commonly referred to as the Rouse model. The equations for the two-time correlation and response functions are derived within the Gaussian variational approximation. In order to implement this approximation faithfully, we employ the supersymmetric representation of the Martin-Siggia-Rose dynamical action. For a short-ranged correlated random potential the equations are solved analytically in the limit of large times using certain assumptions concerning the asymptotic behavior. Two possible dynamical behaviors are identified depending upon the time separation: a stationary regime and an aging regime. In the stationary regime time translation invariance holds and so does the fluctuation dissipation theorem. The aging regime which occurs for large time separations of the two-time correlation functions is characterized by a history dependence and the breakdown of certain equilibrium relations. The large-time limit of the equations yields equations among the order parameters that are similar to the equations obtained in statics using replicas. In particular the aging solution corresponds to the broken replica solution. But there is a difference in one equation that leads to important consequences for the solution. The stationary regime corresponds to the motion of the polymer inside a local minimum of the random potential, whereas in the aging regime the polymer hops between different minima. As a by-product we also solve exactly the dynamics of a chain in a random potential with quadratic correlations.

Semiflexible polymers in a random environment

We present using simple scaling arguments and one step replica symmetry breaking a theory for the localization of semiflexible polymers in a quenched random environment. In contrast to completely flexible polymers, localization of semiflexible polymers depends not only on the details of the disorder but also on the ease with which polymers can bend. The interplay of these two effects can lead to the delocalization of a localized polymer with an increase in either the disorder density or the stiffness. Our theory provides a general criterion for the delocalization of polymers with varying degrees of flexibility and allows us to propose a phase diagram for the highly folded (localized) states of semiflexible polymers as a function of the disorder strength and chain rigidity.

Localization and freezing of a Gaussian chain in a quenched random potential

The Gaussian chain in a quenched random potential (which is characterized by the disorder strength Delta) is investigated in the d-dimensional space by the replicated variational method. The general expression for the free energy within so-called one-step-replica symmetry breaking (1-RSB) scenario has been systematically derived. We have shown that the replica symmetrical (RS) limit of this expression can describe the chain center-of-mass localization and collapse. The critical disorder when the chain becomes localized scales as Delta(c) approximately b(d)N(-2+d/2) (where b is the length of the Kuhn segment length and N is the chain length) whereas the chain gyration radius R(g) approximately b(b(d)/Delta)(1/(4-d)). The freezing of the internal degrees of freedom follows to the 1-RSB-scenario and is characterized by the beads localization length D(2). It was demonstrated that the solution for D(2) appears as a metastable state at Delta=Delta(A) and behaves similarly to the corresponding frozen states in heteropolymers or in p-spin random spherical model.

Functional renormalization group for anisotropic depinning and relation to branching processes

Using the functional renormalization group, we study the depinning of elastic objects in presence of anisotropy. We explicitly demonstrate how the Kardar-Parisi-Zhang (KPZ) term is always generated, even in the limit of vanishing velocity, except where excluded by symmetry. This mechanism has two steps. First a nonanalytic disorder-distribution is generated under renormalization beyond the Larkin length. This nonanalyticity then generates the KPZ term. We compute the beta function to one loop taking properly into account the nonanalyticity. This gives rise to additional terms, missed in earlier studies. A crucial question is whether the nonrenormalization of the KPZ coupling found at 1-loop order extends beyond the leading one. Using a Cole-Hopf-transformed theory we argue that it is indeed uncorrected to all orders. The resulting flow equations describe a variety of physical situations: We study manifolds in periodic disorder, relevant for charge density waves, as well as in nonperiodic disorder. Further the elasticity of the manifold can either be short range (SR) or long range (LR). A careful analysis of the flow yields several nontrivial fixed points. All these fixed points are transient since they possess one unstable direction towards a runaway flow, which leaves open the question of the upper critical dimension. The runaway flow is dominated by a Landau-ghost mode. For LR elasticity, relevant for contact line depinning, we show that there are two phases depending on the strength of the KPZ coupling. For SR elasticity, using the Cole-Hopf transformed theory we identify a nontrivial 3-dimensional subspace which is invariant to all orders and contains all above fixed points as well as the Landau mode. It belongs to a class of theories which describe branching and reaction-diffusion processes, of which some have been mapped onto directed percolation.

Localization of a polymer in random media: relation to the localization of a quantum particle

In this paper we consider in detail the connection between the problem of a polymer in a random medium and that of a quantum particle in a random potential. We are interested in a system of finite volume where the polymer is known to be localized inside a low minimum of the potential. We show how the end-to-end distance of a polymer that is free to move can be obtained from the density of states of the quantum particle using extreme value statistics. We give a physical interpretation to the recently discovered one-step replica-symmetry-breaking solution for the polymer [Phys. Rev. E 61, 1729 (2000)] in terms of the statistics of localized tail states. Numerical solutions of the variational equations for chains of different length are performed and compared with quenched averages computed directly by using the eigenfunctions and eigenenergies of the Schrödinger equation for a particle in a one-dimensional random potential. The quantities investigated are the radius of gyration of a free Gaussian chain, its mean square distance from the origin and the end-to-end distance of a tethered chain. The probability distribution for the position of the chain is also investigated. The glassiness of the system is explained and is estimated from the variance of the measured quantities.

Extremal-point densities of interface fluctuations in a quenched random medium

We give a number of exact, analytical results for the stochastic dynamics of the density of local extrema (minima and maxima) of linear Langevin equations and solid-on-solid lattice growth models driven by spatially quenched random noise. Such models can describe nonequilibrium surface fluctuations in a spatially quenched random medium, diffusion in a random catalytic environment, and polymers in a random medium. In spite of the nonuniversal character for the quantities studied, their behavior against the variation of the microscopic length scale can present generic features, characteristic of the macroscopic observables of the system.