A new ''static'' renormalization group approach to stochastic models of fluctuating surfaces with spatially quenched noise is proposed in which only time-independent quantities are involved. As examples, quenched versions of the Kardar-Parisi-Zhang model and its Pavlik's modification, the Hwa-Kardar model of self-organized criticality, and Pastor-Satorras-Rothman model of landscape erosion are studied. It is shown that the upper critical dimension in the quenched models is shifted by two units upwards in comparison to their counterparts with white in-time noise. Possible scaling regimes associated with fixed points of the renormalization group equations are found and the critical exponents are derived to the leading order of the corresponding epsilon-expansions. Some exact values and relations for these exponents are obtained.
We study the stochastically driven conserved Kardar-Parisi-Zhang (CKPZ) equation with quenched disorders. Short-ranged quenched disorders is found to be a relevant perturbation on the pure CKPZ equation at one dimension, and as a result, a new universality class different from pure CKPZ equation appears to emerge. At higher dimensions, quenched disorder turns out to be ineffective to influence the universal scaling. This results in the asymptotic long wavelength scaling to be given by the linear theory, a scenario identical with the pure CKPZ equation. For sufficiently long-ranged quenched dis…
Quenched disorder affects significantly the behavior of phase transitions. The Imry-Ma-Aizenman-Wehr-Berker argument prohibits first-order or discontinuous transitions and their concomitant phase coexistence in low-dimensional equilibrium systems in the presence of random fields. Instead, discontinuous transitions become rounded or even continuous once disorder is introduced. Here we show that phase coexistence and first-order phase transitions are also precluded in nonequilibrium low-dimensional systems with quenched disorder: discontinuous transitions in two-dimensional systems with absorbin…
We investigate the effects of quenched disorder on the coherent response in a driven system of coupled oscillators. In particular, the interplay between quenched noise and periodic driving is explored, with particular attention to the possibility of resonance. The phase velocity is examined as the response of the system; revealed is the enhancement of the fraction of oscillators locked to the periodic driving, displaying resonance behavior. It is thus concluded that resonance behavior may also be induced by quenched disorder which does not have time-dependent correlations.
We elucidate the effects of chiral quenched disorder on the scaling properties of pure systems by considering a reduced model that is a variant of the quenched disordered cubic anisotropic O(N) model near its second order phase transition. A generic short-ranged Gaussian disorder distribution is considered. For distributions not invariant under spatial inversion ({hence chiral}), the scaling exponents are found to depend continuously on a model parameter that describes the extent of inversion symmetry breaking. Experimental and phenomenological implications of our results are discussed.
Inspired by the recent results on totally asymmetric simple exclusion processes on a periodic lattice with short-ranged quenched hopping rates [A. Haldar, A. Basu, Phys Rev Research 2, 043073 (2020)], we study the universal scaling properties of the Kardar-Parisi-Zhang (KPZ) equation with short-ranged quenched columnar disorder in general d-dimensions. We show that there are generic propagating modes in the system that have their origin in the quenched disorder and make the system anisotropic. We argue that the presence of the propagating modes actually make the effects of the quenched disorde…
We revisit the effects of short-ranged random quenched disorder on the universal scaling properties of the classical $N$-vector model with cubic anisotropy. We set up the nonconserved relaxational dynamics of the model, and study the universal dynamic scaling near the second order phase transition. We extract the critical exponents and the dynamic exponent in a one-loop dynamic renormalisation group calculation with short-ranged isotropic disorder. We show that the dynamics near a critical point is generically slower when the quenched disorder is relevant than when it is not, independent of wh…
We consider the A + A --> emptyset reaction, where the transport of the particles is given by Levy flights in a quenched random potential. With a common literature model of the disorder, the random potential can only increase the rate of reaction. With a model of the disorder that obeys detailed balance, however, the rate of reaction initially increases and then decreases as a function of the disorder strength. The physical behavior obtained with this second model is in accord with that for reactive turbulent flow, indicating that Levy flight statistics can model aspects of turbulent fluid tra…
It is investigated how a spatial quenched disorder modifies the dynamics of coupled map lattices. The disorder is introduced via the presence or absence of coupling terms among lattice sites. Two nonlinear maps have been considered embodying two paradigmatic dynamics. The Miller and Huse map can be associated with an Ising-like dynamics, whereas the logistic coupled maps is a prototype of a non trivial collective dynamics. Various indicators quantifying the overall behavior, demonstrates that even a small amount of spatial disorder is capable to alter the dynamics found for purely ordered case…
We extend to quenched disordered systems the variational scheme for real space renormalization group calculations that we recently introduced for homogeneous spin Hamiltonians. When disorder is present our approach gives access to the flow of the renormalized Hamiltonian distribution, from which one can compute the critical exponents if the correlations of the renormalized couplings retain finite range. Key to the variational approach is the bias potential found by minimizing a convex functional in statistical mechanics. This potential reduces dramatically the Monte Carlo relaxation time in la…
Recent experimental findings on anomalous diffusion have demanded novel models that combine annealed (temporal) and quenched (spatial or static) disorder mechanisms. The comb-model is a simplified description of diffusion on percolation clusters, where the comb-like structure mimics quenched disorder mechanisms and yields a subdiffusive regime. Here we extend the comb-model to simultaneously account for quenched and annealed disorder mechanisms. To do so, we replace usual derivatives in the comb diffusion equation by different fractional time-derivative operators and the conventional comb-like…
We study the equilibrium properties of an Ising model on a disordered random network where the disorder can be quenched or annealed. The network consists of four-fold coordinated sites connected via variable length one-dimensional chains. Our emphasis is on nonuniversal properties and we consider the transition temperature and other equilibrium thermodynamic properties, including those associated with one dimensional fluctuations arising from the chains. We use analytic methods in the annealed case, and a Monte Carlo simulation for the quenched disorder. Our objective is to study the differenc…
In this lecture we present an overview of the physics of irreversible fractal growth process, with particular emphasis on a class of models characterized by {\em quenched disorder}. These models exhibit self-organization, with critical properties developing spontaneously, without the fine tuning of external parameters. This situation is different from the usual critical phenomena, and requires the introduction of new theoretical methods. Our approach to these problems is based on two concepts, the Fixed Scale Transformation, and the quenched-stochastic transformation, or Run Time Statistics (R…
We develop the hydrodynamic theory for number conserving asymmetric exclusion processes with short-range random quenched disordered hopping rates, which is one-dimensional Kardar-Parisi- Zhang (KPZ) equation with quenched columnar disorder. We show that when the system is away from half-filling, the universal spatio-temporal scaling of the density fluctuations is indistinguishable from its pure counterpart, with the model belonging to the one-dimensional Kardar-Parisi-Zhang universality class. In contrast, close to half-filling, the quenched disorder is relevant, leading to a new universality …
We study the effects of distinct types of quenched disorder in the contact process (CP) with a competitive dynamics on bipartite sublattices. In the model, the particle creation depends on its first and second neighbors and the extinction increases according to the local density. The clean (without disorder) model exhibits three phases: inactive (absorbing), active symmetric and active asymmetric, where the latter exhibits distinct sublattice densities. These phases are separated by continuous transitions; the phase diagram is reentrant. By performing mean field analysis and Monte Carlo simula…
We comment on a recent Letter by Braunstein and Buceta [PRL vol.81, 630 (1998)], in which a novel equation has been proposed to describe the dynamics of interfaces in the presence of quenched disorder. We argue that the ansatz Braunstein and Buceta presented is just an effective description of the very short times regime and has no significance for the macroscopic (large scale) behaviour of the interface. We calculate an expression for the short time behaviour of the interface that is valid for any q (and not only at q_c).
Local diffusion coefficients in disordered materials such as living cells are highly heterogeneous. Quenched disorder is utilized substantially to study such complex systems, whereas its analytical treatment is difficult to handle. We consider finite systems with quenched disorder in order to investigate the effects of sample disorder fluctuations and confinement on single-particle diffusivity. While the system is ergodic in a single disorder realization, the time-averaged mean squared displacement depends on the disorder, i.e., the system is ergodic but non-self-averaging. We find that the in…
A single sandpile model with quenched random toppling matrices captures the crucial features of different models of self-organized criticality. With symmetric matrices avalanche statistics falls in the multiscaling BTW universality class. In the asymmetric case the simple scaling of the Manna model is observed. The presence or absence of a precise toppling balance between the amount of sand released by a toppling site and the total quantity the same site receives when all its neighbors topple once determines the appropriate universality class.
The Binder cumulant (BC) has been widely used for locating the phase transition point accurately in systems with thermal noise. In systems with quenched disorder, the BC may show subtle finite-size effects due to large sample-to-sample fluctuations. We study the globally coupled Kuramoto model of interacting limit-cycle oscillators with random natural frequencies and find an anomalous dip in the BC near the transition. We show that the dip is related to non-self-averageness of the order parameter at the transition. Alternative definitions of the BC, which do not show any anomalous behavior reg…
The stability of antiferromagnetic long-range order against quenched disorder is considered. A simple model of an antiferromagnet with a spatially varying Neel temperature is shown to possess a nontrivial fixed point corresponding to long-range order that is stable unless either the order parameter or the spatial dimensionality exceeds a critical value. The instability of this fixed point corresponds to the system entering a random-singlet phase. The stabilization of long-range order is due to quantum fluctuations, whose role in determining the phase diagram is discussed.
With dynamic Monte Carlo simulations, we investigate the continuous phase transition in the three-dimensional three-state random-bond Potts model. We propose a useful technique to deal with the strong corrections to the dynamic scaling form. The critical point, static exponents $β$ and $ν$, and dynamic exponent $z$ are accurately determined. Particularly, the results support that the exponent $ν$ satisfies the lower bound $ν\geqslant 2/d$.
We study by Monte Carlo simulations the effect of quenched orientational disorder in systems of interacting classical dipoles on a square lattice. Each dipole can lie along any of two perpendicular axes that form an angle psi with the principal axes of the lattice. We choose psi at random and without bias from the interval [-Delta, Delta] for each site of the lattice. For 0<Delta <~ pi/4 we find a thermally driven second order transition between a paramagnetic and a dipolar antiferromagnetic order phase and critical exponents that change continously with Delta. Near the case of maximum disorde…
We elucidate the nature of universal scaling in disordered driven models. We in particularly explore the intriguing possibility of whether coupling with quenched disorders can lead to continuously varying universality classes. We examine this question in the context of the Kardar-Parisi-Zhang (KPZ) equation, with and without a conservation law, coupled with quenched disorders of appropriate structures. By using a renormalisation group (RG) framework, we show when the disorder is relevant in the RG sense, the scaling exponents can depend continuously on a dimensionless parameter that defines th…
A trapping mechanism is observed and proposed as the origin of the anomalous behavior recently discovered in transport properties of overdamped ratchets subject to external oscillatory drive in the presence of quenched noise. In particular, this mechanism is shown to appear whenever the quenched disorder strength is greater than a threshold value. The minimum disorder strength required for the existence of traps is determined by studying the trap structure in a disorder configuration space. An approximation to the trapping probability density function in a disordered region of finite length in…
We consider the isotropic-to-nematic transition in liquid crystals confined to aerogel hosts, and assume that the aerogel acts as a random field. We generally find that self-averaging is violated. For a bulk transition that is weakly first-order, the violation of self-averaging is so severe, even the correlation length becomes non-self-averaging: no phase transition remains in this case. For a bulk transition that is more strongly first-order, the violation of self-averaging is milder, and a phase transition is observed.
Through large-scale numerical simulations, we study the phase ordering kinetics of the $2d$ Ising Model after a zero-temperature quench from a high-temperature homogeneous initial condition. Analysing the behaviour of two important quantities -- the winding angle and the pair-connectedness -- we reveal the presence of a percolating structure in the pattern of domains. We focus on the pure case and on the random field and random bond Ising Model.
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: $t\to t/Λ^{1/α}$ for transient process on translationally invariant lattice, in the sub-diffusive regime, is sufficient for asymptotic mapping. Exact form of the constant $Λ^{1/α}$ is established. Disorder averaged position probability density function for quenched trap model is obtained and analytic expressions for the diffusion coefficient and drift are provided.
The tensor renormalization-group method, developed by Levin and Nave, brings systematic improvability to the position-space renormalization-group method and yields essentially exact results for phase diagrams and entire thermodynamic functions. The method, previously used on systems with no quenched randomness, is extended in this study to systems with quenched randomness. Local magnetizations and correlation functions as a function of spin separation are calculated as tensor products subject to renormalization-group transformation. Phase diagrams are extracted from the long-distance behavior …
We present an experimental study of the influence of quenched disorder on the distribution of flux avalanches in type-II superconductors. In the presence of much quenched disorder, the avalanche sizes are power-law distributed and show finite size scaling, as expected from self-organized criticality (SOC). Furthermore, the shape of the avalanches is observed to be fractal. In the absence of quenched disorder, a preferred size of avalanches is observed and avalanches are smooth. These observations indicate that a certain minimum amount of disorder is necessary for SOC behavior. We relate these …
The effect of quenched (frozen) orientational disorder on the collective motion of active particles is analyzed. We find that, as with annealed disorder (Langevin noise), active polar systems are far more robust against quenched disorder than their equilibrium counterparts. In particular, long ranged order (i.e., the existence of a non-zero average velocity $\langle {\bf v} \rangle$) persists in the presence of quenched disorder even in spatial dimensions $d=3$, while it is destroyed even by arbitrarily weak disorder in $d \le 4$ in equilibrium systems. Furthermore, in $d=2$, quasi-long-ranged…
With quenched disorder, we introduce two-dimensional active nematics suspended in an incompressible fluid. We write the coarse-grained hydrodynamic equations of motion for slow variables, viz. density, orientation and flow fields. The quenched disorder is introduced such that it interacts with the local orientation at every point with some strength. Disorder strength is tuned from zero to large values. We numerically study the defect dynamics and system's kinetics and find that the finite disorder slows the ordering. The presence of fluid induces large fluctuation in the orientation field, fur…
We investigate the effects of quenched randomness on the universal properties of a two-temperature lattice gas. The disorder modifies the dynamical transition rates of the system in an anisotropic fashion, giving rise to a new fixed point. We determine the associated scaling form of the structure factor, quoting critical exponents to two-loop order in an expansion around the upper critical dimension d$_c=7$. The close relationship with another quenched disorder fixed point, discovered recently in this model, is discussed.
The roughening behavior of a one-dimensional interface fluctuating under quenched disorder growth is examined while keeping an anchored boundary. The latter introduces detailed balance conditions which allows for a thorough analysis of equilibrium aspects at both macroscopic and microscopic scales. It is found that the interface roughens linearly with the substrate size only in the vicinity of special disorder realizations. Otherwise, it remains stiff and tilted.
The effect of quenched (frozen) disorder on the collective motion of active particles is analyzed. We find that active polar systems are far more robust against quenched disorder than equilibrium ferromagnets. Long ranged order (a non-zero average velocity $\langle{\bf v}\rangle$) persists in the presence of quenched disorder even in spatial dimensions $d=3$; in $d=2$, quasi-long-ranged order (i.e., spatial velocity correlations that decay as a power law with distance) occurs. In equilibrium systems, only quasi-long-ranged order in $d=3$ and short ranged order in $d=2$ are possible. Our theore…
We study critical behaviour of disordered magnets near four dimensions. We consider the system with explicit cubic anisotropy and scalar disorder and that with random direction of anisotropy axis. The quenched disorder is taken into account by replica method. Using the method of operator product expansion, we derive in the first order to $ε$ approximation the renormalization group equations taking into account possible replica symmetry breaking.
We explore the effects that quenched disorder has on discontinuous nonequilibrium phase transitions into absorbing states. We focus our analysis on the Naming Game model, a nonequilibrium low-dimensional system with different absorbing states. The results obtained by means of the finite-size scaling analysis and from the study of the temporal dynamics of the density of active sites near the transition point evidence that the spatial quenched disorder does not destroy the discontinuous transition.
We show that spatial quenched disorder affects polar active matter in ways more complex and far-reaching than believed heretofore. Using simulations of the 2D Vicsek model subjected to random couplings or a disordered scattering field, we find in particular that ergodicity is lost in the ordered phase, the nature of which we show to depend qualitatively on the type of quenched disorder: for random couplings, it remains long-range ordered, but qualitatively different from the pure (disorderless) case. For random scatterers, polar order varies with system size but we find strong non-self-averagi…
We show that incompressible polar active fluids can exhibit an ordered, coherently moving phase even in the presence of quenched disorder in two dimensions. Unlike such active fluids with annealed (i.e., time-dependent) disorder only, which behave like equilibrium ferromagnets with long-range interactions, this robustness against quenched disorder is a fundamentally non-equilibrium phenomenon. The ordered state belongs to a new universality class, whose scaling laws we calculate using three different renormalization group schemes, which all give scaling exponents within 0.02 of each other, ind…
We evaluate the mean escape time of overdamped particles over potential barriers in short-correlated quenched Gaussian disorder potentials in one dimension at low temperature. The thermally activated escape is very sensitive to the form of the \emph{tail} of the potential barrier probability distribution. We evaluate this tail by using the optimal fluctuation method. For monotone decreasing autocovariances we reproduce the tail obtained by Lopatin and Vinokur (2001). However, for nonmonotonic autocovariances of the disorder potential we show that the tail changes. It is much higher when the di…
The extinction of the contact process in lattice models with quenched disorder is analysed in the limit of small density of infected sites. It is shown that the problem in such a regime can be mapped to the quantum-mechanical one characterized by the Anderson Hamiltonian for an electron in a random lattice. It is demonstrated both analytically (self-consistent mean-field) and numerically (by direct diagonalization of the Hamiltonian and by means of cellular automata simulations) that disorder enhances the contact process given the mean values of random parameters are not influenced by disorder…
We study the effect of quenched disorder on nonequilibrium systems of interacting particles, specifically, driven diffusive lattice gases with spatially disordered jump rates. The exact steady-state measure is found for a class of models evolving by drop-push dynamics, allowing several physical quantities to be calculated. Dynamical correlations are studied numerically in one dimension. We conjecture that the relevance of quenched disorder depends crucially upon the speed of the kinematic waves in the system. Time-dependent correlation functions, which monitor the dissipation of kinematic wave…
We introduce a two-dimensional active nematic with quenched disorder. We write the coarse-grained hydrodynamic equations of motion for slow variables, viz. density, and orientation. Disorder strength is tuned from zero to large values. Results from the numerical solution of equations of motion as well as the calculation of two-point orientation correlation function using linear approximation show that the ordered steady-state follows a disorder-dependent crossover from quasi long-range order (QLRO) to short-range order (SRO). Such crossover is due to the pinning of +1/2 and -1/2 topological de…
Critical behavior in short-time dynamics is investigated by a Monte Carlo study for the random-bond Potts ferromagnet with a trinary distribution of quenched disorders on two-dimensional triangular lattices. The dynamic scaling is verified and applied to estimate critical exponents $θ$, $z$ and $β/ν$ for several realizations of the trinary distribution. Our critical scaling analysis strongly indicates that the bond randomness influences the critical universality.
We introduce a model for fractures in quenched disordered media. This model has a deterministic extremal dynamics, driven by the energy function of a network of springs (Born Hamiltonian). The breakdown is the result of the cooperation between the external field and the quenched disorder. This model can be considered as describing the low temperature limit for crack propagation in solids. To describe the memory effects in this dynamics, and then to study the resistance properties of the system we realized some numerical simulations of the model. The model exhibits interesting geometric and dyn…
We examine the dynamics and stripe formation in a system with competing short and long range interactions in the presence of both an applied dc drive and quenched disorder. Without disorder, the system forms stripes organized in a labyrinth state. We find that, when the disorder strength exceeds a critical value, an applied dc drive can induce a dynamical stripe ordering transition to a state that is more ordered than the originating undriven, unpinned pattern. We show that signatures in the structure factor and transport properties correspond to this dynamical reordering transition, and we pr…
We analyze the combined effect of a Laplacian field and quenched disorder for the generation of fractal structures with a study, both numerical and theoretical, of the quenched dielectric breakdown model (QDBM). The growth dynamics is shown to evolve from the avalanches of invasion percolation (IP) to the smooth growth of Laplacian fractals, i. e. diffusion limited aggregation (DLA) and the dielectric breakdown model (DBM). The fractal dimension is strongly reduced with respect to both DBM and IP, due to the combined effect of memory and field screening. This implies a specific relation betwee…
Two-dimensional systems in which there is a competition between long-range repulsion and short range attraction exhibit a remarkable variety of patterns such as stripes, bubbles, and labyrinths. Such systems include magnetic films, Langmuir monolayers, polymers, gels, water-oil mixtures, and two-dimensional electron systems. In many of these systems quenched disorder from the underlying substrate may be present. We examine the dynamics and stripe formation in the presence of both an applied dc drive and quenched disorder. When the disorder strength exceeds a critical value, an applied dc drive…
Very diluted Bose gas placed into a disordered environment falls into a fragmented localized state. At some critical density the repulsion between particles overcomes the disorder. The gas transits into a coherent superfluid state. In this article the geometrical and energetic characteristics of the localized state at zero temperature and the critical density at which the quantum phase transition from the localized to the superfluid state proceeds are found.
We discuss mean field theory of glasses without quenched disorder focusing on the justification of the replica approach to thermodynamics. We emphasize the assumptions implicit in this method and discuss how they can be verified. The formalism is applied to the long range Ising model with orthogonal coupling matrix. We find the one step replica-symmetry breaking solution and show that it is stable in the intermediate temperature range that includes the glass state but excludes very low temperatures. At very low temperatures this solution becomes unstable and this approach fails.
The process of protein synthesis in biological systems resembles a one dimensional driven lattice gas in which the particles (ribosomes) have spatial extent, covering more than one lattice site. Realistic, nonuniform gene sequences lead to quenched disorder in the particle hopping rates. We study the totally asymmetric exclusion process with large particles and quenched disorder via several mean field approaches and compare the mean field results with Monte Carlo simulations. Mean field equations obtained from the literature are found to be reasonably effective in describing this system. A num…
We investigate the effect of quenched spatial disordered hopping rates on the characteristics of the asymmetric simple exclusion process (ASEP) with open boundaries both numerically and by extensive simulations. Disorder averages of the bulk density and current are obtained in terms of various input and output rates. We study the binary and uniform distributions of disorder. It is verified that the effect of spatial inhomogeneity is generically to enlarge the size of the maximal current phase. This is in accordance with the mean field results obtained by Harris and Stinchcombe. Furthermore, we…
We study the effect of quenched spatial disorder on the steady states of driven systems of interacting particles. Two sorts of models are studied: disordered drop-push processes and their generalizations, and the disordered asymmetric simple exclusion process. We write down the exact steady-state measure, and consequently a number of physical quantities explicitly, for the drop-push dynamics in any dimensions for arbitrary disorder. We find that three qualitatively different regimes of behaviour are possible in 1-$d$ disordered driven systems. In the Vanishing-Current regime, the steady-state …
The problem of random walk is considered in one dimension in the simultaneous presence of a quenched random force field and long-range connections the probability of which decays with the distance algebraically as p_l ~ βl^{-s}. The dynamics are studied mainly by a numerical strong disorder renormalization group method. According to the results, for s>2 the long-range connections are irrelevant and the mean-square displacement increases as <x^2(t)> ~ (ln t)^{2/ψ} with the barrier exponent ψ=1/2, which is known in one-dimensional random environments. For s<2, instead, the quenched disorder is f…
We consider the influence of quenched spatial disorder on phase transitions in classical and quantum systems. We show that rare strong disorder fluctuations can have dramatic effects on critical points. In classical systems with sufficiently correlated disorder or in quantum systems with overdamped dynamics they can completely destroy the sharp phase transition by smearing. This is caused by effects similar to but stronger than Griffiths phenomena: True static order can develop on a rare region while the bulk system is still in the disordered phase. We discuss the thermodynamic behavior in the…
A simple model of cyclically competing species on a directed graph with quenched disorder is proposed as an extension of the rock-paper-scissors model. By assuming that the effects of loops in a directed random graph can be ignored in the thermodynamic limit, it is proved for any finite disorder that the system fixates to a frozen configuration when the species number $s$ is larger than the spatial connectivity $c$, and otherwise stays active. Nontrivial lower and upper bounds for the persistence probability of a site never changing its state are also analytically computed. The obtained bounds…
