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Alés A, López JM. Roughening of the anharmonic elastic interface in correlated random media. Phys Rev E 2021; 104:044108. [PMID: 34781530 DOI: 10.1103/physreve.104.044108] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/19/2021] [Accepted: 09/21/2021] [Indexed: 11/07/2022]
Abstract
We study the roughening properties of the anharmonic elastic interface in the presence of temporally correlated noise. The model can be seen as a generalization of the anharmonic Larkin model, recently introduced by Purrello, Iguain, and Kolton [Phys. Rev. E 99, 032105 (2019)2470-004510.1103/PhysRevE.99.032105], to investigate the effect of higher-order corrections to linear elasticity in the fate of interfaces. We find analytical expressions for the critical exponents as a function of the anharmonicity index n, the noise correlator range θ∈[0,1/2], and dimension d. In d=1 we find that the interface becomes faceted and exhibits anomalous scaling for θ>1/4 for any degree of anharmonicity n>1. Analytical expressions for the anomalous exponents α_{loc} and κ are obtained and compared with a numerical integration of the model. Our theoretical results show that anomalous roughening cannot exist for this model in dimensions d>1.
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Affiliation(s)
- Alejandro Alés
- Instituto de Física de Materiales Tandil, Facultad de Ciencias Exactas, Universidad Nacional del Centro de la Provincia de Buenos Aires, Pinto 399, 7000 Tandil, Argentina.,Consejo Nacional de Investigaciones Científicas y Técnicas, Godoy Cruz 2290, Buenos Aires, Argentina
| | - Juan M López
- Instituto de Física de Cantabria, CSIC-Universidad de Cantabria, 39005 Santander, Spain
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2
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Bauer M, Krapivsky PL, Mallick K. Random walk through a fertile site. Phys Rev E 2021; 103:022114. [PMID: 33736009 DOI: 10.1103/physreve.103.022114] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/20/2020] [Accepted: 01/15/2021] [Indexed: 11/07/2022]
Abstract
We study the dynamics of random walks hopping on homogeneous hypercubic lattices and multiplying at a fertile site. In one and two dimensions, the total number N(t) of walkers grows exponentially at a Malthusian rate depending on the dimensionality and the multiplication rate μ at the fertile site. When d>d_{c}=2, the number of walkers may remain finite forever for any μ; it surely remains finite when μ≤μ_{d}. We determine μ_{d} and show that 〈N(t)〉 grows exponentially if μ>μ_{d}. The distribution of the total number of walkers remains broad when d≤2, and also when d>2 and μ>μ_{d}. We compute 〈N^{m}〉 explicitly for small m, and show how to determine higher moments. In the critical regime, 〈N〉 grows as sqrt[t] for d=3, t/lnt for d=4, and t for d>4. Higher moments grow anomalously, 〈N^{m}〉∼〈N〉^{2m-1}, in the critical regime; the growth is normal, 〈N^{m}〉∼〈N〉^{m}, in the exponential phase. The distribution of the number of walkers in the critical regime is asymptotically stationary and universal, viz., it is independent of the spatial dimension. Interactions between walkers may drastically change the behavior. For random walks with exclusion, if d>2, there is again a critical multiplication rate, above which 〈N(t)〉 grows linearly (not exponentially) in time; when d≤d_{c}=2, the leading behavior is independent on μ and 〈N(t)〉 exhibits a sublinear growth.
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Affiliation(s)
- Michel Bauer
- Institut de Physique Théorique, Université Paris-Saclay, CEA and CNRS, 91191 Gif-sur-Yvette, France.,Départment de Mathématiques et Applications, ENS-Paris, 75005 Paris, France
| | - P L Krapivsky
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
| | - Kirone Mallick
- Institut de Physique Théorique, Université Paris-Saclay, CEA and CNRS, 91191 Gif-sur-Yvette, France
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3
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Goychuk I. Viscoelastic subdiffusion in a random Gaussian environment. Phys Chem Chem Phys 2018; 20:24140-24155. [PMID: 30206605 DOI: 10.1039/c8cp05238g] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/27/2022]
Abstract
Viscoelastic subdiffusion governed by a fractional Langevin equation is studied numerically in a random Gaussian environment modeled by stationary Gaussian potentials with decaying spatial correlations. This anomalous diffusion is archetypal for living cells, where cytoplasm is known to be viscoelastic and a spatial disorder also naturally emerges. We obtain some first important insights into it within a model one-dimensional study. Two basic types of potential correlations are studied: short-range exponentially decaying and algebraically slow decaying with an infinite correlation length, both for a moderate (several kBT, in the units of thermal energy), and strong (5-10kBT) disorder. For a moderate disorder, it is shown that on the ensemble level viscoelastic subdiffusion can easily overcome the medium's disorder. Asymptotically, it is not distinguishable from the disorder-free subdiffusion. However, a strong scatter in single-trajectory averages is nevertheless seen even for a moderate disorder. It features a weak ergodicity breaking, which occurs on a very long yet transient time scale. Furthermore, for a strong disorder, a very long transient regime of logarithmic, Sinai-type diffusion emerges. It can last longer and be faster in the absolute terms for weakly decaying correlations as compared with the short-range correlations. Residence time distributions in a finite spatial domain are of a generalized log-normal type and are reminiscent also of a stretched exponential distribution. They can be easily confused for power-law distributions in view of the observed weak ergodicity breaking. This suggests a revision of some experimental data and their interpretation.
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Affiliation(s)
- Igor Goychuk
- Institute for Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, 14476 Potsdam-Golm, Germany.
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Goychuk I, Kharchenko VO, Metzler R. Persistent Sinai-type diffusion in Gaussian random potentials with decaying spatial correlations. Phys Rev E 2017; 96:052134. [PMID: 29347675 DOI: 10.1103/physreve.96.052134] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/06/2017] [Indexed: 06/07/2023]
Abstract
Logarithmic or Sinai-type subdiffusion is usually associated with random force disorder and nonstationary potential fluctuations whose root-mean-squared amplitude grows with distance. We show here that extremely persistent, macroscopic logarithmic diffusion also universally emerges at sufficiently low temperatures in stationary Gaussian random potentials with spatially decaying correlations, known to exist in a broad range of physical systems. Combining results from extensive simulations with a scaling approach we elucidate the physical mechanism of this unusual subdiffusion. In particular, we explain why with growing temperature and/or time a first crossover occurs to standard, power-law subdiffusion, with a time-dependent power-law exponent, and then a second crossover occurs to normal diffusion with a disorder-renormalized diffusion coefficient. Interestingly, the initial, nominally ultraslow diffusion turns out to be much faster than the universal de Gennes-Bässler-Zwanzig limit of the renormalized normal diffusion, which realistically cannot be attained at sufficiently low temperatures and/or for strong disorder. The ultraslow diffusion is also shown to be nonergodic and it displays a local bias phenomenon. Our simple scaling theory not only explains our numerical findings but qualitatively also has a predictive character.
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Affiliation(s)
- Igor Goychuk
- Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - Vasyl O Kharchenko
- Institute of Applied Physics, National Academy of Sciences of Ukraine, 40030 Sumy, Ukraine
| | - Ralf Metzler
- Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
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5
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Hanes RDL, Egelhaaf SU. Dynamics of individual colloidal particles in one-dimensional random potentials: a simulation study. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2012; 24:464116. [PMID: 23114221 DOI: 10.1088/0953-8984/24/46/464116] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
Using Monte Carlo simulations, individual Brownian particles have been investigated in a one-dimensional random energy landscape whose energy levels are selected from a Gaussian distribution. The standard deviation of the distribution determines the roughness of the noise-like potential and was varied in the simulations. After initialization, which was done by an instantaneous or infinitely slow (annealed) quench, the particle dynamics were followed. They were characterized by a number of parameters, such as the mean squared displacement, the time dependent diffusion coefficient, the non-Gaussian parameter, and the van Hove function. The dynamics exhibit different regimes: at very short times superdiffusion, followed by normal diffusion, and subsequently an extended period of subdiffusive dynamics due to localization within the minima of the potential, and finally, after a very slow approach towards the long-time limit, again diffusion with a significantly reduced diffusion coefficient. The long-time diffusion coefficient is consistent with theoretical predictions while no predictions exist for the intermediate times. Nevertheless, over the whole time range, the simulation results are in agreement with recent experimental findings on colloidal particles in a random potential created by a holographic optical setup.
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Szendro IG, López JM, Rodríguez MA. Dynamics of perturbations in disordered chaotic systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:036202. [PMID: 18851116 DOI: 10.1103/physreve.78.036202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/27/2008] [Indexed: 05/26/2023]
Abstract
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.
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Affiliation(s)
- Ivan G Szendro
- Instituto de Física de Cantabria (IFCA), CSIC-UC, E-39005 Santander, Spain.
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7
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Szendro IG, López JM, Rodríguez MA. Localization in disordered media, anomalous roughening, and coarsening dynamics of faceted surfaces. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:011603. [PMID: 17677460 DOI: 10.1103/physreve.76.011603] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/26/2007] [Revised: 05/30/2007] [Indexed: 05/16/2023]
Abstract
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.
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Affiliation(s)
- Ivan G Szendro
- Instituto de Física de Cantabria (IFCA), CSIC-UC, E-39005 Santander, Spain.
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Hansen A, Kertész J. Phase diagram of optimal paths. PHYSICAL REVIEW LETTERS 2004; 93:040601. [PMID: 15323745 DOI: 10.1103/physrevlett.93.040601] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/17/2004] [Indexed: 05/24/2023]
Abstract
We show that by choosing appropriate distributions of the randomness the search for optimal paths links diverse problems of disordered media, such as directed percolation, invasion percolation, and directed and nondirected spanning polymers. We also introduce a simple and efficient algorithm, which solves the d-dimensional model numerically in O(N(1+df/d)) steps, where df is the fractal dimension of the path. Using extensive simulations in two dimensions, we identify the phase boundaries of the directed polymer universality class. A new strong-disorder phase occurs where the optimum paths are self-affine with parameter-dependent scaling exponents. Furthermore, the phase diagram contains directed and nondirected percolation as well as the directed random walk models at specific points and lines.
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Affiliation(s)
- Alex Hansen
- Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway.
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Chang R, Jagannathan K, Yethiraj A. Diffusion of hard sphere fluids in disordered media: a molecular dynamics simulation study. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:051101. [PMID: 15244802 DOI: 10.1103/physreve.69.051101] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/06/2004] [Indexed: 05/24/2023]
Abstract
Molecular dynamic simulations are reported for the static and dynamic properties of hard sphere fluids in matrices (or media) composed of quenched hard spheres. The effect of fluid and matrix density, matrix structure, and fluid to matrix sphere size ratio on the static and dynamic properties is studied using discontinuous molecular dynamics. The matrix density has a stronger effect on the self-diffusion coefficient than the fluid density, especially at high matrix densities where the geometric constraints due to the quenched spheres are significant. When the ratio of the size of the fluid spheres to that of the matrix spheres is equal to or greater than one, the diffusion increases as the fluid density is increased, at constant total volume fraction. This trend is however reversed if the ratio is smaller than one. Different methods of generating the matrix have a very strong effect on the dynamic properties even though the static correlations are similar. An analysis of the single-chain structure factor of the particle trajectories shows a change in the particle diffusive behavior at different time scales, suggestive of a hopping mechanism, although normal diffusion is recovered at long times. At high matrix densities, there is considerable heterogeneity in the diffusion of the fluid particles. The simulations demonstrate that the correlations in the matrix play a significant role on the diffusion of fluid spheres. For example, the diffusion constant in matrices constructed by different methods can be an order of magnitude different even though the pair correlation functions are almost identical.
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Affiliation(s)
- Rakwoo Chang
- Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin, Madison, Wisconsin 53706, USA
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10
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Shiferaw Y, Goldschmidt YY. Localization of a polymer in random media: relation to the localization of a quantum particle. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:051803. [PMID: 11414926 DOI: 10.1103/physreve.63.051803] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/03/2000] [Indexed: 05/23/2023]
Abstract
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.
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Affiliation(s)
- Y Shiferaw
- Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennyslvania 15260, USA
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11
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Lam PM, Tan S. Extremal-point densities of interface fluctuations in a quenched random medium. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:6246-52. [PMID: 11101956 DOI: 10.1103/physreve.62.6246] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/03/2000] [Indexed: 11/07/2022]
Abstract
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.
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Affiliation(s)
- PM Lam
- Fachbereich Physik, Universitat-Gesamthochschule Essen, D-45117 Essen, Germany
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12
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Nieuwenhuizen TM. Disorder-induced first-order wetting transitions in two dimensions. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/21/10/007] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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13
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Bender CM, Boettcher S, Meisinger PN. Spherically symmetric random walks. II. Dimensionally dependent critical behavior. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:112-126. [PMID: 9965053 DOI: 10.1103/physreve.54.112] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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14
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Dunlap DH, LaViolette RA, Parris PE. The spatial evolution of particles diffusing in the presence of randomly placed traps. J Chem Phys 1994. [DOI: 10.1063/1.467261] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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15
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Arsenin I, Halpin-Healy T, Krug J. Competing effects of point versus columnar defects on the roughening of directed polymers in random media. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:R3561-R3564. [PMID: 9961784 DOI: 10.1103/physreve.49.r3561] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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16
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Chakraborty AK, Bratko D, Chandler D. Diffusion of ionic penetrants in charged disordered media. J Chem Phys 1994. [DOI: 10.1063/1.466632] [Citation(s) in RCA: 36] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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17
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Deutsch JM. Generic behavior in linear systems with multiplicative noise. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 48:R4179-R4182. [PMID: 9961182 DOI: 10.1103/physreve.48.r4179] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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18
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Kolomeisky AB, Kolomeisky EB. Replica-scaling analysis of diffusion in quenched correlated random media. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1992; 45:R5327-R5330. [PMID: 9907721 DOI: 10.1103/physreva.45.r5327] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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19
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Tao R. Path-integral approach to diffusion in random media. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1991; 43:5284-5288. [PMID: 9904840 DOI: 10.1103/physreva.43.5284] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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20
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Tao R. Critical volume in diffusion through random media. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1990; 42:994-996. [PMID: 9904116 DOI: 10.1103/physreva.42.994] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Jayannavar AM, Köhler J. Density of states in the band tails and random multiplication in a diffusive medium. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1990; 41:3391-3394. [PMID: 9903502 DOI: 10.1103/physreva.41.3391] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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23
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Nattermann T, Renz W. Diffusion in a random catalytic environment, polymers in random media, and stochastically growing interfaces. PHYSICAL REVIEW. A, GENERAL PHYSICS 1989; 40:4675-4681. [PMID: 9902712 DOI: 10.1103/physreva.40.4675] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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24
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Tao R, Widom A, Webman I. Multidimensional diffusion in random potentials. PHYSICAL REVIEW. A, GENERAL PHYSICS 1989; 39:3748-3750. [PMID: 9901694 DOI: 10.1103/physreva.39.3748] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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25
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Tao R. Exact solution for diffusion in a random potential. PHYSICAL REVIEW LETTERS 1988; 61:2405-2408. [PMID: 10039109 DOI: 10.1103/physrevlett.61.2405] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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26
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27
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Teitel S. Transition to anomalous relaxation: Localization in a hierarchical potential. PHYSICAL REVIEW LETTERS 1988; 60:1154-1157. [PMID: 10037955 DOI: 10.1103/physrevlett.60.1154] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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28
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Cates M, Ball R. Statistics of a polymer in a random potential, with implications for a nonlinear interfacial growth model. ACTA ACUST UNITED AC 1988. [DOI: 10.1051/jphys:0198800490120200900] [Citation(s) in RCA: 98] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022]
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29
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Engel A, Ebeling W. Comment on "Diffusion in a random potential: Hopping as a dynamical consequence of localization". PHYSICAL REVIEW LETTERS 1987; 59:1979. [PMID: 10035384 DOI: 10.1103/physrevlett.59.1979] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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