1
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Li Y, Suleiman K, Xu Y. Anomalous diffusion, non-Gaussianity, nonergodicity, and confinement in stochastic-scaled Brownian motion with diffusing diffusivity dynamics. Phys Rev E 2024; 109:014139. [PMID: 38366530 DOI: 10.1103/physreve.109.014139] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/14/2023] [Accepted: 12/07/2023] [Indexed: 02/18/2024]
Abstract
Scaled Brownian motions (SBMs) with power-law time-dependent diffusivity have been used to describe various types of anomalous diffusion yet Gaussian observed in granular gases kinetics, turbulent diffusion, and molecules mobility in cells, to name a few. However, some of these systems may exhibit non-Gaussian behavior which can be described by SBM with diffusing diffusivity (DD-SBM). Here, we numerically investigate both free and confined DD-SBM models characterized by fixed or stochastic scaling exponent of time-dependent diffusivity. The effects of distributed scaling exponent, random diffusivity, and confinement are considered. Different regimes of ultraslow diffusion, subdiffusion, normal diffusion, and superdiffusion are observed. In addition, weak ergodic and non-Gaussian behaviors are also detected. These results provide insights into diffusion in time-fluctuating diffusivity landscapes with potential applications to time-dependent temperature systems spreading in heterogeneous environments.
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Affiliation(s)
- Yongge Li
- School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710072, China
| | - Kheder Suleiman
- School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710072, China
| | - Yong Xu
- School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710072, China
- MOE Key Laboratory for Complexity Science in Aerospace, Northwestern Polytechnical University, Xi'an 710072, China
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2
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Wang X, Chen Y. Random diffusivity processes in an external force field. Phys Rev E 2022; 106:024112. [PMID: 36109990 DOI: 10.1103/physreve.106.024112] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/03/2022] [Accepted: 07/21/2022] [Indexed: 06/15/2023]
Abstract
Brownian yet non-Gaussian processes have recently been observed in numerous biological systems, and corresponding theories have been constructed based on random diffusivity models. Considering the particularity of random diffusivity, this paper studies the effect of an external force acting on two kinds of random diffusivity models whose difference is embodied in whether the fluctuation-dissipation theorem is valid. Based on the two random diffusivity models, we derive the Fokker-Planck equations with an arbitrary external force, and we analyze various observables in the case with a constant force, including the Einstein relation, the moments, the kurtosis, and the asymptotic behaviors of the probability density function of particle displacement at different timescales. Both the theoretical results and numerical simulations of these observables show a significant difference between the two kinds of random diffusivity models, which implies the important role of the fluctuation-dissipation theorem in random diffusivity systems.
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Affiliation(s)
- Xudong Wang
- School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, People's Republic of China
| | - Yao Chen
- College of Sciences, Nanjing Agricultural University, Nanjing 210095, People's Republic of China
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3
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Poncet A, Grabsch A, Bénichou O, Illien P. Exact time dependence of the cumulants of a tracer position in a dense lattice gas. Phys Rev E 2022; 105:054139. [PMID: 35706275 DOI: 10.1103/physreve.105.054139] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/18/2022] [Accepted: 05/03/2022] [Indexed: 06/15/2023]
Abstract
We develop a general method to calculate the exact time dependence of the cumulants of the position of a tracer particle in a dense lattice gas of hardcore particles. More precisely, we calculate the cumulant-generating function associated with the position of a tagged particle at arbitrary time, and at leading order in the density of vacancies on the lattice. In particular, our approach gives access to the short-time dynamics of the cumulants of the tracer position, a regime in which few results are known. The generality of our approach is demonstrated by showing that it goes beyond the case of a symmetric 1D random walk and covers the important situations of (1) a biased tracer, (2) comblike structures, and (3) d-dimensional situations.
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Affiliation(s)
- Alexis Poncet
- Univ. Lyon, ENS de Lyon, Univ. Claude Bernard, CNRS, Laboratoire de Physique, 69342 Lyon, France
| | - Aurélien Grabsch
- Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée (LPTMC), 4 Place Jussieu, 75005 Paris, France
| | - Olivier Bénichou
- Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée (LPTMC), 4 Place Jussieu, 75005 Paris, France
| | - Pierre Illien
- Sorbonne Université, CNRS, Physicochimie des Electrolytes et Nanosystèmes Interfaciaux (PHENIX), 4 Place Jussieu, 75005 Paris, France
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4
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Rizkallah P, Sarracino A, Bénichou O, Illien P. Microscopic Theory for the Diffusion of an Active Particle in a Crowded Environment. PHYSICAL REVIEW LETTERS 2022; 128:038001. [PMID: 35119883 DOI: 10.1103/physrevlett.128.038001] [Citation(s) in RCA: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/09/2021] [Accepted: 12/13/2021] [Indexed: 06/14/2023]
Abstract
We calculate the diffusion coefficient of an active tracer in a schematic crowded environment, represented as a lattice gas of passive particles with hardcore interactions. Starting from the master equation of the problem, we put forward a closure approximation that goes beyond trivial mean field and provides the diffusion coefficient for an arbitrary density of crowders in the system. We show that our approximation is accurate for a very wide range of parameters, and that it correctly captures numerous nonequilibrium effects, which are the signature of the activity in the system. In addition to the determination of the diffusion coefficient of the tracer, our approach allows us to characterize the perturbation of the environment induced by the displacement of the active tracer. Finally, we consider the asymptotic regimes of low and high densities, in which the expression of the diffusion coefficient of the tracer becomes explicit, and which we argue to be exact.
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Affiliation(s)
- Pierre Rizkallah
- Sorbonne Université, CNRS, Laboratoire de Physico-Chimie des Électrolytes et Nanosystèmes Interfaciaux (PHENIX), 4 Place Jussieu, 75005 Paris, France
| | - Alessandro Sarracino
- Dipartimento di Ingegneria, Università della Campania "Luigi Vanvitelli", 81031 Aversa (CE), Italy
| | - Olivier Bénichou
- Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée (LPTMC), 4 Place Jussieu, 75005 Paris, France
| | - Pierre Illien
- Sorbonne Université, CNRS, Laboratoire de Physico-Chimie des Électrolytes et Nanosystèmes Interfaciaux (PHENIX), 4 Place Jussieu, 75005 Paris, France
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5
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Sarracino A, Vulpiani A. On the fluctuation-dissipation relation in non-equilibrium and non-Hamiltonian systems. CHAOS (WOODBURY, N.Y.) 2019; 29:083132. [PMID: 31472486 DOI: 10.1063/1.5110262] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/15/2019] [Accepted: 08/08/2019] [Indexed: 06/10/2023]
Abstract
We review generalized fluctuation-dissipation relations, which are valid under general conditions even in "nonstandard systems," e.g., out of equilibrium and/or without a Hamiltonian structure. The response functions can be expressed in terms of suitable correlation functions computed in the unperturbed dynamics. In these relations, typically, one has nontrivial contributions due to the form of the stationary probability distribution; such terms take into account the interaction among the relevant degrees of freedom in the system. We illustrate the general formalism with some examples in nonstandard cases, including driven granular media, systems with a multiscale structure, active matter, and systems showing anomalous diffusion.
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Affiliation(s)
- A Sarracino
- Dipartimento di Ingegneria, Università della Campania "L. Vanvitelli," via Roma 29, 81031 Aversa (CE), Italy
| | - A Vulpiani
- Dipartimento di Fisica, Università Sapienza-p.le A. Moro 2, 00185 Roma, Italy
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6
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Bénichou O, Illien P, Oshanin G, Sarracino A, Voituriez R. Tracer diffusion in crowded narrow channels. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2018; 30:443001. [PMID: 30211693 DOI: 10.1088/1361-648x/aae13a] [Citation(s) in RCA: 19] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
We summarise different results on the diffusion of a tracer particle in lattice gases of hard-core particles with stochastic dynamics, which are confined to narrow channels-single-files, comb-like structures and quasi-one-dimensional channels with the width equal to several particle diameters. We show that in such geometries a surprisingly rich, sometimes even counter-intuitive, behaviour emerges, which is absent in unbounded systems. This is well-documented for the anomalous diffusion in single-files. Less known is the anomalous dynamics of a tracer particle in crowded branching single-files-comb-like structures, where several kinds of anomalous regimes take place. In narrow channels, which are broader than single-files, one encounters a wealth of anomalous behaviours in the case where the tracer particle is subject to a regular external bias: here, one observes an anomaly in the temporal evolution of the tracer particle velocity, super-diffusive at transient stages, and ultimately a giant diffusive broadening of fluctuations in the position of the tracer particle, as well as spectacular multi-tracer effects of self-clogging of narrow channels. Interactions between a biased tracer particle and a confined crowded environment also produce peculiar patterns in the out-of-equilibrium distribution of the environment particles, very different from the ones appearing in unbounded systems. For moderately dense systems, a surprising effect of a negative differential mobility takes place, such that the velocity of a biased tracer particle can be a non-monotonic function of the force. In some parameter ranges, both the velocity and the diffusion coefficient of a biased tracer particle can be non-monotonic functions of the density. We also survey different results obtained for a tracer particle diffusion in unbounded systems, which will permit a reader to have an exhaustively broad picture of the tracer diffusion in crowded environments.
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Affiliation(s)
- O Bénichou
- Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée (UMR 7600), 4 Place Jussieu, 75252 Paris Cedex 05, France
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7
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Poncet A, Bénichou O, Démery V, Oshanin G. N-tag probability law of the symmetric exclusion process. Phys Rev E 2018; 97:062119. [PMID: 30011439 DOI: 10.1103/physreve.97.062119] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/25/2018] [Indexed: 11/07/2022]
Abstract
The symmetric exclusion process (SEP), in which particles hop symmetrically on a discrete line with hard-core constraints, is a paradigmatic model of subdiffusion in confined systems. This anomalous behavior is a direct consequence of strong spatial correlations induced by the requirement that the particles cannot overtake each other. Even if this fact has been recognized qualitatively for a long time, up to now there has been no full quantitative determination of these correlations. Here we study the joint probability distribution of an arbitrary number of tagged particles in the SEP. We determine analytically its large-time limit for an arbitrary density of particles, and its full dynamics in the high-density limit. In this limit, we obtain the time-dependent large deviation function of the problem and unveil a universal scaling form shared by the cumulants.
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Affiliation(s)
- Alexis Poncet
- LPTMC, CNRS/Sorbonne Université, 4 Place Jussieu, F-75005 Paris, France
| | - Olivier Bénichou
- LPTMC, CNRS/Sorbonne Université, 4 Place Jussieu, F-75005 Paris, France
| | - Vincent Démery
- Gulliver, CNRS, ESPCI Paris, PSL Research University, 10 rue Vauquelin, Paris, France.,Univ Lyon, ENS de Lyon, Univ Claude Bernard Lyon 1, CNRS, Laboratoire de Physique, F-69342 Lyon, France
| | - Gleb Oshanin
- LPTMC, CNRS/Sorbonne Université, 4 Place Jussieu, F-75005 Paris, France
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8
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Illien P, Bénichou O, Oshanin G, Sarracino A, Voituriez R. Nonequilibrium Fluctuations and Enhanced Diffusion of a Driven Particle in a Dense Environment. PHYSICAL REVIEW LETTERS 2018; 120:200606. [PMID: 29864325 DOI: 10.1103/physrevlett.120.200606] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2017] [Revised: 01/02/2018] [Indexed: 06/08/2023]
Abstract
We study the diffusion of a tracer particle driven out of equilibrium by an external force and traveling in a dense environment of arbitrary density. The system evolves on a discrete lattice and its stochastic dynamics is described by a master equation. Relying on a decoupling approximation that goes beyond the naive mean-field treatment of the problem, we calculate the fluctuations of the position of the tracer around its mean value on a lattice of arbitrary dimension, and with different boundary conditions. We reveal intrinsically nonequilibrium effects, such as enhanced diffusivity of the tracer induced by both the crowding interactions and the external driving. We finally consider the high-density and low-density limits of the model and show that our approximation scheme becomes exact in these limits.
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Affiliation(s)
- Pierre Illien
- Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3NP, United Kingdom
- Department of Chemistry, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
| | - Olivier Bénichou
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS UMR 7600, Université Pierre-et-Marie-Curie, 4 Place Jussieu, 75005 Paris, France
| | - Gleb Oshanin
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS UMR 7600, Université Pierre-et-Marie-Curie, 4 Place Jussieu, 75005 Paris, France
| | - Alessandro Sarracino
- Istituto dei Sistemi Complessi-CNR, P.le Aldo Moro 2, 00185 Rome, Italy
- Dipartimento di Fisica, Università di Roma Sapienza, P.le Aldo Moro 2, 00185 Rome, Italy
| | - Raphaël Voituriez
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS UMR 7600, Université Pierre-et-Marie-Curie, 4 Place Jussieu, 75005 Paris, France
- Laboratoire Jean Perrin, CNRS UMR 8237, Université Pierre-et-Marie-Curie, 4 Place Jussieu, 75005 Paris, France
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9
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Choi Y, Kim HJ. Nonstationary Markovian replication process causing diverse diffusions. Phys Rev E 2017; 96:042144. [PMID: 29347487 DOI: 10.1103/physreve.96.042144] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/29/2017] [Indexed: 06/07/2023]
Abstract
We introduce a single generative mechanism that can be used to describe diverse nonstationary diffusions. A nonstationary Markovian replication process for steps is considered for which we derive analytically the time evolution of the probability distribution of the walker's displacement and the generalized telegrapher equation with time-varying coefficients, and we find that diffusivity can be determined by temporal changes of replication of an immediate step. By controlling the replications, we realize diverse diffusions such as alternating diffusion, superdiffusion, subdiffusion, and marginal diffusion, which originate from oscillating, increasing, decreasing, and slowly increasing or decreasing replications with time, respectively.
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Affiliation(s)
- Yichul Choi
- Department of Physics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0435, USA
| | - Hyun-Joo Kim
- Department of Physics Education, Korea National University of Education, Chungbuk 363-791, Korea
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10
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Huang J, Zhao H. Ultraslow diffusion and weak ergodicity breaking in right triangular billiards. Phys Rev E 2017; 95:032209. [PMID: 28415210 DOI: 10.1103/physreve.95.032209] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/18/2016] [Indexed: 11/07/2022]
Abstract
We investigate the diffusion behavior of a right triangular billiard system by transforming its dynamics to a two-dimensional piecewise map. We find that the diffusion in the momentum space is ultraslow, i.e., the mean squared displacement grows asymptotically as the square of the logarithm of time. The mechanism of the ultraslow diffusion behavior is explained and numerical evidence corroborating our conclusion is provided. The weak ergodicity breaking of the system is also discussed.
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Affiliation(s)
- Junxiang Huang
- Department of Physics and Institute of Theoretical Physics and Astrophysics, Xiamen University, Xiamen 361005, Fujian, China
| | - Hong Zhao
- Department of Physics and Institute of Theoretical Physics and Astrophysics, Xiamen University, Xiamen 361005, Fujian, China
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11
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Prostnev AS, Shub BR. Simulation of the diffusion of atoms in a dense adsorbed layer with a hexagonal structure. RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B 2017. [DOI: 10.1134/s1990793116060087] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
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12
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Prostnev AS, Shub BR. Diffusion of atoms in a dense adsorbed layer with a hexagonal structure. RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B 2016. [DOI: 10.1134/s1990793116030076] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
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13
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Bénichou O, Illien P, Oshanin G, Sarracino A, Voituriez R. Diffusion and Subdiffusion of Interacting Particles on Comblike Structures. PHYSICAL REVIEW LETTERS 2015; 115:220601. [PMID: 26650285 DOI: 10.1103/physrevlett.115.220601] [Citation(s) in RCA: 27] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/02/2015] [Indexed: 06/05/2023]
Abstract
We study the dynamics of a tracer particle (TP) on a comb lattice populated by randomly moving hard-core particles in the dense limit. We first consider the case where the TP is constrained to move on the backbone of the comb only. In the limit of high density of the particles, we present exact analytical results for the cumulants of the TP position, showing a subdiffusive behavior ∼t^{3/4}. At longer times, a second regime is observed where standard diffusion is recovered, with a surprising nonanalytical dependence of the diffusion coefficient on the particle density. When the TP is allowed to visit the teeth of the comb, based on a mean-field-like continuous time random walk description, we unveil a rich and complex scenario with several successive subdiffusive regimes, resulting from the coupling between the geometrical constraints of the comb lattice and particle interactions. In this case, remarkably, the presence of hard-core interactions asymptotically speeds up the TP motion along the backbone of the structure.
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Affiliation(s)
- O Bénichou
- Sorbonne Universités, UPMC Université Paris 06, UMR 7600, LPTMC, F-75005 Paris, France and CNRS, UMR 7600, Laboratoire de Physique Théorique de la Matière Condensée, F-75005 Paris, France
| | - P Illien
- Sorbonne Universités, UPMC Université Paris 06, UMR 7600, LPTMC, F-75005 Paris, France and CNRS, UMR 7600, Laboratoire de Physique Théorique de la Matière Condensée, F-75005 Paris, France
| | - G Oshanin
- Sorbonne Universités, UPMC Université Paris 06, UMR 7600, LPTMC, F-75005 Paris, France and CNRS, UMR 7600, Laboratoire de Physique Théorique de la Matière Condensée, F-75005 Paris, France
| | - A Sarracino
- Sorbonne Universités, UPMC Université Paris 06, UMR 7600, LPTMC, F-75005 Paris, France and CNRS, UMR 7600, Laboratoire de Physique Théorique de la Matière Condensée, F-75005 Paris, France
| | - R Voituriez
- Sorbonne Universités, UPMC Université Paris 06, UMR 7600, LPTMC, F-75005 Paris, France and CNRS, UMR 7600, Laboratoire de Physique Théorique de la Matière Condensée, F-75005 Paris, France
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14
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Bénichou O, Illien P, Oshanin G, Sarracino A, Voituriez R. Microscopic theory for negative differential mobility in crowded environments. PHYSICAL REVIEW LETTERS 2014; 113:268002. [PMID: 25615388 DOI: 10.1103/physrevlett.113.268002] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2014] [Indexed: 06/04/2023]
Abstract
We study the behavior of the stationary velocity of a driven particle in an environment of mobile hard-core obstacles. Based on a lattice gas model, we demonstrate analytically that the drift velocity can exhibit a nonmonotonic dependence on the applied force, and show quantitatively that such negative differential mobility (NDM), observed in various physical contexts, is controlled by both the density and diffusion time scale of the obstacles. Our study unifies recent numerical and analytical results obtained in specific regimes, and makes it possible to determine analytically the region of the full parameter space where NDM occurs. These results suggest that NDM could be a generic feature of biased (or active) transport in crowded environments.
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Affiliation(s)
- O Bénichou
- Sorbonne Universités, UPMC Univ Paris 06, UMR 7600, LPTMC, F-75005 Paris, France and CNRS, UMR 7600, Laboratoire de Physique Théorique de la Matiére Condensée, F-75005 Paris, France
| | - P Illien
- Sorbonne Universités, UPMC Univ Paris 06, UMR 7600, LPTMC, F-75005 Paris, France and CNRS, UMR 7600, Laboratoire de Physique Théorique de la Matiére Condensée, F-75005 Paris, France
| | - G Oshanin
- Sorbonne Universités, UPMC Univ Paris 06, UMR 7600, LPTMC, F-75005 Paris, France and CNRS, UMR 7600, Laboratoire de Physique Théorique de la Matiére Condensée, F-75005 Paris, France
| | - A Sarracino
- Sorbonne Universités, UPMC Univ Paris 06, UMR 7600, LPTMC, F-75005 Paris, France and CNRS, UMR 7600, Laboratoire de Physique Théorique de la Matiére Condensée, F-75005 Paris, France
| | - R Voituriez
- Sorbonne Universités, UPMC Univ Paris 06, UMR 7600, LPTMC, F-75005 Paris, France and CNRS, UMR 7600, Laboratoire de Physique Théorique de la Matiére Condensée, F-75005 Paris, France
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15
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Illien P, Bénichou O, Oshanin G, Voituriez R. Velocity anomaly of a driven tracer in a confined crowded environment. PHYSICAL REVIEW LETTERS 2014; 113:030603. [PMID: 25083625 DOI: 10.1103/physrevlett.113.030603] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/10/2014] [Indexed: 06/03/2023]
Abstract
We consider a discrete model in which a tracer performs a random walk biased by an external force, in a dense bath of particles performing symmetric random walks constrained by hard-core interactions. We reveal the emergence of a striking velocity anomaly in confined geometries: in quasi-1D systems such as stripes or capillaries, the velocity of the tracer displays a long-lived plateau before ultimately dropping to a lower value. We develop an analytical solution that quantitatively accounts for this intriguing behavior. Our analysis suggests that such a velocity anomaly could be a generic feature of driven dynamics in quasi-1D crowded systems.
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Affiliation(s)
- Pierre Illien
- Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600), Université Pierre et Marie Curie, 4 Place Jussieu, 75255 Paris Cedex, France
| | - Olivier Bénichou
- Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600), Université Pierre et Marie Curie, 4 Place Jussieu, 75255 Paris Cedex, France
| | - Gleb Oshanin
- Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600), Université Pierre et Marie Curie, 4 Place Jussieu, 75255 Paris Cedex, France
| | - Raphaël Voituriez
- Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600), Université Pierre et Marie Curie, 4 Place Jussieu, 75255 Paris Cedex, France and Laboratoire Jean Perrin, FRE 3231 CNRS /UPMC, 4 Place Jussieu, 75255 Paris Cedex, France
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16
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Prostnev AS, Shub BR. Diffusion of atoms incorporated in the surface layer of the fcc(111) face. RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B 2014. [DOI: 10.1134/s1990793114030245] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
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17
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Metzler R, Jeon JH, Cherstvy AG, Barkai E. Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys Chem Chem Phys 2014; 16:24128-64. [DOI: 10.1039/c4cp03465a] [Citation(s) in RCA: 1046] [Impact Index Per Article: 104.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/26/2022]
Abstract
This Perspective summarises the properties of a variety of anomalous diffusion processes and provides the necessary tools to analyse and interpret recorded anomalous diffusion data.
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Affiliation(s)
- Ralf Metzler
- Institute of Physics and Astronomy
- University of Potsdam
- Potsdam-Golm, Germany
- Physics Department
- Tampere University of Technology
| | - Jae-Hyung Jeon
- Physics Department
- Tampere University of Technology
- Tampere, Finland
- Korean Institute for Advanced Study (KIAS)
- Seoul, Republic of Korea
| | - Andrey G. Cherstvy
- Institute of Physics and Astronomy
- University of Potsdam
- Potsdam-Golm, Germany
| | - Eli Barkai
- Physics Department and Institute of Nanotechnology and Advanced Materials
- Bar-Ilan University
- Ramat Gan, Israel
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18
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Bénichou O, Bodrova A, Chakraborty D, Illien P, Law A, Mejía-Monasterio C, Oshanin G, Voituriez R. Geometry-induced superdiffusion in driven crowded systems. PHYSICAL REVIEW LETTERS 2013; 111:260601. [PMID: 24483787 DOI: 10.1103/physrevlett.111.260601] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/30/2013] [Indexed: 06/03/2023]
Abstract
Recent molecular dynamics simulations of glass-forming liquids revealed superdiffusive fluctuations associated with the position of a tracer particle (TP) driven by an external force. Such an anomalous response, whose mechanism remains elusive, has been observed up to now only in systems close to their glass transition, suggesting that this could be one of its hallmarks. Here, we show that the presence of superdiffusion is in actual fact much more general, provided that the system is crowded and geometrically confined. We present and solve analytically a minimal model consisting of a driven TP in a dense, crowded medium in which the motion of particles is mediated by the diffusion of packing defects, called vacancies. For such nonglass-forming systems, our analysis predicts a long-lived superdiffusion which ultimately crosses over to giant diffusive behavior. We find that this trait is present in confined geometries, for example long capillaries and stripes, and emerges as a universal response of crowded environments to an external force. These findings are confirmed by numerical simulations of systems as varied as lattice gases, dense liquids, and granular fluids.
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Affiliation(s)
- Olivier Bénichou
- Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600), Université Pierre et Marie Curie, 4 Place Jussieu, 75255 Paris Cedex, France
| | - Anna Bodrova
- Department of Physics, Moscow State University, Moscow 119991, Russia
| | - Dipanjan Chakraborty
- Max-Planck-Institut für Intelligente Systeme, Heisenbergstr. 3, 70569 Stuttgart, Germany, and IV Institut für Theoretische Physik, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
| | - Pierre Illien
- Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600), Université Pierre et Marie Curie, 4 Place Jussieu, 75255 Paris Cedex, France
| | - Adam Law
- Max-Planck-Institut für Intelligente Systeme, Heisenbergstr. 3, 70569 Stuttgart, Germany, and IV Institut für Theoretische Physik, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
| | - Carlos Mejía-Monasterio
- Laboratory of Physical Properties, Technical University of Madrid, Avenida Complutense s/n, 28040 Madrid, Spain, and Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, FIN-00014 Helsinki, Finland
| | - Gleb Oshanin
- Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600), Université Pierre et Marie Curie, 4 Place Jussieu, 75255 Paris Cedex, France
| | - Raphaël Voituriez
- Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600), Université Pierre et Marie Curie, 4 Place Jussieu, 75255 Paris Cedex, France and Laboratoire Jean Perrin, FRE 3231 CNRS/UPMC, 4 Place Jussieu, 75255 Paris Cedex, France
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Prostnev AS, Shub BR. Percolation mechanism of the diffusion of impurity atoms in dense surface layers. RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B 2013. [DOI: 10.1134/s1990793113040180] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
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Froemberg D, Barkai E. No-go theorem for ergodicity and an Einstein relation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:024101. [PMID: 24032966 DOI: 10.1103/physreve.88.024101] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/11/2013] [Indexed: 05/13/2023]
Abstract
We provide a simple no-go theorem for ergodicity and the generalized Einstein relation for anomalous diffusion processes. The theorem states that either ergodicity in the sense of equal time and ensemble averaged mean squared displacements (MSD) is broken, and/or the generalized Einstein relation for time averaged diffusivity and mobility is invalid, which is in complete contrast to normal diffusion processes. We also give a general relation for the time averages of drift and MSD for ergodic (in the MSD sense) anomalous diffusion processes, showing that the ratio of these quantities depends on the measurement time. The Lévy walk model is used to exemplify the no-go theorem.
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Affiliation(s)
- D Froemberg
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar Ilan University, Ramat-Gan 52900, Israel
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Illien P, Bénichou O, Mejía-Monasterio C, Oshanin G, Voituriez R. Active transport in dense diffusive single-file systems. PHYSICAL REVIEW LETTERS 2013; 111:038102. [PMID: 23909364 DOI: 10.1103/physrevlett.111.038102] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/04/2013] [Indexed: 06/02/2023]
Abstract
We study a minimal model of active transport in crowded single-file environments which generalizes the emblematic model of single-file diffusion to the case when the tracer particle (TP) performs either an autonomous directed motion or is biased by an external force, while all other particles of the environment (bath) perform unbiased diffusions. We derive explicit expressions, valid in the limit of high density of bath particles, of the full distribution P((n))(X) of the TP position and of all its cumulants, for arbitrary values of the bias f and for any time n. Our analysis reveals striking features, such as the anomalous scaling [proportionality] √[n] of all cumulants, the equality of cumulants of the same parity characteristic of a Skellam distribution and a convergence to a Gaussian distribution in spite of asymmetric density profiles of bath particles. Altogether, our results provide the full statistics of the TP position and set the basis for a refined analysis of real trajectories of active particles in crowded single-file environments.
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Affiliation(s)
- P Illien
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS UMR 7600, case courrier 121, Université Paris 6, 4 Place Jussieu, 75255 Paris Cedex, France
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Bénichou O, Mejía-Monasterio C, Oshanin G. Anomalous field-induced growth of fluctuations in dynamics of a biased intruder moving in a quiescent medium. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:020103. [PMID: 23496439 DOI: 10.1103/physreve.87.020103] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/23/2012] [Indexed: 06/01/2023]
Abstract
We present exact results on the dynamics of a biased, by an external force F, intruder (BI) in a two-dimensional lattice gas of unbiased, randomly moving hard-core particles. Going beyond the usual analysis of the force-velocity relation, we study the probability distribution P(R(n)) of the BI displacement R(n) at time n. We show that despite the fact that the BI drives the gas to a nonequilibrium steady state, P(R(n)) converges to a Gaussian distribution as n→∞. We find that the variance σ(x)(2) of P(R(n)) along F exhibits a weakly superdiffusive growth σ(x)(2)~ν(1)nln(n), and a usual diffusive growth, σ(y)(2)~ν(2)n, in the perpendicular direction. We determine ν(1) and ν(2) exactly for arbitrary bias, in the lowest order in the density of vacancies, and show that ν(1)~|F|(2) for small bias, which signifies that superdiffusive behavior emerges beyond the linear-response approximation. We also present analytical arguments predicting a striking field-induced superdiffusive behavior σ(x)(2)~n(3/2) for two-dimensional stripes and three-dimensional capillaries, which is confirmed by Monte Carlo simulations.
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Affiliation(s)
- Olivier Bénichou
- Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600), Université Pierre et Marie Curie (Paris 6), 4 Place Jussieu, 75252 Paris, France.
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Prostnev AS, Shub BR. Kinetics of the diffusion of an impurity atom on a fcc(111) surface. RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B 2012. [DOI: 10.1134/s1990793112010113] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
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Bénichou O, Meunier N, Redner S, Voituriez R. Non-Gaussianity and dynamical trapping in locally activated random walks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:021137. [PMID: 22463182 DOI: 10.1103/physreve.85.021137] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/03/2011] [Indexed: 05/31/2023]
Abstract
We propose a minimal model of locally activated diffusion, in which the diffusion coefficient of a one-dimensional Brownian particle is modified in a prescribed way-either increased or decreased-upon each crossing of the origin. Such a local mobility decrease arises in the formation of atherosclerotic plaques due to diffusing macrophage cells accumulating lipid particles. We show that spatially localized mobility perturbations have remarkable consequences on diffusion at all scales, such as the emergence of a non-Gaussian multipeaked probability distribution and a dynamical transition to an absorbing static state. In the context of atherosclerosis, this dynamical transition can be viewed as a minimal mechanism that causes macrophages to aggregate in lipid-enriched regions and thereby to the formation of atherosclerotic plaques.
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Affiliation(s)
- O Bénichou
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS UMR 7600, case courrier 121, Université Paris 6, 4 Place Jussieu, FR-75255 Paris Cedex, France
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Prostnev AS, Shub BR. Kinetics of the diffusion of an impurity atom on a square lattice. RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B 2011. [DOI: 10.1134/s1990793111020370] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
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Denisov SI, Kantz H. Continuous-time random walk with a superheavy-tailed distribution of waiting times. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:041132. [PMID: 21599140 DOI: 10.1103/physreve.83.041132] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/13/2011] [Revised: 03/23/2011] [Indexed: 05/30/2023]
Abstract
We study the long-time behavior of the probability density associated with the decoupled continuous-time random walk which is characterized by a superheavy-tailed distribution of waiting times. It is shown that, if the random walk is unbiased (biased) and the jump distribution has a finite second moment, then the properly scaled probability density converges in the long-time limit to a symmetric two-sided (an asymmetric one-sided) exponential density. The convergence occurs in such a way that all the moments of the probability density grow slower than any power of time. As a consequence, the reference random walk can be viewed as a generic model of superslow diffusion. A few examples of superheavy-tailed distributions of waiting times that give rise to qualitatively different laws of superslow diffusion are considered.
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Affiliation(s)
- S I Denisov
- Sumy State University, Rimsky-Korsakov Street 2, UA-40007 Sumy, Ukraine.
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Seki K, Wojcik M, Tachiya M. Effects of excluded volume interaction and dimensionality on diffusion-mediated reactions. J Chem Phys 2011; 134:094506. [DOI: 10.1063/1.3560419] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Eab CH, Lim SC. Fractional Langevin equations of distributed order. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:031136. [PMID: 21517483 DOI: 10.1103/physreve.83.031136] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/26/2010] [Indexed: 05/30/2023]
Abstract
Distributed-order fractional Langevin-like equations are introduced and applied to describe anomalous diffusion without unique diffusion or scaling exponent. It is shown that these fractional Langevin equations of distributed order can be used to model the kinetics of retarding subdiffusion whose scaling exponent decreases with time and the strongly anomalous ultraslow diffusion with mean square displacement which varies asymptotically as a power of logarithm of time.
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Affiliation(s)
- C H Eab
- Department of Chemistry, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand.
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Prostnev AS, Shub BR. Migration of vacancies and impurities on the (111) surface of a face-centered cubic lattice. RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B 2009. [DOI: 10.1134/s1990793109050200] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
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Seki K, Tachiya M. Reaction under vacancy-assisted diffusion at high quencher concentration. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:041120. [PMID: 19905286 DOI: 10.1103/physreve.80.041120] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/16/2009] [Indexed: 05/28/2023]
Abstract
The theory of diffusion-mediated reactions is already established for the target problem in the dilute limit, where the immobile target is surrounded by many quenchers. For lattice random walks in the crowded situation, each quencher is surrounded by other quenchers differently. As a result, each quencher migrates differently in the presence of site blocking effects. However, in the conventional theory, such difference is ignored and quenchers are assumed to move independently of each other. In this paper, theory of diffusion-mediated reactions of target problem is developed by taking into account the site blocking effects for quencher migration and the difference in the configuration of quenchers around each quencher. Our result interpolates between those in high and low limits of quencher concentrations and is a lower bound of the survival probability. In the static limit, the exact result is reproduced for a localized sink. In the presence of diffusion, the approximation is better when intrinsic reaction rates are low.
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Affiliation(s)
- Kazuhiko Seki
- National Institute of Advanced Industrial Science and Technology, Tsukuba Central 5, Tsukuba, Ibaraki 305-8565, Japan.
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