1
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Grabsch A, Bénichou O. Tracer Diffusion beyond Gaussian Behavior: Explicit Results for General Single-File Systems. PHYSICAL REVIEW LETTERS 2024; 132:217101. [PMID: 38856256 DOI: 10.1103/physrevlett.132.217101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/24/2024] [Accepted: 04/24/2024] [Indexed: 06/11/2024]
Abstract
Single-file systems, in which particles diffuse in narrow channels while not overtaking each other, is a fundamental model for the tracer subdiffusion observed in confined geometries, such as in zeolites or carbon nanotubes. Twenty years ago, the mean squared displacement of a tracer was determined at large times, for any diffusive single-file system. Since then, for a general single-file system, even the determination of the fourth cumulant, which probes the deviation from Gaussianity, has remained an open question. Here, we fill this gap and provide an explicit formula for the fourth cumulant of an arbitrary single-file system. Our approach also allows us to quantify the perturbation induced by the tracer on its environment, encoded in the correlation profiles. These explicit results constitute a first step towards obtaining a closed equation for the correlation profiles for arbitrary single-file systems.
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Affiliation(s)
- 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
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2
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Krajenbrink A, Le Doussal P. Weak noise theory of the O'Connell-Yor polymer as an integrable discretization of the nonlinear Schrödinger equation. Phys Rev E 2024; 109:044109. [PMID: 38755892 DOI: 10.1103/physreve.109.044109] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/25/2023] [Accepted: 02/15/2024] [Indexed: 05/18/2024]
Abstract
We investigate and solve the weak noise theory for the semidiscrete O'Connell-Yor directed polymer. In the large deviation regime, the most probable evolution of the partition function obeys a classical nonlinear system which is a nonstandard discretization of the nonlinear Schrödinger equation with mixed initial-final conditions. We show that this system is integrable and find its general solution through an inverse scattering method and a non-standard Fredholm determinant framework that we develop. This allows us to obtain the large deviation rate function of the free energy of the polymer model from its conserved quantities and to study its convergence to the large deviations of the Kardar-Parisi-Zhang equation. Our model also degenerates to the classical Toda chain, which further substantiates the applicability of our Fredholm framework.
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Affiliation(s)
- Alexandre Krajenbrink
- Quantinuum, Terrington House, 13-15 Hills Road, Cambridge CB2 1NL, United Kingdom and Le Lab Quantique, 58 rue d'Hauteville, 75010 Paris, France
| | - Pierre Le Doussal
- Laboratoire de Physique de l'École Normale Supérieure, CNRS, ENS & PSL University, Sorbonne Université, Université de Paris, 75005 Paris, France
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3
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Hartmann AK, Krajenbrink A, Le Doussal P. Probing the large deviations for the beta random walk in random medium. Phys Rev E 2024; 109:024122. [PMID: 38491613 DOI: 10.1103/physreve.109.024122] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/08/2023] [Accepted: 01/11/2024] [Indexed: 03/18/2024]
Abstract
We consider a discrete-time random walk on a one-dimensional lattice with space- and time-dependent random jump probabilities, known as the beta random walk. We are interested in the probability that, for a given realization of the jump probabilities (a sample), a walker starting at the origin at time t=0 is at position beyond ξsqrt[T/2] at time T. This probability fluctuates from sample to sample and we study the large-deviation rate function, which characterizes the tails of its distribution at large time T≫1. It is argued that, up to a simple rescaling, this rate function is identical to the one recently obtained exactly by two of the authors for the continuum version of the model. That continuum model also appears in the macroscopic fluctuation theory of a class of lattice gases, e.g., in the so-called KMP model of heat transfer. An extensive numerical simulation of the beta random walk, based on an importance sampling algorithm, is found in good agreement with the detailed analytical predictions. A first-order transition in the tilted measure, predicted to occur in the continuum model, is also observed in the numerics.
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Affiliation(s)
| | - Alexandre Krajenbrink
- Quantinuum, Terrington House, 13-15 Hills Road, Cambridge CB2 1NL, United Kingdom
- Le Lab Quantique, 58 rue d'Hauteville, 75010, Paris, France
| | - Pierre Le Doussal
- Laboratoire de Physique de l'École Normale Supérieure, PSL University, CNRS, Sorbonne Universités, 24 rue Lhomond, 75231 Paris, France
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4
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Grabsch A, Berlioz T, Rizkallah P, Illien P, Bénichou O. From Particle Currents to Tracer Diffusion: Universal Correlation Profiles in Single-File Dynamics. PHYSICAL REVIEW LETTERS 2024; 132:037102. [PMID: 38307067 DOI: 10.1103/physrevlett.132.037102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/23/2023] [Revised: 09/19/2023] [Accepted: 11/27/2023] [Indexed: 02/04/2024]
Abstract
Single-file transport refers to the motion of particles in a narrow channel, such that they cannot bypass each other. This constraint leads to strong correlations between the particles, described by correlation profiles, which measure the correlation between a generic observable and the density of particles at a given position and time. They have recently been shown to play a central role in single-file systems. Up to now, these correlations have only been determined for diffusive systems in the hydrodynamic limit. Here, we consider a model of reflecting point particles on the infinite line, with a general individual stochastic dynamics. We show that the correlation profiles take a simple universal form, at arbitrary time. We illustrate our approach by the study of the integrated current of particles through the origin, and apply our results to representative models such as Brownian particles, run-and-tumble particles and Lévy flights. We further emphasise the generality of our results by showing that they also apply beyond the 1D case, and to other observables.
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Affiliation(s)
- Aurélien Grabsch
- Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée (LPTMC), 4 Place Jussieu, 75005 Paris, France
| | - Théotim Berlioz
- Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée (LPTMC), 4 Place Jussieu, 75005 Paris, France
| | - Pierre Rizkallah
- Sorbonne Université, CNRS, Physico-Chimie des Électrolytes et Nanosystèmes Interfaciaux (PHENIX), 4 Place Jussieu, 75005 Paris, France
| | - Pierre Illien
- Sorbonne Université, CNRS, Physico-Chimie des Électrolytes et Nanosystèmes Interfaciaux (PHENIX), 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
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5
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Agranov T, Cates ME, Jack RL. Tricritical Behavior in Dynamical Phase Transitions. PHYSICAL REVIEW LETTERS 2023; 131:017102. [PMID: 37478424 DOI: 10.1103/physrevlett.131.017102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/15/2022] [Revised: 03/31/2023] [Accepted: 05/23/2023] [Indexed: 07/23/2023]
Abstract
We identify a new scenario for dynamical phase transitions associated with time-integrated observables occurring in diffusive systems described by the macroscopic fluctuation theory. It is characterized by the pairwise meeting of first- and second-order bias-induced phase transition curves at two tricritical points. We formulate a simple, general criterion for its appearance and derive an exact Landau theory for the tricritical behavior. The scenario is demonstrated in three examples: the simple symmetric exclusion process biased by an activity-related structural observable; the Katz-Lebowitz-Spohn lattice gas model biased by its current; and in an active lattice gas biased by its entropy production.
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Affiliation(s)
- Tal Agranov
- DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom
| | - Michael E Cates
- DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom
| | - Robert L Jack
- DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom
- Yusuf Hamied Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
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6
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Dandekar R, Krapivsky PL, Mallick K. Dynamical fluctuations in the Riesz gas. Phys Rev E 2023; 107:044129. [PMID: 37198790 DOI: 10.1103/physreve.107.044129] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/13/2022] [Accepted: 02/10/2023] [Indexed: 05/19/2023]
Abstract
We consider an infinite system of particles on a line performing identical Brownian motions and interacting through the |x-y|^{-s} Riesz potential, causing the overdamped motion of particles. We investigate fluctuations of the integrated current and the position of a tagged particle. We show that for 0<s<1, the standard deviations of both quantities grow as t^{s/2(1+s)}. When s>1, the interactions are effectively short-ranged, and the universal subdiffusive t^{1/4} growth emerges with only amplitude depending on the exponent s. We also show that the two-time correlations of the tagged-particle position have the same form as for fractional Brownian motion.
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Affiliation(s)
- Rahul Dandekar
- Institut de Physique Théorique, CEA Paris-Saclay, F-91191 Gif-sur-Yvette Cedex, France
| | - P L Krapivsky
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
- Santa Fe Institute, Santa Fe, New Mexico 87501, USA
| | - Kirone Mallick
- Institut de Physique Théorique, CEA Paris-Saclay, F-91191 Gif-sur-Yvette Cedex, France
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7
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Grabsch A, Rizkallah P, Poncet A, Illien P, Bénichou O. Exact spatial correlations in single-file diffusion. Phys Rev E 2023; 107:044131. [PMID: 37198815 DOI: 10.1103/physreve.107.044131] [Citation(s) in RCA: 6] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/07/2023] [Accepted: 04/11/2023] [Indexed: 05/19/2023]
Abstract
Single-file diffusion refers to the motion of diffusive particles in narrow channels, so that they cannot bypass each other. This constraint leads to the subdiffusion of a tagged particle, called the tracer. This anomalous behavior results from the strong correlations that arise in this geometry between the tracer and the surrounding bath particles. Despite their importance, these bath-tracer correlations have long remained elusive, because their determination is a complex many-body problem. Recently, we have shown that, for several paradigmatic models of single-file diffusion such as the simple exclusion process, these bath-tracer correlations obey a simple exact closed equation. In this paper, we provide the full derivation of this equation, as well as an extension to another model of single-file transport: the double exclusion process. We also make the connection between our results and the ones obtained very recently by several other groups and which rely on the exact solution of different models obtained by the inverse scattering method.
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Affiliation(s)
- Aurélien Grabsch
- Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée (LPTMC), 4 Place Jussieu, 75005 Paris, France
| | - Pierre Rizkallah
- Sorbonne Université, CNRS, Laboratoire de Physico-Chimie des Électrolytes et Nanosystèmes Interfaciaux (PHENIX), 4 Place Jussieu, 75005 Paris, France
| | - Alexis Poncet
- Université de Lyon, ENS de Lyon, Université Claude Bernard, CNRS, Laboratoire de Physique, F-69342, Lyon, France
| | - Pierre Illien
- Sorbonne Université, CNRS, Laboratoire de Physico-Chimie des Électrolytes et Nanosystèmes Interfaciaux (PHENIX), 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
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8
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Grabsch A, Rizkallah P, Illien P, Bénichou O. Driven Tracer in the Symmetric Exclusion Process: Linear Response and Beyond. PHYSICAL REVIEW LETTERS 2023; 130:020402. [PMID: 36706397 DOI: 10.1103/physrevlett.130.020402] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/28/2022] [Accepted: 12/07/2022] [Indexed: 06/18/2023]
Abstract
Tracer dynamics in the symmetric exclusion process (SEP), where hard-core particles diffuse on an infinite one-dimensional lattice, is a paradigmatic model of anomalous diffusion. While the equilibrium situation has received a lot of attention, the case where the tracer is driven by an external force, which provides a minimal model of nonequilibrium transport in confined crowded environments, remains largely unexplored. Indeed, the only available analytical results concern the means of both the position of the tracer and the lattice occupation numbers in its frame of reference and higher-order moments but only in the high-density limit. Here, we provide a general hydrodynamic framework that allows us to determine the first cumulants of the bath-tracer correlations and of the tracer's position in function of the driving force, up to quadratic order (beyond linear response). This result constitutes the first determination of the bias dependence of the variance of a driven tracer in the SEP for an arbitrary density. The framework presented here can be applied, beyond the SEP, to more general configurations of a driven tracer in interaction with obstacles in one dimension.
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Affiliation(s)
- Aurélien Grabsch
- Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée (LPTMC), 4 Place Jussieu, 75005 Paris, France
| | - Pierre Rizkallah
- Sorbonne Université, CNRS, Physico-Chimie des Électrolytes et Nanosystèmes Interfaciaux (PHENIX), 4 Place Jussieu, 75005 Paris, France
| | - Pierre Illien
- Sorbonne Université, CNRS, Physico-Chimie des Électrolytes et Nanosystèmes Interfaciaux (PHENIX), 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
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9
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Rana J, Sadhu T. Large deviations of a tracer position in the dense and the dilute limits of single-file diffusion. Phys Rev E 2023; 107:L012101. [PMID: 36797963 DOI: 10.1103/physreve.107.l012101] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/03/2022] [Accepted: 01/04/2023] [Indexed: 06/18/2023]
Abstract
We apply the macroscopic fluctuation theory to analyze the long-time statistics of the position of a tracer in the dense and the dilute limits of diffusive single-file systems. Our explicit results are about the corresponding large deviation functions for an initial step density profile with the fluctuating (annealed) and the fixed (quenched) initial conditions. These hydrodynamic results are applicable for a general single-file system and they agree with recent exact results obtained by microscopic solutions for specific model systems.
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Affiliation(s)
- Jagannath Rana
- Department of Theoretical Physics, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Mumbai 400005, India
| | - Tridib Sadhu
- Department of Theoretical Physics, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Mumbai 400005, India
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10
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Banerjee T, Jack RL, Cates ME. Role of initial conditions in one-dimensional diffusive systems: Compressibility, hyperuniformity, and long-term memory. Phys Rev E 2022; 106:L062101. [PMID: 36671167 DOI: 10.1103/physreve.106.l062101] [Citation(s) in RCA: 6] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/23/2022] [Accepted: 11/04/2022] [Indexed: 06/17/2023]
Abstract
We analyze the long-lasting effects of initial conditions on dynamical fluctuations in one-dimensional diffusive systems. We consider the mean-squared displacement of tracers in homogeneous systems with single-file diffusion, and current fluctuations for noninteracting diffusive particles. In each case we show analytically that the long-term memory of initial conditions is mediated by a single static quantity: a generalized compressibility that quantifies the density fluctuations of the initial state. We thereby identify a universality class of hyperuniform initial states whose dynamical variances coincide with the quenched cases studied previously, alongside a continuous family of other classes among which equilibrated (or annealed) initial conditions are but one member. We verify our predictions through extensive Monte Carlo simulations.
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Affiliation(s)
- Tirthankar Banerjee
- DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
| | - Robert L Jack
- DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
- Yusuf Hamied Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
| | - Michael E Cates
- DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
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11
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Smith NR. Exact short-time height distribution and dynamical phase transition in the relaxation of a Kardar-Parisi-Zhang interface with random initial condition. Phys Rev E 2022; 106:044111. [PMID: 36397488 DOI: 10.1103/physreve.106.044111] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/21/2022] [Accepted: 09/23/2022] [Indexed: 06/16/2023]
Abstract
We consider the relaxation (noise-free) statistics of the one-point height H=h(x=0,t), where h(x,t) is the evolving height of a one-dimensional Kardar-Parisi-Zhang (KPZ) interface, starting from a Brownian (random) initial condition. We find that, at short times, the distribution of H takes the same scaling form -lnP(H,t)=S(H)/sqrt[t] as the distribution of H for the KPZ interface driven by noise, and we find the exact large-deviation function S(H) analytically. At a critical value H=H_{c}, the second derivative of S(H) jumps, signaling a dynamical phase transition (DPT). Furthermore, we calculate exactly the most likely history of the interface that leads to a given H, and show that the DPT is associated with spontaneous breaking of the mirror symmetry x↔-x of the interface. In turn, we find that this symmetry breaking is a consequence of the nonconvexity of a large-deviation function that is closely related to S(H), and describes a similar problem but in half space. Moreover, the critical point H_{c} is related to the inflection point of the large-deviation function of the half-space problem.
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Affiliation(s)
- Naftali R Smith
- Department of Solar Energy and Environmental Physics, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, 8499000, Israel
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12
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Mallick K, Moriya H, Sasamoto T. Exact Solution of the Macroscopic Fluctuation Theory for the Symmetric Exclusion Process. PHYSICAL REVIEW LETTERS 2022; 129:040601. [PMID: 35939019 DOI: 10.1103/physrevlett.129.040601] [Citation(s) in RCA: 18] [Impact Index Per Article: 9.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/20/2022] [Accepted: 06/24/2022] [Indexed: 06/15/2023]
Abstract
We present the first exact solution for the time-dependent equations of the macroscopic fluctuation theory (MFT) for the symmetric simple exclusion process by combining a generalization of the canonical Cole-Hopf transformation with the inverse scattering method. For the step initial condition with two densities, we obtain exact and compact formulas for the optimal density profile and the response field that produce a required fluctuation, both at initial and final times. The large deviation function of the current is derived and coincides with the formula obtained previously by microscopic calculations. This provides the first analytic confirmation of the validity of the MFT for an interacting model in the time-dependent regime.
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Affiliation(s)
- Kirone Mallick
- Institut de Physique Théorique, CEA, CNRS, Université Paris-Saclay, F-91191 Gif-sur-Yvette cedex, France
| | - Hiroki Moriya
- Department of physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan
| | - Tomohiro Sasamoto
- Department of physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan
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13
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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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