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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Kumar V, Pal A, Shpielberg O. Emerging universality classes in thermally assisted activation of interacting diffusive systems: A perturbative hydrodynamic approach. J Chem Phys 2024; 160:134107. [PMID: 38563303 DOI: 10.1063/5.0195570] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/03/2024] [Accepted: 03/01/2024] [Indexed: 04/04/2024] Open
Abstract
Thermal activation of a particle from a deep potential trap follows the Arrhenius law. Recently, this result has been generalized for interacting diffusive particles in the trap, revealing two universality classes-the Arrhenius class and the excluded volume class. The result was demonstrated with the aid of numerical analysis. Here, we present a perturbative hydrodynamic approach to analytically validate the existence and range of validity for the two universality classes.
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Affiliation(s)
- Vishwajeet Kumar
- The Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600113, India
- Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India
| | - Arnab Pal
- The Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600113, India
- Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India
| | - Ohad Shpielberg
- Department of Mathematics and Physics, University of Haifa at Oranim, Kiryat Tivon 3600600, Israel
- Haifa Research Center for Theoretical Physics and Astrophysics, University of Haifa, Abba Khoushy Ave. 199, Haifa 3498838, Israel
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3
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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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4
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Kumar V, Pal A, Shpielberg O. Arrhenius law for interacting diffusive systems. Phys Rev E 2024; 109:L032101. [PMID: 38632768 DOI: 10.1103/physreve.109.l032101] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/22/2023] [Accepted: 02/08/2024] [Indexed: 04/19/2024]
Abstract
Finding the mean time it takes for a particle to escape from a metastable state due to thermal fluctuations is a fundamental problem in physics, chemistry, and biology. Here, we consider the escape rate of interacting diffusive particles, from a deep potential trap within the framework of the macroscopic fluctuation theory-a nonequilibrium hydrodynamic theory. For systems without excluded volume, our investigation reveals adherence to the well-established Arrhenius law. However, in the presence of excluded volume, a universality class emerges, fundamentally altering the escape rate. Remarkably, the modified escape rate within this universality class is independent of the interactions at play. The universality class, demonstrating the importance of excluded volume effects, may bring insights to the interpretation of escape processes in the realm of chemical physics.
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Affiliation(s)
- Vishwajeet Kumar
- The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India
- Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India
| | - Arnab Pal
- The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India
- Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India
| | - Ohad Shpielberg
- Department of Mathematics and Physics, University of Haifa at Oranim, Kiryat Tivon 3600600, Israel
- Haifa Research Center for Theoretical Physics and Astrophysics, University of Haifa, Abba Khoushy Avenue 199, Haifa 3498838, Israel
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5
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Tang Y, Liu J, Zhang J, Zhang P. Learning nonequilibrium statistical mechanics and dynamical phase transitions. Nat Commun 2024; 15:1117. [PMID: 38321012 PMCID: PMC10847122 DOI: 10.1038/s41467-024-45172-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/31/2023] [Accepted: 01/15/2024] [Indexed: 02/08/2024] Open
Abstract
Nonequilibrium statistical mechanics exhibit a variety of complex phenomena far from equilibrium. It inherits challenges of equilibrium, including accurately describing the joint distribution of a large number of configurations, and also poses new challenges as the distribution evolves over time. Characterizing dynamical phase transitions as an emergent behavior further requires tracking nonequilibrium systems under a control parameter. While a number of methods have been proposed, such as tensor networks for one-dimensional lattices, we lack a method for arbitrary time beyond the steady state and for higher dimensions. Here, we develop a general computational framework to study the time evolution of nonequilibrium systems in statistical mechanics by leveraging variational autoregressive networks, which offer an efficient computation on the dynamical partition function, a central quantity for discovering the phase transition. We apply the approach to prototype models of nonequilibrium statistical mechanics, including the kinetically constrained models of structural glasses up to three dimensions. The approach uncovers the active-inactive phase transition of spin flips, the dynamical phase diagram, as well as new scaling relations. The result highlights the potential of machine learning dynamical phase transitions in nonequilibrium systems.
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Affiliation(s)
- Ying Tang
- Institute of Fundamental and Frontier Sciences, University of Electronic Sciences and Technology of China, Chengdu, 611731, China.
- International Academic Center of Complex Systems, Beijing Normal University, Zhuhai, 519087, China.
| | - Jing Liu
- CAS Key Laboratory for Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, China
- School of Systems Science, Beijing Normal University, Beijing, 100875, China
| | - Jiang Zhang
- School of Systems Science, Beijing Normal University, Beijing, 100875, China
- Swarma Research, Beijing, 102308, China
| | - Pan Zhang
- CAS Key Laboratory for Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, China.
- School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou, 310024, China.
- Hefei National Laboratory, Hefei, 230088, China.
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6
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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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7
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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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8
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Jose S, Rosso A, Ramola K. Generalized disorder averages and current fluctuations in run and tumble particles. Phys Rev E 2023; 108:L052601. [PMID: 38115454 DOI: 10.1103/physreve.108.l052601] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/15/2023] [Accepted: 11/07/2023] [Indexed: 12/21/2023]
Abstract
We present exact results for the fluctuations in the number of particles crossing the origin up to time t in a collection of noninteracting run and tumble particles in one dimension. In contrast to passive systems, such active particles are endowed with two inherent degrees of freedom, positions and velocities, which can be used to construct density and magnetization fields. We introduce generalized disorder averages associated with both these fields and perform annealed and quenched averages over various initial conditions. We show that the variance σ^{2} of the current in annealed versus quenched magnetization situations exhibits a surprising difference at short times, σ^{2}∼t vs σ^{2}∼t^{2}, respectively, with a sqrt[t] behavior emerging at large times. Our analytical results demonstrate that in the strictly quenched scenario, where both the density and magnetization fields are initially frozen, the fluctuations in the current are strongly suppressed. Importantly, these anomalous fluctuations cannot be obtained solely by freezing the density field.
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Affiliation(s)
- Stephy Jose
- Tata Institute of Fundamental Research, Hyderabad 500046, India
| | - Alberto Rosso
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Kabir Ramola
- Tata Institute of Fundamental Research, Hyderabad 500046, India
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9
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Doyon B, Perfetto G, Sasamoto T, Yoshimura T. Emergence of Hydrodynamic Spatial Long-Range Correlations in Nonequilibrium Many-Body Systems. PHYSICAL REVIEW LETTERS 2023; 131:027101. [PMID: 37505951 DOI: 10.1103/physrevlett.131.027101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/10/2022] [Revised: 05/07/2023] [Accepted: 06/20/2023] [Indexed: 07/30/2023]
Abstract
At large scales of space and time, the nonequilibrium dynamics of local observables in extensive many-body systems is well described by hydrodynamics. At the Euler scale, one assumes that each mesoscopic region independently reaches a state of maximal entropy under the constraints given by the available conservation laws. Away from phase transitions, maximal entropy states show exponential correlation decay, and independence of fluid cells might be assumed to subsist during the course of time evolution. We show that this picture is incorrect: under ballistic scaling, regions separated by macroscopic distances "develop long-range correlations as time passes." These correlations take a universal form that only depends on the Euler hydrodynamics of the model. They are rooted in the large-scale motion of interacting fluid modes and are the dominant long-range correlations developing in time from long-wavelength, entropy-maximized states. They require "the presence of interaction" and "at least two different fluid modes" and are of a fundamentally different nature from well-known long-range correlations occurring from diffusive spreading or from quasiparticle excitations produced in far-from-equilibrium quenches. We provide a universal theoretical framework to exactly evaluate them, an adaptation of the macroscopic fluctuation theory to the Euler scale. We verify our exact predictions in the hard-rod gas, by comparing with numerical simulations and finding excellent agreement.
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Affiliation(s)
- Benjamin Doyon
- Department of Mathematics, King's College London, Strand, London WC2R 2LS, United Kingdom
| | - Gabriele Perfetto
- Institut für Theoretische Physik, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany
| | - Tomohiro Sasamoto
- Department of Physics, Tokyo Institute of Technology, Ookayama 2-12-1, Tokyo 152-8551, Japan
| | - Takato Yoshimura
- All Souls College, Oxford OX1 4AL, United Kingdom
- Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, United Kingdom
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10
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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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11
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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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12
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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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13
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Cates ME, Nardini C. Classical Nucleation Theory for Active Fluid Phase Separation. PHYSICAL REVIEW LETTERS 2023; 130:098203. [PMID: 36930897 DOI: 10.1103/physrevlett.130.098203] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/11/2022] [Accepted: 02/10/2023] [Indexed: 06/18/2023]
Abstract
Classical Nucleation Theory (CNT), linking rare nucleation events to the free-energy landscape of a growing nucleus, is central to understanding phase-change kinetics in passive fluids. Nucleation in nonequilibrium systems is much harder to describe because there is no free energy, but instead a dynamics-dependent quasipotential that typically must be found numerically. Here we extend CNT to a class of active phase-separating systems governed by a minimal field-theoretic model (Active Model B+). In the small noise and supersaturation limits that CNT assumes, we compute analytically the quasipotential, and hence, nucleation barrier, for liquid-vapor phase separation. Crucial to our results, detailed balance, although broken microscopically by activity, is restored along the instanton trajectory, which in CNT involves the nuclear radius as the sole reaction coordinate.
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Affiliation(s)
- M E Cates
- DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
| | - C Nardini
- Service de Physique de l'Etat Condensé, CEA, CNRS Université Paris-Saclay, CEA-Saclay, 91191 Gif-sur-Yvette, France
- Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée, 75005 Paris, France
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14
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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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15
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Krajenbrink A, Le Doussal P. Crossover from the macroscopic fluctuation theory to the Kardar-Parisi-Zhang equation controls the large deviations beyond Einstein's diffusion. Phys Rev E 2023; 107:014137. [PMID: 36797871 DOI: 10.1103/physreve.107.014137] [Citation(s) in RCA: 7] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/26/2022] [Accepted: 12/21/2022] [Indexed: 01/30/2023]
Abstract
We study the crossover from the macroscopic fluctuation theory (MFT), which describes one-dimensional stochastic diffusive systems at late times, to the weak noise theory (WNT), which describes the Kardar-Parisi-Zhang (KPZ) equation at early times. We focus on the example of the diffusion in a time-dependent random field, observed in an atypical direction which induces an asymmetry. The crossover is described by a nonlinear system which interpolates between the derivative and the standard nonlinear Schrodinger equations in imaginary time. We solve this system using the inverse scattering method for mixed-time boundary conditions introduced by us to solve the WNT. We obtain the rate function which describes the large deviations of the sample-to-sample fluctuations of the cumulative distribution of the tracer position. It exhibits a crossover as the asymmetry is varied, recovering both MFT and KPZ limits. We sketch how it is consistent with extracting the asymptotics of a Fredholm determinant formula, recently derived for sticky Brownian motions. The crossover mechanism studied here should generalize to a larger class of models described by the MFT. Our results apply to study extremal diffusion beyond Einstein's theory.
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Affiliation(s)
| | - 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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16
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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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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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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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