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Causer L, Bañuls MC, Garrahan JP. Optimal Sampling of Dynamical Large Deviations in Two Dimensions via Tensor Networks. PHYSICAL REVIEW LETTERS 2023; 130:147401. [PMID: 37084432 DOI: 10.1103/physrevlett.130.147401] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/17/2022] [Accepted: 03/20/2023] [Indexed: 05/03/2023]
Abstract
We use projected entangled-pair states (PEPS) to calculate the large deviation statistics of the dynamical activity of the two-dimensional East model, and the two-dimensional symmetric simple exclusion process (SSEP) with open boundaries, in lattices of up to 40×40 sites. We show that at long times both models have phase transitions between active and inactive dynamical phases. For the 2D East model we find that this trajectory transition is of the first order, while for the SSEP we find indications of a second order transition. We then show how the PEPS can be used to implement a trajectory sampling scheme capable of directly accessing rare trajectories. We also discuss how the methods described here can be extended to study rare events at finite times.
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Affiliation(s)
- Luke Causer
- School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, United Kingdom
- Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, United Kingdom
| | - Mari Carmen Bañuls
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany
- Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 München
| | - Juan P Garrahan
- School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, United Kingdom
- Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, United Kingdom
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2
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Strand NE, Vroylandt H, Gingrich T. Using tensor network states for multi-particle Brownian ratchets. J Chem Phys 2022; 156:221103. [DOI: 10.1063/5.0097332] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
The study of Brownian ratchets has taught how time-periodic driving supports a time-periodic steady state that generates nonequilibrium transport. When a single particle is transported in one dimension, it is possible to rationalize the current in terms of the potential, but experimental efforts have ventured beyond that single-body case to systems with many interacting carriers. Working with a lattice model of volume-excluding particles in one dimension, we analyze the impact of interactions on a flashing ratchet's current. To surmount the many-body problem, we employ the time-dependent variational principle (TDVP) applied to binary tree tensor networks (BTTN). Rather than propagating individual trajectories, the tensor network approach propagates a distribution over many-body configurations via a controllable variational approximation. The calculations, which reproduce Gillespie trajectory sampling, identify and explain a shift in the frequency of maximum current to higher driving frequency as the lattice occupancy increases.
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Affiliation(s)
- Nils E Strand
- Chemistry, Northwestern University, United States of America
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Lazarescu A, Cossetto T, Falasco G, Esposito M. Large deviations and dynamical phase transitions in stochastic chemical networks. J Chem Phys 2019. [DOI: 10.1063/1.5111110] [Citation(s) in RCA: 27] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/27/2023] Open
Affiliation(s)
| | - Tommaso Cossetto
- Complex Systems and Statistical Mechanics, Physics and Material Science Research Unit, University of Luxembourg, L-1511 Luxembourg, Luxembourg
| | - Gianmaria Falasco
- Complex Systems and Statistical Mechanics, Physics and Material Science Research Unit, University of Luxembourg, L-1511 Luxembourg, Luxembourg
| | - Massimiliano Esposito
- CPHT, CNRS, École Polytechnique, IP Paris, F-91128 Palaiseau, France
- Complex Systems and Statistical Mechanics, Physics and Material Science Research Unit, University of Luxembourg, L-1511 Luxembourg, Luxembourg
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Pérez-Espigares C, Hurtado PI. Sampling rare events across dynamical phase transitions. CHAOS (WOODBURY, N.Y.) 2019; 29:083106. [PMID: 31472495 DOI: 10.1063/1.5091669] [Citation(s) in RCA: 16] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/04/2019] [Accepted: 07/16/2019] [Indexed: 06/10/2023]
Abstract
Interacting particle systems with many degrees of freedom may undergo phase transitions to sustain atypical fluctuations of dynamical observables such as the current or the activity. In some cases, this leads to symmetry-broken space-time trajectories which enhance the probability of such events due to the emergence of ordered structures. Despite their conceptual and practical importance, these dynamical phase transitions (DPTs) at the trajectory level are difficult to characterize due to the low probability of their occurrence. However, during the last decade, advanced computational techniques have been developed to measure rare events in simulations of many-particle systems that allow the direct observation and characterization of these DPTs. Here we review the application of a particular rare-event simulation technique, based on cloning Monte Carlo methods, to characterize DPTs in paradigmatic stochastic lattice gases. In particular, we describe in detail some tricks and tips of the trade, paying special attention to the measurement of order parameters capturing the physics of the different DPTs, as well as to the finite-size effects (both in the system size and in the number of clones) that affect the measurements. Overall, we provide a consistent picture of the phenomenology associated with DPTs and their measurement.
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Affiliation(s)
- Carlos Pérez-Espigares
- Departamento de Electromagnetismo y Física de la Materia, and Institute Carlos I for Theoretical and Computational Physics, Universidad de Granada, Granada 18071, Spain
| | - Pablo I Hurtado
- Departamento de Electromagnetismo y Física de la Materia, and Institute Carlos I for Theoretical and Computational Physics, Universidad de Granada, Granada 18071, Spain
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Garrido PL, Hurtado PI, Tizón-Escamilla N. Infinite family of universal profiles for heat current statistics in Fourier's law. Phys Rev E 2019; 99:022134. [PMID: 30934242 DOI: 10.1103/physreve.99.022134] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/25/2018] [Indexed: 06/09/2023]
Abstract
Using tools from large deviation theory, we study fluctuations of the heat current in a model of d-dimensional incompressible fluid driven out of equilibrium by a temperature gradient. We find that the most probable temperature fields sustaining atypical values of the global current can be naturally classified in an infinite set of curves, allowing us to exhaustively analyze their topological properties and to define universal profiles onto which all optimal fields collapse. We also compute the statistics of empirical heat current, where we find remarkable logarithmic tails for large current fluctuations orthogonal to the thermal gradient. Finally, we determine explicitly a number of cumulants of the current distribution, finding interesting relations between them.
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Affiliation(s)
- P L Garrido
- Departamento de Electromagnetismo y Física de la Materia, and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, E-18071 Granada. Spain
| | - P I Hurtado
- Departamento de Electromagnetismo y Física de la Materia, and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, E-18071 Granada. Spain
| | - N Tizón-Escamilla
- Departamento de Electromagnetismo y Física de la Materia, and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, E-18071 Granada. Spain
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Tizón-Escamilla N, Hurtado PI, Garrido PL. Structure of the optimal path to a fluctuation. Phys Rev E 2017; 95:032119. [PMID: 28415174 DOI: 10.1103/physreve.95.032119] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/08/2016] [Indexed: 11/07/2022]
Abstract
Macroscopic fluctuations have become an essential tool to understand physics far from equilibrium due to the link between their statistics and nonequilibrium ensembles. The optimal path leading to a fluctuation encodes key information on this problem, shedding light on, e.g., the physics behind the enhanced probability of rare events out of equilibrium, the possibility of dynamic phase transitions, and new symmetries. This makes the understanding of the properties of these optimal paths a central issue. Here we derive a fundamental relation which strongly constrains the architecture of these optimal paths for general d-dimensional nonequilibrium diffusive systems, and implies a nontrivial structure for the dominant current vector fields. Interestingly, this general relation (which encompasses and explains previous results) makes manifest the spatiotemporal nonlocality of the current statistics and the associated optimal trajectories.
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Affiliation(s)
- N Tizón-Escamilla
- Departamento de Electromagnetismo y Física de la Materia, and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, E-18071 Granada, Spain
| | - P I Hurtado
- Departamento de Electromagnetismo y Física de la Materia, and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, E-18071 Granada, Spain
| | - P L Garrido
- Departamento de Electromagnetismo y Física de la Materia, and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, E-18071 Granada, Spain
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Shpielberg O, Don Y, Akkermans E. Numerical study of continuous and discontinuous dynamical phase transitions for boundary-driven systems. Phys Rev E 2017; 95:032137. [PMID: 28415355 DOI: 10.1103/physreve.95.032137] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/05/2017] [Indexed: 06/07/2023]
Abstract
The existence and search for thermodynamic phase transitions is of unfading interest. In this paper, we present numerical evidence of dynamical phase transitions occurring in boundary-driven systems with a constrained integrated current. It is shown that certain models exhibit a discontinuous transition between two different density profiles and a continuous transition between a time-independent and a time-dependent profile. We also verified that the Kipnis-Marchioro-Presutti model exhibits no phase transitions in a range much larger than previously explored.
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Affiliation(s)
- Ohad Shpielberg
- Laboratoire de Physique Théorique, École Normale Supérieure and CNRS, 75005 Paris, France
- Physics Department, Technion-Israel Institute of Technology, 3200003 Haifa, Israel
| | - Yaroslav Don
- Physics Department, Technion-Israel Institute of Technology, 3200003 Haifa, Israel
| | - Eric Akkermans
- Physics Department, Technion-Israel Institute of Technology, 3200003 Haifa, Israel
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Baek Y, Kafri Y, Lecomte V. Dynamical Symmetry Breaking and Phase Transitions in Driven Diffusive Systems. PHYSICAL REVIEW LETTERS 2017; 118:030604. [PMID: 28157352 DOI: 10.1103/physrevlett.118.030604] [Citation(s) in RCA: 30] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/21/2016] [Indexed: 06/06/2023]
Abstract
We study the probability distribution of a current flowing through a diffusive system connected to a pair of reservoirs at its two ends. Sufficient conditions for the occurrence of a host of possible phase transitions both in and out of equilibrium are derived. These transitions manifest themselves as singularities in the large deviation function, resulting in enhanced current fluctuations. Microscopic models which implement each of the scenarios are presented, with possible experimental realizations. Depending on the model, the singularity is associated either with a particle-hole symmetry breaking, which leads to a continuous transition, or in the absence of the symmetry with a first-order phase transition. An exact Landau theory which captures the different singular behaviors is derived.
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Affiliation(s)
- Yongjoo Baek
- Department of Physics, Technion, Haifa 32000, Israel
| | - Yariv Kafri
- Department of Physics, Technion, Haifa 32000, Israel
| | - Vivien Lecomte
- LIPhy, Université Grenoble Alpes and CNRS, F-38042 Grenoble, France
- Laboratoire Probabilités et Modèles Aléatoires, UMR7599 CNRS, Sorbonne Paris Cité, Université Pierre et Marie Curie and Université Paris Diderot, F-75013 Paris, France
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Shpielberg O, Akkermans E. Le Chatelier Principle for Out-of-Equilibrium and Boundary-Driven Systems: Application to Dynamical Phase Transitions. PHYSICAL REVIEW LETTERS 2016; 116:240603. [PMID: 27367375 DOI: 10.1103/physrevlett.116.240603] [Citation(s) in RCA: 18] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/20/2015] [Indexed: 06/06/2023]
Abstract
A stability analysis is presented for boundary-driven and out-of-equilibrium systems in the framework of the hydrodynamic macroscopic fluctuation theory. A Hamiltonian description is proposed which allows us to thermodynamically interpret the additivity principle. A necessary and sufficient condition for the validity of the additivity principle is obtained as an extension of the Le Chatelier principle. These stability conditions result from a diagonal quadratic form obtained using the cumulant generating function. This approach allows us to provide a proof for the stability of the weakly asymmetric exclusion process and to reduce the search for stability to the solution of two coupled linear ordinary differential equations instead of nonlinear partial differential equations. Additional potential applications of these results are discussed in the realm of classical and quantum systems.
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Affiliation(s)
- O Shpielberg
- Department of Physics, Technion Israel Institute of Technology, Haifa 32000, Israel
| | - E Akkermans
- Department of Physics, Technion Israel Institute of Technology, Haifa 32000, Israel
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Pietzonka P, Barato AC, Seifert U. Universal bounds on current fluctuations. Phys Rev E 2016; 93:052145. [PMID: 27300867 DOI: 10.1103/physreve.93.052145] [Citation(s) in RCA: 95] [Impact Index Per Article: 11.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/25/2015] [Indexed: 05/18/2023]
Abstract
For current fluctuations in nonequilibrium steady states of Markovian processes, we derive four different universal bounds valid beyond the Gaussian regime. Different variants of these bounds apply to either the entropy change or any individual current, e.g., the rate of substrate consumption in a chemical reaction or the electron current in an electronic device. The bounds vary with respect to their degree of universality and tightness. A universal parabolic bound on the generating function of an arbitrary current depends solely on the average entropy production. A second, stronger bound requires knowledge both of the thermodynamic forces that drive the system and of the topology of the network of states. These two bounds are conjectures based on extensive numerics. An exponential bound that depends only on the average entropy production and the average number of transitions per time is rigorously proved. This bound has no obvious relation to the parabolic bound but it is typically tighter further away from equilibrium. An asymptotic bound that depends on the specific transition rates and becomes tight for large fluctuations is also derived. This bound allows for the prediction of the asymptotic growth of the generating function. Even though our results are restricted to networks with a finite number of states, we show that the parabolic bound is also valid for three paradigmatic examples of driven diffusive systems for which the generating function can be calculated using the additivity principle. Our bounds provide a general class of constraints for nonequilibrium systems.
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Affiliation(s)
- Patrick Pietzonka
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Andre C Barato
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany
| | - Udo Seifert
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
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Pérez-Espigares C, Garrido PL, Hurtado PI. Weak additivity principle for current statistics in d dimensions. Phys Rev E 2016; 93:040103. [PMID: 27176236 DOI: 10.1103/physreve.93.040103] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/27/2015] [Indexed: 06/05/2023]
Abstract
The additivity principle (AP) allows one to compute the current distribution in many one-dimensional nonequilibrium systems. Here we extend this conjecture to general d-dimensional driven diffusive systems, and validate its predictions against both numerical simulations of rare events and microscopic exact calculations of three paradigmatic models of diffusive transport in d=2. Crucially, the existence of a structured current vector field at the fluctuating level, coupled to the local mobility, turns out to be essential to understand current statistics in d>1. We prove that, when compared to the straightforward extension of the AP to high d, the so-called weak AP always yields a better minimizer of the macroscopic fluctuation theory action for current statistics.
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Affiliation(s)
- C Pérez-Espigares
- University of Modena and Reggio Emilia, via G. Campi 213/b, 41125 Modena, Italy
| | - P L Garrido
- Institute Carlos I for Theoretical and Computational Physics and Departamento de Electromagnetismo y Física de la Materia, Universidad de Granada, 18071 Granada, Spain
| | - P I Hurtado
- Institute Carlos I for Theoretical and Computational Physics and Departamento de Electromagnetismo y Física de la Materia, Universidad de Granada, 18071 Granada, Spain
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