1
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Yu H, Jack RL. Competition between lanes and transient jammed clusters in driven binary mixtures. Phys Rev E 2024; 109:024123. [PMID: 38491710 DOI: 10.1103/physreve.109.024123] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/31/2023] [Accepted: 01/23/2024] [Indexed: 03/18/2024]
Abstract
We consider mixtures of oppositely driven particles, showing that their nonequilibrium steady states form lanes parallel to the drive, which coexist with transient jammed clusters where particles are temporarily immobilized. We analyze the interplay between these two types of nonequilibrium pattern formation, including their implications for macroscopic demixing perpendicular to the drive. Finite-size scaling analysis indicates that there is no critical driving force associated with demixing, which appears as a crossover in finite systems. We attribute this effect to the disruption of long-ranged order by the transient jammed clusters.
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Affiliation(s)
- Honghao Yu
- Yusuf Hamied Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
| | - Robert L Jack
- Yusuf Hamied Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
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2
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Bouchet F, Tribe R, Zaboronski O. Sample-path large deviations for stochastic evolutions driven by the square of a Gaussian process. Phys Rev E 2023; 107:034111. [PMID: 37073000 DOI: 10.1103/physreve.107.034111] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/05/2021] [Accepted: 01/30/2023] [Indexed: 04/20/2023]
Abstract
Recently, a number of physical models have emerged described by a random process with increments given by a quadratic form of a fast Gaussian process. We find that the rate function which describes sample-path large deviations for such a process can be computed from the large domain size asymptotic of a certain Fredholm determinant. The latter can be evaluated analytically using a theorem of Widom which generalizes the celebrated Szegő-Kac formula to the multidimensional case. This provides a large class of random dynamical systems with timescale separation for which an explicit sample-path large-deviation functional can be found. Inspired by problems in hydrodynamics and atmosphere dynamics, we construct a simple example with a single slow degree of freedom driven by the square of a fast multivariate Gaussian process and analyze its large-deviation functional using our general results. Even though the noiseless limit of this example has a single fixed point, the corresponding large-deviation effective potential has multiple fixed points. In other words, it is the addition of noise that leads to metastability. We use the explicit answers for the rate function to construct instanton trajectories connecting the metastable states.
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Affiliation(s)
- Freddy Bouchet
- Laboratoire de Physique ENS de Lyon and CNRS, 46 Alley d'Italie, F-69364 Lyon Cedex 07, France
| | - Roger Tribe
- Department of Mathematics, University of Warwick, Coventry CV4 7AL, United Kingdom
| | - Oleg Zaboronski
- Department of Mathematics, University of Warwick, Coventry CV4 7AL, United Kingdom
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3
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Gagrani P, Smith E. Action functional gradient descent algorithm for estimating escape paths in stochastic chemical reaction networks. Phys Rev E 2023; 107:034305. [PMID: 37072963 DOI: 10.1103/physreve.107.034305] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/17/2022] [Accepted: 02/21/2023] [Indexed: 04/20/2023]
Abstract
We first derive the Hamilton-Jacobi theory underlying continuous-time Markov processes, and then we use the construction to develop a variational algorithm for estimating escape (least improbable or first passage) paths for a generic stochastic chemical reaction network that exhibits multiple fixed points. The design of our algorithm is such that it is independent of the underlying dimensionality of the system, the discretization control parameters are updated toward the continuum limit, and there is an easy-to-calculate measure for the correctness of its solution. We consider several applications of the algorithm and verify them against computationally expensive means such as the shooting method and stochastic simulation. While we employ theoretical techniques from mathematical physics, numerical optimization and chemical reaction network theory, we hope that our work finds practical applications with an inter-disciplinary audience including chemists, biologists, optimal control theorists and game theorists.
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Affiliation(s)
- Praful Gagrani
- Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA
- Wisconsin Institute for Discovery, University of Wisconsin-Madison, Madison, Wisconsin 53715, USA
| | - Eric Smith
- Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA
- Department of Biology, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
- Earth-Life Science Institute, Tokyo Institute of Technology, Tokyo 152-8550, Japan
- Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
- Ronin Institute, 127 Haddon Place, Montclair, New Jersey 07043, USA
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4
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Onuki Y. Irreversible energy extraction from negative-temperature two-dimensional turbulence. Phys Rev E 2022; 106:064131. [PMID: 36671150 DOI: 10.1103/physreve.106.064131] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/12/2022] [Accepted: 11/09/2022] [Indexed: 06/17/2023]
Abstract
The formation and transition of patterns of two-dimensional turbulent flows observed in various geophysical systems are commonly explained in terms of statistical mechanics. Different from ordinary systems, for a two-dimensional flow, the absolute temperature defined for a statistical equilibrium can take negative values. In a state of negative temperature, the second law of thermodynamics predicts that energy in microscopic fluctuations is irreversibly converted to a macroscopic form. This study explores the possibility of this one-way energy conversion in a two-dimensional flow using a basic conceptual model. We consider an inviscid incompressible fluid contained in a bounded domain, the shape of which is distorted by an externally imposed force. Unlike the usual fixed boundary cases, the flow energy within the domain is exchanged with the external system via pressure work through the moving lateral boundary. Concurrently, the flow field remains constrained by vorticity conservation. Beginning from a state of Kraichnan's grand-canonical ensemble, when the domain shape is distorted from one shape to another in a finite time, the Jarzynski equality is established. This equality states that, on average, the direction of a net energy flow through the boundary during a cycle of domain distortion changes with the sign of the initial temperature of the system. Numerical experiments are carried out to verify this theoretical argument and to investigate the parameter dependence of the energy exchange rate.
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Affiliation(s)
- Yohei Onuki
- Research Institute for Applied Mechanics, Kyushu University, 6-1 Kasuga-koen, Kasuga, Fukuoka, Japan
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5
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Fuchs A, Herbert C, Rolland J, Wächter M, Bouchet F, Peinke J. Instantons and the Path to Intermittency in Turbulent Flows. PHYSICAL REVIEW LETTERS 2022; 129:034502. [PMID: 35905359 DOI: 10.1103/physrevlett.129.034502] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/14/2021] [Revised: 02/23/2022] [Accepted: 06/10/2022] [Indexed: 06/15/2023]
Abstract
Processes leading to anomalous fluctuations in turbulent flows, referred to as intermittency, are still challenging. We consider cascade trajectories through scales as realizations of a stochastic Langevin process for which multiplicative noise is an intrinsic feature of the turbulent state. The trajectories are conditioned on their entropy exchange. Such selected trajectories concentrate around an optimal path, called instanton, which is the minimum of an effective action. The action is derived from the Langevin equation, estimated from measured data. In particular instantons with negative entropy pinpoint the trajectories responsible for the emergence of non-Gaussian statistics at small scales.
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Affiliation(s)
- A Fuchs
- Institute of Physics and ForWind, University of Oldenburg, Küpkersweg 70, 26129 Oldenburg, Germany
| | - C Herbert
- Ens de Lyon, CNRS, Laboratoire de Physique, F-69364 Lyon, France
| | - J Rolland
- Université de Lille, CNRS, ONERA, Arts et Métiers Institute of Technology, Centrale Lille, UMR 9014 - LMFL - Laboratoire de Mécanique des fluides de Lille - Kampé de Fériet, F-59000 Lille, France
| | - M Wächter
- Institute of Physics and ForWind, University of Oldenburg, Küpkersweg 70, 26129 Oldenburg, Germany
| | - F Bouchet
- Ens de Lyon, CNRS, Laboratoire de Physique, F-69364 Lyon, France
| | - J Peinke
- Institute of Physics and ForWind, University of Oldenburg, Küpkersweg 70, 26129 Oldenburg, Germany
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6
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Gomé S, Tuckerman LS, Barkley D. Extreme events in transitional turbulence. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2022; 380:20210036. [PMID: 35527637 DOI: 10.1098/rsta.2021.0036] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/22/2021] [Accepted: 01/04/2022] [Indexed: 06/14/2023]
Abstract
Transitional localized turbulence in shear flows is known to either decay to an absorbing laminar state or to proliferate via splitting. The average passage times from one state to the other depend super-exponentially on the Reynolds number and lead to a crossing Reynolds number above which proliferation is more likely than decay. In this paper, we apply a rare-event algorithm, Adaptative Multilevel Splitting, to the deterministic Navier-Stokes equations to study transition paths and estimate large passage times in channel flow more efficiently than direct simulations. We establish a connection with extreme value distributions and show that transition between states is mediated by a regime that is self-similar with the Reynolds number. The super-exponential variation of the passage times is linked to the Reynolds number dependence of the parameters of the extreme value distribution. Finally, motivated by instantons from Large Deviation theory, we show that decay or splitting events approach a most-probable pathway. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'.
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Affiliation(s)
- Sébastien Gomé
- Laboratoire de Physique et Mécanique des Milieux Hétérogènes, CNRS, ESPCI Paris, PSL Research University, Sorbonne Université, Université de Paris, Paris 75005, France
| | - Laurette S Tuckerman
- Laboratoire de Physique et Mécanique des Milieux Hétérogènes, CNRS, ESPCI Paris, PSL Research University, Sorbonne Université, Université de Paris, Paris 75005, France
| | - Dwight Barkley
- Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
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Yuan S, Blömker D, Duan J. Stochastic turbulence for Burgers equation driven by cylindrical Lévy process. STOCH DYNAM 2021. [DOI: 10.1142/s0219493722400044] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
This work is devoted to investigating stochastic turbulence for the fluid flow in one-dimensional viscous Burgers equation perturbed by Lévy space-time white noise with the periodic boundary condition. We rigorously discuss the regularity of solutions and their statistical quantities in this stochastic dynamical system. The quantities include moment estimate, structure function and energy spectrum of the turbulent velocity field. Furthermore, we provide qualitative and quantitative properties of the stochastic Burgers equation when the kinematic viscosity [Formula: see text] tends towards zero. The inviscid limit describes the strong stochastic turbulence.
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Affiliation(s)
- Shenglan Yuan
- Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany
- Center for Mathematical Sciences, Huazhong University of Science and Technology, 430074 Wuhan, P. R. China
| | - Dirk Blömker
- Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany
| | - Jinqiao Duan
- Center for Mathematical Sciences, Huazhong University of Science and Technology, 430074 Wuhan, P. R. China
- Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Illinois 60616, USA
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Galfi VM, Lucarini V. Fingerprinting Heatwaves and Cold Spells and Assessing Their Response to Climate Change Using Large Deviation Theory. PHYSICAL REVIEW LETTERS 2021; 127:058701. [PMID: 34397226 DOI: 10.1103/physrevlett.127.058701] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/07/2020] [Accepted: 06/30/2021] [Indexed: 06/13/2023]
Abstract
Extreme events provide relevant insights into the dynamics of climate and their understanding is key for mitigating the impact of climate variability and climate change. By applying large deviation theory to a state-of-the-art Earth system model, we define the climatology of persistent heatwaves and cold spells in key target geographical regions by estimating the rate functions for the surface temperature, and we assess the impact of increasing CO_{2} concentration on such persistent anomalies. Hence, we can better quantify the increasing hazard due to heatwaves in a warmer climate. We show that two 2010 high impact events-summer Russian heatwave and winter Dzud in Mongolia-are associated with atmospheric patterns that are exceptional compared to the typical ones but typical compared to the climatology of extremes. Their dynamics is encoded in the natural variability of the climate. Finally, we propose and test an approximate formula for the return times of large and persistent temperature fluctuations from easily accessible statistical properties.
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Affiliation(s)
- Vera Melinda Galfi
- Department of Earth Sciences, Uppsala University, Uppsala, 75236, Sweden
| | - Valerio Lucarini
- Department of Mathematics and Statistics & Centre for the Mathematics of Planet Earth, University of Reading, Reading, RG66AX, United Kingdom
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Wang Q, Verzicco R, Lohse D, Shishkina O. Multiple States in Turbulent Large-Aspect-Ratio Thermal Convection: What Determines the Number of Convection Rolls? PHYSICAL REVIEW LETTERS 2020; 125:074501. [PMID: 32857539 DOI: 10.1103/physrevlett.125.074501] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/03/2020] [Accepted: 07/20/2020] [Indexed: 06/11/2023]
Abstract
Wall-bounded turbulent flows can take different statistically stationary turbulent states, with different transport properties, even for the very same values of the control parameters. What state the system takes depends on the initial conditions. Here we analyze the multiple states in large-aspect ratio (Γ) two-dimensional turbulent Rayleigh-Bénard flow with no-slip plates and horizontally periodic boundary conditions as model system. We determine the number n of convection rolls, their mean aspect ratios Γ_{r}=Γ/n, and the corresponding transport properties of the flow (i.e., the Nusselt number Nu), as function of the control parameters Rayleigh (Ra) and Prandtl number. The effective scaling exponent β in Nu∼Ra^{β} is found to depend on the realized state and thus Γ_{r}, with a larger value for the smaller Γ_{r}. By making use of a generalized Friedrichs inequality, we show that the elliptical shape of the rolls and viscous damping determine the Γ_{r} window for the realizable turbulent states. The theoretical results are in excellent agreement with our numerical finding 2/3≤Γ_{r}≤4/3, where the lower threshold is approached for the larger Ra. Finally, we show that the theoretical approach to frame Γ_{r} also works for free-slip boundary conditions.
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Affiliation(s)
- Qi Wang
- Physics of Fluids Group and Max Planck Center for Complex Fluid Dynamics, MESA+ Institute and J. M. Burgers Centre for Fluid Dynamics, University of Twente, P.O. Box 217, 7500AE Enschede, Netherlands
- Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027, China
| | - Roberto Verzicco
- Physics of Fluids Group and Max Planck Center for Complex Fluid Dynamics, MESA+ Institute and J. M. Burgers Centre for Fluid Dynamics, University of Twente, P.O. Box 217, 7500AE Enschede, Netherlands
- Dipartimento di Ingegneria Industriale, University of Rome "Tor Vergata", Via del Politecnico 1, 00133 Roma, Italy
- Gran Sasso Science Institute-Viale F. Crispi, 767100 L'Aquila, Italy
| | - Detlef Lohse
- Physics of Fluids Group and Max Planck Center for Complex Fluid Dynamics, MESA+ Institute and J. M. Burgers Centre for Fluid Dynamics, University of Twente, P.O. Box 217, 7500AE Enschede, Netherlands
- Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany
| | - Olga Shishkina
- Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany
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Woillez E, Bouchet F. Instantons for the Destabilization of the Inner Solar System. PHYSICAL REVIEW LETTERS 2020; 125:021101. [PMID: 32701322 DOI: 10.1103/physrevlett.125.021101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/24/2019] [Revised: 05/25/2020] [Accepted: 06/17/2020] [Indexed: 06/11/2023]
Abstract
For rare events, path probabilities often concentrate close to a predictable path, called instanton. First developed in statistical physics and field theory, instantons are action minimizers in a path integral representation. For chaotic deterministic systems, where no such action is known, shall we expect path probabilities to concentrate close to an instanton? We address this question for the dynamics of the terrestrial bodies of the Solar System. It is known that the destabilization of the inner Solar System might occur with a low probability, within a few hundred million years, or billion years, through a resonance between the motions of Mercury and Jupiter perihelia. In a simple deterministic model of Mercury dynamics, we show that the first exit time of such a resonance can be computed. We predict the related instanton and demonstrate that path probabilities actually concentrate close to this instanton, for events which occur within a few hundred million years. We discuss the possible implications for the actual Solar System.
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Affiliation(s)
- Eric Woillez
- Université Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France
- Department of Physics, Technion, Haifa 32000, Israel
| | - Freddy Bouchet
- Université Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France
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Bouchet F, Rolland J, Wouters J. Rare Event Sampling Methods. CHAOS (WOODBURY, N.Y.) 2019; 29:080402. [PMID: 31472513 DOI: 10.1063/1.5120509] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2019] [Accepted: 07/18/2019] [Indexed: 06/10/2023]
Affiliation(s)
- Freddy Bouchet
- Laboratoire de Physique, Ens de Lyon, Univ Claude Bernard, Univ Lyon, CNRS, F-69342 Lyon, France
| | - Joran Rolland
- Laboratoire de Physique, Ens de Lyon, Univ Claude Bernard, Univ Lyon, CNRS, F-69342 Lyon, France
| | - Jeroen Wouters
- Department of Mathematics and Statistics, University of Reading, Reading, United Kingdom
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