Shocks, Rarefaction Waves, and Current Fluctuations for Anharmonic Chains

Mendl, Christian B.; Spohn, Herbert

doi:10.1007/s10955-016-1626-5

Shocks, Rarefaction Waves, and Current Fluctuations for Anharmonic Chains

Journal Article · Tue Oct 04 00:00:00 EDT 2016 · Journal of Statistical Physics

DOI:https://doi.org/10.1007/s10955-016-1626-5· OSTI ID:1353110

^[1]; Spohn, Herbert ^[2]

Stanford Univ., CA (United States). Geballe Lab. for Advanced Materials; SLAC National Accelerator Lab., Menlo Park, CA (United States). Stanford Institute for Materials and Energy Science (SIMES)
Technische Univ. Munchen, Garching (Germany). Zentrum Mathematik and Physik Dept.

The nonequilibrium dynamics of anharmonic chains is studied by imposing an initial domain-wall state, in which the two half lattices are prepared in equilibrium with distinct parameters. Here, we analyse the Riemann problem for the corresponding Euler equations and, in specific cases, compare with molecular dynamics. Additionally, the fluctuations of time-integrated currents are investigated. In analogy with the KPZ equation, their typical fluctuations should be of size t^1/3 and have a Tracy–Widom GUE distributed amplitude. The proper extension to anharmonic chains is explained and tested through molecular dynamics. Our results are calibrated against the stochastic LeRoux lattice gas.

View Accepted Manuscript (DOE)

Research Organization:: SLAC National Accelerator Laboratory (SLAC), Menlo Park, CA (United States)

Sponsoring Organization:: USDOE; Alexander von Humboldt Foundation

Grant/Contract Number:: AC02-76SF00515

OSTI ID:: 1353110

Journal Information:: Journal of Statistical Physics, Journal Name: Journal of Statistical Physics Journal Issue: 3-4 Vol. 166; ISSN 0022-4715

Publisher:: SpringerCopyright Statement

Country of Publication:: United States

Language:: English

References (29)

Solutions in the large for nonlinear hyperbolic systems of equations Glimm, James Communications on Pure and Applied Mathematics, Vol. 18, Issue 4 https://doi.org/10.1002/cpa.3160180408	journal	November 1965
Steady States and Universal Conductance in a Quenched Luttinger Model Langmann, Edwin; Lebowitz, Joel L.; Mastropietro, Vieri Communications in Mathematical Physics, Vol. 349, Issue 2 https://doi.org/10.1007/s00220-016-2631-x	journal	May 2016
Equilibrium time-correlation functions for one-dimensional hard-point systems Mendl, Christian B.; Spohn, Herbert Physical Review E, Vol. 90, Issue 1 https://doi.org/10.1103/physreve.90.012147	journal	July 2014
Emergent Hydrodynamics in Integrable Quantum Systems Out of Equilibrium Castro-Alvaredo, Olalla A.; Doyon, Benjamin; Yoshimura, Takato Physical Review X, Vol. 6, Issue 4 https://doi.org/10.1103/physrevx.6.041065	journal	December 2016
Hyperbolic systems of conservation laws II Lax, P. D. Communications on Pure and Applied Mathematics, Vol. 10, Issue 4 https://doi.org/10.1002/cpa.3160100406	journal	January 1957
Hyperbolic Conservation Laws: An Illustrated Tutorial Bressan, Alberto Lecture Notes in Mathematics https://doi.org/10.1007/978-3-642-32160-3_2	book	September 2012
Level-spacing distributions and the Airy kernel Tracy, Craig A.; Widom, Harold Communications in Mathematical Physics, Vol. 159, Issue 1 https://doi.org/10.1007/BF02100489	journal	January 1994
Derivation of the Leroux System as the Hydrodynamic Limit of a Two-Component Lattice Gas Fritz, József; Tóth, Bálint Communications in Mathematical Physics, Vol. 249, Issue 1 https://doi.org/10.1007/s00220-004-1103-x	journal	May 2004
Asymptotics in ASEP with Step Initial Condition Tracy, Craig A.; Widom, Harold Communications in Mathematical Physics, Vol. 290, Issue 1 https://doi.org/10.1007/s00220-009-0761-0	journal	February 2009
Shape Fluctuations and Random Matrices Johansson, Kurt Communications in Mathematical Physics, Vol. 209, Issue 2 https://doi.org/10.1007/s002200050027	journal	February 2000
Coupled Kardar-Parisi-Zhang Equations in One Dimension Ferrari, Patrik L.; Sasamoto, Tomohiro; Spohn, Herbert Journal of Statistical Physics, Vol. 153, Issue 3 https://doi.org/10.1007/s10955-013-0842-5	journal	September 2013
Nonlinear Fluctuating Hydrodynamics for Anharmonic Chains Spohn, Herbert Journal of Statistical Physics, Vol. 154, Issue 5 https://doi.org/10.1007/s10955-014-0933-y	journal	February 2014
The Riemann problem for materials with nonconvex equations of state I: Isentropic flow Wendroff, Burton Journal of Mathematical Analysis and Applications, Vol. 38, Issue 2 https://doi.org/10.1016/0022-247X(72)90103-5	journal	May 1972
Level-spacing distributions and the Airy kernel Tracy, Craig A.; Widom, Harold Physics Letters B, Vol. 305, Issue 1-2 https://doi.org/10.1016/0370-2693(93)91114-3	journal	May 1993
Global solution of the cauchy problem for a class of 2 × 2 nonstrictly hyperbolic conservation laws Temple, Blake Advances in Applied Mathematics, Vol. 3, Issue 3 https://doi.org/10.1016/S0196-8858(82)80010-9	journal	September 1982
Thermalization and its mechanism for generic isolated quantum systems Rigol, Marcos; Dunjko, Vanja; Olshanii, Maxim Nature, Vol. 452, Issue 7189 https://doi.org/10.1038/nature06838	journal	April 2008
Weak and strong chaos in Fermi–Pasta–Ulam models and beyond Pettini, Marco; Casetti, Lapo; Cerruti-Sola, Monica Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 15, Issue 1 https://doi.org/10.1063/1.1849131	journal	March 2005
Spontaneous breaking of translational invariance in one-dimensional stationary states on a ring Arndt, Peter F.; Heinzel, Thomas; Rittenberg, Vladimir Journal of Physics A: Mathematical and General, Vol. 31, Issue 2 https://doi.org/10.1088/0305-4470/31/2/001	journal	January 1998
Current fluctuations for anharmonic chains in thermal equilibrium Mendl, Christian B.; Spohn, Herbert Journal of Statistical Mechanics: Theory and Experiment, Vol. 2015, Issue 3 https://doi.org/10.1088/1742-5468/2015/03/P03007	journal	March 2015
Nonequilibrium thermal transport and its relation to linear response Karrasch, C.; Ilan, R.; Moore, J. E. Physical Review B, Vol. 88, Issue 19 https://doi.org/10.1103/PhysRevB.88.195129	journal	November 2013
Nonequilibrium steady states in the quantum XXZ spin chain Sabetta, Thiago; Misguich, Grégoire Physical Review B, Vol. 88, Issue 24 https://doi.org/10.1103/PhysRevB.88.245114	journal	December 2013
Equilibrium time-correlation functions for one-dimensional hard-point systems Mendl, Christian B.; Spohn, Herbert Physical Review E, Vol. 90, Issue 1 https://doi.org/10.1103/PhysRevE.90.012147	journal	July 2014
Searching for the Tracy-Widom distribution in nonequilibrium processes Mendl, Christian B.; Spohn, Herbert Physical Review E, Vol. 93, Issue 6 https://doi.org/10.1103/PhysRevE.93.060101	journal	June 2016
Strong and Weak Thermalization of Infinite Nonintegrable Quantum Systems Bañuls, M. C.; Cirac, J. I.; Hastings, M. B. Physical Review Letters, Vol. 106, Issue 5 https://doi.org/10.1103/PhysRevLett.106.050405	journal	February 2011
Dynamic Correlators of Fermi-Pasta-Ulam Chains and Nonlinear Fluctuating Hydrodynamics Mendl, Christian B.; Spohn, Herbert Physical Review Letters, Vol. 111, Issue 23 https://doi.org/10.1103/PhysRevLett.111.230601	journal	December 2013
Expansion Potentials for Exact Far-from-Equilibrium Spreading of Particles and Energy Vasseur, Romain; Karrasch, Christoph; Moore, Joel E. Physical Review Letters, Vol. 115, Issue 26 https://doi.org/10.1103/PhysRevLett.115.267201	journal	December 2015
Breakdown of Hydrodynamics in a Simple One-Dimensional Fluid Hurtado, Pablo I. Physical Review Letters, Vol. 96, Issue 1 https://doi.org/10.1103/PhysRevLett.96.010601	journal	January 2006
The Riemann problem for fluid flow of real materials Menikoff, Ralph; Plohr, Bradley J. Reviews of Modern Physics, Vol. 61, Issue 1 https://doi.org/10.1103/RevModPhys.61.75	journal	January 1989
Current fluctuations for TASEP: A proof of the Prähofer–Spohn conjecture Ben Arous, Gérard; Corwin, Ivan The Annals of Probability, Vol. 39, Issue 1 https://doi.org/10.1214/10-aop550	journal	January 2011

Cited By (2)

Dynamics of hard rods with initial domain wall state Doyon, Benjamin; Spohn, Herbert Journal of Statistical Mechanics: Theory and Experiment, Vol. 2017, Issue 7 https://doi.org/10.1088/1742-5468/aa7abf	journal	July 2017
Bethe-Boltzmann hydrodynamics and spin transport in the XXZ chain Bulchandani, Vir B.; Vasseur, Romain; Karrasch, Christoph Physical Review B, Vol. 97, Issue 4 https://doi.org/10.1103/physrevb.97.045407	journal	January 2018

Similar Records

Nonlinear Fluctuating Hydrodynamics for the Classical XXZ Spin Chain

Journal Article · Tue Oct 22 00:00:00 EDT 2019 · Journal of Statistical Physics · OSTI ID:1802885

Harvesting thermal fluctuations: Activation process induced by a nonlinear chain in thermal equilibrium

Journal Article · Thu Jun 15 00:00:00 EDT 2000 · Journal of Chemical Physics · OSTI ID:20216748

Random paths and current fluctuations in nonequilibrium statistical mechanics

Journal Article · Tue Jul 15 00:00:00 EDT 2014 · Journal of Mathematical Physics · OSTI ID:22306196

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
Fluid dynamics
Hydrodynamic waves
Nonequilibrium thermodynamics
Statistical mechanics of classical fluids

Shocks, Rarefaction Waves, and Current Fluctuations for Anharmonic Chains

Citation Formats

References (29)

Cited By (2)

Similar Records

Related Subjects