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Herbert Spohn

Publications and source records attributed to Herbert Spohn.

At least 55 records · Page 3Linked to original sources
arXiv2017

Dynamics of hard rods with initial domain wall state

Inspired by recent results on the non-equilibrium dynamics of many-body quantum systems, we study the classical hard rod problem in one dimension with initial domain wall condition. Hard rods are an integrable system, in the sense that for each velocity the density of particles is locally conserved. It was proven by Boldrighini, Dobrushin and Sukhov (1983) that on the hydrodynamic space-time scale, the fluid of hard rods satisfies Euler-type equations which comprise all conservation laws. We provide the general solution to these equations on the line, with an initial condition where the left and right halves are, asymptotically, in different states. The solution is interpreted as being composed of a continuum of contact discontinuities, one for each velocity. This is a classical counterpart of the transport problem solved recently in quantum integrable systems. We provide two independent derivations, one paralleling that in the quantum case, the other using explicitly the microscopic dynamics. Finally, we discuss the Navier-Stokes (viscous) corrections, and study its effect on the broadening of the contact discontinuity and on entropy production.

cond-mat.stat-mech↗
arXiv2017

Reflected Brownian motions in the KPZ universality class

This book presents a detailed study of a system of interacting Brownian motions in one dimension. The interaction is point-like such that the $n$-th Brownian motion is reflected from the Brownian motion with label $n-1$. This model belongs to the Kardar-Parisi-Zhang (KPZ) universality class. In fact, because of the singular interaction, many universal properties can be established with rigor. They depend on the choice of initial conditions. Discussion addresses packed and periodic initial conditions, stationary initial conditions, and mixtures thereof. The suitably scaled spatial process will be proven to converge to an Airy process in the long time limit. A chapter on determinantal random fields and another one on Airy processes are added to have the notes self-contained. This book serves as an introduction to the KPZ universality class, illustrating the main concepts by means of a single model only. It will be of interest to readers from interacting diffusion processes and non-equilibrium statistical mechanics.

math-ph↗
arXiv2016

Coupled Kardar-Parisi-Zhang Equations in One Dimension

Over the past years our understanding of the scaling properties of the solutions to the one-dimensional KPZ equation has advanced considerably, both theoretically and experimentally. In our contribution we export these insights to the case of coupled KPZ equations in one dimension. We establish equivalence with nonlinear fluctuating hydrodynamics for multi-component driven stochastic lattice gases. To check the predictions of the theory, we perform Monte Carlo simulations of the two-component AHR model. Its steady state is computed using the matrix product ansatz. Thereby all coefficients appearing in the coupled KPZ equations are deduced from the microscopic model. Time correlations in the steady state are simulated and we confirm not only the scaling exponent, but also the scaling function and the non-universal coefficients.

cond-mat.stat-mech↗
arXiv2016

On Time Correlations for KPZ Growth in One Dimension

Time correlations for KPZ growth in 1+1 dimensions are reconsidered. We discuss flat, curved, and stationary initial conditions and are interested in the covariance of the height as a function of time at a fixed point on the substrate. In each case the power laws of the covariance for short and long times are obtained. They are derived from a variational problem involving two independent Airy processes. For stationary initial conditions we derive an exact formula for the stationary covariance with two approaches: (1) the variational problem and (2) deriving the covariance of the time-integrated current at the origin for the corresponding driven lattice gas. In the stationary case we also derive the l arge time behavior for the covariance of the height gradients.

math-ph↗
arXiv2016

Shocks, rarefaction waves, and current fluctuations for anharmonic chains

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. 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.

cond-mat.stat-mech↗
arXiv2016

Nonlinear fluctuating hydrodynamics for anharmonic chains

With focus on anharmonic chains, we develop a nonlinear version of fluctuating hydrodynamics, in which the Euler currents are kept to second order in the deviations from equilibrium and dissipation plus noise are added. The required model-dependent parameters are written in such a way that they can be computed numerically within seconds, once the interaction potential, pressure, and temperature are given. In principle the theory is applicable to any one-dimensional system with local conservation laws. The resulting nonlinear stochastic field theory is handled in the one-loop approximation. Some of the large scale predictions can still be worked out analytically. For more details one has to rely on numerical simulations of the corresponding mode-coupling equations. In this way we arrive at detailed predictions for the equilibrium time correlations of the locally conserved fields of an anharmonic chain.

cond-mat.stat-mech↗
arXiv2016

Searching for the Tracy-Widom distribution in nonequilibrium processes

While originally discovered in the context of the Gaussian Unitary Ensemble, the Tracy-Widom distribution also rules the height fluctuations of growth processes. This suggests that there might be other nonequilibrium processes in which the Tracy-Widom distribution plays an important role. In our contribution we study one-dimensional systems with domain wall initial conditions. For an appropriate choice of parameters the profile develops a rarefaction wave, while maintaining the initial equilibrium states far to the left and right, which thus serve as infinitely extended thermal reservoirs. For a Fermi-Pasta-Ulam type anharmonic chain we will demonstrate that the time-integrated current has a deterministic contribution, linear in time $t$, and fluctuations of size $t^{1/3}$ with a Tracy-Widom distributed random amplitude.

cond-mat.stat-mech↗
arXiv2016

Fluctuating hydrodynamics approach to equilibrium time correlations for anharmonic chains

Linear fluctuating hydrodynamics is a useful and versatile tool for describing fluids, as well as other systems with conserved fields, on a mesoscopic scale. In one spatial dimension, however, transport is anomalous, which requires to develop a nonlinear extension of fluctuating hydrodynamics. The relevant nonlinearity turns out to be the quadratic part of the Euler currents when expanding relative to a uniform background. We outline the theory and compare with recent molecular dynamics simulations.

cond-mat.stat-mech↗
arXiv2016

The Kardar-Parisi-Zhang equation - a statistical physics perspective

The article covers the one-dimensional Kardar-Parisi-Zhang equation, weak drive limit, universality, directed polymers in a random medium, replica solutions, statistical mechanics of line ensembles, and its generalization to several components which is used to study equilibrium time correlations of anharmonic chains and of the discrete nonlinear Schrödinger equation. The notes are an extended version of my lectures at Les Houches July 2015.

cond-mat.stat-mech↗
arXiv2015

Low temperature dynamics of the one-dimensional discrete nonlinear Schrödinger equation

We study equilibrium time correlations for the discrete nonlinear Schrödinger equation on a one-dimensional lattice and unravel three dynamical regimes. There is a high temperature regime with density and energy as the only two conserved fields. Their correlations have zero velocity and spread diffusively. In the low temperature regime umklapp processes are rare with the consequence that phase differences appear as an additional (almost) conserved field. In an approximation where all umklapp is suppressed, while the equilibrium state remains untouched, one arrives at an anharmonic chain. Using the method of nonlinear fluctuating hydrodynamics we establish that the DNLS equilibrium time correlations have the same signature as a generic anharmonic chain, in particular KPZ broadening for the sound peaks and Lévy 5/3 broadening for the heat peak. In the, so far not sharply defined, ultra-low temperature regime the integrability of the dynamics becomes visible. As an illustration we simulate the completely integrable Ablowitz-Ladik model and confirm ballistic broadening of the time correlations.

cond-mat.stat-mech↗
arXiv2015

Scaling limit for Brownian motions with one-sided collisions

We consider Brownian motions with one-sided collisions, meaning that each particle is reflected at its right neighbour. For a finite number of particles a Schütz-type formula is derived for the transition probability. We investigate an infinite system with periodic initial configuration, that is, particles are located at the integer lattice at time zero. The joint distribution of the positions of a finite subset of particles is expressed as a Fredholm determinant with a kernel defining a signed determinantal point process. In the appropriate large time scaling limit, the fluctuations in the particle positions are described by the Airy$_1$ process.

math-ph↗
arXiv2015

The one-dimensional KPZ equation and its universality class

Our understanding of the one-dimensional KPZ equation, \textit{alias} noisy Burgers equation, has advanced substantially over the past five years. We provide a non-technical review, where we limit ourselves to the stochastic PDE and lattice type models approximating it.

math-ph↗
arXiv2015

Fluctuating hydrodynamics for a discrete Gross-Pitaevskii equation: mapping to Kardar-Parisi-Zhang universality class

We show that several aspects of the low-temperature hydrodynamics of a discrete Gross-Pitaevskii equation (GPE) can be understood by mapping it to a nonlinear version of fluctuating hydrodynamics. This is achieved by first writing the GPE in a hydrodynamic form of a continuity and an Euler equation. Respecting conservation laws, dissipation and noise due to the system's chaos are added, thus giving us a nonlinear stochastic field theory in general and the Kardar-Parisi-Zhang (KPZ) equation in our particular case. This mapping to KPZ is benchmarked against exact Hamiltonian numerics on discrete GPE by investigating the non-zero temperature dynamical structure factor and its scaling form and exponent. Given the ubiquity of the Gross-Pitaevskii equation (a.k.a. nonlinear Schrodinger equation), ranging from nonlinear optics to cold gases, we expect this remarkable mapping to the KPZ equation to be of paramount importance and far reaching consequences.

cond-mat.quant-gas↗
arXiv2015

Brownian Motions with One-Sided Collisions: The Stationary Case

We consider an infinite system of Brownian motions which interact through a given Brownian motion being reflected from its left neighbor. Earlier we studied this system for deterministic periodic initial configurations. In this contribution we consider initial configurations distributed according to a Poisson point process with constant intensity, which makes the process space-time stationary. We prove convergence to the Airy process for stationary the case. As a byproduct we obtain a novel representation of the finite-dimensional distributions of this process. Our method differs from the one used for the TASEP and the KPZ equation by removing the initial step only after the limit $t\to\infty$. This leads to a new universal cross-over process.

math-ph↗
arXiv2014

Quantum Boltzmann equation for spin-dependent reactions in the kinetic regime

We derive and analyze an effective quantum Boltzmann equation in the kinetic regime for the interactions of four distinguishable types of fermionic spin-$\frac{1}{2}$ particles, starting from a general quantum field Hamiltonian. Each particle type is described by a time-dependent, $2 \times 2$ spin-density ("Wigner") matrix. We show that density and energy conservation laws as well as the H-theorem hold, and enumerate additional conservation laws depending on the interaction. The conserved quantities characterize the $t \to \infty$ thermal (Fermi-Dirac) equilibrium state. We illustrate the approach to equilibrium by numerical simulations in the isotropic three-dimensional setting.

math-ph↗
arXiv2014

Current fluctuations for anharmonic chains in thermal equilibrium

We study the total current correlations for anharmonic chains in thermal equilibrium, putting forward predictions based on the second moment sum rule and on nonlinear fluctuating hydrodynamics. We compare with molecular dynamics simulations for hard collision models. For the first time we investigate the full statistics of time-integrated currents. Generically such a quantity has Gaussian statistics on a scale $\sqrt{t}$. But if the time integration has its endpoint at a moving sound peak, then the fluctuations are suppressed and only of order $t^{1/3}$. The statistics is governed by the Baik-Rains distribution, known already from the fluctuating Burgers equation.

cond-mat.stat-mech↗