arXiv subjects

Herbert Spohn

Publications and source records attributed to Herbert Spohn.

At least 19 recordsLinked to original sources

arXiv2026

Quasiparticle Diffusion for the Toda Fluid in Equilibrium

Many-body integrable systems can be understood as a gas of quasiparticles. They propagate ballistically and drive large-scale transport. However, with the exception of the hard rods system, no tools have been available to numerically track such quasiparticles. Focusing on the Toda fluid, whose integrability relies on the availability of a Lax pair, we present a numerical scheme to track quasiparticle trajectories as determined by the time-dependent eigenvectors of the Lax matrix. Simulating the Toda fluid in thermal equilibrium, this tracking scheme is used to numerical confirm Brownian motion of a quasiparticle. Simulated is also the motion of a tagged particle. Our numerical results for the diffusion constant matches with a novel TBA prediction. We believe our numerical scheme can be extended to other classical many-particle models possessing a Lax matrix.

cond-mat.stat-mech↗

arXiv2026

On the asymptotic behavior at the kinetic time of a weakly interacting Fermi gas

This paper is devoted to the dynamics of a weakly interacting Fermi gas at the kinetic time regime $t\sim \lambda^{-2}$ where $\lambda \ll 1$ is the strength of the interaction potential. We prove that if the initial state is close to equilibrium, then the two-point time correlation function of the many-body quantum dynamics can be computed effectively. In fact, we show that its leading order behavior is determined completely by the collisional frequency of the Boltzmann-Nordheim collision operator at equilibrium. This settles a prediction by Lukkarinen-Spohn, and thus gives a justification of the quantum Boltzmann equation from many-body quantum mechanics.

math-ph↗

arXiv2025

Stochastic dynamics of quasiparticles in the hard rod gas

We consider a one-dimensional gas of hard rods, one of the simplest examples of an interacting integrable model. It is well known that the hydrodynamics of such integrable models can be understood by viewing the system as a gas of quasiparticles. Here, we explore the dynamics of individual quasiparticles for a variety of initial conditions of the background gas. The mean, variance, and two-time correlations are computed exactly and lead to a picture of quasiparticles as drifting Brownian particles. For the case of a homogeneous background, we show that the motion of two tagged quasiparticles is strongly correlated, and they move like a rigid rod at late times. Apart from a microscopic derivation based on the mapping to point particles, we provide an alternate derivation which emphasizes that quasiparticle fluctuations are related to initial phase-space fluctuations, which are carried over in time by Euler scale dynamics. For the homogeneous state, we use the Brownian motion picture to develop a Dean-Kawasaki-type fluctuating hydrodynamic theory, formally having the same structure as that derived recently by Ferrari and Olla. We discuss differences with existing proposals on the hydrodynamics of hard rods and some puzzles.

cond-mat.stat-mech↗

arXiv2025

The Popkov-Sch\"{u}tz two-lane lattice gas: Universality for general jump rates

We consider the asymmetric version of the Popkov-Sch\"{u}tz two-lane lattice gas with general jump rates, subject to the stationary measure being of product form. This still leaves five free parameters. At density 1/2 the eigenvalues of the flux Jacobian are degenerate. We compute the second order expansion of the average fluxes at density 1/2 and thereby identify the universality classes.

cond-mat.stat-mech↗

arXiv2025

Fixed Points and Universality Classes in Coupled Kardar-Parisi-Zhang Equations

We study coupled KPZ equations with three control parameters $X,Y,T$. These equations are used in the context of stretched polymers in a random medium, for the spacetime spin-spin correlator of the isotropic quantum Heisenberg chain, and for exciton-polariton condensates. In an earlier article we investigated merely the diagonal $X=Y$, $T=1$. Then the stationary measure is delta-correlated Gaussian and the dynamical exponent is obtained numerically to be close to $z = \tfrac{3}{2}$. We observed that the scaling functions of the dynamic correlator change smoothly when varying $X$. In this contribution, the analysis is extended to the whole $X$-$Y$-$T$ plane. Solutions are stable only if $XY \geq 0$. Based on numerical simulations, the static correlator still has rapid decay. We argue that the parameter space is foliated into distinct universality classes. They are labeled by $X$ and consist of half-planes parallel to the $Y$-$T$ plane containing the point $(X,X,1)$.

cond-mat.stat-mech↗

arXiv2024

Lax random matrices from Calogero systems

We study a class of random matrices arising from the Lax matrix structure of classical integrable systems, particularly the Calogero family of models. Our focus is the density of eigenvalues for these random matrices. The problem can be mapped to analyzing the density of eigenvalues for generalized versions of conventional random matrix ensembles, including a modified form of the log-gas. The mapping comes from the underlying integrable structure of these models. Such deep connection is confirmed by extensive Monte-Carlo simulations. Thereby we move forward not only in terms of understanding such class of random matrices arising from integrable many-body systems, but also by providing a building block for the generalized hydrodynamic description of integrable systems.

cond-mat.stat-mech↗

arXiv2024

Currents of the Ablowitz-Ladik Chain

While the currents of the defocusing Ablowitz-Ladik chain are unambiguously defined, no explicit formulas for their densities have been obtained yet. We close this gap. As a consequence, we also complete the argument which establishes the anticipated expression for average currents, averaged over a generalized Gibbs ensemble.

cond-mat.stat-mech↗

arXiv2024

Particle Scattering and Fusion for the Ablowitz-Ladik Chain

The Ablowitz-Ladik chain is an integrable discretized version of the nonlinear Schr\"{o}dinger equation. We report on a novel underlying Hamiltonian particle system with properties similar to the ones known for the classical Toda chain and Calogero fluid with $1/\sinh^2$ pair interaction. Boundary conditions are imposed such that, both in the distant past and future, particles have a constant velocity. We establish the many-particle scattering for the Ablowitz-Ladik chain and obtain properties known for generic integrable many-body systems. For a specific choice of the chain, real initial data remain real in the course of time. Then, asymptotically, particles move in pairs with a velocity-dependent size and scattering shifts are governed by the fusion rule.

cond-mat.stat-mech↗

arXiv2024

Universality in coupled stochastic Burgers systems with degenerate flux Jacobian

In our contribution we study stochastic models in one space dimension with two conservation laws. One model is the coupled continuum stochastic Burgers equation, for which each current is a sum of quadratic non-linearities, linear diffusion, and spacetime white noise. The second model is a two-lane stochastic lattice gas. As distinct from previous studies, the two conserved densities are tuned such that the flux Jacobian, a $2 \times 2$ matrix, has coinciding eigenvalues. In the steady state, investigated are spacetime correlations of the conserved fields and the time-integrated currents at the origin. For a particular choice of couplings the dynamical exponent 3/2 is confirmed. Furthermore, at these couplings, continuum stochastic Burgers equation and lattice gas are demonstrated to be in the same universality class.

cond-mat.stat-mech↗

arXiv2023

Thermalization and hydrodynamics in an interacting integrable system: the case of hard rods

We consider the relaxation of an initial non-equilibrium state in a one-dimensional fluid of hard rods. Since it is an interacting integrable system, we expect it to reach the Generalized Gibbs Ensemble (GGE) at long times for generic initial conditions. Here we show that there exist initial conditions for which the system does not reach GGE even at very long times and in the thermodynamic limit. In particular, we consider an initial condition of uniformly distributed hard-rods in a box with the left half having particles with a singular velocity distribution (all moving with unit velocity) and the right half particles in thermal equilibrium. We find that the density profile for the singular component does not spread to the full extent of the box and keeps moving with a fixed effective speed at long times. We show that such density profiles can be well described by the solution of the Euler equations almost everywhere except at the location of the shocks, where we observe slight discrepancies due to dissipation arising from the initial fluctuations of the thermal background. To demonstrate this effect of dissipation analytically, we consider a second initial condition with a single particle at the origin with unit velocity in a thermal background. We find that the probability distribution of the position of the unit velocity quasi-particle has diffusive spreading which can be understood from the solution of the Navier-Stokes equation of the hard rods. Finally, we consider an initial condition with a spread in velocity distribution for which we show convergence to GGE. Our conclusions are based on molecular dynamics simulations supported by analytical arguments.

cond-mat.stat-mech↗

arXiv2023

Nonequilibrium spin transport in integrable and non-integrable classical spin chains

Anomalous transport in low dimensional spin chains is an intriguing topic that can offer key insights into the interplay of integrability and symmetry in many-body dynamics. Recent studies have shown that spin-spin correlations in spin chains, where integrability is either perfectly preserved or broken by symmetry-preserving interactions, fall in the Kardar-Parisi-Zhang (KPZ) universality class. Similarly, energy transport can show ballistic or diffusive-like behaviour. Although such behaviour has been studied under equilibrium conditions, no results on nonequilibrium spin transport in classical spin chains has been reported so far. In this work, we investigate both spin and energy transport in classical spin chains (integrable and non-integrable) when coupled to two reservoirs at two different temperatures/magnetization. In both the integrable case and broken-integrability (but spin-symmetry preserving), we report anomalous scaling of spin current with system size ($\mathbb{J}^s \propto L^{-\mu}$) with an exponent, $\mu \approx 2/3$, falling under the KPZ universality class. On the other hand, it is noteworthy that energy current remains ballistic ($\mathbb{J}^e \propto L^{-\eta}$ with $\eta \approx 0$) in the purely integrable case and there is departure from ballistic behaviour ($\eta > 0$) when integrability is broken regardless of spin-symmetry. Under nonequilibrium conditions, we have thoroughly investigated spatial profiles of local magnetization and energy. We find interesting nonlinear spatial profiles which are hallmarks of anomalous transport. We also unravel subtle striking differences between the equilibrium and nonequilibrium steady state through the lens of spatial spin-spin correlations.

cond-mat.stat-mech↗

arXiv2023

Domain wall fluctuations of the six-vertex model at the ice point

We report on Monte-Carlo simulations of the six-vertex model with domain wall boundary conditions. In thermal equilibrium such boundary conditions force a fluctuating line separating the disordered region from the perfectly ordered ones. Specifically we study the ice point at which all vertex weights are equal. With high precision the one-point fluctuations of the line are confirmed to be of order $N^{1/3}$ and governed by the Tracy-Widom distribution. Furthermore, the non-universal scaling coefficients are computed for a wide range of interaction strengths. A draft of this paper was completed in January 2019. We improved the presentation and updated references.

cond-mat.stat-mech↗

arXiv2023

Hydrodynamic scales of integrable many-particle systems

The lecture notes cover the emergence of generalized hydrodynamics for the classical and quantum Toda chain, the classical Calogero fluid, the Ablowitz-Ladik discretization of the non-linear Schroedinger equation, and the Lieb-Liniger delta-Bose gas. Such diverse models have structurally identical hydrodynamic equations. Added is a comparison with conventional soliton gases.

cond-mat.stat-mech↗

arXiv2023

Equilibrium Spacetime Correlations of the Toda Lattice on the Hydrodynamic Scale

We report on molecular dynamics simulations of spacetime correlations of the Toda lattice in thermal equilibrium. The correlations of stretch, momentum, and energy are computed numerically over a wide range of pressure and temperature. Our numerical results are compared with the predictions from linearized generalized hydrodynamics on the Euler scale. The system size is N=3000,4000 and time t =600, at which ballistic scaling is well confirmed. With no adjustable parameters, the numerically obtained scaling functions agree with the theory within a precision of less than 3.5%.

cond-mat.stat-mech↗

arXiv2022

Hard Rod Hydrodynamics and the Levy Chentsov Field

We study the hydrodynamics of the hard rod model proposed by Boldrighini, Dobrushin and Soukhov by describing the displacement of each quasiparticle with respect to the corresponding ideal gas particle as a height difference in a related field. Starting with a family of nonhomogeneous Poisson processes contained in the position-velocity-length space $\mathbb{R}^3$, we show laws of large numbers for the quasiparticle positions and the length fields, and the joint convergence of the quasiparticle fluctuations to a Levy Chentsov field. We allow variable rod lengths, including negative lengths.

math.PR↗

arXiv2022

Robustness of Kardar-Parisi-Zhang scaling in a classical integrable spin chain with broken integrability

Recent investigations have observed superdiffusion in integrable classical and quantum spin chains. An intriguing connection between these spin chains and Kardar-Parisi-Zhang (KPZ) universality class has emerged. Theoretical developments (e.g. generalized hydrodynamics) have highlighted the role of integrability as well as spin-symmetry in KPZ behaviour. However understanding their precise role on superdiffusive transport still remains a challenging task. The widely used quantum spin chain platform comes with severe numerical limitations. To circumvent this barrier, we focus on a classical integrable spin chain which was shown to have deep analogy with the quantum spin-$\frac{1}{2}$ Heisenberg chain. Remarkably, we find that KPZ behaviour prevails even when one considers integrability-breaking but spin-symmetry preserving terms, strongly indicating that spin-symmetry plays a central role even in the non-perturbative regime. On the other hand, in the non-perturbative regime, we find that energy correlations exhibit clear diffusive behaviour. We also study the classical analog of out-of-time-ordered correlator (OTOC) and Lyapunov exponents. We find significant presence of chaos for the integrability-broken cases even though KPZ behaviour remains robust. The robustness of KPZ behaviour is demonstrated for a wide class of spin-symmetry preserving integrability-breaking terms.

cond-mat.stat-mech↗

arXiv2021

Hydrodynamic equations for the Ablowitz-Ladik discretization of the nonlinear Schroedinger equation

Ablowitz and Ladik discovered a discretization which preserves the integrability of the nonlinear Schroedinger equation in one dimension. We compute the generalized free energy of this model and determine the GGE averaged fields and currents. They are linked to the mean-field circular unitary matrix ensemble (CUE). The resulting hydrodynamic equations follow the pattern already known from other integrable many-body systems. Studied is also the discretized modified Korteweg-de-Vries equation which turns out to be related to the beta Jacobi log gas.

cond-mat.stat-mech↗

arXiv2021

Form factors and generalized hydrodynamics for integrable systems

Our review covers microscopic foundations of generalized hydrodynamics (GHD). As one generic approach we develop form factor expansions, for ground states and generalized Gibbs ensembles (GGE). In the latter case the so obtained results are compared with predictions from GHD. One cornerstone of GHD are the GGE averaged microscopic currents, which can also be obtained through employing form factors. Discussed is a second, completely orthogonal approach based on the availability of a self-conserved current.

cond-mat.stat-mech↗