Density profiles in open superdiffusive systems

Heat conduction in low-dimensional electron gases without and with a magnetic field

We investigate the behavior of heat conduction in two-dimensional (2D) electron gases without and with a magnetic field. We perform simulations with the multiparticle collision approach, suitably adapted to account for the Lorenz force acting on the particles. For zero magnetic field, we find that the heat conductivity κ diverges with the system size L following the logarithmic relation κ∼lnL (as predicted for 2D systems) for small L values; however, in the thermodynamic limit the heat conductivity tends to follow the relation κ∼L^{1/3}, as predicted for one-dimensional (1D) fluids. This suggests the presence of a dimensional-crossover effect in heat conduction in electronic systems that adhere to standard momentum conservation. Under the magnetic field, time-reversal symmetry is broken and the standard momentum conservation in the system is no longer satisfied but the pseudomomentum of the system is still conserved. In contrast with the zero-field case, both equilibrium and nonequilibrium simulations indicate a finite heat conductivity independent on the system size L as L increases. This indicates that pseudomomentum conservation can exhibit normal diffusive heat transport, which differs from the abnormal behavior observed in low-dimensional coupled charged harmonic oscillators with pseudomomentum conservation in a magnetic field. These findings support the validity of the hydrodynamic theory in electron gases and clarify that pseudomomentum conservation is not enough to ensure the anomalous behavior of heat conduction.

Universal Kardar-Parisi-Zhang transient diffusion in nonequilibrium anharmonic chains

The well known nonlinear fluctuating hydrodynamics theory has grouped diffusions in anharmonic chains into two universality classes: one is the Kardar-Parisi-Zhang (KPZ) class for chains with either asymmetric potential or nonzero static pressure and the other is the Gaussian class for chains with symmetric potential at zero static pressure, such as Fermi-Pasta-Ulam-Tsingou (FPUT)-β chains. However, little is known of the nonequilibrium transient diffusion in anharmonic chains. Here, we reveal that the KPZ class is the only universality class for nonequilibrium transient diffusion, manifested as the KPZ scaling of the side peaks of momentum correlation (corresponding to the sound modes correlation), which was completely unexpected in equilibrium FPUT-β chains. The underlying mechanism is that the nonequilibrium soliton dynamics cause nonzero transient pressure so that the sound modes satisfy approximately the noisy Burgers equation, in which the collisions of solitons was proved to yield the KPZ dynamic exponent of the soliton dispersion. Therefore, the unexpected KPZ universality class is obtained in the nonequilibrium transient diffusion in FPUT-β chains and the corresponding carriers of nonequilibrium transient diffusion are attributed to solitons.

Heat conduction in a three-dimensional momentum-conserving fluid

Size dependence of energy transport and the effects of reduced dimensionality on transport coefficients are of key importance for understanding nonequilibrium properties of matter on the nanoscale. Here, we perform nonequilibrium and equilibrium simulations of heat conduction in a three-dimensional (3D) fluid with the multiparticle collision dynamics, interacting with two thermal walls. We find that the bulk 3D momentum-conserving fluid has a finite nondiverging thermal conductivity. However, for large aspect ratios of the simulation box, a crossover from 3D to one-dimensional (1D) abnormal behavior of the thermal conductivity occurs. In this case, we demonstrate a transition from normal to abnormal transport by a suitable decomposition of the energy current. These results not only provide a direct verification of Fourier's law, but also further confirm the validity of existing theories for 3D fluids. Moreover, they indicate that abnormal heat transport persists also for almost 1D fluids over a large range of sizes.

Too Close to Integrable: Crossover from Normal to Anomalous Heat Diffusion

Energy transport in one-dimensional chains of particles with three conservation laws is generically anomalous and belongs to the Kardar-Parisi-Zhang dynamical universality class. Surprisingly, some examples where an apparent normal heat diffusion is found over a large range of length scales were reported. We propose a novel physical explanation of these intriguing observations. We develop a scaling analysis that explains how this may happen in the vicinity of an integrable limit, such as, but not only, the famous Toda model. In this limit, heat transport is mostly supplied by quasiparticles with a very large mean free path ℓ. Upon increasing the system size L, three different regimes can be observed: a ballistic one, an intermediate diffusive range, and, eventually, the crossover to the anomalous (hydrodynamic) regime. Our theoretical considerations are supported by numerical simulations of a gas of diatomic hard-point particles for almost equal masses and of a weakly perturbed Toda chain. Finally, we discuss the case of the perturbed harmonic chain, which exhibits a yet different scenario.

Lévy walks on finite intervals: A step beyond asymptotics

A Lévy walk of order β is studied on an interval of length L, driven out of equilibrium by different-density boundary baths. The anomalous current generated under these settings is nonlocally related to the density profile through an integral equation. While the asymptotic solution to this equation is known, its finite-L corrections remain unstudied despite their importance in the study of anomalous transport. Here a perturbative method for computing such corrections is presented and explicitly demonstrated for the leading correction to the asymptotic transport of a Lévy walk of order β=5/3, which represents a broad universal class of anomalous transport models. Surprisingly, many other physical problems are described by similar integral equations, to which the method introduced here can be directly applied.

Nonlocal transport in bounded two-dimensional systems: An iterative method

The concept of transport mediated through the dynamics of "jumping" particles is used to develop an iterative method for obtaining steady-state solutions to the nonlocal transport equation in two dimensions. The technique is self-adjoint and capable of correctly treating spatially nonuniform, asymmetric systems. An appropriate reduced version of the iteration method is used to compare with results obtained with a self-adjoint one-dimensional transport matrix approach [Maggs and Morales, Phys. Rev. E 94, 053302 (2016)10.1103/PhysRevE.94.053302]. The transport "jump" probability distribution functions are based on Lévy α-stable distributions. The technique can handle the entire Lévy α-parameter range from one (Lorentz distributions) to two (Gaussian distributions). Cases with α=2 (standard diffusion) are used to establish the validity of the iterative method. The capabilities of the iterative method are demonstrated by presenting examples from systems with various source configurations, boundary shapes, boundary conditions, and spatial variations in parameters.

Derivation of fluctuating hydrodynamics and crossover from diffusive to anomalous transport in a hard-particle gas

A recently developed nonlinear fluctuating hydrodynamics theory has been quite successful in describing various features of anomalous energy transport. However, the diffusion and the noise terms present in this theory are not derived from microscopic descriptions but rather added phenomenologically. We here derive these hydrodynamic equations with explicit calculation of the diffusion and noise terms in a one-dimensional model. We show that in this model the energy current scales anomalously with system size L as ∼L^{-2/3} in the leading order with a diffusive correction of order ∼L^{-1}. The crossover length ℓ_{c} from diffusive to anomalous transport is expressed in terms of microscopic parameters. Our theoretical predictions are verified numerically.

Solitons as candidates for energy carriers in Fermi-Pasta-Ulam lattices

Currently, effective phonons (renormalized or interacting phonons) rather than solitary waves (for short, solitons) are regarded as the energy carriers in nonlinear lattices. In this work, by using the approximate soliton solutions of the corresponding equations of motion and adopting the Boltzmann distribution for these solitons, the average velocities of solitons are obtained and are compared with the sound velocities of energy transfer. Excellent agreements with the numerical results and the predictions of other existing theories are shown in both the symmetric Fermi-Pasta-Ulam-β lattices and the asymmetric Fermi-Pasta-Ulam-αβ lattices. These clearly indicate that solitons are suitable candidates for energy carriers in Fermi-Pasta-Ulam lattices. In addition, the root-mean-square velocity of solitons can be obtained from the effective phonons theory.

Energy Current Cumulants in One-Dimensional Systems in Equilibrium

A recent theory based on fluctuating hydrodynamics predicts that one-dimensional interacting systems with particle, momentum, and energy conservation exhibit anomalous transport that falls into two main universality classes. The classification is based on behavior of equilibrium dynamical correlations of the conserved quantities. One class is characterized by sound modes with Kardar-Parisi-Zhang scaling, while the second class has diffusive sound modes. The heat mode follows Lévy statistics, with different exponents for the two classes. Here we consider heat current fluctuations in two specific systems, which are expected to be in the above two universality classes, namely, a hard particle gas with Hamiltonian dynamics and a harmonic chain with momentum conserving stochastic dynamics. Numerical simulations show completely different system-size dependence of current cumulants in these two systems. We explain this numerical observation using a phenomenological model of Lévy walkers with inputs from fluctuating hydrodynamics. This consistently explains the system-size dependence of heat current fluctuations. For the latter system, we derive the cumulant-generating function from a more microscopic theory, which also gives the same system-size dependence of cumulants.

Self-adjoint integral operator for bounded nonlocal transport

An integral operator is developed to describe nonlocal transport in a one-dimensional system bounded on both ends by material walls. The "jump" distributions associated with nonlocal transport are taken to be Lévy α-stable distributions, which become naturally truncated by the bounding walls. The truncation process results in the operator containing a self-consistent, convective inward transport term (pinch). The properties of the integral operator as functions of the Lévy distribution parameter set [α,γ] and the wall conductivity are presented. The integral operator continuously recovers the features of local transport when α=2. The self-adjoint formulation allows for an accurate description of spatial variation in the Lévy parameters in the nonlocal system. Spatial variation in the Lévy parameters is shown to result in internally generated flows. Examples of cold-pulse propagation in nonlocal systems illustrate the capabilities of the methodology.

Numerical test of hydrodynamic fluctuation theory in the Fermi-Pasta-Ulam chain

Recent work has developed a nonlinear hydrodynamic fluctuation theory for a chain of coupled anharmonic oscillators governing the conserved fields, namely, stretch, momentum, and energy. The linear theory yields two propagating sound modes and one diffusing heat mode, all three with diffusive broadening. In contrast, the nonlinear theory predicts that, at long times, the sound mode correlations satisfy Kardar-Parisi-Zhang scaling, while the heat mode correlations have Lévy-walk scaling. In the present contribution we report on molecular dynamics simulations of Fermi-Pasta-Ulam chains to compute various spatiotemporal correlation functions and compare them with the predictions of the theory. We obtain very good agreement in many cases, but also some deviations.

Dissipation and entropy production in deterministic heat conduction of quasi-one-dimensional systems

We explore the consequences of a deterministic microscopic thermostat-reservoir contact mechanism. With different temperature reservoirs at each end of a two-dimensional system, a heat current is produced and the system has an anomalous thermal conductivity. The microscopic form for the local heat flux vector is derived and both the kinetic and potential contributions are calculated. The total heat flux vector is shown to satisfy the continuity equation. The properties of this nonequilibrium steady state are studied as functions of system size and temperature gradient, identifying key scaling relations for the local fluid properties and separating bulk and boundary effects. The local entropy density calculated from the local equilibrium distribution is shown to be a very good approximation to the entropy density calculated directly from the velocity distribution even for systems that are far from equilibrium. The dissipation and kinetic entropy production and flux are compared quantitatively and the differing mechanisms discussed within the Bhatnagar-Gross-Krook approximation. For equal-temperature reservoirs the entropy production near the reservoir walls is shown to be proportional to the local phase space contraction calculated from the tangent space dynamics. However, for unequal temperatures, the connection between local entropy production and local phase space contraction is more complicated.

Space-time velocity correlation function for random walks

Space-time correlation functions constitute a useful instrument from the research toolkit of continuous-media and many-body physics. Here we adopt this concept for single-particle random walks and demonstrate that the corresponding space-time velocity autocorrelation functions reveal correlations which extend in time much longer than estimated with the commonly employed temporal correlation functions. A generic feature of considered random-walk processes is an effect of velocity echo identified by the existence of time-dependent regions where most of the walkers are moving in the direction opposite to their initial motion. We discuss the relevance of the space-time velocity correlation functions for the experimental studies of cold atom dynamics in an optical potential and charge transport on micro- and nanoscales.

Exact solution of a Lévy walk model for anomalous heat transport

The Lévy walk model is studied in the context of the anomalous heat conduction of one-dimensional systems. In this model, the heat carriers execute Lévy walks instead of normal diffusion as expected in systems where Fourier's law holds. Here we calculate exactly the average heat current, the large deviation function of its fluctuations, and the temperature profile of the Lévy walk model maintained in a steady state by contact with two heat baths (the open geometry). We find that the current is nonlocally connected to the temperature gradient. As observed in recent simulations of mechanical models, all the cumulants of the current fluctuations have the same system-size dependence in the open geometry. For the ring geometry, we argue that a size-dependent cutoff time is necessary for the Lévy walk model to behave like mechanical models. This modification does not affect the results on transport in the open geometry for large enough system sizes.

Random walk in chemical space of Cantor dust as a paradigm of superdiffusion

We point out that the chemical space of a totally disconnected Cantor dust K(n) [Symbol: see text E(n) is a compact metric space C(n) with the spectral dimension d(s) = d(ℓ) = n > D, where D and d(ℓ) = n are the fractal and chemical dimensions of K(n), respectively. Hence, we can define a random walk in the chemical space as a Markovian Gaussian process. The mapping of a random walk in C(n) into K(n) [Symbol: see text] E(n) defines the quenched Lévy flight on the Cantor dust with a single step duration independent of the step length. The equations, describing the superdiffusion and diffusion-reaction front propagation ruled by the local quenched Lévy flight on K(n) [Symbol: see text] E(n), are derived. The use of these equations to model superdiffusive phenomena, observed in some physical systems in which propagators decay faster than algebraically, is discussed.