We derive the thermal conductivities of one-dimensional harmonic and anharmonic lattices with self-consistent heat baths (BRV lattice) from the Single-Mode Relaxation Time (SMRT) approximation. For harmonic lattice, we obtain the same result as previous works. However, our approach is heuristic and reveals phonon picture explicitly within the heat transport process. The results for harmonic and anharmonic lattices are compared with numerical calculations from Green-Kubo formula. The consistency between derivation and simulation strongly supports that effective (renormalized) phonons are energy carriers in anharmonic lattices although there exist some other excitations such as solitons and breathers.
For students familiar with equilibrium statistical mechanics, the notion of a positive specific heat, being intimately related to the idea of stability, is both intuitively reasonable and mathematically provable. However, for system in non-equilibrium stationary states, coupled to more than one energy reservoir (e.g., thermal bath), negative specific heat is entirely possible. In this paper, we present a ``minimal'' system displaying this phenomenon. Being in contact with two thermal baths at different temperatures, the (internal) energy of this system may increase when a thermostat is turned …
We investigate the stationary nonequilibrium states of a quasi one-dimensional system of heavy particles whose interaction is mediated by purely elastic collisions with light particles, in contact at the boundary with two heat baths with fixed temperatures $T^+$ and $T^-$. It is shown that Fourier law is satisfied with a thermal conductivity proportional to $\sqrt{T(x)}$ where $T(x)$ is the local temperature. Entropy flux and entropy production are also investigated.
In a recent paper [Phys. Rev. Lett. 86, 63 (2001)], Li et al have reported that the nonequilibrium heat conducting steady state of a disordered harmonic chain is not unique. In this comment we point out that for a large class of stochastic heat baths the uniqueness of the steady state can be proved, and therefore the findings of Li et al could be either due to their use of deterministic heat baths or insufficient equilibration times in the simulations. We give a simple example where the uniquness of the steady state can be explicitly demonstrated.
Under many conditions, macroscopic grains flow like a fluid; kinetic theory pred icts continuum equations of motion for this granular fluid. In order to test the theory, we perform event driven molecular simulations of a two-dimensional gas of inelastic hard disks, driven by contact with a heat bath. Even for strong dissipation, high densities, and small numbers of particles, we find that continuum theory describes the system well. With a bath that heats the gas homogeneously, strong velocity correlations produce a slightly smaller energy loss due to inelastic collisions than that predicted by…
We explore transport properties in a disordered nonlinear chain of classical harmonic oscillators and thereby identify a regime exhibiting behavior analogous to that seen in quantum many-body-localized systems. Through extensive numerical simulations of this system connected at its ends to heat baths at different temperatures, we computed the heat current and the temperature profile in the nonequilibrium steady state as a function of system size $N$, disorder strength $Δ$, and temperature $T$. The conductivity $κ_N$, obtained for finite length ($N$) systems, saturates to a value $κ_\infty >0$ …
We consider chains consisting of several identical subsystems weakly coupled by various types of next neighbor interactions. At both ends the chain is coupled to a respective heat bath with different temperature modeled by a Lindblad formalism. The temperature gradient introduced by this environment is then treated as an external perturbation. We propose a method to evaluate the heat current and the local temperature profile of the resulting stationary state as well as the heat conductivity in such systems. This method is similar to Kubo techniques used e.g. for electrical transport but extend…
The harmonic Fibonacci chain, which is one of a quasiperiodic chain constructed with a recursion relation, has a singular continuous frequency-spectrum and critical eigenstates. The validity of the Fourier law is examined for the harmonic Fibonacci chain with stochastic heat baths at both ends by investigating the system size N dependence of the heat current J and the local temperature distribution. It is shown that J asymptotically behaves as (ln N)^{-1} and the local temperature strongly oscillates along the chain. These results indicate that the Fourier law does not hold on the harmonic Fib…
We consider two types of strongly disordered one-dimensional Hamiltonian systems coupled to baths (energy or particle reservoirs) at the boundaries: strongly disordered quantum spin chains and disordered classical harmonic oscillators. These systems are believed to exhibit localization, implying in particular that the conductivity decays exponentially in the chain length $L$. We ask however for the profile of the (very slowly) transported quantity in the steady state. We find that this profile is a step-function, jumping in the middle of the chain from the value set by the left bath to the val…
Chains of particles coupled through anharmonic interactions and subject to non-Gaussian baths can exhibit paradoxical outcomes such as heat currents flowing from colder to hotter reservoirs. Aiming to explore the role of generic non-harmonicities in mediating the contributions of non-Gaussian fluctuations to the direction of heat propagation, we consider a chain of power-law oscillators, with interaction potential $V(x) \propto |x|^α$, subject to Gaussian and Poissonian baths at its ends. Performing numerical simulations and addressing heuristic considerations, we show that a deformable potent…
Momentum-conserving one-dimensional models are known to exhibit anomalous Fourier's law, with a thermal conductivity varying as a power law of the system size. Here we measure, by numerical simulations, several cumulants of the heat flux of a one-dimensional hard particle gas. We find that the cumulants, like the conductivity, vary as power laws of the system size. Our results also indicate that cumulants higher than the second follow different power laws when one compares the ring geometry at equilibrium and the linear case in contact with two heat baths (at equal or unequal temperatures). ke…
We take an ordered, anharmonic crystal in the form of slab geometry in three dimensions. Apart from attaching baths of Langevin type to the extreme surfaces, we also attach baths of same type to the intermediate surfaces of the slab to simulate the environment surrounding the system. We assume noise functions to be Gaussian and their widths to be site dependent. We find that the radiated heat from the slab does not receive any correction at the leading order of anharmonic coupling and the Newton's law of cooling holds for an appropriate choice of the widths. We observe that in the steady state…
We analytically study heat conduction in a chain with interparticle interaction V(x)=lambda[1-cos(x)] and harmonic on-site potential. We start with each site of the system connected to a Langevin heat bath, and investigate the case of small coupling for the interior sites in order to understand the behavior of the system with thermal reservoirs at the boundaries only. We study, in a perturbative analysis, the heat current in the steady state of the one-dimensional system with weak interparticle potential. We obtain an expression for the thermal conductivity, compare the low and high temperatur…
A general formulation is developed to study heat conduction in disordered harmonic chains with arbitrary heat baths that satisfy the fluctuation-dissipation theorem. A simple formal expression for the heat current J is obtained, from which its asymptotic system-size (N) dependence is extracted. It is shown that the ``thermal conductivity'' depends not just on the system itself but also on the spectral properties of the fluctuation and noise used to model the heat baths. As special cases of our heat baths we recover earlier results which reported that for fixed boundaries $J \sim 1/N^{3/2}$, wh…
We investigate directed thermal heat flux across 1D homogenous nonlinear lattices when no net thermal bias is present on average. A nonlinear lattice of Fermi-Pasta-Ulam-type or Lennard-Jones-type system is connected at both ends to thermal baths which are held at the same temperature on temporal average. We study two different modulations of the heat bath temperatures, namely: (i) a symmetric, harmonic ac-driving of temperature of one heat bath only and (ii) a harmonic mixing drive of temperature acting on both heat baths. While for case (i) an adiabatic result for the net heat transport can …
We investigate thermal transport in square ice, a two-dimensional analogue of spin ice, exploring the role played by emergent magnetic monopoles in transporting energy. Using kinetic Monte Carlo simulations based on energy preserving extensions of single-spin-flip dynamics, we explicitly compute the (longitudinal) thermal conductivity, $κ$, over a broad range of temperatures. We use two methods to determine $κ$: a measurement of the energy current between thermal baths at the boundaries, and the Green-Kubo formula, yielding quantitatively consistent values for the thermal conductivity. We inte…
We study the thermal transport properties of the one dimensional Fermi-Pasta-Ulam model ($β$-type) with long-range interactions. The strength of the long-range interaction decreases with the (shortest) distance between the lattice sites as ${distance}^{-δ}$, where $δ\ge 0$.Two Langevin heat baths at unequal temperatures are connected to the ends of the one dimensional lattice via short-range harmonic interactions that drive the system away from thermal equilibrium. In the nonequilibrium steady state the heat current, thermal conductivity and temperature profiles are computed by solving the equ…
Self-organization and nonequilibrium phase transitions are well known to occur in two- and three- dimensional dissipative systems. Here, instead, we provide numerical evidence that these phenomena also occur in a one-dimensional Hamiltonian system. To this end, we calculate the heat conductivity by coupling the two ends of our system to two heat baths at different temperatures. It is found that when the temperature difference is smaller than a critical value, the heat conductivity increases with the system size in power law with an exponent considerably smaller than 1. However, as the temperat…
We examine numerically the full spatio-temporal correlation functions for all hydrodynamic quantities for the random collision model introduced recently. The autocorrelation function of the heat current, through the Kubo formula, gives a thermal conductivity exponent of 1/3 in agreement with the analytical prediction and previous numerical work. Remarkably, this result depends crucially on the choice of boundary conditions: for periodic boundary conditions (as opposed to open boundary conditions with heat baths) the exponent is approximately 1/2. This is expected to be a generic feature of sys…
We have numerically studied heat conduction in a few one-dimensional momentum-conserving lattices with asymmetric interparticle interactions by the nonequilibrium heat bath method, the equilibrium Green-Kubo method, and the heat current power spectra analysis. Very strong finite-size effects are clearly observed. Such effects make the heat conduction obey a Fourier-like law in a wide range of lattice lengths. However, in yet longer lattice lengths, the heat conductivity regains its power-law divergence. Therefore the power-law divergence of the heat conductivity in the thermodynamic limit is v…
We consider a d-dimensional crystal with an arbitrary harmonic interaction and an anharmonic on-site potential, with stochastic Langevin heat bath at each site. We develop an integral formalism for the correlation functions that is suitable for the study of their relaxation (time decay) as well as their behavior in space. Furthermore, in a perturbative analysis, for the one-dimensional system with weak coupling between the sites and small quartic anharmonicity, we investigate the steady state and show that the Fourier's law holds. We also obtain an expression for the thermal conductivity (for …
We study the heat current flowing between two baths consisting of harmonic oscillators interacting with a qubit through a spin-boson coupling. An explicit expression for the generating function of the total heat flowing between the hot and cold baths is derived by evaluating the corresponding Feynman-Vernon path integral under the non-interacting blip approximation (NIBA). This generating function satisfies the Gallavotti-Cohen fluctuation theorem, both before and after performing the NIBA. We also verify that the heat conductivity is proportional to the variance of the heat current, retrievin…
We investigate a chain of oscillators with anharmonic on-site potentials, with long range interparticle interactions, and coupled both to external and internal stochastic thermal reservoirs of Ornstein-Uhlenbeck type. We develop an integral representation, a la Feynman-Kac, for the correlations and the heat current. We assume the approximation of discrete times in the integral formalism (together with a simplification in a subdominant part of the harmonic interaction) in order to develop a suitable polymer expansion for the model. In the regime of strong anharmonicity, strong harmonic pinning,…
In one-dimensional electronic systems with strong repulsive interactions, charge excitations propagate much faster than spin excitations. Such systems therefore have an intermediate temperature range [termed the "spin-incoherent Luttinger liquid'" (SILL) regime] where charge excitations are "cold" (i.e., have low entropy) whereas spin excitations are "hot." We explore the effects of charge-sector disorder in the SILL regime in the absence of external sources of equilibration. We argue that the disorder localizes all charge-sector excitations; however, spin excitations are protected against ful…
We consider heat transport in one-dimensional harmonic chains attached at its ends to Langevin heat baths. The harmonic chain has mass impurities where the separation $d$ between any two successive impurities is randomly distributed according to a power-law distribution $P(d)\sim 1/d^{α+1}$, being $α>0$. In the regime where the first moment of the distribution is well defined ($1<α<2$) the thermal conductivity $κ$ scales with the system size $N$ as $κ\sim N^{(α-3)/α}$ for fixed boundary conditions, whereas for free boundary conditions $κ\sim N^{(α-1)/α}$ if $N\gg1$. When $α=2$, the inverse loc…
We consider a chain composed of $N$ coupled harmonic oscillators in contact with heat baths at temperature $T_\ell$ and $T_r$ at sites 1 and $N$ respectively. The oscillators are also subjected to non-momentum conserving bulk stochastic noises. These make the heat conductivity satisfy Fourier's law. Here we describe some new results about the hydrodynamical equations for typical macroscopic energy and displacement profiles, as well as their fluctuations and large deviations, in two simple models of this type.
We address the general problem of heat conduction in one dimensional harmonic chain, with correlated isotopic disorder, attached at its ends to white noise or oscillator heat baths. When the low wavelength $μ$ behavior of the power spectrum $W$ (of the fluctuations of the random masses around their common mean value) scales as $W(μ)\sim μ^β$, the asymptotic thermal conductivity $κ$ scales with the system size $N$ as $κ\sim N^{(1+β)/(2+β)}$ for free boundary conditions, whereas for fixed boundary conditions $κ\sim N^{(β-1)/(2+β)}$; where $β>-1$, which is the usual power law scaling for one dime…
We present the computer simulation results of a chain of hard point particles with alternating masses interacting on its extremes with two thermal baths at different temperatures. We found that the system obeys Fourier's law at the thermodynamic limit. This result is against the actual belief that one dimensional systems with momentum conservative dynamics and nonzero pressure have infinite thermal conductivity. It seems that thermal resistivity occurs in our system due to a cooperative behavior in which light particles tend to absorb much more energy than the heavier ones.
We investigate a class of anharmonic crystals in $d$ dimensions, $d\ge 1$, coupled to both external and internal heat baths of the Ornstein-Uhlenbeck type. The external heat baths, applied at the boundaries in the 1-direction, are at specified, unequal, temperatures $\tlb$ and $\trb$. The temperatures of the internal baths are determined in a self-consistent way by the requirement that there be no net energy exchange with the system in the non-equilibrium stationary state (NESS). We prove the existence of such a stationary self-consistent profile of temperatures for a finite system and show it…
Studies of thermal transport in long-range (LR)interacting systems are currently particularly challenging. The main difficulties lie in the choice of boundary conditions and the definition of heat current when driving systems in an out-of-equilibrium state by the usual thermal reservoirs. Here, by employing a reverse type of thermal baths that can overcome such difficulties, we reveal the intrinsic features of thermal transport underlying a LR interacting Fermi-Pasta-Ulam chain. We find that under an appropriate range value of LR exponent $σ=2$, while a \emph{nonballistic} power-law length ($L…
A framework for studying the effect of the coupling to the heat bath in models exhibiting anomalous heat conduction is described. The framework is applied to the harmonic chain with momentum exchange model where the non-trivial temperature profile is calculated. In this approach one first uses the hydrodynamic (HD) equations to calculate the equilibrium current-current correlation function in large but finite chains, explicitly taking into account the BCs resulting from the coupling to the heat reservoirs. Making use of a linear response relation, the anomalous conductivity exponent $α$ and an…
First-principles density-functional calculations are performed to investigate the thermal transport properties in graphene nanoribbons (GNRs). The dimensional crossover of thermal conductance from one to two dimensions (2D) is clearly demonstrated with increasing ribbon width. The thermal conductance of GNRs in a few nanometer width already exhibits an approximate low-temperature dependence of $T^{1.5}$, like that of 2D graphene sheet which is attributed to the quadratic nature of dispersion relation for the out-of-plane acoustic phonon modes. Using a zone-folding method, we heuristically deri…
We describe a mechanism to control energy and magnetisation currents in an artificial spin-chain, consisting of an array of Permalloy nano-disks coupled through the magneto-dipolar interaction. The chain is kept out of equilibrium by two thermal baths with different temperatures connected to its ends, which control the current propagation. Transport is enhanced by applying a uniform radio frequency pump field resonating with some of the spin-wave modes of the chain. Moreover, the two currents can be controlled independently by tuning the static field applied on the chain. Thus we describe two …
A diathermal wall between two heat baths at different temperatures can be mimicked by a layer of independent spin pairs with some internal energy and where each spin $σ_a$ is flipped by thermostat $a$ ($a=1,2$). The transition rates are determined from the modified detailed balance discussed in Ref.[1]. Generalized heat capacities, excess heats, the housekeeping entropy flow and the thermal conductivity in the steady state are calculated. The joint probability distribution of the heat cumulated exchanges at any time is computed explicitly. We obtain the large deviation function of heat transfe…
Local master equations are a widespread tool to model open quantum systems, especially in the context of many-body systems. These equations, however, are believed to lead to thermodynamic anomalies and violation of the laws of thermodynamics. In contrast, here we rigorously prove that local master equations are consistent with thermodynamics and its laws without resorting to a microscopic model, as done in previous works. In particular, we consider a quantum system in contact with multiple baths and identify the relevant contributions to the total energy, heat currents and entropy production r…
We investigate the steady state heat current in two and three dimensional disordered harmonic crystals in a slab geometry, connected at the boundaries to stochastic white noise heat baths at different temperatures.The disorder causes short wavelength phonon modes to be localized so the heat current in this system is carried by the extended phonon modes which can be either diffusive or ballistic. Using ideas both from localization theory and from kinetic theory we estimate the contribution of various modes to the heat current and from this we obtain the asymptotic system size dependence of the …
