We present an exact solution for the heat conductance along a harmonic chain connecting two reservoirs at different temperatures. In this model, the end points correspond to Brownian particles with different damping coefficients. Such analytical expression for the heat conductance covers its behavior from mesoscopic to very long one-dimensional quantum chains, and validates the ballistic nature of the heat transport in the latter example. This implies the absence of the Fourier law for classical and quantum harmonic chains. We also provide a thorough analysis of the normal modes of system which helps us to satisfactorily interpret these results.
We study heat conduction mediated by longitudinal phonons in one dimensional disordered harmonic chains. Using scaling properties of the phonon density of states and localization in disordered systems, we find non-trivial scaling of the thermal conductance with the system size. Our findings are corroborated by extensive numerical analysis. We show that a system with strong disorder, characterized by a `heavy-tailed' probability distribution, and with large impedance mismatch between the bath and the system satisfies Fourier's law. We identify a dimensionless scaling parameter, related to the t…
We have determined the thermal conductance of a system consisting of a two-level atom coupled to two quantum harmonic oscillators in contact with heat reservoirs at distinct temperatures. The calculation of the heat flux as well as the atomic population and the rate of entropy production are obtained by the use of a quantum Fokker-Planck-Kramers equation and by a Lindblad master equation. The calculations are performed for small values of the coupling constant. The results coming from both approaches show that the conductance is proportional to the coupling constant squared and that, at high t…
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 …
This paper proposes a new methodological framework within which the heat conductance in 1D lattices can be studied. The total process of heat conductance is separated into two parts where the first one is the equilibrium process at equal temperatures $T$ of both ends and the second one -- non-equilibrium with the temperature $ΔT$ of one end and zero temperature of the other. This approach allows significant decrease of computational time at $ΔT \to 0$. The threshold temperature $T_{\rm thr}$ is found which scales $T_{\rm thr}(N) \sim N^{-3}$ with the lattice size $N$ and by convention separate…
We study the Kondo effect in heat transport via a local two-state system. This system is described by the spin-boson Hamiltonian with Ohmic dissipation, which can be mapped onto the Kondo model with anisotropic exchange coupling. We calculate thermal conductance by the Monte Carlo method based on the exact formula. Thermal conductance has a scaling form κ= (k_B^2 T_K/\hbar) f(α,T/T_K ), where T_K and αindicate the Kondo temperature and dimensionless coupling strength, respectively. Temperature dependence of conductance is classified by the Kondo temperature as κ\propto (T/T_K )^3 for T\ll T_K …
Thermal rectification and negative differential thermal conductance were realized in harmonic chains in this work. We used the generalized Caldeira-Leggett model to study the heat flow. In contrast to the most previous studies considering only the linear system-bath coupling, we considered the nonlinear system-bath coupling based on recent experiment [A. Eichler \emph{et al.}, Nat. Nanotech. \textbf{6}, 339 (2011)]. When the linear coupling constant is weak, the multiphonon processes induced by the nonlinear coupling allow more phonons transport across the system-bath interface and hence the h…
An analytical model of high frequency oscillations of the kinetic and potential energies in a one-dimensional harmonic crystal with a substrate potential is obtained by introducing the nonlocal energies [1]. A generalization of the kinetic temperature (nonlocal temperature) is adopted to derive a closed equation determining the heat propagation processes in the harmonic crystal with a substrate potential.
We have analyzed the spectral density of fluctuations of the energy flux through a mesoscopic constriction between two equilibrium reservoirs. It is shown that at finite frequencies, the fluctuating energy flux is not related to the thermal conductance of the constriction by the standard fluctuation-dissipation theorem, but contains additional noise. The main physical consequence of this extra noise is that the fluctuations do not vanish at zero temperature together with the vanishing thermal conductance.
We work out the non-equilibrium steady state properties of a harmonic lattice which is connected to heat reservoirs at different temperatures. The heat reservoirs are themselves modeled as harmonic systems. Our approach is to write quantum Langevin equations for the system and solve these to obtain steady state properties such as currents and other second moments involving the position and momentum operators. The resulting expressions will be seen to be similar in form to results obtained for electronic transport using the non-equilibrium Green's function formalism. As an application of the fo…
We study the heat transport properties of a chain of coupled quantum harmonic oscillators in contact at its ends with two heat reservoirs at distinct temperatures. Our approach is based on the use of an evolution equation for the density operator which is a canonical quantization of the classical Fokker-Planck-Kramers equation. We set up the evolution equation for the covariances and obtain the stationary covariances at the stationary states from which we determine the thermal conductance in closed form when the interparticle interaction is small. The conductance is finite in the thermodynamic…
Unsteady heat transfer in a harmonic chain is analyzed. Two types of thermal perturbations are considered: 1) initial instant temperature perturbation, 2) external heat supply. Closed equations describing the heat propagation are obtained and their analytical solution is constructed.
Quantum physics predicts that there is a fundamental maximum heat conductance across a single transport channel, and that this thermal conductance quantum $G_Q$ is universal, independent of the type of particles carrying the heat. Such universality, combined with the relationship between heat and information, signals a general limit on information transfer. We report on the quantitative measurement of the quantum limited heat flow for Fermi particles across a single electronic channel, using noise thermometry. The demonstrated agreement with the predicted $G_Q$ establishes experimentally this …
We present an exact solution of the Langevin's equation in the steady state limit in a three dimensional, harmonic crystal of slab geometry whose boundary surfaces along its length are connected to two stochastic, white noise heat baths at different temperatures. We show that the heat trasport obeys the Fourier's law in the continuum limit.
An analysis of efficiency and its bounds at maximum work output for Carnot-like heat engines is conducted. The heat transfer processes are described by the linear law with time-dependent heat conductance. The upper bound of efficiency is found to be the CA efficiency,and is independent of the time duration completing each process and the time-dependent conductance. We prove that even the working medium exchanges heat sufficiently with the heat reservoirs, the work which could be extracted is finite and limited. The optimal temperature profiles in the heat exchanging processes are also analyzed…
Thermodynamic quantities, like heat, entropy, or work, are random variables, in stochastic systems. Here, we investigate the statistics of the heat exchanged by a Brownian particle subjected to a logarithm-harmonic potential. We derive analytically the characteristic function and its moments for the heat. Through numerical integration, and numerical simulation, we calculate the probability distribution as well, characterizing fully the statistical behavior of the heat. The results are also investigated in the asymptotic limit, where we encounter the characteristic function in terms of hypergeo…
We study vibrational energy transport in a quasi 1-D harmonic chain with both longitudinal and transverse vibrations. We demonstrate via both numerical simulation and theoretic analysis that for 1-D atomic chain connected by 3D harmonic springs, the coefficient of heat conduction changes it continuously with its lattice constant, indicating the qualitative difference from the corresponding 1-D case where the coefficient is independent of the lattice constant.
In this work exact solutions for the equation that describes anomalous heat propagation in 1D harmonic lattices are obtained. Rectangular, triangular, and sawtooth initial perturbations of the temperature field are considered. The solution for an initially rectangular temperature profile is investigated in detail. It is shown that the decay of the solution near the wavefront is proportional to $1/ \sqrt{t}$. In the center of the perturbation zone the decay is proportional to $1/t$. Thus the solution decays slower near the wavefront, leaving clearly visible peaks that can be detected experiment…
An analytical model of unsteady heat transfer in a one-dimensional harmonic crystal is presented. A nonlocal temperature is introduced as a generalization of the kinetic temperature. A closed equation determining unsteady thermal processes in terms of the nonlocal temperature is derived. For an instantaneous heat perturbation a time-reversible equation for the kinetic temperature is derived and solved. The resulting constitutive law for the heat flux in the considered system is obtained. This law significantly differs from Fourier's law and it predicts a finite velocity of the heat front and i…
Heat fluctuations of a harmonic oscillator in contact with a thermostat and driven out of equilibrium by an external deterministic force are studied experimentally and theoretically within the context of Fluctuation Theorems. We consider the case of a periodic forcing of the oscillator, and we calculate the analytic probability density function of heat fluctuations. The limit of large time is discussed and we show that heat fluctuations satisfy the conventional fluctuation theorem, even if a different fluctuation relation exists for this quantity. Experimental results are also given for a tran…
We consider a simple model of a stochastic heat engine, which consists of a single Brownian particle moving in a one-dimensional periodically breathing harmonic potential. Overdamped limit is assumed. Expressions of second moments (variances and covariances ) of heat and work are obtained in the form of integrals, whose integrands contain functions satisfying certain differential equations. The results in the quasi-static limit are simple functions of temperatures of hot and cold thermal baths. The coefficient of variation of the work is suggested to give an approximate probability for the wor…
We investigate the occurrence of heat leakages in overdamped Brownian harmonic systems. We exactly compute the underdamped and overdamped stochastic heats exchanged with the bath for a sudden frequency or temperature switch. We show that the underdamped heat reduces to the corresponding overdamped expression in the limit of large friction for the isothermal process. However, we establish that this is not the case for the isochoric transformation. We microscopically derive the additionally generated heat leakage and relate its origin to the initial relaxation of the velocity of the system. Our …
In the recent paper by Sokolov et al. (Int. J. of Heat and Mass Transfer 176, 2021, 121442) ballistic heat propagation in 1D harmonic crystal is considered and the properties of the exact discrete solution and the solution of the ballistic heat equation introduced by Krivtsov are numerically compared. The aim of this note is to demonstrate that the latter continuum fundamental solution can be formally obtained as the slow time-varying component of the large-time asymptotics for the exact discrete solution on a moving point of observation.
We consider unsteady heat transfer in a one-dimensional harmonic crystal surrounded by a viscous environment and subjected to an external heat supply. The basic equations for the crystal particles are stated in the form of a system of stochastic differential equations. We perform a continualization procedure and derive an infinite set of linear partial differential equations for covariance variables. An exact analytic solution describing unsteady ballistic heat transfer in the crystal is obtained. It is shown that the stationary spatial profile of the kinetic temperature caused by a point sour…
We study the heat transport in systems of coupled oscillators driven out of equilibrium by Gaussian heat baths. We illustrate with a few examples that such systems can exhibit ``strange'' transport phenomena. In particular, {\em circulation} of heat flux may appear in the steady state of a system of three oscillators only. This indicates that the direction of the heat fluxes can in general not be "guessed" from the temperatures of the heat baths. Although we primarily consider harmonic couplings between the oscillators, we explain why this strange behavior persists under weak anharmonic pertur…
We consider heat transfer across an arbitrary harmonic network connected to two heat baths at different temperatures. The network has $N$ positional degrees of freedom, of which $N_L$ are connected to a bath at temperature $T_L$ and $N_R$ are connected to a bath at temperature $T_R$. We derive an exact formula for the cumulant generating function for heat transfer between the two baths. The formula is valid even for $N_L \ne N_R$ and satisfies the Gallavotti-Cohen fluctuation symmetry. Since harmonic crystals in three dimensions are known to exhibit different regimes of transport such as balli…
The formalism of Kundu et al. [J. Stat. Mech. (2011) P03007], for computing the large deviations of heat flow in harmonic systems, is applied to the case of single Brownian particle in a harmonic trap and coupled to two heat baths at different temperatures. The large-t form of the moment generating function <exp[-sQ]> ~ g(s) exp[t m(s)], of the total heat flow Q from one of the baths to the particle in a given time interval t, is studied and exact explicit expressions are obtained for both m(s) and g(s). For a special case of the single particle problem that corresponds to the work done by an …
We study the problem of heat conduction in a mass-disordered two-dimensional harmonic crystal. Using two different stochastic heat baths, we perform simulations to determine the system size (L) dependence of the heat current (J). For white noise heat baths we find that J ~ 1/L^a with $ a \approx 0.59 $ while correlated noise heat baths gives $ a \approx 0.51 $. A special case with correlated disorder is studied analytically and gives a=3/2 which agrees also with results from exact numerics.
We consider a thermal quantum harmonic oscillator weakly coupled to a heat bath at a different temperature. We analytically study the quantum heat exchange statistics between the two systems using the quantum-optical master equation. We exactly compute the characteristic function of the heat distribution and show that it verifies the Jarzynski-Wójcik fluctuation theorem. We further evaluate the heat probability density in the limit of long thermalization times, both in the low and high temperature regimes, and investigate its time evolution by calculating its first two cumulants.
We provide an explicit method for solving general markovian master equations for quadratic bosonic Hamiltonians with linear bath operators. As an example we consider a one-dimensional quantum harmonic oscillator chain coupled to thermal reservoirs at both ends of the chain. We derive an analytic solution of the Redfield master equation for homogeneous harmonic chain and recover classical results, namely, vanishing temperature gradient and constant heat current in the thermodynamic limit. In the case of the disordered gapped chains we observe universal heat current scaling independent of the ba…
We study heat rectification in a minimalistic model composed of two masses subjected to on-site and coupling linear forces in contact with effective Langevin baths induced by laser interactions. Analytic expressions of the heat currents in the steady state are spelled out. Asymmetric heat transport is found in this linear system if both the bath temperatures and the temperature dependent bath-system couplings are also exchanged.
We consider heat conduction across an ordered oscillator chain with harmonic interparticle interactions and also onsite harmonic potentials. The onsite spring constant is the same for all sites excepting the boundary sites. The chain is connected to Ohmic heat reservoirs at different temperatures. We use an approach following from a direct solution of the Langevin equations of motion. This works both in the classical and quantum regimes. In the classical case we obtain an exact formula for the heat current in the limit of system size N to infinity. In special cases this reduces to earlier resu…
We investigate the mechanism of heat conduction in ordered and disordered harmonic onedimensional chains within the quantum mechanical Langevin method. In the case of the disordered chains we find indications for normal heat conduction which means that there is a finite temperature gradient but we cannot clearly decide whether the heat resistance increases linearly with the chain length. Furthermore we observe characteristic quantum mechanical features like Bose-Einstein statistics of the occupation numbers of the normal modes, freezing of the heat conductivity and the influence of the entangl…
In d-dimensional lattices of coupled quantum harmonic oscillators, we analyze the heat current caused by two thermal baths of different temperature, which are coupled to opposite ends of the lattice, with focus on the validity of Fourier's law of heat conduction. We provide analytical solutions of the heat current through the quantum system in the non-equilibrium steady state using the rotating-wave approximation and bath interactions described by a master equation of Lindblad form. The influence of local dephasing in the transition of ballistic to diffusive transport is investigated.
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…
