Recent results on theoretical studies of heat conduction in low-dimensional systems are presented. These studies are on simple, yet nontrivial, models. Most of these are classical systems, but some quantum-mechanical work is also reported. Much of the work has been on lattice models corresponding to phononic systems, and some on hard particle and hard disc systems. A recently developed approach, using generalized Langevin equations and phonon Green's functions, is explained and several applications to harmonic systems are given. For interacting systems, various analytic approaches based on the Green-Kubo formula are described, and their predictions are compared with the latest results from simulation. These results indicate that for momentum-conserving systems, transport is anomalous in one and two dimensions, and the thermal conductivity κ, diverges with system size L, as κ ∼ L α . For one dimensional interacting systems there is strong numerical evidence for a universal exponent α = 1/3, but there is no exact proof for this so far. A brief discussion of some of the experiments on heat conduction in nanowires and nanotubes is also given. IntroductionIt is now about two hundred years since Fourier first proposed the law of heat conduction that goes by his name. Consider a macroscopic system subjected to different temperatures at its boundaries. One assumes that it is possible to have a * Corresponding author. Email: dabhi@rri.res.in November 19, 2008 19:56 Advances in Physics reva 2 coarse-grained description with a clear separation between microscopic and macroscopic scales. If this is achieved, it is then possible to define, at any spatial point x in the system and at time t, a local temperature field T (x, t) which varies slowly both in space and time (compared to microscopic scales). One then expects heat currents to flow inside the system and Fourier argued that the local heat current density J(x, t) is given bywhere κ is the thermal conductivity of the system. If u(x, t) represents the local energy density then this satisfies the continuity equation ∂u/∂t + ∇.J = 0. Using the relation ∂u/∂T = c, where c is the specific heat per unit volume, leads to the heat diffusion equation:Thus, Fourier's law implies diffusive transfer of energy. In terms of a microscopic picture, this can be understood in terms of the motion of the heat carriers, i.e. , molecules, electrons, lattice vibrations(phonons), etc., which suffer random collisions and hence move diffusively. Fourier's law is a phenomological law and has been enormously succesful in providing an accurate description of heat transport phenomena as observed in experimental systems. However there is no rigorous derivation of this law starting from a microscopic Hamiltonian description and this basic question has motivated a large number of studies on heat conduction in model systems. One important and somewhat surprising conclusion that emerges from these studies is that Fourier's law is probably not valid in one and two dimensional systems, except when the ...

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 formalism we discuss heat conduction in a harmonic chain connected to self-consistent reservoirs. We obtain a temperature dependent thermal conductivity which, in the high-temperature classical limit, reproduces the exact result on this model obtained recently by Bonetto, Lebowitz and Lukkarinen.

We investigate the motion of a run-and-tumble particle (RTP) in one dimension. We find the exact probability distribution of the particle with and without diffusion on the infinite line, as well as in a finite interval. In the infinite domain, this probability distribution approaches a Gaussian form in the long-time limit, as in the case of a regular Brownian particle. At intermediate times, this distribution exhibits unexpected multi-modal forms. In a finite domain, the probability distribution reaches a steady-state form with peaks at the boundaries, in contrast to a Brownian particle. We also study the relaxation to the steady-state analytically. Finally we compute the survival probability of the RTP in a semi-infinite domain with an absorbing boundary condition at the origin. In the finite interval, we compute the exit probability and the associated exit times. We provide numerical verification of our analytical results. arXiv:1711.08474v3 [cond-mat.stat-mech]

A general formulation is developed to study heat conduction in disordered harmonic chains with arbitrary heat baths, satisfying 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 "thermal conductivity" depends not just on the system itself but also on the spectral properties of the heat baths. As special cases we recover earlier results that gave J approximately 1/N(3/2) for fixed boundaries and J approximately 1/N(1/2) for free boundaries. Other choices give other power laws including the "Fourier behavior" J approximately 1/N.

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We study the nonequilibrium state of heat conduction in a one-dimensional system of hard point particles of unequal masses interacting through elastic collisions. A BBGKY-type formulation is presented and some exact results are obtained from it. Extensive numerical simulations for the two-mass problem indicate that, even for arbitrarily small mass differences, a nontrivial steady state is obtained. This state exhibits local thermal equilibrium and has a temperature profile as predicted by kinetic theory. The temperature jumps typically seen in such studies are shown to be finite-size effects. The thermal conductivity appears to have a very slow divergence with system size, different from that seen in most other systems.

The non-equilibrium Green's function formalism for infinitely extended reservoirs coupled to a finite system can be derived by solving the equations of motion for a tight-binding Hamiltonian.While this approach gives the correct density for the continuum states, we find that it does not lead, in the absence of any additional mechanisms for equilibration, to a unique expression for the density matrix of any bound states which may be present. Introducing some auxiliary reservoirs which are very weakly coupled to the system leads to a density matrix which is unique in the equilibrium situation, but which depends on the details of the auxiliary reservoirs in the non-equilibrium case.

We consider steady-state heat conduction across a quantum harmonic chain connected to reservoirs modeled by infinite collection of oscillators. The heat, Q, flowing across the oscillator in a time interval tau is a stochastic variable and we study the probability distribution function P(Q). We compute the exact generating function of Q at large tau and the large deviation function. The generating function has a symmetry satisfying the steady-state fluctuation theorem without any quantum corrections. The distribution P(Q) is non-Gaussian with clear exponential tails. The effect of finite tau and nonlinearity is considered in the classical limit through Langevin simulations. We also obtain the prediction of quantum heat current fluctuations at low temperatures in clean wires.