Anomalous heat conduction and anomalous diffusion in low dimensional nanoscale systems

Heat conduction is an important energy transport process in nature. Phonon is the major energy carrier for heat in semiconductors and dielectric materials. In analogy to Ohm’s law of electrical conduction, Fourier’s law is the fundamental law of heat conduction in solids. Although Fourier’s law has received great success in describing macroscopic heat conduction in the past two hundred years, its validity in low dimensional systems is still an open question. Here we give a brief review of the recent developments in experimental, theoretical and numerical studies of heat conduction in low dimensional systems, including lattice models and low dimensional nanostructures such as nanowires, nanotubes and graphene. We will demonstrate that phonons transport in low dimensional systems superdiffusively, which leads to a size dependent thermal conductivity. In other words, Fourier’s law is not applicable in low dimensional structures.     

Anomalous heat conduction in one-dimensional momentum-conserving systems

We show that for one-dimensional fluids the thermal conductivity generically diverges with system size L as L(1/3), as a result of momentum conservation. Our results are consistent with the largest-scale numerical studies of two-component hard-particle systems. We suggest explanations for the apparent disagreement with studies on Fermi-Pasta-Ulam chains.     

Anomalous energy diffusion and heat conduction in one-dimensional system

We propose a new concept, the centre of energy, to study energy diffusion and heat conduction in one-dimensional hard-point model. For diatom model, we find an anomalous energy diffusion as $\langle x^2 \rangle\sim t^\beta$ with $\beta=1.33$, which is independent of initial condition and mass rate. The present model can be viewed as the model composed by independent quasi-particles, the centre of energy. In this way, heat current can be calculated. Based on theory of dynamic billiard, the divergent exponent of heat conductivity is estimated to be $\alpha=0.33$, which is confirmed by a simple numerical calculation.     

Convective heat conduction and diffusion in one-dimensional hydrodynamics

The impurity concentration in localized structures is described on the basis of analytic solutions of model equations for convective diffusion in the one-dimensional hydrodynamic approximation without pressure. The simplicity of the derivation of the analytic results depends on the ratio of the kinetic coefficients of the liquid (the Prandtl numbers). For the same kinetic coefficients, any time-dependent problem can be reduced to problems for the conventional heat conduction equation. For integer Prandtl numbers the problem of time-dependent convective diffusion in the flow field of a uniformly moving shock wave likewise reduces to problems for the heat conduction equation. Relations are established between problems whose Prandtl numbers differ by an integer. Various representations of the Green’s functions for the equations of convective diffusion are analyzed. For integer Prandtl numbers they can be expressed in terms of error functions. The asymptotic character of the solutions depends strongly on the satisfaction of global conservation laws. For global conservation of the impurity mass, coalescence of shock waves corresponds to merging of impurity solitons, i.e., clustering. Zh. éksp. Teor. Fiz. 116, 1616–1629 (November 1999)     

Anomalous heat conduction and anomalous diffusion in nonlinear lattices, single walled nanotubes, and billiard gas channels

We study anomalous heat conduction and anomalous diffusion in low-dimensional systems ranging from nonlinear lattices, single walled carbon nanotubes, to billiard gas channels. We find that in all discussed systems, the anomalous heat conductivity can be connected with the anomalous diffusion, namely, if energy diffusion is sigma(2)(t)=2Dt(alpha) (01) implies an anomalous heat conduction with a divergent thermal conductivity (beta>0), and more interestingly, a subdiffusion (alpha<1) implies an anomalous heat conduction with a convergent thermal conductivity (beta<0), consequently, the system is a thermal insulator in the thermodynamic limit. Existing numerical data support our theoretical prediction.     

Theory of one-dimensional and quasi-one-dimensional heat conduction

It is shown that the heat conduction process in a one-dimensional flow of a fluid moving with a velocity V in a constant temperature field follows a law that is considerably more complicated than an “ordinary” exponential law. It is demonstrated that in the quasi-one-dimensional case the heat conduction process in an abstract space of dimension 1+ɛ, where ɛ varies from zero to unity, is described by a modified Fourier equation. Its solution for an infinite space is found. Zh. Tekh. Fiz. 67, 8–12 (July 1997)     

Normal and anomalous heat transport in one-dimensional classical lattices

We present analytic and numerical results on several models of one-dimensional (1D) classical lattices with the goal of determining the origins of anomalous heat transport and the conditions for normal transport in these systems. Some of the recent results in the literature are reviewed and several original "toy" models are added that provide key elements to determine which dynamical properties are necessary and which are sufficient for certain types of heat transport. We demonstrate with numerical examples that chaos in the sense of positivity of Lyapunov exponents is neither necessary nor sufficient to guarantee normal transport in 1D lattices. Quite surprisingly, we find that in the absence of momentum conservation, even ergodicity of an isolated system is not necessary for the normal transport. Specifically, we demonstrate clearly the validity of the Fourier law in a pseudo-integrable particle chain.     

阻尼作用下一维体系热传导性质的研究

利用分子动力学方法，研究了阻尼对一维体系热传导性质的影响.研究结果表明，当体系的可积性因阻尼的引入而被破坏时，在一维谐振子晶格体系中也能够形成线性的温度分布.弱阻尼对非可积体系的温度分布和热导率的影响是一种微扰，而对于可积体系是一种性态的改变.在强阻尼条件下，由于体系能量的过度耗散，体系中呈现出凹形的温度分布和收敛的热导率. 关键词： 非线性动力学 一维体系 能量传输 阻尼     

Current progress on heat conduction in one-dimensional gas channels

We give a brief review of the past development of model studies on one-dimensional heat conduction. Particularly, we describe recent achievements on the study of heat conduction in one-dimensional gas models including the hard-point gas model and billiard gas channel. For a one-dimensional gas of elastically colliding particles of unequal masses, heat conduction is anomalous due to momentum conservation, and the divergence exponent of heat conductivity is estimated as α≈0.33 in k ∼ L ^α . Moreover, in billiard gas models, it is found that exponent instability is not necessary for normal heat conduction. The connection between heat conductivity and diffusion is investigated. Some new progress is reported. A recently proposed model with a quantized degree of freedom to study the heat transport in quasi-one dimensional systems is illustrated in which three distinct temperature regimes of heat conductivity are manifested. The establishment of local thermal equilibrium (LTE) in homogeneous and heterogeneous systems is also discussed. Finally, we give a summary with an outlook for further study about the problem of heat conduction.     

Exact results for anomalous transport in one-dimensional hamiltonian systems

Anomalous transport in one-dimensional translation invariant hamiltonian systems with short range interactions is shown to belong in general to the Kardar-Parisi-Zhang universality class. Exact asymptotic forms for density-density and current-current time correlation functions and their Fourier transforms are given in terms of the Pr?hofer-Spohn scaling functions, obtained from their exact solution for the polynuclear growth model. The exponents of corrections to scaling are found as well, but not so the coefficients. Mode coupling theories developed previously are found to be adequate for weakly nonlinear chains but in need of corrections for strongly anharmonic interparticle potentials. A simple condition is given under which Kardar-Parisi-Zhang behavior does not apply, sound attenuation is only logarithmically superdiffusive, and heat conduction is more strongly superdiffusive than under Kardar-Parisi-Zhang behavior.     

Anomalous scaling in systems partially controlled by diffusion

We study the nature of anomalous scaling in several systems partially controlled by diffusion. We quantify the departure from Fickian scaling by means of an apparent exponent governing the scaling of long-time behavior with system size. We find that anomalous scaling should be expected whenever complex geometries, higher dimensionality, or time-dependent boundary conditions are encountered.     

Oscillatory variation of anomalous diffusion in pendulum systems

Numerical studies of anomalous diffusion in undamped but periodically-driven and parametrically-driven pendulum systems are presented. When the frequency of the periodic driving force is varied, the exponent μ, which is the rate of divergence of the mean square displacement with time, is found to vary in an oscillatory manner. We show the presence of such a variation in other statistical measures such as variance of position, kurtosis, and exponents in the power-exponential law of probability distribution of position.     

Heat conduction in one-dimensional systems with hard-point interparticle interactions

Results of extensive and accurate numerical studies on heat transfer in a system of particles with unequal masses, interacting through hard-point potentials with two types of symmetry, are reported. The particles are confined in a one-dimensional box with fixed ends coupled to heat reservoirs at different temperatures. The study aims to throw light upon recent controversial results on thermal conductivity in one-dimensional systems. When the particles interact through elastic hard-point collisions (a standard asymmetric case), the system is shown to have always infinite (anomalous) thermal conductivity as follows from the Prosen-Campbell theorem.     

一维均匀Morse晶格体系的热流棘齿效应

本文系统研究了系统两端无平均温差时一维均匀Morse晶格中的热流棘齿效应. Morse晶格的两端分别与两个热浴相接触, 其中一端热浴温度周期调制,另一端热浴温度保持不变, 两端热浴温度长时平均相等. 数值结果表明, 当对一端热浴温度进行周期调制时, 系统中便会有稳定的定向热流产生. 通过改变调制频率和强度, 可以控制热流的大小及方向. 在合适的频率范围内, 可观察到一种非常有趣的现象——非定态负热导现象, 即系统中产生的定向热流逆着系统温度梯度方向由低温端流向高温端. 通过热波动力学分析(分析热流及温度分     

Relation between anomalous and normal diffusion in systems with memory

We present a simple criterion based on the Einstein relation for determining whether diffusion in systems governed by a generalized Langevin equation with long-range memory is normal, superdiffusive, or subdiffusive. We support our analysis with numerical simulations.     

Anomalous quantum heat transport in a one-dimensional harmonic chain with random couplings

We investigate quantum heat transport in a one-dimensional harmonic system with random couplings. In the presence of randomness, phonon modes may normally be classified as ballistic, diffusive or localized. We show that these modes can roughly be characterized by the local nearest-neighbor level spacing distribution, similarly to their electronic counterparts. We also show that the thermal conductance G(th) through the system decays rapidly with the system size (G(th)?～?L(-α)). The exponent α strongly depends on the system size and can change from α??1 with increasing system size, indicating that the system undergoes a transition from a heat conductor to a heat insulator. This result could be useful in thermal control of low-dimensional systems.     

Surface diffusion of k-mers in one-dimensional systems

The surface diffusion of interacting k-mers is studied both through analytical and Monte Carlo simulation methods in one-dimensional systems. Adsorption isotherms, jump diffusion coefficients and collective diffusion coefficients are obtained for attractive and repulsive k-mers, showing a variety of behaviors as a function of the size of particles, k. The following main results are found: (a) diffusion coefficients increase with k for non-interacting particles; (b) for fixed k, diffusion coefficients increase as the interaction energy increases from negative (attractive) to positive (repulsive) values; (c) for attractive interactions diffusion coefficients increase with k in the whole range of coverage; (d) for repulsive interactions diffusion coefficients decrease with k up to moderately high coverage and increase with k at high coverage. Results are rationalized in terms of the behavior of the vacancy probability distribution.     

Periodic oscillation of quantum diffusion in coupled one-dimensional systems

We analytically show that quantum diffusion in the coupled system composed of two identical chains exhibits a well-defined periodic oscillation in both transverse and longitudinal directions with a frequency determined by the interchain hopping strength, no matter whether the chains are periodic or non-periodic. We illustrate the result through numerical work on the coupled periodic chains and the quasiperiodic Aubry-Andre-Harper(AAH) chains with various modulations of onsite potentials supporting extended, critical, and localized states. We further numerically show that quantum diffusion in the coupled chains of different degrees of disorder W exhibits an exponential decay oscillation similar to the behavior of an underdamped harmonic oscillator, with a decay time inversely proportional to the square of W and a slight frequency change proportional to the square of W. Moreover, quantum diffusions in the coupled systems composed of two different chains are numerically studied, including periodic/disordered chains, periodic/AAH chains, and two different AAH chains, which exhibit the same behavior of underdamped periodic oscillation if the onsite potential difference between two chains is smaller than the interchain hoping strength.Existence of this universal periodic oscillation is a result of spectral splitting of the iso-spectra of two chains determined by interchain hopping, independent of system size, boundary condition, and intrachain onsite potentials. Because the oscillation frequency and spreading distance of wavepacket can be tuned separately by interchain hopping and intrachain potentials, the periodic oscillation of quantum diffusion in coupled chains is expected to find applications in control of quantum states and in designing nanoscale quantum devices.     

Heat-transfer mechanisms in solar flares. 1: Classical and anomalous heat conduction

In the context of the problem of energy transport in solar flares, simplified analytical models have been developed that describe plasma heating in the solar atmosphere by heat fluxes from the super-hot (T _e ≳ 10⁸ K) reconnecting current layer. It is shown that the applicability conditions of common heat conduction produced by Coulomb collisions of electrons in plasma are not fulfilled in solar flares. The heat flux calculated using the classical Fourier’s law proves to be significantly higher than the real energy fluxes known from modern multi-wavelength observations of flares. The so called anomalous flux produced by interaction of free electrons with ion acoustic waves in a plasma is critically analyzed. The question of what the dominant mechanism of heat transfer in solar flares is requires additional consideration [1].