Heat Conduction in the α−β Fermi–Pasta–Ulam Chain

Recent simulation results on heat conduction in a one-dimensional chain with an asymmetric inter-particle interaction potential and no onsite potential found non-anomalous heat transport in accordance to Fourier’s law. This is a surprising result since it was long believed that heat conduction in one-dimensional systems is in general anomalous in the sense that the thermal conductivity diverges as the system size goes to infinity. In this paper we report on detailed numerical simulations of this problem to investigate the possibility of a finite temperature phase transition in this system. Our results indicate that the unexpected results for asymmetric potentials is a result of insufficient chain length, and does not represent the asymptotic behavior.     

Controlling the heat flow: now it is possible

We discuss the problem of heat conduction in 1D nonlinear chains in relation to the dynamical properties of the system. We provide convincing numerical evidence for the validity of Fourier law of heat conduction in linear mixing systems. Therefore, deterministic diffusion and normal heat transport which are usually associated with full hyperbolicity, actually take place in systems without exponential instability. We then show that, acting on the parameter which controls the strength of the on site potential inside a segment of the chain, we induce a transition from conducting to insulating behavior in the whole system. The control of heat conduction by nonlinearity opens the possibility to propose new devices such as a thermal rectifier.     

The time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems

In this paper a time fractional Fourier law is obtained from fractional calculus. According to the fractional Fourier law, a fractional heat conduction equation with a time fractional derivative in the general orthogonal curvilinear coordinate system is built. The fractional heat conduction equations in other orthogonal coordinate systems are readily obtainable as special cases. In addition, we obtain the solution of the fractional heat conduction equation in the cylindrical coordinate system in terms of the generalized H-function using integral transformation methods. The fractional heat conduction equation in the case 0<α≤1 interpolates the standard heat conduction equation (α=1) and the Localized heat conduction equation (α→0). Finally, numerical results are presented graphically for various values of order of fractional derivative.     

Heat conduction in the Frenkel-Kontorova model

Heat conduction is an old yet important problem. Since Fourier introduced the law bearing his name almost 200 years ago, a first-principle derivation of this simple law from statistical mechanics is still lacking. Worse still, the validity of this law in low dimensions, and the necessary and sufficient conditions for its validity are far from clear. In this paper we will review recent works on heat conduction in a simple nonintegrable model called the Frenkel-Kontorova model. The thermal conductivity of this model has been found to be finite. We will study the dependence of the thermal conductivity on the temperature and other parameters of the model such as the strength and the periodicity of the external potential. We will also discuss other related problems such as phase transitions and finite-size effects. The study of heat conduction is not only of theoretical interest but also of practical interest. We will show various recent designs of thermal rectifiers and thermal diodes by coupling nonlinear chains together. The study of heat conduction in low dimensions is also important to the understanding of the thermal properties of carbon nanotubes.     

Thermoelasticity of thin shells based on the time-fractional heat conduction equation

The time-nonlocal generalizations of Fourier’s law are analyzed and the equations of the generalized thermoelasticity based on the time-fractional heat conduction equation with the Caputo fractional derivative of order 0 < α ≤ 2 are presented. The equations of thermoelasticity of thin shells are obtained under the assumption of linear dependence of temperature on the coordinate normal to the median surface of a shell. The conditions of Newton’s convective heat exchange between a shell and the environment have been assumed. In the particular case of classical heat conduction (α = 1) the obtained equations coincide with those known in the literature.     

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.     

热质的运动与传递-微尺度导热中的热质动能效应

基于热质(热量的当量动质量)的概念,通过建立和分析热质的运动方程得到了反映热质动能变化的稳态导热微分方程,表明Fourier导热定律只有在热质的动能变化相对热质势能变化很小而可以忽略时才成立;在高热流密度和低温的情况下热质的动能变化不可忽略,这种动能效应表现为热流密度和温度梯度不再成线性关系.动能效应也导致Fourier导热定律不能通过热流和温度梯度准确地获得物体的导热系数,本文基于热质运动方程给出了导热系数动能效应的修正式.最后针对高热流密度和低温一维稳态导热进行了分子动力学模拟验证.     

Transport Processes from Mechanics: Minimal and Simplest Models

We review the current state of a fundamental problem of rigorous derivation of transport processes in classical statistical mechanics from classical mechanics. Such derivations for diffusion and momentum transport (viscosities) were obtained for minimal models of these processes involving one and two particles respectively. However, a minimal model which demonstrates heat conductivity contains three particles. Its rigorous analysis is currently out of reach for existing mathematical techniques. The gas of localized balls is widely accepted as a basis for a simplest model for derivation of Fourier’s law. We suggest a modification of the localized balls gas and argue that this gas of localized activated balls is a good candidate to rigorously prove Fourier’s law. In particular, hyperbolicity is derived for a reduced version of this model.     

A microscopic view of the Fourier law

The Fourier law of heat conduction describes heat diffusion in macroscopic systems. This physical law has been experimentally tested for a large class of physical systems. A natural question is to know whether it can be derived from the microscopic models using the fundamental laws of mechanics.     

Energy transport and the Fourier heat law in classical systems

The energy transport in one-dimensional nonlinear systems is discussed. By numerically studying a model system, we verify the Fourier heat law on purely dynamical grounds and we compute the coefficient of thermal conductivity K. The same value ofK is independently obtained by use of the Green-Kubo formalism.     

Derivation of the modification heating conduction equation of a kind of laser thermal effect by quantum mechanics

The Fourier equation of heat conduction predicts a paradox that the effect of a thermal impulse (e.g. the thermal effect in pulse laser) in an infinite medium; i.e., a thermal impulse is propagated in an infinite velocity. In order to solve the thermal transport paradox, C. W. Ulbrich and M. Chester have proposed the modification heat conduction equation respectively from different macroscopic viewpoint. This paper derived the modification heat conduction equation according to phonon model and quantum mechanics from microscopic viewpoint.     

Numerical Simulation of the Non-Fourier Heat Conduction in a Solid-State Laser Medium

A hyperbolic model of non-Fourier heat conduction with non-uniform heat source is used to simulate the transient heat transfer in a high-pulse-pumped solid-state laser medium. The temperature fields are numerically analysed using the finite difference method combined with the TDMA algorithm for different pump power densities, pulse durations, thermal relaxation time and cooling intensities, respectively. The calculated results are compared with those predicted by the parabolic heat conduction model based on the Fourier law. The results indicate that the non-Fourier heat conduction phenomenon in laser media should be considered when the pump power density exceeds 104 W/m^2 or under low pulse duration. In addition, the conditions of non-Fourier effects and their influencing factors are analysed.     

Benard-Marangoni instability in a Maxwell-Cattaneo fluid

The effects resulting from the substitution of the classical Fourier law of heat conduction by the Maxwell-Cattaneo law in Bénard's and Marangoni's problems are examined.     

Benard-Marangoni instability in a Maxwell-Cattaneo fluid

The effects resulting from the substitution of the classical Fourier law of heat conduction by the Maxwell-Cattaneo law in Bénard's and Marangoni's problems are examined.     

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.     

Energy-transport by radiative diffusion

A new derivation of the general-relativistic Fourier equation is given for radiation transport by using the principle of conservation of momentum plus some rather simple assumptions. The Fourier equation at which I arrive is not the usual one but has an additional term. For this reason it leads to a hyperbolic equation for heat conduction, thus avoiding the paradox of infinite velocity of heat propagation, which is a consequence of the usual Fourier equation, as the latter one leads to a parabolic equation for heat conduction. The new Fourier equation is compared with the one that was given by Kranys by using ad hoc assumptions.     

Role of conserved quantities in Fourier's law for diffusive mechanical systems

Energy transport can be influenced by the presence of other conserved quantities. We consider here diffusive systems where energy and the other conserved quantities evolve macroscopically on the same diffusive space–time scale. In these situations, the Fourier law depends also on the gradient of the other conserved quantities. The rotor chain is a classical example of such systems, where energy and angular momentum are conserved. We review here some recent mathematical results about the diffusive transport of energy and other conserved quantities, in particular for systems where the bulk Hamiltonian dynamics is perturbed by conservative stochastic terms. The presence of the stochastic dynamics allows us to define the transport coefficients (thermal conductivity) and in some cases to prove the local equilibrium and the linear response argument necessary to obtain the diffusive equations governing the macroscopic evolution of the conserved quantities. Temperature profiles and other conserved quantities profiles in the non-equilibrium stationary states can be then understood from the non-stationary diffusive behavior. We also review some results and open problems on the two step approach (by weak coupling or kinetic limits) to the heat equation, starting from mechanical models with only energy conserved.     

A high-efficiency aerothermoelastic analysis method

In this paper,a high-efficiency aerothermoelastic analysis method based on unified hypersonic lifting surface theory is established.The method adopts a two-way coupling form that couples the structure,aerodynamic force,and aerodynamic thermo and heat conduction.The aerodynamic force is first calculated based on unified hypersonic lifting surface theory,and then the Eckert reference temperature method is used to solve the temperature field,where the transient heat conduction is solved using Fourier’s law,and the modal method is used for the aeroelastic correction.Finally,flutter is analyzed based on the p-k method.The aerothermoelastic behavior of a typical hypersonic low-aspect ratio wing is then analyzed,and the results indicate the following:(1)the combined effects of the aerodynamic load and thermal load both deform the wing,which would increase if the flexibility,size,and flight time of the hypersonic aircraft increase;(2)the effect of heat accumulation should be noted,and therefore,the trajectory parameters should be considered in the design of hypersonic flight vehicles to avoid hazardous conditions,such as flutter.     

The physical defects of the hyperbolic heat conduction equation

In this paper the HHCE is inspected on a microscopic level from a physical point of view. Starting from the Boltzmann transport equations we study the underlying approximations. We find that the hyperbolic approach to the heat current density violates the fundamental law of energy conservation. As a consequence, the HHCE predicts physically impossible solutions with a negative local heat content. This behaviour is demonstrated in detail for a standard problem in heat conduction, the solution for a point source. Received: 29 October 1997/Accepted: 17 February 1998     

Fourier's Law confirmed for a class of small quantum systems

Within the Lindblad formalism we consider an interacting spin chain coupled locally to heat baths. We investigate the dependence of the energy transport on the type of interaction in the system as well as on the overall interaction strength. For a large class of couplings we find a normal heat conduction and confirm Fourier's Law. In a fully quantum mechanical approach linear transport behavior appears to be generic even for small quantum systems.Received: 14 May 2003, Published online: 11 August 2003PACS: 05.60.Gg Quantum transport - 05.30.-d Quantum statistical mechanics - 05.70.Ln Nonequilibrium and irreversible thermodynamics