We propose a stochastic order parameter equation for describing phase coexistence in steady heat conduction near equilibrium. By analyzing the stochastic dynamics with a non-equilibrium adiabatic boundary condition, where total energy is conserved over time, we derive a variational principle that determines thermodynamic properties in non-equilibrium steady states. The resulting variational principle indicates that the temperature of the interface between the ordered region and the disordered region becomes greater (less) than the equilibrium transition temperature in the linear response regime when the thermal conductivity in the ordered region is less (greater) than that in the disordered region. This means that a super-heated ordered (super-cooled disordered) state appears near the interface, which was predicted by an extended framework of thermodynamics proposed in [N. Nakagawa and S.-i. Sasa, Liquid-gas transitions in steady heat conduction, Phys. Rev. Lett. {\bf 119}, 260602, (2017).]
The experimental basis and theoretical background of non-Fourier heat conduction is shortly reviewed from the point of view of non-equilibrium thermodynamics. The performance of different theories is compared in case of heat pulse experiments.
Dual phase lag equation for heat conduction is analyzed from the point of view of non-equilibrium thermodynamics. Its first order Taylor series expansion is consistent with the second law as long as the two relaxation times are not negative.
We investigate the relationship between a diffusive motion of an interface, heat conduction, and the roughening transition in the three-dimensional Ising model. We numerically compute the thermal conductivity and the diffusion constant and find that the diffusion constant shows a crossover in its temperature dependence. The crossover temperature is equal to the roughening transition temperature in equilibrium and deviates from it when heat flows in the system. From these results, we discuss the possibility that heat conduction causes a shift of the roughening transition temperature.
Thermodynamics and Dynamic Stability: Extended Theories of Heat Conduction
HUN-REN Wigner Research Centre for Physics（匈牙利研究与教育网络维格纳物理研究中心）；Eötvös Loránd University（厄特沃什·罗兰大学）；Montavid Thermodynamic Research Group（蒙塔维德热力学研究组）；University of Messina（墨西拿大学）；Budapest University of Technology and Economics（布达佩斯技术与经济大学）
There are various situations where the classical Fourier's law for heat conduction is not applicable, such as heat conduction in heterogeneous materials or for modeling low-temperature phenomena. In such cases, heat flux is not directly proportional to temperature gradient, hence, the role -- and both the analytical and numerical treatment -- of boundary conditions becomes nontrivial. Here, we address this question for finite difference numerics via a shifted field approach. Based on this ground,implicit schemes are presented and compared to each other for the Guyer--Krumhansl generalized heat…
We propose the concept of global temperature for spatially non-uniform heat conduction systems. With this novel quantity, we present an extended framework of thermodynamics for the whole system such that the fundamental relation of thermodynamics holds, which we call "global thermodynamics" for heat conduction systems. Associated with this global thermodynamics, we formulate a variational principle for determining thermodynamic properties of the liquid-gas phase coexistence in heat conduction, which corresponds to the natural extension of the Maxwell construction for equilibrium systems. We qu…
The ding-a-ling model is a kind of half lattice and half hard-point-gas (HPG) model. The original ding-a-ling model proposed by Casati {\it et.al} does not conserve total momentum and has been found to exhibit normal heat conduction behavior. Recently, a modified ding-a-ling model which conserves total momentum has been studied and normal heat conduction has also been claimed. In this work, we propose a full lattice ding-a-ling model without hard point collisions where total momentum is also conserved. We investigate the heat conduction and energy diffusion of this full lattice ding-a-ling mod…
Second sound phenomenon and ballistic heat conduction, the two wave like propagation modes of heat, are the two most prominent, experimentally observed non-Fourier effects of heat conduction. In this paper we compare three related theories by quantitatively analyzing the crucial NaF experiments of Jackson, Walker and McNelly, where these effects were observed together. We conclude that with the available information the best comparison and insight is provided by non-equilibrium thermodynamics with internal variables. However, the available data and information is not the best, and further, new…
A novel equation of heat conduction is derived with the help of a generalized entropy current and internal variables. The obtained system of constitutive relations is compatible with the momentum series expansion of the kinetic theory. The well known Fourier, Maxwell-Cattaneo-Vernotte, Guyer-Krumhansl, Jeffreys-type, and Cahn-Hilliard type equations are derived as special cases. Some remarkable properties of solutions of the general equation are demonstrated with heat pulse initial and boundary conditions. A simple numerical method is developed and its stability is proved. Apparent faster than…
Some analogies between different nonequilibrium heat conduction models, particularly, random walk, discrete variable model, and Boltzmann transport equation with the single relaxation time approximation, have been discussed. We show that under an assumption of a finite value of the heat carriers velocity, these models lead to the hyperbolic heat conduction equation and the modified Fourier law with the relaxation term. Corresponding effective temperature and entropy have been introduced and analyzed. It has been demonstrated that the effective temperature, defined as a geometric mean of the ki…
In the study of 1D nonlinear Hamiltonian lattices, the conserved quantities play an important role in determining the actual behavior of heat conduction. Besides the total energy, total momentum and total stretch could also be conserved quantities. In microcanonical Hamiltonian dynamics, the total energy is always conserved. It was recently argued by Das and Dhar that whenever stretch (momentum) is not conserved in a 1D model, the momentum (stretch) and energy fields exhibit normal diffusion. In this work, we will systematically investigate the stretch diffusions for typical 1D nonlinear latti…
We study liquid-gas transitions of heat conduction systems in contact with two heat baths under constant pressure in the linear response regime. On the basis of local equilibrium thermodynamics, we propose an equality with a global temperature, which determines the volume near the equilibrium liquid-gas transition. We find that the formation of the liquid-gas interface is accompanied by a discontinuous change in the volume when increasing the mean temperature of the baths. A super-cooled gas near the interface is observed as a stable steady state.
The paper considers heat conduction in a model chain of composite particles with hard core and elastic external shell. Such model mimics three main features of realistic interatomic potentials - hard repulsive core, quasilinear behavior in a ground state and possibility of dissociation. It has become clear recently, that this latter feature has crucial effect on convergence of the heat conduction coefficient in thermodynamic limit. We demonstrate that in one-dimensional chain of elastic particles with hard core the heat conduction coefficient also converges, as one could expect. Then we explor…
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…
Local Temperature and Universal Heat Conduction in FPU chains
数值结果表明，对于Fermi Pasta Ulam（FPU）链交替质量和热浴在略有不同的温度在两端，局部温度（LT）在小尺度上表现出自相矛盾的稳态。这扩展了FPU链平衡的长期建立的问题。对于质量相等的链，表现良好的LT似乎可以实现;热导率随链长N发散为N^（1/3），这与备受争议的一维热传导普遍性问题有关。解释了为什么早期的模拟系统地获得更高的指数。
I revisit the exactly solvable Kipnis--Marchioro--Presutti model of heat conduction [J. Stat. Phys. 27 65 (1982)] and describe, for one-dimensional systems of arbitrary sizes whose ends are in contact with thermal baths at different temperatures, a systematic characterization of their non-equilibrium stationary states. These arguments avoid resorting to the analysis of a dual process and yield a straightforward derivation of Fourier's law, as well as higher-order static correlations, such as the covariant matrix. The transposition of these results to families of gradient models generalizing th…
Results of heat pulse experiments in various artificial and natural materials are reported in the paper. The experiments are performed at room temperature with macroscopic samples. It is shown that temperature evolution does not follow the Fourier's law but well explained by the Guyer-Krumhansl equation. The observations confirm the ability of non-equilibrium thermodynamics to formulate universal constitutive relations for thermomechanical processes.
We present a rigorous approach that leads, from a many-particle description, to a nonlinear, stochastic constitutive relation for the modeling of transient heat conduction processes at nanoscale. By enforcing statistical consistency, in that the statistics of the local energy is consistent with that from an all-atom description, we identify the driving force as well as the model parameters in these generalized constitutive models.
Among the three heat conduction modes, the ballistic propagation is the most difficult to model. In the present paper, we discuss its physical interpretations and showing different alternatives to its modeling. We highlight two of them: a thermo-mechanical model in which the generalized heat equation - the so-called Maxwell-Cattaneo-Vernotte equation - is coupled to thermal expansion. At the same time, the other one uses internal variables. For the first one, we developed a numerical solution and tested on the heat pulse experiment performed by McNelly et al.~on NaF samples. For the second one…
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.
Heat and energy are conceptually different, but often are assumed to be the same without justification. An effective method for investigating diffusion properties in equilibrium systems is discussed. With this method, we demonstrate that for one-dimensional systems, using the indices of particles as the space variable , which has been accepted as a convention, may lead to misleading conclusions. We then show that though in one-dimensional systems there is no general connection between energy diffusion and heat conduction, however, a general connection between heat diffusion and heat conduction…
We study the thermal conduction behaviors of one-dimensional lattice models with asymmetry harmonic interparticle interactions in this paper. Normal thermal conductivity independent of the system size is observed when the lattice chains are long enough. Because only the harmonic interactions are involved, the result confirms without ambiguous interpretation that the asymmetry plays key role in resulting in the normal thermal conduction in one-dimensional momentum conserving lattices. Both equilibrium and nonequilibrium simulations are performed to support the conclusion.
