Normal Heat Conduction Research Articles

Observation of thermal transport close to diffusive in a weak long-ranged Fermi-Pasta-Ulam-Tsingou-type model

We explore thermal transport in a one-dimensional Fermi-Pasta-Ulam-Tsingou-type (FPUT-type) system with long-range (LR) interactions. In such a system, the harmonic part of the potential is nearest-neighbor coupled, while the strength of the quartic part of the potential between two lattice sites decays as a power σ of the inverse of their distance, demonstrating the LR feature of the system. The relevant strong LR model (0≤σ≤1) has been considered in detail in our recent study [Xiong and Wang, ], and here, we focus on the weak LR regime (1≤σ≤3). We show that the thermal transport behaviors in this regime are quite unexpected. We discovered a subregime (1≤σ≤1.5) wherein the thermal transport behaviors are very close to diffusive. This suggests that even a momentum-conserving system with appropriate LR interactions can still exhibit normal heat conduction. By meticulously analyzing the space-time scaling properties of equilibrium heat correlations, we also determine the center of this diffusive thermal transport at approximately σ≃1.25. These discoveries, together with our prior results, provide a comprehensive understanding of thermal transport of this kind of LR-FPUT-type model. Published by the American Physical Society 2024

Fluctuation Relation for the Dissipative Flux: The Role of Dynamics, Correlations and Heat Baths.

The fluctuation relation stands as a fundamental result in nonequilibrium statistical physics. Its derivation, particularly in the stationary state, places stringent conditions on the physical systems of interest. On the other hand, numerical analyses usually do not directly reveal any specific connection with such physical properties. This study proposes an investigation of such a connection with the fundamental ingredients of the derivation of the fluctuation relation for the dissipation, which includes the decay of correlations, in the case of heat transport in one-dimensional systems. The role of the heat baths in connection with the system's inherent properties is then highlighted. A crucial discovery of our research is that different lattice models obeying the steady-state fluctuation relation may do so through fundamentally different mechanisms, characterizing their intrinsic nature. Systems with normal heat conduction, such as the lattice ϕ4 model, comply with the theorem after surpassing a certain observational time window, irrespective of lattice size. In contrast, systems characterized by anomalous heat conduction, such as Fermi-Pasta-Ulam-Tsingou-β and harmonic oscillator chains, require extended observation periods for theoretical alignment, particularly as the lattice size increases. In these systems, the heat bath's fluctuations significantly influence the entire lattice, linking the system's fluctuations with those of the bath. Here, the current autocorrelation function allows us to discern the varying conditions under which different systems satisfy with the fluctuation relation. Our findings significantly expand the understanding of the stationary fluctuation relation and its broader implications in the field of nonequilibrium phenomena.

Correlation functions and their universal connection during an extremely slow equilibration process.

We study the equilibration process of a one-dimensional lattice with transverse motions and external magnetic field. Starting from certain initial states, the system commonly reaches a metastable transient state shortly and then stays there for an extremely long time before it finally arrives in the ergodic equilibrium state. The relaxation time T_{eq} diverges even much more rapidly than exponential, which, compared with the widely reported power-law or even exponential divergence in many other systems, implies much higher stability of the transient state. Two correlation functions, the spatiotemporal correlation of the local energy and the autocorrelation of global heat current, are studied in both the metastable transient and the final equilibrium states. It is revealed that the correlations behave entirely differently in the two different states. In the former case they suggest normal heat diffusion and normal heat conduction; whereas in the latter one they indicate super heat diffusion and anomalous heat conduction. More importantly, we confirm that a general relation which connects the two correlations keeps valid not only in the equilibrium state but in the metastable transient state as well. The universality of the connection is, thus, extended.

Approach to Phonon Relaxation Time and Mean Free Path in Nonlinear LatticesSupported by the National Natural Science Foundation of China (Grant Nos. 12075199 and 11675133).

Based on the self-consistent phonon theory, the spectral energy density is calculated by the canonical transformation and the Fourier transformation. Through fitting the spectral energy density by the Lorentzian profile, the phonon frequency as well as the phonon relaxation time is obtained in one-dimensional nonlinear lattices, which is validated in the Fermi–Pasta–Ulam-β (FPU-β) and ϕ 4 lattices at different temperatures. The phonon mean free path is then evaluated in terms of the phonon relaxation time and phonon group velocity. The results show that, in the FPU-β lattice, the phonon mean free path as well as the phonon relaxation time displays divergent power-law behavior. The divergent exponent coincides well with that derived from the Peierls–Boltzmann theory at weak anharmonic nonlinearity. The value of the divergent exponent expects a power-law divergent heat conductivity with system size, which violates Fourier’s law. For the ϕ 4 lattice, both the phonon relaxation time and mean free path are finite, which ensures normal heat conduction.

Coupling turbulent natural convection-radiation-conduction in differentially heated cavity with high aspect ratio

In this paper, the effect of a several combined heat transfer processes: turbulent natural convection, surface radiation and conduction, are investigated numerically in an air-filled three-dimensional (3D) differentially heated tall cavity. The objectives of this work is to analyse the effect of coupled of different heat exchanges on flow structures of turbulent natural convection, and to distinguish the most realistic thermal boundary conditions of internal walls providing thus more accurate results.The cavity is parallelepiped and extended with sides length 0.076 × 2.18 × 0.52 m. A temperature gradient between the lateral walls is subjected of ΔT=19.9°C., inducing a turbulent flow due to the high Rayleigh number (RaH = 2.03 × 1010). A three dimensional unsteady numerical simulations are performed by the finite volume method using the Shear Stress Transport (SST k-ω) model and using the P-1 thermal radiation model.The effect of surface thermal radiation on heat transfer and dynamic structure is examined through a different values of internal surface emissivity of ε = [0, 0.6] and the air is assumed non-participating medium. Also, different previously works concluded that the heat conduction of passive walls must be considered for more accurate prediction. However, the heat conduction is considered in the likewise direction by imposing a temperature distribution along adiabatic walls. In the present study, the thickness of horizontal walls is specified in order to consider the normal heat conduction as a new approach in physical modelling of an enclosed cavity.The present numerical results are compared to available experimental databases for the same flow configuration. A numerical predictions show that the emissivity of the internal walls influences essentially the flow structures close to the horizontal walls, without any effects on the flow in the core of the cavity. On the other hand, somewhat accurate numerical predictions are obtained with combined different heat transfer along internal walls of differentially heated cavity.

Energy and momentum diffusion in one-dimensional periodic and asymmetric nonlinear lattices with momentum conservation.

The energy diffusion in one-dimensional (1D) momentum conserving nonlinear lattices usually exhibits anomalous superdiffusion, except the coupled rotator lattice with symmetric and periodic interacting potential which has normal energy diffusion corresponding to normal heat conduction. For nonperiodic 1D lattices with momentum conservation, it has been argued that the asymmetric potential can induce normal heat conduction. Later results indicate the observed normal behavior might be the finite size effect and the anomalous behavior will appear in the thermodynamical limit. Here we propose asymmetric and periodic 1D nonlinear lattices with momentum conservation. The energy and momentum diffusion behaviors will be investigated in detail and the same normal diffusion behaviors for both energy and momentum can be observed. These results confirm that the periodicity is the key for normal transport behavior in 1D momentum conserving lattices, whether the potential is symmetric or asymmetric.

Pressure-induced recovery of Fourier's law in one-dimensional momentum-conserving systems.

We report the two typical models of normal heat conduction in one-dimensional momentum-conserving systems. They show the Arrhenius and the non-Arrhenius temperature dependence. We construct the two corresponding phenomenologies, transition-state theory of thermally activated dissociation and the pressure-induced crossover between two fixed points in fluctuating hydrodynamics. Compressibility yields the ballistic fixed point, whose scaling is observed in Fermi-Pasta-Ulam (FPU) β lattices.

Development of Inverse Analysis Method of Heat Conduction and Thermal Stress for Elbow—Part II

An inverse heat conduction analysis method for piping elbow was developed to estimate the temperature and stress distribution on the inner surface by measuring the outer surface temperature. In the paper, the accuracy for the thermal stress calculation using the inverse heat conduction analysis method was confirmed by comparing with the reference results from normal FE heat conduction and thermal stress analyses. In the case of the measured-basis fluid temperature input from a high temperature–pressure test, the inverse analysis method estimated the maximum stress change by 7% conservative comparing with the normal FE analyses.

Triggering wave-domain heat conduction in graphene

Using non-equilibrium molecular dynamics simulations, we systematically investigate the non-Fourier heat conduction in graphene under steady high heat flux. The results show that if two triggering factors, i.e. steady high heat flux and tensile stress, are satisfied simultaneously, a low-frequency mechanical wave and corresponding wave-like energy profile can be observed, which are distinctly different from ripples and linear temperature profile of the normal Fourier heat conduction. This mechanical wave provides an additional channel of heat transport and renders graphene more conductive without changing its pristine thermal conductivity. What's more, as the heat flux or original bond length increases, its frequency increases and energy transported by this mechanical wave is also on the rise. Further analyses show that such anomalous phenomenon is not arising from the high-energy or high-frequency pulses and also not artifacts of the velocity-exchange method. It is a dissipative structure, a new order state far from thermodynamic equilibrium, and the corresponding nonlinear relationship between the gradient of the wave-like kinetic temperature and the heat flux enables more efficient heat transport in graphene.

Heat conduction and energy diffusion in momentum-conserving one-dimensional full-lattice ding-a-ling model.

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 etal. 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 model with three different nonlinear inter-particle potential forms. For symmetrical potential lattices, the thermal conductivities diverges with lattice length and their energy diffusions are superdiffusive signaturing anomalous heat conduction. For asymmetrical potential lattices, although the thermal conductivity seems to converge as the length increases, the energy diffusion is definitely deviating from normal diffusion behavior indicating anomalous heat conduction as well. No normal heat conduction behavior can be found for the full-lattice ding-a-ling model.

Normal heat conductivity in two-dimensional scalar lattices

The paper revisits recent counterintuitive results on the divergence of the heat conduction coefficient in two-dimensional lattices. It was reported that in certain lattices with on-site potential, for which a one-dimensional chain has convergent conductivity, the latter diverges in the 2D counterpart. We demonstrate that this conclusion is an artifact caused by the insufficient size of the simulated system. To overcome computational restrictions, a ribbon of relatively small width is simulated instead of a more traditional square specimen. It is further demonstrated that the heat conduction coefficient in the “long” direction of the ribbon ceases to depend on the width, as the latter achieves only 10 to 20 chains. So, one can consider the dynamics of much longer systems, than in the traditional setting, and still can gain reliable information regarding the 2D lattice. It turns out that for all considered models, for which the conductivity is convergent in the 1D case, it is convergent also in the 2D case. At the same time, however, the length of the system, necessary to reveal the convergence in the 2D case, may be much bigger than in its 1D counterpart.

Temperature dependence of thermal conductivities of coupled rotator lattice and the momentum diffusion in standard map

In contrary to other 1D momentum-conserving lattices such as the Fermi-Pasta-Ulam $\beta$ (FPU-$\beta$) lattice, the 1D coupled rotator lattice is a notable exception which conserves total momentum while exhibits normal heat conduction behavior. The temperature behavior of the thermal conductivities of 1D coupled rotator lattice had been studied in previous works trying to reveal the underlying physical mechanism for normal heat conduction. However, two different temperature behaviors of thermal conductivities have been claimed for the same coupled rotator lattice. These different temperature behaviors also intrigue the debate whether there is a phase transition of thermal conductivities as the function of temperature. In this work, we will revisit the temperature dependent thermal conductivities for the 1D coupled rotator lattice. We find that the temperature dependence follows a power law behavior which is different with the previously found temperature behaviors. Our results also support the claim that there is no phase transition for 1D coupled rotator lattice. We also give some discussion about the similarity of diffusion behaviors between the 1D coupled rotator lattice and the single kicked rotator also called the Chirikov standard map.

Recovery of normal heat conduction in harmonic chains with correlated disorder

We consider heat transport in one-dimensional harmonic chains with isotopic disorder, focusing our attention mainly on how disorder correlations affect heat conduction. Our approach reveals that long-range correlations can change the number of low-frequency extended states. As a result, with a proper choice of correlations one can control how the conductivity κ scales with the chain length N. We present a detailed analysis of the role of specific long-range correlations for which a size-independent conductivity is exactly recovered in the case of fixed boundary conditions. As for free boundary conditions, we show that disorder correlations can lead to a conductivity scaling as , with the scaling exponent ε being arbitrarily small (although not strictly zero), so that normal conduction is almost recovered even in this case.

Normal heat conductivity in chains capable of dissociation

The paper considers the highly debated problem of convergence of heat conductivity in one-dimensional chains with asymmetric nearest-neighbor potential. We conjecture that the convergence may be promoted not by the mere asymmetry of the potential, but due to the possibility that the chain dissociates. In other terms, the attractive part of the potential function should approach a finite value as the distance between the neighbors grows. To clarify this point, we study the simplest model of this sort —a chain of linearly elastic rods with finite size. If the distance between the rod centers exceeds their size, the rods cease to interact. Formation of gaps between the rods is the only possible mechanism for scattering of the elastic waves. Heat conduction in this system turns out to be convergent. Moreover, an asymptotic behavior of the heat conduction coefficient for the case of large densities and relatively low temperatures obeys a simple Arrhenius-type law. In the limit of low densities, the heat conduction coefficient converges due to triple rod collisions. Numeric observations in both limits are grounded by analytic arguments. In a chain with Lennard-Jones nearest-neighbor potential the heat conductivity also saturates in a thermodynamic limit and the coefficient also scales according to the Arrhenius law for low temperatures. This finding points on a universal role played by the possibility of dissociation, as convergence of the heat conduction coefficient is considered.

Interactions of heat and mass transfer in steam reheating of starchy foods

Steaming is a common practice in both household and industrial food processing; however, the process of heat transfer during steaming has not been fully explored. In this study, mathematical models that consider coupled energy, water and vapor transport were developed, and were solved using finite element software (COMSOL Multiphysics) for their application to the steam reheating of non-porous starch gel radish cake and steamed buns with porous matrices. Our simulation results indicated that the reheating of non-porous starch gel is a heat conduction process. During the reheating of the porous starch matrix, the infusion of condensate and evaporation–condensation mechanism contribute to steep sigmoid-shaped temperature profiles that are different from those of normal heat conduction. The validity of our mathematical model is corroborated by our experimental work of the steam reheating of samples with and without PE film wraps in a rice cooker. It shows a good agreement between the simulation and experimental results in terms of temperature and increased moisture. The implications of these models can provide a basis for future elucidation of more complicated food steaming processes.

Normal heat conduction in one-dimensional momentum conserving lattices with asymmetric interactions

We study heat conduction behavior of one-dimensional lattices with asymmetric, momentum conserving interparticle interactions. We find that with a certain degree of interaction asymmetry, the heat conductivity measured in nonequilibrium stationary states converges in the thermodynamical limit. Our analysis suggests that the mass gradient resulting from asymmetric interactions may provide a phonon scattering mechanism in addition to that caused by nonlinear interactions.

Stability and Normal Zone Propagation in a 50 Tesla Solenoid Wound of YBCO Coated Conductor Tape—FEM Modeling

We present an analysis of stability, heating and quench propagation in a 50 Tesla solenoid immersed in a liquid helium bath at 4.2 K, obtained by numerical Finite Element Method (FEM) modeling. The solenoid was assumed to be wound of a cable made of YBCO coated conductor tapes. Full heat transfer curve in the whole temperature range (from 4.2 K up to 300 K) for liquid helium, taken from experiment, was used. An anisotropic continuum model of the winding for thermal propagation, with input data taken from experiments, was developed and adopted in computations. Magnetic field and temperature dependent parameters, such as electrical and thermal conductivities, heat capacities etc., also taken from experiments, were considered. We modeled a single-coil 50 T solenoid as well as a segment approach, in which a 34 T winding, segmented radially, is the inner-most part of a system of and NbTi solenoids providing a background field of 16 Tesla. Stress-strain modeling showed that due to a strong anisotropy of in YBCO film, the critical current of the solenoid is not limited by mechanical forces, but by the radial magnetic field component in the winding, i.e. by the field component parallel to c-axis of YBCO film. We found quite a high degree of stability of the 50 Tesla solenoid with existing stagnant normal zones and significant heat conduction across the winding.

Nonstationary heat conduction in one-dimensional models with substrate potential

The paper investigates nonstationary heat conduction in one-dimensional models with substrate potential. To establish universal characteristic properties of the process, we explore three different models: Frenkel-Kontorova (FK), phi4+ (φ(4)+), and phi4- (φ(4)-). Direct numeric simulations reveal in all these models a crossover from oscillatory decay of short-wave perturbations of the temperature field to smooth diffusive decay of the long-wave perturbations. Such behavior is inconsistent with the parabolic Fourier equation of heat conduction and clearly demonstrates the necessity for hyperbolic corrections in the phenomenological description of the heat conduction process. The crossover wavelength decreases with an increase in the average temperature. The decay patterns of the temperature field almost do not depend on the amplitude of the perturbations, so the use of linear evolution equations for the temperature field is justified. In all models investigated, the relaxation of thermal perturbations is exponential, contrary to a linear chain, where it follows a power law. The most popular lowest-order hyperbolic generalization of the Fourier law, known as the Cattaneo-Vernotte or telegraph equation, is also not valid for the description of the observed behavior of the models with the substrate potential, since the characteristic relaxation time in an oscillatory regime strongly depends on the excitation wavelength. For some of the models, this dependence seems to obey a simple scaling law.

Heat Conduction in a Three-Dimensional Momentum-Conserving Anharmonic Lattice

Heat conduction in three-dimensional anharmonic lattices was numerically studied by the Green-Kubo theory. For a given lattice width W, a dimensional crossover is generally observed to occur at a W-dependent threshold of the lattice length. Lattices shorter than W will display a 3D behavior while lattices longer than W will display a 1D behavior. In the 3D regime, the heat current autocorrelation function was found to show a power-law decay as a function of the time lag τ as τ^{β} with β=-1.2. This indicates normal heat conduction. However, the decay exponent deviates significantly from the conventional theoretical value of β=-1.5. A flat power spectrum S(ω) of the global heat current in the low-frequency limit was also observed in the 3D regime. This provides not only an alternative verification of normal heat conduction but also a clear physical insight into its origin.

Quantum mechanical heat transport in disordered harmonic chains

We investigate the mechanism of heat conduction in ordered and disordered harmonic one-dimensional chains within the quantum mechanical Langevin method. In the case of 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 the Bose-Einstein statistics of the occupation numbers of the normal modes, freezing of the heat conductivity, and influence of the entanglement within the chain on the current. For the ordered chain we recover some classical results like a vanishing temperature gradient and a heat flux independent of the length of the chain.