Anomalous Heat Conduction Research Articles

Adaptive fractional physics-informed neural networks for solving forward and inverse problems of anomalous heat conduction in functionally graded materials

In this paper, adaptive fractional physics-informed neural networks (PINNs) are employed to solve the forward and inverse problems of anomalous heat conduction in three-dimensional functionally graded materials. In adaptive fractional PINNs, the finite difference L1 scheme is employed to discretize the time fractional derivatives in anomalous heat conduction problems. By combining the finite difference L1 scheme with automatic differentiation technique, adaptive fractional PINNs minimize a loss function constructed based on the governing equation, initial and boundary conditions to obtain solutions to heat conduction problems. To avoid competition among various loss terms during the training process, an adaptive loss balancing algorithm is adopted to balance the interactions among different terms of the loss functions. In addition, polynomial basis functions are used to expand unknown functions characterizing material parameters or heat sources, aiming to enhance the performance of adaptive fractional PINNs in resolving inverse problems. Five numerical examples involving forward, inverse, and nonlinear problems related to anomalous heat conduction are carried out to demonstrate the feasibility and effectiveness of adaptive fractional PINNs.

Subdiffusive and superdiffusive phonon transport induced significant thermal rectification across a one-dimensional Kapitza interface

Anomalous heat conduction in low dimensional materials shows divergent thermal conductivity with increasing characteristic length, which brings many opportunities in thermal management applications and unexplored phonon transport mechanisms. The molecular heterojunctions, consisting of an interface that is formed by different lattices, exhibit unique physical properties especially thermal rectification phenomenon. In this work, we investigate the heat conduction across a one-dimensional heterojunction bridging a carbon nanotube (CNT) and a boron nitride nanotube (BNNT). The molecular dynamic (MD) simulation shows that the rectification ratio is as high as 8.9 in a 300-nm system at/above the room temperature, which is due to the asymmetric interfacial thermal conductance in the opposite directions of heat flow. The interfacial thermal conductance (G) obeys the length dependence of G ∝ Lβ, presenting a divergent and convergent behaviors of anomalous heat conduction that was only reported for homogeneous materials. By analyzing the length-dependent and temperature-dependent phonon density of state, this anomalous interfacial heat conduction can be explained by the Kapitza resistance model and the population of frequency-dependent phonons. Further calculation of mean square displacement of atoms validates the superdiffusion and subdiffusion behavior of the interfacial conductance. This result demonstrates the superb capability of the one-dimensional heterojunction to control phonon transport, which could be used as the basic element of high-efficient and intelligent thermal management devices.

Electron–phonon coupling and non-equilibrium thermal conduction in ultra-fast heating systems

A discrete unified gas kinetic scheme is developed to solve the Boltzmann transport equation (BTE) with coupled electron and phonon thermal transport, whose time step is not limited by the relaxation time. The key of the present scheme is that the electron/phonon advection, scattering and electron–phonon interactions are coupled together within one time step by solving the BTE again at the cell interface. Numerical results show that the present scheme can correctly predict the electron–phonon coupling effects, and is in agreement with typical two-temperature model and experimental results in existing literatures and our performed time-domain thermoreflectance technique. It can also capture the ballistic effects when the characteristic length is comparable to or smaller than the mean free path where the two-temperature model fails. In addition, two anomalous heat conduction phenomena are predicted by the present scheme, namely, heat flows from phonon to electron in transient thermal grating geometry and re-rise reflected signal appears in Au/Pt bilayer metal structure. The present work can provide theoretical guidance for the study of electron–phonon coupling and offer a useful tool for multi-scale thermal management.

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.

Thermal conductivity in one-dimensional electronic fluids

We study thermal conductivity in one-dimensional electronic fluids combining kinetic [R. Samanta, I. V. Protopopov, A. D. Mirlin, and D. B. Gutman, Thermal transport in one-dimensional electronic fluid, Phys. Rev. Lett. 122, 206801 (2019)] and hydrodynamic [I. V. Protopopov, R. Samanta, A. D. Mirlin, and D. B. Gutman, Anomalous hydrodynamics in one-dimensional electronic fluid, Phys. Rev. Lett. 126, 256801 (2021)] theories. The kinetic approach is developed by partitioning the Hilbert space into bosonic and fermionic sectors. We focus on the regime where the long-living thermal excitations are fermions and compute thermal conductivity. From the kinetic theory standpoint, the fermionic part of thermal conductivity is normal, while the bosonic one is anomalous, that scales as ω–1/3 and thus dominates in the infrared limit. The multi-mode hydrodynamic theory is obtained by projecting the fermionic kinetic equation on the zero modes of its collision integral. On a bare level, both theories agree and the thermal conductivity computed in hydrodynamic theory matches the result of the kinetic equation. The interaction between hydrodynamic modes leads to renormalization and consequently to anomalous scaling of the transport coefficients. In a four-mode regime, all modes are ballistic and the anomaly manifests itself in Kardar-Parisi-Zhang-like broadening with asymmetric power-law tails. “Heads” and “tails” of the pulses contribute equally to thermal conductivity, leading to ω–1/3 scaling of heat conductivity. In the three-mode regime, the system is in the universality class of a classical viscous fluid [Herbert Spohn, Nonlinear fluctuating hydrodynamics for anharmonic chains, J. Stat. Phys. 154, 1191 (2014); O. Narayan and S. Ramaswamy, Anomalous heat conduction in one-dimensional momentum-conserving systems, Phys. Rev. Lett. 89, 200601 (2002)].

Comparison of Heuristic Algorithms in Identification of Parameters of Anomalous Diffusion Model Based on Measurements from Sensors.

In recent times, fractional calculus has gained popularity in various types of engineering applications. Very often, the mathematical model describing a given phenomenon consists of a differential equation with a fractional derivative. As numerous studies present, the use of the fractional derivative instead of the classical derivative allows for more accurate modeling of some processes. A numerical solution of anomalous heat conduction equation with Riemann-Liouville fractional derivative over space is presented in this paper. First, a differential scheme is provided to solve the direct problem. Then, the inverse problem is considered, which consists in identifying model parameters such as: thermal conductivity, order of derivative and heat transfer. Data on the basis of which the inverse problem is solved are the temperature values on the right boundary of the considered space. To solve the problem a functional describing the error of the solution is created. By determining the minimum of this functional, unknown parameters of the model are identified. In order to find a solution, selected heuristic algorithms are presented and compared. The following meta-heuristic algorithms are described and used in the paper: Ant Colony Optimization (ACO) for continous function, Butterfly Optimization Algorithm (BOA), Dynamic Butterfly Optimization Algorithm (DBOA) and Aquila Optimize (AO). The accuracy of the presented algorithms is illustrated by examples.

Phononic thermal conduction and thermal regulation in low-dimensional micro-nano scale systems: Nonequilibrium statistical physics problems from chip heat dissipation

“Heat death”, namely, overheating, which will deteriorate the function of chips and eventually burn the device and has become an obstacle in the roadmap of the semiconductor industry. Therefore, heat dissipation becomes a key issue in further developing semiconductor. Heat conduction in chips encompasses the intricate dynamics of phonon conduction within one-dimensional, two-dimensional materials, as well as the intricate phonon transport through interfaces. In this paper, the research progress of the complexities of phonon transport on a nano and nanoscale in recent three years, especially the size dependent phonon thermal transport and the relationship between anomalous heat conduction and anomalous diffusion are summarized. Further discussed in this paper is the fundamental question within non-equilibrium statistical physics, particularly the necessary and sufficient condition for a given Hamiltonian whose macroscopic transport behavior obeys Fourier’s law. On the other hand, the methods of engineering the thermal conduction, encompassing nanophononic crystals, nanometamaterials, interfacial phenomena, and phonon condensation are also introduced. In order to comprehensively understand the phononic thermal conduction, a succinct overview of phonon heat transport phenomena, spanning from thermal quantization and the phonon Hall effect to the chiral phonons and their intricate interactions with other carriers is presented. Finally, the challenges and opportunities, and the potential application of phonons in quantum information are also discussed.

Parametric dependencies of anomalous ion heat conductivity in T-10 plasma with Ohmic heating

Ion heat conductivity in deuterium plasma of the circular limiter tokamak T-10 is analyzed in more than 100 Ohmic shots. Four data scans: plasma density, effective charge, plasma current, and toroidal magnetic field are presented. Each scan is formed with the preservation of other parameters. It is shown that at the plasma periphery, r/a=0.6−0.8, the ion heat conductivity exceeds neoclassical values up to three to eight times. The anomalous part χian increases with plasma density and effective charge, while it decreases with plasma current and does not depend on the toroidal magnetic field. The highest value of χian is observed in high-density low-current regimes. More negative values of plasma electric potential correlate with higher values of χian. The empirical scaling of χian is derived. The transport model based on this scaling allows us to describe experimental profiles of ion temperature Ti(r) within experimental error ±15%.

Numerical study of anomalous heat conduction in absorber plate of a solar collector using time-fractional single-phase-lag model

The primary purpose of this work is to analyze the intermediate behaviors in the thermal response of an absorber plate of a flat-plate solar collector, considering the non-Fourier effects. The Caputo fractional derivative is applied and the time-fractional single-phase-lag (FSPL) model is presented to study the non-diffusive process in the heat conduction through the absorber plates. In addition, the effect of variations in different parameters such as Vernotte number, dimensionless time, and thermo-physical parameter are studied on the temperature response of absorber plates. It has been apparent that the thermo-physical parameter can significantly affect the thermal behaviors of absorber plates, and that for small quantities of this parameter, the temperature distribution follows the wave-like behaviors. To validate this model, the fractional model is presented for a thin metal film, and non-diffusive heat conduction has been investigated in detail, and the results of the fractional model are compared to those of the dual-phase-lag (DPL) model, which proves the precision of the fractional model proposed in this work. Also, the effect of variation in the time rate of the initial condition has been conducted on the thermal distribution of metal films. It has been discussed that the fractional model proposed can describe a wide range of behavior in thermal response, including diffusive-like and wave-like behaviors. In addition, it has been found that the non-Fourier effects are quite remarkable at earlier times, so it would be important to consider such effects when the time instances are smaller than the relaxation time.

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.

A meshless collocation method for solving the inverse Cauchy problem associated with the variable-order fractional heat conduction model under functionally graded materials

A localized meshless collocation method, namely the generalized finite difference method (GFDM), is introduced to cope with the inverse Cauchy problem associated with the fractional heat conduction model under functionally graded materials (FGMs). The variable-order time-fractional heat conduction equation under the Caputo definition is employed to describe anomalous heat conduction problems. In the present numerical framework, temporal-discretization is implemented by using the standard implicit finite difference method with the spatial-discretization through the GFDM. On the basis of moving least squares and Taylor series expansion, the GFDM is capable of avoiding the ill-posedness in inverse Cauchy problems and solving the time fractional heat conduction equations (TFHCE) under FGMs. To give evidence of the efficiency and accuracy of the proposed approach for solving inverse heat conduction of FGMs, three numerical experiments are considered in the results and discussions section.

A localized collocation scheme with fundamental solutions for long-time anomalous heat conduction analysis in functionally graded materials

This paper makes the first attempt to develop a novel localized collocation scheme based on fundamental solutions for long-time anomalous heat conduction analysis in functionally graded materials. In the present numerical framework, the Laplace transformation and Fixed Talbot numerical inverse Laplace transformation are employed to deal with the time fractional derivative in anomalous time fractional heat conduction model. The localized collocation scheme based on fundamental solutions is applied to solve Laplace-transformed time-independent partial differential equations (PDEs), and the generalized reciprocity method (GRM) coupled with the derived problem-dependent generalized fundamental solution is proposed to deal with the inhomogeneous term in Laplace-transformed time-independent PDEs. Numerical tests demonstrate that the developed computational strategy can be considered as one of the competitive alternatives in the simulation of anomalous heat conduction behavior of functionally graded materials. Furthermore, the effect of time fractional derivative order on the behavior of anomalous heat conduction is numerically investigated.

High precision simulation and analysis of non-Fourier heat transfer during laser processing

With the rapid development of ultrashort-pulse laser heating and nanomaterials in science and engineering, the research on non-Fourier models to predict the anomalous heat conduction has attracted more and more attention. Among the existing non-Fourier models, the dual-phase-lag model can provide the best performance and it is more suitable for a short duration of heating. Considering the complexity of the equation, an efficient and accurate numerical method is proposed and applied to the analysis of the non-Fourier problems with complex geometry. The Cattaneo-Vernotte model and the dual-phase-lag model are selected and analyzed by the proposed method in the process of laser heating. Numerical simulations are made conducted for 1-D, 2-D, and 3-D problems in single-phase media exposed to laser pulses. The numerical results represent that the proposed method is accurate and stable to predict the thermal behavior of the plate irradiated by laser and other non-Fourier heat transfer problems.

Anomalous Hydrodynamics in a One-Dimensional Electronic Fluid.

We construct multimode viscous hydrodynamics for one-dimensional spinless electrons. Depending on the scale, the fluid has six (shortest lengths), four (intermediate, exponentially broad regime), or three (asymptotically long scales) hydrodynamic modes. Interaction between hydrodynamic modes leads to anomalous scaling of physical observables and waves propagating in the fluid. In the four-mode regime, all modes are ballistic and acquire Kardar-Parisi-Zhang (KPZ)-like broadening with asymmetric power-law tails. "Heads" and "tails" of the waves contribute equally to thermal conductivity, leading to ω^{-1/3} scaling of its real part. In the three-mode regime, the system is in the universality class of a classical viscous fluid [O. Narayan and S. Ramaswamy, Anomalous Heat Conduction in One-Dimensional Momentum-Conserving Systems, Phys. Rev. Lett. 89, 200601 (2002).PRLTAO0031-900710.1103/PhysRevLett.89.200601, H. Spohn, Nonlinear fluctuating hydrodynamics for anharmonic chains, J. Stat. Phys. 154, 1191 (2014).JSTPBS0022-471510.1007/s10955-014-0933-y]. Self-interaction of the sound modes results in a KPZ-like shape, while the interaction with the heat mode results in asymmetric tails. The heat mode is governed by Levy flight distribution, whose power-law tails give rise to ω^{-1/3} scaling of heat conductivity.

Modelling charge carrier transport with anomalous diffusion and heat conduction in amorphous semiconductors using fractional calculus

An ordered semiconductor has a crystalline lattice in which charge carriers move around by the Gaussian process of normal diffusion. The mean square displacement (MSD) of these charge carriers is proportional to time. On the contrary, the movement of carriers in a material with a non-crystalline structure such as amorphous semiconductors is considered to be non-Gaussian in nature. In this case, MSD is proportional to some power of time. Diffusion in this type of transport mechanism is classified as anomalous diffusion. The usual drift-diffusion equation (DDE) cannot adequately describe this process because it has non-Gaussian and dispersive transport mechanisms. Fractional calculus has been used to generalize the standard DDE to a time fractional equation in order to include the hereditary effects of the carrier transport. For power devices, the distribution and conduction of heat is the primary criteria considered when making a device. Therefore, an equation for heat conduction is added to the model for inclusion of variable temperature. The coupled system is solved using a Numerical scheme wherein Finite Difference method has been employed to discretize the Riemann - Liouville time derivative of order α and the space variable. The effects of different physical factors such as light intensity, heat and applied electric field are discussed with the help of graphical illustrations.

Universal scaling for recovery of Fourier's law in low-dimensional solids under momentum conservation.

Dynamic renormalization group (RG) of fluctuating viscoelastic equations is investigated to clarify the cause for numerically reported disappearance of anomalous heat conduction (recovery of Fourier's law) in low-dimensional momentum-conserving systems. RG flow is obtained explicitly for simplified two model cases: a one-dimensional continuous medium under low pressure and incompressible viscoelastic medium of arbitrary dimensions. Analyses of these clarify that the inviscid fixed point of contributing the anomalous heat conduction becomes unstable under the RG flow of nonzero elastic-wave speeds. The dynamic RG analysis further predicts a universal scaling of describing the crossover between the growth and saturation of observed heat conductivity, which is confirmed through the numerical experiments of Fermi-Pasta-Ulam β (FPU-β) lattices.

A boundary collocation method for anomalous heat conduction analysis in functionally graded materials

This paper applies a semi-analytical boundary collocation method, the singular boundary method (SBM), in conjunction with the dual reciprocity method (DRM) and Laplace transformation technique to solve anomalous heat conduction problems under functionally graded materials (FGMs). In this study, transient heat conduction equation with Caputo time fractional derivative is considered to describe anomalous heat conduction phenomena. In the present numerical implementation, Laplace transformation and numerical inverse Laplace transformation are used to deal with the Caputo time fractional derivative, which avoid the effect of time step on the computational efficiency of the time fractional derivation approximation. The SBM in conjunction with the DRM is employed to obtain the high accurate results in the solution of Laplace-transformed time-independent nonhomogeneous problems. To demonstrate the effectiveness of the proposed method for anomalous heat conduction analysis under functionally graded materials, three numerical examples are considered and the present results are compared with known analytical solutions and COMSOL simulation.

Green-Kubo algorithm in the calculation of anomalous heat conduction for models with and without sound mode

Green-Kubo algorithm is an effective method for the calculation of transport coefficients in terms of integral of the current correlation function. In this paper, we investigate two important issues of this algorithm in the calculation of anomalous heat conduction. Firstly, since the correlation function should be calculated in the thermodynamic limit which is never possible in practise, the necessary size of system should be determined. Secondly and more importantly, in the anomalous heat conduction cases, in order to work out a length-N-dependent heat conductivity κ(N), we need to set a cutoff time τN as a upper limit of the integral. τN is commonly set to be proportional to N. However it has been observed very recently that in a model without sound mode τN is not that simple but grows as Nν with ν = 1.5 instead. We apply the algorithm to two typical one-dimensional models with and without sound modes, and find that the necessary size is extremely small for the model without sound mode, but relatively large for the model with strong sound modes. By studying the heat diffusion process via the local energy correlation, the value of τN can also be quantitatively determined.

A superstatistical model for anomalous heat conduction and diffusion

We propose a superstatistical model for anomalous heat conduction and diffusion, which is formulated by the thermal conductivity distribution, overall temperature and heat flux distributions. Our model obeys Fourier's law and the continuity equation at the individual level. The evolution of the thermal conductivity distribution is described by an advection-diffusion equation. We show that the superstatistical model predict anomalous behaviors including the time-dependent effective thermal conductivity and slow long-time asymptotics. The time-dependence of the effective thermal conductivity is determined by the mean square displacement (MSD), which coincides with existing investigations. The superstatistical structure can also be extended into other non-Fourier models including the Cattaneo and fractional-order heat conduction models.

Anomalous heat equations based on non-Brownian descriptions

The Brownian description of heat conduction proposed by Razi-Naqvi and Waldenstrøm (2005) is generalized into non-Brownian anomalous heat conduction. It is found that there exists an entropic relation between the heat equation and Fokker–Planck equation (FPE) of energy fluctuations. Based on the entropic relation and fractional Brownian motion (FBM), we propose an anomalous heat equation (AHE), which is able to perform Brownian and non-Brownian long-time asymptotics of the mean-square displacement (MSD), Δx2∝tβ. The AHE predicts a power-law length-dependence of the effective thermal conductivity κeff connected to the MSD, namely, κeff∝Lβ−1 with L denoting the system length. This scaling connection has been observed in the Lévy Walk (LW) model and linear response theory. Due to the coincidences with existing studies, the AHE can be considered as phenomenological models for anomalous heat conduction.