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Super heat diffusion in one-dimensional momentum-conserving nonlinear lattices
Abstract
Heat diffusion processes in various one-dimensional total-momentum-conserving nonlinear lattices with symmetric interaction and asymmetric interaction are systematically studied. It is revealed that the asymmetry of interaction largely enhances the heat diffusion; while according to our existing studies for heat conduction in the same lattices, it slows the divergence of heat conductivity in a wide regime of system size. These findings violate the proposed relations that connect anomalous heat conduction and super heat diffusion. The generality of those expectations is thus questioned.
... From the connection theory [49], the thermal conductivity will diverge as κ ∝ N α with α = β − 1 exhibiting anomalous heat conduction behavior. The connection theory has been verified quantitatively by numerical simulations for lattices with symmetrical potential [49,64] but fails for lattices with asymmetrical potential [64]. However, one can still use the diffusion method to determine whether the heat conduction is normal or anomalous qualitatively. ...
... From the connection theory [49], the thermal conductivity will diverge as κ ∝ N α with α = β − 1 exhibiting anomalous heat conduction behavior. The connection theory has been verified quantitatively by numerical simulations for lattices with symmetrical potential [49,64] but fails for lattices with asymmetrical potential [64]. However, one can still use the diffusion method to determine whether the heat conduction is normal or anomalous qualitatively. ...
... The energy density E = 1 and the lattice length N = 1001. relation between α and β here deviates from the connection theory as noticed in a recent work [64]. It might be that the asymptotic length is very large for asymmetric lattices. ...
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 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 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.
... An immediate consequence of this relation is β = α + 1. However, numerical simulations in some one-dimensional (1D) nonlinear lattices suggested that such a relation is valid only in those lattices with symmetric interactions and clear discrepancy can be observed in those with asymmetric interactions [13]. ...
... Such definitions guarantee the continuity equation discussed below [13]. Periodic boundary conditions are applied and the initial states are randomly extracted from the microcanonical ensemble with fixed energy density that corresponds to the desired temperature [12]. ...
... Equation (34) is, however, found to be valid only in systems without temperature pressure, i.e., lattices with only symmetric interactions. In lattices with nonzero temperature pressure, e.g., the FPU-αβ lattices, clear discrepancies can be observed in numerical simulation [13]. To clarify it, we reproduce the proof for discrete lattices. ...
In this paper, we study systematically a serial of correlation functions in some one-dimensional nonlinear lattices. Due to the energy conservation law, they are implicitly interdependent. Various transport coefficients are thus also connected. In the studies of the autocorrelations of local energy density and of local heat current, a general relation between diverging heat conduction and super heat diffusion has been proposed recently. We clarify that such a relation is valid only in systems without temperature pressure. In those with temperature pressure, a constant but nontrivial term appears. This term explains a previously observed fact that heat diffusion in such systems is always ballistic but heat conduction can diverge very slowly. Such a result not only disproves the existence of any general relation between diverging heat conduction and super heat diffusion, but it also breaks the long-term presumption that ballistic heat conduction and diffusion always coexist.
... gives us the leading behavior σ 2 (t) ∼ t 3−1/γ which then leads to the relation β = 3 − 1/γ. Observations from several other numerical simulations have confirmed the super-diffusive behavior [8,[54][55][56][57][58][59]. ...
... The relaxation part satisfies the equations given in Eqs. (55,56,57), while the steady state part satisfies these equations but with ∂ τ T ss (v) = 0. The boundary conditions for the steady state part are given by [34] C ss (u, z → 0) = 0, C ss (u → ∞, z) = 0, C ss (u = 0, z) = J/2. ...
It has been observed in many numerical simulations, experiments and from various theoretical treatments that heat transport in one-dimensional systems of interacting particles cannot be described by the phenomenological Fourier's law. The picture that has emerged from studies over the last few years is that Fourier's law gets replaced by a spatially non-local linear equation wherein the current at a point gets contributions from the temperature gradients in other parts of the system. Correspondingly the usual heat diffusion equation gets replaced by a non-local fractional-type diffusion equation. In this review, we describe the various theoretical approaches which lead to this framework and also discuss recent progress on this problem.
... gives us the leading behavior σ 2 (t) ∼ t 3−1/γ which then leads to the relation β = 3 − 1/γ . Observations from several other numerical simulations have confirmed the superdiffusive behavior [8,[54][55][56][57][58][59]. ...
... We note that under this transformation, the "anti-diffusion" Equation (55), becomes a diffusion equation, with v as the time variable and z the space variable. The relaxation part satisfies the equations given in Equations (55,56,57), while the steady state part satisfies these equations but with ∂ τ T ss (v) = 0. The boundary conditions for the steady state part are given by Priyanka et al. [34] C ss (u, z → 0) = 0, C ss (u → ∞, z) = 0, C ss (u = 0, z) = J/2. ...
It has been observed in many numerical simulations, experiments and from various theoretical treatments that heat transport in one-dimensional systems of interacting particles cannot be described by the phenomenological Fourier's law. The picture that has emerged from studies over the last few years is that Fourier's law gets replaced by a spatially non-local linear equation wherein the current at a point gets contributions from temperature gradients in other parts of the system. Correspondingly the usual heat diffusion equation gets replaced by a non-local fractional-type diffusion equation. In this review, we describe the various theoretical approaches which lead to this framework and also discuss recent progress on this problem.
... for momentum-conserving lattices [9,13], where v s (≡ v k=0 = dω dk | k=0 ) is the sound velocity which will be calculated numerically [14,15] in this Rapid Communication; and ...
... The first one is the FPU-α2β lattice. Our previous work suggests that the asymmetric interaction [V (x) = V (−x)] can enhance the diffusion while reducing the heat conduction [15]. But the underlying mechanism behind this is still not clear. ...
Based on the linear response theory, we propose a resonance phonon (r-ph) approach to study the renormalized phonons in a few one-dimensional nonlinear lattices. Compared with the existing anharmonic phonon (a-ph) approach, the dispersion relations derived from this approach agree with the expectations of the effective phonon (e-ph) theory much better. The application is also largely extended, i.e., it is applicable in many extreme situations, e.g., high frequency, high temperature, etc., where the existing one can hardly work. Furthermore, two separated phonon branches (one acoustic and one optical) with a clear gap in between can be observed by the r-ph approach in a diatomic anharmonic lattice. While only one combined branch can be detected in the same lattice with both the a-ph approach and the e-ph theory.
... where x = n − 0, A n (t ) = A n (t ) − A n is the fluctuation of A of the nth particle at time t, and · · · denotes an average over the initial conditions. The energy of the nth particle is chosen as the symmetric form E n = p 2 n /2 + [V (r n−1 ) + V (r n )]/2 to calculate C E [39]. The temperature T = 0.5 is used to calculate (≈0.4474) for FPUT-β chains. ...
The well known nonlinear fluctuating hydrodynamics theory has grouped diffusions in anharmonic chains into two universality classes: one is the Kardar-Parisi-Zhang (KPZ) class for chains with either asymmetric potential or nonzero static pressure and the other is the Gaussian class for chains with symmetric potential at zero static pressure, such as Fermi-Pasta-Ulam-Tsingou (FPUT)-β chains. However, little is known of the nonequilibrium transient diffusion in anharmonic chains. Here, we reveal that the KPZ class is the only universality class for nonequilibrium transient diffusion, manifested as the KPZ scaling of the side peaks of momentum correlation (corresponding to the sound modes correlation), which was completely unexpected in equilibrium FPUT-β chains. The underlying mechanism is that the nonequilibrium soliton dynamics cause nonzero transient pressure so that the sound modes satisfy approximately the noisy Burgers equation, in which the collisions of solitons was proved to yield the KPZ dynamic exponent of the soliton dispersion. Therefore, the unexpected KPZ universality class is obtained in the nonequilibrium transient diffusion in FPUT-β chains and the corresponding carriers of nonequilibrium transient diffusion are attributed to solitons.
... Some studies also suggested normal heat conduction in some 1D momentum-conserving models with proper asymmetric interactions [12,13]. Interestingly, heat diffusion in these lattices is quite close to ballistic [14]. In the limiting case k 3 = 0, the lattice reduces to a FPU β lattice, i.e., V (x) = 1 2 x 2 + 1 4 x 4 and U (x) = 0. ...
Nonstationary heat conduction in a few one-dimensional nonlinear lattices is studied numerically based on the Maxwell-Cattaneo (MC) law. We simulate the relaxation process and calculate the magnitudes of the temperature oscillation A_{T}(t) and the local heat current oscillation A_{j}(t). A phase difference between A_{T}(t) and A_{j}(t) is observed, which not only verifies the existence of the time lag τ in the MC law but also provides a better way of determining the critical wavelength L^{*} that separates between oscillatory and diffusive relaxation modes. However, clear deviations from the MC law are observed. Not only do the decay exponents differ from the theoretical expectations, but, more importantly, suboscillation in the diffusive regime, which is not expected by the MC law, is found in the lattices with asymmetric interactions as well. These findings imply that higher-order effects must be considered in order to well describe the nonstationary heat conduction process in these systems.
... The time dependence of the MSD of the excess energy has been plotted in Fig. 8(b) and a superdiffusion with ∆x 2 (t) E ∝ t 1.80 has been obtained. The relation between α and β here deviates from the connection theory as noticed in recent work [54]. It might be that the asymptotic length is very large for asymmetric lattices. ...
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 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.
It is expected that the energy-diffusion propagator in a one-dimensional nonlinear lattice with three conserved quantities: energy, momentum, and stretch, consists of a central heat mode and two sound modes. The heat mode follows a Lévy distribution. Consequently, the heat diffusion is super, i.e., the second moment of the diffusion propagator diverges as tβ with β>1; and the heat conduction is anomalous, i.e., the heat conductivity is size dependent and diverges with size N by Nα, with α>0. In this paper, we study a one-dimensional lattice with two-dimensional transverse motions, in which the total angular momentum also conserves. More importantly, the diffusion of this conserved quantity is ballistic. Surprisingly, the above pictures and the values of the mentioned power exponents keep unchanged. The universality of the scalings is then further extended. On the other hand, the detailed strengths of heat transports are largely enhanced. Such a counterintuitive finding can be explained by the change of the phonon mean-free path of the lattices.
The energy diffusion process in a few two-dimensional Fermi-Pasta-Ulam-type lattices is numerically simulated via the equilibrium local energy spatiotemporal correlation. Just as the nonlinear fluctuating hydrodynamic theory suggested, the diffusion propagator consists of a bell-shaped central heat mode and a sound mode extending with a constant speed. The profiles of the heat and sound modes satisfy the scaling properties from a random-walk-with-velocity-fluctuation process very well. An effective phonon approach is proposed, which expects the frequencies of renormalized phonons as well as the sound speed with quite good accuracy. Since many existing analytical and numerical studies indicate that heat conduction in such two-dimensional momentum-conserving lattices is divergent and the thermal conductivity κ increases logarithmically with lattice length, it is expected that the mean-square displacement of energy diffusion grows as tlnt. Discrepancies, however, are noticeably observed.
The interactive risks of different devices, serving as clients or servers, have increasingly attracted huge attention in various communication systems, such as wireless sensor networks, wireless communication networks, and mobile crowd‐sensing. At present, lots of countermeasures had been proposed and deployed accordingly. Nevertheless, the investigation on the interaction risks between different devices is still very limited to date. In this paper, we propose a novel adverse effect inference mechanism TAEffect for malicious behaviors of devices emerged in various decentralized and open communication systems/networks through network‐percolation theory. At first, four typical malicious interactional behaviors are mapped into four topologies, then, upon which a network influence‐inspired approach is employed to quantify the adverse effect. Finally, multifacet experiments using five real‐world datasets and a synthetic testbed are performed to validate the efficiency and effectiveness. The experimental results show our proposed approach is significant and rational to quantitatively calculate and qualitatively mirror the four kinds of malicious interactional behaviors in diverse misbehavior‐emerged communication systems. This work aims to explore how to evaluate the adverse effect for various malicious participants in an objective manner. First, four typical malicious interactional behaviors are defined and mapped into four corresponding network topologies, then, upon which an effective method to quantitatively calculate the adverse effect is figured out using network‐percolation theory.
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 Teq 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.
Anomalous energy transport is a widely investigated character in one-dimensional lattice and both phonons and solitons are candidate energy carriers responsible for the thermal conductivity divergence before their mean-free paths. However, it was long believed the contribution of solitons could be neglected even close to the melting point in a real crystalline solid. In this paper, we show that a crossover of wave-packet dynamics, from a phonon- to a soliton-dominated state, occurs in a quasi-one-dimensional molybdenum disulfide (MoS2) sheet at high temperatures far below its melting point by nonequilibrium and equilibrium molecular dynamics simulations. Divergent sound speed variation and the corresponding heat capacity peaks are observed in the transition-temperature region that is related to a second-order phase transition. We also find that its anomalous energy transport falls into a universality class with thermal conductivity divergence exponent α=2/5 at a finite length scale within 2000 nm when solitons are excited above 600 K, and the scaling relations derived from the Lévy walk of energy carriers is fulfilled as α=2−1/γ=β−1 in the soliton-dominated state above 1800 K. Our results reveal the peculiar solitonlike contribution to thermal conduction at high temperatures in the low-dimensional crystalline solids.
The property of total momentum conservation is a key issue in determining the energy diffusion behavior for 1d nonlinear lattices. The super-diffusion of energy has been found for 1d momentum conserving nonlinear lattices with the only exception of 1d coupled rotator model. However, for all the other 1d momentum non-conserving nonlinear lattices studied so far, the energy diffusion is normal. Here we investigate the energy diffusion in a 1d nonlinear lattice model with inverse couplings. For the standard definition of momentum, this 1d inverse coupling model does not preserve the total momentum while it exhibits energy super-diffusion behavior. In particular, with a parity transformation, this 1d inverse coupling model can be mapped into the well-known 1d FPU-β model although they have different phonon dispersion relations. In contrary to the 1d FPU-β model where the long-wave length phonons are responsible for the super-diffusion behavior, the short-wave length phonons contribute to the super-diffusion of energy in the 1d inverse coupling model.
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.
We present a chain-of-atoms model where heat is rectified, with different fluxes from the hot to the cold baths located at the chain boundaries when the temperature bias is reversed. The chain is homogeneous except for boundary effects and a local modification of the interactions at one site, the "impurity". The rectification mechanism is due here to the localized impurity, the only asymmetrical element of the structure, apart from the externally imposed temperature bias, and does not rely on putting in contact different materials or other known mechanisms such as grading or long-range interactions. The effect survives if all interaction forces are linear except the ones for the impurity.
We extend a previously proposed resonance phonon approach that is based on the linear response theory. By studying the complex response function in depth, we work out the phonon relaxation time besides the oscillating frequency of the phonons in a few one-dimensional nonlinear lattices. The results in the large wave-number-k regime agree with the expectations of the effective phonon theory. However, in the small-k limit they follow different scaling laws. The phonon mean free path can also be calculated indirectly. It coincides well with that derived from the anharmonic phonon approach. A power-law divergent heat conduction, i.e., the heat conductivity κ depends on lattice length N by κ∼Nβ with β>0, then is supported for the momentum-conserving lattices. Furthermore, this approach can be applied to diatomic lattices. So obtained relaxation time quantitatively agrees with that from the effective phonon theory. As for the mean free path, the resonance phonon approach can detect both the acoustic and the optical branches, whereas the anharmonic phonon approach can only detect a combined branch, i.e., the acoustic branch for small k and the optical branch for large k.
We show through calculations on typical lattice models that thermal conductance σ can well describe the near-equilibrium thermal transport property of one-dimensional materials of finite length, which presents a situation often met in the application of nanoscale devices. The σ generally contains contributions from the material itself and those from the thermal reservoirs. The intrinsic σ of the material, i.e., the one with the fewest external influences, can be efficiently calculated with the help of the “blackbody”-like nonreflective thermal reservoir, either through the nonequilibrium method or through the Green-Kubo-type formula. σ thus calculated would be helpful to guide the design of thermal management and heat control in nanoscale devices.
We study numerically the process of dynamical thermalization in the Frenkel-Kontorova (FK) model with weak nonlinearity. The total energy has initially equidistributed among some of the lowest frequency linear modes. It is found that the energy transfers continuously to the high-frequency modes and finally evolves towards energy equipartition in the FK model. However, the metastable state, which was found in Fermi-Pasta-Ulam (FPU) model and φ4 model in a relatively short time scale, is not found in the FK model. We further perform a very accurate systematic study of the equipartition time Teq as functions of the particle number N, the nonlinear parameter β, and the energy density ɛ. In the thermodynamic limit, the dependence of Teq on β and ɛ is found to display a power law behavior: Teq∝βaɛb. The exponents a and b are numerically found to be approximately −2.0 and 1.43. This scaling law is also quite different from those of the FPU-β model and φ4 model.
The study of heat transport in low-dimensional oscillator lattices presents a
formidable challenge. Theoretical efforts have been made trying to reveal the
underlying mechanism of diversified heat transport behaviors. In lack of a
unified rigorous treatment, approximate theories often may embody controversial
predictions. It is therefore of ultimate importance that one can rely on
numerical simulations in the investigation of heat transfer processes in
low-dimensional lattices. The simulation of heat transport using the
non-equilibrium heat bath method and the Green-Kubo method will be introduced.
It is found that one-dimensional (1D), two-dimensional (2D) and
three-dimensional (3D) momentum-conserving nonlinear lattices display power-law
divergent, logarithmic divergent and constant thermal conductivities,
respectively. Next, a novel diffusion method is also introduced. The heat
diffusion theory connects the energy diffusion and heat conduction in a
straightforward manner. This enables one to use the diffusion method to
investigate the objective of heat transport. In addition, it contains
fundamental information about the heat transport process which cannot readily
be gathered otherwise.
The Green-Kubo formula provides a mathematical expression for heat conductivity in terms of integrals of the heat-current correlation function, which should be calculated in the thermodynamic limit. In finite systems this function generally decreases, i.e., it decays faster than it does in infinite systems. We compared the values of the correlation function in a one-dimensional purely quartic lattice with various lengths, and found that this loss is much smaller than is conventionally estimated. By studying the heat diffusion process in this lattice, we found that, in contrast to the conventional belief, the collisions between sound modes do not noticeably affect the current correlation function. Therefore, its loss being surprisingly small can be well understood. This finding allows one to calculate the heat conductivity in a very large system with desirable accuracy by performing simulations in a system with much smaller size, and thus greatly broadens the application of the Green-Kubo method.
Two typographical errors were contained in Table 1 on page 425. Use of the values for a73 and b7′ will invalidate the RKN8(6)9FM formula pair. The correct values for these parameters are
The authors are grateful to Steve Stalos of Laurel, MD, USA, for reporting these errors.
As recently proposed, the long-time behavior of equilibrium time-correlation functions for one-dimensional systems are expected to be captured by a nonlinear extension of fluctuating hydrodynamics. We outline the predictions from the theory aimed at the comparison with molecular dynamics. We report on numerical simulations of a fluid with a hard-shoulder potential and of a hard-point gas with alternating masses. These models have in common that the collision time is zero and their dynamics amounts to iterating collision by collision. The theory is well confirmed, with the twist that the nonuniversal coefficients are still changing at longest accessible times.
Recent work has developed a nonlinear hydrodynamic fluctuation theory for a
chain of coupled anharmonic oscillators governing the conserved fields, namely
stretch, momentum, and energy. The linear theory yields two propagating sound
modes and one diffusing heat mode. In contrast, the nonlinear theory predicts
that, at long times, the sound mode correlations satisfy Kardar-Parisi-Zhang
(KPZ) scaling, while the heat mode correlations satisfies Levy-walk scaling. In
the present contribution we report on molecular dynamics simulations of
Fermi-Pasta-Ulam chains to compute various spatiotemporal correlation functions
and compare them with the predictions of the theory. We find very good
agreement in many cases, but also some deviations.
We study the equilibrium time correlations for the conserved fields of classical anharmonic chains and argue that their dynamic correlator can be predicted on the basis of nonlinear fluctuating hydrodynamics. In fact, our scheme is more general and would also cover other one-dimensional Hamiltonian systems, for example, classical and quantum fluids. Fluctuating hydrodynamics is a nonlinear system of conservation laws with noise. For a single mode, it is equivalent to the noisy Burgers equation, for which explicit solutions are available. Our focus is the case of several modes. No exact solution has been found so far, and we rely on a one-loop approximation. The resulting mode-coupling equations have a quadratic memory kernel and describe the time evolving 3×3 correlator matrix of the locally conserved fields. Long time asymptotics is computed analytically, and finite time properties are obtained through a numerical simulation of the mode-coupling equations.
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 may exist. Relaxation behavior of local energy current fluctuations and that of local heat current fluctuations are also studied. We find that they are significantly different, though the global energy current equals the globe heat current.
We introduce a fractional Fokker-Planck equation describing the stochastic evolution of a particle under the combined influence of an external, nonlinear force and a thermal heat bath. For the force-free case, a subdiffusive behavior is recovered. The equation is shown to obey generalized Einstein relations, and its stationary solution is the Boltzmann distribution. The relaxation of single modes is shown to follow a Mittag-Leffler decay. We discuss the example of a particle in a harmonic potential.
With focus on anharmonic chains, we develop a nonlinear version of
fluctuating hydrodynamics, in which the Euler currents are kept to second order
in the deviations from equilibrium and dissipation and noise are added. The
required model-dependent parameters are written in such a way that they can be
computed numerically within seconds, once the interaction potential, pressure,
and temperature are given. In principle the theory is applicable to any
one-dimensional system with local conservation laws. The resulting nonlinear
stochastic field theory is handled in the one-loop approximation. Its large
scale predictions can still be worked out analytically. For more details one
has to rely on numerical simulations of the corresponding mode-coupling
equations. In this way we arrive at detailed predictions for the equilibrium
time correlations of the locally conserved fields of an anharmonic chain.
Relaxation is crucial for understanding various nonequilibrium processes.
However, in general how the fluctuations of a physical quantity may evolve is
still elusive. Here we show by examples that fluctuations of heat, energy,
momentum and mass may have dramatically distinct diffusion behaviors in a
system and vary from system to system. As an important consequence, the
diffusion of fluctuations of a given physical quantity can not be explored by
studying that of other physical quantities. In particular, recent efforts
trying to establish a general connection between heat conduction and energy
diffusion, and those trying to explore heat waves by investigating the
diffusion processes of quantities other than heat, are questionable.
Nevertheless, we show that for the same physical quantity, a universal
connection may exist to link its relaxation and transport such as heat
diffusion and heat conduction.
Consider anomalous energy spread in solid phases, i.e., , as induced by a small initial excess
energy perturbation distribution away from equilibrium. The
associated total thermal equilibrium heat flux autocorrelation function
is shown to obey rigorously the intriguing relation, , where c is the specific volumetric heat capacity.
Its integral assumes a time-local Helfand-moment relation; i.e. , where the chosen
cut-off time is determined by the maximal signal velocity for heat
transfer. Given the premise that the averaged nonequilibrium heat flux is
governed by an anomalous heat conductivity, energy diffusion scaling
necessarily determines a corresponding anomalous thermal conductivity scaling
behaviour.
We investigate anomalous energy transport processes in the Fermi-Pasta-Ulam
lattice. They are determined by the maximum sound velocity of the
relevant weakly damped energy carriers. That velocity can be numerically
resolved by measuring the propagating fronts of the correlation functions of
energy/momentum fluctuations at different times. The numerical results are
compared with the predictions for solitons and effective (renormalized)
phonons, respectively. Excellent agreement has been found for the prediction of
effective long wavelength phonons, giving strong evidence that the energy
carriers should be effective phonons rather than solitons.
Under quite general conditions, we prove that for classical many-body lattice Hamiltonians in one dimension (1D) total momentum conservation implies anomalous conductivity in the sense of the divergence of the Kubo expression for the coefficient of thermal conductivity, \kappa. Our results provide rigorous confirmation and explanation of many of the existing ``surprising'' numerical studies of anomalous conductivity in 1D classical lattices, including the celebrated Fermi-Pasta-Ulam problem.
Heat conduction in three types of 1D channels is studied. The channels consist of two parallel walls, right triangles as scattering obstacles, and noninteracting particles. The triangles are placed along the walls in three different ways: (i) periodic, (ii) disordered in height, and (iii) disordered in position. The Lyapunov exponents in all three models are zero because of the flatness of triangle sides. It is found numerically that the temperature gradient can be formed in all three channels, but the Fourier heat law is observed only in two disordered ones. The results show that there might be no direct connection between chaos (in the sense of positive Lyapunov exponent) and normal thermal conduction.
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.
We present analytical and numerical results on the heat conduction in a linear mixing system. In particular we consider a quasi-one-dimensional channel with triangular scatterers with internal angles which are irrational multiples of [Formula presented] and we show that the system obeys the Fourier law of heat conduction. Therefore, deterministic diffusion and normal heat transport which are usually associated with full hyperbolicity, actually take place in systems without exponential instability.
We study in momentum-conserving systems, how nonintegrable dynamics may
affect thermal transport properties. As illustrating examples, two
one-dimensional (1D) diatomic chains, representing 1D fluids and lattices,
respectively, are numerically investigated. In both models, the two species of
atoms are assigned two different masses and are arranged alternatively. The
systems are nonintegrable unless the mass ratio is one. We find that when the
mass ratio is slightly different from one, the heat conductivity may keep
significantly unchanged over a certain range of the system size and as the mass
ratio tends to one, this range may expand rapidly. These results establish a
new connection between the macroscopic thermal transport properties and the
underlying dynamics.
We study why the calculation of current correlation functions (CCFs) still suffers from finite-size effects even when the periodic boundary condition is taken. Two important one-dimensional, momentum-conserving systems are investigated as examples. Intriguingly, it is found that the state of a system recurs in the sense of microcanonical ensemble average, and such recurrence may result in oscillations in CCFs. Meanwhile, we find that the sound mode collisions induce an extra time decay in a current so that its correlation function decays faster (slower) in a smaller (larger) system. Based on these two unveiled mechanisms, a procedure for correctly evaluating the decay rate of a CCF is proposed, with which our analysis suggests that the global energy CCF decays as ̃t-2/3 in the diatomic hard-core gas model and in a manner close to ̃t-1/2 in the Fermi-Pasta-Ulam-β model.
We provide molecular-dynamics simulation of heat transport in one-dimensional molecular chains with different interparticle pair potentials. We show that the thermal conductivity is finite in the thermodynamic limit in chains with the potentials that allow for bond dissociation. The Lennard-Jones, Morse, and Coulomb potentials are such potentials. The convergence of the thermal conductivity is provided by phonon scattering on the locally strongly stretched loose interatomic bonds at low temperature and by the many-particle scattering at high temperature. On the other hand, chains with a confining pair potential, which does not allow for bond dissociation, possess anomalous thermal conductivity, diverging with the chain length. We emphasize that chains with a symmetric or asymmetric Fermi-Pasta-Ulam potential or with combined potentials, containing a parabolic and/or a quartic confining potential, all exhibit anomalous heat transport.
We have numerically studied heat conduction in a few one-dimensional momentum-conserving lattices with asymmetric interparticle interactions by the nonequilibrium heat bath method, the equilibrium Green-Kubo method, and the heat current power spectra analysis. Very strong finite-size effects are clearly observed. Such effects make the heat conduction obey a Fourier-like law in a wide range of lattice lengths. However, in yet longer lattice lengths, the heat conductivity regains its power-law divergence. Therefore, the power-law divergence of the heat conductivity in the thermodynamic limit is verified, as is expected by many existing theories.
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.
We numerically study heat conduction in a few one-dimensional Fermi-Pasta-Ulam (FPU)-type lattices by both nonequilibrium heat bath and equilibrium Green-Kubo algorithms. In those lattices, heat conductivity κ is known to diverge with length N as Nα. It is commonly expected that the running exponent α should monotonously decreases with N and a recent study has shown that α for the FPU-β lattice saturates to 1/3 as N~104. However, our calculations clearly show that α changes its behaviour, increasing towards the asymptotic value 2/5 for yet larger N values. As for the purely quartic lattice, α=2/5 is clearly observed in four orders of magnitude of N ranging from 102 to 106. This unexpected reversal phenomenon can be observed more clearly in a much shorter FPU-αβ lattice.
The heat conduction behavior of one dimensional momentum conserving lattice
systems with asymmetric interparticle interactions is numerically investigated.
It is found that with certain degree of interaction asymmetry, the heat
conductivity measured in nonequilibrium stationary states converges in the
thermodynamical limit, in clear contrast to the well accepted viewpoint that
Fourier's law is generally violated in low dimensional momentum conserving
systems. It suggests in nonequilibrium stationary states the mass gradient
resulted from the asymmetric interactions may provide an additional phonon
scattering mechanism other than that due to the nonlinear interactions.
The propagation of an initially localized perturbation via an interacting many-particle Hamiltonian dynamics is investigated. We argue that the propagation of the perturbation can be captured by the use of a continuous-time random walk where a single particle is traveling through an active, fluctuating medium. Employing two archetype ergodic many-particle systems, namely, (i) a hard-point gas composed of two unequal masses and (ii) a Fermi-Pasta-Ulam chain, we demonstrate that the corresponding perturbation profiles coincide with the diffusion profiles of the single-particle Lévy walk approach. The parameters of the random walk can be related through elementary algebraic expressions to the physical parameters of the corresponding test many-body systems.
We establish a connection between anomalous heat conduction and anomalous diffusion in one-dimensional systems. It is shown that if the mean square of the displacement of the particle is <Deltax(2)>=2Dt(alpha)(0<alpha</=2), then the thermal conductivity can be expressed in terms of the system size L as kappa=cL(beta) with beta=2-2/alpha. This result predicts that normal diffusion (alpha=1) implies normal heat conduction obeying the Fourier law (beta=0) and that superdiffusion (alpha>1) implies anomalous heat conduction with a divergent thermal conductivity (beta>0). More interestingly, subdiffusion (alpha<1) implies anomalous heat conduction with a convergent thermal conductivity (beta<0), and, consequently, the system is a thermal insulator in the thermodynamic limit. Existing numerical data support our results.
A class of dynamical heat conductors was intorduced. This class includes as particular cases recently proposed Hamiltonian billiard channels. In the absence of interactions between the particles and the independence of the dynamics on particle energy, the proposed approach goes beyond hamiltonian dynamics and allows one to express heat conductivity in terms of channel diffusion properties. Thus, the Hamiltonian character of dynamics becomes essential when interactions between particles were introduced or a dependence of dynamics on particle energy was included.
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) (0<alpha< or =2), then the thermal conductivity can be expressed in terms of the system size L as kappa=cL(beta) with beta=2-2/alpha. This result predicts that a normal diffusion (alpha=1) implies a normal heat conduction obeying the Fourier law (beta=0), a superdiffusion (alpha>1) 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.
In this Letter, I propose that a properly rescaled spatiotemporal correlation function of the energy density fluctuations may be applied to characterize the equilibrium diffusion processes in lattice systems with finite temperature. Applying this function, the diffusion processes in three one-dimensional nonlinear lattices are studied. The diffusion exponent is shown to be related to the diverging exponent of the thermal conductivity of a lattice through the relation , as has been proved based on the Lévy walk assumption. The diffusion behavior is explained in terms of solitons and phonons.
Deriving macroscopic phenomenological laws of irreversible thermodynamics from simple microscopic models is one of the tasks of non-equilibrium statistical mechanics. We consider stationary energy transport in crystals with reference to simple mathematical models consisting of coupled oscillators on a lattice. The role of lattice dimensionality on the breakdown of the Fourier's law is discussed and some universal quantitative aspects are emphasized: the divergence of the finite-size thermal conductivity is characterized by universal laws in one and two dimensions. Equilibrium and non-equilibrium molecular dynamics methods are presented along with a critical survey of previous numerical results. Analytical results for the non-equilibrium dynamics can be obtained in the harmonic chain where the role of disorder and localization can be also understood. The traditional kinetic approach, based on the Boltzmann-Peierls equation is also briefly sketched with reference to one-dimensional chains. Simple toy models can be defined in which the conductivity is finite. Anomalous transport in integrable nonlinear systems is briefly discussed. Finally, possible future research themes are outlined.
Recent results on theoretical studies of heat conduction in low-dimensional systems are presented. These studies are on simple, yet nontrivial, models. Most of these are classical systems, but some quantum-mechanical work is also reported. Much of the work has been on lattice models corresponding to phononic systems, and some on hard particle and hard disc systems. A recently developed approach, using generalized Langevin equations and phonon Green's functions, is explained and several applications to harmonic systems are given. For interacting systems, various analytic approaches based on the Green-Kubo formula are described, and their predictions are compared with the latest results from simulation. These results indicate that for momentum-conserving systems, transport is anomalous in one and two dimensions, and the thermal conductivity kappa, diverges with system size L, as kappa ~ L^alpha. For one dimensional interacting systems there is strong numerical evidence for a universal exponent alpha =1/3, but there is no exact proof for this so far. A brief discussion of some of the experiments on heat conduction in nanowires and nanotubes is also given. Comment: 78 pages, 25 figures, Review Article (revised version)