Ballistic resonance and thermalization in the Fermi-Pasta-Ulam-Tsingou chain at finite temperature

Molecular heat transport across a time-periodic temperature gradient

The time-periodic modulation of a temperature gradient can alter the heat transport properties of a physical system. Oscillating thermal gradients give rise to behaviors such as modified thermal conductivity and controllable time-delayed energy storage that are not present in a system with static temperatures. Here, we examine how the heat transport properties of a molecular lattice model are affected by an oscillating temperature gradient. We use analytical analysis and molecular dynamics simulations to investigate the vibrational heat flow in a molecular lattice system consisting of a chain of particles connected to two heat baths at different temperatures, where the temperature difference between baths is oscillating in time. We derive expressions for heat currents in this system using a stochastic energetics framework and a nonequilibrium Green's function approach that is modified to treat the nonstationary average energy fluxes. We find that emergent energy storage, energy release, and thermal conductance mechanisms induced by the temperature oscillations can be controlled by varying the frequency, waveform, and amplitude of the oscillating gradient. The developed theoretical approach provides a general framework to describe how vibrational heat transmission through a molecular lattice is affected by temperature gradient oscillations.

Nonequilibrium thermal rectification at the junction of harmonic chains

A thermal diode or rectifier is a system that transmits heat or energy in one direction better than in the opposite direction. We investigate the influence of the distribution of energy among wave numbers on the diode effect for the junction of two dissimilar harmonic chains. An analytical expression for the diode coefficient, characterizing the difference between heat fluxes through the junction in two directions, is derived. It is shown that the diode coefficient depends on the distribution of energy among wave numbers. For an equilibrium energy distribution, the diode effect is absent, while for non-equilibrium energy distributions the diode effect is observed even though the system is harmonic. We show that the diode effect can be maximized by varying the energy distribution and relative position of spectra of the two harmonic chains. Conditions are formulated under which the system acts as an ideal thermal rectifier, i.e., transmits heat only in one direction. The results obtained are important for understanding the heat transfer in heterogeneous low-dimensional nanomaterials.

Recurrence recovery in heterogeneous Fermi-Pasta-Ulam-Tsingou systems

The computational investigation of Fermi, Pasta, Ulam, and Tsingou (FPUT) of arrays of nonlinearly coupled oscillators has led to a wealth of studies in nonlinear dynamics. Most studies of oscillator arrays have considered homogeneous oscillators, even though there are inherent heterogeneities between individual oscillators in real-world arrays. Well-known FPUT phenomena, such as energy recurrence, can break down in such heterogeneous systems. In this paper, we present an approach-the use of structured heterogeneities-to recover recurrence in FPUT systems in the presence of oscillator heterogeneities. We examine oscillator variabilities in FPUT systems with cubic nonlinearities, and we demonstrate that centrosymmetry in oscillator arrays may be an important source of recurrence.

Acoustic transparency of the chain-chain interface

We study propagation of wave packets through the interface between two dissimilar harmonic chains with on-site potentials (e.g., chains lying on elastic foundations). An expression for the transmission coefficient, relating energies of the incident and transmitted wave packets is derived using two different approaches. Without elastic foundation, the transmission coefficient monotonically decreases with increasing wave frequency. We show that by adding elastic foundations, one can qualitatively change this dependence and make it nonmonotonic or even increasing. Moreover, in some cases, the interface is totally transparent (the transmission coefficient is equal to unity at some frequency) if at least one of the chains has the elastic foundation. Presented results may serve for manipulation of the transmission coefficient and corresponding interfacial thermal resistance in low-dimensional nanosystems.

Motion of localized disturbances in scalar harmonic lattices

We present analytical and numerical investigations of energy propagation in systems of massive particles that interact via harmonic (linear) forces. The particle motion is described by a scalar displacement, and the particles are arranged in a simple crystal lattice. For the systems under consideration we prove the conservation of the total energy flux analytically. Then, using a sample case of a square lattice, we confirm the analytical results numerically. We create disturbances of a special kind which can move with a predefined velocity with a minor change in their shape. We show that a clot of energy, associated with each disturbance, moves similarly to a free body of matter in classical mechanics. We also numerically study a simultaneous propagation of a number of energy clots as an analogy to the motion of point masses in the conventional mechanics of particles. The obtained results demonstrate that an energy flow in lattices can be described in terms of numerous separated energy bodies, making a step towards a linkage between lattice dynamics and the kinetic theory of heat transfer in solids.

Anomalous conduction in one-dimensional particle lattices: Wave-turbulence approach

One-dimensional particle chains are fundamental models to explain anomalous thermal conduction in low-dimensional solids such as nanotubes and nanowires. In these systems the thermal energy is carried by phonons, i.e., propagating lattice oscillations that interact via nonlinear resonance. The average energy transfer between the phonons can be described by the wave kinetic equation, derived directly from the microscopic dynamics. Here we use the spatially nonhomogeneous wave kinetic equation of the prototypical β-Fermi-Pasta-Ulam-Tsingou model, to study thermal conduction in one-dimensional particle chains on a mesoscale description. By means of numerical simulations, we study two complementary aspects of thermal conduction: in the presence of thermostats setting different temperatures at the two ends and propagation of a temperature perturbation over an equilibrium background. Our main findings are as follows. (i) The anomalous scaling of the conductivity with the system size, in close agreement with the known results from the microscopic dynamics, is due to a nontrivial interplay between high and low wave numbers. (ii) The high-wave-number phonons relax to local thermodynamic equilibrium transporting energy diffusively, in the manner of Fourier. (iii) The low-wave-number phonons are nearly noninteracting and transfer energy ballistically. These results present perspectives for the applicability of the full nonhomogeneous wave kinetic equation to study thermal propagation.

Analysis of ballistic transport and resonance in the α-Fermi-Pasta-Ulam-Tsingou model

Ballistic transport and resonance phenomena are elucidated in the one-dimensional α-Fermi-Pasta-Ulam-Tsingou (FPUT) model using an approach of computing thermal response functions. The existence of periodic oscillations in spatially sinusoidal temperature profiles seen in previous studies is confirmed. However, the results obtained using response functions enable a more complete understanding. In particular, it is shown that resonance involves beats between normal modes which tend to reinforce in a one-dimensional chain. Anharmonic scattering acts to destroy phase coherence across the statistical ensemble, and with increasing anharmonicity, transport is driven toward the diffusive regime. These results provide additional insight into anomalous heat transport in low-dimensional systems. Normal-mode scattering is also explored using time correlation functions. Interestingly, these calculations, in addition to demonstrating loss of phase coherence across an ensemble of simulations, appear to show evidence of so-called q-breathers in conditions of strong anharmonicity. Finally, we describe how the approach outlined here could be developed to include quantum statistics and also also first-principles estimates of phonon scattering rates to elucidate second sound and ballistic transport in realistic materials at low temperatures.

Unsteady two-temperature heat transport in mass-in-mass chains

We investigate the unsteady heat (energy) transport in an infinite mass-in-mass chain with a given initial temperature profile. The chain consists of two sublattices: the β-Fermi-Pasta-Ulam-Tsingou (FPUT) chain and oscillators (of a different mass) connected to each FPUT particle. Initial conditions are such that initial kinetic temperatures of the FPUT particles and the oscillators are equal. Using the harmonic theory, we analytically describe evolution of these two temperatures in the ballistic regime. In particular, we derive a closed-form fundamental solution and solution for a sinusoidal initial temperature profile in the case when the oscillators are significantly lighter than the FPUT particles. The harmonic theory predicts that during the heat transfer the temperatures of sublattices are significantly different, while initially and finally (at large times) they are equal. This may look like an artifact of the harmonic approximation, but we show that it is not the case. Two distinct temperatures are also observed in the anharmonic case, even when the heat transport regime is no longer quasiballistic. We show that the value of the nonlinearity coefficient required to equalize the temperatures strongly depends on the particle mass ratio. If the oscillators are much lighter than the FPUT particles, then a fairly strong nonlinearity is required for the equalization.

Unsteady ballistic heat transport in two-dimensional harmonic graphene lattice

We study the evolution of initial temperature profiles in a two-dimensional isolated harmonic graphene lattice. Two heat transfer problems are solved analytically and numerically. In the first problem, the evolution of a spatially sinusoidal initial temperature profile is considered. This profile is usually generated in real experiments based on the transient thermal grating technique. It is shown that at short times the amplitude of the profile decreases by an order magnitude and then it performs small decaying oscillations. A closed-form solution, describing the decay of the amplitude at short times is derived. It shows that due to symmetry of the lattice, the anisotropy of the ballistic heat transfer is negligible at short times, while at large times it is significant. In the second problem, a uniform spatial distribution of the initial temperature in a circle is specified. The profile is the simplest model of graphene heating by an ultrashort localized laser pulse. The corresponding solution has the symmetry of the lattice and many local maxima. Additionally, we show that each atom has two distinct temperatures corresponding to motions in zigzag and armchair directions. Presented results may serve for proper statement and interpretation of laboratory experiments and molecular dynamics simulations of unsteady heat transfer in graphene.

Finite-time fluctuation theorem for oscillatory lattices driven by a temperature gradient

The finite-time fluctuation theorem (FT) for the master functional, total entropy production, and medium entropy is studied in the one-dimensional Fermi-Pasta-Ulam-Tsingou-β (FPUT-β) chain coupled with two heat reservoirs at different temperatures. Through numerical simulations and theoretical analysis, we find that the nonequilibrium steady-state distribution of the one-dimensional FPUT-β chain violates the time-reversal symmetry. Thus, unlike the master functional, the total entropy production fails to satisfy the fluctuation relation for finite time. Meanwhile, we discuss the range of medium entropy production which obeys the conventional steady-state fluctuation theorem (SSFT) in the infinite time limit. Furthermore, we find that the generalized SSFT for medium entropy monotonically approaches the conventional SSFT as the time interval increases, irrespective of temperature difference, anharmonicity, and system size. Interestingly, the medium entropy production rate shows a nonmonotonic variation with anharomonicity, which comes from a competition mechanism of the phonon transport. Correspondingly, the difference between the generalized SSFT and the conventional SSFT shows similar nonmonotonic behaviors.

Kapitza thermal resistance in linear and nonlinear chain models: Isotopic defect

Kapitza resistance in the chain models with internal defects is considered. For the case of the linear chain, the exact analytic solution for the boundary resistance is derived for arbitrary linear time-independent conservative inclusion or defect. A simple case of isolated isotopic defects is explored in more detail. Contrary to the bulk conductivity in the linear chain, the Kapitza resistance is finite. However, the universal thermodynamic limit does not exist in this case. In other terms, the exact value of the resistance is not uniquely defined, and depends on the way of approaching the infinite lengths of the chain fragments. By this reason, and also due to the explicit dependence on the parameters of the thermostats, the resistance cannot be considered as a local property of the defect. Asymptotic scaling behavior of the heat flux in the case of very heavy defect is explored and compared to the nonlinear counterparts; similarities in the scaling behavior are revealed. For the lightweight isotopic defect in the linear chain, one encounters a typical dip of the temperature profile, related to weak excitation of the localized mode in the attenuation zone. If the nonlinear interactions are included, this dip can still appear at a relatively short timescale, with subsequent elimination due to the nonlinear interactions. This observation implies that even in the nonlinear chains, the linear dynamics can predict the main features of the short-time evolution of the thermal profile if the temperature is low enough.

Transition to thermal equilibrium in a crystal subjected to instantaneous deformation

An adiabatic transition between two equilibrium states corresponding to different stiffnesses in an infinite chain of particles is studied. Initially, the particles have random displacements and random velocities corresponding to uniform initial temperature distributions. An instantaneous change in the parameters of the chain initiates a transitional process. Analytical expressions for the chain temperature as a function of time are obtained from statistical analysis of the dynamic equations. It is shown that the transition process is oscillatory and that the temperature converges non-monotonically to a new equilibrium state, in accordance with what is usually unexpected for thermal processes. The analytical results are supplemented by numerical simulations.

Equilibration of sinusoidal modulation of temperature in linear and nonlinear chains

The equilibration of sinusoidally modulated distribution of the kinetic temperature is analyzed in the β-Fermi-Pasta-Ulam-Tsingou chain with different degrees of nonlinearity and for different wavelengths of temperature modulation. Two different types of initial conditions are used to show that either one gives the same result as the number of realizations increases and that the initial conditions that are closer to the state of thermal equilibrium give faster convergence. The kinetics of temperature equilibration is monitored and compared to the analytical solution available for the linear chain in the continuum limit. The transition from ballistic to diffusive thermal conductivity with an increase in the degree of anharmonicity is shown. In the ballistic case, the energy equilibration has an oscillatory character with an amplitude decreasing in time, and in the diffusive case, it is monotonous in time. For smaller wavelength of temperature modulation, the oscillatory character of temperature equilibration remains for a larger degree of anharmonicity. For a given wavelength of temperature modulation, there is such a value of the anharmonicity parameter at which the temperature equilibration occurs most rapidly.

Equilibration of kinetic temperatures in face-centered cubic lattices

We study thermal equilibration in face-centered cubic lattices with harmonic and anharmonic (Lennard-Jones) interactions. Initial conditions are chosen such that the kinetic temperatures, corresponding to three spatial directions, are different. We show that in the anharmonic case the approach to thermal equilibrium has two time scales. The first time scale is the period of atomic vibration. At times of the order of several atomic periods, the approach to equilibrium is accompanied by decaying high frequency oscillations of the temperatures. The oscillations are described analytically using the harmonic approximation. In particular, the characteristic frequencies of the oscillations are calculated. It is shown that the oscillations decay in time more slowly than expected. The second time scale, presented in the anharmonic case only, depends on the initial temperature of the system. Normalizing time by this scale, we obtain numerically a universal curve describing equilibration in the Lennard-Jones crystal over a wide range of temperatures.