Anharmonic On-site Potential Research Articles

Thermal conductivity in one-dimensional nonlinear disordered lattices: two kinds of scattering effects of hard-type and soft-type anharmonicities

Amorphous solids can be theoretically modeled by anharmonic disordered lattices, but most of the theoretical studies on thermal conductivity in such lattices only consider hard-type (HT) anharmonicity. In this study, we investigate the thermal conductivity κ of one-dimensional disordered lattices with both HT and soft-type (ST) anharmonic on-site potentials. Our results from molecular dynamics simulations and the quasi-harmonic Green–Kubo (QHGK) method show that while the HT model exhibits non-monotonic dependence of κ on anharmonicity, the ST model shows a monotonically increasing trend. This trend provides a novel approach to enhancing thermal conductivity in disordered systems. Additionally, the QHGK predictions for κ in the HT model are consistent with simulation results over a wide range of anharmonicity values; however, for the ST model, deviations appear as the anharmonicity becomes softer. This peculiar feature may be attributed to delocalization effects being dominant in contrast to the competing roles played by both delocalization and localization effects observed in the HT model.

Coarse-grained intermolecular interactions on quantum processors

Variational quantum algorithms (VQAs) are increasingly being applied in simulations of strongly bound (covalently bonded) systems using full molecular orbital basis representations. The application of quantum computers to the weakly bound intermolecular and noncovalently bonded regime, however, has remained largely unexplored. In this work, we develop a coarse-grained representation of the electronic response that is ideally suited for determining the ground state of weakly interacting molecules using a VQA. We require qubit numbers that grow linearly with the number of molecules and derive scaling behavior for the number of circuits and measurements required, which compare favorably to traditional variational quantum eigensolver methods. We demonstrate our method on IBM superconducting quantum processors and show its capability to resolve the dispersion energy as a function of separation for a pair of nonpolar molecules---thereby establishing a means by which quantum computers can model Van der Waals interactions directly from zero-point quantum fluctuations. Within this coarse-grained approximation, we conclude that current-generation quantum hardware is capable of probing energies in this weakly bound but nevertheless chemically ubiquitous and biologically important regime. Finally, we perform experiments on simulated and real quantum computers for systems of three, four, and five oscillators as well as oscillators with anharmonic onsite binding potentials; the consequences of the latter are unexamined in large systems using classical computational methods but can be incorporated here with low computational overhead.

Prethermalization in a classical phonon field: Slow relaxation of the number of phonons

We investigate the emergence of an astonishingly long pre-thermal plateau in a classical phonon field, here a harmonic chain with on-site pinning. Integrability is broken by a weak anharmonic on-site potential with strength $\lambda$. In the small $\lambda$ limit, the approach to equilibrium of a translation invariant initial state is described by kinetic theory. However, when the phonon band becomes narrow, we find that the (non-conserved) number of phonons relaxes on much longer time scales than kinetic. We establish rigorous bounds on the relaxation time, and develop a theory that yields exact predictions for the dissipation rate in the limit $\lambda \to 0$. We compare the theoretical predictions with data from molecular dynamics simulations and find good agreement. Our work shows how classical systems may exhibit phenomena which at the first glance appear to require quantization.

Travelling heteroclinic waves in a Frenkel–Kontorova chain with anharmonic on-site potential

The Frenkel–Kontorova model for dislocation dynamics from 1938 is given by a chain of atoms, where neighbouring atoms interact through a linear spring and are exposed to a smooth periodic on-site potential. A dislocation moving with constant speed corresponds to a heteroclinic travelling wave, making a transition from one well of the on-site potential to another. The ensuing system is nonlocal, nonlinear and nonconvex. We present an existence result for a class of smooth nonconvex on-site potentials. Previous results in mathematics and mechanics have been limited to on-site potentials with harmonic wells. To overcome this restriction, we propose a novel approach: we first develop a global centre manifold theory for anharmonic wave trains, then parametrise the centre manifold to obtain asymptotically correct approximations to the solution sought, and finally obtain the heteroclinic wave via a fixed point argument.

Heat controlling in the mass-graded harmonic lattices with double-well on-site potential

Using nonequilibrium molecular dynamics simulations, we investigate thermal rectification in mass-graded lattices with a new type on-site potential which has a physical picture of the double-well. By adjusting the ratio of harmonic on-site potential and anharmonic on-site potential, we could obtain the optimal heat transport and the best thermal rectification. In addition, we observe the reversal thermal rectification by changing the ratio of on-site potential and analyzes the mechanism of thermal rectification through the power spectrum. At last, we also study the heat flux and thermal rectification in a different case of average temperature and mass gradient.

Heat flow in anharmonic crystals with internal and external stochastic baths: a convergent polymer expansion for a model with discrete time and long range interparticle interaction

We investigate a chain of oscillators with anharmonic on-site potentials, with long range interparticle interactions, and coupled both to external and internal stochastic thermal reservoirs of Ornstein–Uhlenbeck type. We develop an integral representation, a` la Feynman–Kac, for the correlations and the heat current. We assume the approximation of discrete times in the integral formalism (together with a simplification in a subdominant part of the harmonic interaction) and develop a suitable polymer expansion for the model. In the regime of strong anharmonicity, strong harmonic pinning, and for the interparticle interaction with integrable polynomial decay, we prove the convergence of the polymer expansion uniformly in volume (number of sites and time). We also show that the two-point correlation decays in space such as the interparticle interaction. The existence of a convergent polymer expansion is of practical interest: it establishes a rigorous support for a perturbative analysis of the heat flow problem and for the computation of the thermal conductivity in related anharmonic crystals, including those with inhomogeneous potentials and long range interparticle interactions. To show the usefulness and trustworthiness of our approach, we compute the thermal conductivity of a specific anharmonic chain, and make a comparison with related numerical results presented in the literature.

Breathers and surface modes in oscillator chains with Hertzian interactions

We study localized waves in chains of oscillators coupled by Hertzian interactions and trapped in local potentials. This problem is originally motivated by Newton's cradle, a mechanical system consisting of a chain of touching beads subject to gravity and attached to inelastic strings. We consider an unusual setting with local oscillations and collisions acting on similar time scales, a situation corresponding e.g. to a modified Newton's cradle with beads mounted on stiff cantilevers. Such systems support static and traveling breathers with unusual properties, including double exponential spatial decay, almost vanishing Peierls-Nabarro barrier and spontaneous direction-reversing motion. We prove analytically the existence of surface modes and static breathers for anharmonic on-site potentials and weak Hertzian interactions.

Thermalization in Harmonic Particle Chains with Velocity Flips

We propose a new mathematical tool for the study of transport properties of models for lattice vibrations in crystalline solids. By replication of dynamical degrees of freedom, we aim at a new dynamical system where the "local" dynamics can be isolated and solved independently from the "global" evolution. The replication procedure is very generic but not unique as it depends on how the original dynamics are split between the local and global dynamics. As an explicit example, we apply the scheme to study thermalization of the pinned harmonic chain with velocity flips. We improve on the previous results about this system by showing that after a relatively short time period the average kinetic temperature profile satisfies the dynamic Fourier's law in a local microscopic sense without assuming that the initial data is close to a local equilibrium state. The bounds derived here prove that the above thermalization period is at most of the order L^(2/3), where L denotes the number of particles in the chain. In particular, even before the diffusive time scale Fourier's law becomes a valid approximation of the evolution of the kinetic temperature profile. As a second application of the dynamic replica method, we also briefly discuss replacing the velocity flips by an anharmonic onsite potential.

Thermal conductivity of anharmonic crystals with self-consistent baths: Analytical computation with discrete time

We analytically compute the thermal conductivity of anharmonic crystals with self-consistent stochastic reservoirs. We develop an integral representation for the heat current, assume the approximation of discrete times, and in a perturbative analysis that is rigorously supported by a convergent cluster expansion compute the thermal conductivity for a chain with quartic anharmonic on-site potential. In the high anharmonicity regime, the result for the dependence of the conductivity on temperature is the same as for the system without inner reservoirs. The presented formalism is quite general and is also valid for inhomogeneous systems in any space dimension.

Nontrivial properties of heat flow: Analytical study of some anharmonic lattice microscopic models

We address the analytical investigation of nontrivial properties of heat flow, starting from microscopic models of matter. We present an integral representation for the expression of the heat flow, by taking as our crystal model a self-consistent chain of anharmonic oscillators, precisely, a chain of oscillators with harmonic interparticle interactions, anharmonic on-site potentials, thermal reservoirs at the boundaries, and still with some residual inner stochastic baths. We propose an approximative scheme to make the integral formalism analytically treatable: we test the approximations in harmonic models and analyze some anharmonic systems. For the case of graded anharmonic models with weak interparticle interactions, we derive an expression for the thermal conductivity, and show the existence of thermal rectification and also nonnegative differential thermal resistance. The details of our formalism allow us to note the ingredients behind these phenomena, and the generality of our results (i.e., the results will be valid for other interactions and regimes) shows that rectification in graded materials is a ubiquitous property.

Anharmonic on-site potential implies asymmetry thermal transport in 1D mass-graded harmonic lattice

Exploring the thermal rectification in one-dimensional (1D) lattice model is of considerable practical importance. Here we consider the 1D mass-graded harmonic lattice with on-site potential. By the nonequilibrium molecular dynamics simulation, it is uncovered that the temperature gradient is formed regardless of the fact that the on-site potential is harmonic or anharmonic. Differently, the thermal rectification rises in the model with anharmonic on-site potential whereas disappears with harmonic one. Moreover, we find that the heat flux of 1D mass-graded Frenkel–Kontorova (FK) model in the direction of mass increasing is larger than that in the direction of mass decreasing. Our results further emphasize that nonlinearity is necessary for thermal rectification.

Symmetry of heat conductivity in inhomogeneous quantum chains

We address the problem of the existence of thermal rectification in inhomogeneous (in particular, graded) chains. Searching for analytical results, we investigate the symmetry properties of the thermal conductivity of the quantum inhomogeneous harmonic chain of oscillators with self-consistent reservoirs, an analytically treatable effective anharmonic model (the inner reservoirs mimic the anharmonic on-site potentials). Considering the linear response regime, i.e., for the system submitted to a small gradient of temperature, we show that, even with quantum effects in the conductivity, there is no thermal rectification. Moreover, as a secondary result, we analyze an old expression derived for the thermal conductivity of pure harmonic chains (i.e., a chain with baths at the boundaries only), and prove that there is no thermal rectification in such inhomogeneous systems, as suggested by numerical simulations in previous works.

Heat conduction in a weakly anharmonic chain: an analytical approach

The analytical study of heat conduction in an anharmonic chain is considered here. We investigate an one-dimensional system (directly related to the Frenkel-Kontorova model) with anharmonic cosine on-site potential and harmonic interparticle interaction. We start with a stochastic thermal reservoir connected to each site of the system, and analyse the behaviour of the conductivity in the steady state with all the heat baths as we turn off the interior reservoirs, i.e., as we keep the heat baths at the boundaries only. For a weak interparticle potential and small anharmonicity, in a perturbative computation, we derive an analytic expression for the heat conductivity which indicates that the Fourier's law holds only when each site is connected to a heat bath. To show the trustworthiness of our perturbative computation, we also derive an expression for the conductivity by starting from the exact solution of the linear part of the dynamics and compare with the result which comes from the previous perturbative analysis.

Travelling Breathers in Klein-Gordon Lattices as Homoclinic Orbits to p-Tori

Klein-Gordon chains are one-dimensional lattices of nonlinear oscillators in an anharmonic on-site potential, linearly coupled with their first neighbors. In this paper, we study the existence in such networks of spatially localized solutions, which appear time periodic in a referential in translation at constant velocity. These solutions are called travelling breathers. In the case of travelling wave solutions, the existence of exact solutions has been obtained by Iooss and Kirchgassner. Formal multiscale expansions have been used by Remoissenet to derive approximate solutions of travelling breathers in the form of modulated plane waves. James and Sire have studied the existence of specific travelling breather solutions, consisting in pulsating travelling waves which are exactly translated of 2 lattice sites after a fixed propagation time T. In this paper, we generalize this approach to pulsating travelling waves which are exactly translated of p≥ 3 sites after a given time Tp being arbitrary. By formulating the problem as a dynamical system, one is able to reduce the system locally to a finite dimensional set of ordinary differential equations (ODE), whose dimension depends on the parameter values of the problem. We prove that the principal part of this system of ODE admits homoclinic connections to p-tori under general conditions on the potential. One can obtain leading order approximations of these homoclinic connections and these orbits should correspond, for the oscillator chain, to small amplitude travelling breather solutions superposed on an exponentially small quasi-periodic tail.

Numerical computation of travelling breathers in Klein–Gordon chains

We numerically study the existence of travelling breathers in Klein–Gordon chains, which consist of one-dimensional networks of nonlinear oscillators in an anharmonic on-site potential, linearly coupled to their nearest neighbors. Travelling breathers are spatially localized solutions having the property of being exactly translated by p sites along the chain after a fixed propagation time T (these solutions generalize the concept of solitary waves for which p = 1 ). In the case of even on-site potentials, the existence of small amplitude travelling breathers superposed on a small oscillatory tail has been proved recently [G. James, Y. Sire, Travelling breathers with exponentially small tails in a chain of nonlinear oscillators, Commun. Math. Phys., 2005, in press (available online at http://www.springerlink.com)], the tail being exponentially small with respect to the central oscillation size. In this paper, we compute these solutions numerically and continue them into the large amplitude regime for different types of even potentials. We find that Klein–Gordon chains can support highly localized travelling breather solutions superposed on an oscillatory tail. We provide examples where the tail can be made very small and is difficult to detect at the scale of central oscillations. In addition, we numerically observe the existence of these solutions in the case of non-even potentials.

Travelling Breathers with Exponentially Small Tails in a Chain of Nonlinear Oscillators

We study the existence of travelling breathers in Klein-Gordon chains, which consist of one-dimensional networks of nonlinear oscillators in an anharmonic on-site potential, linearly coupled to their nearest neighbors. Travelling breathers are spatially localized solutions which appear time periodic in a referential in translation at constant velocity. Approximate solutions of this type have been constructed in the form of modulated plane waves, whose envelopes satisfy the nonlinear Schrodinger equation (M. Remoissenet, Phys. Rev. B33, n.4, 2386 (1986), J. Giannoulis and A. Mielke, Nonlinearity17, p. 551–565 (2004)). In the case of travelling waves (where the phase velocity of the plane wave equals the group velocity of the wave packet), the existence of nearby exact solutions has been proved by Iooss and Kirchgassner, who have obtained exact solitary wave solutions superposed on an exponentially small oscillatory tail (G. Iooss, K. Kirchgassner, Commun. Math. Phys.211, 439–464 (2000)). However, a rigorous existence result has been lacking in the more general case when phase and group velocities are different. This situation is examined in the present paper, in a case when the breather period and the inverse of its velocity are commensurate. We show that the center manifold reduction method introduced by Iooss and Kirchgassner is still applicable when the problem is formulated in an appropriate way. This allows us to reduce the problem locally to a finite dimensional reversible system of ordinary differential equations, whose principal part admits homoclinic solutions to quasi-periodic orbits under general conditions on the potential. For an even potential, using the additional symmetry of the system, we obtain homoclinic orbits to small periodic ones for the full reduced system. For the oscillator chain, these orbits correspond to exact small amplitude travelling breather solutions superposed on an exponentially small oscillatory tail. Their principal part (excluding the tail) coincides at leading order with the nonlinear Schrodinger approximation.

Nonequilibrium statistical mechanics of anharmonic crystals with self-consistent stochastic reservoirs

We consider a d -dimensional crystal with an arbitrary harmonic interaction and an anharmonic on-site potential, with a stochastic Langevin heat bath at each site. We develop an integral formalism for the correlation functions that is suitable for the study of their relaxation (time decay) as well as their behavior in space. Furthermore, in a perturbative analysis, for the one-dimensional system with weak coupling between the sites and small quartic anharmonicity, we investigate the steady state and show that Fourier's law holds. We also obtain an expression for the thermal conductivity (for arbitrary next-neighbor interactions) and give the temperature profile in the steady state.

Dependence of thermal conductivity on discrete breathers in lattices.

We study the properties of heat conduction in chains of coupled particles subjected to different anharmonic on-site potentials. Particular emphasis is placed on the role of breathers in saturation of the thermal conductivity for chains with hard anharmonicity. When the chain particles are subject to on-site potentials with soft anharmonicity, we find a characteristic temperature, below which the conductivity decreases but while above which it increases.

Effective kink-kink interaction in a one-dimensional model mediated by phonon exchange.

The general 1D double-well model with anharmonic interaction is considered in the displacive limit. Expansion of the Hamiltonian around a multikink state results in a phonon-kink Hamiltonian. It is shown that at rather low temperatures and short wave lengths the phonon-kink interaction can be treated in Born approximation, leading to a decomposition of the multikink-phonon Hamiltionian. Elimination of the phonons results in an effective potential for the kink-kink interaction, which corresponds to the one-dimensional analog of the RKKY interaction. This long-range interaction is inherent only for models with anharmonic on-site potentials and not in case of a double-parabola model.

Kinetic-Ising-model description of Newtonian dynamics: A one-dimensional example.

We show that the Newtonian dynamics of a chain of particles with an anharmonic on-site potential and harmonic nearest-neighbor interactions can be described by a one-dimensional kinetic Ising model with most general Glauber transition rates, provided the temperature is low enough compared to the minimum barrier height. The transition rates are calculated by use of the transition-state theory. At higher temperatures, memory effects occur which invalidate this kinetic description. These memory effects are due to the appearance of dynamically correlated clusters of particles performing periodic oscillations over a certain time scale.