Energy transport in the presence of long-range interactions

Abstract

We study energy transport in the paradigmatic Hamiltonian mean-field (HMF) model and other related long-range interacting models using molecular dynamics simulations. We show that energy diffusion in the HMF model is subdiffusive in nature, which confirms a recently obtained intriguing result that, despite being globally interacting, this model is a thermal insulator in the thermodynamic limit. Surprisingly, when additional nearest-neighbor interactions are introduced to the HMF model, an energy superdiffusion is observed. We show that these results can be consistently explained by studying energy localization due to thermally generated intrinsic localized excitation modes (discrete breathers) in nonlinear discrete systems. Our analysis for the HMF model can also be readily extended to more generic long-range interacting models where the interaction strength decays algebraically with the (shortest) distance between two lattice sites. This reconciles many of the apparently counterintuitive results presented recently [C. Olivares and C. Anteneodo, Phys. Rev. E 94, 042117 (2016)2470-004510.1103/PhysRevE.94.042117; D. Bagchi, Phys. Rev. E 95, 032102 (2017)2470-004510.1103/PhysRevE.95.032102] concerning energy transport in two such long-range interacting models.

A family of fractional diffusion equations derived from stochastic harmonic chains with long-range interactions

“…where local spectral density of energy W ǫ depends on the position y ∈ R along the chain, the wave number k ∈ T = [− 1 2 , 1 2 ) and time t ≥ 0. ω(k) = α(k) is the dispersion relation and L is a scattering operator on T. In [4,9], under the weak noise assumption γ = ǫγ 0 , the authors showed that the scaled solution of the Boltzmann equation converges to a soluiton of a fractional diffusion equation with the exponent 3 4 by a two-step procedure. As a first step, by taking a kinetic limit with time scale t ǫ , the so-called Boltzmann equation is derived:…”

Subdiffusive Energy Transport and Antipersistent Correlations Due to the Scattering of Phonons and Discrete Breathers

“…This latter type of LRIs, together with the harmonic NN interaction, thus favor the excitation of both phonons and DBs. However, the HMF model with both NN and LRIs always supports superdiffusive transport [19]. Fortunately, in the follow- ing we are able to reveal the subdiffusive motion.…”

“…This raises an interesting question if energy subdiffusion can exist in lattices [17]. Recently, this question was partially answered by studying subdiffusive energy transport in a special Hamiltonian meanfield (HMF) model [18,19], where some of the signatures of subdiffusion like the vanishing of heat current have been revealed. However, other significant features such as the antipersistent [20] correlations that were fre-quently observed in the subdiffusion of particles have not yet been explored.…”

“…So κ is finite (divergent) for diffusive (superdiffusive) transport. For subdiffusive transport, however, κ should be vanishing in the thermodynamic limit, indicating the system to be a thermal insulator [16][17][18][19]. To support this, mathematically C JJ (t) has to change sign at least once in order to cause the Green-Kubo integral vanishing since it is a continuous function of t. Physically, particularly in the framework of particle models for subdiffusion [11][12][13], this means the property of antipersistence, i.e., a period of energy transport with positive heat current is typically followed by a period of transport with negative current.…”

“…Indeed, phonons, as the main heat carriers, are originated from the harmonic NN coupling in the Hamiltonian (1). DBs, supposed as the viscoelastic elements [6], come from the anharmonic LRIs [19]. As the phonon waves move through DBs, which is regarded as a scatter, phonons will be partially reflected, partially pass through, and accompanied with energy loss [31,32].…”