Abstract
Transport and the spread of heat in Hamiltonian one dimensional momentum conserving nonlinear systems is commonly thought to proceed anomalously. Notable exceptions, however, do exist of which the coupled rotator model is a prominent case. Therefore, the quest arises to identify the origin of manifest anomalous energy and momentum transport in those low dimensional systems. We develop the theory for both, the statistical densities for momentum- and energy-spread and particularly its momentum-/heat-diffusion behavior, as well as its corresponding momentum/heat transport features. We demonstrate that the second temporal derivative of the mean squared deviation of the momentum spread is proportional to the equilibrium correlation of the total momentum flux. Subtracting the part which corresponds to a ballistic momentum spread relates (via this integrated, subleading momentum flux correlation) to an effective viscosity, or equivalently, to the underlying momentum diffusivity. We next put forward the intriguing hypothesis: normal spread of this so adjusted excess momentum density causes normal energy spread and alike normal heat transport (Fourier Law). Its corollary being that an anomalous, superdiffusive broadening of this adjusted excess momentum density in turn implies an anomalous energy spread and correspondingly anomalous, superdiffusive heat transport. This hypothesis is successfully corroborated within extensive molecular dynamics simulations over large extended time scales. Our numerical validation of the hypothesis involves four distinct archetype classes of nonlinear pair-interaction potentials: (i) a globally bounded pair interaction (the noted coupled rotator model), (ii) unbounded interactions acting at large distances (the coupled rotator model amended with harmonic pair interactions), (iii) the case of a hard point gas with unbounded square-well interactions and (iv) a pair interaction potential being unbounded at short distances while displaying an asymptotic free part (Lennard–Jones model). We compare our findings with recent predictions obtained from nonlinear fluctuating hydrodynamics theory.
Highlights
The investigation of heat conduction in low dimensional nonlinear lattices has attracted ever increasing attention in the statistical physics community [1, 2, 3]
We demonstrate that the second temporal derivative of the mean squared deviation of the momentum spread is proportional to the equilibrium correlation of the total momentum flux
We put forward the intriguing hypothesis: normal spread of this so adjusted excess momentum density causes normal energy spread and alike normal heat transport (Fourier Law)
Summary
The investigation of heat conduction in low dimensional nonlinear lattices has attracted ever increasing attention in the statistical physics community [1, 2, 3]. Most recently, relying on numerical simulations, Savin and Kosevich [23] showed that thermal conduction obeys Fourier’s law for 1D momentum-conserving lattices with a 1D Lennard-Jones interaction, a Morse interaction, and as well a Coulomb-like interaction. In addition we discuss the cases with a hard point gas and a LennardJones pair interaction These detailed numerical MD studies for these four nonlinear lattice systems support the fact that it is not the mere presence or absence of the symmetry of momentum conservation but rather the presence or absence of a fluidlike behavior, as characterized with normal spread of the momentum excess density, which we speculate to be at the source for the validity or the breakdown of Fourier’s law behavior. Additional conclusions and remaining open issues are presented with Section 5
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