Fourier heat conduction characteristics of a lattice chain arising from considerations in intermolecular potentials and the Second law

Abstract

Two aspects of conductive heat are considered here (i) its definition and characterization and (ii) the intermolecular potentials that induces both energy flow and the temperature profile at the steady state for a 1-D lattice chain. It is found from NEMD simulations and theory, such as that of Rieder, Lebowitz and Lieb (RLL) that the fundamental presuppositions maintained by persons like Joseph Fourier and others do not always obtain locally for some potentials since the Fourier Principle (FP), with stated form Jq·∇T≤0 in standard terminology and which is sometimes interpreted as reflecting the local form of the Second law, is violated. This result necessitates here a discussion of the nature of heat energy as defined in the First and Second laws of thermodynamics. A variational principle of form δS|Traj=0 for conductive heat is proposed in general, which also accounts for those regions that seem to violate the FP and local statements of the Second law. The 1-D simulations were conducted without recourse to synthetic algorithms utilizing coupling coefficients under very large temperature gradients leading to results that differ somewhat from some theories including that of RLL for harmonic potentials, that feature single particle thermostats with coupling coefficients. Here, we define control volumes at the right and left ends of the chain of 200 particles for each control volume and apply the non-synthetic thermostatting algorithm there for total chain length of 1000 particles. If the method used here is considered feasible, then a re-evaluation of some of the standard theoretical methodology would prove beneficial in order to design and implement more extended paradigms with enrichment of the axiomatic basis. The sinusoidal temperature profiles observed here and perhaps elsewhere for the harmonic lattice suggests that thermal integrated circuits with several thermal PN junctions may be constructed at the coordinate positions of the peaks and troughs of the profile which serve as sources and sinks of thermal energy, which might open the possibility of creating more complex thermal transistor circuits. The simulation results indicate that the presence of an anharmonic term per se does not guarantee diffusive behavior or the Fourier law obtaining because for any strength of an anharmonic symmetric contribution |bh|, a high enough value of the harmonic parameter |kh| induces “ballistic” behavior with the breakdown of the conventional Fourier law. It is noted that current research in the lattice problem subjected to temperature forces has perhaps unfortunately veered away from the original trajectory of FPU's research that focused on dynamics, and which also spawned much of the entire field of KdV and KAM analysis.