Last modified: 2020-01-19

#### Abstract

The RLL [1] harmonic lattice chain potential solution to heat conduction has been taken to be the standard result for anomalous heat conduction where the Fourier Law and Inequality $ \mathbf{J_q}.\nabla T \leq 0 $ breaks down and where the solution temperature profile has been strenously verified via computer simulations at low temperature gradients where “ballistic” non-diffusive energy flow unlike that which characterizes Fourier heat conduction is said to occur, obscuring the nature of the type of energy flow. Raising the temperature gradient fourfold [2] yields sinusoidally varying temperature profiles at the steady state of a non-RLL type, where it is conjectured that the location of the peaks and troughs of the temperature profile could act as sources and sinks for heat transfer in thermal circuits of interest to material science. Recently, a a single-particle representation theory of thermal energy transfer was developed for the anharmonic lattice chain [3] where Fourier’s law obtains and where a zero entropy trajectory is constructed from the tautology $-k +k =0$ where heat is described as an optimized process at maximum Carnot efficiency and where the specific transfer of energy down the lattice chain is a work term. Such a view of heat is not standard in thermodynamic theory, where for the entire edifice of thermodynamics, the energy of the system $U$ exists in two forms, heat $Q$ and work $W$ leading to the First law perfect differential for the energy where $dU=dQ + dW$ , where in the theory described here, the two forms are concatenated, so that the transfer of heat is a work term. In the theory presented in [3], two temperatures are associated with each particle along the chain, representing the temperature before and after energy transfer of the particle to an adjacent particle. This idea is an extension of a theory of thermal desorption of particles [4] from a surface described as a zero-entropy, Carnot optimized trajetory. The next and final step in completing the model is to associate one mean temperature to each particle and to describe the trajectory as having entropy invariance. The results of ongoing NEMD simulations to validate these final objectives will be discussed. There seems to be no restriction to the theory developed here that can be applied to anomalous heat transfer, and future and ongoing research will focus on testing the theory for anomalous steady state heat conduction, such as for RLL systems. If the test yields positive outcomes, then a variational principle may be stated that conventional Second law formulations over the two centuries were unable to accomodate.

**References**

[1] Rieder, Z.; Lebowitz, J. L. & Lieb, E. (1967), *J. Math. Phys* **8**(5), 1073-1078.[2] Jesudason, C. G. (2017*), Int J Therm Sci* **120**,491-507.[3] Jesudason, C. G. (2016), *PLoS ONE* 11(1), e0145026. [4] Jesudason, C. G. (1991), *Indian J. Pure Ap. Phy.* **29**(3), 163-182.