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.