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# One-Particle Representation of Heat Conduction Described within the Scope of the Second Law

^{1}Chemistry Department and Center for Theoretical and Computation Physics, University of Malaya, 50603 Kuala Lumpur, Malaysia

PLoS ONE, 2016;11(1):e0145026.

PMID: 26760507
DOI: 10.1371/journal.pone.0145026
The Carnot cycle and its deduction of maximum conversion efficiency of heat inputted and outputted isothermally at different temperatures necessitated the construction of isothermal and adiabatic pathways within the cycle that were mechanically "reversible", leading eventually to the Kelvin-Clausius development of the entropy function S with differential dS = dq/T such that [symbol: see text]C dS = 0 where the heat absorption occurs at the isothermal paths of the elementary Carnot cycle. Another required condition is that the heat transfer processes take place infinitely slowly and "reversibly", implying that rates of transfer are not explicitly featured in the theory. The definition of 'heat' as that form of energy that is transferred as a result of a temperature difference suggests that the local mode of transfer of "heat" in the isothermal segments of the pathway implies a Fourier-like heat conduction mechanism which is apparently irreversible, leading to an increase in entropy of the combined reservoirs at either end of the conducting material, and which is deemed reversible mechanically. These paradoxes are circumvented here by first clarifying the terms used before modeling heat transfer as a thermodynamically reversible but mechanically irreversible process and applied to a one dimensional atomic lattice chain of interacting particles subjected to a temperature difference exemplifying Fourier heat conduction. The basis of a "recoverable trajectory" i.e. that which follows a zero entropy trajectory is identified. The Second Law is strictly maintained in this development. A corollary to this zero entropy trajectory is the generalization of the Zeroth law for steady state non-equilibrium systems with varying temperature, and thus to a statement about "equilibrium" in steady state non-thermostatic conditions. An energy transfer rate term is explicitly identified for each particle and agrees quantitatively (and independently) with the rate of heat absorbed at the reservoirs held at different temperatures and located at the two ends of the lattice chain in MD simulations, where all energy terms in the simulation refer to a single particle interacting with its neighbors. These results validate the theoretical model and provides the necessary boundary conditions (for instance with regard to temperature differentials and force fields) that thermodynamical variables must comply with to satisfy the conditions for a recoverable trajectory, and thus determines the solution of the differential and integral equations that are used to model these processes. These developments and results, if fully pursued would imply that not only can the Carnot cycle be viewed as describing a local process of energy-work conversion by a single interacting particle which feature rates of energy transfer and conversion not possible in the classical Carnot development, but that even irreversible local processes might be brought within the scope of this cycle, implying a unified treatment of thermodynamically (i) irreversible (ii) reversible (iii) isothermal and (iv) adiabatic processes by conflating the classically distinct concept of work and heat energy into a single particle interactional process. A resolution to the fundamental and long-standing conjecture of Benofy and Quay concerning the Fourier principle is one consequence of the analysis.

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