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On the Definition of Energy Flux in One-Dimensional Chains of Particles

Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy
Entropy 2019, 21(11), 1036; https://doi.org/10.3390/e21111036
Received: 18 September 2019 / Revised: 15 October 2019 / Accepted: 21 October 2019 / Published: 25 October 2019
(This article belongs to the Special Issue Recent Developments in Dissipative Phenomena)
We review two well-known definitions present in the literature, which are used to define the heat or energy flux in one dimensional chains. One definition equates the energy variation per particle to a discretized flux difference, which we here show it also corresponds to the flux of energy in the zero wavenumber limit in Fourier space, concurrently providing a general formula valid for all wavelengths. The other relies somewhat elaborately on a definition of the flux, which is a function of every coordinate in the line. We try to shed further light on their significance by introducing a novel integral operator, acting over movable boundaries represented by the neighboring particles’ positions, or some combinations thereof. By specializing to the case of chains with the particles’ order conserved, we show that the first definition corresponds to applying the differential continuity-equation operator after the application of the integral operator. Conversely, the second definition corresponds to applying the introduced integral operator to the energy flux. It is, therefore, an integral quantity and not a local quantity. More worryingly, it does not satisfy in any obvious way an equation of continuity. We show that in stationary states, the first definition is resilient to several formally legitimate modifications of the (models of) energy density distribution, while the second is not. On the other hand, it seems peculiar that this integral definition appears to capture a transport contribution, which may be called of convective nature, which is altogether missed by the former definition. In an attempt to connect the dots, we propose that the locally integrated flux divided by the inter-particle distance is a good measure of the energy flux. We show that the proposition can be explicitly constructed analytically by an ad hoc modification of the chosen model for the energy density. View Full-Text
Keywords: energy flux; heat flux; Fermi-Pasta-Ulam-Tsingou chain; nonequilibrium steady states; dissipative systems; transport properties energy flux; heat flux; Fermi-Pasta-Ulam-Tsingou chain; nonequilibrium steady states; dissipative systems; transport properties
MDPI and ACS Style

De Gregorio, P. On the Definition of Energy Flux in One-Dimensional Chains of Particles. Entropy 2019, 21, 1036.

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