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Nonlocal and nonlinear effects in hyperbolic heat transfer in a two-temperature model. (English) Zbl 1460.80001

Manufacturing of micro- and nanodevices makes it necessary to analyze heat transfer on nanoscale. It is impossible to use classical Fourier law in this case because characteristic length scale of the problem is comparable with mean free path of heat carreers. For example, these difficulties take place while modelling of electrons and phonons passing through a crystal lattice.
In order to describe the physical system mentioned above, the authors use a two-temperature model that takes into account nonlocal and nonlinear effects as well. In particular, the authors introduce relaxation times and mean free paths of electrons and phonons as important model parameters in addition to the “classical” parameters such as specific heats and heat conductivities.
Though this model is not obtained by rigorous microscopic derivation, the authors discuss its consistency with statistics of Bose-Einstein and Fermi-Dirac as well as with the 2nd law of thermodynamics.
The techniques of acceleration waves is then used to describe the heat propagation in the two-temperature medium. The dependence of heat wave speeds and amplitudes in electronic and phononic gases on the physical system characteristics is found. In particular, conditions are derived that show when the waves under discussion become shock waves.

MSC:

80A05 Foundations of thermodynamics and heat transfer
80A19 Diffusive and convective heat and mass transfer, heat flow
35Q49 Transport equations
74J40 Shocks and related discontinuities in solid mechanics
35L67 Shocks and singularities for hyperbolic equations
81V73 Bosonic systems in quantum theory
81V74 Fermionic systems in quantum theory
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