Change of entropy for the one-dimensional ballistic heat equation: Sinusoidal initial perturbation

Abstract

This work presents a thermodynamic analysis of the ballistic heat equation from two viewpoints: classical irreversible thermodynamics (CIT) and extended irreversible thermodynamics (EIT). A formula for calculating the entropy within the framework of EIT for the ballistic heat equation is derived. The entropy is calculated for a sinusoidal initial temperature perturbation by using both approaches. The results obtained from CIT show that the entropy is a non-monotonic function and that the entropy production can be negative. The results obtained for EIT show that the entropy is a monotonic function and that the entropy production is nonnegative. A comparison between the entropy behaviors predicted for the ballistic, for the ordinary Fourier-based, and for the hyperbolic heat equation is made. A crucial difference of the asymptotic behavior of the entropy for the ballistic heat equation is shown. It is argued that mathematical time reversibility of the partial differential ballistic heat equation is not consistent with its physical irreversibility. The processes described by the ballistic heat equation are irreversible because of the entropy increase. *

Steady-state kinetic temperature distribution in a two-dimensional square harmonic scalar lattice lying in a viscous environment and subjected to a point heat source

“…In particular, a heat transfer equation was obtained that differs from the extended heat transfer equations suggested earlier [40][41][42][43]; however, it is in an excellent agreement with molecular dynamics simulations and previous analytical estimates [31]. The properties of the solutions describing heat transfer in a one-dimensional harmonic crystal were discussed in [44][45][46]. Later this approach was generalized [37,38,[47][48][49][50][51]] to a number of systems, namely, to an infinite one-dimensional crystal on an elastic substrate [47], to an infinite one-dimensional diatomic harmonic crystal [52], to a finite one-dimensional crystal [51], and to two and three-dimensional infinite harmonic lattices [48][49][50].…”

Equilibration of energies in a two-dimensional harmonic graphene lattice

“…In the case of non-uniform initial temperature profile, an additional physical process, notably the ballistic heat transport, should be considered. The heat transport is much slower than equilibration of energies considered above [30,[50][51][52][53][54]. Therefore, at short times behaviour of kinetic energy at each spatial point can be approximately described by formula (6.5).…”

Transition to thermal equilibrium in a crystal subjected to instantaneous deformation

“…In the present paper, we study an adiabatic non-equilibrium process analogous to those considered previously [22][23][24][25][26][27]. This process is, however, initiated by instantaneous loading instead of instantaneous heating.…”

“…In the pioneering paper by Klein and Prigogine [20], the equations of atomic motion for a onedimensional harmonic crystal were solved directly and it was shown that the energy oscillations following an instantaneous thermal perturbation were described by the Bessel function of the first kind. In later work [21], this problem was solved by analysing the dynamics equations of the velocity covariances, which allowed generalization of these results to more complex systems, including multidimensional crystals [22][23][24][25][26][27][28].…”