Transition to thermal equilibrium in a crystal subjected to instantaneous deformation

Abstract

An adiabatic transition between two equilibrium states corresponding to different stiffnesses in an infinite chain of particles is studied. Initially, the particles have random displacements and random velocities corresponding to uniform initial temperature distributions. An instantaneous change in the parameters of the chain initiates a transitional process. Analytical expressions for the chain temperature as a function of time are obtained from statistical analysis of the dynamic equations. It is shown that the transition process is oscillatory and that the temperature converges non-monotonically to a new equilibrium state, in accordance with what is usually unexpected for thermal processes. The analytical results are supplemented by numerical simulations.

Dynamics of matter and energy

“…Formula (8) shows that the center of the cloud moves uniformly. From Equation (54), it follows that there is a moment of time (𝑡 = 𝑡 * ) when the cloud reaches its minimum size, described by constant 𝜌 min . Until the time 𝑡 * , the cloud shrinks, afterwards it expands infinitely-see the solid curve in Figure 1.…”

“…Such evolution is obvious if a cloud containing only two particles is considered: the faster particle overtakes the slower one, therefore the size of the cloud first shrinks up to zero, then it expands to infinity, in full compliance with formula (57). If there are more than two particles then they do not necessary meet in one point, therefore the minimum size of the cloud is not zero and the evolution is described by general formula (54). Suppose the speed of dispersion is much less than the speed of transfer:…”

“…In consistence with the latter equality, 𝜌 can be also referred as the radius of energy. Since according to Equation (75), it fulfils that θ𝑐 = const, then the time dependence of radius 𝜌 is described by formula (54):…”

“…Wave energy transfer is responsible for heat transfer in crystalline solids, where nondiffusive ballistic and anomalous processes of heat energy transfer are common at micro/nanolevel, especially in low dimensions, which is shown by analytical [23][24][25][26][27][28][29][30][31][32][33][34][35][36] and experimental [3,[37][38][39][40][41][42][43][44] investigations. Recently an extensive research for description of unsteady ballistic heat transfer processes was performed by our scientific group [45][46][47][48][49][50][51][52][53][54][55][56], and the obtained results together with some ideas inspired by Zhilin [57] and Kunin [58] played a motivation role for this paper.…”