Thermal equilibration in infinite harmonic crystals


We study transient thermal processes in infinite harmonic crystals having a unit cell with an arbitrary number of particles. Initially, particles have zero displacements and random velocities such that spatial distribution of temperature is uniform. Initial kinetic and potential energies are different and therefore the system is far from thermal equilibrium. Time evolution of kinetic temperatures, corresponding to different degrees of freedom of the unit cell, is investigated. It is shown that the temperatures oscillate in time and tend to generally different equilibrium values. The oscillations are caused by two physical processes: equilibration of kinetic and potential energies and redistribution of temperature among degrees of freedom of the unit cell. An exact formula describing these oscillations is obtained. At large times, a crystal approaches thermal equilibrium, i.e., a state in which the temperatures are constant in time. A relation, referred to as the non-equipartition theorem, between equilibrium values of the temperatures and initial conditions is derived. For illustration, transient thermal processes in a diatomic chain and graphene lattice are considered. Analytical results are supported by numerical solution of lattice dynamics equations.

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The author is deeply grateful to A.M. Krivtsov, S.V. Dmitriev, M.A. Guzev, D.A. Indeitsev, E.A. Ivanova, S.N. Gavrilov, I.E. Berinskii, and A.S. Murachev for useful discussions. The work was financially supported by the Russian Science Foundation under Grant No. 17-71-10213.

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Kuzkin, V.A. Thermal equilibration in infinite harmonic crystals. Continuum Mech. Thermodyn. 31, 1401–1423 (2019).

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  • Thermal equilibrium
  • Stationary state
  • Approach to equilibrium
  • Polyatomic crystal
  • Complex lattice
  • Kinetic temperature
  • Harmonic crystal
  • Transient processes
  • Equipartition theorem
  • Non-equipartition theorem
  • Temperature matrix