Thermal equilibration in a one-dimensional damped harmonic crystal

Abstract

The features for the unsteady process of thermal equilibration ("the fast motions") in a onedimensional harmonic crystal lying in a viscous environment (e.g., a gas) are under investigation. It is assumed that initially the displacements of all the particles are zero and the particle velocities are random quantities with zero mean and a constant variance, thus, the system is far away from the thermal equilibrium. It is known that in the framework of the corresponding conservative problem the kinetic and potential energies oscillate and approach the equilibrium value that equals a half of the initial value of the kinetic energy. We show that the presence of the external damping qualitatively changes the features of this process. The unsteady process generally has two stages. At the first stage oscillations of kinetic and potential energies with decreasing amplitude, subjected to exponential decay, can be observed (this stage exists only in the underdamped case). At the second stage (which always exists), the oscillations vanish, and the energies are subjected to a power decay. The large-time asymptotics for the energy is proportional to t −3/2 in the case of the potential energy and to t −5/2 in the case the kinetic energy. Hence, at large values of time the total energy of the crystal is mostly the potential energy. The obtained analytic results are verified by independent numerical calculations. arXiv:1904.11902v6 [cond-mat.stat-mech]

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.…”

“…The initial value problem ( 15) and ( 16) describes the stochastic dynamics of the particles. The kinetic temperature of the crystal as a function of time can then be found using the covariance analysis approach [21,25], by introducing the generalized energies…”

“…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].…”

First breeding record of the black‐headed penduline tit (Remiz macronyx) in China

“…The ssaposhnikowi subspecies is rare and variable in morphology, whether it should be recognized as a separate subspecies is controversial (Bot & Van Dijk, 2009; Svensson & Shirihai, 2018; Vaurie, 1957). To date, most records of this subspecies are found in the Ili River basin, and the easternmost known distribution in the Ili River basin is Burundysu (Figure 1; 43°42′N 78°39′E), Almaty Province, Kazakhstan (Gavrilov & Gavrilov, 2005). We speculated that this subspecies may occur in Ili, Xinjiang, China, where the Ili River originates and a similar habitat occurs.…”

“…Nigricans are distributed in southeastern Iran and southwestern Afghanistan, but they are possibly extinct (Barani‐Beiranvand & Aliabadian, 2012). The ssaposhnikowi is known to only occur around lakes (Balkhash, Sasykkol, Alakol, and Zaysan) in southeastern Kazakhstan (Bot & Van Dijk, 2009; Gavrilov & Gavrilov, 2005; Madge, 2020). The distribution of the black‐headed penduline tit is disjointed and fragmented, and it only occurs along lakes or rivers surrounded by extensive reedbeds (Figure 1; Madge, 2020).…”