Equilibration of energies in a two-dimensional harmonic graphene lattice

Abstract

<jats:p>We study dynamical phenomena in a harmonic graphene (honeycomb) lattice, consisting of equal particles connected by linear and angular springs. Equations of in-plane motion for the lattice are derived. Initial conditions typical for molecular dynamic modelling are considered. Particles have random initial velocities and zero displacements. In this case, the lattice is far from thermal equilibrium. In particular, initial kinetic and potential energies are not equal. Moreover, initial kinetic energies (and temperatures), corresponding to degrees of freedom of the unit cell, are generally different. The motion of particles leads to equilibration of kinetic and potential energies and redistribution of kinetic energy among degrees of freedom. During equilibration, the kinetic energy performs decaying high-frequency oscillations. We show that these oscillations are accurately described by an integral depending on dispersion relation and polarization matrix of the lattice. At large times, kinetic and potential energies tend to equal values. Kinetic energy is partially redistributed among degrees of freedom of the unit cell. Equilibrium distribution of the kinetic energies is accurately predicted by the non-equipartition theorem. Presented results may serve for better understanding of the approach to thermal equilibrium in graphene.</jats:p> <jats:p>This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.</jats:p>

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].…”

Unsteady ballistic heat transport in two-dimensional harmonic graphene lattice

“…Displacements of atoms from the unit cell {n, m} are denoted as 2D vectors U n,m , V n,m . Following [27], we use the harmonic approximation for the potential energy of the lattice. Each atom is connected with three nearest neighbors by linear springs (bonds) with stiffness c. Additionally, the nearest bonds between particles are connected by angular springs with stiffness g. Equations of motion for this system were obtained using the Euler-Lagrange formalism in paper [27].…”

Mechanical Properties of Cubene Crystals

“…Such crystals support rotobreathers that are dynamical regimes, in which one molecule (or particle) rotates, while others oscillate [ 31 , 32 , 33 , 34 , 35 ]. Properties of graphene can be described using particles with rotational degrees of freedom [ 36 , 37 ].…”