Boltzmann Factor for Frenkel-Kontorova Atomic Chain from Conservative Dynamics to Dissipative Ones
Project no. 22512 · Status: Kapanmış · 2021
Summary
<p>Cannonical probability distribution that is defined by Boltzmann factor is one of the most effective tolls often used in statistical mechanics. By obtaining the possible microscobic energy states of any system, which are related to the Boltzmann factor, macroscobic thermodynamical properties at constant temperature such as mean energy, entropy, pressure, latent heat, susceptibility can be determined. In other words, determination of the Boltzmann factor enables us to understand their macroscobic properties from collective properties of microscobic systems by the statistical mechanic point of view. In this project, the Frenkel-Kontorova chain, that is a one dimensional atom chain in a static equilibrium and is a model of many real applications in surface/material physics such as dislocations in a crystal structure, absorbed layers on a crystal surface, waves of charge density, Josephon junction in superconductivity and thermal diods, has been used. The Boltzmann factor has been calculated for both conservative and dissipative cases of the model. Separately, the two cases have been investigated for two distinct regimes which the atomic positions in the chain are linear and chaotic, and for every cases, the probability distributions that depend on the energy states of the atomic chain has been estimated. The regimes in which the cannonical equilibrium in the exponential form and meta-equilibrium distributions in the power law form appear in the Frenkel-Kontorova model have been determined. Moreover, it has been revealed how the distributions change with the increasing number of atoms in the chain and effect of dissipation. Finally, it has been found how the Boltzmann factor obtained for each equilibrium distribution scales with dissipation and number of atoms in the system. All these analysis say that the statistical mechanical background to be used for the calculation of the macroscobic properties of the Frenkel-Kontorova model, which has a place in many applications, should be establised based on either Gibbsian or Tsallis distribution, depending on the regime the system is in.</p><p><br></p><p><strong>Keywords: </strong>Boltzmann factor, Frenkel-Kontorova model, scaling relations.</p>