Effect of lattice discreteness on the statistical mechanics of a dilute gas of kinks

Abstract

We apply the collective-variable projection-operator approach based on the Dirac-bracket theory of constrained Hamiltonian systems to the calculation of the Helmholtz free energy of discrete nonlinear Klein-Gordon systems in the ideal-kink-gas limit. The kinks in the continuum nonlinear Klein-Gordon systems in the ideal-gas limit behave as free particles and the phonon density of states changes due to the presence of the kink. In discrete nonlinear Klein-Gordon systems the kinks are no longer free but see a potential ${\mathit{V}}_{\mathit{K}}$(X) where X is the center of mass of the kink. The phonons satisfy the discrete-lattice dispersion law with a discrete density of states, which differs from the continuum density of states, which also changes in the presence of the discrete kink. We find that when ${\mathit{l}}_{0}$ (the effective size of the kink) is greater than about five lattice spacings that the effect of discreteness is to lower the rest energy of the kink by less than 1%. For ${\mathit{l}}_{0}$\ensuremath{\sim}\ensuremath{\pi} the rest energy of the kink is lowered further and the potential ${\mathit{V}}_{\mathit{K}}$(X) starts to make a contribution giving rise to the presence of the periodic Peierls-Nabarro potential well. For ${\mathit{l}}_{0}$Peierls-Nabarro well becomes deeper and the kinks start to become trapped and when ${\mathit{l}}_{0}$\ensuremath{\le}2 all kinks with velocities less than one third of the speed of sound become trapped. In the strong-trapping region the rest energy of the kink is reduced by about 8--10 %.