Thermal energy transport in harmonic systems

One and two-dimensional oscillator systems are constructed so that individual oscillators are bound to their home positions and coupled to nearest neighbors by linear forces; both systems are treated for the case of weak coupling. The two-dimensional system has a tensor binding force that does not permit it to be decoupled into two noninteracting systems.Segments and strips are treated as thermodynamic systems in contact with a heat bath and Gaussian initial probability densities are used to compute correlations. Expressions for entropy and “temperature” are computed by relating initial system and bath variances to equilibrium temperature.Asymptotic relations for “temperature” are found that show the degree of coupling between dimensions affects the rate of approach to equilibrium, but only through the coefficient of time and not its power. Both systems approach equilibrium as t^?1 rather than $\text{t}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}$ as predicted by continuum theory.