Heat bath dynamics harmonic oscillators

A simple example of vibrational relaxation is the relaxation of harmonic oscillators in a heat bath. Dynamic studies on the interaction of a molecule, simulated by a harmonic oscillator, with a structurless molecule of the heat bath shows (see Section IV. 13) that transitions occur only between neighbouring vibrational states with mean transition probabilities (per one gas-kinetic collision)... 

The simple class of models just discussed is of interest because it is possible to characterize the decay of correlations rather completely. However, these models are rather far from reality since they take no account of interparticle forces. A next step in our examination of the decay of initial correlations is to find an interacting system of comparable simplicity whose dynamics permit us to calculate at least some of the quantities that were calculated for the noninteracting systems. One model for which reasonably complete results can be derived is that of an infinite chain of harmonic oscillators in which initial correlations in momentum are imposed. Since the dynamics of the system can be calculated exactly, one can, in principle, study the decay of correlations due solely to internal interactions (as opposed to interactions with an external heat bath). We will not discuss the most general form of initial correlations but restrict our attention to those in which the initial positions and momenta have a Gaussian distribution so that two-particle correlations characterize the initial distribution completely. Let the displacement of oscillator j from its equilibrium position be denoted by qj and let the momentum of oscillator j be pj. On the assumption that the mass of each oscillator is equal to 1, the momentum is related to displacement by pj =. We shall study... 

Fig. 7 Molecular dynamic trajectories of a particle coupled to a heat bath on the potential-energy surface of Fig. 1. The saddle point is indicated by the dashed line, (a) and (b) are for relatively high friction and (c) low friction. A = 0.05 eV. The unit of time has been chosen such that the period of the harmonic oscillator is 2jr, and the friction coefficient y is given in terms of this unit.

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