From Determinism to Stochasticity.
The question of compatibility between the deterministic classical mechanics and the probabilistic statistical physics is studied. First, the motion of a hard sphere between two walls under the influence of a periodic square wave force is analyzed. It is shown that the existence of a special set of KAM curves depends on the fact that any segment of them can not be transformed by the dynamics into a "folded" structure. It is also shown that these KAM curves are the most stable KAM curves in their respective neighborhoods, and hence they constitute the boundaries of the global (strong) stochastic regions of the system. The present criterion for the existence of the KAM curves can be applied to a number of other systems. The relationship between the exponential divergence of close trajectories in the phase space and the appearance of irreversible behavior in dynamical systems is studied. The degree of divergence of trajectories is described quantitatively by the local exponent k((')q,(')p,(tau)), the value of which depends on the region of the phase space where the trajectories initiated. The local exponent is related to the rate at which the autocorrelation function of a coordinate or other macroscopic quantities of a system, described as phase averages, approach their equilibrium value. The exponent k((')q,(')p,(tau)) can also be associated with transport properties of a system such as its thermal conductivity. Finally, the thermal conductivity of one-dimensional and two-dimensional lattices is studied. Numerical experiments are conducted for one-dimensional and two-dimensional nonlinear lattices with various anharmonicities and diatomic mass ratios. The dependence of their thermal conductivity of the number of particles in the lattice and the mass ratio is studied. The transition from infinite to normal thermal conductivity, which occurs as the value of the mass ratio is changed, is examined. This transition is related to the rate at which pulses traveling through the lattice decay. The relationship between the thermal conductivity and the local exponents is also discussed.
- Pub Date:
- Physics: General