Macroscopic determinism in noninteracting systems using large deviation theory

We consider a general system of n noninteracting identical particles which evolve under a given dynamical law and whose initial microstates are a priori independent. The time evolution of the n-particle average of a bounded function on the particle microstates is then examined in the large n limit. Using the theory of large deviations, we show that if the initial macroscopic average is constrained to be near a given value, y, then the macroscopic average at time t converges in probability as n → ∞ to a value, ψt(y), given explicitly in terms of a canonical expectation. Some general features of the graph of ψt(y) versus t are examined, particularly in regard to continuity, symmetry, and convergence.