Quasi Markovian behavior in mixing maps

Details Quasi Markovian behavior in mixing maps Quasi Markovian behavior in mixing maps

1999-02-08

arxiv.org/abs/cond-mat/9902105v1

Physica D 133 (1999) 309-320

Physics

Condensed Matter

Statistical Mechanics

16 pages, 1 figure, uses elsart.sty, to be published in Physica D Special Issue on Predictability: Quantifying Uncertainty in

Scientific paper

10.1016/S0167-2789(99)00087-1

We consider the time dependent probability distribution of a coarse grained observable Y whose evolution is governed by a discrete time map. If the map is mixing, the time dependent one-step transition probabilities converge in the long time limit to yield an ergodic stochastic matrix. The stationary distribution of this matrix is identical to the asymptotic distribution of Y under the exact dynamics. The nth time iterate of the baker map is explicitly computed and used to compare the time evolution of the occupation probabilities with those of the approximating Markov chain. The convergence is found to be at least exponentially fast for all rectangular partitions with Lebesgue measure. In particular, uniform rectangles form a Markov partition for which we find exact agreement.

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