Chaos - The Interplay Between Stochastic and Deterministic by Piotr Garbaczewski, Marek Wolf, Aleksander Weron

By Piotr Garbaczewski, Marek Wolf, Aleksander Weron

The research of chaotic behaviour of dynamical platforms has caused new efforts to reconcile deterministic and stochastic methods, in addition to classical and quantum physics. Efforts are being made to appreciate complicated and unpredictable behaviour. this article is an outline of those actions.

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P ( x n 1 x n ) p ( x n ). qtn 1 ,tn ( x n 1 , x n )q ( x n ), where qti 1 ,ti ( xi 1 , xi ) is the transition probability density, 1 i  n and t i 1  t i . e. qti 1 ,ti ( xi 1 , xi )  For deriving density the stochastic 1 e iu ( xi 1 , xi ) Ee iu ( xi 1  xi ) du.  2 equation, we consider the conditional (11) probability p ( x1 x 2 ), where p ( x1 , x 2 )  p ( x1 x 2 ) p ( x 2 ). After integrating over the variable x 2 , the above equation leads to p ( x1 )   q t1 ,t2 ( x1 , x 2 ) p ( x 2 )dx 2 .

The Fokker-Planck equation is also known as the Kolmogorov forward equation. The Fokker-Planck equation is a special case of the stochastic equation (kinetic equation) as well. The stochastic equation is about the evolution of the conditional probability for given initial states for non-Markov processes. The stochastic equation is an infinite series. Here, we explain how the Fokker-Planck equation becomes a special case of the stochastic equation. , x n ).... p ( x n 1 x n ) p ( x n ). , x n )  p ( x1 x 2 ) p ( x 2 x3 )....

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