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# Stochastic dynamics Ⅱ: Finite random dynamical systems, linear representation, and entropy production

• * Corresponding author: Felix X.-F. Ye
• We study finite state random dynamical systems (RDS) and their induced Markov chains (MC) as stochastic models for complex dynamics. The linear representation of deterministic maps in RDS is a matrix-valued random variable whose expectation corresponds to the transition matrix of the MC. The instantaneous Gibbs entropy, Shannon-Khinchin entropy of a step, and the entropy production rate of the MC are discussed. These three concepts, as key anchoring points in applications of stochastic dynamics, characterize respectively the uncertainties of a system at instant time $t$, the randomness generated in a step in the dynamics, and the dynamical asymmetry with respect to time reversal. The stationary entropy production rate, expressed in terms of the cycle distributions, has found an expression in terms of the probability of the deterministic maps with single attractor in the maximum entropy RDS. For finite RDS with invertible transformations, the non-negative entropy production rate of its MC is bounded above by the Kullback-Leibler divergence of the probability of the deterministic maps with respect to its time-reversal dual probability.

Mathematics Subject Classification: Primary: 28D20, 60J10; Secondary: 37H05.

 Citation: • • Figure 1.  Random maps are generated via this Markov chain. This is the illustration of state transition diagram for the Markov chain

Figure 2.  A 3-state completely connected MC. The transition matrix of derived chain dynamics is in (34)

Figure 3.  A 4-state MC with initial state 1

Figure 4.  $e(\mathcal{T}_1) = M_{21}M_{31}, e(\mathcal{T}_2) = M_{12}M_{31}, e(\mathcal{T}_3) = M_{21}M_{13}$

Table 1.  The derived chain $\eta_t$ and the cycles formed for this sample trajectory $X_t$ 

 $t$ 0 1 2 3 4 5 6 7 8 $X_t(\omega)$ 2 1 3 1 3 1 1 2 2 $\eta_t(\omega)$  [2, 1] [2, 1, 3] [2, 1] [2, 1, 3] [2, 1] [2, 1]   $cycles$ (1, 3) (3, 1) (1) (1, 2) (2)
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