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

Ye, Felix X.-F.; Qian, Hong

doi:10.3934/dcdsb.2019122

Cited by 6 publications

(6 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, the general structure of the proof may well be extended to more general chemical reaction networks, also with multiple reactants and corresponding random periodic orbits. Note that the works [16,35,36] mentioned earlier also deal with synchronization of RDSs for Markov chains, and in [16] even partial synchronization is considered. However, the latter approach is focused on linear cocycles for random networks, using the theory of Lyapunov exponents.…”

Section: Resultsmentioning

confidence: 99%

“…For a discrete-time system on a finite state space the maps are given by deterministic transitions matrices (containing only entries zero and one), and the expectation of the matrix-valued random variable of transitions maps agrees with the stochastic transition matrix of the corresponding Markov chain. The relation between such finite-state RDS and the related Markov chains has been studied by F. Ye et al [35,36]. Among other things, it has been found that a given finite-state RDS induces a unique Markov chain, while one Markov chain might be compatible with several RDS [36], as already discussed in a general context by Kifer [18].…”

Section: Background and Related Workmentioning

confidence: 99%

“…a weak attractor if for each compact set B ⊂ X lim n→∞ dist(ϕ n q (B), A θ n q ) = 0 in probability, (35) cf. e.g.…”

Section: General Properties Of Random Attractors In Discrete Time And...mentioning

confidence: 99%

See 2 more Smart Citations

Synchronization and random attractors for reaction jump processes

Engel¹,

Olicón-Méndez²,

Unger³

et al. 2022

Preprint

View full text Add to dashboard Cite

This work explores a synchronization-like phenomenon induced by common noise for continuous-time Markov jump processes given by chemical reaction networks. A corresponding random dynamical system is formulated in a two-step procedure, at first for the states of the embedded discrete-time Markov chain and then for the augmented Markov chain including also random jump times. We uncover a time-shifted synchronization in the sense that -after some initial waiting time -one trajectory exactly replicates another one with a certain time delay. Whether or not such a synchronization behaviour occurs depends on the combination of the initial states. We prove this partial time-shifted synchronization for the special setting of a birth-death process by analyzing the corresponding two-point motion of the embedded Markov chain and determine the structure of the associated random attractor. In this context, we also provide general results on existence and form of random attractors for discrete-time, discrete-space random dynamical systems.

show abstract

Section: Resultsmentioning

confidence: 99%

Section: Background and Related Workmentioning

confidence: 99%

See 1 more Smart Citation

Synchronization and random attractors for reaction jump processes

Engel¹,

Olicón-Méndez²,

Unger³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…In our change-of-measure formalism, we could extend (Ω, F) to include all possible transition matrices. Such future work could be conducted by considering the theories of random dynamical system for Markov chains [66,67]. very definition: (1) containing the empty set ∅ where nothing happen, (2) being closed under countable union, and (3) being closed under complement.…”

Section: Housekeeping Heat and Non-adiabatic Entropy Productionmentioning

confidence: 99%

Unified formalism for entropy production and fluctuation relations

Yang

Qian

2020

Phys. Rev. E

Self Cite

View full text Add to dashboard Cite

Stochastic entropy production, which quantifies the difference between the probabilities of trajectories of a stochastic dynamics and its time reversals, has a central role in nonequilibrium thermodynamics. In the theory of probability, the change in the statistical properties of observables due to reversals can be represented by a change in the probability measure. We consider operators on the space of probability measure that induce changes in the statistical properties of a process, and formulate entropy productions in terms of these change-of-probability-measure (CPM) operators. This mathematical underpinning of the origin of entropy productions allows us to achieve an organization of various forms of fluctuation relations: All entropy productions have a non-negative mean value, admit the integral fluctuation theorem, and satisfy a rather general fluctuation relation. Other results such as the transient fluctuation theorem and detailed fluctuation theorems then are derived from the general fluctuation relation with more constraints on the operator of a entropy production. We use a discrete-time, discrete-state-space Markov process to draw the contradistinction among three reversals of a process: time reversal, protocol reversal and the dual process. The properties of their corresponding CPM operators are examined, and the domains of validity of various fluctuation relations for entropy productions in physics and chemistry are revealed. We also show that our CPM operator formalism can help us rather easily extend other fluctuations relations for excess work and heat, discuss the martingale properties of entropy productions, and derive the stochastic integral formulas for entropy productions in constant-noise diffusion process with Girsanov theorem. Our formalism provides a general and concise way to study the properties of entropy-related quantities in stochastic thermodynamics and information theory. *

show abstract

“…Physics, engineering, and sciences of complex systems and processes often encounter network dynamics that are subject to uncertainties from "individuals" in a population but also to random influences from the surrounding environment, referred to respectively as intrinsic and extrinsic noise. While the former is often due to internal complexities of the individuals one studies, the latter reflects the unpredictable world one lives in [9,15,16]. This paper proposes and develops the theory of Markov random networks as an appropriate conceptual framework that distinguishes intrinsic and extrinsic noise and incorporates them into a dynamic theory.…”

Section: Introductionmentioning

confidence: 99%

Intermittent Synchronization in finite-state random networks under Markov Perturbations

Berger,

Qian,

Wang

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

By introducing extrinsic noise as well as intrinsic uncertainty into a network with stochastic events, this paper studies the dynamics of the resulting Markov random network and characterizes a novel phenomenon of intermittent synchronization and desynchronization that is due to an interplay of the two forms of randomness in the system. On a finite state space and in discrete time, the network allows for unperturbed (or "deterministic") randomness that represents the extrinsic noise but also for small intrinsic uncertainties modelled by a Markov perturbation. It is shown that if the deterministic random network is synchronized (resp., uniformly synchronized), then for almost all realizations of its extrinsic noise the stochastic trajectories of the perturbed network synchronize along almost all (resp., along all) time sequences after a certain time, with high probability. That is, both the probability of synchronization and the proportion of time spent in synchrony are arbitrarily close to one. Under smooth Markov perturbations, high-probability synchronization and low-probability desynchronization occur intermittently in time. If the perturbation is C m (m ≥ 1) in ε, where ε is a perturbation parameter, then the relative frequencies of synchronization with probability 1 − O(ε ) and of desynchronization with probability O(ε ) can both be precisely described for 1 ≤ ≤ m via an asymptotic expansion of the invariant distribution. Existence and uniqueness of invariant distributions are established, as well as their convergence as ε → 0. An explicit asymptotic expansion is derived. Ergodicity of the extrinsic noise dynamics is seen to be crucial for the characterization of (de)synchronization sets and their respective relative frequencies. An example of a smooth Markov perturbation of a synchronized probabilistic Boolean network is provided to illustrate the intermittency between high-probability synchronization and low-probability desynchronization.

show abstract

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

Cited by 6 publications

References 42 publications

Synchronization and random attractors for reaction jump processes

Synchronization and random attractors for reaction jump processes

Unified formalism for entropy production and fluctuation relations

Intermittent Synchronization in finite-state random networks under Markov Perturbations

Contact Info

Product

Resources

About