Contrasting chaotic with stochastic dynamics via ordinal transition networks

Dense fluids and liquids molecules are in constant interaction; hence, they do not fit into the Boltzmann’s picture of a clearcut separation between free-streaming and collisional interactions. Since the interactions are soft and do not involve large scattering angles, an effective way of describing dense fluids is to formulate stochastic models of particle motion, as pioneered by Einstein’s theory of Brownian motion and later extended by Paul Langevin. Besides its practical value for the study of the kinetic theory of dense fluids, Brownian motion bears a central place in the historical development of kinetic theory. Among others, it provided conclusive evidence in favor of the atomistic theory of matter. This chapter introduces the basic notions of stochastic dynamics and its connection with other important kinetic equations, primarily the Fokker–Planck equation, which bear a complementary role to the Boltzmann equation in the kinetic theory of dense fluids.

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Stochastic Dynamics in Computational Biology

10.1007/978-3-030-62387-6 ◽

2020 ◽

Author(s):

Stefanie Winkelmann ◽

Christof Schütte

Keyword(s):

Computational Biology ◽

Stochastic Dynamics

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Neural network aided approximation and parameter inference of non-Markovian models of gene expression

Nature Communications ◽

10.1038/s41467-021-22919-1 ◽

2021 ◽

Vol 12 (1) ◽

Author(s):

Qingchao Jiang ◽

Xiaoming Fu ◽

Shifu Yan ◽

Runlai Li ◽

Wenli Du ◽

...

Keyword(s):

Neural Network ◽

Stochastic Dynamics ◽

Stochastic Simulations ◽

Parameter Inference ◽

Biochemical Processes ◽

Markovian Models ◽

The Neural Network ◽

Large Numbers ◽

Small Set ◽

Noisy Measurements

AbstractNon-Markovian models of stochastic biochemical kinetics often incorporate explicit time delays to effectively model large numbers of intermediate biochemical processes. Analysis and simulation of these models, as well as the inference of their parameters from data, are fraught with difficulties because the dynamics depends on the system’s history. Here we use an artificial neural network to approximate the time-dependent distributions of non-Markovian models by the solutions of much simpler time-inhomogeneous Markovian models; the approximation does not increase the dimensionality of the model and simultaneously leads to inference of the kinetic parameters. The training of the neural network uses a relatively small set of noisy measurements generated by experimental data or stochastic simulations of the non-Markovian model. We show using a variety of models, where the delays stem from transcriptional processes and feedback control, that the Markovian models learnt by the neural network accurately reflect the stochastic dynamics across parameter space.

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Generalized fractional Poisson process and related stochastic dynamics

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2020-0034 ◽

2020 ◽

Vol 23 (3) ◽

pp. 656-693 ◽

Cited By ~ 2

Author(s):

Thomas M. Michelitsch ◽

Alejandro P. Riascos

Keyword(s):

Poisson Process ◽

Continuous Time ◽

Counting Process ◽

Stochastic Dynamics ◽

Waiting Times ◽

Fractional Diffusion ◽

Diffusion Limit ◽

Fractional Operators ◽

Fractional Poisson Process ◽

Erlang Process

AbstractWe survey the ‘generalized fractional Poisson process’ (GFPP). The GFPP is a renewal process generalizing Laskin’s fractional Poisson counting process and was first introduced by Cahoy and Polito. The GFPP contains two index parameters with admissible ranges 0 < β ≤ 1, α > 0 and a parameter characterizing the time scale. The GFPP involves Prabhakar generalized Mittag-Leffler functions and contains for special choices of the parameters the Laskin fractional Poisson process, the Erlang process and the standard Poisson process. We demonstrate this by means of explicit formulas. We develop the Montroll-Weiss continuous-time random walk (CTRW) for the GFPP on undirected networks which has Prabhakar distributed waiting times between the jumps of the walker. For this walk, we derive a generalized fractional Kolmogorov-Feller equation which involves Prabhakar generalized fractional operators governing the stochastic motions on the network. We analyze in d dimensions the ‘well-scaled’ diffusion limit and obtain a fractional diffusion equation which is of the same type as for a walk with Mittag-Leffler distributed waiting times. The GFPP has the potential to capture various aspects in the dynamics of certain complex systems.

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Stochastic dynamics of populations with refuge in polluted turbidostat

Chaos Solitons & Fractals ◽

10.1016/j.chaos.2021.110963 ◽

2021 ◽

Vol 147 ◽

pp. 110963

Author(s):

Yu Mu ◽

Wing-Cheong Lo

Keyword(s):

Stochastic Dynamics ◽

Dynamics Of Populations

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Effect of Energy Degeneracy on the Transition Time for a Series of Metastable States

Journal of Statistical Physics ◽

10.1007/s10955-021-02788-0 ◽

2021 ◽

Vol 184 (1) ◽

Author(s):

Gianmarco Bet ◽

Vanessa Jacquier ◽

Francesca R. Nardi

Keyword(s):

Metastable State ◽

Stochastic Dynamics ◽

Transition Probabilities ◽

Mixing Time ◽

Stable State ◽

Temperature Limit ◽

Metastable States ◽

Transition Time ◽

Sharp Estimates ◽

Model Independent

AbstractWe consider the problem of metastability for stochastic dynamics with exponentially small transition probabilities in the low temperature limit. We generalize previous model-independent results in several directions. First, we give an estimate of the mixing time of the dynamics in terms of the maximal stability level. Second, assuming the dynamics is reversible, we give an estimate of the associated spectral gap. Third, we give precise asymptotics for the expected transition time from any metastable state to the stable state using potential-theoretic techniques. We do this in a general reversible setting where two or more metastable states are allowed and some of them may even be degenerate. This generalizes previous results that hold for a series of only two metastable states. We then focus on a specific Probabilistic Cellular Automata (PCA) with configuration space $${\mathcal {X}}=\{-1,+1\}^\varLambda $$ X = { - 1 , + 1 } Λ where $$\varLambda \subset {\mathbb {Z}}^2$$ Λ ⊂ Z 2 is a finite box with periodic boundary conditions. We apply our model-independent results to find sharp estimates for the expected transition time from any metastable state in $$\{\underline{-1}, {\underline{c}}^o,{\underline{c}}^e\}$$ { - 1 ̲ , c ̲ o , c ̲ e } to the stable state $$\underline{+1}$$ + 1 ̲ . Here $${\underline{c}}^o,{\underline{c}}^e$$ c ̲ o , c ̲ e denote the odd and the even chessboard respectively. To do this, we identify rigorously the metastable states by giving explicit upper bounds on the stability level of every other configuration. We rely on these estimates to prove a recurrence property of the dynamics, which is a cornerstone of the pathwise approach to metastability.

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