Physically consistent numerical solver for time-dependent Fokker-Planck equations

Holubec, Viktor; Kroy, Klaus; Steffenoni, Stefano

doi:10.1103/physreve.99.032117

Cited by 38 publications

(53 citation statements)

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“…We solve this formula numerically using the method described in Refs. [71,72]. The steady state value of the transition rate predicted using Bullerjahn's method reads…”

Section: Long-time Behavior and Kramers' Methodsmentioning

confidence: 99%

Brownian molecules formed by delayed harmonic interactions

2019

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A time-delayed response of individual living organisms to information exchange within flocks or swarms leads to the formation of complex collective behaviors. A recent experimental setup by Khadka et al., Nat. Commun. 9, 3864 (2018), employing synthetic microswimmers, allows to realize and study such behavior in a controlled way, in the lab. Motivated by these experiments, we study a system of N Brownian particles interacting via a retarded harmonic interaction. For N ≤ 3, we characterize its collective behavior analytically via linear stochastic delay-differential equations, and for N > 3 by Brownian dynamics simulations. The particles form nonequilibrium molecule-like structures which become unstable with increasing number of particles, delay time, and interaction strength. We evaluate the entropy fluxes in the system and develop an approximate time-dependent transition-state theory to characterize transitions between different isomers of the molecules. For completeness, we include a comprehensive discussion of the analytical solution procedure for systems of linear stochastic delay differential equations in finite dimension, and new results for covariance and time-correlation matrices.

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“…We solve this formula numerically using the method described in Refs. [71,72]. The steady state value of the transition rate predicted using Bullerjahn's method reads…”

Section: Long-time Behavior and Kramers' Methodsmentioning

confidence: 99%

Brownian molecules formed by delayed harmonic interactions

2019

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“…The term F ( r ) = −∇ U , where U is the pseudo-potential obtained from experiment data, the pseudo-temperature T was set as , where k B is the Boltzmann constant. With these inputs, we solved the equations numerically 40 .…”

Section: Methodsmentioning

confidence: 99%

Epithelial-to-mesenchymal transition proceeds through directional destabilization of multidimensional attractor

Wang

Poe

Yang

et al. 2020

Preprint

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A problem ubiquitous in almost all scientific areas is escape from a metastable state, or relaxation from one stationary distribution to a new one 1 . More than a century of studies lead to celebrated theoretical and computational developments such as the transition state theory and reactive flux formulation. Modern transition path sampling and transition path theory focus on an ensemble of trajectories that connect the initial and final states in a state space 2, 3 . However, it is generally unfeasible to experimentally observe these trajectories in multiple dimensions and compare to theoretical results. Here we report and analyze single cell trajectories of human A549 cells undergoing TGF-β induced epithelial-to-mesenchymal transition (EMT) in a combined morphology and protein texture space obtained through time lapse imaging. From the trajectories we identify parallel reaction paths with corresponding reaction coordinates and quasi-potentials. Studying cell phenotypic transition dynamics will provide testing grounds for nonequilibrium reaction rate theories. AUTHOR CONTRIBUTIONSJ. X. designed the research; W. W. and J. X. performed the research and wrote the paper. SUPPLEMENTAL FIGURE CAPTIONFigure S1 Different views of the trajectory shown in Fig. 2E. (A) Representation in the Morphology PC1-Vimentin Haralick PC1 space. (B) Representation in the Vimentin Haralick PC3-Vimentin Haralick PC4 space.Figure S2 Iterative procedure of the finite temperature string method. (A) Flow chart of the procedure. (B) Example RC curves obtained at different iteration cycle.

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“…Numerical integration of these equations yields trajectories of the stochastic process that bare complete information about its dynamics and energetics. Furthermore, for overdamped systems, the continuous state-space can be discretized in a thermodynamically consistent way [42]. Then the Fokker-Planck equation transforms into a matrix Master equation, which can be solved numerically by evaluation of the (time ordered) matrix exponential.…”

Section: Continuous State Spacementioning

confidence: 99%

“…and similarly for w out (t). At long times the PDF ρ(w out , q in ) can be approximated using the so-called large deviation theory [42,50]. Specifically, this theory postulates that up to sub-exponential contributions the PDF is given by…”

Section: Heat and Work In Steady-state Heat Enginesmentioning

confidence: 99%

Fluctuations in heat engines

Holubec,

Ryabov

2021

Preprint

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Investigation of fundamental limitations of heat to work conversion at the macroscale gave birth to the second law of classical thermodynamics. Recent years have witnessed an enormous effort to develop and describe microscopic heat engines based on systems as small as individual colloidal particles or quantum dots. At such scales, fluctuations are an inherent part of dynamics and thermodynamics. The fundamentally stochastic character of the second law then yields interesting implications on statistics of output power and efficiency of the small heat engines. The general properties, symmetries, and limitations of these statistics have become revealed, understood, and tested only during the last few years. Will their complete understanding ignite a similar revolution as the discovery of the second law? Here, we review the known general results concerning fluctuations in the performance of small heat engines. To make the discussion more transparent, we illustrate the main abstract findings on exactly solvable models and provide a thorough theoretical introduction for newcomers to the field.

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Physically consistent numerical solver for time-dependent Fokker-Planck equations

Cited by 38 publications

References 50 publications

Brownian molecules formed by delayed harmonic interactions

Brownian molecules formed by delayed harmonic interactions

Epithelial-to-mesenchymal transition proceeds through directional destabilization of multidimensional attractor

Fluctuations in heat engines

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