Effective Hamiltonians and Lagrangians for Conditioned Markov Processes at Large Volume

Chabane, Lydia; Lazarescu, Alexandre; Verley, Gatien

doi:10.1007/s10955-022-02898-3

“…In this section, we provide an effective Langevin equation to generate surviving trajectories in an efficient manner. The construction presented in this section is known as a Doob transform in the probability literature [58,59] and has generated recent interest in the physics community due to its various applications [37,[60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76][77][78]. The goal is to generate only trajectories of surviving particles up to a final time t f : the absorption event can possibly occur at a later time t t f .…”

Section: Effective Langevin Equationmentioning

confidence: 99%

Transport properties of diffusive particles conditioned to survive in trapping environments

Pozzoli¹,

Bruyne²

2022

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We consider a one-dimensional Brownian motion with diffusion coefficient D in the presence of n partially absorbing traps with intensity β, separated by a distance L and evenly spaced around the initial position of the particle. We study the transport properties of the process conditioned to survive up to time t. We find that the surviving particle first diffuses normally, before it encounters the traps, then undergoes a period of transient anomalous diffusion, after which it reaches a final diffusive regime. The asymptotic regime is governed by an effective diffusion coefficient D eff, which is induced by the trapping environment and is typically different from the original one. We show that when the number of traps is finite, the environment enhances diffusion and induces an effective diffusion coefficient that is systematically equal to D eff = 2D, independently of the number of the traps, the trapping intensity β and the distance L. On the contrary, when the number of traps is infinite, we find that the environment inhibits diffusion with an effective diffusion coefficient that depends on the traps intensity β and the distance L through a non-trivial scaling function D eff = D F ( β L / D ) , for which we obtain a closed-form. Moreover, we provide a rejection-free algorithm to generate surviving trajectories by deriving an effective Langevin equation with an effective repulsive potential induced by the traps. Finally, we extend our results to other trapping environments.

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“…In this sense, when dealing with non-equilibrium stationary jump processes, there is a dynamical freedom in the way of imposing the fundamental forces that has no consequences on the fluctuations of fundamental currents. This dynamical freedom is in fact a gauge freedom since in the framework of path probability the Doob transform relating two elements in an equivalence class is in fact a gauge change [26].…”

Section: Dynamical Equivalence Classesmentioning

confidence: 99%

Dynamical equivalence classes for Markov jump processes

Verley¹

2022

J. Stat. Mech.

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2

0

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Two different Markov jump processes driven out of equilibrium by constant thermodynamic forces may have identical current fluctuations in the stationary state. The concept of dynamical equivalence classes emerges from this statement as proposed by Andrieux for discrete-time Markov chains on simple graphs. We define dynamical equivalence classes in the context of continuous-time Markov chains on multigraphs using the symmetric part of the rate matrices that define the dynamics. The freedom on the skew-symmetric part is at the core of the freedom inside a dynamical equivalence class. It arises from different splittings of the thermodynamic forces onto the system’s transitions.

show abstract

“…In this sense, when dealing with non-equilibrium stationary jump processes, there is a dynamical freedom in the way of imposing the fundamental forces that has no consequences on the fluctuations of fundamental currents. This dynamical freedom is in fact a gauge freedom since in the framework of path probability the Doob transform relating two elements in an equivalence class is in fact a gauge change [21].…”

Section: Non-equilibrium Equivalence Classesmentioning

confidence: 99%

Dynamical equivalence classes for Markov jump processes

Verley

2021

Preprint

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0

View full text Add to dashboard Cite

Two different Markov jump processes driven out of equilibrium by constant thermodynamic forces may have identical current fluctuations in the stationary state. The concept of equivalence classes emerges from this statement as proposed by Andrieux for discrete-time Markov chains on simple graphs. We define equivalence classes in the context of continuous-time Markov chains on multigraphs using the symmetric part of the rate matrices that define the dynamics. The freedom on the skew symmetric part is at the core of the dynamical freedom inside an equivalence class. It arises from different splittings of the thermodynamic forces onto the system's transitions.

show abstract

Effective Hamiltonians and Lagrangians for Conditioned Markov Processes at Large Volume

Cited by 19 publications

References 80 publications

Transport properties of diffusive particles conditioned to survive in trapping environments

Transport properties of diffusive particles conditioned to survive in trapping environments

Dynamical equivalence classes for Markov jump processes

Dynamical equivalence classes for Markov jump processes

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