Langevin formulation of a subdiffusive continuous-time random walk in physical time

Cairoli, Andrea; Baule, Adrian

doi:10.1103/physreve.92.012102

Cited by 15 publications

(20 citation statements)

References 79 publications

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“…We consider the stochastic process Y (t) in the comoving frame S, whose dynamics is described by the LE˙ Y (t) = −v 0 + ξ(t), where the noise ξ is defined by its hierarchy of correlation functions; specifically, the odd ones are null, i.e., 1+2N j=1 ξ(t j ) = 0, while the even ones are [59] 2N j=1…”

Section: Appendix F: Derivation Of the Nonlocal Advection-diffusion Ementioning

confidence: 99%

“…What is now the corresponding Langevin dynamics of the anomalous diffusive process described by (13)? The key is to describe the CTRW directly in physical time rather than in the widely used subordination picture [51,54,57]. In the physical time representation a CTRW in S is given asẎ (t) = ξ(t), where ξ is the derivative of a subordinated Brownian motion [59]. This is equivalently written as the formal definition…”

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confidence: 99%

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Weak Galilean invariance as a selection principle for coarse-grained diffusive models

Cairoli

Klages

Baule

2018

Proc. Natl. Acad. Sci. U.S.A.

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How does the mathematical description of a system change in different reference frames? Galilei first addressed this fundamental question by formulating the famous principle of Galilean invariance. It prescribes that the equations of motion of closed systems remain the same in different inertial frames related by Galilean transformations, thus imposing strong constraints on the dynamical rules. However, real world systems are often described by coarse-grained models integrating complex internal and external interactions indistinguishably as friction and stochastic forces. Since Galilean invariance is then violated, there is seemingly no alternative principle to assess a priori the physical consistency of a given stochastic model in different inertial frames. Here, starting from the Kac-Zwanzig Hamiltonian model generating Brownian motion, we show how Galilean invariance is broken during the coarse-graining procedure when deriving stochastic equations. Our analysis leads to a set of rules characterizing systems in different inertial frames that have to be satisfied by general stochastic models, which we call "weak Galilean invariance." Several well-known stochastic processes are invariant in these terms, except the continuous-time random walk for which we derive the correct invariant description. Our results are particularly relevant for the modeling of biological systems, as they provide a theoretical principle to select physically consistent stochastic models before a validation against experimental data.

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Section: Appendix F: Derivation Of the Nonlocal Advection-diffusion Ementioning

confidence: 99%

mentioning

confidence: 99%

Weak Galilean invariance as a selection principle for coarse-grained diffusive models

Cairoli

Klages

Baule

2018

Proc. Natl. Acad. Sci. U.S.A.

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“…where the friction force becomes g ( ) s t t d d , and ( ( )) B s t t d d can be regarded as a new noise x ( ) t [41]. For any specific realization of the inverse subordinator s(t), the correlation function of the new noise x ( ) t reads as where the average á ñ  is only taken over Brownian motion.…”

Section: Fluctuation-dissipation Theorem (Fdt) For the Case G ¹mentioning

confidence: 99%

Lévy-walk-like Langevin dynamics

2019

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Continuous-time random walks and Langevin equations are two classes of stochastic models used to describe the dynamics of particles in the natural world. While some of the processes can be conveniently characterized by both of them, more often one model has significant advantages (or has to be used) compared with the other one. In this paper, we consider the weakly damped Langevin system coupled with a new subordinator-α-dependent subordinator with 1<α<2. We pay attention to the diffusive behavior of the stochastic process described by this coupled Langevin system, and find the super-ballistic diffusion phenomenon for the system with an unconfined potential on velocity but sub-ballistic superdiffusion phenomenon with a confined potential, which is like Lévy walk for long times. One can further note that the two-point distribution of inverse subordinator affects mean square displacement of this coupled weakly damped Langevin system in essential. 0 , model (1) yields subdiffusion. Nowadays, subordinator is a very effective tool to characterize various complex dynamical systems, especially some real-life data in biology [8]

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“…When F = 0, the Langevin picture (9) is coincided with the free-force case [22]. The constant force F multiplied by Lévy noise η(s) is meant to affect the stochastic process for all physical time t after making the subordination [39,40]. Otherwise, the force F is only effective over operational time s, similar to the case that the external force only affects on the instant of jump in CTRW model, and it is invalid during the trap event with the constant s [41,42].…”

Section: Lévy Walk In a Constant Force Fieldmentioning

confidence: 99%

Langevin picture of Lévy walk in a constant force field

2019

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Lévy walk is a practical model and has wide applications in various fields. Here we focus on the effect of an external constant force on the Lévy walk with the exponent of the power-law distributed flight time α ∈ (0, 2). We add the term F η(s) (η(s) is the Lévy noise) on a subordinated Langevin system to characterize such a constant force, being effective on the velocity process for all physical time after the subordination. We clearly show the effect of the constant force F on this Langevin system and find this system is like the continuous limit of the collision model. The first moments of velocity processes for these two models are consistent. In particular, based on the velocity correlation function derived from our subordinated Langevin equation, we investigate more interesting statistical quantities, such as the ensemble-and time-averaged mean squared displacements. Under the influence of constant force, the diffusion of particles becomes faster. Finally, the super-ballistic diffusion and the non-ergodic behavior are verified by the simulations with different α.

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Langevin formulation of a subdiffusive continuous-time random walk in physical time

Cited by 15 publications

References 79 publications

Weak Galilean invariance as a selection principle for coarse-grained diffusive models

Weak Galilean invariance as a selection principle for coarse-grained diffusive models

Lévy-walk-like Langevin dynamics

Langevin picture of Lévy walk in a constant force field

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