Diffusion in a Medium with Nonlinear Friction

Lisý, V.; Tóthová, Jana; Glod, Lukáš

doi:10.1007/s10765-013-1501-4

Cited by 5 publications

(6 citation statements)

References 21 publications

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“…[23], and Refs. [24][25][26][27] for other approaches). Encouraged by the observation that the experimental MCDs are linear in x 0 (Methods), we proceed by considering a linear equation for x,…”

Section: Figures 1dmentioning

confidence: 99%

“…Ref.[23], and Refs. [24][25][26][27] for other approaches). Encouraged by the observation that the experimental MCDs are linear in x 0 (Methods), we proceed by considering a linear equation for x,(1)Formally, Γ (v) is the functional derivative of G around the non-equilibrium steady state,and similarly for the noise f (v) , which is then independent ofẋ, and f (t) = 0.…”

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confidence: 99%

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Oscillating modes of driven colloids in overdamped systems

Berner¹,

Müller²,

Gomez-Solano³

et al. 2018

Nat Commun

134

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Microscopic colloidal particles suspended in liquids are a prominent example of an overdamped system where viscous forces dominate over inertial effects. Frequently, colloids are used as sensitive probes, e.g., in biophysical applications from which molecular forces are inferred. The interpretation of such experiments rests on the assumption that, even when the particles are driven, the liquid remains in equilibrium. Here we experimentally demonstrate that this is not valid for particles in viscoelastic fluids. Even at small driving forces, we observe particle oscillations with several tens of seconds. They are attributed to non-equilibrium fluctuations of the fluid being excited by the particle’s motion. The oscillatory dynamics is in quantitative agreement with an overdamped Langevin equation with negative friction-memory term being equivalent to a stochastically driven underdamped oscillator. Such oscillatory modes are expected to widen the use of colloids as model systems but must also be considered in colloidal probe experiments.

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Section: Figures 1dmentioning

confidence: 99%

mentioning

confidence: 99%

Oscillating modes of driven colloids in overdamped systems

Berner¹,

Müller²,

Gomez-Solano³

et al. 2018

Nat Commun

134

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“…( 1) becomes the standard memoryless LE with the white noise force and the purely viscous Stokes friction force (t). To our opinion, the most simple method of solving the GLE equations, which goes back to the old work by Vladimirsky [42,43], is as follows [44][45][46][47][48]. If we are interested in finding the mean square displacement (MSD) of the particle,…”

Section: Harmonically Bounded Particle Described By the Fractional La...mentioning

confidence: 99%

“…must hold. For more details see the above cited articles [42][43][44][45][46][47][48]. The new equation,…”

Section: Harmonically Bounded Particle Described By the Fractional La...mentioning

confidence: 99%

NMR signals within the generalized Langevin model for fractional Brownian motion

Lisý

Tóthová

2018

Physica A: Statistical Mechanics and its Applications

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The methods of Nuclear Magnetic Resonance belong to the best developed and often used tools for studying random motion of particles in different systems, including soft biological tissues. In the longtime limit the current mathematical description of the experiments allows proper interpretation of measurements of normal and anomalous diffusion. The shorter-time dynamics is however correctly considered only in a few works that do not go beyond the standard memoryless Langevin description of the Brownian motion (BM). In the present work, the attenuation function S(t) for an ensemble of spinbearing particles in a magnetic-field gradient, expressed in a form applicable for any kind of stationary stochastic dynamics of spins with or without a memory, is calculated in the frame of the model of fractional BM. The solution of the model for particles trapped in a harmonic potential is obtained in an exceedingly simple way and used for the calculation of S(t). In the limit of free particles coupled to a fractal heat bath, the results compare favorably with experiments acquired in human neuronal tissues. The effect of the trap is demonstrated by introducing a simple model for the generalized diffusion coefficient of the particle.

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“…In that case, a functional limit theorem requires the convergence in non-standard Skorokhod topologies such as the M 1 -Skorokhod topology. Non-linear (β = 1) stochastic systems of the type (1.6) driven by Brownian motion, Z = B, have been studied in recent years both in physical and mathematical literature, see Lindner (2007Lindner ( , 2008Lindner ( , 2010; Lisý et al (2014) for the analysis for β = 1, 2, 3, 5, Baule and Sollich (2012); Touchette et al (2010); de Gennes (2005); Hayakawa (2005); Kawarada and Hayakawa (2004); Mauger (2006) for the important case of dry (Coulomb) friction β = 0, and Goohpattader and Chaudhury (2010) for experiments and simulations for the dry friction β = 0 and irregular friction β = 0.4. The main goal of these papers was to determine on the physical level of rigour how the so-called effective diffusion coefficient, which is roughly speaking the variance of the particle's position, depends on ε.…”

Section: Introductionmentioning

confidence: 99%

Non-Gaussian limit theorem for non-linear Langevin equations driven by Lévy noise

Kulik¹,

Pavlyukevich²

2019

Ann. Inst. H. Poincaré Probab. Statist.

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In this paper, we study the small noise behaviour of solutions of a non-linear second order Langevin equationẍ ε t + |ẋ ε t | β =Ż ε εt , β ∈ R, driven by symmetric non-Gaussian Lévy processes Z ε . This equation describes the dynamics of a one-degree-of-freedom mechanical system subject to non-linear friction and noisy vibrations. For a compound Poisson noise, the process x ε on the macroscopic time scale t/ε has a natural interpretation as a non-linear filter which responds to each single jump of the driving process. We prove that a system driven by a general symmetric Lévy noise exhibits essentially the same asymptotic behaviour under the principal condition α+2β < 4, where α ∈ [0, 2] is the "uniform" Blumenthal-Getoor index of the family {Z ε }ε>0.

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Diffusion in a Medium with Nonlinear Friction

Cited by 5 publications

References 21 publications

Oscillating modes of driven colloids in overdamped systems

Oscillating modes of driven colloids in overdamped systems

NMR signals within the generalized Langevin model for fractional Brownian motion

Non-Gaussian limit theorem for non-linear Langevin equations driven by Lévy noise

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