A mechanical model of Brownian motion in half-space

Calderoni, P.; Dürr, Detlef; Kusuoka, Shigeo

doi:10.1007/bf01041603

Cited by 12 publications

(13 citation statements)

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“…The restriction to a single large particle is a necessary assumption for the argument here and in [3,6,7], since it allows us to estimate the distribution of velocities of the bath atoms that collide with the particle. In particular, fast moving bath atoms can collide at most once with the large particle, whereas in the multi-particle case these atoms could bounce between large particles and possibly collide with them many times.…”

Section: Rapide Note Highlightmentioning

confidence: 99%

Derivation of Langevin dynamics in a nonzero background flow field

et al. 2013

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Abstract.We propose a derivation of a nonequilibrium Langevin dynamics for a large particle immersed in a background flow field. A single large particle is placed in an ideal gas heat bath composed of point particles that are distributed consistently with the background flow field and that interact with the large particle through elastic collisions. In the limit of small bath atom mass, the large particle dynamics converges in law to a stochastic dynamics. This derivation follows the ideas of Mathematics Subject Classification. 82C05, 82C31.

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Section: Rapide Note Highlightmentioning

confidence: 99%

Derivation of Langevin dynamics in a nonzero background flow field

et al. 2013

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“…We have derived two expressions for F 0 (0) • F 0 (t) , Eqs. (20) and (65), by using ensemble averaging and the ray representation approach, respectively. From the property that the dynamics of each solvent particle is decoupled from those of the other solvent particles in the frozen dynamics of the Rayleigh model, these expressions are given in terms of one-particle trajectory, which enables one to obtain the desired quantity without performing MD simulations.…”

Section: Summary and Discussionmentioning

confidence: 99%

“…An explicit expression of the friction coefficient γ in the Brownian limit has been derived by various methods [12][13][14][15], and the velocity autocorrelation function and the Mori memory kernel in the NBL regime have also been investigated [16]. Mathematically rigorous results have been available for the one-dimensional model with various limiting procedures toward the Brownian limit [17], and some multidimensional extensions [18] and subsequent generalizations to introduce rotational motion to the Brownian particle [19] and to impose a reflecting boundary to the system [20] have followed. Some progress in the generalization of the interaction structure other than elastic collisions and multiple Brownian particles has been recently made by Kusuoka and Liang [21].…”

Section: B Rayleigh Modelmentioning

confidence: 99%

“…Since the momentum of the Brownian particle obeys the Ornstein-Uhlenbeck process without scaling under this limit, this limit is more favorable for presenting mathematically rigorous statements and proofs; it has also been used in Refs. [18][19][20]. However, we can also use the infinite mass limit with the ray representation approach, and the main results under the infinite mass limit are presented below in Sec.…”

Section: Ray Representation Approachmentioning

confidence: 99%

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Microscopic theory of Brownian motion revisited: The Rayleigh model

Kim

Karniadakis

2013

Phys. Rev. E

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We investigate three force autocorrelation functions F(0) • F(t) , F + (0) • F + (t) , and F 0 (0) • F 0 (t) and the friction coefficient γ for the Rayleigh model (a massive particle in an ideal gas) by analytic methods and molecular-dynamics (MD) simulations. Here, F and F + are the total force and the Mori fluctuating force, respectively, whereas F 0 is the force on the Brownian particle under the frozen dynamics, where the Brownian particle is held fixed and the solvent particles move under the external potential due to the presence of the Brownian particle. By using ensemble averaging and the ray representation approach, we obtain two expressions for F 0 (0) • F 0 (t) in terms of the one-particle trajectory and corresponding expressions for γ by the time integration of these expressions. Performing MD simulations of the near-Brownian-limit (NBL) regime, we investigate the convergence of F(0) • F(t) and F + (0) • F + (t) and compare them with F 0 (0) • F 0 (t). We show that for a purely repulsive potential between the Brownian particle and a solvent particle, both expressions for F 0 (0) • F 0 (t) produce F + (0) • F + (t) in the NBL regime. On the other hand, for a potential containing an attractive component, the ray representation expression produces only the contribution of the nontrapped solvent particles. However, we show that the net contribution of the trapped particles to γ disappears, and hence we confirm that both the ensemble-averaged expression and the ray representation expression for γ are valid even if the potential contains an attractive component. We also obtain a closed-form expression of γ for the square-well potential. Finally, we discuss theoretical and practical aspects for the evaluation of F 0 (0) • F 0 (t) and γ .

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“…We study here the simplest model which consists of only one massive particle interacting with a scalar wave field, and the whole system is in dimension 1. This article is along the same line as [1]: we derive Brownian motion as a Brownian limit of a classical mechanical system consisting of massive particle(s) and an ideal environment. This type of model, called a mechanical model of Brownian motion, was first introduced and studied by Holley [2], and extended by, e.g., Dürr et al [3][4][5], Calderoni et al [6], Kusuoka and Liang [1], and others. In all these former articles, the environment is given by an ideal gas, that is, a system consists of infinite "light" particles with its initial distribution given by Poisson point process.…”

Section: Introductionmentioning

confidence: 99%

A Classical Mechanical Model of Brownian Motion with One Particle Coupled to a Random Wave Field

Kusuoka¹,

Song²

2012

Stochastic Analysis and Applications

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We consider the problem of deriving Brownian motions from classical mechanical systems. Specifically, we consider a system with one massive particle coupling to an ideal random wave field, evolved according to classical mechanical principles. We prove the almost sure existence and uniqueness of the solution of the considered dynamics, prove the convergence of the solution under a certain scaling limit and give the precise expression of the limiting process, a diffusion process.

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A mechanical model of Brownian motion in half-space

Cited by 12 publications

References 5 publications

Derivation of Langevin dynamics in a nonzero background flow field

Derivation of Langevin dynamics in a nonzero background flow field

Microscopic theory of Brownian motion revisited: The Rayleigh model

A Classical Mechanical Model of Brownian Motion with One Particle Coupled to a Random Wave Field

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