Langevin equation Brownian motion

In his treatment of Brownian motion, Langevin began by writing down the equation of motion of a Brownian particle in a suspension. He assumed the forces acting on it could be divided into two parts ... 

We further discuss how quantities typically measured in the experiment (such as a rate constant) can be computed with the new formalism. The computations are based on stochastic path integral formulation [6]. Two different sources for stochasticity are considered. The first (A) is randomness that is part of the mathematical modeling and is built into the differential equations of motion (e.g. the Langevin equation, or Brownian dynamics). The second (B) is the uncertainty in the approximate numerical solution of the exact equations of motion. 

Depending on the desired level of accuracy, the equation of motion to be numerically solved may be the classical equation of motion (Newton s), a stochastic equation of motion (Langevin s), a Brownian equation of motion, or even a combination of quantum and classical mechanics (QM/MM, see Chapter 11). 

The friction coefficient is one of the essential elements in the Langevin description of Brownian motion. The derivation of the Langevin equation from the microscopic equations of motion provides a Green-Kubo expression for this transport coefficient. Its computation entails a number of subtle features. Consider a Brownian (B) particle with mass M in a bath of N solvent molecules with mass m. The generalized Langevin equation for the momentum P of the B... 

Brownian motion theory may be generalized to treat systems with many interacting B particles. Such many-particle Langevin equations have been investigated at a molecular level by Deutch and Oppenheim [58], A simple system in which to study hydrodynamic interactions is two particles fixed in solution at a distance Rn- The Langevin equations for the momenta P, (i = 1,2)... 

Constraints may be introduced either into the classical mechanical equations of motion (i.e., Newton s or Hamilton s equations, or the corresponding inertial Langevin equations), which attempt to resolve the ballistic motion observed over short time scales, or into a theory of Brownian motion, which describes only the diffusive motion observed over longer time scales. We focus here on the latter case, in which constraints are introduced directly into the theory of Brownian motion, as described by either a diffusion equation or an inertialess stochastic differential equation. Although the analysis given here is phrased in quite general terms, it is motivated primarily by the use of constrained mechanical models to describe the dynamics of polymers in solution, for which the slowest internal motions are accurately described by a purely diffusive dynamical model. 

After the work of Einstein and Smoluchowski an alternative treatment of Brownian motion was initiated by Langevin.Consider the velocity of the Brownian particle, as in VIII.4. When the mass is taken to be unity it obeys the equation of motion... 

In another paper, R. Kuho (Kcio University, Japan) illustrates in a rather technical and mathematical fashion tire relationship between Brownian motion and non-equilibrium statistical mechanics, in this paper, the author describes the linear response theory, Einstein s theory of Brownian motion, course-graining and stochastization, and the Langevin equations and their generalizations. 

Before discussing other results it is informative to first consider some correlation and memory functions obtained from a few simple models of rotational and translational motion in liquids. One might expect a fluid molecule to behave in some respects like a Brownian particle. That is, its actual motion is very erratic due to the rapidly varying forces and torques that other molecules exert on it. To a first approximation its motion might then be governed by the Langevin equations for a Brownian particle 61... 

Brownian Motion in a Force Field Consider the Langevin equation for a particle in a force field ... 

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