Langevin description

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... 

In order to examine the nature of the friction coefficient it is useful to consider the various time, space, and mass scales that are important for the dynamics of a B particle. Two important parameters that determine the nature of the Brownian motion are rm = (m/M) /2, that depends on the ratio of the bath and B particle masses, and rp = p/(3M/4ttct3), the ratio of the fluid mass density to the mass density of the B particle. The characteristic time scale for B particle momentum decay is xB = Af/ , from which the characteristic length lB = (kBT/M)i lxB can be defined. In derivations of Langevin descriptions, variations of length scales large compared to microscopic length but small compared to iB are considered. The simplest Markovian behavior is obtained when both rm << 1 and rp 1, while non-Markovian descriptions of the dynamics are needed when rm << 1 and rp > 1 [47]. The other important times in the problem are xv = ct2/v, the time it takes momentum to diffuse over the B particle radius ct, and Tp = cr/Df, the time it takes the B particle to diffuse over its radius. 

In the Langevin description, one assumes that the degrees of freedom within the system that are not explicitly considered in the simulation, exert, on average, a damping force that is linear in velocity y,-f, along with additional random forces Ti t). This leads to the following equation of motion for particle number i ... 

Some of the ideas behind the Langevin description of Brownian motion were mentioned in Chap. 8, Sect. 2.4 when the Kramers [67] theory of reactions in solution was discussed. Their ideas are expanded further below and follow Chandrasekhar quite closely [271]. 

In the previous section, the phenomenological description of Brownian motion was presented. The Langevin analysis leads to a velocity autocorrelation function which decays exponentially with time. This is characteristic of a Markovian process, as Doobs has shown (see ref. 490). Since it is known heyond question that the velocity autocorrelation function is far from such an exponential function, the effect that the solvent structure has on the progress of a chemical reaction cannot be assessed very reliably by means of phenomenological Langevin description. Since the velocity of a solute is correlated with its velocity a while before, a description which fails to consider solute and solvent velocities can hardly be satisfactory. Necessarily, the analysis requires a modification of the Langevin or Fokker—Plank description. In this section, some comments are made on this new and exciting area of research. 

The simplicity of the Langevin description commends itself strongly. A simple extension of the Langevin equations allows any velocity autocorrelation function to be described. By writing... 

At this point it has to be emphasized the links of the Langevin description with the diffusion processes. By comparing the transition density functions (4.121) and (4.130), it is clear that the Langevin equation (4.126) is equivalent to the Ornstein-Uhlenbeck process. Equation (4.130) satisfies the following one-dimensional Fokker-Planck... 

In the more refined Langevin description, the free Brownian particle equation of motion contains an inertial term and reads... 

Fig. 4.22. Comparison between NMF as measured by FCS [46] and simulation using Langevin description [67] with a 2-loop cycle...

We have already noted the difference between the Langevin description of stochastic processes in terms of the stochastic variables, and the master or Fokker-Planck equations that focus on their probabilities. Still, these descriptions are equivalent to each other when applied to the same process and variables. It should be possible to extract information on the dynamics of stochastic variables from the time evolution of their probabihty distribution, for example, the Fokker-Planck equation. Here we show that this is indeed so by addressing the passage time distribution associated with a given stochastic process. In particular we will see (problem 14.3) that the first moment of this distribution, the mean first passage time, is very useful for calculating rates. 

The problem can also be approached by considering the Langevin equation rather than the Fokker-Planck equation.In the Langevin description, the motion of the particle is given by the stochastic equations... 

The friction coefficient is the inverse particle s relaxation time, jS = 9py/(2pp ), where py is the fluid s dynamic viscosity. Since the Langevin equations are linear, particle velocity and position may be formally solved as functionals of the random force, and in the diffusive limit f >> i. e., for times much larger than the particle relaxation time, they allow for the analytical evaluation of ensemble averaged products of particle position and velocity and two-point correlation functions, in terms of the random-force strength q. The authors carefully justify why they use the classical (equilibrium) form of the fluctuation-dissipation theorem (FDT) in a Langevin description the time scale of the white noise is considered to be much shorter than the time scale of the imjxjsed flow. Thus, the non-equilibrium corrections would be of the order of the ratio of the fluid molecular relaxation time to the time scale of the imposed shear and may be neglected. In this case both the time scales are clearly separated and q may be determined solely from the classical form of the FDT,... 

As in the Langevin description, the dynamic description of noninteractmg Brownian particles moving in a fluid in stationary flow, demands a mesoscopic treatment in terms of the probability density /(r,u, f). The evolution in time of this quantity is governed by the continuity equation... 

Rodriguez, R. R, e. a. (1983). Fokker-Planck and Langevin descriptions of fluctuations in uniform shear flow,. Stat. Phys. 32 279-298. 

These general considerations suggest a Langevin description (stochastic differential equation) for the time evolution of the bead positions r, ... 

The unified statistical model and its generalizations discussed in section III and the generalized Langevin description developed in section IV may find utility as computational techniques and/or conceptual frameworks for understanding various aspects of reaction dynamics in polyatomic systems. 

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