The Langevin description of Brownian motion

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]. 

This is the Langevin equation [271, 490]. The fluctuating force, f, varies so rapidly that it cannot be described as a function of time directly. Instead, only certain statistical properties can be defined. 

Providing the ergodic principle may be used, the time average of the fluctuating force is also zero and it will be true if the time scale of collision events is very small compared with the time interval of interest. 

The constant term T is, in part, by way of a normalising constant. The rapid and abrupt loss of memory about the magnitude and direction of this force is characteristic of a Markovian process. At a time f, the state of a system only depends on that at an infinitesimally short time earlier. 

The velocity relaxation time is m/ (and this is mD/kB T 0.01— 0.1 ps at room temperature for small molecules). It is much longer than the time scale of fluctuations of the force f(t). The time average of f(f ) in the integral above is zero. Taking the average over the ensemble gives 

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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 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. 

In fact, the validity of Eqs. (90) and (91) is not restricted to the simple (i.e., nonretarded) Langevin model as defined by Eq. (73). These formulas can be applied in other classical descriptions of Brownian motion in which a time-dependent diffusion coefficient can be defined. This is for instance, the case in the presence of non-Ohmic dissipation, in which case the motion of the Brownian particle is described by a retarded Langevin equation (see Section V). 

In Chapters 3, 6 and 7, the two equivalent descriptions of Brownian motion the Langevin and Smoluchowski equations for an entanglement-free system have been studied in the cases where analytic solutions are obtainable the time-correlation function of the end-to-end vector of a Rouse chain and the constitutive equation of the Rouse model. When the Brownian motion of a more complicated model is to be studied, where an analytical solution cannot be obtained, the Monte Carlo simulation becomes a useful tool. Unlike the Monte Carlo simulation that is employed to calculate static properties using the Metropolis criterion, the simulation based on the Langevin equation can be used to calculate both static and dynamic quantities. 

An alternative description of Brownian motion is to study the equation of motion of the Brownian particle writing the random force f(t) explicitly in the Langevin form ... 

Although Gaussian random functions are commonly encountered, their representation in terms of functional integrals is often unnecessary the averages (6.27) are usually sufficient for the development. A familiar example occurs in the description of Brownian motion by means of the Langevin equation - ... 

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

General Linearized Brownian Motion. Our discussion has not so far passed beyond the Langevin approximation, which must be modified for realistic description of the motion of molecules not vastly more massive than the molecules of the solvent medium. This is particularly necessary for the discussion of pure substances. 

Classic Brownian motion has been widely applied in the past to the interpretation of experiments sensitive to rotational dynamics. ESR and NMR measurements of T and Tj for small paramagnetic probes have been interpreted on the basis of a simple Debye model, in which the rotating solute is considered a rigid Brownian rotator, sueh that the time scale of the rotational motion is much slower than that of the angular momentum relaxation and of any other degree of freedom in the liquid system. It is usually accepted that a fairly accurate description of the molecular dynamics is given by a Smoluchowski equation (or the equivalent Langevin equation), that can be solved analytically in the absence of external mean potentials. 

Expression (11.56) is obtained under the assumption, that particles are involved completely in relative motion by pulsations of scale L Therefore the formula (11.56) can be used only if particles are relatively far apart. However, when the particles approach each other so that the clearance S between them is about radius of a smaller particle, then the velocity of their approach will be affected by the force of hydrodynamic resistance, which, as was emphasized earlier in section 8.1, grows unboundedly at 0. To account for this force, one should apply the approach, which is used in statistical physics for description of the Brownian motion of a particle under action of random external force and based on Langevin equation [37, 38] (see also section 8.2). 

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