Langevin equation rotational motion

The first objective of this review is to describe a method of solution of the Langevin equations of motion of the itinerant oscillator model for rotation about a fixed axis in the massive cage limit, discarding the small oscillation approximation in the context of dielectric relaxation of polar molecules, this solution may be obtained using a matrix continued fraction method. The second... 

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... 

Here (Oe and co are delivered by the corresponding Langevin equations of the theory of the rotational Brownian motion. In order to obtain these equations, one must include in the dynamic equations (4.308) and (4.310) the random thermal torques. We do that in the following way ... 

Thus the Debye equation [Eq. (1)] may be satisfactorily explained in terms of the thermal fluctuations of an assembly of dipoles embedded in a heat bath giving rise to rotational Brownian motion described by the Fokker-Planck or Langevin equations. The advantage of a formulation in terms of the Brownian motion is that the kinetic equations of that theory may be used to extend the Debye calculation to more complicated situations [8] involving the inertial effects of the molecules and interactions between the molecules. Moreover, the microscopic mechanisms underlying the Debye behavior may be clearly understood in terms of the diffusion limit of a discrete time random walk on the surface of the unit sphere. 

We illustrate this by referring to the motion of a fixed axis rotator, the normal Brownian rotation of which is described by the Langevin equation (see Ref. 8, Chapters 4 and 10) ... 

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. 

The Langevin approach has been used by many authors in order to treat nonlinear systems. This is of importance to us since the equations of rotational motion are intrinsically nonlinear. The concept of a nonlinear Langevin equation is also subject to a number of criticisms. These have been discussed extensively by van Kampen [58] (Chapters 8 and 14). In our calculations, we shall encounter stochastic differential equations of the form... 

The rotational Brownian motion can also be described by the Langevin equation, but it is rarely used in the problem of rodlike polymers because it is less convenient for calculation than the Smoluchowski equation. 

The time-dependent classical statistical mechanics of systems of simple molecules is reviewed. The Liouville equation is derived the relationship between the generalized susceptibility and time-correlation function of molecular variables is obtained and a derivation of the generalized Langevin equation from the Liouville equation is given. The G.L.E. is then simplified and/or approximated by introducing physical assumptions that are appropriate to the problem of rotational motion in a dense fluid. Finally, the well-known expressions for spectral intensity of infrared and Raman vibration-rotation bands are reformulated in terms of time correlation functions. As an illustration, a brief discussion of the application of these results to the analysis of spectral data for liquid benzene is presented. 

With the proper choice of the operator F in (3), motional constraints in the form of an ordering potential can also he introduced into these discrete models. In a different model of a single particle undergoing rotational diffusion in the mean field of an ordering potential, the BD trajectories are obtained hy direct numerical solution of the Langevin equation ... 

The Mean-Square Displacement of a Brownian Particle Langevin s Method Applied to Rotational Relaxation Application of Langevin s Method to Rotational Brownian Motion The Fokker-Planck Equation Method (Intuitive Treatment) Brown s Intuitive Derivation of the Fokker-Planck Equation... 

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