Rotational Brownian equations

We call the correlation time it is equal to 1/6 Dj, where Dj is the rotational diffusion coefficient. The correlation time increases with increasing molecular size and with increasing solvent viscosity, equation Bl.13.11 and equation B 1.13.12 describe the rotational Brownian motion of a rigid sphere in a continuous and isotropic medium. With the Lorentzian spectral densities of equation B 1.13.12. it is simple to calculate the relevant transition probabilities. In this way, we can use e.g. equation B 1.13.5 to obtain for a carbon-13... 

According to the theory developed by Smoluchowski and by Einstein, if a spherical particle of radius r rotates in a liquid of viscosity i), in a short time A/, by an angle Aa, then the mean value of angular rotation A is given by the Brownian equation for rotational motion ... 

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 ... 

Chapter 8 by W. T. Coffey, Y. P. Kalmykov, and S. V. Titov, entitled Fractional Rotational Diffusion and Anomalous Dielectric Relaxation in Dipole Systems, provides an introduction to the theory of fractional rotational Brownian motion and microscopic models for dielectric relaxation in disordered systems. The authors indicate how anomalous relaxation has its origins in anomalous diffusion and that a physical explanation of anomalous diffusion may be given via the continuous time random walk model. It is demonstrated how this model may be used to justify the fractional diffusion equation. In particular, the Debye theory of dielectric relaxation of an assembly of polar molecules is reformulated using a fractional noninertial Fokker-Planck equation for the purpose of extending that theory to explain anomalous dielectric relaxation. Thus, the authors show how the Debye rotational diffusion model of dielectric relaxation of polar molecules (which may be described in microscopic fashion as the diffusion limit of a discrete time random walk on the surface of the unit sphere) may be extended via the continuous-time random walk to yield the empirical Cole-Cole, Cole-Davidson, and Havriliak-Negami equations of anomalous dielectric relaxation from a microscopic model based on a... 

Here, V, the volume of the spherical molecule, t] is the viscosity of the solution, k is the Boltzmann constant, and T is the thermodynamic temperature. This equation is based on the theory of rotational Brownian motion of rigid spheres in a viscous fluid. 

Debye extended the foregoing arguments in order to establish the Smoluchowski equation [Eq. (5)] for the rotational Brownian motion of a dipolar particle about a... 

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. 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

The above stochastic collision model then leads to a generalization, Eq. (253), of the Fokker-Planck equation for the evolution of the phase distribution function for mechanical particles, where the velocities acquire a fractional character [30], rather than both the displacements and the velocities as in Eq. (235). In the present context, all these comments apply, of course, to rotational Brownian motion. 

The calculation of orientational autocorrelation functions from the free rotator Eq. (14) which describes the rotational Brownian motion of a sphere is relatively easy because Sack [19] has shown how the one-sided Fourier transform of the orientational autocorrelation functions (here the longitudinal and transverse autocorrelation functions) may be expressed as continued fractions. The corresponding calculation from Eq. (15) for the three-dimensional rotation in a potential is very difficult because of the nonlinear relation between and p [33] arising from the kinematic equation, Eq. (7). 

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... 

Assuming a spherical shape for the fluorescent molecule, the degree of change in the rotational Brownian motion is given by Eq. (3.25), where v is the volume of the spherical molecule, r)0 is the solvent viscosity, r is the fluorescence lifetime of the chromophore, and T is the temperature. The values of r0 and r/v can be obtained from a plot of Mr versus T/rj0. Thus, if the fluorescence lifetime of the chromophore is known, it is possible to determine the hydrodynamic volume of the rotating molecule and its rotational diffusion constant D,. This data treatment is known as the Perrin-Weber approximation,25 after the two scientists who first derived the equations in the case of protein chromophores. 

CONCEPTS More about the effect of collisions on distribution functions microscopic theory of dielectric loss The Debye theory can define a distribution function which obeys a rotational diffusion equation. Debye [22, 23] has based his theory of dispersion on Einstein s theory of Brownian motion. He supposed that rotation of a molecule because of an applied field is constantly interrupted by collisions with neighbors, and the effect of these collisions can be described by a resistive couple proportional to the angular velocity of the molecule. This description is well adapted to liquids, but not to gases. 

Equation (8.1) is correct only for 1. To discuss the general case, we have to study the Smoluchowski equation for the rotational Brownian motion. This equation can be derived straightforwardly according to the Kirkwood theory described in Section 3.8. Such a derivation is given in Appendix 8.1. Here we derive it by an elementary method to clarify the underlying physics. 

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. 

Equation (8.173) indicates the analogy between the change of (.y) of the Kratky-Porod model and the time evolution of u(i) in rotational Brownian motion both processes are Gaussian with the constraint 1,2 = 1,35 pjqjj gqn (g.i74) it can be shown that for small s... 

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... 

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