Stochastic torques

We are thus led to assume that V and the fiuctuation-dissipation process driving the virtual body depends on the variable 17 so as to simulate the efiects of the H-bond dynamics. For example, a strong interaction potential V accompanied by friction and stochastic torques (forces) of weak intensity simulates the solidlike properties of the environment when the tt ed molecule is characterized by four hydrogen bonds. In this case a reasonable approximation is to simulate such an environment by ice Ih. In the opposite limit, a weak potential V with strong friction and stochastic torques (forces) simulates the difiusional properties of the unbounded molecule (liquid water at high temperature or in very dilute nonpolar solution). 

The random motion of the stochastic variable 17 will also explore the states with intermediate values of potential and stochastic torques (forces). The macroscopic properties of the system will depend on suitable averages on all these states. 

Let the crystallographic axes now rotate with angular velocity ) corresponding to physical rotation of the ferrofltiid particle due to the stochastic torques... 

In principle, the presence of slow stochastic torques directly affecting the solute reorientational motion can be dealt with in the framework of generalized stochastic Fokker-Planck equations including frequency-dependent frictional terms. However, the non-Markovian nature of the time evolution operator does not allow an easy treatment of this kind of model. Also, it may be difficult to justify the choice of frequency dependent terms on the basis of a sound physical model. One would like to take advantage of some knowledge of the physical system under... 

The quantities subscripted by 1 refer to the inner dipole while those subscripted by 2 refer to the outer cage, /, /x,., i = 1,2, are the corresponding moments of inertia and dipole moments, and A, are the stochastic torques and is the angular position. A,(t) is as usual a white noise driving torque with correlation function... 

The phenomenological Langevin Eqs. (227) and (228) are only applicable to a very restricted class of physical processes. In particular, they are only valid when the stochastic forces and torques have infinitely short correlation times, i.e., their autocorrelation functions are proportional to Dirac delta functions. As was shown in the previous section, these restrictions can be removed by a suitable generalization of these Langevin equations. As we saw in the particular case of the velocity, the modified Langevin equation is... 

In principle, since the potential V is not linear, Eqs. (4.1) can be used to simulate the non-Gaussian non-Markovian behavior of the variables < > and V as well as the rototranslational phenomena. Note that this non-Gaussian, non-Markovian behavior depends on the presence of the virtual body and the nonlinear nature of the potential V in spite of the Markovian-Gaussian character of the fluctuation-dissipation process governing the stodiastic torques and the stochastic forces Note also... 

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