Multivariate diffusion equation

NORDIO - According to the Rotational Isomeric State approximation, the geometry dependent interactions are simply expressed by additional terms in the total energy. On the other hand, the hydrodynamic interactions give rise to configuration dependent friction coefficients. Under these assumptions, the stationary distribution function is uniquely defined, but the construction of the site functions requires identification of a reactive path. This is done by quadratic expansion of the multivariate diffusion equation about the saddle point connecting two stable conformers, followed by a normal mode analysis. 

So far we only considered transport of particles by diffusion. As mentioned in 1 the continuous description was not strictly necessary, because diffusion can be described as jumps between cells and therefore incorporated in the multivariate master equation. Now consider particles that move freely and should therefore be described by their velocity v as well as by their position r. The cells A are six-dimensional cells in the one-particle phase space. As long as no reaction occurs v is constant but r changes continuously. As a result the probability distribution varies in a way which cannot be described as a succession of jumps but only in terms of a differential operator. Hence the continuous description is indispensable, but the method of compounding moments can again be used. 

Nicolis, G. Malek-Mansour, M. (1980). Systematic analysis of the multivariate master equation for a reaction-diffusion system. J. Stat. Phys., 22, 495-512. 

The theoretical method developed here provides a rigorous approach to the description of the internal dynamics of flexible aliphatic tails. The treatment is able to link the master equations used in connection with the RIS approximation to the multivariate Fokker Planck or diffusive equations, avoiding loosely defined phenomenological parameters. 

A stepwise multivariate LR analysis of the log PS values of the 23 diffusion compounds and 50 descriptors yielded a linear equation that consisted of 10 descriptors. After considering the relevance in physical meaning of each descriptor and statistical significance, the 10-descriptor model was reduced to a 3-descriptor model (Eq. 68) ... 

Consider a n-dimensional diffusion process X(t) = X (t),...,X (t) defined by a multivariate stochastic differential equation of the form of [3]... 

When the term in e is neglected in eq.(17) one obtains a Multivariate Fokker-Planck Equation which has been widely used to study the stochastic reaction-diffusion problem approximately. At long times however, this... 

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