Multivariate Fokker-Planck equation

The nonlinear multivariate Fokker-Planck equation (6.1) will be derived and studied in chapter X. 

We now apply the temperature expansion to the multivariate Fokker-Planck equation (4.1) or (4.7). First one must display the temperature dependence explicitly. In principle this requires a further specification of the system, but in all our examples Ff is independent of 6 and Btj is proportional to it Bij(x) = dbij(x). It is then convenient to start from (4.17),... 

Equation (11.10) arises from inserting equation (11.52) into a = Uo ria/N and then expanding c(a) in powers of fl. The matrix (gij) coincides with the matrix of the linearized macroscopic equations (11.5). The fluctuations enter through the matrix (hij). The Hurwitz criterion [15] assures that this matrix is positive semi definite, which means that equation (11.7) is a linear multivariate Fokker-Planck equation. 

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... 

This is a multivariate linear Fokker-Planck equation of the type solved in VIII.6. We use it to determine the moments of c and i/. 

Let us suppose that the liquid system is described by a MFPKE in N + 1 rigid bodies (the solute, or body 1 and N rotational solvent modes or bodies ), each characterized by inertia and friction tensors I and a set of Euler angles ft , and an angular momentum vector L (n = 1,..., N -I-1) plus K fields, each defined by a generalized mass tensor and friction tensor and a position vector and the conjugate linear momentum k = 1,..., K). The time evolution of the joint conditional probability x", L , P° 11, X, L, P, t) (where ft, X, etc. stand for the collection of Euler angles, field coordinates etc.) for the system is governed by the multivariate Fokker-Planck-Kramers equation... 

We shall return to the question of the truncation at n = 2 later. Since we will be dealing with the multivariable form of the Fokker-Planck equation it is necessary to quote the form of that equation for many dimensions. The multivariable form of the Fokker-Planck equation [31] is... 

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. 

Clearly, a general theory able to naturally include other solvent modes in order to simulate a dissipative solute dynamics is still lacking. Our aim is not so ambitious, and we believe that an effective working theory, based on a self-consistent set of hypotheses of microscopic nature is still far off. Nevertheless, a mesoscopic approach in which one is not limited to the one-body model, can be very fruitful in providing a fairly accurate description of the experimental data, provided that a clever choice of the reduced set of coordinates is made, and careful analytical and computational treatments of the improved model are attained. In this paper, it is our purpose to consider a description of rotational relaxation in the formal context of a many-body Fokker-Planck-Kramers equation (MFPKE). We shall devote Section I to the analysis of the formal properties of multivariate FPK operators, with particular emphasis on systematic procedures to eliminate the non-essential parts of the collective modes in order to obtain manageable models. Detailed computation of correlation functions is reserved for Section II. A preliminary account of our approach has recently been presented in two Letters which address the specific questions of (1) the Hubbard-Einstein relation in a mesoscopic context [39] and (2) bifurcations in the rotational relaxation of viscous liquids [40]. 

---