Fokker-Planck-Kramers equation

Many-Body Fokker-Planck-Kramers Equations... 

Three-Body Fokker-Planck-Kramers Equation... 

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]. 

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... 

In the first section we have discussed a general methodology for the theoretical description of rotational dynamics of rigid solute molecules in complex solvents. Many-body Fokker-Planck-Kramers equations (MFPKE), including collective solvent degrees of freedom (either rotational ones, i.e., rigid bodies, or translational ones, i.e., vector fields), and their conjugate momenta, have been described as convenient tools to reproduce (or simulate) the complexity of an actual liquid system. 

Both the SRLS and the FT inertial models were discussed in the context of the Hubbard-Einstein relation, that is, the relation between the momentum correlation time Tj and the rotational correlation time (second rank) for a stochastic Brownian rotator [39]. It was shown that both models can cause a substantial departure from the simple expression predicted by a one-body Fokker-Planck-Kramers equation ... 

Here we show how to implement the TTOC procedure for eliminating in a single step a set of harmonic degrees of freedom together with their conjugate momenta from an initial MFPKE. This technique is applied in Section I.C to project out the fast field X and its momentum P from the initial three body Fokker-Planck-Kramers equation. 

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