Coordinates natural bifurcation

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]. 

A large part of the computational work has been influenced by the introduction of curvilinear coordinates, designed to take advantage of the topography of potential surfaces. These coordinates allow for a smooth change from reactant to product conformations and in effect transform the rearrangement problem into the much simpler one of inelastic collisions. The various treatments have employed reaction-path (or natural collision) coordinates less restricted reaction coordinates atom-transfer coordinates, somewhat analogous to those used for electron-transfer and, for planar and spatial motion, bifurcation coordinates. 

Let X be a set of parameter values for which a solution of eqs. (2), referred to as reference state, loses its stability and gives rise to new branches o7 sol uti ons by a bifurcation mechanism. We want to see how the solution of the master equation, eq. (1), behaves under these conditions, and how this behavior depends on small changes of the parameters X around The answer to this question depends on the kind of bifurcation considered, on the nature of the reference state, and on the number of variables involved in the dynamics. The simplest case is, by far, the pitchfork bifurcation occurring as a first transition from a previously stable spatially uniform stationary state. This transition is characterized by a remarkable universality. First, whatever the number of variables present initially, it is always possible to cast the stochastic dynamics in terms of a single, "critical" variable. This is the probabilistic analog of adiabatic elimination or, in more modern terms, of the center manifold theorem [4,8-10]. Second, the stationary probability distribution of the critical variable can be cast in the form (we set 6X = (p-Xg, rstands for the spatial coordinates) ... 

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