Particle dynamics Hamiltonian formulation

In this chapter size effects in encounter and reaction dynamics are evaluated using a stochastic approach. In Section IIA a Hamiltonian formulation of the Fokker-Planck equation (FPE) is develojjed, the form of which is invariant to coordinate transformations. Theories of encounter dynamics have historically concentrated on the case of hard spheres. However, the treatment presented in this chapter is for the more realistic case in which the particles interact via a central potential K(/ ), and it will be shown that for sufficiently strong attractive forces, this actually leads to a simplification of the encounter problem and many useful formulas can be derived. These reduce to those for hard spheres, such as Eqs. (1.1) and (1.2), when appropriate limits are taken. A procedure is presented in Section IIB by which coordinates such as the center of mass and the orientational degrees of freedom, which are often characterized by thermal distributions, can be eliminated. In the case of two particles the problem is reduced to relative motion on the one-dimensional coordinate R, but with an effective potential (1 ) given by K(l ) — 2fcTln R. For sufficiently attractive K(/ ), a transition state appears in (/ X this feature that is exploited throughout the work presented. The steady-state encounter rate, defined by the flux of particles across this transition state, is evaluated in Section IIC. 

Another apparent difference between the various free energy methods lies in the treatment of order parameters. In the original formulation of a number of methods, order parameters were dynamical variables - i.e., variables that can be expressed in terms of the Cartesian coordinates of the particles - whereas in others, they were parameters in the Hamiltonian. This implies a different treatment of the order parameter in the equations of motion. If one, however, applies the formalism of metadynamics, or extended dynamics, in which any parameter can be treated as a dynamical variable, most conceptual differences between these two cases vanish. 

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