Hamiltonian systems dividing surfaces

In the vicinity of tire dividing surface, it is assumed that the Hamiltonian for the system may be separated into the two parts... 

The hypersurface fomied from variations in the system s coordinates and momenta at//(p, q) = /Tis the microcanonical system s phase space, which, for a Hamiltonian with 3n coordinates, has a dimension of 6n -1. The assumption that the system s states are populated statistically means that the population density over the whole surface of the phase space is unifomi. Thus, the ratio of molecules at the dividing surface to the total molecules [dA(qi, p )/A]... 

Gray, Rice, and Davis [12] developed an alternative RRKM (ARRKM) theory in an attempt to simplify the Davis-Gray theory for van der Waals predissociation reactions. Specifically, they replaced the exact separatrix with an approximate phase space dividing surface by dropping a number of small terms in the system Hamiltonian, and they replaced the exact mapping that defines the flux across the tme separatrix with an analytic treatment of the flux across the approximate separatrix. This simplification is schematically presented in Fig. 18. 

We also found [41] that besides total energy, the velocity across the transition state plays a major role in many-dof systems to migrate the reaction bottleneck outward from the naive dividing surface S(qi = 0). A similar picture has been observed by Pechukas and co-workers [37] in 2D Hamiltonian systems, that is, as energy increases, pairs of the periodic orbit dividing surfaces (PODSs) appearing on each reactant and product side migrate outwards, toward reactant and product state, and the outermost... 

Notice also that the system of surfaces, even if it has many components, divides space everywhere into an inside (I) and an outside (O). This definition is imique up to an overall sign thus, once a given point has been chosen and labelled as I the classification of all other points is determined. If the two sides of the fiuid film are equivalent, then the Hamiltonian of the entire system is unaffected by the interchange I 0. This symmetry is exact for bilayers. For microemulsions the two sides of the film are inequivalent, and the I and O regions are filled with different solvents, so the symmetry is an approximate one at best. 

To present briefly the different possible scenarios for the growth of multilayer films on a homogeneous surface, it is very convenient to use a simple lattice gas model language [168]. Assuming that the surface is a two-dimensional square lattice of sites and that also the entire space above the surface is divided into small elements, forming a cubic lattice such that each of the cells can be occupied by one adsorbate particle at the most, the Hamiltonian of the system can be written as [168,169]... 

Most microscopic theories of adsorption and desorption are based on the lattice gas model. One assumes that the surface of a sohd can be divided into two-dimensional cells, labelled i, for which one introduces microscopic variables Hi = 1 or 0, depending on whether cell i is occupied by an adsorbed gas particle or not. (The connection with magnetic systems is made by a transformation to spin variables cr, = 2n, — 1.) In its simplest form a lattice gas model is restricted to the submonolayer regime and to gas-solid systems in which the surface structure and the adsorption sites do not change as a function of coverage. To introduce the dynamics of the system one writes down a model Hamiltonian which, for the simplest system of a one-component adsorbate with one adsorption site per unit cell, is... 

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