Phase space theory Hamiltonian

The intennolecular Hamiltonian of the product fragments is used to calculate the sum of states of the transitional modes, when they are treated as rotations. The resulting model [28] is nearly identical to phase space theory [29],... 

In Section VII we conclude our results and discuss several issues arising from our proposals. We revisit our original motivation—that is, to find a simple model, in the sense of dynamical systems, that captures several common aspects of slow dynamics in liquid water, or more generally supercooled liquids or glasses. Our attempt is to make clear the relation and compatibility between the potential energy landscape picture and phase space theories in the Hamiltonian dynamics. Importance of heterogeneity of the system is discussed in several respects. Unclarified and unsolved points that still remain but should be considered as crucial issues in slow dynamics in molecular systems are listed. 

Chesnavich and Bowers (1977a,b 1979) modified the phase space theory model by assuming (a) an orbiting transition state located at the centrifugal barrier, and (b) that orbital rotational energy at this transition state is converted into relative translational energy of the products. The Hamiltonian used for this orbiting transition state/phase... 

The "unified" statistical model was introduced originally in order to reconcile, or unify, two different kinds of statistical theories, transition state theory which is appropriate for reactions proceeding via a direct" reaction mechanism, and the phase space theory of Lightand Nikitin which is designed to describe reactions proceeding via a long-lived collision complex. It is particularly straightforward to apply this theory within the framework of the reaction path Hamiltonian. 

Evans and Baranyai [51, 52] have explored what they describe as a nonlinear generalization of Prigogine s principle of minimum entropy production. In their theory the rate of (first) entropy production is equated to the rate of phase space compression. Since phase space is incompressible under Hamilton s equations of motion, which all real systems obey, the compression of phase space that occurs in nonequilibrium molecular dynamics (NEMD) simulations is purely an artifact of the non-Hamiltonian equations of motion that arise in implementing the Evans-Hoover thermostat [53, 54]. (See Section VIIIC for a critical discussion of the NEMD method.) While the NEMD method is a valid simulation approach in the linear regime, the phase space compression induced by the thermostat awaits physical interpretation even if it does turn out to be related to the rate of first entropy production, then the hurdle posed by Question (3) remains to be surmounted. 

A theory for nonequilibrium quantum statistical mechanics can be developed using a time-dependent, Hermitian, Hamiltonian operator Hit). In the quantum case it is the wave functions [/ that are the microstates analogous to a point in phase space. The complex conjugate / plays the role of the conjugate point in phase space, since, according to Schrodinger, it has equal and opposite time derivative to v /. 

So far, the discussion of the dynamics and the associated phase-space geometry has been restricted to the linearized Hamiltonian in eq. (5). However, in practice the linearization will rarely be sufficiently accurate to describe the reaction dynamics. We must then generalize the discussion to arbitrary nonlinear Hamiltonians in the vicinity of the saddle point. Fortunately, general theorems of invariant manifold theory [88] ensure that the qualitative features of the dynamics are the same as in the linear approximation for every energy not too high above the energy of the saddle point, there will be a NHIM with its associated stable and unstable manifolds that act as separatrices between reactive and nonreactive trajectories in precisely the manner that was described for the harmonic approximation. 

To achieve the desired separation of the reactive degree of freedom from the bath modes, we use time-dependent normal form theory [40,99]. As a first step, the phase space is extended through the addition of two auxiliary variables a canonical coordinate x, which takes the same value as time t, and its conjugate momentum PT. The dynamics on the extended phase space is described by the Hamiltonian... 

A powerful tool for analyzing fluctuations in a nonequilibrium systems is based on the Hamiltonian [57] theory of fluctuations or alternatively on a path-integral approach to the problem [44,58-62]. The analysis requires the solution of two closely interrelated problems. The first is the evaluation of the probability density for a system to occupy a state far from the stable state in the phase space. In the stationary regime, the tails of this probability are determined by the probabilities of large fluctuations. 

The first reason that led Latora and Baranger to evaluate the time evolution of the Gibbs entropy by means of a bunch of trajectories moving in a phase space divided into many small cells is the following In the Hamiltonian case the density equation must obey the Liouville theorem, namely it is a unitary transformation, which maintains the Gibbs entropy constant. However, this difficulty can be bypassed without abandoning the density picture. In line with the advocates of decoherence theory, we modify the density equation in such a way as to mimic the influence of external, extremely weak fluctuations [141]. It has to be pointed out that from this point of view, there is no essential difference with the case where these fluctuations correspond to a modified form of quantum mechanics [115]. 

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. 

The phase space structure of classical molecular dynamics is extensively used in developing classical reaction rate theory. If the quanmm reaction dynamics can also be viewed from a phase-space perspective, then a quantum reaction rate theory can use a significant amount of the classical language and the quantum-classical correspondence in reaction rate theory can be closely examined. This is indeed possible by use of, for example, the Wigner function approach. For simplicity let us consider a Hamiltonian system with only one DOF. Generalization to many-dimensional systems is straightforward. The Wigner function associated with a density operator /)( / is defined by... 

It is a phase space rather than configuration space theory, so it can treat Hamiltonian systems containing unconserved angular momenta like Coriolis interactions which prevent the Hamiltonian from being written as a sum of the kinetic and potential energies [6,18]. The resulting hypersurfaces are dynamical in that they involve momenta as well as coordinates. 

Most realistic Hamiltonians with simple saddles do not appear in this form. In what follows, we show how to transform such a Hamiltonian into this form using Normal-Form theory [13]. The phase-space structures that form the subject of this review will then be expressed in terms of the normal-form coordinates (qi,..., q ,pi,..., p ). Therefore, before analyzing Eq. (2) we show that any Hamiltonian vector field in the neighborhood of an equilibrium point of saddle (g) center (8> center type can be transformed to the form of Eq. (2). 

One explanation for anomalous diffusion in Hamiltonian dynamics is the presence of self-similar invariant sets or hierarchical structures formed in phase space that play the role of partial barriers. They slow down the normal diffusion. A different explanation for intermittent behavior is given by the existence of deformed and approximate adiabatic invariants in phase space. They are shown in terms of elaborated perturbation theories such as the KAM and Nekhoroshev theorems. 

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