Time scales phase-space transition states

The Allan variance analyses of energy fluctuations can tell us about the nature of the configurational energy landscapes and the existence of (multidimensional) cooperativity among individual DOFs, enhanced for an intermediate time scale at the transition temperature. However, what can we learn or deduce from an (observed) scalar time series about the geometrical aspects of the underlying multidimensional state (or phase) space buried in the observations The so-called embedding theorems attributed to Whitney [75] and Takens [76] provide us with a clue to the answer of such a question (see also Section VI.A and Refs. 77-80). 

Transition state theory (TST) (4) is a well-known method used to calculate the kinetics of infrequent events. The rate constant of the process of interest may be factored into two terms, a TST rate constant based on a knowledge of an equilibrium phase space distribution of the system, and a dynamical correction factor (close to unity) used to correct for errors in the TST rate constant. The correction factor can be evaluated from dynamical information obtained over a short time scale. 

In contrast, conventional reaction rate theory replaces the dynamics within the potential well by fluctuations at equilibrium. This replacement is made possible by the assumption of local equilibrium, in which the characteristic time scale of vibrational relaxation is supposed to be much shorter than that of reaction. Furthermore, it is supposed that the phase space within the potential well is uniformly covered by chaotic motions. Thus, only information concerning the saddle regions of the potential is taken into account in considering the reaction dynamics. This approach is called the transition state theory. 

In order to simplify the description of this system one neglects the fast dynamics in the potential wells and considers only the transitions from one well to the other which happen on a much slower time scale. Under the assumption that the potential barrier AU between the two wells is large compared to the noise strength D and the relaxation in the wells is fast compared to the time scale of the jumps between the wells, the transitions can be considered as a rate process. Such a rate process has a probability per unit time to cross the barrier, which is independent on the time which has elapsed since the last crossing event. The resulting dynamics in the reduced discrete phase space which consists just of two discrete states left and right is thus still a Markovian one, i.e. the present state determines the future evolution to a maximal extent. 

Electrons residing in molecular clusters can be viewed as microscopic probes of both the local liquid structure and the molecular dynamics of liquids, and as such their transitory existence becomes a theoretical and experimental metaphor for one of the major fundamental and contemporary problems in chemical and molecular physics, that is, how to describe the transition between the microscopic and macroscopic realms of physical laws in the condensed phase. Since this chapter was completed in the Spring of 1979, several new and important observations have been made on the dynamics and structure of e, which, as a fundamental particle interacting with atoms and molecules in a fundamental way, serves to assist that transformation for electronic states in disordered systems. In a sense, disorder has become order on the subpicosecond time-scale, as we study events whose time duration is shorter than, or comparable to, the period during which the atoms or molecules retain some memory of the initial quantum state, or of the velocity or phase space correlations of the microscopic system. This approach anticipated the new wave of theoretical and experimental interest in developing microscopic theories of... 

Variations of the evaporation rate constant of the (H2O)50 cluster, as predicted by phase space theory (PST) in its orbiting transition state version, and values of the rate constant obtained from statistical molecular dynamics (MD) trajectories at high energies. The inset shows the decay of the number of clusters N(t) having resisted evaporation as a function of time, at three internal energies denoted next to the curves and in logarithmic scale. 

Systems in which problems of multiple time scales and of "bottlenecks in phase space" occur are hardly exceptional, and promising methods of theory and discrete-event simulation have been developed recently in several important areas. In a marriage of molecular dynamics (and Monte Carlo methods) to transition state theory (2 j, for example, Bennett (22J has developed a general simulation method for treating arbitrarily infrequent dynamical events (e.g., an enzyme-catalyzed reaction process). 

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