Quantum ergodic theory

Conditions on system properties, for example, potential surfaces, state densities, masses, and so forth, which are necessary for relaxation and which emerge from quantum ergodic theory, would be used to identify properties of molecular systems necessary and sufficient to ensure statistical reaction dynamics. 

The number of difficulties associated with carrying out the preceding program is formidable. First and foremost, there is no satisfactory theory of chaotic motion in quantum ergodic theory, either for isolated bound or unbound, systems. Indeed, the qualitative concept of relaxation that we traditionally associate with statistical theories is only consistent with results for ideal systems in the classical ergodic theory of bound systems. Second,... 

S. Nordholm and S. A. Rice, A quantum ergodic theory approach to unimolecular fragmentation, J. Chem. Phys. 62 157 (1975). 

Vibrational Relaxation. Stochastic processes, including vibrational relaxation in condensed media, have been considered from a theoretical standpoint in an extensive review,502 and a further review has considered measurement of such processes also.503 Models have been presented for vibrational relaxation in diatomic liquids 504 and in condensed media,505 using a master-equation approach. An extensive development of quantum ergodic theory for relaxation processes has been published,506 and quantum resonance effects in electronic to vibrational energy transfer have been considered.507 A paper has also considered the coupling between vibrational relaxation and molecular electronic transitions.508 A theory has also been outlined for the time-resolved electronic absorption spectrum of a molecule undergoing collisional vibrational relaxation.509... 

The implimentation of quantum statistical ensemble theory applied to physically real systems presents the same problems as in the classical case. The fundamental questions of how to define macroscopic equilibrium and how to construct the density matrix remain. The ergodic theory and the hypothesis of equal a priori probabilities again serve to forge some link between the theory and working models. 

Section II provides a summary of Local Random Matrix Theory (LRMT) and its use in locating the quantum ergodicity transition, how this transition is approached, rates of energy transfer above the transition, and how we use this information to estimate rates of unimolecular reactions. As an illustration, we use LRMT to correct RRKM results for the rate of cyclohexane ring inversion in gas and liquid phases. Section III addresses thermal transport in clusters of water molecules and proteins. We present calculations of the coefficient of thermal conductivity and thermal diffusivity as a function of temperature for a cluster of glassy water and for the protein myoglobin. For the calculation of thermal transport coefficients in proteins, we build on and develop further the theory for thermal conduction in fractal objects of Alexander, Orbach, and coworkers [36,37] mentioned above. Part IV presents a summary. 

The clearest results have been obtained for classical relaxation in bound systems where the full machinery of classical ergodic theory may be utilized. These concepts have been carried over empirically to molecular scattering and decay, where the phase space is not compact and hence the ergodic theory is not directly applicable. This classical approach is the subject of Section II. Less complete information is available on the classical-quantum correspondence, which underlies step 4. This is discussed in Section III where we introduce the Liouville approach to correspondence, which, we believe, provides a unified basis for future studies in this area. Finally, the quantum picture is beginning to emerge, and Section IV summarizes a number of recent approaches relevant for a quantum-mechanical understanding of relaxation phenomena and statistical behavior in bound systems and scattering. 

The classical unimolecular dynamics is ergodic for molecules like NO2 and D2CO, whose resonance states are highly mixed and unassignable. As described above, their unimolecular dynamics is identified as statistical state specific. The classical dynamics for these molecules are intrinsically RRKM and a microcanonical ensemble of phase space points decays exponentially in accord with Eq. (3). The correspondence found between statistical state specific quantum dynamics and quantum RRKM theory is that the average of the N resonance rate constants fe,) in an energy window E + AE approximates the quantum RRKM rate constant k E) [27,90]. Because of the state specificity of the resonance rates, the decomposition of an ensemble of the A resonances is non-exponential, i.e. 

The conclusion one reaches is that quantum RRKM theory is an incomplete model for unimolecular decomposition. It does not describe fluctuations in state-specific resonance rates, which arise from the nature of the couplings between the resonance states and the continuum. It also predicts steps in k E), which appear to be inconsistent with the actual quantum dynamics as determined from computational chemistry. However, for molecules whose classical unimolecular dynamics is ergodic and intrinsically RRKM and/or whose resonance rates are statistical state specific (see Section 15.2.4), the quantum RRKM k E) gives an accurate average rate constant for an energy interval E E + AE [47]. 

Upon using the GNS construction it is easily checked that if is a C-inv state on there exists in 2 a (strongly) continuous unitary representation U (g) geG of G such that (1) t/ (g)d) = 4> (where 1> is the GNS cyclic vector associated to ), and (2)iT (a [A]) U g) x for all g in C and A in j/. [The proof is trivially obtained by defining t/ (g)0 = O, (fl) defined as in Section IIB).] The reader familiar with classical ergodic theory will recognize in this result an extension to quantum situations of the Koopman formalism for ciassiail mechanics. 

The preceding result is well-known in the case where s/ is abelian (classical ergodic theory). The point here is that it is valid as well for quantum systems. It represents the proper adaptations of the first result of Section VIE to a much more general situation. 

MSN.85. I. Prigogine and A. P. Grecos, Kinetic theory and ergodic properties in quantum mechanics, in 75 Jahre Quantenmechanik, Akademie-Verlag, Berlin, 1977, pp. 57-68. 

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