Relaxation dynamics master equations

The introduction of statistical features in the basic molecular models is considered in Section II,E,2. It is argued that, in most cases, at least one part of the nonradiant molecular manifold is unknown and should be treated under statistical assumptions. By means of a partially random representation of the zero-order Hamiltonian, of the kind introduced by Wigner and others in statistical nuclear theory (see Bloch, 1966, 1969), we define a general dynamical model in which both the quantal and statistical properties of the molecular excitations are combined. Special attention is given to the nature of the statistical limit and irreversible radiationless transitions for the molecular excited states. We also discuss the relationship between this concept and similar concepts in quantum statistical theory of relaxation and master equations (Zwanzig, 1961). 

Immediately when the dynamic interpretation of Monte Carlo sampling in terms of the master equation, Eq. (31), was realized an application to study the critical divergence of the relaxation time in the two-dimensional Ising nearest-neighbor ferromagnet was attempted . For kinetic Ising and Potts models without any conservation laws, the consideration of dynamic universality classespredicts where z is the dynamic exponent , but the... 

Pfadenhauer and McCain (179) and Tarasov and Buslaev (180) studied the hexafluoroscandate ion in terms of two dynamic processes affecting the line-shape of the resonance, namely chemical exchange involving fluoride ions on the one hand and " Sc relaxation on the other hand. The latter two authors tackled the problem by solving the Kubo-Sack master equation for chemical exchange using... 

These results leave several basic questions open How to derive a non-Markovian master equation (ME) for arbitrary time-dependent driving and modulation of a thermally relaxing two-level system Would the two-level system (TLS) model hold at all for modulation rates, that are comparable to the TLS transition frequency u)a (between its states e) and g)) which may invalidate the standard rotating-wave approximation (RWA), [to hen-Tannoudji 1992] Would temperature effects, which are known to incur upward g) —> e) transitions, [Lifshitz 1980], further complicate the dynamics and perhaps hinder the suppression of decay How to control decay in an efficient, optimal fashion We address these questions by outlining the derivation of a ME of a TLS that is coupled to an arbitrary bath and is driven by an arbitrary time-dependent field. 

The glassy curves at e > 0, panels (b,c), exhibit a shift of the final relaxation with Ty from (26) and asymptotically approach a scaling function <5+ (t/fy). The master equation for the yielding scaling functions 0+ in the ISHSM can be obtained from eliminating the short-time dynamics in (18a). After a partial integration, the equation with dtqll) = 0 is solved by the scaling functions ... 

In this example the master equation formalism is appliedto the process of vibrational relaxation of a diatomic molecule represented by a quantum harmonic oscillator In a reduced approach we focus on the dynamics of just this oscillator, and in fact only on its energy. The relaxation described on this level is therefore a particular kind of random walk in the space of the energy levels of this oscillator. It should again be emphasized that this description is constructed in a phenomenological way, and should be regarded as a model. In the construction of such models one tries to build in all available information. In the present case the model relies on quantum mechanics in the weak interaction limit that yields the relevant transition matrix elements between harmonic oscillator levels, and on input from statistical mechanics that imposes a certain condition (detailed balance) on the transition rates. 

Although we are unable to use the master equation approach to examine the dynamics of relaxation from the liquid-like state because a statistical method that can account for the dynamical contribution from the huge number of liquid-like minima has not yet been developed, we are able to illustrate the type of processes that would operate by examining relaxation from the highest-energy minima in our sample (Fig. 1.14). The probability... 

The previous sections indicate that the full quantum dynamical treatment of IVR in an intermediate size molecule even under conditions of coherent excitation shows phenomena reminiscent of relaxation and equilibration. This suggests that, in general, at very high excitations in large polyatomic molecules with densities of states easily exceeding the order of 10 cm (or about 10 molecular states in an energy interval corresponding to 1 J moP ), a statistical master equation treatment may be possible [38, 122]. Such an approach has been justified by quantum simulations in model systems as well as analytical considerations... 

Master equations have been used to describe relaxation and kinetics of clusters. The first approaches were extremely approximate, and served primarily as proof-of-principle. ° Master equations had been used to describe relaxation in models of proteins somewhat earlier and continue to be used in that context. " More elaborate master-equation descriptions of cluster behavior have now appeared. These have focused on how accurate the rate coefficients must be in order that the master equation s solutions reproduce the results of molecular dynamics simulations and then on what constitutes a robust statistical sample of a large master equation system, again based on both agreement with molecular dynamics simulations and on the results of a full master equation.These are only indications now of how master equations may be used in the future as a way to describe and even control the behavior of clusters and nanoscale systems of great complexity. ... 

Understanding the structure and function of biomolecules requires insight into both thermodynamic and kinetic properties. Unfortunately, many of the dynamical processes of interest occur too slowly for standard molecular dynamics (MD) simulations to gather meaningful statistics. This problem is not confined to biomolecular systems, and the development of methods to treat such rare events is currently an active field of research. - If the kinetic system can be represented in terms of linear rate equations between a set of M states, then the complete spectrum of M relaxation timescales can be obtained in principle by solving a memoryless master equation. This approach was used in the last century for a number of studies involving atomic... 

The temperature dependency of the apparent energy of activation of the bleeching process as exhibited by the curvature in the Arrhenius plot (Fig. 3) is typically found for, e. g., dynamic mechanical relaxation processes (17), which leads to the connection whith the free volume theory. The latter processes are best described by the WLF-equation (18), log a =Cj(T-Tg)/(C +T-Tg), i. e., a master plot is obtained when plotting the logarithm of... 

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