Stochastic theory principles

The origin of each element composing the nuclear-ensemble approach can be traced back to decades ago, first with the works of Heller, Wilson and others in the 1980s, where absorption bands were computed based on molecular dynamics [9]. It is also influenced by the works of Skinner [10], which provided a useful link between Kubo s stochastic theory of the line shape [11] and molecular dynamics, and by the reflection principle [12], which approaches bound to continuum transitions from the nuclear-ensemble perspective. The intuitive character of the nuclear-ensemble approach has created a situation where although the method is frequently employed, there is no clear derivation of its formalism. This information gap makes difficult to understand the reasons for its limitations and to propose ways to improve the method. In this contribution, we derive equations for absorption cross sections and radiative decay rates based on the nuclear-ensemble method. The main approximations are made explicit, and improvements on the method are proposed, in particular ways to get rid of arbitrary parameters. 

Rytov S. M. Principles of Statistical Radiophysics Vol. I Elements of Random Process Theory. Springer, Berlin. (1987) [Stochastic Processes. Moscow, Nauka (1976)]. 

Nonequilibrium statistical mechanics Green-Kubo theory, 43-44 microstate transitions, 44-51 adiabatic evolution, 44—46 forward and reverse transitions, 47-51 stationary steady-state probability, 47 stochastic transition, 464-7 steady-state probability distribution, 39—43 Nonequilibrium thermodynamics second law of basic principles, 2-3 future research issues, 81-84 heat flow ... 

Somewhat closer to the designation of a microscopic model are those diffusion theories which model the transport processes by stochastic rate equations. In the most simple of these models an unique transition rate of penetrant molecules between smaller cells of the same energy is determined as function of gross thermodynamic properties and molecular structure characteristics of the penetrant polymer system. Unfortunately, until now the diffusion models developed on this basis also require a number of adjustable parameters without precise physical meaning. Moreover, the problem of these later models is that in order to predict the absolute value of the diffusion coefficient at least a most probable average length of the elementary diffusion jump must be known. But in the framework of this type of microscopic model, it is not possible to determine this parameter from first principles . 

Various methods have been developed that interpolate between the coherent and incoherent regimes (for reviews see, e.g. (3)-(5)). Well-known approaches use the stochastic Liouville equation, of which the Haken-Strobl-Reineker (3) model is an example, and the generalized master equation (4). A powerful technique, which in principle deals with all aspects of the problem, uses the reduced density matrix of the exciton subsystem, which is obtained by projecting out all degrees of freedom (the bath) from the total statistical operator (6). This reduced density operator obeys a closed non-Markovian (integrodifferential) equation with a memory kernel that includes the effects of (multiple) interactions between the excitons and the bath. In practice, one is often forced to truncate this kernel at the level of two interactions. In the Markov approximation, the resulting description is known as Redfield theory (7). 

The derivation of the error variance requires some theory from different fields. For the convenience of the reader a very short overview will be given, including some basic principles and definitions of the required theory. Another reason to give some textbook theory is that the definition of several quantities can differ literature is not very consistent in that respect. A detailed description of signal theory, system theory, stochastic processes and of course mathematics can be found in several textbooks (1-5). 

This section opened with an example of the macroscopic theory which is based, of course, on the conservation laws. The "mesoscopic" description (a term due to VAN KAMPEN [2.93) permits knowledge not only of the average behavior of an aerosol but also of its stochastic behavior through so-called master equations. However, this mesoscopic level of description may require (in complex systems) some physical assumptions as to the transition probabilities between states describing the system. Finally, the microscopic approach attempts to develop the theory of an aerosol from "first principles"—that is, through study of the dynamics of molecular motion in a suitable phase space. Master equations and macroscopic theory appear from the microscopic theory by the reduction of the complete dynamical description of the system in a suitable phase space to small subsets of chosen variables. 

It was logical to extend the results of the thermostatic theory to temporal changes. Stochastic thermostatics adopts an intermediate level between statistical and phenomenological thermodynamics. Analogously, in principle the stochastic treatment of thermodynamics processes has an intermediate character between nonequilibrium statistical mechanics and phenomenolog-... 

Stochastic chance-constrained programming is proposed by Charnes and Cooper [3] in 1959, which is an optimization theory in terms of probability. It is mainly for constraint conditions including random variables and the decisions must be made before random variables are observed. A principle is adopted with consideration that the decisions are made in the event of adverse situations which may not satisfy the constraints decisions are allowed not to meet the constraints in some degree, but the probability of constraints being satisfied should be kept not less than a confidence level a [1, 2]. 

Coupling electronic structure theory with dynamics (direct dynamics), kinetics from first-principles— rate constants, stochastic simulation theory... 

Nowadays, computer simulations are treated as the third fundamental discipline of interface research in addition to the two classieal ones, namely theory and experiment. Based direetly on a microscopie model of the system, eomputer simulations can, in principle at least, provide an exact solution of any physicochemical problem. By far the most common methods of studying adsorption systems by simulations are the Monte Carlo (MC) technique and the molecular dynamics (MD) method. In this ehapter, a description of simidation methods will be omitted because several textbooks and review artieles on the subject are available [274-277]. The present discussion will be restricted to elementary aspects of simulation methods. In the deterministic MD method, the moleeular trajectories are eomputed by solving Newton s equations, and a time-correlated sequenee of configurations is generated. The main advantage of this technique is that it permits the study of time-dependent processes. In MC simulation, a stochastic element is an essential part of the method the trajectories are generated by random walk in configuration space. Struetural and thermodynamic properties are accessible by both methods. 

In this paper we shall begin by a short historical overview of the different theories of electron transfer. This overview will be of course limited, in order to emphasize the physical principles involved in electron transfer. For additional details, exhaustive reviews of the different theoretical treatments can be found in refs. Moreover, we shall restrict ourselves to theories of quantum mechanical nature. Thus, stochastic models (cf for instance the Kramers model) are not discussed here, but they are treated elsewhere in this book. We shall then focus on recent developments in intramolecular electron transfer and its solvent influence. 

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