Self stochastic

Weissbuch I, Leiserowitz L, Lahav M (2005) Stochastic Mirror Symmetry Breaking via Self-Assembly, Reactivity and Amplification of Chirality Relevance to Abiotic Conditions. 259 123-165... 

Interesting variants on the simplest star formation laws include stochastic self-propagating star formation (Gerola Seiden 1978 Dopita 1985), self-regulating star formation (Arimoto 1989 Hensler Burkert 1990), stochastic star-formation bursts (Matteucci Tosi 1985), separate laws for the halo and disk, the latter including terms that account for cloud collisions and induced star formation from interactions between massive stars and clouds (Ferrini et al. 1992, 1994), and the existence of a threshold surface gas density for star formation (Kennicutt 1989 Chamcham, Pitts Tayler 1993). 

The goal of this chapter is twofold. First we wish to critically compare—from both a conceptional and a practical point of view—various classical and mixed quantum-classical strategies to describe non-Born-Oppenheimer dynamics. To this end. Section II introduces five multidimensional model problems, each representing a specific challenge for a classical description. Allowing for exact quantum-mechanical reference calculations, aU models have been used as benchmark problems to study approximate descriptions. In what follows, Section III describes in some detail the mean-field trajectory method and also discusses its connection to time-dependent self-consistent-field schemes. The surface-hopping method is considered in Section IV, which discusses various motivations of the ansatz as well as several variants of the implementation. Section V gives a brief account on the quantum-classical Liouville description and considers the possibility of an exact stochastic realization of its equation of motion. 

The theory of Brownian motion for a constrained system is more subtle than that for an unconstrained system of pointlike particles, and has given rise to a substantial, and sometimes confusing, literamre. Some aspects of the problem, involving equilibrium statistical mechanics and the diffusion equation, have been understood for decades [1-8]. Other aspects, particularly those involving the relationships among various possible interpretations of the corresponding stochastic differential equations [9-13], remain less thoroughly understood. This chapter attempts to provide a self-contained account of the entire theory. 

In this section, we consider the description of Brownian motion by Markov diffusion processes that are the solutions of corresponding stochastic differential equations (SDEs). This section contains self-contained discussions of each of several possible interpretations of a system of nonlinear SDEs, and the relationships between different interpretations. Because most of the subtleties of this subject are generic to models with coordinate-dependent diffusivities, with or without constraints, this analysis may be more broadly useful as a review of the use of nonlinear SDEs to describe Brownian motion. Because each of the various possible interpretations of an SDE may be defined as the limit of a discrete jump process, this subject also provides a useful starting point for the discussion of numerical simulation algorithms, which are considered in the following section. 

III, Stochastic Model for Restricted Self-Avoiding Chains with Nearest-Neighbor... 

HI. STOCHASTIC MODEL FOR RESTRICTED SELF-AVOIDING CHAINS WITH NEAREST-NEIGHBOR INTERACTIONS... 

So far, the effects of the chain ends were neglected in our stochastic model for the restricted chain. Therefore, n must be much larger than the number of steps needed to form the largest excluded polygon. The partition function, which incorporates the chain-end effects and which could be also employed for exact statistical description of short non-self-intersecting chains can be obtained as follows Assume, as before, that we eliminate only lowest-order polygons of t steps. Therefore, the first t — 1 steps in the chain are described as a sequence of independent events. Eq (9), then, will be replaced by... 

Shortcomings of the above described approach are self-evident the fluctuations entering equation (2.2.2) are independent of the deterministic motion, the passage from the deterministic description given by equation (2.1.1) to the stochastic one needs a large number of additional phenomenological parameters determining Gij. To define them, the fluctuation-dissipative theorem should be used. 

Stochastic aggregation does not emerge for oppositely charged particles, when electroneutrality holds due to conditions nk(t) = nB(f) = n(f), particle charge ea = — eB = e. Let us introduce, following the Debye-Hiickel method, the self-consistent potential (J> through Poisson equation... 

This statement is not self-evident and needs some comments. A role of concentration degrees of freedom in terms of the formally-kinetic description was discussed in Section 2.1.1. Stochastic approach adds here a set of equations for the correlation dynamics where the correlation functions are field-type values. Due to very complicated form of the complete set of these equations, the analytical analysis of the stationary point stability is hardly possible. In its turn, a numerical study of stability was carried out independently for the correlation dynamics with the fixed particle concentrations. 

Therefore, the study of the stochastic Lotka and Lotka-Volterra models carried out in Chapter 8, has demonstrated that the traditional estimates of the complexity of the system necessary for its self-organisation are not correct. Incorporation of the fluctuation effects and thus introduction of a continuous number of degrees of freedom prove their ability for self-organisation and thus put them into a class of the basic models for the study of the autowave processes. 

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