Academic literature on the topic 'Mean-field stochastic differential equations (SDE)'

Abstract We derive the stochastic dynamics of the complex-valued amplitude resulting from coherent scattering from a random population of scatterers when this becomes asymptotically large. Considerations of a random walk model, introduced by Jakeman, are used to derive stochastic differential equations (SDEs) for the amplitude and corresponding intensity and phase stochastic processes. An analysis of the correlation structure in the fluctuations is provided and interpreted geometrically in terms of the gauge invariant properties of the field and the Markov property. A Fokker–Planck description for the evolution of the probability density is given and the equilibrium and detailed balance conditions are shown to hold. Expressions for the intensity autocorrelation function and power spectral density are provided in closed form. The practical implications of the stochastic theory are discussed.

We present a scenario that is useful for describing hierarchies within classes of many-component systems. Although this scenario may be quite general, it will be illustrated in the case of many-body systems whose space-time evolution can be described by a class of stochastic parabolic nonlinear partial differential equations. The stochastic component we will consider is in the form of additive noise, but other forms of noise such as multiplicative noise may also be incorporated. It will turn out that hierarchical behavior is only one of a class of asymptotic behaviors that can emerge when an out-of-equilibrium system is coarse grained. This phenomenology can be analyzed and described using the renormalization group (RG) [6, 15]. It corresponds to the existence of complex fixed points for the parameters characterizing the system. As is well known (see, for example, Hochberg and Perez-Mercader [8] and Onuki [12] and the references cited there), parameters such as viscosities, noise couplings, and masses evolve with scale. In other words, their values depend on the scale of resolution at which the system is observed (examined). These scaledependent parameters are called effective parameters. The evolutionary changes due to coarse graining or, equivalently, changes in system size, are analyzed using the RG and translate into differential equations for the probability distribution function [8] of the many-body system, or the n-point correlation functions and the effective parameters. Under certain conditions and for systems away from equilibrium, some of the fixed points of the equations describing the scale dependence of the effective parameters can be complex; this translates into complex anomalous dimensions for the stochastic fields and, therefore, the correlation functions of the field develop a complex piece. We will see that basic requirements such as reality of probabilities and maximal correlation lead, in the case of complex fixed points, to hierarchical behavior. This is a first step for the generalization of extensive behavior as described by real power laws to the case of complex exponents and the study of hierarchical behavior.

Analysis of local entropy generation is an effective means to investigate sources of efficiency loss in turbulent combustion from the standpoint of the second law of thermodynamics. A methodology, termed the entropy filtered density function (En-FDF), is developed for large eddy simulation (LES) of turbulent reacting flows to include the transport of entropy, which embodies the complete statistical information about entropy variations within the subgrid scale. The modeled En-FDF contains a stochastic differential equation (SDE) for entropy which is solved by a Lagrangian Monte Carlo method. In