Evolution equation stochastic

As a final note, it has to be stressed out that Eqs. (4.49) and (4.50) and Eqs. (4.52) and (4.53) hold for an arbitrary stochastic process. These evolution equations cannot give any information about whether or not the process is Markovian.135 The master equation concept has been used to analyze some examples of multistate relaxation processes.139... 

As discussed in Section 8.1, a phenomenological stochastic evolution equation can be constructed by using a model to describe the relevant states of the system and the transition rates between them. For example, in the one-dimensional random walk problem discussed in Section 7.3 we have described the position of the walker by equally spaced points nAx (n = —cx3,..., oo) on the real axis. Denoting by Pin, Z) the probability that the particle is at position n at time Z and by kr and ki the probabilities per unit time (i.e. the rates) that the particle moves from a given... 

Not only is the probability of discharge d of interest, but also the current evolution with time. From the description of the current evolution with time one can deduce the mean current, which is easily accessible experimentally and is important to evaluate the quantity of heat generated by the process. From the description of the stochastic process of the discharge activity given previously, the current can be computed. In order to obtain the current evolution equation, an auxiliary random variable n(t) defined by the time derivation of the random variable N(t) is introduced ... 

Stochastic differential equations then obviously describe the evolution of stochastic processes in terms of a dependent random variable X(t). 

One of the most extensively discussed topics of the theory of stochastic physics is whether the evolution equations of the discrete state-space stochastic processes, i.e. the master equations of the jump processes, can be approximated asymptotically by Fokker-Planck equations when the volume of the system increases. We certainly do not want to deal with the details of this problem, since the literature is comprehensive. Many opinions about this question have been expressed in a discussion (published in Nicolis et ai, 1984). However, some comments have to be made. 

The methods start from an evolution equation describing the internal fluctuation. In the next step a fixed value of an external parameter, namely the infinitesimal transition probability, is substituted by a stochastic process. This procedure can be done in the master equation or in the equation for the generating function. The first case leads to an integrodifferential equation, while the second model leads to a stochastic partial differential equation. 

Arnold, L. Kotelenez, P. (1981). Linear stochastic evolution equation models for chemical reactions. In Stochastic nonlinear systems, eds L. Arnold R. Lefever, pp. 174-84. Springer Verlag, Berlin. 

Arnold, L., Curtain, R. F. Kotelenez, P. (1980). Nonlinear stochastic evolution equations in Hilbert space. Report No. 17 (June 1980), Forschungsschwerpunkt Dynamische Systeme, Bremen University. 

Dawson, D. A. (1972). Stochastic evolution equations. Mathematical Biosciences, 15, 287-316. 

Ichikawa, A. (1982). Stability and optimal control of stochastic evolution equations. In Distributed parameter control systems. Theory and application, ed. S. G. Tsafestas, pp. 147-77. Pergamon Press, Oxford. 

The basic idea of the MC approach lies in the discrete representation of the joint PDF by an ensemble of stochastic particles. Each particle carries an array of properties denoting position, velocity and scalar composition. During a fractional time stepping procedure [6] the particles are submitted to certain deterministic and stochastic processes changing each particle s set of properties in accordance with the different terms in the PDF evolution equation. Afterwards the statistical moments may be derived in the simplest case by averaging from the ensemble of particles. 

Instead of solving the evolution equation in terms of the orientation tensor, one can simulate the stochastic equation such as Eq. 5.7 for the orientation vector p without the need of closure approximations, using the numerical technique for the simulation of stochastic processes (Ottinger 1996) known as the Brownian dynamics simulation. Once trajectories for aU fibers are obtained, the orientation tensor can be calculated in terms of the ensemble average of the discrete form ... 

The vector of dynamical variables x(t) which characterizes the bistable system obeys a stochastic evolution equation of the form... 

Generalized density evolution equation Global reliability Nonlinear stochastic dynamics PDEM Stochastic harmonic function Stochastic response... 

Chen JB, Li J (2007) The extreme value distribution and dynamic reliability analysis of nonlinear structures with uncertain parameters. Stmct Saf 29 77-93 Chen JB, Li J (2009) A note on the principle of preservation of probability and probability density evolution equation. Probab Eng Mech 24(l) 51-59 Chen JB, Li J (2010) Stochastic seismic response analysis of structures exhibiting high nonlinearity. Comput Struct 88(7-8) 395 12... 

Li J, Chen JB (2008) The principle of preservation of probability and the generalized density evolution equation. Struct Saf 30 65-77 Li J, Chen JB (2009) Stochastic dynamics of structures. Wiley, Singapore... 

An approximate method for the response variability calculation of dynamical systems with uncertain stiffness and damping ratio can be found in Papadimitriou et al. (1995). This approach is based on complex mode analysis where the variability of each mode is analyzed separately and can efficiently treat a variety of probability distributions assumed for the system parameters. A probability density evolution method (PDEM) has also been developed for the dynamic response analysis of linear stochastic structures (Li and Chen 2004). In this method, a probability density evolution equation (PDEE) is derived according to the principle of preservation of probability. With the state equation expression, the PDEE is further reduced to a one-dimensional partial differential equation from which the instantaneous probability density function (PDF) of the response and its evolution are obtained. Finally, variability response functions have been recently proposed as an alternative to direct MCS for the accurate and efficient computation of the dynamic response of linear structural systems with rmcertain Young modulus (Papadopoulos and Kokkinos 2012). 

Instead of describing the stochastic dynamics via a Langevin equation, we use the Fokker-Planck equation, which is the evolution equation for the probability density in phase space. For an Al-particle system. 

The Fokker-Planck equation accurately captures the time evolution of stochastic processes whose probahihty distribution can be completely determined by its average and variance. For example, stochastic processes with Gaussian probahihty distributions, such as the random walk, can be completely described with a Fokker-Planck equation. 

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

The stochastic problem is to describe properly the time evolution of the Heisenberg operator d(t) averaged over all the realizations of collisional process in the interval (0,t). The averaging, performed in the impact theory, results in the phenomenological kinetic equation [170, 158]... 

Detailed modeling study of practical sprays has a fairly short history due to the complexity of the physical processes involved. As reviewed by O Rourke and Amsden, 3l() two primary approaches have been developed and applied to modeling of physical phenomena in sprays (a) spray equation approach and (b) stochastic particle approach. The first step toward modeling sprays was taken when a statistical formulation was proposed for spray analysis. 541 Even with this simplification, however, the mathematical problem was formidable and could be analyzed only when very restrictive assumptions were made. This is because the statistical formulation required the solution of the spray equation determining the evolution of the probability distribution function of droplet locations, sizes, velocities, and temperatures. The spray equation resembles the Boltzmann equation of gas dynamics[542] but has more independent variables and more complex terms on its right-hand side representing the effects of nucleations, collisions, and breakups of droplets. 

Third, Eq. (31) shows that A is nondistributive, and determines fluctuations. Since there is a flucmation, we can expect that the time evolution in Eq. (34) may be related to a stochastic process. Indeed, one can show that the time evolution (34) is identical to the time evolution generated by the set of Langevin equations for the stochastic operators aj(r), a (r) (see Ref. 14) ... 

More complicated case of standing waves emerges in the regime of chaotic oscillations. Here the equations for the correlation dynamics are able to describe auto-oscillations (for d — 3). However, a noise in concentrations changes stochastically the amplitude and period of the standing waves. It results finally in the correlation functions with non-monotonous behaviour. Despite the fact that the motion of both concentrations and of the correlation functions is aperiodic, the time evolution of the correlation functions reveals several distinctive distributions shown in Fig. 8.6. 

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