M. A. Rodriguez

Density fluctuations in simple stochastic reactor models

Abstract We present a method to study density fluctuations in a nuclear reactor simultaneously taking into account intrinsic fluctuations usually modelled by a master equation and parametric noise usually modelled by stochastic kinetic equations. The mean density and density-fluctuations variance are calculated for a simple point-reactor model in the presence of Gaussian white-noise fluctuations in the fission, capture and source event rates. They are also calculated in the case of dichotomous noise fluctuations in the source event rate. We find ‘crossed fluctuation’ contributions to the density fluctuations that only appear when considering simultaneously intrinsic and parametric noise. These contributions allow us to distinguish between reactivity fluctuations due to fission-rate and capture-rate fluctuations.

Master Equation Description of External Poisson White Noise in Finite Systems

We consider finite systems with random control parameters. A theory for a unified description of internal fluctuations and external noise is presented. Internal fluctuations are modeled by a one-step Markovian master equation. External noise is introduced by random parameters in the master equation. It is modeled by a Poisson white noise. The unified description of fluctuations features a Markovian master equation with nonvanishing transition probabilities for all steps in the state space. Alternative formulations are given in terms of the generating function, Poisson representation and the equations for the factorial moments. An expansion around the thermodynamic limit is considered. The theory permits the calculation of finite-size effects. It predicts the existence of a coupling of the two types of fluctuations leading to “crossed-fluctuation” contributions. Two examples are considered: (i) a Poisson counting process with fluctuating parameter, (ii) a creation and annihilation process with source terms and fluctuations in each of the creation, annihilation, and source parameters. In the second example a complete analysis is given for the stationary distribution and associated moments for a finite system and also in the thermodynamic limit. The different role of the fluctuations of the three parameters is discussed. Explicit “crossed-fluctuations” contributions are found. The effect of the system size on the type of transitions induced by external noise in the thermodynamic limit is discussed.

Random Walk in Dynamically Disordered Chains: Poisson White Noise Disorder

Exact solutions are given for a variety of models of random walks in a chain with time-dependent disorder. Dynamic disorder is modeled by white Poisson noise. Models with site-independent (global) and site-dependent (local) disorder are considered. Results are described in terms of an affective random walk in a nondisordered medium. In the cases of global disorder the effective random walk contains multistep transitions, so that the continuous limit is not a diffusion process. In the cases of local disorder the effective process is equivalent to usual random walk in the absence of disorder but with slower diffusion. Difficulties associated with the continuous-limit representation of random walk in a disordered chain are discussed. In particular, we consider explicit cases in which taking the continuous limit and averaging over disorder sources do not commute.

Theory of fluctuations and parametric noise in a point nuclear reactor model

Abstract We present a joint description of internal fluctuations and parametric noise in a point nuclear reactor model in which delayed neutrons and a detector are considered. We obtain kinetic equations for the first moments and define effective kinetic parameters which take into account the effect of parametric Gaussian white noise. We comment on the validity of Langevin approximations for this problem. We propose a general method to deal with weak but otherwise arbitrary non-white parametric noise. Exact kinetic equations are derived for Gaussian non-white noise.

Positivity requirements on fluctuating parameters

Abstract The problem of the requirement of positivity of random parameters in the phenomenological modeling of external noise is addressed. We establish the connections of positivity requirements with stability of the system and with changes of the probability distribution, through the explicit solution of models incorporating noise with bounded realizations from below (Poisson white noise). In particular, we analyze gain and loss fluctuations in dye laser models. Earlier studies of this system which violate physical conditions are justified as a valid approximation close to the instability point.

Correlation functions of non-linear nuclear reactor models

Abstract The steady-state correlation function of a non-linear point nuclear reactor model with instantaneous temperature feedback, reactivity fluctuations and neglection of delayed neutrons is calculated by means of a continued fraction expansion method. Near criticality, where other methods fail, this method gives good results for early times. A first-order truncation gives a spectral density with two corner frequencies. Results are given for a non-vanishing independent neutron source. In this case no exact results are known. For a vanishing source our results are compared with exact ones and with those of a linear analysis and a system size expansion. The necessity of a proper treatment of non-Markovian effects for non-white reactivity fluctuations is pointed out. The existence, in the spectral density, of a corner frequency associated with the correlation time of the noise is shown.

Unified description of internal fluctuations and external white poisson noise

Abstract A theory for a unified description of internal fluctuations and external noise is presented. External noise is introduced by random parameters in a master equation by using a Poisson white noise. The theory predicts the existence of a coupling of the two types of fluctuations leading to “crossed-fluctuation” contributions. The effect of the system size in a type of transition induced by external noise in the thermodynamic limit is discussed by considering a birth-death process with source term and fluctuations in the annihilation parameter.

Reactant Segregation: The Effect of Strong Space Disorder in Diffusion-Limited Bimolecular Reactions

Quite recently, the non-classical kinetics of the diffusion-limited reaction processes for two species annihilation A+B→C (C: inert specie), has attracted considerable attention [1–7]. Such a process has been studied as a model of several different physical and chemical systems (ionic, electron-hole and defect recombination, matter-antimatter annihilation, etc.). An aspect pointed out almost a decade ago by Zeldovich and coworkers was the possibility of macroscopic segregation [8]. Several recent reviews have analyzed this and related aspects of such systems [9–12]. Among other problems, authors have been concerned with the influence of dimensionality, initial conditions, sources, conservation laws, etc. [1, 5, 6, 12]. Also the form of the kinetic or rate equations have received some attention [4, 7, 11 12 ,13].