Stochastic difference equations and a stochastic partial differential equation for neutron transport

Stochastic difference equations and a stochastic partial differential equation (SPDE) are simultaneously derived for the time-dependent neutron angular density in a general three-dimensional medium where the neutron angular density is a function of position, direction, energy and time. Special cases of the equations are given, such as transport in one-dimensional plane geometry with isotropic scattering and transport in a homogeneous medium. The stochastic equations are derived from basic principles, i.e. from the changes that occur in a small time interval. Stochastic difference equations of the neutron angular density are constructed, taking into account the inherent randomness in scatters, absorptions and source neutrons. As the time interval decreases, the stochastic difference equations lead to a system of Itô stochastic differential equations. As the energy, direction, and position intervals decrease, an SPDE is derived for the neutron angular density. Comparisons between numerical solutions of the stochastic difference equations and independently formulated Monte Carlo calculations support the accuracy of the derivations.

[1]  E. Allen.  Derivation of stochastic partial differential equations for size- and age-structured populations , 2009, Journal of biological dynamics.

[2]  Edward J. Allen,et al.  Stochastic point kinetics equations in nuclear reactor dynamics. , 2005 .

[3]  Imre Pázsit,et al.  Theory of Neutron Noise in a Temporally Fluctuating Multiplying Medium , 2007 .

[4]  D. Gillespie.  The chemical Langevin equation , 2000 .

[5]  E. Richard Cohen,et al.  Neutron Transport Theory , 1959 .

[6]  Giuseppe Da Prato,et al.  Stochastic Partial Differential Equations and Applications - VII , 2005 .

[7]  Paul F. Zweifel,et al.  Neutron Transport Theory , 1967 .

[8]  E. Larsen,et al.  Neutron Transport in Spatially Random Media: Eigenvalue Problems , 2001 .

[9]  Linda J. S. Allen,et al.  Construction of Equivalent Stochastic Differential Equation Models , 2008 .

[10]  E. M. Cabaña,et al.  The vibrating string forced by white noise , 1970 .

[11]  G. I. Bell.  Probability distribution of neutrons and precursors in a multiplying assembly , 1963 .

[12]  E. Allen.  Derivation of Stochastic Partial Differential Equations , 2008 .

[13]  E. Lewis,et al.  Computational Methods of Neutron Transport , 1993 .

[14]  Stig Larsson,et al.  Introduction to stochastic partial differential equations , 2008 .

[15]  Bernt Øksendal,et al.  Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach , 1996 .

[16]  Simon M. J. Lyons.  Introduction to stochastic differential equations , 2011 .

[17]  M. Williams.  The Effect of Random Geometry on the Criticality of a Multiplying System IV: Transport Theory , 2003 .

[18]  William Ranson,et al.  Pipe-anchor discontinuity analysis utilizing power series solutions, Bessel functions, and Fourier series , 2003 .

[19]  M. A. Schultz.  Dynamics of Nuclear Reactors , 1971 .

[20]  A. Yamamoto,et al.  Some properties of zero power neutron noise in a time-varying medium with delayed neutrons , 2008 .

[21]  W. Ebeling.  Stochastic Processes in Physics and Chemistry , 1995 .

[22]  E. Allen.  Modeling with Itô Stochastic Differential Equations , 2007 .

[23]  Zhimin Zhang,et al.  Finite element and difference approximation of some linear stochastic partial differential equations , 1998 .

[24]  L. Allen.  An introduction to stochastic processes with applications to biology , 2003 .

[25]  Edward J. Allen,et al.  Stochastic neutron transport equations for rod and plane geometries , 2000 .