Stochastic models for Boltzmann-type equations

[Next]:

Research Group Stochastic Algorithms and Nonparametric

[Up]:

Projects

[Previous]:

Branching processes in random media

[Contents]

[Index]

Stochastic models for Boltzmann-type equations

Collaborator: W. Wagner

Cooperation with: K. Aoki (Kyoto University, Japan), H. Babovsky (Technische Universität Ilmenau), A. Garcia (San José State University, USA), M. Kraft (University of Cambridge, UK), C. Lécot (Université de Savoie, Chambéry, France), O. Muscato (University of Catania, Italy), S. Rjasanow (Universität des Saarlandes, Saarbrücken), K.K. Sabelfeld (WIAS: Research Group 6)

Supported by: DFG: ``Einfluss räumlicher Fluktuationen auf das Gelationsverhalten von Koagulationsprozessen'' (Influence of spatial fluctuations on the gelation behavior of coagulation processes)

Description:

Rarefied gas flows play an important role in applications like aerospace design (space shuttle reentry), vacuum engineering (material processing, pumps), or, more recently, nanotechnology. Mathematically such flows are described (in the simplest case of a monatomic gas) by the Boltzmann equation

$\displaystyle {\frac{{\partial}}{{\partial t}}}$ f (t, x, v)+(v, $\displaystyle \nabla_{x}^{}$ ) f (t, x, v) =
		$\displaystyle \int_{{{\cal{R}}^3}}^{}$ dw $\displaystyle \int_{{{\cal{S}}^2}}^{}$ de B(v, w, e) $\displaystyle \Big[$ f (t, x, v^) f (t, x, w^) - f (t, x, v) f (t, x, w) $\displaystyle \Big]$ ,	(1)

where v^* = v + e (e, w - v) , w^* = w + e (e, v - w), and $\cal {S}$ ² denotes the unit sphere in the Euclidean space $\cal {R}$ ³ . The solution f (t, x, v) represents the relative amount of gas molecules with velocity v at position x and time t . The quadratic nonlinearity in (1) corresponds to the pairwise interaction between gas particles, which consists in the change of velocities of two particles. The collision kernel B contains information about the assumed microscopic interaction potential.

A nonlinear equation of similar structure as equation (1) is Smoluchowski's coagulation equation

$\displaystyle {\frac{{\partial}}{{\partial t}}}$ c(t, x)

$\displaystyle {\frac{{1}}{{2}}}$ $\displaystyle \sum_{{y=1}}^{{x-1}}$ K(x - y, y) c(t, x - y) c(t, y) - $\displaystyle \sum_{{y=1}}^{{\infty}}$ K(x, y) c(t, x) c(t, y) ,

(2)

where t $\ge$ 0 and x = 1, 2,... . It describes the time evolution of the average concentration of particles of a given size in some spatially homogeneous physical system. The concentration of particles of size x increases as a result of coagulation of particles of sizes x - y and y . It decreases if particles of size x merge with any other particles. The intensity of the process is governed by the coagulation kernel K representing properties of the physical medium. The phenomenon of coagulation occurs in a wide range of applications, e.g., in physics (aggregation of colloidal particles, growth of gas bubbles), meteorology (merging of drops in atmospheric clouds, aerosol transport), chemistry (reacting polymers, soot formation), and astrophysics (formation of stars and planets).

The purpose of the project is to study the relationship between stochastic interacting particle systems and solutions of equations of type (1) or (2). On the one hand, results on the asymptotic behavior of the particle system (when the number of particles increases) provide insight into properties of the solution. On the other hand, appropriate stochastic particle systems are used for the numerical treatment of the macroscopic equation.

Significant progress in the development and justification of stochastic algorithms for the Boltzmann equation (1) has been achieved in recent years. Various aspects of this development are described in [1], which is based on an invited lecture presented at the 24th International Symposium on Rarefied Gas Dynamics (Bari, Italy, 07/2004). This paper concentrates on applied and numerical issues. Theoretical and convergence aspects are emphasized in [2], which is an extended version of a plenary talk given at the Fifth International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (Singapore, 11/2002). First steps towards another field of application were taken in [3], where a Boltzmann-type equation describing charge transport in semiconductors is considered. In particular, the time-step discretization error of several direct simulation Monte Carlo algorithms is studied.

The topic of studying stochastic models for Smoluchowski's coagulation equation (2) has attracted much interest in recent years. A challenging direction of research is the phenomenon of gelation, which occurs for sufficiently fast increasing coagulation kernels. At the level of the macroscopic equation, the gelation effect is represented by a loss of mass of the solution. An appropriate interpretation of this phenomenon in terms of stochastic particle systems is of both theoretical and practical interest. Some conjectures based on detailed numerical observations have been stated in [4]. Interesting new results contributing to this direction of research are obtained in [5]. First we establish explosion criteria for jump processes with an arbitrary locally compact separable metric state space. Then these results are applied to two stochastic coagulation-fragmentation models---the direct simulation model and the mass flow model. In the pure coagulation case, there is almost sure explosion in the mass flow model for arbitrary homogeneous coagulation kernels with exponent bigger than 1 . In the case of pure multiple fragmentation with a continuous size space, explosion occurs in both models provided the total fragmentation rate grows sufficiently fast at zero. However, an example shows that the explosion properties of both models are not equivalent. Finally, some numerical issues related to Smoluchowski's coagulation equation are studied in [6]. Convergence for a quasi-Monte Carlo scheme is proved and error estimates are obtained.

References:

W. WAGNER, Monte Carlo methods and numerical solutions, WIAS Preprint no. 954, 2004.
, Stochastic models and Monte Carlo algorithms for Boltzmann type equations, in: Monte Carlo and Quasi-Monte Carlo Methods 2002, H. Niederreiter, ed., Springer, New York, 2004, pp. 129-153.
O. MUSCATO, W. WAGNER, Time step truncation error in direct simulation Monte Carlo for semiconductors, WIAS Preprint no. 915, 2004.
W. WAGNER, Stochastic, analytic and numerical aspects of coagulation processes, Math. Comput. Simulation, 62 (2003), pp. 265-275.
, Explosion phenomena in stochastic coagulation-fragmentation models, Preprint no. NI04006-IGS, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, 2004.
C. LÉCOT, W. WAGNER, A quasi-Monte Carlo scheme for Smoluchowski's coagulation equation, Math. Comp., 73 (2004), pp. 1953-1966.