Uniform asymptotic expansions in transport theory with small mean free paths, and the diffusion approximation

Habetler, G. J.; Matkowsky, B. J.

doi:10.1063/1.522618

Cited by 119 publications

(97 citation statements)

References 7 publications

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“…In section 3 we introduce the parabolic scaling and formally derive the parabolic limit equation. Since the parabolic limit is the outer solution in singular perturbations terms, these higher approximations depend only on the initial values for the parabolic limit problem, which is also true for the Chapman-EnskogHilbert expansion of Boltzmann's equation, and which is known as Hilbert's paradox in that context [12,30,17]. We also derive several equivalent conditions on the turn angle distribution under which the diffusion matrix is a scalar multiple of the identity.…”

Section: ∂ ∂T P(x V T) + V · ∇P(x V T) = −λP(x V T) + λmentioning

confidence: 94%

“…This ansatz gives the "outer" solution in the sense of singular perturbations and is similar to what is used in the context of the Boltzmann equation, where it is called a Hilbert expansion (see, e.g., [12,30,17]). Comparing terms of equal order in ε, we obtain the following system of equations:…”

Section: The Shift Operator a = −V · ∇ The Shift Operator A = −(V · mentioning

confidence: 99%

“…From the mathematical point of view the transport equation (1.1) is similar to the Boltzmann equation, in which the right-hand side is an integral operator that describes collision of two particles, and is therefore quadratic in p [4]. The kernel of the integral operator is specified by the dynamics, and it is known that an appropriate scaling of space and time leads at least formally to a diffusion process [30,17]. As we show here, this also holds for velocityjump processes as described by (1.1) for one particle and for more general transport processes which also include acceleration terms and other forces (see also Hersh [22], Papanicolaou [39], or Patlak [40]).…”

Section: ∂ ∂T P(x V T) + V · ∇P(x V T) = −λP(x V T) + λmentioning

confidence: 99%

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The Diffusion Limit of Transport Equations Derived from Velocity-Jump Processes

Othmer

Hillen

2000

SIAM J. Appl. Math.

328

110

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Abstract. In this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a Poisson process. We show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations can be approximated by the solution of a diffusion equation obtained via a regular perturbation expansion. In general the resulting diffusion tensor is anisotropic, and we give necessary and sufficient conditions under which it is isotropic. We also give a method to construct approximations of arbitrary high order for large times. In a second paper (Part II) we use this approach to systematically derive the limiting equation under a variety of external biases imposed on the motion. Depending on the strength of the bias, it may lead to an anisotropic diffusion equation, to a drift term in the flux, or to both. Our analysis generalizes and simplifies previous derivations that lead to the classical Patlak-Keller-Segel-Alt model for chemotaxis.

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Section: ∂ ∂T P(x V T) + V · ∇P(x V T) = −λP(x V T) + λmentioning

confidence: 94%

Section: The Shift Operator a = −V · ∇ The Shift Operator A = −(V · mentioning

confidence: 99%

Section: ∂ ∂T P(x V T) + V · ∇P(x V T) = −λP(x V T) + λmentioning

confidence: 99%

See 1 more Smart Citation

The Diffusion Limit of Transport Equations Derived from Velocity-Jump Processes

Othmer

Hillen

2000

SIAM J. Appl. Math.

328

110

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“…The kernel of the integral operator is specified by the dynamics, and it is well known that an appropriate scaling of space and time leads at least formally from the Boltzmann equation to a diffusion process [25,15]. This also holds for transport equations and more general transport processes (see e.g.…”

mentioning

confidence: 99%

From Individual to Collective Behavior in Bacterial Chemotaxis

Erban¹,

Othmer²

2004

SIAM J. Appl. Math.

264

380

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Bacterial chemotaxis is widely studied from both the microscopic (cell) and macroscopic (population) points of view, and here we connect these different levels of description by deriving the classical macroscopic description for chemotaxis from a microscopic model of the behavior of individual cells. The analysis is based on the velocity jump process for describing the motion of individuals such as bacteria, wherein each individual carries an internal state that evolves according to a system of ordinary differential equations forced by a time-and/or space-dependent external signal. In the problem treated here the turning rate of individuals is a functional of the internal state, which in turn depends on the external signal. Using moment closure techniques in one space dimension, we derive and analyze a macroscopic system of hyperbolic differential equations describing this velocity jump process. Using a hyperbolic scaling of space and time we obtain a single second-order hyperbolic equation for the populations density, and using a parabolic scaling we obtain the classical chemotaxis equation, wherein the chemotactic sensitivity is now a known function of parameters of the internal dynamics. Numerical simulations show that the solutions of the macroscopic equations agree very well with the results of Monte Carlo simulations of individual movement.

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“…The asymptotic diffusion analysis of the transport equation in the continuous form yields that the leading-order solution meets the Dirichlet boundary condition [20,21,32] …”

Section: Asymptotic Diffusion Analysismentioning

confidence: 99%