The BGK Equation as the Limit of an N -Particle System
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The BGK Equation as the Limit of an N-Particle System Dawan Mustafa1 · Bernt Wennberg2,3 Received: 3 March 2020 / Accepted: 15 June 2020 © The Author(s) 2020
Abstract The spatially homogeneous BGK equation is obtained as the limit of a model of a many particle system, similar to Mark Kac’s charicature of the spatially homogeneous Boltzmann equation. Keywords BGK equation · Particle system · Kinetic theory Mathematics Subject Classification 35Q70 · 35Q20 · 60K35 · 82C22
1 Introduction The BGK equation is named after Bhatnagar, Gross, and Krook, who first presented it in an influential paper published in 1954 [3]. In its original form it is ∂f eE ∂ f n n2 +v·∇ f − = − f + . ∂t m ∂v σ σ
(1)
Here f = f (x, v, t) gives the number density of particles in phase-space (x, v) ∈ R3 × R3 . The constant σ > 0 controls collision rate of particles, and = q,T is the Maxwellian distribution (x, v, t) = q,T =
m (2πkT (x, t))3/2
m exp − (v − q(x, t))2 2kT (x, t)
,
(2)
Communicated by Eric A. Carlen.
B
Bernt Wennberg [email protected] Dawan Mustafa [email protected]
1
University of Borås, 50190 Borås, Sweden
2
Department of Mathematical Sciences, Chalmers University of Technology, 41296 Göteborg, Sweden
3
Department of Mathematical Sciences, University of Gothenburg, 41296 Göteborg, Sweden
123
D. Mustafa, B. Wennberg
where n(x, t), q(x, t), and T (x, t) represent the local number density, mean velocity and temperature respectively: n(x, t) =
f (v, x, t) dv, 1 q(x, t) = v f (v, x, t) dv, n(x, t) 3kT (x, v) 1 = (v − q(x, t))2 f (v, x, t)dv. m n(x, t)
(3)
The same kind of equation was formulated independently by Welander [25]. In [3], one considers charged particles, and E is the electric field computed from the particle density. It is a model of the kinetic Boltzmann equation with the purpose of providing a numerically tractable model, while retaining the most important aspects of the original Boltzmann equation: conservation of mass, momentum and energy, convergence to a unique equilibrium state, monotonicity of entropy, etc. And while easier from a computational point of view, it is considerably more difficult to analyse mathematically, and most theoretical results concerning existence and uniqueness of solutions to the BGK eqution actually hold for a modified version where the right hand side is replaced by −
1 n f + , σ σ
(4)
i.e. where the collision frequency is constant [20,21]. There are also results concerning solutions close to a global equilibrium, which hold also for density and temperature dependent collision frequencies [27,28]. There is a rather large litterature concerning various aspects of the BGK-equation dealing, for example, with methods for numerical treatment of rarefied gases (some recent examples are [2,13,26]), their fluid dynamical limits (see for example [9,10,22]), or models accounting for polyatomic gases or mixtures of different gases (for example in [1,4,12]), to give a few examples. A paper attempting to find a well-motivated approximation of the collisio
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