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Boltzmann equation and hydrodynamic fluctuations

Matteo Colangeli, Martin Kröger^*, and Hans Christian Öttinger

Polymer Physics, Department of Materials, ETH Zürich, CH-8093 Zürich, Switzerland

^*mk@mat.ethz.ch; http://www.mat.ethz.ch

Phys. Rev. E 80, 051202 – Published 10 November, 2009

DOI: https://doi.org/10.1103/PhysRevE.80.051202

Abstract

We apply the method of invariant manifolds to derive equations of generalized hydrodynamics from the linearized Boltzmann equation and determine exact transport coefficients, obeying Green-Kubo formulas. Numerical calculations are performed in the special case of Maxwell molecules. We investigate, through the comparison with experimental data and former approaches, the spectrum of density fluctuations and address the regime of finite Knudsen numbers and finite frequencies hydrodynamics.

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References (31)

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Adopting the notation in [27,28] the distribution function is written as a sum over $n$ -fold contracted products of $n th$ rank tensors, $f (c) = f_{0} (c) \sum_{k, n = 0}^{\infty} ⟨ ϕ_{k}^{n} ⟩ ⊙^{n} ϕ_{k}^{n} (c)$ with $⟨ ϕ_{k}^{n} ⟩ = \int f (c) ϕ_{k}^{n} d^{3} c$ and base functions $ϕ_{k}^{n} (c) = l_{k}^{n} L_{k}^{n + 1 / 2} (c^{2}) \overset{⎴}{\otimes^{n} c}$ , where $L_{k}^{n}$ are the associated Laguerre ( $k th$ order) polynomials [29], $\otimes^{n} c$ denotesthe $n$ -fold tensor product, and $\overset{⎴}{a}$ denotes the irreducible part of a tensor $a$ . For the explicit construction of $n th$ rank irreducible tensors $\overset{⎴}{\otimes^{n} c}$ see page 160 of [28]. The normalization coefficients evaluate as $l_{k}^{n} = {(\sqrt{π} k! (1 + 2 n)!! / [2 (k + n + 1 / 2)! n!])}^{1 / 2}$ . The base function $ϕ_{k}^{n} (c)$ is thus a $(2 k + n) th$ order polynomial in $c$ . The lowest-order base functions read $ϕ_{0}^{0} = 1$ , $ϕ_{0}^{1} = \sqrt{2} c$ , $ϕ_{1}^{0} = \sqrt{2 / 3} (3 / 2 - c^{2})$ , $ϕ_{1}^{1} = (2 / \sqrt{5}) (5 / 2 - c^{2}) c$ , and $ϕ_{0}^{2} = \sqrt{2} \overset{⎴}{c c}$ . Density, velocity, temperature, heat flux, and stress tensor are related to the moments as follows: $\tilde{n} = ⟨ ϕ_{0}^{0} ⟩$ , $\tilde{u} = ⟨ ϕ_{0}^{1} ⟩ / \sqrt{2}$ , $\tilde{T} = ⟨ ϕ_{1}^{0} ⟩ \sqrt{3 / 2}$ , $q = ⟨ ϕ_{1}^{1} ⟩$ , and $σ = ⟨ ϕ_{0}^{2} ⟩ / \sqrt{2}$ . The distribution function is then split into (orthogonal) parts as $f (c) = f^{LM} (c) + δ f^{Grad} (c) + δ f^{rest} (c)$ with $f^{LM} (c) \equiv f_{0} (c) (⟨ ϕ_{0}^{0} ⟩ ϕ_{0}^{0} + ⟨ ϕ_{0}^{1} ⟩ ϕ_{0}^{1} + ⟨ ϕ_{1}^{0} ⟩ ϕ_{1}^{0})$ and $δ f^{Grad} (c) \equiv f_{0} (c) (⟨ ϕ_{1}^{1} ⟩ ϕ_{1}^{1} + ⟨ ϕ_{0}^{2} ⟩ ϕ_{0}^{2})$ , while the sum in $δ f^{rest} (c) = \sum_{k, n} ⟨ ϕ_{k}^{n} ⟩ ⊙^{n} ϕ_{k}^{n} (c)$ extends over the remaining $(k, n)$ –pairs. Density, velocity, and temperature are therefore determined by $f^{LM}$ alone, and $δ f$ automatically obeys constrains such as orthogonality requirement $\int δ f (c) ϕ_{1}^{0} d^{3} c = 0$ and also $\int δ f (c) ξ (c) d^{3} c = 0$ , as mentioned in the text part. These conditions become redundant ones calculations are performed using the particular basis $ϕ_{k}^{n}$ . For Maxwell molecules, the dependence on the polar angle $ϕ$ can be included by replacing $P_{l} (z)$ by $e^{i m ϕ} P_{l}^{m} (z)$ involving the associated Legendre polynomials [29], and the eigenvalues are independent of $m$ . Then, these base function reduce to the eigenfunctions $Ψ_{r, l} (c, z)$ [Eq. (B6)] of the Maxwell gas.
The integrals listed in Table II obey the following decoupling rules, $\int (c_{∥}^{2} - \frac{1}{3} c^{2}) δ X_{n} d^{3} c \propto 1 - δ_{n, 4},$ $\int c_{∥} c_{⊥} δ X_{n} d^{3} c \propto δ_{n, 4},$ $\int c_{∥} (c^{2} - \frac{5}{2}) δ X_{n} d^{3} c \propto 1 - δ_{n, 4},$ $\int c_{⊥} (c^{2} - \frac{5}{2}) δ X_{n} d^{3} c \propto δ_{n, 4} .$

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Boltzmann equation and hydrodynamic fluctuations

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