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Kinetic solutions of the Boltzmann-Peierls equation and its moment systems

Collaborator: W. Dreyer , M. Herrmann  

Cooperation with: M. Kunik (Otto-von-Guericke-Universität Magdeburg), M. Junk (Universität Kaiserslautern)

Supported by: DFG: Priority Program ``Analysis und Numerik von Erhaltungsgleichungen'' (ANumE - Analysis and numerics for conservation laws)

Description:

At low temperatures the evolution of heat in crystalline solids is carried by phonons. In particular, the classical Fourier theory of heat fails to describe heat conduction at low temperatures, where the evolution of a phonon gas is governed by the Boltzmann-Peierls equation which is a kinetic equation for the phase density of phonons. For more details we refer to [4] and [5]. In [4] we have shown that the full Boltzmann-Peierls equation can be reduced to the simplified kinetic equation  

$$\frac{\partial \,\varphi }{\partial {\,t}}(t,\,\mathbf{x},\,... ...\,\mathbf{n})+(\Psi _{N}\varphi )(t,\,\mathbf{x},\,\mathbf{n}).$$ (1)

In (1), $\varphi$ is a reduced phase density depending on time t, space $\mathbf{x}=(x_{1},\,x_{2})$, and a normal vector $\mathbf{n}=(n_{1},\,n_{2})\,\in {S}^{1}$; and the positive constant c_D is the Debye speed. For simplicity we consider (1) in two space dimensions only. In addition to the reduced equation (1) we introduce a reduced definition of the (entropy density h / entropy flux $\Phi _{i}$ ) pair according to  

$$h(\varphi ):=\mu \int\limits_{S^{1}}{\varphi (\mathbf{n})^{... ...{S^{1}}{n_{i} \varphi (\mathbf{n})^{\frac{2}{3}}\,d\mathbf{n}},$$ (2)

where $\mu$ is a given constant. We have proved that these definitions and the kinetic equation (1) imply the entropy inequality  

$$\frac{\partial }{\partial {\,t}}h(\varphi )(t,\,\mathbf{x})+... ...\partial {\,x_{i}}}{\Phi }_{i}(\varphi )(t,\,\mathbf{x})\geq 0.$$ (3)

As well as the phase density, also the reduced phase density $\varphi$ can be used to define moments. The following moments have an immediate physical interpretation

$$\begin{eqnarray} e(\varphi )=\hbar {c_{D}}\int\limits_{S^{1}}\varphi (\mathbf{n}... ...}\int\limits_{S^{1}}n_{i}n_{j}\varphi (\mathbf{n})\,d{\mathbf{n}}.\end{eqnarray}$$

The fields e, p_i, Q_i, and N_ij denote the energy density, the momentum density, the heat flux, and the momentum flux, respectively. From a physical point of view, the moments of $\varphi$are more important than $\varphi$ itself. On the right-hand side of (1) there are two collision operators $\Psi _{R}$ and $\Psi _{N}$ which are simplified models of two different kinds of interaction processes, which are called R-processes and N-processes. The N-processes describe phonon-phonon interactions and conserve the energy as well as the momentum, while the R-processes indicate interactions of phonons with lattice impurities. The latter ones only conserve the energy. The Callaway ansatz for the collision operators reads

$$\Psi _{\alpha }\varphi =\frac{1}{\tau _{\alpha }}(\Theta _{\... ...\varphi -\varphi ),\quad \alpha \,\in \left\{ {R,\,N}\right\} .$$ (4)

Here $\tau _{R}$ and $\tau _{N}$ are two relaxation times, and $\Theta _{R}$ and $\Theta _{N}$ describe the phase density in the limiting cases $\tau _{R}\rightarrow {0}$ and $\tau _{N}\rightarrow {0}$,respectively. In particular, $\Theta _{R}\varphi$ and $\Theta _{N}\varphi$ maximize the entropy according to

  $$\begin{eqnarray} h(\Theta _{R}\varphi ) &=&\max\limits_{\varphi ^{\prime }}\Big\... ...)=e(\varphi ),\;\;p_{i}(\varphi ^{\prime })=p_{i}(\varphi )\Big\}.\end{eqnarray}$$

The reduced Boltzmann-Peierls equation (1) induces an infinite number of balance equations. For any vector $\vec{m}=\vec{m}(\mathbf{n})$ of moment weights, there is a corresponding vector of densities $\vec{u}$, fluxes $\vec{F}_{1}$,$\vec{F}_{2}$, and productions P, according to

$$\begin{eqnarray} \vec{u}(\varphi ) &=&\int\limits_{S^{1}}\vec{m}(\mathbf{n})\var... ...)(\Psi _{R}\varphi +\Psi _{N}\varphi )(\mathbf{n})\,d{\mathbf{n}}.\end{eqnarray}$$

With these definitions, the kinetic equation implies  

$$\frac{\partial }{\partial {\,t}}\vec{u}(\varphi )(t,\,\mathb... ...i}(\varphi )(t,\,\mathbf{x})=\vec{P}(\varphi )(t,\,\mathbf{x}).$$ (5)

Physicists and engineers are not interested in the phase density itself, but in balance equations for moments as (12). It is a common strategy in Extended Thermodynamics (cf. [4, 1, 3, 5]) to consider in (12) the densities $\vec{u}(\varphi )$ as variables. However, since the fluxes and the productions are in general not functions of $\vec{u}(\varphi )$ there arises the so-called closure problem. An often used and powerful closure principle is the Maximum Entropy Principle. It has the advantages that

1.

the resulting moment systems are hyperbolic, after a suitable transformation, they are even symmetric hyperbolic;

2.

there exists a corresponding entropy inequality.

The Maximum Entropy Principle (MEP) can be formulated as follows. We start with a fixed vector of densities $\vec{u}$ with N components. In the first step, we determine the corresponding MEP projector $\Theta _{M}$so that, for any given $\varphi$, there holds

$$h(\Theta _{M}\varphi )=\max\limits_{\varphi ^{\prime }}\Big\... ...rphi )\,:\,\vec{u}(\varphi ^{\prime })=\vec{u}(\varphi )\Big\}.$$

In the second step we use $\Theta _{M}$ in order to solve the closure problem of the system (12). When we replace formally the phase density $\varphi$ by $\Theta _{M}\varphi$, we obtain the resulting MEP system  

$$\frac{\partial }{\partial {\,t}}\vec{u}(\Theta _{M}\varphi )... ...)(t,\,\mathbf{x})=\vec{P}(\Theta _{M}\varphi )(t,\,\mathbf{x}),$$ (6)

which is indeed a closed system for the variables $\vec{u}$.

In the year 2001 we have studied the following problems:

1.

Derivation of the reduced equation and the corresponding reduced entropy inequality
We have proved that the reduced equation contains all important information on the solutions of the original Boltzmann-Peierls equation. Furthermore, we have derived the reduced entropy (2) from the original Bose entropy so that

(a)

there holds the entropy inequality (3),

(b)

the MEP applied to the original Boltzmann-Peierls equation and to the reduced equation (1) lead to identical moment systems.

2.

Existence of MEP-operators
The existence of MEP operators is a crucial problem. Junk has shown in [3] that in the most prominent application of the Maximum Entropy Principle, which is the Boltzmann equation for monatomic gases, MEP operators fail to exist within a reasonable domain of definition. In [1] we have proved that in the case of the Boltzmann-Peierls equation, the Maximum Entropy Principle will lead to well-defined MEP operators $\Theta _{M}$, if it is applied to the reduced entropy.

3.

Kinetic approximations and kinetic schemes
Corresponding to the MEP system (13) there exists the following modified kinetic equation

$$\frac{\partial \,\varphi }{\partial {\,t}}+c_{D}n_{i}\frac{\... ..._{N}\varphi +\frac{1}{\tau _{M}}(\Theta _{M}\varphi -\varphi ),$$

where $\Theta _{M}$ is the MEP operator corresponding to $\vec{u}$(cf. [1]) and $\tau _{M}$ is an artificial relaxation time. The moment system (13) formally results in the limit $\tau _{M}\rightarrow \infty$. This approach illustrates the close connection between the kinetic equation and the MEP system.
Furthermore there exists a class of consistent kinetic schemes containing schemes for the kinetic equation as well as for the moment systems. For the details we refer to [1].

4.

Numerical simulations
A numerical example shall illustrate that the MEP systems induce a sequence of approximations of the kinetic equation. To this end we study the evolution of an initial energy pulse as shown in Fig. 1.

 

Fig. 1: Initial energy pulse

  For simplicity we set $\tau _{R}=0$, and we assume that all quantities do not depend on the x₂ direction. Furthermore we set $\tau _{N}=0.7$, c_D=0.5. The initial data $\phi _{0}$ are determined by an equilibrium assumption, i.e. $\Theta _{R}\phi _{0}=\phi _{0}$. The evolution of the energy pulse according to the kinetic equation is depicted in Fig. 2. The figures show the corresponding solutions of various MEP systems.

 

Fig. 2: Solution of a MEP system with 81 moments

 

References:

1.   W. DREYER, M. HERRMANN, M. KUNIK, Kinetic schemes and initial boundary value problems for the Euler System,
   WIAS Preprint no. 607, 2000, Transport Theory Statist. Phys., in print.
2.   , Kinetic solutions of the Boltzmann-Peierls equation and its moment systems,
   WIAS Preprint no. 709, 2001.
3.   W. DREYER, M. JUNK, M. KUNIK, On the approximation of kinetic equations by moment systems, Nonlinearity, 14 (2001), pp. 881-906.
4.   W. DREYER, M. KUNIK, Kinetische Behandlung von ausgewählten hyperbolischen Anfangs- und Randwertproblemen, DFG Project DR 401/2-2.
5.   W. DREYER, H. STRUCHTRUP, Heat pulse experiments revisited, Cont. Mech. Thermodyn., 5 (1993), pp. 3-50.

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