The definition of the linearized collision operator and the results of the above Chapman Enskog procedure above as well as the BGK model suggest a decomposition of f of the form (3) rather than (2). In this case, the central element e contains all macroscopic moments, but no contribution to the closure moments. In view of the balance equations described above with source term ${\left(v_x{\partial }_{x}e\right)}_{\perp }$, we propose an alternative decomposition. For simplicity we restrict in the following to spatially one-dimensional problems (with space variable x) and with bulk velocities $\overline{v}={({\overline{v}}_x,0)}^T$ resp. $\overline{v}={({\overline{v}}_x,0,0)}^T$.

with the same macroscopic moments $[\overline{\rho },{\overline{v}}_x,\overline{T}]$ as $\overline{e}$. It is a two-dimensional manifold in $\mathcal{EM}$ and takes the form with $({\gamma }^{\left(1\right)},{\gamma }^{\left(2\right)})$ and $({\widehat{v}}_x,\widehat{T})$ connected to $({\overline{v}}_x,\overline{T})$ via Elements of $\mathcal{T}\left[\overline{e}\right]$ are addressed as The union of tangent spaces is denoted as In the following, the parameter ${\gamma }^{\left(2\right)}$ is not relevant in the case of the full Boltzmann collision operator. Thus we set ${\gamma }^{\left(2\right)}=0$, ${\gamma }^{\left(1\right)}:=\gamma$ and define as the trace of $\overline{e}$ in $\mathcal{TE}$ the mapping $\mathbf{f}\left(\gamma \right)$ has the same macroscopic moments as $\overline{e}$, however it contributes to the closure moments as follows.

Proof: These formulas follow from and Given $\mathbf{f}\left(\gamma \right)$, we denote as $e\left(\gamma \right)$, ${\widehat{\pi }}_x\left(\gamma \right)$ and ${\widehat{\sigma }}_x\left(\gamma \right)$ the Maxwellian and the closure moments corresponding to the parameters ${\widehat{v}}_x=-\gamma$ and $\widehat{T}=\overline{T}+{\gamma }^2/d$. The normed vectors ${\mathbf{n}}_{\pi }$ and ${\mathbf{n}}_{\sigma }$ are for given $\gamma$ to be interpreted according to the formulas Furthermore we define as the local tangent plane the affine linear space Given $\mathbf{z}\in span({\mathbf{n}}_{\pi },{\mathbf{n}}_{\sigma })$, the angle ${\mathrm{\Phi }}_{\mathbf{z}}$ defined by describes the position of $\mathbf{f}\left(\gamma \right)+\mathbf{z}$ in the local tangent plane.

We collect some results concerning the local structure of $\mathbf{f}\left(\gamma \right)$ which can be proved by straightforward calculations and by Taylor expansion.

(b) The leading order approximation of $\mathbf{f}\left(\gamma \right)$ for $\left|\gamma \right|\ll 1$ in the local tangent plane $(\gamma =0)$ is (c) For $\left|\gamma \right|\ll 1$, Thus at $\gamma =0$ the curve $\mathbf{f}\left(\gamma \right)$ is not differentiable and thus not a regular parametrization.

and (b) is obtained by integrating (33) and using $\mathbf{f}\left(0\right)=\overline{e}$ and ${\partial }_{\gamma }\mathbf{f}\left(0\right)=0.\square$

(5.10) Trace of nonlinear operator: (a) With $a_{\pi ,\pi }^J=-\langle J[{\widehat{e}}^{\left(1\right)},{\widehat{e}}^{\left(1\right)}],{\mathbf{n}}_{\pi }\rangle$, (b) Under the approximations of (5.9)(b),(c) for small $\gamma$, and the solution $\mathbf{f}\left(\gamma \right(t\left)\right)$ of the homogeneous trace Boltzmann equation is in lowest order given by Proof: (a) From the definition of $\mathbf{f}\left(\gamma \right)$ (30) and the fact that ${\widehat{e}}^{\left(1\right)}\in ker\left(L_{\widehat{e}}\right)$ follows In a coordinate system centered around $\widehat{v}$, $J[{\widehat{e}}^{\left(1\right)},{\widehat{e}}^{\left(1\right)}]$ is an even function with respect to $\widehat{v}$. Thus (b) follows from inserting the approximations (5.9)(b),(c).$\square$

is centered around $\widehat{e}$ (with the asymptotics $\overline{e}$ “unvisible”), while the Maxwellian $\overline{e}$ appears in the definition of the relaxation operators and thus marks it as their asymptotic end point. For the same reason we find closed formulas for the solutions of the homogenous equations in the relaxation case, while we only can formulate the solution of the homogeneous nonlinear equation in differential form.

Consider the initial value problem with C one of the collision operators discussed above. Depending on $\varphi$, there exists a unique Maxwellian $\overline{e}$ with the same macroscopic moments as $f_0$, which is the asymptotic state of the solution for $t\nearrow \infty$. In the relaxation case we have exponential decay to $\overline{e}$, (33), (34). The dynamics of the nonlinear solution is more detailed, depending on the form of $\varphi$.

Then with For the short time behaviour we may neglect the $\mathcal{O}\left({ϵ}^2\right)$-terms. Thus the initialization phase (initial layer) is dominated by the exponential relaxation of ${\varphi }_{\perp }$ to the “wrong” equilibrium $\widehat{e}$. What follows is a dynamics in $\mathcal{TE}$ which runs on a slower time scale and in which the initial function $\widehat{e}(1+ϵm)$ approaches its new equilibrium $\overline{e}$. The difference between fast exponential decay for the linearized model and of the slower approach to equilibrium for the nonlinear system can be easily verified e.g. running simulations with Discrete Velocity Models (DVM). Figure 1 a, b display the relaxation rates for randomly perturbed initial conditions in the cases 1, $\varphi \in {\mathcal{M}}^{\perp }$, and case 2, $\varphi \in \widehat{e}·\mathcal{M}$. Case 1 exhibits exponential decay, with rates within a narrow band. This situation can be reasonably well imitated with a BGK model (straight line).

Case 2 is more complex and we may recognize three phases. Phase I represents the relaxation of the perturbation $\varphi$ to $(1+ϵ\mathbf{m})\widehat{e}$, phase II (“convex phase”) the transition from $(1+ϵ\mathbf{m})\widehat{e}$ to $\overline{e}$, and phase III the exponential relaxation to $\overline{e}$. In the special case $\widehat{e}·m={\widehat{e}}^{\left(1\right)}$, phase II marks the transition of the trace element $\mathbf{f}\left(ϵ\right)$ to $\mathbf{f}\left(0\right)=\overline{e}$. The slope of phase III is given by the parameter $\lambda$ of (73). As a special feature we notice that the life time of the moment perturbation $ϵ\mathbf{m}\widehat{e}$ is much larger than for a model with relaxation rates based on the mean free path. The BGK model is not capable to cover this situation and we have to decide whether to match to the initial Maxwellian or the final Maxwellian, depending on whether we want to simulate the short-time or the long-time behaviour.

In view of the equations to be solved for the closure relations let’s have a look on solutions of the equation with source, and in particular their asymptotic states $f^{(\infty )}$ for $t\nearrow \infty$ (balanced states). For the relaxation systems we easily find Both solutions satisfy the orthogonality relation This is no surprise, since the source is even with respect to $v_x$, and L maps even into even functions, while ${\widehat{\sigma }}_x$ is odd. So it is remarkable to observe that orthogonality is not given for the nonlinear solution, as can be seen in Figure 2. (This observation was made in [8].) To understand this phenomenon, it helps to look at trace solutions. We easily see that $\gamma =0$ is a balanced solution to the trace version of (36). However, it is not stable in the sense of dynamical systems, and there is another solution. Necessary and sufficient for balancing solutions is the orthogonality relation resp. which in leading order leads to With $\parallel {\partial }_{\gamma }\mathbf{f}\parallel =\mathcal{O}\left(\gamma \right)$ we finally end up with

(5.11) Balanced state (Relaxation with source): Under the smallness assumptions for $\gamma$ in (5.9) and with an appropriate constant $C_{\sigma }>0$, the leading order term of the stable solution is given by The solution $\mathbf{f}\left({\gamma }_{bs}\right)$ has an additonal contribution to the pressure tensor of the order $\mathcal{O}\left({ϵ}^2\right)$ as given in (58) .

(5.12) Balancing condition: Suppose given a source $ϵs$ and a collision operator $Cf$ with traces $\gamma$ marks a balanced state if and only if the orthogonality condition is satisfied which is equivalent to the balancing condition

We consider the steady flow between two totally reflecting walls at different temperatures $T_0=T(x=-1)>T_1=T(x=1)$ with a temperature difference $\Delta T=T_0-T_1>0$. Since $\overline{v}=0$, the moment equations read The closure moments for the solution are determined from the asymptotic solution $f_{corr}$ for the model at hand of In the Navier-Stokes description, $f_{corr}$ contains no contribution to ${\widehat{\pi }}_x$, and from equations ... we find the heat layer solution Comparing the temperature profiles of numerical simulations for the nonlinear and the BGK system (Figure 3) we find that the BGK profile is considerably flatter than that of the nonlinear case. The latter one is close to the constant gradient profile (solid line) linearly connecting the wall temperatures. An explanation for th the difference is given by comparing the closure moments $\langle {\widehat{\pi }}_x,f_{corr}\rangle$ for both models (Figure 4). In Figure 4(a) we find a small almost constant contribution (maybe weakly dependent on the temperature) in the nonlinear case. Zooming into Figure 4(b) confirms this. However, for BGK we find a distinct, almost constant gradient over the whole field of calculation. An explanation for this is again provided by the trace description which allows to draw a rough picture pointing out the differences between nonlinear collision operator and BGK model. A full discussion of the trace solutions for both models in the fluid dynamic limit is provided in [5]. We do not repeat the tedious calculations of [5] but explain by a short argument why the BGK system does not provide the correct temperature gradient in the fluid dynamic limit. Suppose $\mathbf{t}\left[\gamma \right]=\overline{\mathbf{e}}+\mathrm{\Phi }$ is the steady solution of the balance equation for the BGK system, The first term on the left hand side represents in the limit a finite heat flux modeled by some term of the form $B·{\sigma }_x·\overline{\mathbf{e}}$. For small $\gamma$ we may use the approximation (65) for $\mathbf{t}\left[\gamma \right]$ shortly denoted as $(a{\gamma }^2·{\pi }_x+b{\gamma }^3·{\sigma }_x)·\overline{\mathbf{e}}$. Furthermore, $v_x{\partial }_x\mathrm{\Phi }$ yields an unknown contribution $A·{\pi }_x\overline{\mathbf{e}}$ which comes out as a solvability condition for the above equation, Its solution is and Thus under the above assumptions A is singular in the limit $ϵ\searrow 0$ which is not realistic. The only way out is that c and with this B are $ϵ$-dependent and vanish in the limit. This explains qualitatively the flattened curve for the temperature profile of the BGK model. In the limit it is supposed to yield a constant temperature profile.