Burnett and super-Burnett orders

In this paragraph we briefly give a different scheme [62,137] (which has somewhat a better formalization) of the Chapman-Enskog derivation and the definition of the Burnett and super-Burnett orders of the expansion, which will be useful in the discussion of the granular kinetic theories. The scheme includes the same fundamental passage, i.e. the individuation of a reference (``equilibrium'') state which exactly solves the Boltzmann kinetic equation in the absence of perturbation, an expansion of the ``perturbed'' solution around this state in powers of the velocity and an expansion of all the transport equations in gradients of the hydrodynamical fields (above we have given it as a phenomenological fact, in Eq. (2.144)):

• it is observed that a probability density function $P_0$ with three global parameters ($n_0$, $\mathbf{u}_0$ and $T_0$) solves the Boltzmann equation if the external conditions impose as constant the density, average velocity and average square fluctuations of velocity: this solution is a Maxwellian;

• moreover, it is also observed that when different external conditions are imposed (e.g. specific boundaries or particular initial distributions) the same Maxwellian with space-time dependent parameters solves the Boltzmann equation if and only if those parameters satisfy the Euler equations; however this family of solutions is very particular, because it assumes an infinitely rapid relaxation to local equilibrium without any entropy production (i.e. irreversible processes);

• in general it is expected that the true solution $P$ of the Boltzmann equation in situations not too far from mechanical equilibrium (i.e. absence of macroscopic flows) is similar to the space-time dependent Maxwellian, that is the space-time dependence all occurs through the local averages of the first velocity moments: $P(\mathbf{r},\mathbf{v},t)\equiv P(\mathbf{v}\vert n,\mathbf{u},T)$; this passage can be justified by means of the analysis of stability of the kinetic modes in the Boltzmann equations: it can be shown that the short wavelength (order mean free path) modes rapidly decay and only the long wavelength, slow, modes relative to the fluctuations of the conserved quantities (mass, momentum and energy) remains; this separation of scales is the justification of the hydrodynamical limit discussed above (see paragraph (2.2.4))

• then $P$ is formally expanded as a series of powers of a ``uniformity'' parameter $\epsilon$ (which is set to $1$ at the end):

  $$P=P_0+\epsilon P_1+\epsilon^2P_2+...$$ (2.140)

  
  

  and a factor $\epsilon$ is assigned to every gradient operator $\partial/\partial r_i \to \epsilon\partial/\partial r_i$ while different time scales are coupled to the different orders in the gradient expansion: $\partial/\partial t = \partial/\partial t_0 + \epsilon \partial/\partial t_1 + \epsilon^2 \partial/\partial t_2+...$;

• finally this formal expansion of $P(\mathbf{v}\vert n,\mathbf{u},T)$ and of space-time coordinates is inserted into the Boltzmann equation: this results in a set of equations for the different orders terms $P_0$, $P_1$, etc.. At every order these equations contain space derivatives and different time derivatives (for different time scales) of the parameters $n$, $\mathbf{u}$, $T$ plus an integration (due to the collisional term) in the velocity $\mathbf{v}$ (or in $\mathbf{c}=\mathbf{v}-\mathbf{u}$);

• the time derivatives of time scales different from the zero order time scale can be obtained putting the same formal expansion in the equation for the $\mathbf{c}$-moments (Maxwell Eqs. (2.122)) and comparing terms with the same order; in this passage the flux tensors ( $\mathcal{P}$ and $\mathcal{Q}$) enter formally in the Boltzmann equation;

• in this way every $\epsilon^k$-order of the Boltzmann equation becomes an integro-differential equation with first derivatives in the variables $t_0$ and $\mathbf{r}$ and integration in the velocity $\mathbf{v}$ of the $P$ terms of order 0 through $k-1$, while the $k$-th order term ($P_k$) does not appear under space derivatives, so that a solution $P_k$ can be searched in the form $P_k=\mathbf{A}(\mathbf{c}) \cdot \nabla \ln T+\mathbf{B}(\mathbf{c}) \cdot \nabla \ln n+ \mathcal{C}_{ij}(\mathbf{c}) \partial u_j/\partial r_i$;

• this approach leads finally to approximated expressions of the flux tensors as functions of the macroscopic fields $n$, $\mathbf{u}$, $T$, i.e. by truncating the expansion to some order; these expressions close the transport equations (2.122);

• the first order truncation (i.e. second and higher orders in $\epsilon$ are neglected) yields expressions for the flux tensors which are linear in the macroscopic fields, equivalent to Eq. (2.144); so that the transport equations become the Navier-Stokes equations (2.146); this is why the first order truncation of this Chapman-Enskog expansion is called Navier-Stokes order expansion;

• keeping the second order terms in the $\epsilon$ expansion yields the so-called Burnett equations, while adding the third order terms give the so-called super-Burnett equations; Bobylev has shown that the Burnett and super-Burnett hydrodynamics violates the basic physics behind the Boltzmann equation, i.e. sufficiently short acoustic waves are increasing with time instead of decaying, contradicting the H theorem; this remains a central open problem in the kinetic theory;

• the exact functional expression of the transport coefficients needs again a polynomial expansion in $\mathbf{c}$ of the unknown functions $\mathbf{A}(\mathbf{c})$, $\mathbf{B}(\mathbf{v})$ and $\mathcal{C}_{ij}(\mathbf{c})$ and consideration similar to the ones made above.