The simplest choice of the coarse-graining projector is
&(f) = (M(f)M_ (f)) = (mfmJ(f)): (82) For many problems, for example, to investigate the invariance defect (78) it is not necessary to place the manifold M in the initial space E, it is sucient to investigate
_
M#
M:
In the cases when one needs, after all, to have corresponding elements of E, a good choice could be the quasiequilibrium manifold corresponding to &. The quasiequilibrium manifolds have an evident but important property. Let
f m1 !M 1 m2 !M 2 f m1m2 ;! M 2 (83)
be a sequence of linear mappings, where m1 m2 are mappings \on", their images are
whole corresponding spaces. Let, furthermore,
M
1E be a quasiequilibrium manifold inE corresponding tom 1
M
2 be a quasiequilibrium manifold of macro-variables,M2corresponding tom2M
21 bea quasiequilibrium manifold inE corresponding to m2m1. Then
m1(
M
21) =m2 orM
21=m ;11 (
M
21): (84)
For the transition to the quasiequilibrium approximation this property reads simpler:
U2U1 =U21 (85)
whereUi the corresponding tomi procedure of the taking of the quasiequilibrium approx-
imation.
For eachM2 both the point of the quasiequilibrium,M
1(M
2), and the linear manifold,
m;1 2 (M
2), containing this point are dened. For each M1 the quasiequilibrium, f
1(M 1),
and the linear manifold,m;1 1 (M
1) containing this point are dened. As wellf 2(M 2)and containing them (m2m1) ;1(M 2) are dened. Relations: f 2(M 2) = f 1(M 1(M 2)) m;1 1 (m ;1 2 (M 2)) = (m2m1) ;1(M 2): (86) are fullled.
The quasiequilibrium manifold,f 2(M
2)
Eparameterized byM
2lies on the quasiequi-
librium manifold,f 1(M
1)
E parameterized by M
1:For each M2 the set ff 1(M 1) jf 1(M 1) (m 2m1) ;1(M 2) g (87)
forms the quasiequilibrium manifold in (m2m1) ;1(M
2) with the set of macroscopic vari-
ables,m;1 2 (M
2)and the same entropy. For the projector, &(f) = (M(f)M_(f))it means
that for eachM in the linear manifold, on thatm(f) =Mthe quasiequilibrium manifold corresponding to the macroscopic variables _M(f) = mJ(f) (if J(f) is a linear mapping on this manifold) is dened.
The last remark leads us to an important construction named by us \layer-by-layer linearization". The led J(f) could be presented in the form:
JL(f) =J(f m(f)) +DfJ(f) jf m(f)(f ;f m(f)): (88)
The \layer-by-layer quadratic entropy" has special importance for the theory of non- linear equations (88) : SL(f) =S(f m(f)) ;(1=2)hf ;f m(f) jf;f m(f) if m(f): (89)
Let us remind that the bilinear form, hjif
m(f) is generated by the negative second dier-
ential of the entropy at the point f:
The layer-by-layer linearized equations allowed us to add more moment equations and construct the quasiequilibrium approximations using the entropy (89). This is especially important for the moments which are time derivatives _MM$ and so on.
Application of the layer-by-layer linearized equations (85) together with the layer- by-layer quadratic entropy (89) allowed us to construct a thermodynamically consistent theory of the moment equations for the Boltzmann equation 33, 34].
It is convenient to supplement the quasiequilibrium,f
Mwith the quasiequilibrium for
additional macro-variables _M, _
M(f) =m(DfJ(f)jf
m(f)f) (90)
in two stages: i) supplementing by the entropy production, ii) and later by the conserving part of the entropy.
i) We supplement M by the entropy production. In the layer-by-layer linear approxi- mation (f) =hf m(f) jf ;f m(f) if m(f) (91)
(as was already determined, see (45)). The quasiequilibrium manifold corresponding to
in the layer over f
M has the form:
f M =f M + (f)fM hf Mjf Mi fM: (92)
Quasiequilibrium projector in the layer is:
= jf Mihf Mj hf Mjf Mi fM ii) We distinguish in L = DJ(f)jf
M the conservative (conserving the entropy) part
over f
M :
LCM'=LM(';'):
This corresponds to the situation when we have xed (f) = hf
Mj'i and we consider
the motion in the layer for xed : For this:
LM'=LCM'+LMfMhf Mj'i hf Mjf Mif M =LCM'+ (') hf Mjf Mif M: (93)
The quasiequilibrium manifold corresponding to LCM in the layer over f
M could be
constructed in the following way: we search for the kernel of LCM in L (the set of all solutions to equationsLCM'= 0 m'= 0). We dene it asK:The orthogonal complement,
K?toK in the scalar product, hjif
Mis the corresponding manifold. For each point from
the image, LMon L, 2LCM(L) there exists unique '2K ?
fM such that LM'=: We
dene it as '= (LCM);1():As a result, for every
2LCM(L) f M =f M + fM hf Mjf Mif M + (LCM) ;1: (94)
The second and third terms in (94) are reciprocally orthogonal in the scalar product
hjif
M.