Ann. Henri Poincaré 4, Suppl. 2 (2003) S889 – S903
c Birkhäuser Verlag, Basel, 2003
1424-0637/03/02S889-15
DOI 10.1007/s00023-003-0969-z
Annales Henri Poincaré
Half-Space Large Size Discrete Velocity Models
Henri Cornille
Abstract. We recall recent large size ”physical” discrete velocity models (DVMs)
results [1] for a gas in a half-plane (z > 0, x being the other coordinate) and present
new ones. We consider both single-gas and binary mixtures with light mass 1 and
heavy mass M. For the half-space (flow of a semi-infinite expanse of gas in contact
with its condensed phase), the interface is located at z = 0 with only a z spatial
dependence of the gas (densities with (±x, z) are equal) and no velocities parallel
to the x-axis (or z = 0). We construct ”physical” DVMs, (only mass, energy and
momentum along the z-axis invariants, no spurious invariants), filling all integer
coordinates z = 0 of the plane. The main new result for mixtures, is that we present
different M models with the same geometrical structure in the plane with sums of
the moduli of the coordinates either odd or even for heavy or light species.
1 Introduction
1) Recently planar physical DVMs (only binary collisions and physical invariants) DVMs, for binary mixtures (two species ) and for single-gas, (only one
species), tiling all the integer coordinates of the plane, were presented [1] (CornilleCercignani). This was the outcome of recent interest [2] (Bobylev-CercignaniCornille) occurring for the construction of physical DVMs with only a finite number of velocities vi (or momenta). However, from the physical point of view, a
defect was that these models have no constraints in the plane for the flows.
Here we consider both single-gas and binary mixtures half-space models [1]
(Cornille), for which velocities parallel to the interface (x-axis) must be excluded.
The interface is located at z = 0 and the z-axis perpendicular. As in the continuum treatment of the problem and in DVMs [3] (Sone, Cabannes, Gatignol,
d’Almeida, Nicodin, Cornille), the spatial dependence of the gas is assumed to be
one-dimensional depending upon the variable z > 0, associated physical properties
of both the half-space and the two parallel interfaces problems were studied Here
we only construct large size physical half-space models, filling all coordinates of
the plane. In [1] (Cornille), except for M = 2, for other mixtures M = 4, 3/2, 5, one
species was along octagons and the other filling all empty sites. Here, we present
new results for M = 3, 3/2, 4 where, like for M = 2, the 2 species are either
in |x| + |z| odd or even.
2) The three main difficulties of the present work are: only binary collisions, only
physical invariants (no spurious) and half-space for the flows.
In the past, multiple collisions (ternary, collisions of order 4,5. . . ) were very
often used in DVMs. More recently, only binary collisions are acceptable for the
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community. The reason is that we try, for discrete Boltzman equations, collisions
similar to the continuous case, in particular, like here, for DVMs filling all the
plane. Consequently in the determination of physical DVMs models, without spurious invariants, we have more difficulties. For models with spurious invariants and
only binary collisions, very often the introduction of multiple collisions is sufficient
to destroy the nonphysical conservation laws. On the other hand, the determination of physical starting model (like p = 1 square for the single-gas of section2)
with only physical conservation laws is simpler with multiple collisions. Similarly
for the mixtures models of section 3.
We explain the strategy and tools for the construction of physical models
(without spurious invariants). As illustration we recall 2 examples (CercignaniBobylev[2]), for two 13vi and two 25vi mixtures models. For the 13vi , the spurious
invariants were explained geometrically (Cornille: momenta only along 2 perpendicular axes). For the two 25vi with a great number of common collisions, it was
found with powerful computers, that both have spurious (not physical) invariants.
However, starting with more simple models with 11vi , 15vi (Cornille-Cercignani
[2]), geometrical restrictions of these 25vi models, it was shown why one of them
was spurious contrary to the other (containing non common collisions which eliminate the spurious invariants.) Consequently, it was clear that it is better to start
with simple physical preliminary models (easier to check for the nonexistence of
spurious invariants) with few vi or momenta and to try to enlarge the number
of momenta or associated densities with geometrical tools (presented below in 3).
For a model with q physical invariants and p independent densities, it is useful to
start with p− q collisions, to check whether spurious invariants exist, to investigate
their origin and to seek if they can be eliminated with other collisions.
For the single-gas, section 2,the preliminary model 2.1, lemma 1a has 5 independent (ind.) densities and 2 collisions, eqs(2a-b-c),fig1a. For the mixtures, fig3,
for M = 2, section3.3, the preliminary physical model has 7 ind. densities and 3
collisions eqs(4a-b). Similarly for the M = 3, 3/2, sections 3.4-6, the preliminary
physical models have 7 ind. densities and 3 collisions: eqs(5a-b) and eqs(6a-b), but
for M = 4, 3.6a, eq7a, we have 8 ind. densities and 5 collisions.
The most important tools (1 or 2 new densities), for the extensions of physical
models are presented below in 3.1-2-3, figs 1a-b, 2a-b: for one species in squares and
rectangles with 3 known densities or densities symmetric to the bisectors z = ±x.
Sometimes these tools are not sufficient (see below 3.4) and for new densities we
add collisions Xk , some being linked to the old physical densities. We multiply
the new evolution equations of the new densities by arbitrary constants. Adding
to the old physical conservation laws, we must verify that the constants can only
have the correct values of the new conservation laws. We apply this method for
the mixtures models in lemma 3a1, 3.2, eqs(d1-d2) and M = 2, 3/2, eqs(4c-6c) in
sections 3.2-5.
The introduction of half-space means, models more physical, with constraints
for the flows. The technical difficulty is due to the missing of momenta along the
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interface, here x-axis. For instance the starting square p = 1 of section 2 requires
a square of lengths 6 and 42 momenta, more simpler without this constraint.
Furthermore , because we have no momenta with z = 0, x = 0, the study was first
with only z > 0, lemma1a and second we include the z < 0 collisions. For the
extensions into the whole plane the above constraints are not important. However
in section 2.3, for Maxwellians at the interface, we must show how the impinging
densities z < 0 can be deduced from the emerging ones z > 0.
Here the great progress, comparing with [1], is that we could determine mixtures models with the same geometrical structure of the momenta in the plane.
Previously, studying different M mixtures models, we find different structures, like
different octagons and only for M = 2, the heavy and light species in |x|+|z| either
odd or even. Here we find this previous M = 2 structure, fig2c, for all other M
values. Consequently, we have found that for the same M mixture model, different
geometrical structures filling the plane are possible. In fact it is sufficient to prove
this structure for small |x| + |z| = α values and the tools figs 2a-b, lemma 2a-b
in section 3 are sufficient to expand these geometrical structures into the whole
plane. For all M models α = 3 is sufficient for the heavy species but different α
values (the reason being the nonexistence of momenta along the x-axis) for the
light. Consequently if these different M have the same geometrical structure in the
whole plane, on the contrary for small finite sizes (contrary to large sizes) of the
starting models, the structures are different and we must give different proofs for
the physical preliminary models of sections 3.3-4-5-6, fig 3.
3) Tools in order to enlarge physical models.
Starting with a preliminary simple physical model, we can for new ”physical” add
new momenta:
3.1 4 momenta of the same species of a mixture or single-gas physical model being
along rectangles or squares, to 3 (xi , zi ) belonging to the preliminary physical, we
can include the last one (x4 , z4 ): With conservation laws satisfied, in order to
eliminate this collision, we must add the last one.
(xi , zi ) =
i=1,2
(xi , zi ),
i=3,4
i=1,2
x2i + zi2 =
x2i + zi2 → add (x4 , z4 )
(1)
i=3,4
3.2 Along a square with the particle at rest at the center and 2 momenta of the
preliminary model along the diagonals (or medians), we add 2 new ones along the
medians(diagonals).
3.3 For one species of a physical model, we can add the densities symmetric to the
bisectors z = ±x if, for the new densities, exist in the old model other densities
with the same z (see [1]).
3.4 These tools are not always sufficient to enlarge physical models, but in some
cases we can include 2, 3. . . new densities. For a previous physical model we start
with the old conservation laws which are invariants with old densities and old
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collisions. With new densities Ni , i = 1, 2 . . . we have new collisions Xk (satisfying
conservation laws) including some previous densities.
For their evolution equations included into the old conservation laws we deduce
sums of Xk . If these sums (at least for one conservation law) are all zero, then
the new model will have spurious invariants. The reason being that this old conservation is still an invariant with the new collisions, but another invariant exists
with only the new densities. So these sums, for each old conservation law, must be
(j)
different of zero (but this is not sufficient). For these sums, we get
ak Xk with
(j)
ak known numerical constants and (j) = 1, 2, 3 . . . for mass, energy, momentum
conservation laws. Now for the new densities, the evolution equations li (Xk ) give
other sums of Xk with known coefficients. For each physical conservation law (j) we
(j)
(j)
(j)
consider the sum
ak Xk + bi li (Xk ) = 0 where bi are arbitrary unknown
constants. For each collision Xk , the sum of the constants must be zero. Solving
(j)
this linear system we must verify, for the set bi , that only the numerical values
of the new (j) physical conservation law are possible. Otherwise, we have spurious
invariants.
s=q
−
In the sequel, for a set ys , we define yi,j,...q = s=i ys and yi,j
= yi − yj . In
the x, z plane, for all velocities v (±x = 0, z), the associated densities N (±x, z) are
equal. So we write q ind. Ni for the number of really independent Ni . We have
two successive studies:
For single-gas, first for our starting models we write the equations of the evolution
equations with the associated collisions. We must prove that with linear combinations of these collisions, only 3 are independent and only equivalent to the 3
physical conservation laws. We begin by writing the collisions Xk quadratic in
li = (∂t + zi ∂z )Ni (or
the densities Ni . Then we write the evolution equations
zi ∂z Ni ) and write the nonlinear equations li − k λi,k Xk = 0 with λi,k being
known numerical constants. Then we write the Xk as linear combinations of the
lj that we substitute into the above li nonlinear equations and obtain invariants:
j νi,j lj = 0, with νi,j known numerical constants. The sets νi,j = 0 give the
invariants and we prove that they are equivalent to the three conservation laws.
Second with (1), we enlarge the physical models with squares, rectangles with3
densities known (see below 3.1 the most important tool here) adding successively
new densities. For mixtures, the strategy is similar but with light vi = pi , fi and
j , Pj = M V
j , Fj velocities, momenta, densities.
heavy V
With scaling we can generalize to half-integers,. . . . The presented models,
filling all integer coordinates, have a regular grid (mesh-step 1). We can generalize
with a mesh-step h finite: (v → hv , P → hP ) and v 2 → (hv )2 , P 2 → (hP )2
(collisions still valid). For the construction of physical models we present sets
of p-th selected momenta and we know (advantage to the standard numerical
discretizations) that in all intermediate steps we have no spurious invariants.
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2 Physical single-gas DVMs filling all integers (z = 0) of the plane
2.1
Physical p = 1 square, fig 1a, with vi (x, z), x ∈ −3, 3, z ∈ −3, 3, = 0
Lemma 1a. We prove with binary collisions (N0 present or not) that the p = 1
square model with 43 vi (x, z) (42) and 25 (24) ind. Ni is physical.
1) We begin with 7vi (x, z) 5 ind. Ni , i = 1, 3, 5, 6, 7 and x → −x → vk+1 , k = 1, 3:
v1 (1, 1),
v3 (1, 2),
v5 (0, 1),
(2a)
We call Ni (t, z) (or Ni (z)), li the densities, evolution equations associated to
vi (x, z) and Nk+1 = Nk , k = 1, 3 in (2a). With the evolution equations (∂t +
zi ∂z )Ni = li linear combinations of two quadratic collisions Xk and X2 linear
combinations of the li , we deduce 3 invariants [I], [II], [III] which are equivalent to
the mass [M ], energy 2[E] and momentum [J] along z, conservation laws.
X1 = N5 N7 − N32 , X2 = N3 N5 − N6 N1 ,
−
−
, X2 = l1 = l6 /2 = l7,5
/2 = −l3,7
→ l7 = −X1 , −l5 = 2X2 + X1 , l3 = X1,2
−
→ [I] = l6 − 2l1 = 0, [II] = 2l1 + l5,7
= 0, [III] = l1,3,7 = 0
(2b)
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[M ] = l5,6,7 + 2l1,3 = [I] + [II] + 2[III],
[J] = 3l7 + 2l1,6 + 4l3 + l5 = 2[I] + [II] + 4[III],
2[E] = 4l1,6 + 9l7 + 10l3 + l5 = 4[I] + [II] + 10[III]
(2c)
2) We go on, fig 1a, with collisions in squares (3 known) and add successively:
(i) : N (±1, 3) (ii) : N (±2, 2)
(iii) : N (±2, 3), N (±2, 1) (iv) : N (±3, 2) (v) : N (±3, 3), N (±3, 1) (2d)
In x ∈ −3, 3, z ∈ 1, 3 = 0, the N (x, z) belong to a physical model.
3) For a link between the z < 0 and z > 0 densities, we begin with N (0, −1). For
the other z < 0, with collisions in rectangles and squares (3 N (x, z) known), we
successively add a new N (x, z):
(vi) : N (−2, 1)N (2, 1) − N7 N (0, −1) → N (0, −1), (vii) : N (±x, −1),
(viii) : N (−1, −1)N (3, 1) − N (2, 2)N (0, −2) → N (0, −2) (ix) : N (±x, −2),
(x) : N (−1, −2)N (1, −2) − N (0, −1)N (0, −3) → N (0, −3) (xi) : N (±x, −3) (2e)
with x = 1, 2, 3. The Ni with integers x ∈ −3, 3, z ∈ −3, 3 = 0, belong to the
physical p = 1 model. Finally, if we want to add N (0, 0), the collision in the square
N (1, 1)N (−1, 1) − N (0, 2)N (0, 0) is sufficient.
2.2
Physical p model, fig 1b, 2(2p + 3)(p + 2)vi , 2(p + 2)(p + 3) ind.Ni
For the extensions to p = 2, 3, . . . physical models, we use collisions in squares,
rectangles (3 known, without N0 ), adding one N (x, z). First, from the p=1 square,
we construct a physical p = 2 square with sides 8, fig 1b. Then, from the p − 1
physical model, we obtain a physical p-th square, p integer arbitrary, limited by
x = ±(p + 2), y = ±(p + 2), filling (p → ∞) all integer coordinates except the
x-axis. We still write N (x, z) for the densities associated to v (x, z).
Lemma 1b. The p-th model, p integer arbitrary, fig 1b, (sides 2p+4) is physical.
From the p − 1 model we present the new N (x ≥ 0, z > 0): N (i, p + 2), N (p + 2, j),
i = 0, 1, . . . , p + 2, j = 1, 2, . . . , p + 1 which are successively included in squares,
rectangles (3 densities known, add the last one).
(i) : (−1, p + 1) + (1, p + 1) = (0, p) + (0, p + 2)
(ii) : (i − 1, p + 2) + (i, p + 1) = (i − 1, p + 1) + (i, p + 2), i = 1, . . . , p + 1
(iii) : (p + 1, p) + (p + 1, p + 2) = (p, p + 1) + (p + 2, p + 1)
(iv) : (p + 1, p + 2) + (p + 2, p + 1) = (p + 1, p + 1) + (p + 2, p + 2),
(p + 1, j) + (p + 2, j + 1) = (p + 1, j + 1) + (p + 2, j),
j = p, p − 1 . . . 1 → addN (±x, z) :
(0, p + 2), (i, p + 2), (p + 2, p + 1), (p + 2, p + 2), (p + 2, j).
(2f)
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Single-Gas DVMs Half-space [3]
The DVMs is in equilibrium with the surface (wall or condensed phase) when
the microscopic densities Nw (x, z) of the Maxwellian state are associated with
the macroscopic variables at the surface Mw , Jw , Ew (mass, momentum, energy).
At the interface, the particles leaving the condensed phase (emerging) are in
Maxwellian equilibrium with it. At infinity, the gas is assumed to be uniform
and in a Maxwellian equilibrium associated with the macroscopic variables of the
flow. There is evaporation (condensation) when the density of the vapor near the
interface is lower (higher) than the saturation density. Here we discuss only the
Maxwellians at the interface. We start with the emerging densities Nw (x, z > 0)
and with vanishing collision terms and vanishing momentum along the z-axis we
deduce the impinging z < 0 and could obtain all macroscopic quantities Mw , Ew at
the interface. We find, that the Nw with the same |v (x, z)| are equal (for instance
for emerging, impinging densities with opposite velocities or symmetric with the
z = ±x-axes). Furthermore, all Nw , with still Nw (x, z) = Nw (−x, z), are functions
of 2 emerging densities and of |v |. We present results only for p = 1:
First with vanishing collisions in rectangles parallel to the z-axis, z = 1, 2, 3:
x = ±1, ±2, ±3, z = 1, 2, 3, λz = Nw (0, −z)/Nw (0, z) = Nw (x, −z)/Nw (x, z)
Second with 2 squares we show that λ3 = λ31 and with 2 other λ2 = λ21
Nw (0, ∓3)Nw (0, ±1) = Nw2 (2, ∓1) → λ3 = Nw (0, −3)/Nw (0, 3) =
[Nw (0, −1)/Nw (0, 1)][Nw (2, −1)Nw (2, 1)]2 = λ31 , Nw (0, ±2)Nw (2, ±2) =
(2g)
Nw (1, ±3)Nw (1, ±1) → λ22 = λ1 λ3 → λ2 = λ21
Jw = 0, λz = λz , z = 1, 2, 3, x = 1, 2, 3 → λ = 1 and Nw (±x, ±z), |v(x, z)|2 =
x2 + z 2 are equal:
z[Nw (0, z) − Nw (0, −z) + 2
Nw (x, z) − Nw (x, −z)] =
0 = Jw =
z
z
z
z(1 − λ )[Nw (0, z) + 2
x
Nw (x, z)] → λ = 1,
x
Nw (x, z) = Nw (−x, z) = Nw (−x, −z) = Nw (x, −z) and
(2h)
2
Nw (x, z)Nw (−x, −z) = |Nw (x, z)| = Nw (z, x)Nw (−z, x) = |Nw (z, x)|2
deduced from the rectangle (±x, ±z), (±z, ±x), parallel to a bisector. With the
collisions X1 , X2 and (i), (ii), (iii) of (2a-d), we write Nw (x, z) functions of the 3
Nw,i , i = 1, 5, 6 and only 2 with λ = 1, (2g-h), giving a restriction (for brevity we
write Nk for the Nw,k of (2a-b))
N3 = N6 N1 /N5 , N7 = N32 /N5 = N62 N12 /N53 ,
(i)Nw (1, 3) = N3 N7 /N6 = N3 N6 N12 /N53 ,
(ii)Nw (2, 2) = N1 Nw (1, 3)/N6 = N3 N13 /N53 ,
(iii)N3 Nw (2, 1) = N32 = N1 Nw (2, 2) = N14 N3 /N53 → N3 = N14 /N53 =
N6 N1 /N5 → N3w,1 = Nw,6 N2w,5 :
(2i)
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2
Lemma 1c. We define N w (x,z) = Nw (x,z)/N5 ,µ = N1 /N5 , get: Nw = µ|ṽ(x,z)| −1 .
We have N 5 = 1, N 1 = µ and from (2i): N 6 = µ3 , N 3 = µ4 , N 7 = µ8 . For all
other with collisions satisfying I, eq(1) and 3 N w satisfying this relation, we can
extend for the last:
N w (x4 , z4 ) = N w (x1 , z1 )N w (x2 , z2 )/N w (x3 , z3 ) =
µ|v1 |
2.4
2
+|
v2 |2 −2
/µ|v3 |
2
−1
= µ|v4 (x4 ,z4 )|
2
−1
Nicodin p-th squares [3], fig1c, filling |x| , |z| odd integers
aq = 2q − 1, v (x, z) : (η1 aq , η2 as ), (η3 as , η4 as ),
ηi2 = 1, s = 0, 1, . . . q
q = 1, 2, . . . p .
(2j)
Lemma 1d. The p = 2 model is physical, assuming the p − 1 physical, then the p-th
is physical. We write (x, z) for v (x ≥ −1, z ≥ −1) and for the collisions, we must
add (x, z) → (z, x) (for instance below for p-th: (i), (ii)) and other x, z symmetries:
p = 2 X = N (1, 1)N (3, −1) − N (1, −1)N (3, 1) → l(±1, 1) = l(±3, −1) =
−l(±1, −1) = −l(±3, 1) → 3 invariants and we add N (±1, ±3), N (±3, ±3) with
(∓1, ±1) + (±3, ±1)) = (±1, ∓1) + (±1, ±3), (ii)(±1, ±3) + (±3, ±1) = (±1, ±1) +
(±3, ±3)
p-th model with
(i)
(ap−1 , 3) + (ap−1 , −1) = (ap−2 , 1) + (ap , 1)
(ii) (ap , 1) + (ap−1 , aj ) = (ap−1 , 1) + (ap , aj ), j = 2, .p − 1
(iii) (ap−1 , ap ) + (ap , ap−1 ) = (ap−1 , ap−1 ) + (ap , ap ) →
add
N (ap , 1), N (1, ap ), N (ap , aj ), N (aj , ap ), N (ap , ap ) .
(2k)
3 Physical mixtures DVMs filling all integers (z = 0) of the plane
3.1
One species filling (squares,rectangles) |x| + |z| odd or even
We write (x, z) for the momenta and for collisions (1) with 3 (xi , zi ) known we
add the last (x4 , z4 ) We begin, fig 2a, with few (x, z) of one species satisfying
|x| + |z| = 1, 3. We call p = 1 the physical model (mainly parallel to y = −x, first
quadrant x ≥ −1, z ≥ −1) and with squares and rectangles (1), deduce physical
2, 3, . . . p arbitrary with |x| + |z| = 1, 3, . . . 2p + 1, x ≥ −1, z ≥ −1, z = 0.
p = 1,
6 momenta: (0, 1), (0, 3), (±1, 2), (2, ±1)
(3a1)
Lemma 2a. With (1), the p = 2 model deduced from p = 1 (j = 2 in I, adding
momenta |x|+|z| = 5) and . . . the p-th models deduced from p−1 (adding |x|+|z| =
2p + 1) are physical. We successively get new momenta I, II, III, IV :
I: (p− j, p+ j − 1)+ (p− j + 2, p+ j − 3) = (p− j, p+ j − 3)+ (p− j + 2, p+ j − 1), j =
p, p − 1, .2, 1, ≥ 4 − p →
II : (2p − 2, 3) + (2p − 2, 1) = (2p − 3, 2) + (2p −
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1, 2), (2p − 2, 3) + (2p − 2, −1) = (2p − 4, 1) + (2p, 1), (0, 2p − 1) + (2, 2p − 1) =
(1, 2p − 2) + (1, 2p) →
III : (2p, 1) + (2p − 2, −1) = (2p − 2, 1) + (2p, −1), (−1, 2p − 2) + (1, 2p) =
(1, 2p − 2) + (−1, 2p) → IV : (−1, 2p) + (1, 2p) = (0, 2p − 1) + (0, 2p + 1) (3b1)
Geometrically the momenta of the 2p − 3 collisions I are on 3 lines parallel to
y = −x and squares, parallel to the x, z axes and one top for the new p-th model.
We go on, fig 2b, with few momenta (x, z) of one species satisfying |x| + |z| =
2, 4 (geometrically similar to the previous p = 1 model), with x ≥ −1, z ≥ −1,
Similarly we deduce physical p = 2 and p arbitrary models but with |x| + |z| =
2, 4, . . . 2p + 2. We start with p = 1:
p=1:
8 momenta: (0, 2), (0, 4), (±1, 3), (2, 2), (1, 1), (3, ±1)
(3a2)
Lemma 2b. the p = 2 (from p = 1) and p-th (from p − 1) models are physical, with
the same geometrical explanation of the 2p − 2 collisions I as previously:
I: (p − j, p + j) + (p − j + 2, p + j − 2) = (p − j, p + j − 2) + (p − j + 2, p + j), j =
p, p − 1, . . . 2, 1, . ≥ 3 − p →
II : (2p − 1, 1) + (2p − 1, 3) = (2p − 2, 2) +
(2p, 2), (2p − 1, 3) + (2p − 1, −1) = (2p − 3, 1) + (2p + 1, 1), (0, 2p) + (2, 2p) =
(1, 2p − 1) + (1, 2p + 1)
→ III : (2p − 1, −1) + (2p + 1, 1) = (2p − 1, 1) +
(2p + 1, −1), (−1, 2p − 1) + (1, 2p + 1) = (1, 2p − 1) + (−1, 2p + 1)
→ IV :
(−1, 2p + 1) + (1, 2p + 1) = (0, 2p) + (0, 2p + 2) .
(3b2)
In conclusion, fig 2c let us consider any physical binary mixtures where the two
i , fi , li (masses, momenta,
species: heavy M > 1, Pj , Fj , Lj and light: m = 1, p
densities, evolutions equations) contain the starting (3a1-a2), and their x, z symmetric densities (see (3c) and fig 3). Then with Lemmas 2a-b, we cant fill all the
plane with either |x| + |z| odd or even for the 2 species.
p3 , p
7 (1, 3) = −
p9 , p11 (2, 2) = −
p13 , p17 (3, 1) =
light: p0 (0, 0), p1 (1, 1) = −
i (−x, z) → pi+1 , fi+1 = fi i = 1, 3, 7, 9, 11, 13.17, 19, p5 (0, 2) =
−
p19 , p
−
p6 , p15 (0, 4) = −
p16 ,
heavy: P1 (1, 2) = −P3 , P9 (2, 1) = −P11 , Pj (−x, z) → Pj+1 , Fj+1 = Fj j =
5 (0, 1) = −P6 , P7 (0, 3) = −P8 .
1, 11, P
(3c)
In general for M fixed, as we shall see, we add other pi , Pj for the complete proof.
3.2
Tools in order to enlarge one species, fig 3
Lemma 3a1. We assume that one species: Fj , j = 1, 2, 9, 10,(3c) and F17 (1, 4),
F18 (−1, 4), fig3, belongs to a physical model. We can add F5 , F7 , (3c) and F13 (2, 3),
F14 (−2, 3), fig3, for a new physical. With 3 arbitrary a, b, c, we verify that they
can only have the correct values. We write the collisions, Li and verify that the
invariants are equivalent to the conservation laws:
Ω1 = F7 F13 − F1 F17 , Ω2 = F5 F13 − F7 F9 , Ω3 = F5 F7 − F12 → L17 = Ω1 , L1 =
X =
Ω1,3 , L9 = Ω2 , L5 = −2Ω2 − Ω3 , L7 = 2Ω−
2,1 − Ω3 , L13 = −Ω1,2 ,
aL5 +bL7 +cL13 = Ω2 (2(b−a)−c)−Ω1 (2b+c)−Ω3 (a+b),
[ML ] = 2L1,9,17 +X =
Ω1 (4 − 2b − c) + Ω3 (2 − a − b) + Ω2 (2 − 2a + 2b − c) = 0 → a = b = 1, c = 2
Vol. 4, 2003
Half-Space Large Size Discrete Velocity Models
S899
ok
[2E] = 34L17 + 10L1,9 + X = Ω1 (44 − 2b − c) + Ω2 (10 − 2a + 2b − c) +
Ω3 (10 − a − b) → a = 1, b = 9, c = 26 ok
[J] = 8L17 + 4L1 + 2L9 + X =
Ω3 (4 − a − b) + Ω1 (12 − 2b − c) + Ω2 (2 − 2a + 2b − c) = 0 → a = 1, b = 3, c = 6
ok
(3d1)
Lemma 3a2. We assume that one species: Fj , j = 1, 2, 9, 10,(3c) and F13 (2, 3),
F14 (−2, 3), fig 3, belongs to a physical model. We can add F5 , F7 , (3c), for a new
physical. Like previously with 2 arbitrary a, b (only correct values), we write the
Ωi , Li and verify the conservation laws:
→ L13 = Ω2 , L1 = Ω1 , L9 =
Ω1 = F7 F5 − F12 , Ω2 = F7 F9 − F5 F13 ,
X = aL5 +bL7 = 2Ω2 (a−b)−Ω1 (a+b)
−Ω2 , L5 = 2Ω2 −Ω1 , L7 = −2Ω2 −Ω1 ,
[ML ] = 2L1,9,13 + X = −2Ω1 + X → a = b = 1 ok [2E] = 26L13 + 10L1,9 + X =
16Ω2 + 10Ω1 + X → a = 1, b = 9 ok [J] = 6L13 + 4L1 + 2L9 + X = 4Ω−
2,1 + X =
0 → a = 1, b = 3 ok
(3d2)
3.3
Half-Space Physical M = 2, figs 3-2c, with light f0 rest-particle, model
We start with a preliminary physical 5fi , 6Fj model, extend to a starting model
with the heavy (3.1) filling |x| + |z| odd and light (with tools) even.
3.3a: Starting physical model, fig 3: 12Pj (x, z) and 7
pi (x, z) satisfying (3a1):
In (3c), fig 3, we retain:
pi , i = 0, 1, 2, 3, 4, 5, 6,
Pj j = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 .
(4a)
First, we study the 11 Fj , fi model: j = 1, . . . 6 and i = 0, . . . 4, 7 ind. j =
1, 3, 5, 6, i = 0, 1, 3 and 3 collisions, we write the Lj , li and the conservations laws:
[ML ], [Ml ] for the masses (heavy, light species), the energy [E] and [J], the z-axis
momentum. We get 4 invariants, equivalent to the 4 conservation laws.
Γ1 = F6 f1 −F5 f3 , Γ2 = f0 F1 −f1 F5 , Γ3 = f0 F3 −f3 F6 , L1 = (∂t +2∂z )F1 = −Γ2 ,
L3 = (∂t − 2∂z )F3 = −Γ3 , l0 = ∂t f0 = −2Γ2,3 , l1 = (∂t + ∂z )f1 = Γ−
2,1 , l3 =
−
(∂t − ∂z )f3 = Γ1,3 , L5 = (∂t + ∂z )F5 = 2Γ1,2 , L6 = (∂t − ∂z )F6 = 2Γ3,1 , [Ml ] =
l0 +2l1,3 = 0, [ML ] = L5,6 +2L1,3 = 0, 4[E] = 8l1,3 +10L1,3 +L5,6 = 0, [J] =
−
−
(4b)
4L−
1,3 + 2l1,3 + L5,6 = 0 .
For the li , Lj , with arbitrary constants a, b, c, we verify that they can only have
the values for [Ml ], [ML ]. For the li , i = 0, Lj , first we write Γ1 as linear combinations of li , Lj , Γ2 , Γ3 . Second, we eliminate these collisions, get 3 invariants I,
II, III with only 2 new ones [E], [J].
(i) li : 0 = l0 + al1 + bl3 = Γ1 (b − a) + Γ2 (a − 2) + Γ3 (b − 2) → a = b = 2 → [Ml ],
(ii) Lj : 0 = L1 + aL3 + bL5 + cL6 = Γ1 (b − c) + Γ2 (−1 + 2b) + Γ3 (2c − a)
→ a = 1 = 2b = 2c → [ML ]/2. (iii) li , Lj not l0 : Γ1 = l3 − Γ3 = −l1 + Γ2 =
L5 /2 − Γ2 = −L6 /2 + Γ3 → l3 + L3 = −(l1 + L1 ) = L5 /2 + L1 = −L6 /2 − L3 →
[I] : l1,3 +L1,3 = 0 → 8[I] = 4[E]−[ML ],
[II] : 2L1,3 +L5,6 = [ML ],
[III] :
l3 − L5 /2 + L3 − L1 = 0 → 4III = 2[I] − [ML ] − [J] = 0 .
(4c)
S900
H. Cornille
Ann. Henri Poincaré
Second, with collisions in squares and (1), we add the last one Fj , j = 7, 8, 9, 11
fi , i = 5, 6 giving a 19 Fj , fi physical model:
2
F12 − F5 F7 , F32 − F6 F8 , F6 F7 − F92 , F5 F8 − F11
f12 − f0 f5 , f32 − f0 f6 .
(4d)
3.3b Physical models, including all Fj , fi of the plane (except x-axis)
In eq(4a), the Fj , Pj j = 1, 2, 5, 7, 9, 12 are the coordinates of 3.1 eq(3a1), we
can add |x| + |z| odd with (x ≥ −1, z ≥ −1 = 0) for physical models and other
quadrants with x, z symmetries, the heavy fill all coordinates with |x| +|z| odd, z =
0. For the light particles fi , pi , the 3.1, (3a2) momenta (0, 4), (±1, 3), (2, 2), (3, ±1)
are missing. But F (3, 4), P (3, 4), fig3, belongs to the physical model. We start with
the 7fi , i = 0, 1, .6 of (4a) and F (3, 4). With 5 collisions (3 known), we add the
new ones: pi i = 7, 11, 15, 17, 19 written in (3c)
f0 F (3, 4) − f7 F9 , f7 f1 − f5 f11 , f8 f7 − f5 f15 , f4 f7 − f2 f17 , f17 f4 − f1 f19 .
(4e)
We apply 3.1, (3a2), all |x| + |z| even, x ≥ −1, z ≥ −1, = 0 and other quadrants
with x, z symmetries, are filled. All integers z = 0 are filled with either fi or Fj .
3.4
Half-Space Physical M = 3, figs 3-2c, with light f0 rest-particle, model
We start with a preliminary physical 3fi , 8Fj model, extend to a starting model
with the heavy (3.1) filling |x| + |z| odd and light (with tools) even.
pi (x, z)
3.4a. Starting physical model, fig 3: 12Pj (x, z) and 7
pi , i = 0, 1, 2, 3, 4, 5, 6 Pj , j = 1, 2, 3, 4, 9, 10, 11, 12, 17, 18, 19, 20.
(5a)
First, we study the 11 Fj , fi model: j = 1, 2, 3, 4.17, 18, 19, 20 and i = 0, 5, 6 with
7 ind. j = 1, 3, 17, 19 i = 0, 5, 6. With 3 collisions, we write the Lj , li and the
conservation laws. For the li , Lj , with constants a, b, c, they can only have the
values for [Ml ], [ML ]. For the li , i = 0, Lj , we write Γ3 as linear combinations of
li , Lj , get 3 invariants I, II, III but only 2 new [E], [J].
Γ1 = f0 F17 − f5 F1 , Γ2 = f0 F19 − f6 F3 , Γ3 = f6 F1 − f5 F3 , L1 = Γ−
1,3 , L3 =
−
Γ2,3 , L19 = −Γ2 , L17 = −Γ1 , l0 = −2Γ1,2 , l5 = 2Γ1,3 , l6 = 2Γ2,3 , [Ml ] = l0,5,6 =
−
0, [ML ] = 2L1,3,17,19 = 0, 2[E] = 12l5,6 +10L1,3 +34L17,19 = 0, [J] = 4L−
5,7 +2l5,6 +
(i) li : 0 = l0 +al5 +bl6 = 2Γ1 (a−1)+2Γ2 (b−1)+2Γ3(a−b) → a = b =
8L−
1,3 = 0
1 → [Ml ], (ii) Lj : 0 = L1 + aL3 + bL17 + cL19 = Γ1 (1 − b) + Γ2(b − c) + Γ3 (a − 1)
→ a = 1 = b = c → [ML ]/2 . (iii) li , Lj not l0 : Γ3 = −L1,17 = L3,19 = l5 /2+L17 =
−l6 /2 − L19 → [I] = L1,3,17,19 = [ML ]/2 [II] = l5,6 + 2L17,19 → 12[II] + 5[ML ] =
2[E] [III] = 2L3 + 2L−
(5b)
19,17 − l5 → 6[III] − 3[J] = 17[ML ]/2 = [E] .
Second, with collisions in squares (1), we add the last one: f0 f5 − f12 , f0 f6 − f32 →
new fi , i = 1, 2, 3, 4 and also (2.3 same z as f1 , f3 ) we add Fj , j = 9, 10, 11, 12
symmetric z = ±x of j = 1, 2, 3, 4 → 19Fj , fi physical. Finally, from F1 , F9 , F17
and Lemma 3a1, we can add F7 , F5 .
Vol. 4, 2003
Half-Space Large Size Discrete Velocity Models
S901
3.4b Physical models, including all Fj , fi of the plane (except x-axis)
The Fj , Pj j = 1, 2, 5, 7, 9, 12 are the coordinates of 3.1 eq(3a1) and belong to
the 3.4a physical model. We can add |x| + |z| odd with (x ≥ −1, z ≥ −1 = 0)
for physical models and other quadrants with x, z symmetries. The heavy fill
all coordinates with |x| +|z| odd, z = 0 in particular F (1, 8), fig3. For the light
i , with 3 collisions, we add fi , i = 15, 11, 7: 65 = 17+48, f0F (1, 8)−
particles fi , p
2
, f5 f15 − f72 . Now, for the light, we can apply 3.1,(3a2) for the
f15 F17 , f0 f15 − f11
first quadrant and all (x, z symmetries) are filled with light |x| +|z| even.
3.5
Half-Space Physical M = 3/2, figs 3-2c, with light f0 rest-particle
We start with a preliminary physical 5fi , 8Fj model, extend to a starting model
with the heavy (3.1) filling |x| + |z| odd and light (with tools) even.
3.5a. Starting physical model, fig 3: 20 Pj (x, z) and 7 pi (x, z)
fig3 with: pi , i = 0, 11, . . . , 16
21, . . . , 24.
Pj , j = 1, . . . , 4 5, . . . , 8 9, . . . , 12 17, . . . , 20
(6a)
First, we study the 13 Fj , fi model: j = 1, 2, 3, 4, 17, 18, 19, 20 and i = 0, 11, 12, 13,
14 with 7 ind. densities j = 1, 3, 17, 19 i = 0, 11, 13. With 3 collisions, we write
the Lj , li and the conservation laws. For the li , Lj , with arbitrary a, b, c, they
can only have the values for [Ml ], [ML ]. For the li , i = 0, Lj , with Γ3 linear
combinations of 4 li , Lj we get 3 invariants I, II, III but only 2 new [E], [J].
Γ1 = f0 F17 − f11 F1 , Γ2 = f0 F19 − f13 F3 , Γ3 = f13 F1 − f11 F3 , L1 = Γ−
1,3 , L3 =
−
Γ2,3 , L19 = −Γ2 , L17 = −Γ1 , l0 = −2Γ1,2 , l11 = Γ1,3 , l13 = Γ2,3 , [Ml ] = l0 +
2l11,13 = 0, [ML ] = 2L1,3,17,19 = 0, 2[E] = 24l11,13 + 10L1,3 + 34L17,19 =
−
−
0, [J]/4 = L−
(i) li : 0 = l0 + al11 + bl13 = Γ1 (a − 2) +
1,3 + 2L17,19 + l11,13 = 0
Γ2 (b − 2) + Γ3 (a − b) → a = b = 2 → [Ml ], (ii) Lj : 0 = L1 + aL3 + bL17 + cL19 =
Γ1 (1 − b) + Γ2 (a − c) + Γ3 (a − 1) → a = 1 = b = c → [ML ]/2 . (iii) li , Lj
not l0 : Γ3 = −L1,17 = L3,19 = l11 + L17 = −l13 − L19 → I = L1,3,17,19 =
[II] = l11,13 + L17,19 → 12[II] + 5[ML] = 2[E]
[III] = L3 + 2L19 +
[ML ]/2
l13 → 24[III] + 3[J] − 7[ML ]/2 = [E] .
(6b)
Second, we add 8 physical Fj , j = 9, . . . 12 and j = 21, . . . 24 with 4 ind. j =
9, 11, 21, 23
Γ4 = f0 F21 − f11 F11 , Γ5 = f0 F23 − f13 F9 , Ω1 = F23 F9 − F11 F21 , Ω2 = F17 F19 −
F23 F21 , L9 = Γ5 − Ω1 , L11 = Γ4 + Ω1 , L21 = −Γ4 + Ω1,2 , L23 = −Γ5 +
−
−
→ X = j=9,11,21,23 Lj aj = Γ5 (a9 − a23 ) +
Ω−
2,1 , L17,19 = 2Ω2 , l11,13 = Γ4,5
Γ4 (a11 − a21 ) + Ω1 (a11,21 − a9,23 ) + Ω2 a21,23
(i): [ML ] = . . . 2L17,19 + X =
· · · − 4Ω2 + X → 2 = a9 = a11 = a21 = a23 ok
(ii): [2E] = · · · + 34L17,19 +
(iii):
24l11,13 +X = . . . 24Γ4,5 −68Ω2 +X → a9 = a11 = 10, a21 = a23 = 34 ok
−
−
[J] = · · · + 4l11,13
+ 8L−
17,19 + X = · · · + 4Γ4,5 + X → a9 = −a11 = a21 = −a23 = 2
ok
(6c)
S902
H. Cornille
Ann. Henri Poincaré
Third, from F1 , F9 , F17 and Lemma3a1, we can add F7 , F5 . The Fj , Pj j =
1, 2, 5, 7, 9, 12 are the coordinates of 3.1 eq(3a1), we can add |x| + |z| odd with
(x ≥ −1, z ≥ −1 = 0) for physical models (other quadrants with x, z symmetries).
For the light we add
2
,
f0 f15 − f11
2
f0 f16 − f13
→ newfi , i = 15, 16 → 27 Fj , fi physical model. (6d)
3.5b Physical models, figs 3-2c, including all Fj , fi of the plane (except x-axis)
The heavy Fj fill all coordinates with |x| +|z| odd, z = 0 in particular F (4, 7),
fig3. With an energy exchange collision f0 F (4, 7) − f (2, 6)F9 , 65 = 60 + 5, we first
add f (±2, 6). Second in the light square f (±2, 6), f11 , f12 , center f15 belonging
to a physical model, we can add the tops of the medians, in particular f (0, 2) = f5
Third, in the light physical square f0 , f15 , f11 , f12 , center f5 , we add the tops
of the medians (and with the x, z symmetries) fi , i = 1, 2, 3, 4, 7, 8, 9, 10. With
f4 f11 − f0 f17 , f4 f17 − f1 f20 , we add f17 , f20 finally. Now, for the light, we can
apply 3.1, (3a2) for the first quadrant and all (x, z symmetries) are filled with light
satisfying |x| +|z| even.
3.6
Half-Space Physical M=4, fig 3, with light f0 rest-particle, model
We start with a preliminary physical 7fi , 8Fj model, extend to a starting model
with the heavy (3.1) filling |x| + |z| odd and light (with tools) even.
3.6a: Starting physical model, fig 3:
In fig 3, we retain 7 pi , i = 0, 1, 2, 3, 4, 5, 6 and
8 Pj , j = 1, 2, 3, 4, 13, 14, 15, 16
(7a)
First, we study the 15 Fj , fi model (7a): with 8 ind. densities j = 1, 3, 13, 15 i =
0, 1, 3, 5, 6. With 5 collisions, we write the Lj , li and the conservation laws. For
the li , Lj , with arbitrary constants a, b, c, d, we verify that they can only have
the values for [Ml ], [ML ]. For the li , i = 0, Lj , we write Γ3 , Λ1 , Λ2 as linear
combinations of li , Lj and get 4 invariants I,. . . , IV but only 2 new [E], [J].
Γ1 = f0 F13 − f1 F1 , Γ2 = f0 F15 − f3 F3 , Γ3 = f6 F1 − f5 F3 , Λ1 = f0 f5 − f12 , Λ2 =
fo f6 − f32 , L1 = Γ−
1,3 , L13 = −Γ1 , L15 = −Γ2 , L3 = Γ2,3 , l5 = 2Γ3 − Λ1 , l6 =
−2Γ3 −Λ2 , −l0 = 2Γ1,2 +Λ1,2 , l1 = Γ1 +Λ1 , l3 = Γ2 +Λ2 , [Ml ] = l0,5,6 +2l1,3 , [ML ] =
−
−
−
2L1,3,13,15 , 2[E] = 26L13,15 + 10L1,3 + 16l1,3,5,6, [J] = 6L−
13,15 + 4L1,3 + 2l5,6 + 2l1,3
(i) li : 0 = l0 + al1 + bl3 + cl5 + dl6 = Γ1 (a − 2) + Γ2 (b − 2) + 2Γ3 (c − d) +
Λ1 (a − c − 1) + Λ2 (b − d − 1) → a = b = 2, c = d = 1 → [Ml ], (ii) Lj :
0 = L1 + aL3 + bL13 + cL15 = Γ1 (1 − b) + Γ2 (a − c) + Γ3 (a − 1) → a = 1 =
b = c → [ML ]/2. (iii) li , i = 0, Lj : Γ3 = −L1,13 = L3,15 = (l5 + Λ1 )/2 =
−(l6 + Λ2 )/2, l1 + L13 = Λ1 , l3 + L15 = Λ2 → [I] = 2L1,3,13,15 = [ML ], [II] =
−l1,5 + 2L3,15 − L13 → 32[II] + 8[J] + 2[E] = 21[ML] [III] = 2L3 + 3L15 + l3,6 →
32[III] = −8[J] + 11[ML] + 2[E] [IV ] = l1,3,5,6 + L13,15 → 16[IV ] = 2E − 5[ML ]
Vol. 4, 2003
Half-Space Large Size Discrete Velocity Models
S903
Second, we add Fj , j = 9, 10, 11, 12 symmetric z = ±x to j = 1, 2, 3, 4. Finally, we
add F5 , F7 with Lemma 3a2 and with Lemma 2a (x, z symmetries), all |x| + |z|
odd are filled with heavy.
3.6b Physical models, figs 3-2c, including all Fj , fi of the plane (except x-axis)
The heavy F (3, 4), F (4, 7) 3.6a are included into a physical model and we can add
successively fi , i = 7, 11, 15, 17, 20
f0 F (4, 7) − f7 F (3, 4), f1 f7 − f5 f11 , f72 − f5 f15 , f4 f11 − f0 f17 , f4 f17 − f1 f20 .
Finally, we can add all the light densities with |x| +|z| even.
References
[1] H. Cornille, Wascom 2001 W.S.C.169; [T02/16-71]”Half-Space Large Size
DVMs for mixtures”; H. Cornille, C. Cercignani, JPA:Math.Gen. 34, 2985–
98 (2001), 11th-ECMI vol.1, 157–63 (2002)(Spring.Verl.)
[2] C. Cercignani, A.V. Bobylev, TTSP29, 109–16 (2000), including Cornille communication for a spurious 13vi , C. Cercignani, H. Cornille, J. Stat. Phys. 99,
15–40, 947–95 (2000) A.V. Bobylev, C. Cercignani, J. Stat. Phys. 97, 667–85
(1999); H. Cornille, C. Cercignani, World Scient. Sing. 119–27, (2001).
[3] Continuous, recent survey: Y. Sone, TTSP 29, 227–59 (2000) with 55 references.
DVMs: H. Cabannes, Eur. J. Mech. B/Fluids 13, 401–14 (1994); A. d’Almeida,
R. Gatignol, Comput. Fluid Dynamics (Springer-Verlag) 115–30 (1995), Eur.
J. Mech. B/Fluids 16, 401–28 (1997); I. Nicodin, thesis (2001), H. Cornille,
A. d’Almeida, JMP 37, 5470–95 (1996); Eur. J. Mech. B/Fluids 21–3, 335–70
(2002).
Henri Cornille
Service de Physique Théorique
CE Saclay
F-91191 Gif-sur-Yvette
France