Limit of Many Molecules Dynamics
with Rigorous Macroscopic Results
A Dissertation
presented to
the Faculty of the Graduate School
University of Missouri
In Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
by
NICHOLAS C. JACOB
Dr. Stamatis Dostoglou, Dissertation Advisor
July 2013
c
Copyright
by Nicholas Jacob 2013
All Rights Reserved
The undersigned, appointed by the Dean of the Graduate School, have examined the
dissertation entitled
Limit of Many Molecules Dynamics
with Rigorous Macroscopic Results
presented by
Nicholas C. Jacob
a candidate for the degree of Doctor of Philosophy and hereby certify that in their
opinion it is worthy of acceptance.
Professor Mark Ashbaugh
Professor Carmen Chicone
Professor Stamatis Dostoglou
Professor Bahram Mashhoon
ACKNOWLEDGEMENTS
First, I would like to express my gratitude to my dissertation advisor, Stamatis
Dostoglou, for his support throughout the dissertation process. Without him this
dissertation would not have been possible.
I would like to thank Professor Ashbaugh for his advice and patience during
preliminary presentations of some of the material in this thesis and my peer Jianfei
Xue who has always shared his ideas and corrections.
Finally, I would like to thank my family. With everyone’s support, I have had a
chance to fulfill my dream.
All material in this thesis is joint work with S. Dostoglou.
ii
Table of Contents
Acknowledgements
ii
Abstract
iv
1 Introduction
1
2 Molecule dynamics
7
2.1
2.2
Hamiltonian Equations . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.1.1
Equations of Motion . . . . . . . . . . . . . . . . . . . . . . .
7
2.1.2
Effective Radius . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.1.3
Scalings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.1.4
Solutions of the N -system. . . . . . . . . . . . . . . . . . . . .
11
2.1.5
Properties of the Flow . . . . . . . . . . . . . . . . . . . . . .
11
2.1.6
Standing Assumptions . . . . . . . . . . . . . . . . . . . . . .
12
Solutions with Bounds Uniform in N . . . . . . . . . . . . . . . . . . .
13
2.2.1
Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.2.2
The Systems of ODEs . . . . . . . . . . . . . . . . . . . . . .
14
2.2.3
Picard Iteration . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3 Hydrodynamic Equations
21
iii
Letting N → ∞. Previous Results Revisited . . . . . . . . . . . . . .
22
3.1.1
Density and Velocity Measures
. . . . . . . . . . . . . . . . .
22
3.1.2
Weak Convergence in General . . . . . . . . . . . . . . . . . .
23
3.1.3
Weak Limits of Density and Velocity Measures . . . . . . . . .
24
3.1.4
Comparison with Maxwell-Boltzmann Distributions . . . . . .
26
3.1.5
Hydrodynamic Equations . . . . . . . . . . . . . . . . . . . .
27
3.2
Subsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
3.3
Convergence of Second Moments
. . . . . . . . . . . . . . . . . . . .
30
3.4
The Interaction Term in the Momentum Equation . . . . . . . . . . .
32
3.4.1
Tightness of the Pairs Measure . . . . . . . . . . . . . . . . .
32
3.4.2
Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
3.4.3
Examples with Vanishing Limit . . . . . . . . . . . . . . . . .
35
Comparisons with Lanford-Ruelle . . . . . . . . . . . . . . . . . . . .
36
3.1
3.5
4 Energy Equation
38
4.1
Energy Equation for Fixed N . . . . . . . . . . . . . . . . . . . . . .
39
4.2
Converge of the Velocity Terms in the Energy Equation . . . . . . . .
41
4.3
The Interaction Term in the Energy Equation . . . . . . . . . . . . .
43
4.3.1
Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
4.3.2
Tightness of Triple Measures . . . . . . . . . . . . . . . . . . .
47
4.3.3
Other Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.4
. . . . . . . . . . . .
51
5 Disintegration, Conditional Expectation, and Reynolds Averaging
54
5.1
Comparisons with Irving & Kirkwood and Noll
Abstract Reynolds Averaging . . . . . . . . . . . . . . . . . . . . . .
iv
55
5.2
Barycentric Projection as Reynolds Averaging . . . . . . . . . . . . .
56
5.3
Conditional Expectations . . . . . . . . . . . . . . . . . . . . . . . . .
58
5.3.1
The σ-algebra from Reynolds Averaging . . . . . . . . . . . .
58
5.3.2
The σ-algebra from Disintegration . . . . . . . . . . . . . . . .
59
Reynolds Neighbourhood . . . . . . . . . . . . . . . . . . . . . . . . .
60
5.4
6 Landau-Lifshitz Formulas
63
6.1
Hermite Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
6.2
Grad’s Macroscopic Quantities . . . . . . . . . . . . . . . . . . . . . .
65
6.3
Landau-Lifshitz Formulas . . . . . . . . . . . . . . . . . . . . . . . .
69
7 Future Work
72
A The Case of No Heat
75
B Proof of Disintegration
77
Bibliography
79
VITA
83
v
Limit of Many Molecules Dynamics
with Rigorous Macroscopic Results
Nicholas C. Jacob
Stamatis Dostoglou, Dissertation Advisor
ABSTRACT
We obtain and study macroscopic equations as the limit of Hamiltonian equations
for N molecules, with pair interaction potentials, as N becomes arbitrarily large while
the total energy, mass, and moment of inertia stay constant. The construction stays in
physical space and avoids Gibbs measures on phase space. The Boltzmann equation
is not used.
The momentum equation, that was already obtained, is revisited and several assumptions for its validity are removed by constructing a large class of solutions of the
finite Hamiltonian systems with bounds uniform in N . We then obtain the energy
equation following the same methods and compare it with the corresponding equation
of Irving-Kirkwood and Noll.
Our method leads naturally to a rigorous definition of averaging that includes
the Maxwell-Boltzmann averaging as a special case. We argue that the Reynolds
averaging is also captured first by comparing our average with abstract Reynolds
averages and then by showing that a certain approximation of the quadratic velocity
term in the macroscopic equations suffices to obtain the Landau-Lifshitz formula for
thermal fluctuations.
iv
Chapter 1
Introduction
The student of mathematical hydrodynamics soon learns that starting from the incompressible Navier-Stokes equations, periodic or decaying, and with u(t, x) the “velocity of a fluid” at time t at the point x, the resulting energy equation reads
1d
2 dt
Z
Z
2
|u| (t, x)dx = −ν
|∇u|2 (t, x)dx,
(1.0.1)
for ν the viscosity (assumed constant). In other words, the total (kinetic) energy is
not conserved. This is usually explained as energy dissipating to “heat,” the nature
of this “heat” term seldom addressed.
O. Reynolds in [R], trying to provide a theoretical interpretation of his famous
experiment, addressed this:
Theorem 1.0.1. (O. Reynolds, [R], p. 137) The “velocity” u(t, x) on the NavierStokes system can never be the full velocity at x.
Reynolds’s proof is by contradiction. Heat is kinetic energy, too. Therefore if u
were the full velocity the total kinetic energy ought to be conserved.
Reynolds then builds on Maxwell’s definition of u as averaging in [Max], p.68.
Reynolds does not accept the right-hand side of (1.0.1) as necessarily only heat, and
develops (the now called) Reynolds averaging and Reynolds equations to investigate
1
the mechanism of energy dissipation. The lesson from Reynolds is: some of the righthand side of (1.0.1) can become heat, while some can become turbulent energy. One
needs a good decomposition of u, perhaps a second average (if one insists on using
Navier-Stokes), to tell these two apart.
This thesis is a continuation of the project of the thesis’s advisor to understand
with sufficient rigor Maxwell’s and Reynolds’s constructions and obtain hydrodynamic
equations from basic principles with as few, and as physically relevant, assumptions
as possible. The first results in this direction appeared in [D].
It is clear that to understand [R] one needs to understand [Max], i.e., molecular
kinetic theory, at the required level of rigor. Another conundrum awaits us there:
starting with Maxwell, kinetic theory is based on the existence of a distribution
function
f (t, x, v) dx dv,
(1.0.2)
the “number of molecules at x with velocity v” at time t. This is either defined
very heuristically, [Max], or merely postulated. Both Reynolds and Maxwell seem to
interchange discrete and continuous densities freely.
A rigorous definition of f was proposed in [D] and is followed throughout this
thesis. Starting with the Dirac measures of the positions and velocities of N molecules,
one lets N become arbitrarily large to obtain limit measures. This is the suggested
mathematical way of transiting from a discrete structure to a continuum. Using the
disintegration of the limit position-velocity measure with respect to the limit position
measure allows for an equally rigorous definition of u. Dirac measures for finitely
many molecules were used by Klimentovich in [K].
2
The conservation of mass and momentum equations for finite N , as deduced from
the Hamiltonian equations, result in macroscopic (hydrodynamic) equations when N
becomes arbitrarily large. These equations coincide with the corresponding equations of Irving & Kirkwood [IK], and Noll [N], (where N is finite but ensembles of
systems are averaged). This renders our definitions of macroscopic quantities even
more plausible.
We always use the weak form of equations, to be satisfied by any test function
integrated against the correct measures. We were first motivated for this by the work
of Jepsen and ter Haar in [JtH] where they do this for finite N and the complex
exponential as a test function, in effect the characteristic function of the Diracs. The
weak formulation techniques and the measure theoretic tools we use all come from
[AGS], and perhaps constitute our edge over pre-existing literature.
Perhaps our biggest debt is to the work of Morrey [M]. The classical (nonquantum) part of the Jepsen and ter Haar approach is there (and some 13 years
earlier) along with taking N arbitrarily large. We part ways with Morrey in at least
three aspects: he does not use much measure theory (probably because it was not
available), he abandons the single system in favor of an ensemble of systems, and he
is trying to obtain a specific type of limit measure (quasi-Gibbsian). Morrey does not
obtain limit measures, but rather limit density functions in phase space. His definitions of macroscopic quantities are standard, once limit density functions are shown
to exist. His resulting macroscopic equations, after all his assumptions, are of Euler
type, whereas we keep a full stress tensor. We use extensively his powerful analytic
results and refer to his work throughout this thesis. Perhaps the main difference is
3
that we want equations in the correct form, whereas Morrey tries to have the limit
measures in the correct form.
We now outline the thesis.
Chapter 2 first reviews standard facts on the Hamiltonian flow of the system of
N -molecules with general interaction potential. Morrey’s scaling of the potential is
explained in terms of an effective molecule radius and an alternative scaling is introduced and compared to Morrey’s. Standing assumptions on the average positions,
velocity, and energy, for all N are also introduced. We finally present a large class of
examples that satisfy uniform in N bounds for positions and velocities.
Chapter 3 first reviews the results of [D] on the rigorous definitions of macroscopic
density and velocity and the hydrodynamic equations they satisfy. It then improves
on them by providing conditions, satisfied at least by the class of section 2.2, that
imply the assumptions required of [D]. In particular, we show in section 3.2 the
existence of N -subsequences that satisfy the convergence in [D] for all t ∈ [0, T ].
Section 3.3 shows that the convergence of the non-linear term in the momentum
equation holds, provided we have uniform in N bounds. We give an example of
tightness of pair measures, needed for the convergence of the potential term in the
momentum equation for very regular Φ’s, in section 3.4.1. Section 3.4.2 examines the
convergence of the potential term in the momentum equation for more realistic Φ’s.
Recall that the main motivation for the current work is Reynolds’s ideas on the
distribution of energy into thermal and non-thermal components. Chapter 4 makes
the first steps in the direction of a rigorous energy equation that does not omit any
terms. As before, we present first the weak version of the energy equation for finite N
4
and then let N increase to ∞. Three cases are then investigated in detail. The first,
under assumptions similar to section 2.2, shows that limits exist for all terms in the
energy equation but does not identify limit measures. The second, under different
assumptions, shows tightness of measures but works only for regular interactions.
The third, under assumption of tightness, identifies all terms in [IK] and [N] and
is significant in that it considers the interaction as part of the measure and not of
the integrand. (This suggests a similar treatment for the interaction term in the
momentum equation.)
Having defined in section 3.1 the macroscopic average of any microscopic measurement F (xi , ui ) as
Z
F (t, x) =
F (x, v)Mt,x (dv)
(1.0.3)
R3
and given that disintegration is related to conditional expectation, chapter 5 examines the value of these averages as Reynolds averages. In particular, we check that
they satisfy the properties of abstract Reynolds average, see section 5.2. Specifically, the conditioning coming from disintegration ought to be compatible with the
conditioning coming from abstract Reynolds theory. This is addressed in section 5.3.
Finally, section 5.4 returns to our initial motivation, the identification of the Reynolds
neighborhood. Using Besicovich’s Theorem, this is now shown to be the limit over
ever shrinking neighborhoods. The average at x, as defined by the averaging formula
(1.0.3), is now understood as the limit of averages at neighboring points. This fits
Maxwell’s use of “immediate neighborhoods” in his heuristic definition of averages,
see [Max], p. 67. The last section also defines our thinking of (1.0.3): looking at the
microscopic world from afar, one cannot tell what is happening exactly at x (or if any5
thing is happening). We tend to attribute to x our impression of what is happening
somewhere around x.
Chapter 6 gives a proof of how an approximation of a non-equilibrium measure
can account for thermal fluctuations. By appealing to the Hermite polynomials and
expanding a density function about the equilibrium measure it can be shown that
this approximation abides by a Landau-Lifshitz type formula.
The final chapter presents future plans related to this thesis. Two appendices on
measure theoretic results important to this thesis follow.
Our approach throughout is classical and non-relativistic.
6
Chapter 2
Molecule dynamics
This chapter first reviews standard facts on the Hamiltonian flow of the system of
N -molecules with general interaction potential. Morrey’s scaling of the potential is
explained in terms of an effective molecule radius and an alternative scaling is introduced and compared to Morrey’s. Standing assumptions on the average positions,
velocity, and energy, for all N are also introduced. We finally present a large class of
examples that satisfy uniform in N bounds for positions and velocities.
2.1
2.1.1
Hamiltonian Equations
Equations of Motion
We start with the motion of N classical molecules, each of mass mN , without external
forces, and with pair interaction potential energy between a molecule at x and a
molecule at y
m2N ΦN (|x − y|).
(2.1.1)
Assuming always ΦN of negative derivative for small distances, where molecules repulse each other, the force on a molecule at x from a molecule at y is
−m2N ∇x ΦN (|x − y|).
7
(2.1.2)
Newton’s law then gives for the i-th molecule at time t when its position is xi (t)
mN u0i (t)
=
−m2N ∇x
N
X
ΦN (|xi (t) − xj (t)|) ,
(2.1.3)
j=1
j6=i
or, for the acceleration of the i-th molecule at time t,
u0i (t)
= −mN
N
X
∇x ΦN (|xi (t) − xj (t)|) .
(2.1.4)
j=1
j6=i
The total energy of the system consisting of these N molecules when their positions
and velocities are xi (t), ui (t), i = 1, . . . , N , is
N
N
X
X
1
1
EN = mN
|ui (t)|2 + m2N
ΦN (|xi (t) − xj (t)|) .
2
2
i=1
i,j=1
(2.1.5)
i6=j
Indeed, adding the definition of ui as an equation, we can rewrite (2.1.3) as a Hamiltonian system
dxi
= ∇ui EN ,
dt
dui
= −∇xi EN .
mN
dt
mN
(2.1.6)
EN is therefore conserved.
2.1.2
Effective Radius
An effective radius for each of the N molecules in the system can be defined following
[HCB], p. 51, as follows. Let a molecule move with relative velocity v with respect
to some other molecule and on a straight line which is at distance d from the center
of the second molecule. As soon as the molecules start interacting the velocity v and
the trajectory will change. The minimum distance σN between the two molecules
satisfies
1−
1 2
m Φ (σN )
2 N N
1
m v2
2 N
8
−
d
= 0.
σN
(2.1.7)
For a head-on collision d = 0 and the closest distance, a measure of the diameter of
a molecule, satisfies
mN 2 m2N
v =
ΦN (σN ).
2
2
(2.1.8)
Assuming v independent of N (the “mean thermal speed” in [ABGS]), for each N
there ought to be σN such that mN ΦN always gives the same value at σN
mN ΦN (σN ) = constant,
(2.1.9)
A simple way to accommodate this is to set
mN ΦN (r) = Φ
r
σN
.
(2.1.10)
This is what Morrey in [M] adopts.
2.1.3
Scalings
We are interested in a system of molecules of fixed total mass M and energy E, where
each molecule is a center of interaction force of potential Φ, rather an object of finite
dimension, an idea going back at least to Boscovich [Bos]. To achieve this, we take
the limit N → ∞ in the finite N -system described above and, instead of (2.1.10), we
can ask that the ΦN of the finite system is given by
ΦN (r) = Φ
r
τN
,
lim τN = 0.
N →∞
(2.1.11)
As mN = M/N , the comparison between Morrey’s rescaling and ours, for the same
Φ, is
Φ
r
σN
M
=
Φ
N
9
r
τN
.
(2.1.12)
Our construction and Morrey’s will differ in that we will never revert to Gibbsian
ensembles. Morrey does so, leading to different formulas for the change of variables.
An extra factor 1/N would make some of his terms disappear, but for us is necessary
if we are to have convergence of the corresponding terms. In other words, what we
consider as the main object is the interaction Φ in the final macroscopic equations.
We may choose what (rescaled) interactions show up in any approximation scheme,
here the N -molecules Hamiltonian system. Different approximations (non-Gibbsian
vs. Gibbsian) might require different rescaled interactions.
Example 2.1.1. For Φ(r) = 1/rp this gives
p
N σN
= τNp .
(2.1.13)
p
3
, therefore N σN
If σN is the radius of each molecule, mN ∝ σN
∼
τN =
1
N 1/3−1/p
.
1
N p/3−1
for
(2.1.14)
Therefore τN → 0 for p > 3 when σN does. The restriction p > 3, although not
important here, also is compatible with those in [M].
The case Φ(r) = 1/rp is an important one. The first macroscopic results from
molecular kinetic theory in [Max] were for Φ(r) = 1/r4 . Throughout this thesis we
use the case Φ(r) = 1/rp as a test case for our assumptions.
We will need the (2.1.11) rescaling only as a necessary condition for the convergence
of certain pairs measure, see section 3.4.1. Nevertheless, we use (2.1.11) for the rest
of this thesis and rename τN to σN from now on.
10
2.1.4
Solutions of the N -system.
Proposition 2.1.2. If Φ has locally Lipschitz-continuous derivative on (0, ∞) the
initial value problem of (2.1.6) on R3N \ {xi = xj , for some i, j, i 6= j} × R3N has
unique solution on any time interval [0, T ].
Proof. Let D = S × R3N , for S the connected component of R3N \ P where P = {xi =
xj , for some i, j, i 6= j} that contains the initial condition. Then the right-hand side
of the system is continuous on D. Then Theorem 1.3. of [CL] p. 7 applies and shows
that a solution exists.
The estimates (2.1.16) below show that the solution stays in a bounded part of
D for any finite time interval, whereas the conservation of energy shows that the
solution may not approach the singular set, P . Then the solution stays in a domain
where the right-hand side is bounded. Then Theorem 1.3 of [CL], p. 47 applies and
shows that the time interval can in fact be extended.
Finally, the local Lipschitz condition on the right-hand side implies uniqueness of
the solution where it exists, see Theorem 1.8.2 of [C], p. 110.
2.1.5
Properties of the Flow
Assume now that, in addition to the assumptions of Proposition 2.1.2, the potential
Φ satisfies
lim rα Φ(r) = +∞,
r→0+
α>3
lim rα Φ(r) = 0,
r→∞
(2.1.15)
lim rα+1 Φ0 (r) = 0
r→∞
lim sup
r→0+
|rΦ0 (r)|
= δ,
Φ(r)
11
δ > 0.
For example, Φ(r) = r−(α+1) satisfies all the assumptions. Notice that, in general,
only the behavior at 0 and ∞ is prescribed, in particular Φ need not be positive.
Then for positive constants LΦ and K, a solution of the N -system to time T
satisfies:
N
MX
KLΦ M
|ui (t)|2 ≤ 2E +
,
N i=1
N
t ∈ [0, T ]
N
N
2
√
p
MX
MX
2
C + t 2E + KLΦ M/N ,
|xi (0)| ≤ C ⇒
|xi (t)|2 ≤
N i=1
N i=1
t ∈ [0, T ].
(2.1.16)
This follows from corresponding statements in [M], p. 290–291, for our new rescaling.
The first of (2.1.16) is of course trivial for Φ positive. The second follows from the
first after differentiation.
2.1.6
Standing Assumptions
Following [M], we assume in this thesis the following for total mass, energy, and
moment of inertia: there are constants M , E, and C such that for all N ,
N mN = M,
N
MX
|xi |2 = C,
N i=1
1M
2N
N
X
i=1
|ui (t)|2 +
(2.1.17)
N
2 X
1M
2 N2
i,j=1
i6=j
Φ
|xi (t) − xj (t)|
σN
= E.
The assumptions clearly hold for t = 0 for bounded positions and velocities. The
bounds (2.1.16) and conservation of energy show that, once these hold for the initial
conditions then they hold on any finite time interval [0, T ] for constants depending
on T .
We construct in the next section solutions of (2.1.6) for specific initial conditions
12
and choices of σN with uniformly bounded velocities and positions, in addition to the
bounds (2.1.16) on momenta and moments of inertia.
2.2
Solutions with Bounds Uniform in N .
For each fixed N all solutions of the N -system stay bounded on finite time intervals,
as seen from (2.1.16). The same bounds also show that, given uniform bounds on
the energy and the initial conditions, the averages of the solutions stay uniformly
bounded in N .
We will now show that there exists a class of solutions obeying uniform in N
bounds on any finite interval.
2.2.1
Initial Conditions
Fix an infinite bounded sequence {xi (0), ui (0)}i∈N ,
|xi (0)| ≤ X,
|ui (0)| ≤ U,
for all i,
(2.2.1)
and consider for any two distinct i, j the increments of positions and velocities:
Xij := xi (0) − xj (0),
Uij := ui (0) − uj (0).
(2.2.2)
Then
inf |xi (0) − xj (0)| = 0.
i,j∈N
i6=j
(2.2.3)
We also choose the initial positions and velocities to be aligned as follows: For all i, j,
Xij · Uij = (xi (0) − xj (0)) · (ui (0) − uj (0)) ≥ 0.
(2.2.4)
For example, take ui (0) = λxi (0), λ ≥ 0. One might think of this case as “bursts.”
13
2.2.2
The Systems of ODEs
We are interested in uniform in N estimates for the system of ODEs
(N )
dxi
(N )
= ui ,
dt
N
(N )
1 X 1 0
dui
=−
Φ
dt
N j=1 σN
(N )
(N )
|xi (t) − xj (t)|
σN
!
(N )
(N )
(N )
(N )
xi (t) − xj (t)
|xi (t) − xj (t)|
,
1 ≤ i ≤ N.
j6=i
(2.2.5)
In this we have to choose σN → 0. If we are given a sequence {BN }N ∈N we will also
assume that we can choose the σN ’s to satisfy
1
− Φ0
σN
|xi (0) − xj (0)|
σN
≤ BN < B,
(2.2.6)
for and for all i, j up to N , for each N .
Example 2.2.1. For −Φ0 (r) = r−p , p > 3, given any bounded {BN }N ∈N set
1/(p−1)
σN = BN
min |Xij |p/(p−1) ,
1≤i,j≤N
i6=j
(2.2.7)
for example. And (2.2.3) implies, perhaps after relabeling, that σN → 0.
A specific such sequence of BN ’s will be chosen below. It will depend only on
{xi (0), ui (0)}i∈N and T .
We assume for the rest of this section the interaction potential derivative Φ0 (r) to
be eventually decreasing in absolute value.
14
2.2.3
Picard Iteration
For each fixed N we examine the Picard scheme of (2.2.5) for α ∈ N:
Z t
(N,α−1)
(N,α)
ui
(s)ds,
xi
(t) = xi (0) +
0
(N,α)
ui
(t) = ui (0)
N Z
1 X t 1 0
−
Φ
N j=1 0 σN
(N,α−1)
|xi
(N,α−1)
(s) − xj
σN
(s)|
!
(N,α−1)
xi
(N,α−1)
(s) − xj
(N,α−1)
|xi
(s)
−
(s)
(N,α−1)
xj
(s)|
ds,
j6=i
(2.2.8)
starting with
(N,0)
xi
(N,0)
(t) = xi (0),
ui
(t) = ui (0).
(2.2.9)
To obtain uniform bounds we show that they are satisfied by the Picard approximation first. We then show that under our assumptions the Picard scheme converges
uniformly to the unique (by Theorem 2.1.2) solution of (2.2.5).
In general, the unique solution on an arbitrary time interval is not necessarily the
limit of Picard approximations. Indeed, the abstract result on Picard approximations
would only give N -dependent neighborhoods where this holds, see [CL], p. 53. Our
proof that, under our assumptions, this convergence holds on any [0, T ] is a retracing
of the standard proof of the convergence on sufficiently small intervals in [CL], p. 12.
Lemma 2.2.2. Fix T > 0. For BN such that
2 3
0 ≤ (Xij + tUij ) · Uij − (|Xij | + T |Uij |) 2T BN − BN T 2 |Uij | − 2BN
T ,
(2.2.10)
and σN from (2.2.6) for such BN . Then the Picard scheme is well defined for all
t ∈ [0, T ], for all α and satisfy
2
d (N,α)
(N,α)
(t) − xj
(t) ≥ 0,
xi
dt
15
t ∈ [0, T ].
(2.2.11)
Proof. The iterate for α = 1 is well defined since it involves only the given initial
conditions. By (2.2.4), it also satifies
2 1 d (N,1)
(N,1)
(N,1)
(N,1)
xi (t) − xj (t) = xi (t) − xj (t) · (ui (0) − uj (0))
2 dt
= (xi (0) − xj (0) + {ui (0) − uj (0)}t) (ui (0) − uj (0))
= (xi (0) − xj (0)) · (ui (0) − uj (0)) + |ui (0) − uj (0)|2 t ≥ 0.
(2.2.12)
Assume now that the α iterate exists up to time T and that it satisfies
2
d (N,α)
(N,α)
(t)
(t)
−
x
x
≥ 0,
i
j
dt
t ∈ [0, T ].
(2.2.13)
This implies that the terms of the α + 1 iterate involving Φ0 decrease. In particular,
these terms stay bounded by their values at t = 0. Therefore the α + 1 iterate exists.
In addition, using the notation
1
Fij (s) := − Φ0
σN
|xi (s) − xj (s)|
σN
xi (s) − xj (s)
,
|xi (s) − xj (s)|
(2.2.14)
2
1 d (N,α+1)
(N,α+1)
x
(t)
−
x
(t)
j
2 dt i
=
xi (0) − xj (0) + (ui (0) − uj (0))t
·
!
Z Z
Z Z
1 X t s (N,α−2)
1 X t s (N,α−2)
F
(s1 )ds1 ds +
F
(s1 )ds1 ds
−
N k,k6=i 0 0 ik
N l,l6=j 0 0 jl
!
Z
Z
1 X t (N,α−2)
1 X t (N,α−2)
ui (0) − uj (0) −
F
(s)ds +
F
(s)ds
N k,k6=i 0 ik
N l,l6=j 0 jl
(2.2.15)
16
= (Xij + tUij ) · Uij
!
Z
Z
1 X t (N,α−2)
1 X t (N,α−2)
− (Xij + tUij ) ·
F
(s)ds −
F
(s)ds
N k,k6=i 0 ik
N l,l6=j 0 jl
!
Z Z
Z Z
1 X t s (N,α−2)
1 X t s (N,α−2)
F
(s1 )ds1 ds −
F
(s1 )ds1 ds
− Uij ·
N k,k6=i 0 0 ik
N l,l6=j 0 0 jl
!
Z Z
Z Z
1 X t s (N,α−2)
1 X t s (N,α−2)
+
F
F
(s1 )ds1 ds −
(s1 )ds1 ds
N k,k6=i 0 0 ik
N l,l6=j 0 0 jl
!
Z
Z
1 X t (N,α−2)
1 X t (N,α−2)
F
(s)ds −
F
(s)ds .
·
N k,k6=i 0 ik
N l,l6=j 0 jl
(2.2.16)
(N,α−2)
By hypothesis, |Fik
(N,α−2)
|(s) and |Fjl
|(s) are decreasing in s since the distances
are increasing. Therefore
2
1 d (N,α+1)
(N,α+1)
(t)
(t) − xj
xi
2 dt
2 3
≥ (Xij + tUij ) · Uij − (|Xij | + T |Uij |) 2T BN − BN T 2 |Uij | − 2BN
T ≥ 0,
(2.2.17)
by assumption.
For Lip(f, B) the Lipschitz constant of a function f on the set B, let
LN := Lip
1 0
Φ
σN
|r|
σN
r
,
|r|
min |Xi,j (0)|, ∞
< ∞.
1≤i6=j≤N
Lemma 2.2.3. Under the above assumptions on Φ and the initial conditions,
(2.2.18)
converges uniformly on [0, T ] as α → ∞.
Proof. In order to show uniform convergence, use the Weierstrass M-test. For this
examine the differences between α:
Z t
(1)
xi (t) − xi (0) = ui (0)dt ≤ U t ≤ U T,
0
Z t
(1)
(0) ui (t) − ui (0) = Fi,j dt ≤ BN t ≤ BN T,
0
17
(α)
(α)
xi (t), ui (t)
(2.2.19)
T2
(2)
(1)
xi (t) − xi (t) ≤ BN ,
2
T2
(2)
(1)
ui (t) − ui (t) ≤ 2LN U .
2
(2.2.20)
Then the general formula is
Tα
(α)
(α−1)
(t) ≤ max(U, BN ) (2LN )α−3
,
xi (t) − xi
α!
Tα
(α)
(α−1)
.
(t) ≤ max(U, BN ) (2LN )α−2
ui (t) − ui
α!
(2.2.21)
To see this is true assume that it works for α ≤ k, then for k + 1
Z t
(k)
(k+1)
(k−1)
(k)
(t) dt
(t) − xi (t) ≤
ui (t) − ui
xi
0
T k+1
≤ max(U, BN )(2LN )k−2
,
(k + 1)!
Z t
(k+1)
(k)
(k)
(k)
(k−1) (k−1)
(t) − ui (t) ≤
(t)
ui
Fi,j − Fi,j dt ≤ 2LN xi (t) − xi
(2.2.22)
0
≤ max(U, BN ) (2LN )k−1
Since
∞
X
(2LN T )n
n
n!
T k+1
.
(k + 1)!
is convergent the lemma is proven.
Lemma 2.2.4. The limits of the Picards for each N satisfy
(N )
|xi (t)| ≤ X + U T + BN T 2 ,
(N )
|ui (t)| ≤ U + BN T,
(2.2.23)
N
|xN
i (t) − xj (t)| ↑ in t.
Proof. Use uniform convergence.
Lemma 2.2.5. The limit as α → ∞ of the Picards is a solution of the N -system.
(N )
(N )
Proof. Call the limit xi (t), ui (t), t ∈ [0, T ]. Then taking the limit as α → ∞ of
18
(2.2.8) gives
(N )
xi (t)
(N )
ui (t)
t
Z
= xi (0) + lim
α
(N,α−1)
ui
(s)ds
Z
t
(N )
ui (s)ds,
= xi (0) +
0
0
= ui (0)
(N,α−1)
N Z
1 X t 0
− lim
Φ
α N σN
0
j=1
|xi
(N,α−1)
(s) − xj
σN
(s)|
!
(N,α−1)
xi
(N,α−1)
(s) − xj
(N,α−1)
|xi
(s)
−
(s)
(N,α−1)
xj
(s)|
ds.
j6=i
(2.2.24)
Now
N Z t
1 X
1 0
Φ
N
j=1 0 σN
j6=i
(N,α−1)
|xi
(N,α−1)
(s) − xj
σN
N Z
1 X t 1 0
−
Φ
N j=1 0 σN
(s)|
!
(N,α−1)
(s) − xj
(N,α−1)
(s) − xj
xi
|xi
(N )
(N )
|xi (s) − xj (s)|
σN
!
j6=i
N Z
1 X t 1 0
≤
Φ
N j=1 0 σN
(N,α−1)
|xi
(N,α−1)
(s) − xj
σN
(s)|
!
(N,α−1)
(s)
(N,α−1)
(s)|
ds
(N )
(N )
xi (s) − xj (s) ds
(N )
(N )
|xi (s) − xj (s)| (N,α−1)
(s)
(N,α−1)
(s)|
(N,α−1)
(s) − xj
(N,α−1)
(s) − xj
xi
|xi
j6=i
1
− Φ0
σN
Z
≤ LN
t
(N,α−1)
|xi
(N )
(N )
|xi (s) − xj (s)|
σN
(N )
!
(N )
(N )
xi (s) − xj (s) ds
(N )
(N )
|xi (s) − xj (s)| (N,α−1)
(s) − xi (s)| + |xj
(N )
(s) − xj (s)| ds → 0.
0
We have therefore shown:
Theorem 2.2.6. Assume Φ0 decreasing on (0, ∞) and bounded and aligned initial
conditions. Then, provided that there is sequence σN → 0 satisfying
1
− Φ0
σN
|xi (0) − xj (0)|
σN
≤ BN < B,
(2.2.25)
for BN such that
2 3
0 ≤ (Xij + tUij ) · Uij − (|Xij | + T |Uij |) 2T BN − BN T 2 |Uij | − 4BN
T ,
19
(2.2.26)
the solutions of the corresponding Hamiltonian systems satisfy the following uniform
bounds:
(N )
|xi (t)| ≤ X + U T + BN T 2 ,
(N )
|ui (t)| ≤ U + BN T,
t ∈ [0, T ],
t ∈ [0, T ],
(2.2.27)
N
|xN
i (t) − xj (t)| ↑ w.r.t. t ∈ [0, T ],
d
ui (t) ≤ BN , t ∈ [0, T ].
dt
Corollary 2.2.7. For Maxwellian Φ0 (r) = r−p , 3 < p and bounded, aligned initial
conditions there are σN ’s such that the estimates (2.2.27) hold for all N .
20
Chapter 3
Hydrodynamic Equations
We shall first review the results of [D] on the rigorous definitions of macroscopic
density and velocity and the hydrodynamic equations they satisfy. We then improve
them by providing conditions, satisfied at least by the class of section 2.2, that imply
the assumptions required of [D].
In particular, we show in section 3.2 the existence of N -subsequences that satisfy
the convergence in [D] for all t ∈ [0, T ]. Section 3.3 shows that the convergence
of the non-linear term in the momentum equation holds, provided we have uniform
in N bounds. We give an example of tightness of pair measures, needed for the
convergence of the potential term in the momentum equation for very regular Φ’s in
section 3.4.1. Section 3.4.2 examines the convergence of the potential term in the
momentum equation for more realistic Φ’s. (This suggests a similar treatment for the
interaction term in the momentum equation.)
Some comparisons to the work of Lanford and Ruelle on configurations of infinitely
many molecules complete the chapter.
21
3.1
Letting N → ∞. Previous Results Revisited
We now summarize the results of [D]. At the same time, we recall some definitions
and standard results from measure theory. All measures here are probability measures
unless we state otherwise.
3.1.1
Density and Velocity Measures
(N )
(N )
Assume all of (2.1.17) and xi (t), ui (t) solutions of the N -system. We set M = 1
for convenience.
Recall that the Dirac measure δx0 (dx) satisfies
δx0 (B) =
1 if x0 ∈ B
0 if x0 ∈
/ B,
(3.1.1)
for any B Borel subset of RN .
Define for each N the (mass or number density) measures
(N )
µt (dx) =
N
1 X
δx (t) (dx).
N i=1 i
(3.1.2)
Define also the time dependent vector field
uN (t, x) =
x0i (t) if x = xi (t)
0
o/w,
(3.1.3)
Recall that for any measure ν on a space X and measurable map f : X → Y the
push-forward measure f# ν (the “distribution” of f with respect to ν) is defined on
Y by
f# ν(A) = ν(f −1 (A)),
A ⊂ Y.
(3.1.4)
We can then define the (molecule-and-velocity density) measures
(N )
Mt
(N )
(dx, dv) := (Id × uN )# µt (dx, dv).
22
(3.1.5)
3.1.2
Weak Convergence in General
A good source of the material here is chapter 5 of [AGS]. Recall that a sequence of
measures νi weakly converges to ν, written as
νi ⇒ ν,
i → ∞,
(3.1.6)
if
Z
Z
f (x)νi (dx) →
f (x)ν(dx),
i → ∞,
(3.1.7)
for any f continuous and bounded.
A family of measures is called tight if it has weakly convergence subsequence.
Prohorov’s theorem provides a criterion for a family {νi } on X to be tight: For any
there must exist a compact K with ν(X \ K ) < , i.e. tails must have uniformly
small measure.
A positive f is uniformly integrable with respect to a family of measures νi if
Z
f (x)νi (dx) → 0, uniformly in i.
lim
R→∞
(3.1.8)
{f (x)>R}
The following is standard:
Lemma 3.1.1. For any sequence of positive measures µN
Z
sup
p+
|x|
Z
µN (dx) < ∞ ⇒ lim
R→∞
{|x|p >R}
N
|x|p µN (dx) → 0, uniformly in N . (3.1.9)
Proof.
R
Z
{R<|x|}
p
Z
|x| µN (dx) ≤
p+
|x|
{R<|x|}
23
Z
µN (dx) ≤
|x|p+ µN (dx).
Corollary 3.1.2. For any sequence of probability measures µN on Rn and for any
α>0
Z
sup
|x|α µN (dx) < ∞ ⇒ {µN }N ∈N is tight.
(3.1.10)
N
Proof.
α
R µN ({R < |x|}) = R
α
Z
Z
1µN (dx) ≤
{R<|x|}
α
Z
|x| µN (dx) ≤
|x|α µN (dx).
{R<|x|}
(3.1.11)
As balls are compact in Rn the Prohorov theorem implies tightness.
Theorem 3.1.3. The integrals of any continuous function uniformly integrable with
respect to a weakly convergent sequence of measures converge, without passing to a
subsequence.
Proof. Included in Lemma 5.1.7 of [AGS].
3.1.3
Weak Limits of Density and Velocity Measures
Then, as an application of Corollary 3.1.2 for each t, and beacause of (2.1.16), we
have that for each t, up to subsequence,
(N )
µt (dx) ⇒ µt (dx),
(3.1.12)
(N )
Mt (dx, dv)
⇒ Mt (dx, dv).
Recall now the formula for the disintegration of a measure M on X × Y with
respect to (pr1 )# M = µ:
Z
Z
Z
h(x, y)M (dx, dy) =
X×Y
h(x, y)Mx (dy) µ(dx)
X
(3.1.13)
Y
for any integrable h. The probability measure Mx on X × Y is supported entirely on
the fibre pr1−1 ({x}). See [AGS], p. 121 and their references. We provide a proof of
the disintegration theorem in Appendix B.
24
Recall also the definition of barycentric projection with respect to the disintegration µx
Z
y Mx (dy),
y(x) =
(3.1.14)
Y
the fibre-wise average of y, see [AGS], p. 126.
As an application of Theorem 5.4.4., p. 127 of [AGS] for each fixed t and for
Mt,x (dv) the disintegration of Mt (dx, dv) with respect to µt (dx), the random variable
Z
u(t, x) =
vMt,x (dv)
(3.1.15)
on (R3 , B, µt ) is square integrable with respect to µt of (3.1.12) and satisfies for any
test function φ ∈ C0∞ (R3 )
Z
(N )
φ(x)uN (t, x)µt (dx)
Z
→
φ(x)u(t, x)µt (dx).
(3.1.16)
The following simple 1-dimensional example from [D] clarifies what the limits
might look like:
Example 3.1.4. For N = 10n , let
xi =
i
, uN (x) = sin(2πnx), i = 1, . . . , N.
N
(3.1.17)
Then
1
µ = dx on [0, 1], M = dx ⊗ √
dv,
1 − v2
(3.1.18)
cf. [AFP], p. 61, and u ≡ 0:
Z
lim
N
φ(x)uN (x)µN (dx) → 0
(3.1.19)
by Riemann-Lebesque. M is clearly not supported on the graph of u. Notice that the
uN ’s are uniformly bounded.
25
3.1.4
Comparison with Maxwell-Boltzmann Distributions
The significance of u(t, .) is above all that, as the barycentric projection of Mt (dx, dv)
with respect to µt (dx), agrees with the definition of macroscopic velocity of Maxwell
in [Max] and Boltzmann in [B]:
If the final measure were absolutely continuous, i.e. Mt (dx, dv) = ft (x, v)dxdv,
then its first marginal would be
Z
ft (x, v)dv dx.
(3.1.20)
Disintegration is looking for a measure Mt,x (dv) for each x such that
Z
ZZ
h(x, v)Mt,x (dv)
ZZ
ft (x, v)dv dx =
h(x, v)ft (x, v)dxdv.
(3.1.21)
Since trivially
ZZ
h(x, v) Z
ft (x, v)dv
Z
ZZ
ft (x, v)dv dx =
h(x, v)ft (x, v)dxdv,
ft (x, v)dv
(3.1.22)
uniqueness (almost everywhere) of the disintegration gives that the correct measure
on almost all x-fibres is
Mt,x (dv) = Z
ft (x, v)dv
,
ft (x, v)dv
(3.1.23)
and the barycentric projection u(t, x) is
Z
vft (x, v)dv
.
u(t, x) = Z
(3.1.24)
ft (x, v)dv
The formulas (3.1.23), (3.1.24) are precisely the Maxwell-Boltzmann formulas for
probability density and velocity.
26
3.1.5
Hydrodynamic Equations
In addition, u satisfies hydrodynamic equations along with µt (dx) as density. Indeed,
assuming that there is a subsequence for which (3.1.12) holds for all t, the weak
version of the continuity equation holds for µ and u:
T
Z
Z
0
R3
∂
φ(t, x) + ∇x φ(t, x) · u(t, x) µt (dx) dt = 0.
∂t
(3.1.25)
This is a consequence of the fact that the Hamiltonian equations and the elementary
chain rule for derivatives give the continuity equation for each N :
Z
d
1 d X
(N )
φ(x)µt (dx) =
φ(xi (t))
dt
N dt i
1 X
=
∇φ(xi (t)) · ui (t)
N i
Z
(N )
= ∇φ(x) · uN (t, x) µt (dx).
(3.1.26)
The rest follows from a density argument.
In section 3.2 we show we can choose a subsequences for which (3.1.12) holds for
all t.
(N )
Now let νt
(dx, dξ) be the push-forward via the map
x − x0
(x, x ) 7→ x,
σN
0
(3.1.27)
of the probability distribution on R6 of pairs of molecules,
(N )
nt (dx, dx0 ) =
X
δ(xi ,xj ) .
(3.1.28)
t ∈ [0, T ].
(3.1.29)
i6=j
Assumption 3.1.5. The pair measures
1
(N )
νt ⇒ νt ,
N (N − 1)
are tight.
27
We give an example satisfying tightness of pair measures in section 3.4.1.
Assumption 3.1.6. Assume local convergence of moments
Z
(N )
µt (dx)
∇φ(x) · uN (t, x) uN (t, x)
Z
→
∇φ(x) · v v Mt (dx, dv).
(3.1.30)
In section 3.3 we show that local convergence of moments holds for uniformly bounded
velocities. Therefore it holds for the class of section 2.2.
Assumption 3.1.7. Assume further finite range and regularity of pair interactions:
Φ0 (s) = sG(s),
Φ ∈ C01 ([0, ∞)) ,
G ∈ C ([0, ∞)) .
(3.1.31)
We remove this assumption on pair interactions in the following sections and present
results for interactions at least for interactions of the form Φ(r) = r−p .
Under the above assumptions the momentum equation for µt and u holds:
Z TZ
∂
φ(t, x)u(t, x) µt (dx)dt
0
R3 ∂t
Z TZ
∇ φ(t, x) · u(t, x) u(t, x) µt (dx) dt
+
0
R3
(3.1.32)
Z
Z TZ
∇φ(t, x) · (v − u(t, x))(v − u(t, x))Mt,x (dv)µt (dx)dt
=−
0
R3
Z Z
1 T
ξ
+
∇φ(t, x) · ξ Φ0 (|ξ|)
νt (dx, dξ)dt.
2 0 R6
|ξ|
3.2
Subsequences
Proposition 3.2.1. Assume uniformly in N bounded positions and velocities:
(N ) (N ) x
(t)
+
u
(t)
i
i
≤ CT ,
t ∈ [0, T ],
i = 1, . . . , N,
N ∈ N.
(3.2.1)
Then there is a subsequence Ni of N ’s, independent of t, such that for all t in [0, T ],
i
µN
t (dx) ⇒ µt (dx).
28
(3.2.2)
Proof. Examine first the characteristic functions of the µN ’s, cf. [M], p. 291:
Z
χN (t, y) :=
ix·y
e
N
1 X ixj (t)·y
µN (dx) =
e
,
N j=1
N
1 X ixj (t)·y
∂t χN (t, y) =
ie
y · uj (t),
N j=1
(3.2.3)
N
1 X ixj (t)·y
ie
xj (t).
∇y χN (t, y) =
N j=1
Then uniformly in N
|χN (t, y)| ≤ 1,
|∂t χN (t, y)| ≤ C|y|,
(3.2.4)
|∇y χN (t, y)| ≤ C.
In particular, for each fixed T and k ∈ N, there is uniformly convergent subsequence
of χN (t, y) on [0, T ] × Bk (0). Therefore, by taking k → ∞ and diagonalizing, there
is subsequence χNi which converges for all t and y in [0, T ] × R3 (and which still
converges uniformly on any [0, T ] × compact). The limit is, of course, continuous in
t and y as the uniform limit of continuous functions.
Apply now the Levy continuity theorem for any fixed t on this subsequence to find
i
that, without resorting to any further subsequence, µN
t (dx) converges weakly. Fix
this subsequence Ni , rename it to N , and work with it from now on.
Proposition 3.2.2. In addition to uniformly in N bounded positions and velocities,
assume uniformly bounded in N accelerations:
d (N ) u (t) ≤ CT ,
dt i
t ∈ [0, T ],
i = 1, . . . , N,
N ∈ N.
(3.2.5)
Then there is a subsequence Ni of N ’s, independent of t, such that for all t in [0, T ],
MtNi (dx, dv) ⇒ Mt (dx, dv).
29
(3.2.6)
Proof. The characteristic of (Id × uN )# µN , for µN as already chosen, is
Z
ψN (t, y, w) = ei(yx + wv) (Id × uN )# µN (dx, dv)
Z
= ei(yx + wuN (x)) µN (dx)
=
(3.2.7)
1 X i ( yxi (t) + wu(N ) (t) )
i
e
,
N
Then
1 X i ( yxi (t) + wu(N ) (t) )
d (N )
i
∂t ψN (t, y, w) =
e
yui (t) + w
u
(t) ,
N
dt i
1 X i ( yxi (t) + wu(N ) (t) )
(3.2.8)
i
∇y ψN (t, y, w) =
e
xi (t),
N
1 X i ( yxi (t) + wu(N ) (t) ) (N )
i
ui (t).
∇w ψN (t, y, w) =
e
N
Repeat the last part of the proof of the previous Lemma to see that again some
subsequence converges for all t.
Corollary 3.2.3. For the class of examples of section 2.2 there is subsequence of N ’s
independent of t such that for all t
(N )
µt (dx) ⇒ µt (dx),
(N )
(Id × uN )# µt (dx, dv) ⇒ Mt (dx, dv),
Z
Z
(N )
φ(x)uN (t, x)µt (dx) → φ(x)u(t, x)µt (dx).
(3.2.9)
Proof. The weak limits follow immediately from the lemmas above and the bounds
(2.2.23). The third limit is a result of uniform integrability of the second power of u,
see the first of (2.1.16), which, given the convergence of measures involves no further
subsequence, see Theorem 3.1.3.
3.3
Convergence of Second Moments
The discussion here on second moments in the momentum equation parallels the
discussion on third moments in the energy equation, see section 4.2.
30
Theorem 3.3.1. Assume uniformly in N bounded positions, velocities and accelerations. Then there is subsequence of N ’s independent of t such that for all t
(N )
µt (dx) ⇒ µt (dx),
(N )
(Id × uN )# µt (dx, dv) ⇒ Mt (dx, dv),
Z
Z
(N )
φ(x)uN (t, x)µt (dx) → φ(x)u(t, x)µt (dx),
Z
Z
(N )
∇φ(x) · uN (t, x) uN (t, x) µt (dx) → ∇φ(x) · v v Mt (dx, dv).
(3.3.1)
Proof. The first three limits follow from the previous section. For the last limit,
observe that uniformly bounded velocities imply trivially uniformly integrable movements of any order by Lemma 3.1.1. The result then follows from Theorem 3.1.3
(even without a test function).
A few remarks on the meaning of the convergence of the second moments are in
order. First, recall that in general
Z
Z
2
|u| (x)µt (dx) ≤ lim inf
N
(N )
|uN |2 (x)µt (dx)
(3.3.2)
|u|2 (x)µt (dx).
(3.3.3)
and that, in general,
Z
2
|uN |
(N )
(x)µt (dx)
Z
9
In fact, convergence is equivalent to having the limit measure Mt concentrated on the
graph of u(t, .) : R3 → R3 : if
Z
lim
N →∞
p
|uN | (x)µ
(N )
Z
(dx) =
|u|p (x)µ(dx) for p > 1,
(3.3.4)
then M is supported by the graph of u(x). For a proof of this see in [AGS], p. 128,
or Appendix A here.
31
Note that the left-hand side of (3.3.3) is the total kinetic energy (as well as a 2nd
moment) for finitely many particles
Z
(N )
v 2 (Id × uN )# µt (dx, dv),
(3.3.5)
whereas on the right-hand side of (3.3.3) only the kinetic energy of the average motion
u(t, .) shows up. In other words, at the limit of infinitely many particles there is more
kinetic energy than the the kinetic energy of the mean motion. Note that it is not at
all clear that this excess energy is just “heat.”
We think of the limit distribution of v over x as a generalized limit of uN ’s on
points arbitrarily close to x, see section 5.4. It is then reasonable to expect that
Z
2
|uN |
(N )
(x)µt (dx)
Z
=
2
|v| (x)(Id ×
(N )
uN )# µt (dx)
Z
→
|v|2 (x)Mt (dx).
(3.3.6)
This, for the given assumptions, is exactly what the above theorem shows (see last
remark in its proof).
On the other hand, we think of convergence in (3.3.3) as all energy being mean
flow energy without anything remaining at the macroscopic limit, not even “heat.”
3.4
3.4.1
The Interaction Term in the Momentum Equation
Tightness of the Pairs Measure
To show that
1
(N )
ν
N (N − 1) t
32
(3.4.1)
can be tight in N at least at some time t, according to Corollary 3.1.2 it suffices to
show that any α-moment, α > 0 is bounded, in particular it suffices to show:
1
N (N − 1)
Z
(N )
(|x| + |ξ|)νt
(dx, dξ) < C.
(3.4.2)
Assuming that all (single molecule) positions are bounded at the given point in time,
this will follow if
1
N (N − 1)
N
Z
(N )
|ξ|νt
(dx, dξ) =
N
X X |xi − xj |
1
< C.
N (N − 1) i=1 j=1
σN
(3.4.3)
j6=i
To provide an example where this holds, take the molecules to be at (1/i, 0, 0), i ∈ N.
Using the notation
PN
j=1
j −1 = HN , calculate
N
N
N
N
1 X X |xi − xj |
1 1 1 XX
=
|xi − xj |
N 2 i=1 j=1
σN
N σN N i=1 j=1
j6=i
j6=i
X
!
N
N i−1 1 1 1 X X 1 1
1 1
−
−
=
+
N σN N i=1 j=1 j
i
i
j
j=i+1
N 1 1 1 X
i−1 N −i
+
− HN + Hi
=
Hi−1 −
N σN N i=1
i
i
N 1 1 1 X
1 1
N
≤
−1 ≤2
HN .
HN +
N σN N i=1
i
N σN
(3.4.4)
Now choose σN so that for 0 < γ < 1
1 1
< C,
N γ σN
(3.4.5)
(for example, the effective radius choice γ = 1/3 will do), and notice that since HN
grows logarithmically,
1
N 1−γ
HN → 0,
33
N → ∞.
(3.4.6)
3.4.2
Convergence
The potential term from the momentum equation for N particles is
Z
T
0
N
|xi (t) − xj (t)| xi (t) − xj (t)
1 X
0
ϕ(xi , t)Φ
dt.
N 2 σN i,j=1
σN
|xi (t) − xj (t)|
(3.4.7)
i6=j
First note that it is clear that the integrand in (3.4.7) is uniformly bounded under
the assumptions from Theorem 2.2.6. To show uniform convergence of this term,
equi-continuity will be used. Recall the notation
Xi,j (t) = xi (t) − xj (t)
(3.4.8)
Ui,j (t) = ui (t) − uj (t).
Then


N
X

d 
Xi,j (t)
|Xi,j (t)|
0
 1

ϕ(x
,
t)Φ
i
dt  N 2 σN i,j=1
σN
|xi (t) − xj (t)| 
i6=j
N 1 X ∂ϕ
|Xi,j (t)| Xi,j (t)
0
(xi , t) + ∇ϕ(xi , t) · ui (t) Φ
= 2
N σN i,j=1 ∂t
σN
|Xi,j (t)|
i6=j
N
1 X
|Xi,j (t)| Xi,j (t)
Xi,j (t)
00
+ 2 2
ϕ(xi , t)Φ
· Ui,j (t)
N σN i,j=1
σN
|Xi,j (t)|
|Xi,j (t)|
i6=j
N
1 X
|Xi,j (t)|
Ui,j (t)
Xi,j (t) · Ui,j (t)Xi,j (t)
0
+ 2
ϕ(xi , t)Φ
−
.
N σN i,j=1
σN
|Xi,j (t)|
|Xi,j (t)|3
i6=j
(3.4.9)
Then the first line is uniformly bounded by assumption. The second line is bounded
by Theorem 2.2.6 and Lemma 4.3.1. The final line is bounded by assumption and
Lemma 4.3.2.
Thus the desired convergence has been achieved.
34
3.4.3
Examples with Vanishing Limit
For the case −Φ0 (r) = r−p and a more careful choice of σN , still compatible with
section 2.2, we can show that the potential term vanishes at the limit:
Theorem 3.4.1. Assume that
min |xi (0) − xj (0)|(p+2)/(p−1) where Φ0 (r) = r−p .
σN = C
1 0
Then
Φ
σN
(3.4.10)
1≤i6=j≤N
|Xi,j (0)|
σN
≤ BN still holds under “burst” conditions and
N
|xi (t) − xj (t)| xi (t) − xj (t)
1 X
0
ϕ(xi , t)Φ
→ 0.
N 2 σN i,j=1
σN
|xi (t) − xj (t)|
(3.4.11)
i6=j
1
Proof. For (3.4.11) to hold it is only necessary to show that − 2 Φ0
σN
|Xi,j (0)|
σN
≤B
since all other terms are bounded and the introduction of the extra σN would force
the limit to zero.
1
− 2 Φ0
σN
|Xi,j (0)|
σN
p−2
σN
=
|Xi,j (0)|p
≤ |Xi,j (0)|(p
2 −4)/(p−1)−p
(3.4.12)
= |Xi,j (0)|(p−4)/(p−1) .
This quantity is bounded so long as p ≥ 4. For the first claim
1 0
Φ
σN
|Xi,j (0)|
σN
= C|Xi,j (0)|2
(3.4.13)
and
BN = C min Xi,j · Ui,j
= Cλ|Xi,j (0)|2 .
Maxwell in [Max] obtains the same under the assumption of short range Φ0 and
for p = 5. He then argues that the remaining terms, when approximated for the case
of laminar flows, imply the Navier-Stokes equations.
35
3.5
Comparisons with Lanford-Ruelle
Ruelle has introduced spaces of infinitely many molecules, for example see [Ru], and
Lanford has studied dynamics directly on them, not as limit of finite dynamics and
without asuming finite energy or mass, see [L]. A comparison between that work and
this thesis follows.
Without providing full details, the infinite molecule configurations studied by
Lanford and Ruelle (the configurtions supporting “DLR” measures) must have only
finitely many molecules in any bounded set, see [L], p. 18. In particular, the LanfordRuelle configurations may not satisfy (2.1.16): if
N
1 X
|xi | ≤ B,
N i=1
(3.5.1)
and there are only finitely many molecules in the ball of radius B + 1, say NB , then
for N larger than NB
N
1 X
B>
|xi |
N i=1
1
=
N
≥
1
N
NB
X
|xi | +
i=1
NB
X
N
X
|xi | +
i=1
!
|xi |
(3.5.2)
i=NB +1
N − NB + 1
(B + 1).
N
Then let N → ∞ for a contradiction.
Moreover, Lanford-Ruelle configurations will never give tight µ(N ) measures: define
Nn = #{xi ∈ Bn }.
(3.5.3)
Then for each fixed R
µNn (BR 0 ) =
Nn − NR
NR
1
#{xi : R < |xi | ≤ n} =
=1−
.
Nn
Nn
Nn
36
(3.5.4)
Therefore
µNn (BR 0 ) < , ∀n
⇔ 1−<
NR
, ∀n,
Nn
(3.5.5)
and since NR < Nn eventually, this is equivalent for large n to
1−<
NR
< 1 + .
Nn
(3.5.6)
This clearly can only hold if NR increases with n (i.e. we go back to add molecules
to BR ) and
NR
→ 1.
Nn
Finally, note that a lot of effort went into showing that the assumption of uniformly
bounded velocities is a reasonable one, see section 2.2. This should be compared to
relativistic versions of the theory, cf. Lanford’s comments in [L], p. 68.
37
Chapter 4
Energy Equation
Recall here the main motivation of the current work as explained in the introduction,
i.e. Reynolds’s ideas on the distribution of energy into thermal and non-thermal
components. This chapter makes the first steps in the direction of a rigorous energy
equation that does not omit any terms. As before, we present first the weak version
of the energy equation for finite N and then let N increase to ∞.
Three cases are investigated in detail. The first, under assumptions similar to
section 2.2, shows that limits exist for all terms in the energy equation but does not
identify limit measures. The second, under different assumptions, shows tightness of
measures but works only for regular interactions. The third, under assumption of
tightness, identifies all terms in [IK] and [N] and is significant in that it considers the
interaction as part of the measure and not of the integrand. (This suggests a similar
treatment for the interaction term in the momentum equation.)
38
4.1
Energy Equation for Fixed N
First define the energy of the system for each fixed N as
E
(N )
N
N
mN X
m2N X
|xi (t) − xj (t)|
2
(t) =
|uN (xi (t))| +
Φ
2 i=1
2 i,j=1
σN
i6=j
Z
=
(4.1.1)
(N )
et (dx),
R3
(N )
for et (dx) the Dirac measures

(N )
et (dx)
N
N
1 X
1 X
2

=
Φ
|ui (t)| +
2N i=1 
N j=1

|x − xj (t)| 
 δx (t) (dx),
 i
σN
(4.1.2)
j6=i
and for uN as defined in (3.1.15). Note that by setting the total mass to one mN =
1
. Note that again the N dependence of ΦN has been absorbed in Φ by setting
N
x
mN ΦN (x) = mN Φ
. The time invariance of E (N ) is standard:
σN
N
X
d
d (N )
E (t) = mN
uN (xi ) uN (xi )
dt
dt
i=1
N
m2N X 0 |xi − xj | xi − xj
+
Φ
(uN (xi ) − uN (xj ))
2σN i6=j
σN
|xi − xj |


N
N
X

1 X 0 |xi − xj | xi − xj 

= mN
uN (xi ) 
−
Φ
 N σN

σ
|x
−
x
|
N
i
j
i=1
j=1
(4.1.3)
j6=i
N
m2 X 0
+ N
Φ
2σN i6=j
|xi − xj |
σN
xi − x j
(uN (xi ) − uN (xj ))
|xi − xj |
= 0.
To obtain a weak formulation of the evolution of E (N ) , examine its time derivative
against a test function ϕ = ϕ(x):
d
dt
Z
ϕ(x)e(N ) (dx).
R3
39
(4.1.4)
As in [D], this class of test functions is enough. Then
Z
d
ϕ(x)e(N ) (dx)
dt R3
Z
Z
|uN (x)|2 (N )
1
d
|x − y|
(N )
ϕ(x)
µt (dx) +
nt (dx, dy) ,
=
ϕ(x)Φ
dt
2
2N 2 R6
σN
R3
(4.1.5)
for
(N )
nt (dx, dy)
=
N
X
δ(xi (t),xj (t)) (dx, dy).
(4.1.6)
j=1
j6=i
(4.1.5) consists of the the kinetic energy part, (4.1.7) and the potential part, (4.1.8).
For fixed N the equation for the kinetic energy is
Z
1d
(N )
ϕ(x)|uN (x)|2 µt (dx)
2 dt
N
1d 1 X
ϕ(xi )|uN (xi )|2
=
2 dt N i
N
1 X
11 X
(4.1.7)
∇ϕ(xi ) · uN (xi )|uN (xi )|2 +
ϕ(xi )uN (xi ) · u0N (xi )
=
2 N i=1
N i6=j
Z
1
(N )
∇ϕ(x) · uN (x)|uN (x)|2 µt (dx)
=
2
Z
1
|x − y|
x − y (N )
0
− 2
ϕ(x)Φ
uN (x) ·
n (dx, dy).
N σN
σN
|x − y| t
Similarly for fixed N the equation for the potential energy is
N
|xi − xj |
d 1 X
ϕ(xi )Φ
dt 2N 2 i6=j
σN
N
N
|xi − xj |
d
1 X
1 X
|xi − xj |
0
∇ϕ(xi ) · xi Φ
ϕ(xi ) Φ
=
+
2N 2 i6=j
σN
2N 2 i6=j
dt
σN
N
1 X
|xi − xj |
=
∇ϕ(xi ) · uN (xi ) Φ
2N 2 i6=j
σN
N
X
1
|xi − xj | xi − xj
0
0
0
+
·
x
−
x
ϕ(x
)Φ
i
i
j
2N 2 σN i6=j
σN
|xi − xj |
Z
1
|x − y|
(N )
=
∇ϕ(x) · uN (x)Φ
nt (dx, dy)
2N 2
σN
Z
1
|x − y| x − y
(N )
0
+
ϕ(x)Φ
· (uN (x) − uN (y)) nt (dx, dy).
2
2N σN
σN
|x − y|
(4.1.8)
40
4.2
Converge of the Velocity Terms in the Energy
Equation
The discussion here parallels, with more details, the discussion of the convergence of
the non-linear terms in the momentum equation, section 3.3.
Lemma 4.2.1. Assume that u(N ) (t) is uniformly bounded in N and t ∈ [0, T ] by a con(N )
stant U . Then |v|p is uniformly integrable with respect to the measures Mt
given there is k0 such that for k > k0
Z
(N )
|v|p Mt (dx, dv) < ,
(dx, dv):
(4.2.1)
{(x,v):|v|p >k}
for all N .
Proof.
Z
p
|v|
(N )
Mt (dx, dv)
Z
=
(N )
χ{(x,v):|v|p >k} (x, v)|v|p Mt
(dx, dv)
R6
{(x,v):|v|p >k}
Z
=
(N )
(N )
χ{(x,v):|v|p >k} (x, v)|v|p (Id × ut )# µt (dx, dv)
R6
Z
=
(N )
(N )
(N )
χ{(x,v):|v|p >k} (x, ut (x))|ut (x)|p µt (dx).
R3
(4.2.2)
p
For k > U this is identically zero.
Repeating the proof with extra constants gives
Corollary 4.2.2. Under the same assumptions, ϕ(x)|v(x)|2 and ∇ϕ(x) · v(x)|v(x)|2
(N )
are uniformly integrable with respect to the measures Mt
(dx, dv).
Corollary 4.2.3. Under the same assumptions
Z
lim
N →∞
Z
lim
N →∞
∇ϕ(x) ·
Z
(N )
ϕ(x)|uN (x)|2 µt (dx)
(N )
uN (x)|uN (x)|2 µt (dx)
=
Z
=
41
ϕ(x)|v(x)|2 Mt (dx, dv),
(4.2.3)
∇ϕ(x) · v(x)|v(x)|2 Mt (dx, dv). (4.2.4)
Proof. Notice that
Z
Z
(N )
2 (N )
ϕ(x)|uN (x)| µt (dx) = ϕ(x)|v(x)|2 Mt (dx, dv),
Z
Z
(N )
2 (N )
∇ϕ(x) · uN (x)|uN (x)| µt (dx) = ∇ϕ(x) · v(x)|v(x)|2 Mt (dx, dv)
(4.2.5)
and use [AGS], Lemma 5.1.7, p. 110.
The assumption of uniform boundedness holds for the the class of examples of
section 2.2.
For the term
d
dt
Z
ϕ(x)
v 2 (N )
M (dx, dv),
2 t
(4.2.6)
note that since v 2 is uniformly integrable, for η smooth and of compact support in
[0, T ]
0
Z
lim η (t)
n→∞
v 2 (N )
ϕ(x) Mt (dx, dv) = η 0 (t)
2
Z
ϕ(x)
v2
Mt (dx, dv).
2
(4.2.7)
Thus
Z
lim
n→∞
0
T
d
η(t)
dt
v 2 (N )
M (dx, dv)dt
2 t
R6
Z
Z T
v 2 (N )
0
η (t)
ϕ(x) Mt (dx, dv)dt
= − lim
n→∞ 0
2
R6
Z TZ
∂
v 2 (N )
= − lim
(η(t)ϕ(x)) Mt (dx, dv)dt.
n→∞ 0
2
R6 ∂t
Z
ϕ(x)
(4.2.8)
Now by (4.2.3), for each t
Z
R6
∂
v 2 (N )
(η(t)ϕ(x)) Mt (dx, dv) →
∂t
2
Z
R6
∂
v2
(η(t)ϕ(x)) Mt (dx, dv),
∂t
2
(4.2.9)
and by Lebesque Dominated Convergence
Z TZ
Z TZ
∂
v 2 (N )
∂
v2
(η(t)ϕ(x)) Mt (dx, dv)dt →
(η(t)ϕ(x)) Mt (dx, dv)dt,
2
2
0
R6 ∂t
0
R6 ∂t
(4.2.10)
42
for dominating function Cη,ϕ B, where B is any constant satisfying
Z
R6
N
(N )
v 2 (N )
1 X |ui (t)|2
Mt (dx, dv) =
≤ B,
2
N 1
2
t ∈ [0, T ].
(4.2.11)
Now use the density in C0∞ ((0, T ) × R3 ) of test functions of the form η(t)ϕ(x) to show
that (4.2.10) holds for arbitrary test functions φ(t, x). Since in the supremum norm
∂
∂
ηi (t)ϕi (x) → ϕ(t, x),
∂t
∂t
2
Z ∂
∂
v
ηi (t)ϕi (x) − ϕ(t, x)
Mt (dx, dv)dt
∂t
∂t
2
R6
Z T Z
∂
v2
∂
≤ sup η
(t)ϕ
(x)
−
ϕ(t,
x)
Mt (dx, dv)dt.
i
i
∂t
i∈N ∂t
0
R6 2
ηi (t)ϕi (x) → ϕ(t, x) and
Z
T
0
(4.2.12)
Finally, note that
Z
T
Z
R6
0
v2
Mt (dx, dv)dt
2
(4.2.13)
is finite as the limit of the bounded sequence
T
Z
0
Z
R6
v 2 (N )
M (dx, dv)dt.
2 t
(4.2.14)
The same analysis gives the convergence of the first term on the right of (4.1.7) to
the term
Z
T
0
4.3
1
2
Z
∇ϕ(t, x) · v|v|2 Mt (dx, dv)dt.
(4.2.15)
R6
The Interaction Term in the Energy Equation
4.3.1
Convergence
We first show that under certain assumpions, all of which are satisfied by the class of
examples in section 2.2, the interaction terms in the energy equation have limits as
N → ∞.
43
Lemma 4.3.1. Assume that −Φ0 (r) = r−a , |xi (0)| < X, |ui (0)| < U , and that
−
1 0
Φ
σN
|xi (0) − xj (0)| ≤ |xi (t) − xj (t)|,
|xi (t) − xj (t)|
≤ BN = CT,X,U min (xi (0) − xj (0)) · (ui (0) − uj (0)) .
1≤i6=j≤N
σN
(4.3.1)
Then
|xi (t) − xj (t)|
≤ 8CX 2 U,
Φ
σN
1 00 |xi (t) − xj (t)|
Φ
≤ 2CU,
2
σN
σN
(4.3.2)
where C, X, U do not depend on N .
Proof.
|xi (t) − xj (t)|
|xi (0) − xj (0)|
Φ
≤Φ
σN
σN


min |xi (0) − xj (0)|
i6=j≤N

≤ Φ
σN
=
a−1
σN
a−1
min |xi (0) − xj (0)|
i6=j≤N
≤ C min (xi (0) − xj (0)) · (ui (0) − uj (0)) min |xi (0) − xj (0)|
i6=j
i6=j≤N
≤ 8CX 2 U.
(4.3.3)
1 00
Φ
2
σN
|xi (t) − xj (t)|
σN
a−1
σN
|xi (t) − xj (t)|a+1
a−1
σN
≤
|xi (0) − xj (0)|a+1
mini6=j (xi (0) − xj (0)) · (ui (0) − uj (0))
≤C
|xi (0) − xj (0)|
|xi (0) − xj (0)||ui (0) − uj (0)|
≤C
|xi (0) − xj (0)|
=
= C |ui (0) − uj (0)| ≤ 2CU.
44
(4.3.4)
In particular, by evaluating at t = 0,
|Xi,j (0)|
0
−Φ
≤ σN BN ,
σN
1 00 |Xi,j (0)|
Φ
≤ σN 2CU.
σN
σN
(4.3.5)
Lemma 4.3.2. Assume that
|xi (0) − xj (0)| ≤ |xi (t) − xj (t)|,
|xi (0)| ≤ X,
|ui (0)| ≤ U,
(4.3.6)
|ui (t)| ≤ U + 2T C min (xi (0) − xj (0)) · (ui (0) − uj (0)) ,
i6=j
|ui (0) − uj (0)|
≤ λ.
|xi (0) − xj (0)|
Then
ui (t) − uj (t)
(x
(t)
−
x
(t))
·
(u
(t)
−
u
(t))
i
j
i
j
≤ 2λ (1 + 4CT X) .
−
(x
(t)
−
x
(t))
i
j
|xi (t) − xj (t)|
|xi (t) − xj (t)|3
(4.3.7)
Proof. With
Xi,j (t) := xi (t) − xj (t),
(4.3.8)
Ui,j (t) := ui (t) − uj (t),
we have
Ui,j (t)
|Ui,j (t)|
Xi,j (t) · Ui,j (t)Xi,j (t) ≤2
|Xi,j (t)| −
3
|Xi,j (t)|
|Xi,j (t)|
|Ui,j (0)| |Ui,j (t)|
≤2
|Xi,j (0)| |Ui,j (0)|
|Ui,j (0)| |Ui,j (0)| + 2BN T
≤2
|Xi,j (0)|
|Ui,j (0)|
|Ui,j (0)|
≤2
1 + 2CT |Xi,j (0)|
|Xi,j (0)|
|Ui,j (0)| ≤2
1 + 4CT X
|Xi,j (0)|
≤ 2λ 1 + 4CT X .
45
(4.3.9)
We now use these lemmas and the uniform bounds from Lemma 2.2.27 to establish
the convergence of the terms with Φ in the energy equation. We will show uniform
convergence of these terms but will not show that the limit can be expressed in terms
of a limit measure. The terms in question are the last term in (4.1.7) and both terms
on the final right-hand side of (4.1.8), namely:
T
Z
0
N
|xi (t) − xj (t)| xi (t) − xj (t)
1 X
0
ϕ(t, xi )Φ
· uN (xi (t))dt,
N 2 σN i,j=1
σN
|xi (t) − xj (t)|
i6=j
T
Z
0
1
N2
N
X
(4.3.10)
∇ϕ(t, xi ) · uN (xi (t))Φ
i,j=1
i6=j
|xi (t) − xj (t)|
σN
dt.
All of the integrands in (4.3.10) are uniformly bounded by Lemma 4.3.1 and the
uniform bounds on the velocities (2.2.23). The time derivative of the integrands are
also uniformly bounded.


N
X
d 
 1
ϕ(t, xi )Φ0

2
dt N σN i,j=1
|Xi,j (t)|
σN

Xi,j (t)
· ui (t)

|Xi,j (t)|
i6=j
N 1 X ∂ϕ
|Xi,j (t)| Xi,j (t)
0
= 2
(t, xi ) + ∇ϕ(t, xi ) · ui (t) Φ
· ui (t)
N σN i,j=1 ∂t
σN
|Xi,j (t)|
i6=j
N
1 X
|Xi,j (t)| Xi,j (t)
Xi,j (t)
00
+ 2 2
ϕ(t, xi )Φ
· Ui,j (t)
· ui (t)
N σN i,j=1
σN
|Xi,j (t)|
|Xi,j (t)|
i6=j
N
1 X
Ui,j (t)
Xi,j (t) · Ui,j (t)Xi,j (t)
|Xi,j (t)|
0
· ui (t)
+ 2
ϕ(t, xi )Φ
−
N σN i,j=1
σN
|Xi,j (t)|
|Xi,j (t)|3
i6=j
2
N
1 X
|Xi,j (t)|
0
+ 2 2
ϕ(t, xi ) Φ
,
N σN i,j=1
σN
i6=j
(4.3.11)
46


N
X
d 
|Xi,j (t)| 
 1

∇ϕ(t, xi ) · ui (t)Φ

dt  N 2 i,j=1
σN
i6=j
N ∂ϕ
|Xi,j (t)|
1 X
2
∇ (t, xi ) + D ϕ(t, xi )ui (t) · ui (t) Φ
= 2
N i,j=1
∂t
σN
(4.3.12)
i6=j
+
1
N
X
N 2 σN i,j=1
i6=j
∇ϕ(t, xi ) ·
Xi,j (t) 0
Φ
|Xi,j (t)|
|Xi,j (t)|
σN
Φ
|Xi,j (t)|
σN
N
|Xi,j (t)| Xi,j (t)
1 X
0
+ 2
∇ϕ(t, xi ) · ui (t)Φ
· Ui,j (t).
N σN i,j=1
σN
|Xi,j (t)|
i6=j
Then all terms are uniformly bounded by the uniform bounds (2.2.23) and Lemmas
4.3.1 and 4.3.2. Thus both types of potential terms in the energy equation are uniformly convergent in t, up to subsequence. Therefore their integrals on [0, T ] converge.
Using these facts it is also true that the left-hand side of (4.1.8) is uniformly
convergent since the right-hand side is bounded and equi-continuous. Thus
d
dt
Z
0
T
N
1 X
|xi (t) − xj (t)|
dt
ϕ(t, xi )Φ
N 2 i,j=1
σN
(4.3.13)
i6=j
is convergent.
Then all convergences have been achieved.
4.3.2
Tightness of Triple Measures
The discussion here parallels [D] for the convergence of the interaction term in the
momentum equation, informed by the improvements of our section 3.4.1 here. The
main point is to consider the interaction Φ0 as part of the integrand and try to obtain
weak convergence of the remaining terms by considering appropriate measures.
47
Define then
Λ
(N )
Γ(N )
Υ(N )
|x − y|
(N )
x×
× uN (x)
nt
σN
#
|x − y|
1
(N )
=
x×
× uN (x) × uN (y)
nt
N (N − 1)
σN
#
1
|x − y|
(N )
=
x×
nt .
N (N − 1)
σN
#
1
=
N (N − 1)
(4.3.14)
Lemma 4.3.3. Assume (2.1.16) and
N
1 X
|xi (t)| < C
N σN i=1
for all N and t ∈ [0, T ],
N
1 X
|ui (t)| < C for all N and t ∈ [0, T ],
N σN i=1
1 0 |Xi,j (0)|
− Φ
≤ BN .
σN
σN
(4.3.15)
Then, up to a subsequence of N ’s common to all t’s, the measures Λ(N ) , Γ(N ) , and
Υ(N ) converge weakly.
Proof. The characteristic function is
N
X
1
χΛ(N ) (t, y, z, υ) =
ei(xi ·y+ξ·z+v·υ) .
N (N − 1) i,j=1
(4.3.16)
i6=j
This function is bounded by one. Similarly
N
X
i
i(xi ·y+ξ·z+v·υ) e
xi |∇y χΛ(N ) (t, y, z, υ)| = N (N − 1) i,j=1
i6=j
!
N
1 X
≤
|xi | ,
N i=1
N
X
i
|x
−
x
|
i
j
i(x
·y+ξ·z+v·υ)
|∇z χΛ(N ) (t, y, z, υ)| = e i
σN N (N − 1) i,j=1
i6=j
!
N
2 X
≤
|xi | ,
N σN i=1
48
(4.3.17)
(4.3.18)
N
X
i
i(xi ·y+ξ·z+v·υ) |∇υ χΛ(N ) (t, y, z, υ)| = e
ui N (N − 1) i,j=1
i6=j
!
N
1 X
≤
|ui | ,
N i=1
(4.3.19)
and
N
X
∂
1
ui − uj
i(xi ·y+ξ·z+v·υ)
χΛ(N ) (t, y, z, υ) = ui · y +
·z
e
N (N − 1)
∂t
σN
i,j=1
i6=j

N
1 X 0 |xi (t) − xk (t)| Xi,k (t)
 (4.3.20)
−
Φ
· υ 
N σN k=1
σN
|Xi,k (t)|
k6=i
!
!
N
N
1 X
2 X
≤
|ui | |y| +
|ui | |z| + BN |υ|.
N i=1
N σN i=1
Then all terms are uniformly bounded by assumption and therefore convergent. All
other measures are treated in the same way. Moreover one can apply the Levy continuity theorem as these are probability measures. Thus up to subsequence these
measures converge for all t.
To get the desired convergences of the individual terms we must apply restrictions
on the Φ. We ask that Φ is smooth of compact support and that Φ0 (s) = sG(s) where
G is smooth and of compact support. These are exactly the conditions in [D]. Using
the constructions at the end of the next section, (4.3.25) and (4.3.26), allows for the
desired convergences, with the exception that Φ is not inside the measure.
4.3.3
Other Cases
To allow for more realistic, singular Φ’s it is instructive to consider Φ and Φ0 as part
of the measure and not as part of the integrand. For this, and instead of the measures
49
Λ(N ) , Γ(N ) , Υ(N ) , define the measures
|x − y|
|x − y|
1
(N )
(N )
Φ
x×
× uN (x)
nt ,
L
=
N (N − 1)
σN
σN
#
|x
−
y|
1
|x
−
y|
(N )
(N )
0
G
=
Φ
x×
× uN (x) × uN (y)
nt ,
N (N − 1)
σN
σN
#
1
|x − y|
|x − y|
(N )
U(N ) =
Φ
x×
nt ,
N (N − 1)
σN
σN
#
(4.3.21)
and assume them uniformly convergent in t to measures L, G, U respectively.
To deal with the terms on the right-hand side of (4.1.7) and (4.1.8), borrow a
technique from [D]. First combine these two terms into one term resulting in
1
− 2
2N σN
Z
0
ϕ(x)Φ
|x − y|
σN
x−y
(N )
(uN (x) + uN (y)) n2 (dx, dy).
|x − y|
(4.3.22)
Note that because the measure is just summing over all i and j changing the role of
the variables does not change the equation. Indeed
Z
1
|x − y| x − y
(N )
0
− 2
ϕ(x)Φ
· (uN (x) + uN (y)) n2 (dx, dy)
2N σN
σN
|x − y|
Z
1
|y − x| y − x
(N )
0
=− 2
ϕ(y)Φ
· (uN (x) + uN (y)) n2 (dy, dx)
2N σN
σN
|y − x|
Z
1
|x − y| x − y
(N )
0
· (uN (x) + uN (y)) n2 (dx, dy).
=+ 2
ϕ(y)Φ
2N σN
σN
|x − y|
(4.3.23)
Using the change of variables xi − xj = σN ξ,
Z
0
|x − y|
σN
x−y
(N )
ϕ(x)Φ
· (uN (x) + uN (y)) n2 (dx, dy)
|x − y|
Z
1
ξ
(N )
=− 2
ϕ(x)Φ0 (|ξ|)
· (uN (x) + uN (x − σN ξ)) n2 (dx, dξ)
2N σN
|ξ|
Z
1
ϕ(x) − ϕ(x − σN ξ) 0
ξ
(N )
=− 2
Φ (|ξ|)
· (uN (x) + uN (x − σN ξ)) n2 (dx, dξ)
4N
σN
|ξ|
Z
1
ϕ(x) − ϕ(x − σN ξ) ξ
=− 2
· (v + v 0 ) G(N ) (dx, dξ, dv, dv 0 ).
4N
σN
|ξ|
(4.3.24)
1
− 2
2N σN
Then under the assumption of uniform integrability of the velocities and finite range
50
of the potential,
Z
1
|x − y| x − y
(N )
0
lim − 2
· (uN (x) + uN (y)) n2 (dx, dy)
ϕ(x)Φ
N →∞
2N σN
σN
|x − y|
Z
(4.3.25)
0
1
ξ v+v
0
=−
∇ϕ(x) · ξ
G(dx, dξ, dv, dv ).
2
|ξ| 2
Lemma 5.2.1 [AGS] has been used for this convergence.
The remaining terms of (4.1.8), under the assumption of tightness and uniform
integrability, also have limits:
Z
1 d
|x − y|
(N )
lim
nt (dx, dy)
ϕ(x)Φ
N →∞ N 2 dt R6
σN
Z
d
=
ϕ(x)U(dx, dξ),
dt R6
Z
1
|x − y|
(N )
lim
∇ϕ(t, x) · uN (x)Φ
nt (dx, dy)
N →∞ N 2 R6
σN
Z
=
∇ϕ(x) · v L(dx, dξ, dv).
(4.3.26)
R9
Then to introduce the test function’s t dependence, follow the technique used for the
kinetic equation.
4.4
Comparisons with Irving & Kirkwood and Noll
Lemma 4.4.1.
Z
∇ϕ(t, x) · v|v|2 Mt (dx, dv)
Z
= ∇ϕ(t, x) · (v − u(x))|v(x) − u(x)|2 Mt (dx, dv)
Z
+ ∇ϕ(t, x) · u(x)|v|2 Mt (dx, dv)
Z
+ 2 ∇ϕ(t, x) · [(v − u(x)) ⊗ (v − u(x))] · u(x)Mt (dx, dv).
(4.4.1)
Proof. To see this, introduce and remove the barycentre from (4.2.15). It is also
necessary to note that one term will be zero by the definition of the barycentre.
Then, in summary, for realistic potentials Φ the assumptions of section 4.3.1 lead
51
to the energy equation
Z
T
v2
∂
ϕ(t, x) Mt (dx, dv)dt − lim
N →∞
∂t
2
Z
−
0
R6
T
Z
0
N
1 X
|xi − xj |
dt
ϕ(t, xi )Φ
N 2 i6=j
σN
i,j=1
Z
1
∇ϕ(t, x) · (v − u(x))|v(x) − u(x)|2 Mt (dx, dv)
2 R6
Z
1
+
∇ϕ(t, x) · u(x)|v|2 Mt (dx, dv)
2 R6
Z
+
∇ϕ(t, x) · [(v − u(x)) ⊗ (v − u(x))] · u(x)Mt (dx, dv)
=
R6
T
Z
− lim
N →∞
0
N
X
1
|xi (t) − xj (t)| Xi,j (t)
0
ϕ(t, xi )Φ
· (ui (t) + uj (t)) dt
2N 2 σN i,j=1
σN
|Xi,j (t)|
i6=j
Z
T
+ lim
n→∞
0
1
N2
N
X
∇ϕ(t, xi ) · uN (xi (t))Φ
i,j=1
i6=j
|xi (t) − xj (t)|
σN
dt,
(4.4.2)
whereas the assumptions of section 4.3.3 lead to
Z
T
Z
v2
∂
ϕ(t, x) Mt (dx, dv)dt −
∂t
2
Z
T
Z
∂
1
ϕ(t, x) U(dx, dξ)dt
−
2
0
R6 ∂t
0
R6
Z TZ
1
∇ϕ(t, x) · (v(x) − u(x))|v(x) − u(x)|2 Mt (dx, dv)dt
=
2 0 R6
Z Z
1 T
+
∇ϕ(t, x) · u(x)|v(x)|2 Mt (dx, dv)dt
2 0 R6
Z TZ
+
∇ϕ(t, x) · [(v(x) − u(x)) ⊗ (v(x) − u(x))] · u(x)Mt (dx, dv)dt
0
R6
Z Z
1 T
+
∇ϕ(t, x) · u(x)L(dx, dξ)dt
2 0 R6
Z Z
1 T
+
∇ϕ(t, x) · (v − u(x)) L(dx, dξ, dv)dt
2 0 R9
Z TZ
∇ϕ(t, x)
ξ
−
· ξ u(x)
G(dx, dξ)dt
2
|ξ|
0
R6
Z TZ
∇ϕ(t, x)
v + v0
ξ
−
·ξ
− u(x)
G(dx, dξ, dv, dv 0 )dt.
2
2
|ξ|
12
0
R
(4.4.3)
(4.4.4)
(4.4.5)
(4.4.6)
(4.4.7)
(4.4.8)
(4.4.9)
(4.4.10)
The terms (4.4.4), (4.4.5), and (4.4.6) are the result of the convergence of the third
moment and Lemma 4.4.1. (4.4.7) and (4.4.8) come from the convergence in (4.3.26).
52
Fianlly, (4.4.9) and (4.4.10) come from the convergence in (4.3.25) and results from
combining the last terms in (4.1.7) and (4.1.8). The average velocity in this term is
introduced to make the terms in the form of [N].
Comparing these results with [IK] and, especially, [N], (4.4.5) and (4.4.7) combine
to make a term that is weakly of the form div(Eu). Similarly, (4.4.6) and (4.4.9)
combine to make a term that is weakly of the form −div(S · u) where S is the symmetric stress tensor and appears in the momentum equation. Then the remaining
terms are the what Noll calls “heat flux.” The term (4.4.4) is Noll’s (somewhat hastingly named, we believe) “kinetic contribution qK ,” (2.14) of [N]. Noll’s “transport
contribution qT ,” (2.15) of [N], corresponds to (4.4.8) here. And the term (4.4.10)
is Noll’s “interaction contribution qV ,” (2.16) of [N]. In other words, our terms here
combine to a weak version of
d
E = −∇ · (Eu + q − S · u) ,
dt
in agreement with [IK] and [N].
53
(4.4.11)
Chapter 5
Disintegration, Conditional
Expectation, and Reynolds
Averaging
Section 3.1 defined the average velocity u(t, x) in terms of disintegration as
Z
u(t, x) =
vMt,x (dv),
(5.0.1)
R3
and, in general, the macroscopic average of any microscopic measurement F (xi , ui )
as
Z
F (x, v)Mt,x (dv).
F (t, x) =
(5.0.2)
R3
Given that disintegration is related to conditional expectation, see [DM], if these
formulas have value as Reynolds averages, they must fit the properties of abstract
Reynolds averages as described, for example, in [Mo]. This is shown in Section 5.2. In
particular, the conditioning coming from disintegration ought to be compatible with
the conditioning coming from abstract Reynolds theory. This is addressed in Section
5.3.
Finally, Section 5.4 returns to our initial motivation, the identification of the
Reynolds neighborhood. Using Besicovich’s Theorem, this is now shown to be the
54
limit over ever shrinking neighborhoods. And the average at x, as defined by the
averaging formula (5.0.2), is now understood as the limit of averages at neighborhing
points. This fits Maxwell’s use of “immediate neighborhoods” in his heuristic definition of averages, see [Max], p. 67. The last section also defines our thinking of (5.0.2):
looking at the microscopic world from afar, one cannot tell what is happening exactly
at x (or if anything is happening). We tend to attribute to x our impression of what
is happening somewhere around x.
5.1
Abstract Reynolds Averaging
Reynolds averaging was introduced (not very precisely) in [R], p. 134, as a space
average of the form
1
u(x) =
Vol(Sx )
Z
v(y) dy,
(5.1.1)
Sx
for Sx some neighborhood of x. The purpose of the average was to obtain an improvement to the Navier-Stokes equations by including extra terms that possibly describe
turbulence. (The energy of these extra terms allowed Reynolds to give a theoretical interpretation of a critical “Reynolds number.”) The extra terms are now called
Reynolds stresses and the equations are the Reynolds equations.
The properties an average had to satisfy to give the Reynolds equations were later
axiomatized and the abstract setting was studied, see [KdF], [Bir], [Rot]. Here we
follow (the second part of) [Mo].
Let (X, F, µ) a probability space.
Definition 5.1.1. An operator T : L1 (X) → L1 (X) is a Reynolds average, if it satisfies the following:
55
1. T (αf + βg) = αT (f ) + βT (g) for α, β ∈ R.
2. If f is bounded, then T f is bounded.
3. T (f T g) = (T f )(T g) for every pair f, g of bounded functions.
4. T is continuous; that is, if fn → f in L1 then T fn → T f in L1 .
Under the additional assumptions
4. kT f k1 ≤ kf k1 (replacing the previous item), and
5. T 1 = 1,
Moy’s Theorem 2.2, [Mo] p. 61, shows that T is in fact the conditional expectation
with respect to some σ-algebra:
T f = E(f |FT ),
(5.1.2)
for FT the σ-algebra of sets E ∈ F whose characteristic functions χE satisfy
T (χE f ) = χE T f,
(5.1.3)
for all f in L1 (X, P).
5.2
Barycentric Projection as Reynolds Averaging
Starting with M on (R6 , B(R6 )) and µ its first marginal with respect to R6 = R3 ×R3 ,
disintegrate M with respect to µ and define u to be the barycentric projection of this
disintegration:
Z
u(x) =
v Mx (dv).
R3
56
(5.2.1)
This was the construction of (3.1.15). That we start in L1 (M, R6 ) follows from
Z
Z
|v|M (N ) (dx, dv),
|v|M (dx, dv) ≤ lim inf
N →∞
R6
(5.2.2)
R6
as MN converges weakly to M and |v| is continuous and bounded below, see [AGS],
p. 110. Estimate the right-hand side of this by
N
1 X
lim inf
|ui | ≤ lim inf
N →∞ N
N →∞
i=1
N
1 X
|ui |2
N i=1
!1/2
,
(5.2.3)
by Hölder on the discrete measure. The last term is finite by our standing assumptions
(2.1.17).
Define then T : L1 (M, R6 ) → L1 (M, R6 ) as the barycentric projection
Z
T f (x, v) =
f (x, v) Mx (dv).
(5.2.4)
R3
In particular, T f is constant in v. That the right-hand side is indeed in L1 (X) is
implied by boundedness, see below.
Proposition 5.2.1. T satisfies conditions (1)-(5) of Definition 5.1.1.
Proof. Linearity is obvious and
Z
|T f | = Z
f Mx (dv) ≤ sup |f |
6
Mx (dv) = sup |f |
(5.2.5)
R6
R
shows that T takes bounded functions to bounded ones. Property 3 is shown by
noting that the functional T eliminates v:
Z
T (f T g) =
f (x, v)(T g)(x)Mx (dv)
R3
Z
= (T g)(x)
f (x, v)Mx (dv)
R3
= (T f )(T g).
57
(5.2.6)
Since Mx is a probability measure,
Z
T1 =
Mx (dv) = 1.
(5.2.7)
R3
Finally the one-boundedness of the operator norm is seen by
Z Z
f Mx (dv) M (dx, dv)
kT f k1 =
6
3
ZR Z R Z
≤
|f |Mx (dv)Mx (dv)µ(dx)
R3 R3 R3
Z
Z Z
|f |Mx (dv)
Mx (dv) µ(dx)
=
R3
R3
R3
Z Z
|f |Mx (dv) 1µ(dx)
=
3
3
R
R
Z
=
|f |M (dx, dv) = kf k1 .
(5.2.8)
R6
Corollary 5.2.2. The average defined by (5.2.4) is an abstract Reynolds averaging
in the sense of Moy [Mo].
5.3
5.3.1
Conditional Expectations
The σ-algebra from Reynolds Averaging
According to Moy, T is an abstract Reynolds average therefore it is a conditional expectation. We now describe the σ-algebra of this expectation. For this it is necessary
to find all E’s such that
T (χE f ) = χE T f,
(5.3.1)
for all f ∈ L1 . Using the definition of T this is the same as
Z
Z
χE (x, v)f (x, v)Mx (dv) = χE (x, v)
R6
f (x, v)Mx (dv).
(5.3.2)
R6
Setting f equal to 1 leads to
Z
χE (x, v)Mx (dv) = χE (x, v).
R6
58
(5.3.3)
As the left-hand side of (5.3.3) is only a function of x, this implies that the characteristic of E depends only on x, i.e.
χE (x, v) = χE (x, 0),
(5.3.4)
for all v ∈ R3 . Therefore E = (pr1 E) × R3 and
3
∩ B R6 .
FT = pr−1
1 (A) : A ⊂ R
5.3.2
(5.3.5)
The σ-algebra from Disintegration
To close the current circle of ideas we recall briefly the relation of disintegration
to conditioning, following [DM]. A full proof of the disintegration for the measures
considered here is presented in Appendix B.
Starting with the measure Mx (dv) on the tangent space of x, let Ex be its expectation:
Z
Ex (f ) =
f (x, v)Mx (dv).
(5.3.6)
R3
Let
3
σ(pr1 ) = σ pr−1
1 (A)| A ∈ B(R ) .
(5.3.7)
Lemma 5.3.1. The random variable on (R6 , B, M (dx, dv))
Z
(x, v) → Epr1 (x,v) (f ) =
f (x, v)Mx (dv)
(5.3.8)
R3
is a version of the conditional expectation
E( f | σ(pr1 ) ).
59
(5.3.9)
Proof. The σ(pr1 )-measurability of (x, v) → Epr1 (x,v) (f ) follows from the (tautological) measurability of pr1 and the definition of the measurability of the disintegration,
see Appendix B.
It remains to check that the conditional expectation equation is satisfied:
Z Z
f (x, v)χA (pr1 (x, v))M (dx, dv) =
χA (pr1 (x, v))f (x, v)Mx (dx, dv)µ(dx)
3
3
R6
R
R
Z
Z
=
χA (pr1 (x, v))
f (x, v)Mx (dx, dv)µ(dx)
3
3
R
R
Z
=
χA (pr1 (x, v))Epr1 (x,v) (f )µ(dx)
3
ZR
Z
χA (pr1 (x, v))Epr1 (x,v) (f ) 1Mx (dv)µ(dx)
=
3
R
Z Z
=
χA (pr1 (x, v))Epr1 (x,v) (f )Mx (dv)µ(dx)
3
3
R
R
Z
=
χA (pr1 (x, v))Epr1 (x,v) (f )M (dx, dv).
Z
R6
(5.3.10)
As the σ-algebras (5.3.5) and (5.3.7) coincide, obtain:
Proposition 5.3.2. The conditional expectation corresponding to the disintegration
of M coincides with the conditional expectation of Mx as abstract Reynolds averaging.
This provides an alternative proof of Corollary 5.2.2.
5.4
Reynolds Neighbourhood
A more geometric description of our averages is provided by the following:
Theorem 5.4.1. For Bρ = Bρ (x) the ball in R3 of radius ρ centered around x, the
following holds:
Z
f (x, v)M (dx, dv)
Z
R3
f (x, v)Mx (dv) = lim+
(5.4.1)
pr−1
1 (Bρ )
ρ→0
60
µ(Bρ )
.
Proof. For f on R6 let f M denote the measure on R6 that is absolutely continuous
with respect to M with density f . For Mx (dv) as a measure on R6 supported by the
fibre over x, and with slight abuse of notation,
Z
Z
f (x0 , v)Mx (dx0 , dv).
f (x, v)Mx (dv) =
R3
(5.4.2)
R6
Given that Mx (dv) is the disintegration of M with respect to µ = (pr1 )# M the proof
of the disintegration theorem gives, following the notation of [DM], p. 78-III,
Z
Z
f (x, v)Mx (dv) =
R3
f (x0 , v)Mx (dx0 , dv) = df (x)
(5.4.3)
R6
for df (x) the density of the push forward of f M via pr1 with respect to µ:
df (x)µ(dx) = (pr1 )# (f M )(dx).
(5.4.4)
On the other hand, by Besicovich’s theorem, [AFP] page 54,
(pr1 )# (f M )(Bρ )
ρ→0
µ(Bρ )
(f M )(pr−1
1 Bρ )
= lim+
ρ→0
µ(Bρ )
Z
χpr−1
(x, v)f (x, v)M (dx, dv)
1 Bρ
R6
= lim+
.
ρ→0
µ(Bρ )
df (x) = lim+
(5.4.5)
In particular for u defined by (3.1.15) we obtain
Z
1
u(x) = lim+
ρ→0 µ(Bρ )
v M (dx, dv).
(5.4.6)
pr−1
1 (Bρ )
I.e. we have defined u to be the limit on ever shrinking balls of the space average of all
velocities supporting M at points close to x. This is as close as we can currently get
to Reynolds’s claim [R], p. 126:
61
“The mean component velocity (u) of all the molecules in the immediate
neighbourhood of a point, say P, can only be the mean component velocity
of all the molecules in some space (S) enclosing P.”
62
Chapter 6
Landau-Lifshitz Formulas
First we review standard facts on Hermite polynomials. We will expand on this
to develop formulas for a function of two variables in a double Hermite expansion.
Next we will use the Grad thirteen moment expansion to identify certain moments of
the velocity against the Maxwell-Boltzmann distribution with fundamental physical
quantities. Next we prove a Landau-Lifshitz formula for near equilibrium distributions.
6.1
Hermite Polynomials
We first recall standard facts about (Chebyshev-)Hermite polynomials in R3 and R6 .
First set the normalized Gaussian weight function on N dimensions to be
ω(~x) =
1
2
e−x /2 ,
N/2
(2π)
x2 = x21 + · · · + x2N .
(6.1.1)
Now define the nth order Hermite polynomial by
(n)
Hi1 ...in (~x) =
(−1)n
(∂i1 . . . ∂in ) ω(~x),
ω(~x)
63
~x ∈ RN .
(6.1.2)
(n)
Then H(n) is a tensor of order n and each entry Hi1 ...in is a polynomial of order n,
[GH], p. 327. By direct calculation
H(0) = 1
(1)
Hi = xi
(6.1.3)
(2)
Hij
= xi xj − δij
(3)
Hijk = xi xj xk − (xi δjk + xj δik + xk δij ).
To show orthonormality only requires the definition and basic calculus properties, see
[GH], p. 329. Now define the Hermite expansion for a function f (for example in
L2 (RN , ω)) as
f (~x) = ω(~x)
∞
N
X
X
n=0
i1
!
N
X
1 (n)
(n)
ai1 ...in Hi1 ...in (~x) ,
···
n!
=1
i =1
(6.1.4)
n
with
(n)
ai1 ...in
Z
=
(n)
Hi1 ...in (~x)f (~x)d~x,
(6.1.5)
cf. [GH], p. 329, or [GP], p. 269 (noting a slight discrepancy in the Gaussian factor
between the two references).
Having in mind distributions close to the Gaussian, we also make use below of the
lowest order approximation of expressions of the form
Z
(n)
(m)
Hi1 ,...,in (x)Hl1 ...lm (x)f (x)dx.
(6.1.6)
In other words, keep only the first term in the expansion (6.1.4)
f0 := a(0) (f )ω,
64
(6.1.7)
and set
(n,m)
Qi1 ,...,in ,l1 ...lm (f )
Z
:=
(n)
(m)
Hi1 ,...,in (x)Hl1 ,...,lm (x)f0 (x)dx
Z
(n)
(m)
(0)
= a (f ) Hi1 ,...,in (x)Hl1 ...lm (x)ω(x)dx
= a(0) (f )δnm
X
(6.1.8)
δi1 l1 · · · δin lm ,
Sn
where the summation is over the n! permutations of {i1 , . . . , in }. For x in R3 it is
easy to calculate
Q(0,0) = a(0) (f )
(2,2)
Qij,lk = a(0) (f ) (δil δjk + δik δjl )
3
X
(2,2)
Qii,lk
(0)
= a (f )
i=1
3
X
3
X
(δil δik + δik δil ) = 2 a(0) (f )δij
i=1
(2,2)
Qii,jj = a(0) (f )
i,j=1
(3,3)
Qijk,lmn
3 X
3
X
(δij δij + δij δij ) = 2 a(0) (f )
i=1 j=1
3
X
δij = 6 a(0) (f )
i,j=1
(0)
= a (f ) (δil δjm δkn + δil δjn δkm + δim δjl δkn + δim δjn δkl
+ δin δjm δkl + δin δjl δkm )
3
X
(3,3)
Qijj,lmm
j,m=1
(0)
= a (f )
3 X
3
X
(δil δjm δjm + δil δjm δjm + δim δjl δjm + δim δjm δjl
j=1 m=1
+ δim δjm δjl + δim δjl δjm )
= a(0) (f ) δil
3 X
3
X
!
(δjm δjm + δjm δjm ) + 4δil
= 10 a(0) (f )δil .
j=1 m=1
(6.1.9)
6.2
Grad’s Macroscopic Quantities
Grad in [GP], working with the Boltzmann equation and assuming all measures to
be absolutely continuous, has made the point that the Hermite coefficients of the
density functions can be thought to represent certain physical quantities. These
are the analogues of standard thermodynamic quantities, only this time away from
equilibrium.
65
In our setting, and as a continuation of section 3.1.4, assume
Mt (dx, dv) = f (t, x, v)dxdv.
(6.2.1)
From Chapter 2, we know that the first marginal of Mt is µt (dx). Therefore
Z
µt (dx) =
f (t, x, v)dv dx.
(6.2.2)
This precisely corresponds to Grad’s ρ(t, x) dx, the mass dentity:
Z
f (t, x, v)dv.
ρ(t, x) =
(6.2.3)
Similarly, the disintegration formula for u precisely corresponds to
1
~u(t, ~x) =
ρ(t, x)
Z
~v f (t, ~x, ~v )d~v ,
(6.2.4)
cf. (3.1.24). The velocity fluctuations, whose moments we have seen naturally occurring in the hydrodynamic equations, are defined as
c(t, x, v) = v − u(t, x).
(6.2.5)
c(t, x, v)f (t, x, v) dv = 0.
(6.2.6)
As for any fluctuation,
Z
Other moments give the tensors
Z
Pij (t, x) =
ci cj f (t, x, v)dv
(6.2.7)
Z
Sijk (t, x) =
ci cj ck f (t, x, v)dv.
As is standard, a trace free stress tensor is created by first defining the pressure as
p(t, x) =
1X
Pii (t, x)
3 i
66
(6.2.8)
and the stress tensor as
pij (t, x) = Pij (t, x) − p(t, x)δij .
(6.2.9)
In particular, Grad’s terminology pre-supposes that the stress tensor does not have
any contributions from the self interaction Φ0 . This might well be consistant with
Grad’s starting point (the Boltzmann equation, i.e. rarified gases), but we have seen
that, starting from the Hamilton equations, self-interaction contributions may not be
excluded a priori from the stress tensor. In fact, according to Irving & Kirkwood it is
the interaction term that is important for fluids, see [IK] p. 823 for their contention.
Grad’s Pij translates into our
Z
vi − ui (t, x) vj − uj (t, x) Mt,x (dv),
(6.2.10)
appearing, in its weak form, as the first term on the right-hand side of (3.1.32).
Further, Grad defines the heat-flow vector as
qi (t, x) =
X1
r
1
Sirr (t, x) =
2
2
Z
ci c2 f (t, x, v)dv,
(6.2.11)
compare this with the corresponding term in the energy equation (4.4.4).
The internal energy is defined per unit of mass as
1
e(t, x) =
2ρ(t, x)
Z
c2 f (t, x, v)dv.
(6.2.12)
Setting
3
e(t, x) = R T (t, x),
2
(6.2.13)
for R “the gas constant” and T (t, x) a “temperature,” as Grad does, is a way to
recover the standard equation “of state”
p(t, x) = ρ(t, x)R T (t, x).
67
(6.2.14)
Observe that here the dimension of RT is velocity and rescale the probability densities
to
(RT (t, x))3/2
f (t, x, v),
ρ(t, x)
g(t, x, v) =
(6.2.15)
making them dimensionless for x in R3 . Also normalize the velocity fluctuations to
w=√
c
v−u
=√
.
RT
RT
Then simple calculations show that for all t and x
Z
g(w)dw = 1
Z
wg(w)dw = 0
Z
w2 g(w)dw = 3,
(6.2.16)
(6.2.17)
cf. [GK], p. 355. Then (6.1.3) and (6.1.5) for g give
a(0) = 1
(1)
ai = 0
(2)
pij
p
Sijk
,
= √
p RT
(6.2.18)
aij =
(3)
aijk
writing Hermite coefficients as thermodynamic quantities. Up to third order then
wi wj − δij
wi wj wk − wi δjk − wj δik − wk δij
√
g(t, x, w) =ω(w)
~ 1 + pij (t, ~x)
+ Sijk (t, ~x)
2p(t, w)
~
6p(t, ~x) RT
+ higher order terms,
(6.2.19)
or
1
pij (t, ~x)
Sijk (t, ~x)
Si (t, ~x)
w2 /2
√
√
g(t, ~x, w)
~ =
e
1+
wi wj +
wi wj wk −
wi
(2π)3/2
2p(~x, t)
6p(t, ~x) RT
2p(~x, t) RT
+ higher order terms.
(6.2.20)
68
Turning back to f this gives
ρ
2
f (t, ~x, ~c) =
ec /2RT
3/2
(2πRT )
pij (t, ~x)ci cj Sijk (t, ~x)ci cj ck
Si (t, ~x)ci
1+
+
−
2
2
2p(t, ~x)RT
6p(t, ~x)R T
2p(t, ~x)RT
+ higher order terms.
(6.2.21)
In particular, a(0) (f ) = ρ.
6.3
Landau-Lifshitz Formulas
The motivation to examine the coefficients (6.1.8) comes from the work of Tsugé and
Sagara [TS]. Their quite formal considerations are now summarized: For h = h(x, y)
a product density use double series of Hermite polynomials:
h(x, y) = ω(x)ω(y)
∞
X
Qi
1 ,...,iJ ,l1 ...lK
J!K!
J,K
(J)
(K)
Hi1 ,...,iJ (x)Hl1 ...lK (y),
(6.3.1)
where
(J,K)
Qi1 ,...,iJ ,l1 ...lK
Z Z
=
(K)
(J)
Hi1 ,...,iJ (x)Hl1 ...lK (y)h(x, y)dxdy.
(6.3.2)
For h(x, y) = δ(x − y)f (x), i.e. under the assumption that the product density is
concentrated on the diagonal, and keeping only the first term in (6.3.2) gives
Z Z
(J,K)
(J)
(K)
Hi1 ,...,iJ (x)Hl1 ,...,lK (y)δ(x − y)f (x)dxdy
Qi1 ,...,iJ ,l1 ...lK =
Z
(J)
(K)
(0)
= a (f ) Hi1 ,...,iJ (x)Hl1 ...lK (x)ω(x)dx
(6.3.3)
X
= a(0) (f )δnm
δi1 l1 · · · δin lm ,
Sn
where the summation is over the n! permutations of {i1 , . . . , in }.
Tsugé and Sagara work toward a Boltzmann type equation and argue that the first
term captures thermal fluctuations. For this they show that this term captures the
Landau-Lifshitz formulas for such fluctuations. Part of Tsugé-Sagara argument can
69
be reconstructed in our setting (without use of product measures) and is contained
in the following:
Proposition 6.3.1.
Z 1
ci cj − c2
3
1 2
p2
2
cl cm − c f0 (w)dw =
δil δjm + δjl δim − δij δlm . (6.3.4)
3
ρ
3
Proof. From (6.1.3), substituting via (6.2.16),
1 2
1 (2)
c =RT
H (w) + 1
3
3
1 2
1
(2)
(2)
ci cj − c =RT Hij (w) − δij H (w) ,
3
3
(6.3.5)
where
(2)
H (w) =
3
X
(2)
Hii (w).
(6.3.6)
i=1
Write
1 (2)
1
1 (2)
H (w) + 1
H (w) + 1 = H(2) (w)H(2) (w) + 1 + mixed terms.
3
3
9
(6.3.7)
and notice that mixed terms will not contribute after integration by (6.1.8). Then
Z
Z 1 4
1 (2)
1 (2)
2
(0)
a (f )
c ω(w)dw = ρ(RT )
H (w) + 1
H (w) + 1 ω(w)dw
9
3
3
1 (2,2)
(0,0)
=
Q
+Q
(RT )2
9
(6.3.8)
6
2 (0)
=
+ 1 (RT ) a (f )
9
5 p2
.
=
3ρ
Similarly for the stress tensor
Z 1 2
1 2
(0)
a (f )
ci cj − c
cl cm − c ω(w)dw
3
3
Z 1
1
(2)
(2)
2
(2)
(2)
= ρ(RT )
Hij (w) − δij H (w)
Hlm (w) − δlm H (w) ω(w)dw
3
3
1
1
1
(2,2)
(2,2)
(2,2)
2
(2,2)
= (RT ) Qij,lm − δij Q,lm − δlm Qij, + δij δlm Q
3
3
9
2
p
2
=
δil δjm + δjl δim − δij δlm .
ρ
3
(6.3.9)
70
The significance of this is that, as in [TS], we recover well-known formulas for nonturbulent fluctuations. (6.3.8) and (6.3.9) should be compared to formulas (132.13)
of [LL] and (62) of [FU]. Both [LL] and [FU] obtain their formulas macroscopically
by adding random perturbations to the hydrodynamic equations.
([TS] applies a similar process for the heat flow vector. A logical leap is required
to obtain the desired result. First note that the random variable describing the heat
flow vector could be defined as
1 2 (RT )3/2 (3)
(1)
ci c =
Hi (v) + 5Hi (v) .
2
2
(6.3.10)
The logical leap is to consider only the factors that contribute to the heat in expectation. Indeed
(RT )3/2
2
Z
(3)
Hi (w)f (w)dw
Z
(RT )3/2
=
wi w2 − 5wi f (w)dw
2
Z
5(RT )3/2
=qi −
wi f (w)dw
2
(6.3.11)
=qi .
(3,3)
Therefore only the Qi,j
needs to be considered because this part gives the heat flow
vector. Thus
(RT )3 ρ
4
Z
(RT )3 (3,3)
Qi,j
4
5p2 RT
=
δij .
2ρ
H(3) (w)H(3) (w)ω(w)dw =
Then (6.3.12) is the final Landau-Lifshitz formula.)
71
(6.3.12)
Chapter 7
Future Work
• Use an Edgeworth type expansion, see [BM], in section 6.3, instead of Hermite polynomials, for better convergence. Determine if the coefficients are still
related to physical quantities.
• Determine anologues for the work of chapter 6 when the measures are not
absolutely continuous.
• Tsugé’s work indicates that the structure of the limit measures on the diagonal
is of importance. Examine this for the general case.
• Examine exactly what constitutes the heat part of the energy equation and
attempt to use measure theoretic ideas to describe it (as was done for the
average velocity).
• Morrey, in his context, writes a convex function that behaves like entropy. Is
there an analogue of this in our setting?
• Remove as many of the conditions as possible. Especially those for the convergence of the measures containing the potential.
• Eliminate the rescaling σN by keeping the N dependence in the potential ΦN .
72
• Instead of
1 0
Φ
σN
|Xi,j |
σN
< BN ,
(7.0.1)
assume summability in N to repeat the Picard constructions.
• Repeat our argument for potentials that are not strictly decreasing to allow for
Leonard-Jones type potentials.
• For the absolute continuity assumption of chapter 6 to be of value, it should be
preserved throughout the evolution of the system. It is clear that if initially µ0 is
absolutely continuous with respect to the Lebesgue measure and the differential
equation admits a solution operator Xt on [0, T ], then µt is absolutely continuous
with respect to Lebesgue on [0, T ]: indeed, since µ0 (dx) = g(x)dx, then
g(Xt (x))dx = (Xt )# µ0 (dx) = µt (dx).
(7.0.2)
Conditions that ensure the existence of a solution operator can be found in
[AGS], Chapter 8, in particular
Proposition 7.0.2. Let µt be continuous for t ∈ [0, T ] weakly solving the continuity equation,
Z
0
T
Z
∂t ϕ(t, x) + u(x, t) · ∇x ϕ(t, x) µt (dx)dt = 0.
(7.0.3)
Rd
Also ask that u satisfies the conditions
Z
T
0
|u(x, t)|µt (dx)dt < ∞
sup |ut | + Lip(ut , B) dt < ∞.
Rd
0
Z
Z
T
B
73
(7.0.4)
Then for µ0 -a.e. x ∈ Rd the system
Xs (x, s) = x,
d
Xt (x, s) = ut (Xt (x, s))
dt
(7.0.5)
has a globally defined solution Xt (x) on [0, T ] with
µt = (Xt )# µ0 for all t ∈ [0, T ].
(7.0.6)
• Regarding Theorem 5.4.1, and given that M is the weak limit of MN ’s as N →
∞, show that
Z
1
u(x) = lim lim+
N →∞ ρ→0 µN (Bρ )
uN (x) µN (dx).
(7.0.7)
pr−1
1 (Bρ )
As the right-hand side is, by definition,
lim lim+
N →∞ ρ→0
1
#{xi ∈ Bρ }
X
uN (xi ),
(7.0.8)
{i: xi ∈Bρ }
establishing (7.0.7) is the same as reproducing (the limit of) Reynolds’s exact
formulas (and justifying some of Morrey’s definitions).
• Find a rigorous mathematical basis for or against Irving & Kirkwood’s comment
on the insignificance of the “kinetic” part of the stress tensor in the momentum
equation, cf. our comments before (6.2.10).
74
Appendix A
The Case of No Heat
Theorem A.0.3. If µ(N ) weakly converges to µ and (3.3.4) holds, M lies on the
graph of u(x).
Proof. Weak convergence of µn to µ implies
Z
Z
g(u(x))µ(dx) ≤ lim inf
g(un (x))µn (dx),
n
(A.0.1)
for g convex and lower semi-continuous, see [AGS], in particular for g(t) = tp .
If in addition
Z
lim sup
Z
p
kun (x)k µn (dx) ≤
ku(x)kp µ(dx),
(A.0.2)
n
(the “strong convergence” assumption) then
Z
Z
p
ku(x)kp µ(dx).
kun (x)k µn (dx) =
lim
n
(A.0.3)
In this case the limit measure M is on the graph of u. Indeed,
Z
Z
p
|x2 | γ(dx1 , dx2 ) ≤ lim inf
n
X2
|x2 |p γn (dx1 , dx2 )
(A.0.4)
X2
since γn ⇒ γ, and the right-hand side is in fact
Z
lim
n
kun (x)kp µn (dx),
75
(A.0.5)
which, by (A.0.3) is
Z
ku(x)kp µ(dx).
(A.0.6)
Therefore (A.0.4) becomes
Z
Z
p
|x2 | γ(dx1 , dx2 ) ≤
ku(x)kp µ(dx),
(A.0.7)
X2
or
Z Z
Z
p
|x2 | Mx1 (dx2 )µ(dx1 ) ≤
Z Z
ku(x)kp µ(dx)
Z
p
|x2 | Mx1 (dx2 )µ(dx1 ) − ku(x)kp µ(dx) ≤ 0
Z Z
p
p
⇔
|x2 | Mx1 (dx2 ) − ku(x)k µ(dx) ≤ 0
Z
p Z Z
p
⇔
|x2 | Mx1 (dx2 ) −
x2 Mx1 (dx2 )
µ(dx) ≤ 0,
⇔
(A.0.8)
where the second integrand is component wise. Then by Jensen, the whole integrand
is positive, therefore must be 0 for almost all x, i.e.
Z
p
p
Z
|x2 | Mx1 (dx2 ) =
x2 Mx1 (dx2 )
,
(A.0.9)
for almost all x. This is exactly the statement that for these x’s Jensen’s inequality
is in fact equality. Therefore for these x’s Mx1 (dx2 ) is a Dirac measure. Since its
average is u(x), it is the Dirac at u(x).
76
Appendix B
Proof of Disintegration
We present a sketch of the proof of the Disintegration Theorem following [DM], p.
78-III.
Theorem B.0.4 (Disintegration). Let M be a probability measure on R3 × R3 . Let
π1 : R3 × R3 → R3 the first projection of the two copies of R3 and µ = (π1 )# M . Then
there exists a family of measures Mx , x ∈ R3 that is µ-almost everywhere determined
and µ-measurable such that for f (x, y) ∈ L1 (R6 , M )
Z
f (x, y)M (dx, dy) =
R6
Z
Z
f (x, y)Mx (dy) µ(dx).
R3
(B.0.1)
R3
Proof. For f : R6 → R+ the measure f M is, of course, absolutely continuous w.r.t.
M and the measure (π1 )# (f M ) is absolutely continuous w.r.t. µ, i.e. there is density
function df : R3 → R such that
(π1 )# (f M ) = df µ.
(B.0.2)
f 7→ df (x)
(B.0.3)
Then for fixed x ∈ R3 the map
defines a functional. After some density arguments, one finds that the Riesz repre77
sentation theorem applies to give a measure Mx such that
Z
df (x) =
f (x, y)Mx (dx, dy).
(B.0.4)
R6
But then
Z
Z
Z
f (x, y)Mx (dx, dy)µ(dx) =
R3
R6
df (x)µ(dx)
ZR
3
=
(π1 )# (f M )(dx)
3
ZR
=
1(π1 )# (f M )(dx)
(B.0.5)
3
ZR
1(f M )(dx, dy)
=
6
ZR
=
f (x, y)M (dx, dy).
R6
In fact, the support of Mx (dx, dy) in R6 is π1−1 ({x}): for any f vanishing on π1−1 ({x})
the measure (π1 )# (f M ) is supported away form x in R3 , therefore df (x) = 0, or by
definition of Mx ,
Z
f (x, y)Mx (dx, dy) = 0.
Therefore Mx can be identified with a measure on R3 .
78
(B.0.6)
Bibliography
[ABGS] Aoki, K.; Bardos, C.; Golse, F.; Sone, Y. Derivation of Hydrodynamic
Limits from Either the Liouville Equation or Kinetic Models: Study of
an Example. Mathematical analysis of liquids and gases (Kyoto, 1999).
Sūrikaisekikenkyūsho Kokyūroku No.1146 (2000), 154–181.
[AFP] Ambrosio, L., N. Fusco, D. Palara Functions of Bounded Variation and Free
Discontinuity Problems Oxford, 2000.
[AGS] Ambrosio, L.; Gigli, N.; Savaré, G. Gradient Flows in Metric Spaces and
in the Space of Probability Measures. Lectures in Mathematics ETH Zürich.
Birkhäuser Verlag, Basel, 2005. (Second edition 2008. All references here to
first edition.)
[Bir]
Birkhoff, Garrett Lattices in Applied Mathematics. 1961 Proc. Sympos. Pure
Math., Vol. II pp. 155–184 American Mathematical Society, Providence, R.I.
[BM] Blinnikov, S.; Moessner, R. Expansions for Nearly Gaussian Distributions Astron. Astrophys. Suppl. Ser. 130, (1998), 193–205.
[B]
Boltzmann, L. Lectures on Gas Theory University of California Press, 1964.
[Bos] Boscovich, R. Theoria Philosphiae Naturalis, Venice, 1763.
79
[C]
Cartan, H. Differential Calculus Hermann, Paris, 1971.
[CL]
Coddington, E. A.; Levinson, N. Theory of Ordinary Differential Equations
McGraw-Hill 1955.
[DM] Dellacherie, C.; Meyer, P. Probabilities and Potential. Mathematics Studies,
Volume 29, North-Holland Publishing Company, Amsterdam and New York,
1978
[D]
Dostoglou S. On Hydrodynamic Averages. Journal of Mathematical Sciences
Vol. 189, No. 4, (2013), 582-595.
[FU]
Fox, R. F. and Uhlenbeck, G. E. Contributions to Non-Equilibrium Thermodynamics. I. Theory of Hydro-dynamical Fluctuations, Physics of Fluids 13,
(1970), 1893-1902.
[G]
Gibbs, J.W. Elementary Principles in Statistical Mechanics 1902.
[GH] Grad, H. Note on N-dimensional Hermite Polynomials. Comm. Pure Appl.
Math., 2 (1949) 325-330.
[GK] Grad, H. On the Kinetic Theory of Rarefied Gases Comm. Pure Appl. Math.
2, (1949), 331–407.
[GP]
Grad, H., Principles of the Kinetic Theory of Gases Thermodynamik der Gase.
Series: Handbuch der Physik Encyclopedia of Physics, Springer Berlin Heidelberg (Berlin, Heidelberg), Edited by S. Flügge, 3 / 12, pp. 205-294
[HCB] Hirschfelder, J.O.; Curtiss, C.F.; Bird, R. B. Molecular Theory of Gases and
Liquids Wiley, 1954.
80
[IK]
Irving, J.H and J.G. Kirkwood The Statistical Mechanical Theory of Transport
Processes. IV. The Equations of Hydrodynamics J. Chem. Phys. 18 (6), (1950),
817-829.
[JtH] Jepsen D.W., D. ter Haar A Derivation of the Hydrodynamic Equations from
the Equations of Motion of Collective Corrdinates Physica 28, (1968) 70-75.
[K]
Klimentovich, Y.L. The Statistical Theory of Non-equilibrium Processes in a
Plasma MIT Press, 1967.
[KdF] Kampé de Feriet, J. Sur un problem d’algebre abstrait posé par la definition
de la moyenne dans la theorie de la turbulence Ann. Soc. Sci. Bruxelles, 1, 63
(1949), 156- 172.
[LL]
Landau, L. D.; Lifshitz, E.M. Fluid Mechanics Vol 6 (1st ed.). Pergamon Press
1959
[L]
Lanford, O.E. “Time Evolution of Large Classical Systems,” Dynamical Systems and Applications, J. Moser ed., Lecture Notes in Physics, vol 38, (1975),
Springer.
[Max] Maxwell J.C. “On the Dynamical Theory of Gases”, Philosophical Transactions of the Royal Society of London, Vol. 157, (1867), pp. 49–88.
[M]
Morrey, C.B. On the Derivation of the Equations of Hydrodynamics from Statistical Mechanics, Comm. Pure and Applied Math. VIII, (1955), pp. 279–326.
[Mo]
Moy, S.-T. C. Characterizations of Conditional Expectation as a Transformation on Function Spaces. Pacific J. Math. 4, (1954). 47-63.
81
[N]
Noll, W. Die Herleitung der Grundgleichen der Thermomechanik der Kontinua
aus der Statistischen Mechanik J. Rational Mech. and Analysis 4 (1955), 627–
646.
[R]
Reynolds, O. On the Dynamical Theory of Incompressible Viscous Fluids and
the Determination of the Criterion Philosophical Transactions of the Royal
Society, vol. 186, (1895), 123–164.
[Rot] Rota, Gian-Carlo Reynolds Operators. 1964 Proc. Sympos. Appl. Math., Vol.
XVI pp. 70–83 Amer. Math. Soc., Providence, R.I.
[Ru]
Ruelle, D. States of Classical Statistical Mechanics J. Math. Phys. 8 (1967)
1657–1668.
[TS]
Tsugé, S.; Sagara, K. On Master’ Boltzmann Equation Tokyo, University,
Institute of Space and Aeronautical Science, Report no. 516, vol. 39, (Nov.
1974), p. 235-254.
[T]
Tsugé, S. The Boltzmann and Klimentovich Formalisms with Reference to
Turbulence Resolution Proceedings of the 8th International Symposium on
Rarefied Gas Dynamics. Academic Press, 1974.
82
VITA
Nicholas C. Jacob was born July, 1983 in St. Louis Missouri. In 2005 he graduated
from University of Missouri-Columbia with a Bachelor of Science degree in Mathematics and Physics. Then he continued his education at University of Missouri, where
he has been working toward a Ph.D in Mathematics. Starting in Fall 2013, he will
begin at East Central University of Oklahoma as an Assistant Professor.
83