 # H THEOREM AND BEYOND

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H THEOREM AND BEYOND BOLTZMANN'S ENTROPY IN TODAY'S MATHEMATICS Munich, October 11, 2006 Cedric Villani ENS Lyon France

• volume treatise

• vorlesungen uber

• mark kac

• boltzmann's side

• boltzmann

• collision kernel

• ∂f ∂t

• mathematical terms

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##### Boltzmann

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H THEOREM AND BEYOND
BOLTZMANN’S ENTROPY IN
TODAY’S MATHEMATICS
Munich, October 11, 2006
C´edric Villani
ENS Lyon
FrancePunchline
Since their introduction by Boltzmann, entropy and the
H Theorem have been tremendous sources of inspiration
- to understand our world in mathematical terms
- to attack many (purely) mathematical problemsSome quotations by mathematicians
All of us younger mathematicians stood by Boltzmann’s
side.
Arnold Sommerfeld (1895)
Boltzmann’s work on the principles of mechanics suggest
the problem of developing mathematically the limiting
processes (...) which lead from the atomistic view to the
laws of motion of continua.
David Hilbert (1900)
Boltzmann summarized most (but not all) of his work in
a two volume treatise Vorlesungen u¨ber Gastheorie. This
is one of the greatest books in the history of exact
Mark Kac (1959)1872: Boltzmann’s H Theorem
The Boltzmann equation models the dynamics of rareﬁed
gases via the position-velocity density f(x,v):
∂f
+v·∇ f =Q(f,f)
x
∂t
Z
h i
′ ′
Q(f,f) = B f(v)f(v )−f(v)f(v ) dv dσ
∗ ∗

3 2
R ×S
v

B =B(v−v ,σ) collision kernel

According to this model, under ad hoc boundary
conditions, the entropy S is nondecreasing in time:
Z
S(f) =−H(f) :=− f(x,v)logf(x,v)dvdx
3
Ω×R
vThe Entropy Production
A positive amount of entropy is produced, unless f(x,v)
at time t is locally Maxwellian (hydrodynamic):
2
|v−u(x)|

2T(x)
ρ(x)e
f(x,v) =M (v) = .
ρ,u,T
3/2
(2πT(x))
R
−dH/dt = D(f(t,x,·))dx,
Ω
where D(f) =
Z
′ ′
1 f(v)f(v )
′ ′ ∗
B[f(v)f(v )−f(v)f(v )]log ≥ 0

4 f(v)f(v )

v,v ,σ

D(f) = 0⇐⇒f(v) =M (v)
ρuTDisclaimer
130 years later, can we make a rigorous proof of the H
Theorem? Not in full generality
Obstacle: A “slight analytical diﬃculty” (as Euler could
have said): the existence of smooth nice solutions, which
is known only in particular cases.Why is the H Theorem beautiful?
- Starting from a model based on reversible mechanics +
statistics, Boltzmann ﬁnds irreversibility
- This is a theorem — as opposed to a postulate
More “mathematical” reasons:
- Beautiful proof, although not perfectly rigorous
- A priori estimate on a complicated nonlinear equation
- The H functional has a statistical (microscopic)
meaning: how exceptional is the distribution function
- Gives some qualitative information about the evolution
of the (macroscopic) distribution function
These ideas are still crucial in current mathematicsFour remarkable features of the H functional/Theorem
1) A priori estimate on a complicated nonlinear equation
2) Statistical (microscopic) meaning: how exceptional is
the distribution function
3) “Explanation” of the hydrodynamic limit
4) Qualitative information about the evolution of the
(macroscopic) distribution functionThe H Theorem as an a priori estimate
Z Z
t
H(f(t))+ D(f(s))dxds ≤H(f(0)).
0
In fact two a priori estimates!
Finiteness of the entropy is a weak and general way to
prevent concentration (“clustering”).
First important use : Arkeryd (1972) for the spatially
homogeneous Boltzmann equation.
Both estimates are crucial in the DiPerna-Lions stability
theorem (1989): Entropy, entropy production and energy
bounds guarantee that a limit of solutions of the BE is a
solution of the BETheoretical importance of these bounds
• For the full Boltzmann equation, still the only
general nonlinear estimate known to this day!
• Nowadays entropy and entropy production estimates
(robust and physically signiﬁcant) are being used
systematically in PDE/probability theory, for hundreds
of models and problems.

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