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

sciences and the reader is strongly advised to consult it.

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.