The Kac Conjecture

RECENT PROGRESS ON KAC’S PROBABILISTIC APPROACH

TO KINETIC THEORY

International Conference on Particle Systems and PDE’s,

Univ. Minho, December 7, 2012

Maria C Carvalho

This talk presents results of collaboration with Eric Carlen, and Michael Loss.

The Kac Collision Process

For _{N ∈} N, _{p ∈} R³ and _{E > |p|}², let _SN,E,p be the set

consisting of N–tuples ~v = (v₁, . . . , v_N) of vectors v_j in R³ with

1 N

N

X

j=1

|v^j|² = E and 1 N

N

X

j=1

v_j = p .

In what follows, a point _{~v ∈ SN,E,p} specifies the velocities of a collection of N particles with unit mass.

We consider a Markov jump process on _SN,E,p that was introduced by Mark Kac to describe a random binary collision process for the N particles.

When the collision process begins, associated to each pair (i, j), i < j, there is an exponential random variable T_i,j with parameter

λ_i,j = N N 2

!₋₁

|vⁱ − v^j|^α , ^(0.1)

where _{0 ≤ α ≤ 2}, and α = 1 is the case of main interest, corresponding to “hard-sphere collisions”.

T_i,j is the waiting time for particles i and j to collide, and the set of these random times is taken to be independent. The first collision occurs at time

T = min{T } . ^(0.2)

At the time T, the pair (i, j) furnishing the minimum collide:

The state of the process “jumps” from (v₁, . . . , v_N) to (v₁, v₂, . . . , v_i^∗, . . . , v_j^∗, . . . , v_N) ,

where only v_i and v_j have changed. Since the process models momentum and energy conserving collisions, we require that

v_i^∗ + v_j^∗ = v_i + v_j and |vi^∗|² + |vj^∗|² = |vⁱ|² + |v^j|² . Then by the parallelogram law, it follows that

Given v_i and v_j, the kinematically possible collisions of particles i and j may be parameterized in terms of a unit vector _{σ ∈ S}², the unit sphere in R³ as follows:

v_i^∗(σ) = v_i + v_j

2 + |vⁱ − v^j|

2 σ

v_j^∗(σ) = v_i + v_j

2 − |vⁱ − v^j|

2 σ

σ is selected according to the following law: There is a non-negative function b on _{[−1, 1]} such that for any fixed σ^′ ∈ S²,

Z

b(σ · σ^′)dσ = 1

The object of our investigation is the spectral gap for the generator of the Markov semigroup associated to this process. For any continuos function f on _SN,E,p, define

L_N,αf (~v) = 1

h lim

h→0 E{f(~v(h) − f(~v)) | ~v(0) = ~v } . One readily computes that

L_N,αf (~v) = −N N 2

!₋₁

X

i<j

|vⁱ − v^j|^α×

 

Here,

(R_i,j,σ~v)_k =







v_i^∗(σ) k = i v_j^∗(σ) k = j v_k k 6= i, j

.

Introducing the notation

[f ]^(i,j)(~v) :=

Z

S²

b



σ · v_i − v^j

|vⁱ − v^j|



f (R_i,j,σ~v)dσ , we can write the generator more briefly as

L f (~v) = −N N!₋₁

X |v − v |^α h

f (~v) − [f]^(i,j)(~v)i

Note that

cos θ := σ · v_i − v^j

|vⁱ − v^j| = v_i^∗ − v_j^∗

|v_i^∗ − v_j^∗| · v_i − v^j

|vⁱ − v^j| .

This shows that rates for the jump from ~v to R_i,j,σ~v and from R_i,j,σ~v to ~v are equal. This is the property of “detailed

balance” or “microscopic reversibility”. The analytic

expression of this is self-adjointness of the generator L_N,α.

Define the Dirichlet form _EN,α by

E^N,α(f, f ) = −hf, L^N,αf iL²(σN) . A simple computation shows that

E^N,α(f, f ) = N 2

N 2

!₋₁

X

i<j

Z

S^N,E,p

Z

S² |vⁱ − v^j|^α b



σ · v_i − v^j

|vⁱ − v^j|



[f (~v) − f(R^i,j,σ~v)]² dσdσ_N . One sees from this expression that L_N,α is a negative

Provided b is continuous at 1, L_N,αf = 0 if and only if f is constant. We are interested in the spectral gap of the

operator L_N,α on L²(S^N,E,p, σ_N):

∆_N,E,p = inf 

EN,α(f, f ) : hf, 1iL² = 0 and kfk²_L² = 1 . We now investigate the dependence of ∆_N,E,p on N, E and p.

Scaling and dependence on E and p

For fixed N, the dependence of ∆_N,E,p on E and p is quite simple: Consider the “shift and scaling transformation”

φ_E,p(v₁, . . . , v_N) := 1

pE − |p|² (v₁ − p, . . . , v^N − p)

which identifies _SN,E,p with _SN,1,0. This point transformation induces the unitary operator U_E,p from L²(S^N,1,0, σ_N) to

L²(S^N,E,p, σ_N) given by U_E,pf = f ◦ φ^E,p A simple computation then shows that

EN,α(U_E,pf, U_E,pf ) = (E − |p|²)^α/2EN,α(f, f ) .

The dependence of ∆_N,E,p on N is not so simple.

Nonetheless, we have seen that the problem of estimating the quantity ∆_N,E,p is essentially the same as the problem of estimating ∆_N,1,0. We therefore simplify our notation:

DEFINITION 0.1 (Spectral gap). The spectral gap for the N particle Kac model is the quantity

∆_N := ∆_N,1,0 . ^(0.4)

The Kac Conjecture

Kac conjectured that

lim inf

N →∞ ∆_N > 0 .

This has been proved in the case α = 0; see by Carlen, C., and Loss, and later Carlen, Geronimo and Loss proved that lim_{N →∞} ∆_N exists, and computed the exact value of this

limit for many choices of b. The first result in this direction was by Janvresse, who treated a one-dimensional

simplified version of the model that was also discussed by Kac. Her method gave no explicit lower bound.

All results up to now concerned the α = 0 case of uniform jump rates.

Recently, Carlen, C., and Loss have proved:

THEOREM 0.2 (Spectral gap for the Kac Model with 0 ≤ α ≤ 2^{). For}

each function b ^on [−1, 1] that is continuous and strictly positive at 1^,

and for each α ∈ [0, 2], there is a strictly positive constant C ^depending

only on b ^and α, and explicitly computable, so that

∆_N ≥ C > 0

for all N^.

The significance of the spectral gap

The Kac process was not introduced as a model of the actual, physical collision process in a gas of molecules.

Rather, it was introduced as the simplest process

conceivable from which one could deduce, in the limit of a large number of particles, the non-linear spatially

homogeneous Boltzmann equation, which is the basic evolution equation of kinetic theory.

Let

v^′ = v + v_∗

2 + |v − v_∗|σ

2 , v_∗^′ = v + v_∗

2 − |v − v_∗|σ

2 .

The spatially homogeneous Boltzmann equation is a

non-linear equation for the evolution in t > 0 of a probability density n(v, t) for _{v ∈} R³:

∂

∂tn(v, t) = Q(n)(v, t) , where

Q(n)(v, t) = Z

R³×S² B(v − v_∗, σ)(n^′n^′_∗ − nn_∗)dσdv_∗ , n = n(v, t), n^′ = n(v^′, t), n_∗ = n(v_∗, t) and n^′_∗ = n(v_∗^′ , t).

The function B(z, σ) in describes the rate at which the various kinematically possible collisions take place. This rate depends on the interaction between the molecules.

Maxwell determined that if the force law that governs the interaction between pairs of molecules in the dilute gas is an inverse power of the distance separating them, B takes the form

B(z, σ) = b(σ · z/|z|)|z|^α

with the exponent α depending on the power in the interaction.

This is what motivates the particular form we have assumed for law used to select σ in the Kac process. The following ranges of α are usually distinguished:

α < 0 : soft collisions

α = 0 : Maxwellian molecules 0 < α ≤ 1 : hard collisions

α = 2 : super hard collisions

For hard sphere collisions, the most physically significant case of hard collisions,

We are interested in the rate of relaxation to equilibrium, both for the Kac process, and for the Boltzmann equation.

The equilibrium; i.e, steady state solutions of the Boltzmann equation are the Maxwellian distributions:

M (v) =

 1 2πΘ

3/2

e^{−|v−u|2/2Θ}

where Θ is a positive number, and _{u ∈} R³. To see that these are steady states, note that

M (v)M (v_∗) = M (v^′)M (v_∗^′ )

for all v, v and σ, and hence the integrand in _Q vanishes

To quantify the rate of approach to equilibrium for the Boltzmann equation is a mathematically and physically significant problem.

In this regard, one quantity of interest is the spectral gap of the linearized Boltzmann equation .

To linearize the Boltzmann collision operator Q, fix Θ = 1 and u = 0, and write M to denote the corresponding

Maxwellian. Let _H denote the Hilbert space

H = L²(R³_{, M (v)dv)}. Define the linearized Boltzmann operator _L by

Q((1 + ǫh)M, (1 + ǫh)M )(v) = ǫM (v)Lh + O(ǫ²) .

This yields

Lh(v) = Z

R³×S² B(|v−v^′|, cos θ)[h(v^′)+h(v_∗^′ )−h(v)−h(v∗)]M (v_∗)dv_∗dσ , and hence

hg, Lhi_H = −1 4

Z

R³×^R3×S² B(|v−v^′|, cos θ)[g(v^′)+g(v_∗^′ )−g(v)−g(v∗)]×

[h(v^′) + h(v_∗^′ ) − h(v) − h(v∗)]M (v)M (v_∗)dvdv_∗dσ .

The spectral gap of the linearized Boltzmann operator is the quantity

Λ = inf



−hh, Lhi_H

khk²_H : h ∈ (Ker(L))^⊥ .



There is exactly one case in which it is relatively

straightforward to compute Λ: The case of Maxwellian molecules; i.e., α = 0.

In this case, the subspaces _Hn of _H that consist of

polynomials in v₁, v₂ and v₃ of degree n or less are invariant subspaces of _L. Since _L is self-adjoint, this means that the

For the hard-spheres case, or indeed any other case, there is no known method for computing eigenvalues, and for a long time, there were no quantitative estimates whatsoever for the spectral gap Λ in these cases. The first quantitative estimate on Λ for hard collisions is a recent result by

Baranger and Mouhot. It works by making a comparison with the Maxwellian case. Here we shall prove:

THEOREM 0.3 (Spectral gap for Boltzmann via the Kac process). Let b

be continuous on [−1, 1] and strictly positive at 1^{, and let} α ∈ [0, 2]^.

Let Λ be the spectral gap for the corresponding linearized Boltzmann equation, and let ∆_N be the corresponding spectral gap for the Kac process. Then

lim sup

N →∞

∆_N ≤ Λ .

Combining our two theorems yields a quantitative lower bound on Λ.

A word on the proof

The second theorem is much easier than the first. Use a trial function of the form

f (~v) =

N

X

j=1

ϕ_j(v_j)

where ϕ is “built” out of the gap eigenfunction for the linearized collision operator. For such an f, one finds

E^N,α(f, f ) = Λkfk²2 ,

up to small errors (for large N), so the claim follows from the variational definition of ∆_N.

Induction on the number of particles

We now explain how to estimate ∆_N,α in terms of ∆_{N −1,α}. We use a parameterization of _SN in terms of _{SN −1 × B}

where B is the unit ball.

First, for each k = 1, . . . , N, define π_k : S^N → B by π_k(~v) = 1

√N − 1v_k . ^(0.5)

(Note that because of the constraints P_N

j=1 v_j = 0 and PN

j=1 |v^j|² = N, the largest value of _|vk| on _SN is ^√_{N − 1}.)

Define T₁ : S_{N −1} × B → S^N as follows:

T₁(~y, v) =

√

N − 1v , β(v)y¹ − 1

√N − 1v, . . . , β(v)y_{N −1} − 1

√N − 1v

 , where

β²(v) = N

N − 1(1 − |v|²) .

The subscripted 1 in T₁ indicates that the vector v from B went into the first place. We likewise define T₂, . . . , T_N by placing this coordinate in the corresponding position.

Z

S^N

φ(~v)dσ_N = Z

B

Z

SN −1

φ(T_k(~y, v))dσ_{N −1}



dν_N(v) .

where

dν_N(v) = |S^{3N −7}|

|S^{3N −4}|(1 − |v|²)^{(3N −8)/2}dv . Also, for i 6= k, j 6= k,

R_i,j,σ(T_k(~y, v)) = T_k(R_i,j,σ(~y), v) . We now have the means to relate _EN,α to _{EN −1,α}.

Define the projection (conditional expectation) operator

P_kφ(~v) :=

Z

S^{N −1}

φ(T_k(~y, v_k/√

N − 1))dσ_{N −1} ,

Note that

E^N,α(f, f |v^k) = E^N,α(f − P^kf, f − P^kf |v^k) , and then one has

E^N,α(f, f ) ≥ N

N − 1∆_{N −1}× 1 ^N Z 

N 

|v |² α/2 !

Define

P^(α) = 1 N

N

X

k=1

 N N − 1



1 − |vk|² N − 1

α/2

P_k

and

W ^(α) = 1 N

N

X

k=1

 N N − 1



1 − |v^k|² N − 1

α/2

.

LEMMA 0.4 (W^(α) is constant for α = 0 ^and α = 2^{). For all} ~v^, W ⁽⁰⁾(~v) = 1 ^while W⁽²⁾(~v) = 1 − 1

(N − 1)^{2 .}

THEOREM 0.5. For all f ∈ L²(S^{N −1}(√

N )) ^with kfk²₂ = 1 ^{and with} f orthogonal to the constants,

E^N,α(f, f ) ≥ N

N − 1∆_{N −1}

Z

S^N

W^(α)f²dσ_N − hf, P^(α)f iL²(S^N,σN)

 .

Our goal is to prove from this a bound of the type

∆_N ≥



1 − C N²



∆_{N −1} .

The point of this is that for any N₀ we then have, by iteration, lim inf

N →∞ ∆_N ≥

Y∞

n=N⁰



1 − C n²



∆_N0−1 ,

and since 1/n² is summable, the factor on the right is strictly positive.

Since

N

N − 1 = 1 + 1

N + O

 1 N²

 , We need

Z

S^N

W^(α)f²dσ_N − hf, P^(α)f iL²(S^N,σN)



= 1 − 1

N + O

 1 N²

 , where the coefficient of 1/N must be exactly 1.

As we now explain, this is easier to do for α = 0 and α = 2 than for _{α ∈ (0, 2)} since in these cases only we have a

“good” pointwise bound on W^(α).

Lower bound on ∆

for α = 0 and α = 2.

For α = 0, the inductive relation reduces to EN,α(f, f ) ≥ N

N − 1∆_{N −1}

Z

S^N

f²dσ_N − hf, P⁽⁰⁾f iL²(S^N,σN)



Define

µ_N = sup n

hf, P⁽⁰⁾f i_S^N : hf, 1i_S^N = 0 and kfk_S^N = 1 o , so that from the variational characterization of ∆_N,

∆_N ≥ ∆_{N −1} N

N − 1(1 − µN) .

The operator P⁽⁰⁾ is an average over projections onto functions of a single particle’s velocity. That is, any eigenfunction f of P⁽⁰⁾ necessarily has the form

f (~v) =

N

X

j=1

ϕ_j(v_j)

for functions ϕ_j on the ball of radius ^√_{N − 1} in R³.

Determining the ϕ_j is then a problem in R³, no matter how large N is. While there are eigenfunctions of L_N,α that have this simple form, most do not.

Define operators K and K₂ by

Kϕ(v) = E{ϕ(v²) | v¹ = v} , and

K₂ϕ(u, v) = E{ϕ(v³, v₄) | v¹ = u, v₂ = v} . The operators measure correlations on _SN

P⁽⁰⁾





N

X

j=1

ϕ(v_j)



 ₌ 1 N

N

X

j=1

(ϕ(v_j) + (N − 1)Kϕ(v^j))

Thus, the problem of estimating ∆_N is reduced to the

problem of estimating the spectrum of K, and eventually,

∆₂.

LEMMA 0.6 (Spectral gap for P^{(0)). For all} N ≥ 3^, µ_N = 1

N + 5N − 3

3N (N − 1)² . ^(0.6) Applying this, we get:

∆_N ≥ ∆_{N −1} N N − 1



1 − 1

N − 5N − 3 3N (N − 1)²



= ∆_{N −1}



1 − 5N − 3 3(N − 1)³



We obtain the bound

lim inf

N→∞ ∆_N ≥

Y∞

n=3



1 − 5n − 3 3(n − 1)³

!

∆₂

∞ 5n − 3

Numerical computation yields Y∞

n=3



1 − 5n − 3 3(n − 1)³



≥ 0.236 .

Next, it is relatively easy to compute ∆₂:

∆₂ = 10

3 for b(x) = 2|x| .

Moreover, ∆₂ is independent of α. Since v₁ + v₂ = 0,

|v¹ − v²|² = |2v²|² = 2E = 4 .

For α = 2, we use:

LEMMA 0.7. For all N^{, all} 0 ≤ α ≤ 2^{, and all} f ∈ L²(SN)^, hf, P^(α)f i ≤

 N N − 1

α/2

hf, P⁽⁰⁾f i .

In particular, if f is orthogonal to the constants, then

hf, P^(α)f i ≤

 N N − 1

α/2

µ_Nkfk²2 .

Putting it all together, as before, we get

∆_N ≥



1 − O

 1 N²



∆_{N −1}

Evaluating the constants, we obtain that there is a constant C such that C/N² < 1 for all _{N ≥ 4} so that

∆_N ≥



1 − C N²



∆_{N −1}

A direct estimate on ∆₃ then starts the induction.

The difficulty with α ∈ (0, 2).

Let ^e be any unit vector and consider

~v^e :=

_√

N − 1e, 1

√N − 1

e, . . . , 1

√N − 1 e



∈ S^N . Then one readily computes, for α > 0,

W ^(α)(~v^e) =

 N N − 1

_α−1 

1 − 1 N − 1

α/2

= 1 − (1 − α/2) 1

N + O

 1 N²

 .

As before, we can estimate

hf, P^(α)f iL²(S^N,σN) ≤



1 − 1

N + O

 1 N²



kfk²₂ .

Altogether,

Z

S^N

W^(α)f²dσ_N − hf, P^(α)f iL²(S^N,σN)



≤



1 − 2 − α/2

N + O

 1 N²



kfk²2 . ^(0.7)

How to proceed for 0 < α < 2:

Let Π denote the projection onto the space of functions orthogonal to the constants The operator ΠP^(α)Π is self adjoint.

For any f orthogonal to the constants,

hf, P^(α)f i = hf, ΠP^(α)Πf i . ^(0.8) Now, decompose f as f = g + h where h is in the null space of ΠP^(α)Π, and g is in the range. Notice that h and g are

orthogonal, so that

kfk²2 = kgk²2 + khk²2 .

By the definition of h,

hf, P^(α)f i = hg, P^(α)gi , and hence

Z

S^{N −1}(√ N )

W^(α)f²dσ − hf, P^(α)f i_L²_(S^{N −1}₍^√_{N ))} = Z

S^{N −1}(√ N )

W^(α)f²dσ − hg, P^(α)gi_L²_(S^{N −1}₍^√_{N ))} . (The formulas here and in the rest are for a simplified one dimensional model. They fit better on the slides, but the

Z

W ^(α)f²dσ − hf, P^(α)f i

≥



1 − 1 − α/2 N − 1



khk²2



+ 1

N

N

X

k=1

"

Z

S^{N −1}(√ N )

w^(α)(v_k)[g − P^kg]²dσ

#

− 1 − α/2 N

Z ^N X

k=1

1 − v_k² N − 1

²

2ghdσ .

There is enough orthogonality that the last term,

1 − α/2 N

Z ^N X

k=1

1 − v_k² N − 1

²

2ghdσ

is negligible.

Since (1 − α/2) < 1, the multiple of 1/N in



1 − 1 − α/2 N − 1



khk²2



is no problem.

For the remaining term, use the fact that

1 N

N

X

k=1

"

Z

S^{N −1}(√ N )

w^(α)(v_k)[g − Pkg]²dσ

#

together with

g(~v) =

N

X

j=1

ϕ(v_j) and

g − P^kg = X

ϕ(v_j) − (N − 1)Kϕ(v^k) .

Showing cross terms do not matter, as before, we get

1 N − 1

N

X

k=1

Z

S^{N −1}(√ N )

m(v_k)[g − P^kg]²dσ =

N

X

k=1

Z

S^{N −1}(√ N )

m(v_k)[X

j6=k

ϕ(v_j) − (N − 1)Kϕ(vk)]²dσ ^(0.9)

where

m(v) =



1 − (1 − α/2)

1 − v² N − 1

 

1 − v² N − 1



+ 1



> 0 .

w^(α)(v) =

N − v² N − 1

α/2

≥ m(v) =

1 + (α/2)

1 − v² N − 1



− (1 − α/2)

1 − v² N − 1

2

. This reduces the weight to polynomials, facilitating the

remaining estimates.

Altogether, we get, for a computable constant C,

∆_N ≥



1 − C N²



∆_{N −1}

for all _{N ≥ Nα}, where N_α is computable, and small. An

elementary probabilistic argument bounds ∆_N from below, though with a bad N dependence. However, we use this only for the fixed value N = N_α, and then the inductive bound.