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Lecture 1 : Macroscopic laws as laws of large numbers

Raphaël Lefevere1

1Université Paris Diderot (Paris 7)

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Plan of the Lectures

1 Introduction : macroscopic laws are laws of large numbers. Models. Mirrorsmodel.

2 Kac ring model

3 Rings model : diffusion equation

4 Rings model as mirrors model : Fick’s law

5 Back to mirrors model.

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Diffusion equation

8

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Fick’s law

J = κ(ρL − ρR)

L

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Law of large numbers

1023 coins are tossed in 1000 successive games

In each of the 1000 games, the measured proportion of heads is 12± ε,

ε ∼ 10−9.How do we explain this fact ?

Some realizations of the games like HTHTHTHTHTHT . . . yield the0.5± ε, ε ∼ 10−9 proportion of heads and tails.Tosses like HHHHHHHHHHHHHHHH . . . : proportion is equal to 1.

Answer : typicality and law of large numbers

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Law of large numbers

(Weak) law of large numbers

Let (Xi)i∈N an i.i.d sequence such that E[X21 ] 0

limN→∞

P[|SN − µ| > �] = 0

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Law of large numbers

Xi a random variable that takes the value 1 if the i-th coin yields head andzero otherwise.

SN =1N

PNi=1 Xi is a random variable that gives the proportion of heads

that are obtained in N tosses. The “observable” measured equal to 1/2 in theprevious experiment.

Since the coins are symmetric, one sets P[Xi = 0] = P[Xi = 1] = 12 and theexpectation of Xi is E[Xi] = 12 for every i and µ := E[SN ] =

12

.

Thus, one needs to show that in a given experiment, it is extremely likely toobtain a proportion SN that is equal or at least very close to its expectation.

Using the well-known Chebychev’s inequality :

P[|SN − E[SN ]| < δ] ≥ 1−Var[SN ]

δ2

for any strictly positive number δ.

Var[SN ] = E[(SN − E[SN ])2], the average of the square distance of thevariable to its average, is the variance of the proportion SN .

The variance of a sum of independent variables is the sum of the individualvariances. In our case, since the N = 1023 coins tosses are independent, thisyields Var[SN ] = 1/4N .

For a precision δ = 10−9 the probability is 1− 2.5× 10−6

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Law of large numbers : morals

Explanation depends only on symmetry of the coins, which implies theequiprobability of the outcomes and independence.

No need to embark in a detailed study of what is special or common in eachof the 1023 long sequences that have produced the 1/2 proportion of headsand tails.

Similarly, the precise mechanism governing the toss of the coins does not playany role.

The fact that we consider a large number of coins and are interested in a“global” property (i.e. the proportion of heads) makes those points irrelevant.It actually makes the problem simpler and not more complicated.

Keep those simple points in mind while trying to explain macroscopic laws.

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Derivation of macroscopic evolution equations : set-up

A box Λ = [0, L]d with N freely moving distinguishable particles and fixedobstacles of arbitrary shapes.

Λ is a macroscopic volume.

N should be thought to be of the order of magnitude of the Avogadro number6× 1023.For each macroscopic coordinate r = (r1, . . . , rd) ∈ Λ, we define a microscopiccoordinate q = r/�N where �N =

1N1/d

.

The motion of the particles is entirely determined by the law of Newtonianmechanics.

When a particle makes a collision with one of the fixed obstacles or theboundaries of the boxes, its velocity is modified according to the laws ofspecular reflection.

We assume that each particle starts with a speed equal to 1.

Since this property is preserved by the dynamics, the motion of a givenparticle (with label i) is described by a mapt→ (qi(t),pi(t)) ∈ [0, L/�N ]d × Sd−1 where Sd−1 is the unit sphere in ddimensions. (qi,pi) are the microscopic coordinates of the i-th particle.

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Derivation of macroscopic evolution equations : experiment

First experiment : N particles are initially located in a cube Λ′ ⊂ Λ of sidelength L′ < L.

Evolution of the density of the cloud of particles is monitored through a beamof light that crosses the system.

Density : ρ : Λ× [0,∞[→ ρ(x, t) ∈ R+.Initial state : ρ(x, 0) = 1Λ′ (x)/|Λ′|.Empirical fact : 8

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Derivation of macroscopic evolution equations :models

Random Lorentz gas

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Derivation of macroscopic evolution equations :models

Ehrenfest random wind-tree model

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Random Lorentz gas

Random initial distribution with density

f(qi,pi) =�dN

|Λ′||Sd−1|1Λ′ (�Nqi)1Sd−1 (pi), i = 1, . . . , N (2)

P ∼ law of probability of the initial positions and velocities of particles.

P[dq, dp] =NY

i=1

f(qi,pi) dqi dpi

If the position of each particle is chosen independently of the others with thatdensity, the number of particles in a microscopic volume of size of order 1follows a Poisson distribution with finite mean as N →∞.Q the probability distribution on the shape and location of the obstacles in Λ.Number of obstacles is of the same order of magnitude than N .

Q is chosen so that again the number of obstacles in a microscopic volume ofsize 1 follows a distribution with finite mean.

The dynamical system defined in this way is an instance of the randomLorentz gas.

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Macroscopic quantities

Empirical density of particles :

ρN (x, t) =1

N

NXj=1

δ(�N qj(�−2N t)− x) (3)

Scaling of the time variable is fixed here by the fact that the solution of thediffusion equation ρ(x, t) is invariant under the transformation(x, t)→ (λx, λ2t), λ > 0.

ρN (V, t) :=

ZRddx 1V (x)ρN (x, t)

gives the proportion of particles that belong to any V ⊂ Λ at time t.

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Derivation of macroscopic evolution equations : models

Use the notation : 〈h, g〉 =R

Rd dx h(x)g(x) and ρ̂N (t) := ρN (·, t).The goal is to show the

Conjecture

There exists a natural distribution Q, for any t > 0, any bounded function h andany � > 0 :

limN→∞

P× Q[| 〈ρ̂N (t), h〉 − 〈ρ(t), h〉 | > �] = 0. (4)

with 8

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Derivation of macroscopic evolution equations : periodic Lorentz gas

Q = δC

Periodic Lorentz gas

Theorem : Bunimovich-Sinai (CMP 80)

For any t > 0, any bounded function h and any � > 0 :

limN→∞

P× δC [| 〈ρ̂N (t), h〉 − 〈ρ(t), h〉 | > �] = 0. (6)

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“Sketch of proof”

Let h : Rd → R a bounded function. First, one computes :

〈ρ̂N (t), h〉 =1

N

Zdx

NXj=1

δ(�Nqi(�−2N t)− x)h(x) (7)

=1

N

NXj=1

h(�Nqj(�−2N t)). (8)

Thus

E[〈ρ̂N (t), h〉] = E[h(�Nq1(�−2N t))]

And the theorem 2 of Bunimovich-Sinai (CMP 80) says that

limN→∞

E[h(�Nq1(�−2N t))] =Zρ(x, t)h(x) dx (9)

where ρ(x, t) is the solution of the diffusion equation and the expectation is takenwith respect to the the initial uniform distribution of the particles and the secondline is obtained because the N particles are identically distributed. Bunimovichand Sinai rely on the strong chaotic properties of the billard system under study.

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“Sketch of proof”

Next, since {qj(t) : 1 ≤ j ≤ N} are independent , the variance of 〈ρ̂N (t), h〉 is

Var[〈ρ̂N (t), h〉] =1

NVar[h(εNq1(ε

2N t))] = O(

1

N)

since h is bounded.

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Fick’s law

J = κ(ρL − ρR)

L

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Mirrors model

Ruijgrok-Cohen(1990,1991)

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Mirrors model

FilmAvecParticule.mp4Media File (video/mp4)

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Discrete dynamics

Q = the set of midpoints of edges of an hypercube of Zd of side N and withperiodic conditions in all but the first direction : Q =

Sdi=1 Li where

Li =

z +

1

2ei : 0 ≤ z1 ≤ N − 1, (z2, . . . , zd) ∈ (Z/NZ)d−1

ff.

Possible velocities are P = {± e12, . . . ,± ed

2}

Phase space :

M = {x = (q,p) : q ∈ Q,p ∈ P s. t. if q ∈ Li then p = ±ei

2}.

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Dynamical System

Action of a mirror : for any z ∈ Zd, π(z; ·) is a bijection of P into itself thatsatisfies

π(z;−π(z; p)) = −p, ∀z ∈ Zd, ∀p ∈ P

For any (q,p) ∈M :

F (q,p) = (q + p + π(q + p; p), π(q + p; p)) .

The orbit of x ∈M is the set Ox = {y ∈M : ∃t > 0, F t(x) = y}.

F is a bijection on MF−1 = RFR, R(q,p) = (q,−p)For every x ∈M, Ox is a loop.Non-self-intersecting orbits : if y ∈ Ox and y 6= x then F (y) 6= F (x)Non-intersecting among themselves : if Ox 6= Oy , then Ox ∩ Oy = ∅.

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Law of the mirrors

Let

Π = {π(z; ·) : π(z;−π(z; p)) = −p, π(z; p) 6= −p, z ∈ Zd,p ∈ P}

Q is the uniform product measure over Π. For each z fixed, the number of πsatisfying the constraint is (2d− 1)!!.

Orbits are not Markov processes

Known facts on Zd .

Bunimovich-Troubetzkoy prove

@D > 0, such that Q[bx

εc, b

t

ε2c] ∼

εd

(2πDt)d2

exp−x2

Dt, ε→ 0

Kong-Cohen : numerics :

∃D > 0, such that limt→+∞

EQ[x2(t)]t

= D

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Occupation variables

Occupation variables σ(q,p; t) ∈ {0, 1}.{σ(x; 0) : x ∈M} independent Bernoulli parameter ρI ∈ (0, 1)Evolution :

σ(x; t) =

8

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Macroscopic current

Take the hyperplane Ql = {q ∈ Q : q1 = l + 12} , l ∈ {1, . . . , N − 2} as a functionof a configuration σ ∈ {0, 1}M :

J(l, t) =1

Nd+1

Xx∈M

t+N2Xs=t

σ(x, s)∆(x, l)

where ∆(x, l) = 2(p · e1)1q∈Ql ,with x = (q,p).∆(x, l) takes the value +1 (resp. −1) if x crosses the slice Ql from left to right(resp. from right to left).

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Goal

P ∼ law of original state and boundary conditionsQ ∼ law of the mirrors

Theorem : Fick’s law

For any l ∈ {1, . . . , N − 2} and any δ > 0,

limN→∞

limt→∞

P×Q[|NJ(l, t)− κ(ρ− − ρ+)| > δ] = 0,

for some κ > 0.