Upload
others
View
0
Download
0
Embed Size (px)
Citation preview
fsu-logo
Lecture 1 : Macroscopic laws as laws of large numbers
Raphaël Lefevere1
1Université Paris Diderot (Paris 7)
fsu-logo
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.
fsu-logo
Diffusion equation
8
fsu-logo
Fick’s law
J = κ(ρL − ρR)
L
fsu-logo
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
fsu-logo
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
fsu-logo
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
fsu-logo
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.
fsu-logo
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.
fsu-logo
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
fsu-logo
Derivation of macroscopic evolution equations :models
Random Lorentz gas
fsu-logo
Derivation of macroscopic evolution equations :models
Ehrenfest random wind-tree model
fsu-logo
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.
fsu-logo
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.
fsu-logo
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
fsu-logo
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)
fsu-logo
“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.
fsu-logo
“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.
fsu-logo
Fick’s law
J = κ(ρL − ρR)
L
fsu-logo
Mirrors model
Ruijgrok-Cohen(1990,1991)
fsu-logo
Mirrors model
FilmAvecParticule.mp4Media File (video/mp4)
fsu-logo
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}.
fsu-logo
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 = ∅.
fsu-logo
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
fsu-logo
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
fsu-logo
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).
fsu-logo
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.