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fsu-logo Macroscopic Diffusion in random Lorentz gases Rapha¨ el Lefevere 1 1 Universit´ e Paris Diderot (Paris 7)

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    Macroscopic Diffusion in random Lorentz gases

    Raphaël Lefevere1

    1Université Paris Diderot (Paris 7)

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    Joint work with Yann Chiffaudel

    R. L. Fick’s law in a random lattice Lorentz gasArchive for Rational Mechanics and Analysis Volume 216, Issue 3(2015), 983-1008

    Yann Chiffaudel and R.L. The Mirrors Model : Macroscopic DiffusionWithout Noise or Chaos

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    Fick’s law

    J = κ(ρL − ρR)

    L

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    Periodic Lorentz gas

    Bunimovich-Sinai : the rescaled motion of a test particle is diffusive.

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    Random Lorentz gas

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    Ehrenfest random wind-tree model

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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,

    t

    ε2] ∼

    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

    Fick’s law

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

    limN→∞

    P×Q[|NJ(l, t)− κ(ρ− − ρ+)| > δ] = 0,

    for some κ > 0.

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    S± = {x ∈ B± : F 1(x) /∈ B±, . . . , F s−1(x) /∈ B±, F s(x) ∈ B∓ for some s ∈ N∗}.

    N± = the numbers of crossings from B± to B∓ = |S±|

    N+ = N− because every orbit is closed. Set N = N+ = N−.

    Proposition

    Let {σ(x; 0) : x ∈ C} be a set of independent Bernoulli random variables withE[σ(x; 0)] = ρ± ∈ (0, 1) if x ∈ B±, and E[σ(x; 0)] = ρI ∈ (0, 1) if x /∈ B− ∪B+. Forevery δ > 0, any t ≥ |M| and l ∈ {1, . . . , N − 2},

    P»˛̨̨̨J(l, t)−

    NNd−1

    (ρ− − ρ+)˛̨̨̨≥ δ–≤ 2 exp(−4δ2Nd+1).

    N is the central object

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    Theorem

    Fick’s law holds if and only if there exists κ > 0 such that for any � > 0

    limN→∞

    Q»˛̨̨̨N

    Nd−2− κ˛̨̨̨> �

    –= 0

    If d = 2, Kozma-Sidoravicius argument shows that Fick’s law does not hold.

    In d ≥ 3, if Crossing Conditions are satisfied :

    1 There exists κ > 0 such that limN→∞NQ[O ∈ S] = κ.

    2 limN→∞1

    Nd−3P

    x∈B−δ(O, x) = 0 with

    δ(x, y) = Q[x ∈ S, y ∈ S]− Q[x ∈ S]Q[y ∈ S].

    then Fick’s law holds in the stationary state and κ is the conductivity.

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    How to prove that the conditions are verified ?

    If the law of an orbit was “similar” to the law of a random walk, the firstpoint is trivial.

    Recollisions

    Rings model : recollision occurs only after time N , full proof is given if d islarge enough.

    In the mirrors model, recollisions (“loops”) become more and more unlikely asd increases

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    Crossing conditions : d ≥ 3

    Figure: Q(O ∈ S) for N from 5 to 420. κ = 1.535± 0.005

    Figure: δ(O, x) = Q[O ∈ S, x ∈ S]− Q[O ∈ S]Q[x ∈ S] for x = ((1/2, y, 0), e12 ), N=70with a 95% confidence interval.

    Better than independent orbits !!

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    Correlations : cooperation and jamming

    For x ∈ B−,

    δ(O, x) =XO0,Ox

    (Q(O0,Ox)− Q(O0)Q(Ox))

    −XO0,Ox

    ′Q(O0)Q(Ox). (1)

    Both sums run over orbits that cross the box Q.First sum runs over compatible orbits such that O0 and Ox share a mirror.Cooperation

    Second sum runs over incompatible orbits O0,Ox. Jamming

    First sum is positive because Q(O0,Ox) > Q(O0)Q(Ox) when orbits share amirror.

    Jamming wins !