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Large deviations of the current in collisional dynamics

Raphael Lefevere1

1Laboratoire de Probabilites et modeles aleatoires. Universite Paris Diderot (Paris 7).

Paris, December 9 th 2011.

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Collaborations

T. Gilbert, R. Lefevere, Heat conductivity from molecular chaos hypothesis inlocally confined billiard systems. Physical Review Letters (2008) 101, 200601

R.Lefevere, L.Zambotti Hot scatterers and tracers for the transfer of heat incollisional dynamics. Journal of Statistical Physics (2010) 139, 686-713

R. Lefevere, M. Mariani and L. Zambotti, Macroscopic fluctuations theory ofaerogel dynamics. Journal of Statistical Mechanics (2010) L12004

R. Lefevere, M. Mariani and L. Zambotti, Large deviations of the current instochastic collisional dynamics. Journal of Mathematical Physics (2011) 52,033302

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Aerogels

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Local collision dynamics.

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Local collisions dynamics

Introduced by Prosen-Campbell

TL

TL

TR

TR

2b

TL 6= TR, b > 0, b < a < 2b.

(qi, pi)1≤i≤N , qi ∈ [−b, b] et |qi − qi+1| ≤ a.

Ballistic motion+ reflections on the interval’s boundaries.

Interaction if |qi − qi+1| = a (pi = v, pi+1 = v′)→ (pi = v′, pi+1 = v)

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Local collisions dynamics

2b

2b− a

B = {(q1, q2) : q1 ∈ [−b, b], q2 ∈ [−b, b], |q1 − q2| ≤ a}

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Local collision dynamics.

Energy of particle n at time t :

E(n, t)− E(n, 0) = J(n− 1, [0, t])− J(n, ([0, t])

Time-integrated current:

J(n, [0, t]) = − 1

2

Cn(t)Xk=1

[p2n+1(τkn)− p2

n(τkn)]

(τkn)k is the sequence of collision times between particles n and n+ 1. Cn(t)number of collisions up to time t.

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Fourier law

Define TN : [0, 1]× R+ → R+ and JN : [0, 1]× R+ → R.

TN (x, t) =1

εN

Xi∈Bε(x)

E(i, N2t), Bε(x) = {i : |i

N− x| ≤ ε}

JN (x, t) = N ·1

N2∆t· J([Nx], [N2t,N2(t+ ∆t)]), ∆t arbitrary

Want to show :

Fourier law

When N →∞, ∆t→ 0, ε→ 0, (TN ,JN ) converge in L2 to the unique solution

(T , J ) of

∂tT (x, t) = −∂xJ (x, t)

J (x, t) = −κ(T (x, t))∂xT (x, t)

with κ(T ) ∼ (T )12 and suitable b.c.

Much too hard!

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Stochastic collisions dynamics

Evolution of energy :

E(n, s+ ∆s)− E(n, s) = J(n, [s, s+ ∆s])− J(n− 1, [s, s+ ∆s])

Replace by stochastic model :

1 Thermal baths having temperatures T (n, s), piecewise constant in time

2 Tracer particles carrying energy current J(n, [s, s+ ∆s]) and interactingrandomly with the heat baths

T1 T2 T3 T4 T5 T6

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Stochastic collisions dynamics

TL TR

Update of the particle’s speed with the law (β = T−1):

ϕ(v) = βve−βv22 .

T T

Markov process (q(s), p(s)) with invariant measure :

γ(dq, dp) =

rβ

2π1[0,1](q)e

−β p22

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Renewal process

Waiting times distributed with (β± = T−1± )

ψ−(τ) :=β−

τ3exp(−

β−

2τ2) and ψ+(τ) :=

β+

τ3exp(−

β+

2τ2).

Total time elapsed at the k + 1-st collision at side ± :

S±k := S±0 +kXi=1

(τ±i + τ∓i ), τ±i ∼ ψ±

Renewal processes :N±t = sup{k : S±k ≤ t}

with waiting times distributed with ψ+ ∗ ψ−.

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Convergence results.

Let

Sk =kXn=1

τn

andNt = sup{k ≥ 1 : Sk ≤ t}

Strong law of large numbers and renewal theorem

limt→∞

Nt

t=

1

E(τ), a.s.

limt→∞

ENtt

=1

E(τ)

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Average current

J [0, t] =

N−tXk=1

(v−k )2

2−N+

tXk=1

(v+k )2

2

where v±k ∼ ϕ±(v) = β±ve

−β± v22 .

Proposition

limt→∞

J [0, t]

t=

T− − T+

( π2T−

)12 + ( π

2T+)

12

a.s.

where TL and TR are the left and right temperatures.

Proof :

limt→∞

N±tt

=1

E(τ+ + τ−)=

1

( π2TL

)12 + ( π

2TR)

12

:= κ, a.s.

E((v±1 )2

2) = T±

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Fluctuations of the current

Study LDF of the current I(j, τ, T ):

Pτ,T„J [0, t]

t= j

«∼ e−tI(j,τ,T ), t→∞.

Pτ,T stochastic dynamics with a fixed temperature difference τ = TL − TR,

average temperature T = TL+TR2

.

Theorem

If τ 6= 0 then,

limε↓0

ε−2I(εj, ετ, T ) = G(j, τ, T ) =

8>>>>><>>>>>:

(j−κτ)2

4κT2 if jτ > κτ2

0 if jτ ∈ [0, κτ2]

− jτ2T2 if jτ ∈ [−κτ2, 0]

j2+κ2τ2

4κT2 if jτ < −κτ2,

where κ = ( T2π

)12 .

Note : ε will be N in the diffusive scaling limit below.

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-3 -2 -1 0 1 2 3

0.5

1.0

1.5

2.0

Figure: Plot of G as a function of j for κτ = κT 2 = 1

Gallavotti-Cohen symmetry:

G(j, τ, T )− G(−j, τ, T ) =jτ

2T 2.

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Flat part

Origin:

1 Current :

J [0, t] =

N−tXk=1

(v−k )2

2−N+

tXk=1

(v+k )2

2

2 Large deviations of Nt for a renewal process whose renewal times (τk)k∈N aredistributed with density :

ψ(τ) ∼1

τγ, as τ →∞, γ > 2.

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Flat part

Take a i.i.d sequence (τi)i≥1 distributed with density ψ(τ) and compute

ϕ(λ) = limt→∞

1

tlog E(expλNt)

E(expλNt) =∞Xn=0

P(Nt = n) exp(λn)

Note first that

E(expλNt) ≥ P(Nt = 0) = P(τ1 > t) ∼1

tγ−1, as t→∞

so that

ϕ(λ) ≥ lim inft→∞

1

tlog E(expλNt) ≥ lim

t→∞−

1

tlog tγ−1 = 0

But for λ ≤ 0, one has ϕ(λ) ≤ 0. For λ > 0, ϕ(λ) is the unique solution of theimplicit equation for x ∈ R :

expλ

Z ∞0

e−xτψ(τ)dτ = 1

which is strictly positive.

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Dynamics of heat baths and diffusive scaling

Fix ∆t > 0, tk = k∆t, sk = N2tk, k ∈ N,

1

T (n, s) = T (n, sk), sk ≤ s < sk+1, k ∈ N2 Total energy exchanged (stochastic) between the wall n and n+ 1 at fixed

temperatures over the time interval [sk, sk+1]

J(n, [sk, sk+1]) = . . .

J(n, [sk, sk+1]) is computed with {T (n, sk), T (n+ 1, sk)}.3 Evolution of temperatures

T (n, sk+1)− T (n, sk) = J(n, [sk, sk+1])− J(n− 1, [sk, sk+1]), k ∈ N

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Diffusive scaling

Define TN : [0, 1]× R+ → R+ and JN : [0, 1]× R+ → R.

TN (x, t) = T ([Nx], N2t)

JN (x, t) = N ·1

N2∆t· J(n, [N2tk, N

2(tk + ∆t)]), tk ≤ t < tk + ∆t

We can show :

Proposition

When N →∞, ∆t→ 0, (TN ,JN ) converge in L2 to the unique solution (T , J ) of

∂tT (x, t) = −∂xJ (x, t)

J (x, t) = −κ(T (x, t))∂xT (x, t)

with κ(T ) = ( T2π

)12 and suitable b.c.

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Large deviations : thermodynamic potentials in equilibrium

Take the Ising model : spin variables σi = ±1, i ∈ Zd distributed according toBoltzmann-Gibbs at temperature β−1

µ(σΛ) =e−βHΛ(σΛ)

ZΛ(β, h)

with Hamiltonian:

HΛ(σ) = −X

<i,j>∈Λ

σiσj + hXi∈Λ

σi, +b.c.

Large deviations of 1Nd

Pi∈Λ σi

P(1

Nd

Xi∈Λ

σi ∈ dm) ∼ exp(−NdI(m,β))

I(m,β) is the Legendre transform of the Helmholtz free energy :

F (h, β) = − limN→∞

1

NdlogZΛ(h, β)

Compute the cumulants (correlation functions)

See phase transitions (lack of strict convexity of I(m,β))

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Non-equilibrium and large deviations : diffusive description.

Macroscopic fluctuations theoryOnsager-Machlup 1953Bodineau,DerridaBertini, De Sole, Gabrielli, Jona-Lasinio, Landim

For diffusive systems (i.e TN and JN are related to microscopic energy andcurrent by a diffusive scaling):

P ({TN ' T,JN ' j} on[0, 1]× [0, 1]) ∼ exp[−N I(j, T )]

where I(j, T ) is given by

I(j, T ) =

Z 1

0dt

Z 1

0dx G(j(x, t), ∂xT (x, t), T (x, t))

if j and T satisfy ∂sT (x, s) = −∂xj(x, s), and I = +∞ otherwise.“Almost” all known examples :

G(j, τ, T ) =(j − κτ)2

4κT 2.

Local billard dynamics might be different!

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Fluctuations at finite N

Want to look at :

P“{TN ' T ,JN ' j} on[0, 1]× [0, 1]

”∼ exp[−N I(j, T )]

At finite N , and for each (x, t) ∈ [0, 1]× [0, 1], JN (x, t) is a random variable (andso is TN (x, t)).“Independence” over small space-time windows :

P“{TN ' T ,JN ' j} on[0, 1]× [0, 1]

”=Yk,l

P[J(l, [sk, sk+1])

N2∆t=j(xl, tk)

N]

Current computed with temperatures {T (xl, tk), T (xl+1, tk)} and “compatibilityconditions” for temperatures.Compute :

log P“{TN ' T ,JN ' j} on[0, 1]× [0, 1]

”= ∆t ·N2

Xk,l

1

N2∆tlog P[

J(l, [sk, sk+1])

N2∆t=j(xl, tk)

N]

.Remember : the theorem allows to compute:

limε→0

lims→∞

1

ε21

slog P[

J(l, [sk, sk + s]

s= εj(xl, tk)]

= G(j(xl, tk), ∂xT (xl, tk), T (xl, tk))

if T (xl, tk)− T (xl+1, tk) = ε∂xT (xl, tk).

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Macroscopic fluctuation theory

Putting everything together :

P“{TN ' T , jN ' j}

”∼ exp[−N I(j, T )]

where I(j, T ) is given by

I(j, T ) =

Z 1

0dt

Z 1

0dx G(j(x, t), ∂xT (x, t), T (x, t))

if j and T satisfy ∂sT (x, s) = −∂xj(x, s), and I = +∞ otherwise.

G(j, τ, T ) =

8>>>>><>>>>>:

(j−κτ)2

4κT2 if jτ > κτ2

0 if jτ ∈ [0, κτ2]

− jτ2T2 if jτ ∈ [−κτ2, 0]

j2+κ2τ2

4κT2 if jτ < −κτ2,

G(j, τ, T ) 6=(j − κτ)2

4κT 2!

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Conclusions.

Argument based on the fact that under (local) equilibrium distributions thereare slow particles with sufficiently large probability.

Apply to “tracer models” introduced by Larralde, Mejia-Monasterio, Leyvraz

Draw experimental consequences and observe them in numerical simulations.

Applications to continuous time random walk