Rare events in driven one-dimensional models Giacomo Gradenigo - - PowerPoint PPT Presentation
Rare events in driven one-dimensional models Giacomo Gradenigo (LPTMS PARIS 11) Large Deviations in Statistical Physics Stellenbosch 5-11-2014 Simple diffusion may hidden surprises ! h x 2 ( t ) i = 2 Dt h x ( t ) i E = E t
(LPTMS – PARIS 11)
‘’Large Deviations in Statistical Physics’’ Stellenbosch 5-11-2014
Simple diffusion may hidden surprises !
Brownian motion of a colloidal particle in an equilibrium fluid
hx2(t)i = 2Dt hx(t)iE = µEt
hx2(t)i = 2Dt hx(t)iE = µEt
Simple diffusion may hidden surprises !
hx2(t)i = 2Dt hx(t)iE = µEt
Drift induced by external field No Large Deviation Principle
Simple diffusion may hidden surprises !
Average velocity
‘’Non-‑equilibrium ¡fluctua4ons ¡in ¡a ¡driven ¡stochas4c ¡Lorentz ¡gas’’, ¡ ¡
‘’Fluctua4on ¡rela4ons ¡without ¡uniform ¡large ¡devia4ons’’, ¡ ¡
v P(XE/t = v) 6= e−tφ(v) XE
Displacement
Anomalous dynamics heterogeneous substrate
hx2(t)iE hx(t)i2
E ⇠ tγ
γ > 1
Superdiffusion
hx2(t)i = 2Dt hx(t)iE = µEt
‘’Einstein ¡rela4on ¡in ¡superdiffusive ¡system’’, ¡ ¡G. ¡Gradenigo, ¡A. ¡Sarracino, ¡D. ¡Villamaina, ¡A. ¡Vulpiani, ¡JSTAT ¡(2012) ¡ ‘’Rare ¡events ¡and ¡scaling ¡proper4es ¡in ¡field ¡induced ¡anomalous ¡dynamics’’, ¡ ¡
‘’Scaling ¡proper4es ¡of ¡field-‑induced ¡superdiffusion ¡in ¡CTRW’’, ¡ ¡
Simple diffusion may hidden surprises !
Plan of the talk
The driven stochastic Lorentz gas Non-uniform Large Deviations (a constructive role of the Fluctuation-Relation) Continuous Time Random Walk with traps Driven Anomalous (Super-) diffusion Driven Anomalous (Super-) diffusion in a simple glass model Entropy production
Part I Part II
τ v t
γ = (ζ − α)/(1 + ζ) ζ = m/M PS(V ) = Gauss σ2 = T/M
α < 1
Inelastic collisions with random scatterers (independet on |v-V| !!!!)
vi+1 = γvi + (1 − γ)V
Free fligths at constant velocity
Pnc(t) = e−t/τc
tracer ¡ scaQerer ¡
(m, v) (M, V )
Res4tu4on ¡coefficient ¡
v t ¡ ¡ ti ti+1 ¡ t v(t) = vi + (t − ti)E
Inelastic collisions with random scatterers (independet on |v-V| !!!!)
vi+1 = γvi + (1 − γ)V
‘’Free’’ fligths at uniform acceleration
Pnc(t) = e−t/τc
probe ¡ scaQerer ¡
(m, v) (M, V )
τc∂tP(v, t) + τcE∂vP(v, t) = −P(v, t) + 1 1 − γ Z duP(u, t)PS ✓v − γu 1 − γ ◆
Linear Boltzmann equation
α = ζ = m/M γ = 0
Special case exactly solvable
P(v) = 1 τcE r M 2πT Z ∞ du exp ✓ − M 2T (v − u)2 ◆ × exp ✓ − u τcE ◆ τc∂tP(v, t) + τcE∂vP(v, t) = Z 1
1
dv0w(v|v0)P(v0, t) − Z 1
1
dv0w(v0|v)P(v, t)
vi+1 = V
(Renewal process)
α = ζ = m/M γ = 0
Special case exactly solvable
P(v) = 1 τcE r M 2πT Z ∞ du exp ✓ − M 2T (v − u)2 ◆ × exp ✓ − u τcE ◆
P(v) = 1 τcE r M 2πT Z ∞ du exp ✓ − M 2T (v − u)2 ◆ × exp ✓ − u τcE ◆
hx2(t)i ⇠ t hx(t)iε ⇠ t
Diffusion Drift
v t
P[{v(t)}|v0] = Pnc(t − tn)
Nc
Y
j=1
Pnc(ti − ti1) w(v0
i|vi)
P[{˜ v(t)}|˜ v0] = Pnc(t − tn)
Nc
Y
j=1
Pnc(ti − ti1) w(−vi| − v0
i)
Probability of forward and backward trajectories
{v(t)} = (v0, t1, v1, v0
1, t2 − t1, v2, v0 2, . . . , vn, v0 n, t − tn)
{˜ v(t)} = (−vt, t − tn, −v0
n, −vn, tn − tn1, . . . , −v0 1, −v1, t1, −v0)
Forw Back ∆stot(t) = log P[{v(t)}|v0]P(v0) P[{˜ v(t)}|˜ v0]P(˜ v0)
∆stot(t) = log P[{v(t)}|v0]P(v0) P[{˜ v(t)}|˜ v0]P(˜ v0) =
Nc(t)
X
j=1
log w(v0
j|vj)
w(−vj| − v0
j) + log P(v0)
P(−vt)
Total entropy production
P[{v(t)}|v0] = Pnc(t − tn)
Nc
Y
j=1
Pnc(ti − ti1) w(v0
i|vi)
P[{˜ v(t)}|˜ v0] = Pnc(t − tn)
Nc
Y
j=1
Pnc(ti − ti1) w(−vi| − v0
i)
Probability of forward and backward trajectories
∆stot(t) = log P[{v(t)}|v0]P(v0) P[{˜ v(t)}|˜ v0]P(˜ v0) =
Nc(t)
X
j=1
log w(v0
j|vj)
w(−vj| − v0
j) + log P(v0)
P(−vt)
Total entropy production
∆stot(t) = −∆Ecoll(t) θ + log P(v0) P(−vt) W(t) + ∆Ecoll(t) = ∆E(t) = m 2 (v2
t − v2 0)
∆stot(t) = W(t) θ − m 2θ[v2
t − v2 0] + ln P(v(0))
P(−v(t))
θ = θ(α, m, M, T) 6= T
‘’temperature of the probe when E = 0 Energy = work + collisions
Entropy production rate & average velocity
∆stot(t) = mE θ X(t) + B X(t) = O(t) B = O(1) ∆stot(t) = W(t) θ − m 2θ[v2
t − v2 0] + ln P(v(0))
P(−v(t)) W(t) = mEX(t)
P(X(t)/t = s) ∼ e−tI(v)
P(∆stot/t = s) ∼ e−tI(s)+o(t)
Large Deviation Principle Symmetry of the rate function Fluctuation theorem
P(∆stot/t = s) ∼ e−tI(s)+o(t) I(s) − I(−s) = s
But not today for the driven Lorenz gas!!
P(∆stot/t = s) P(∆stot/t = −s) = ets
Fluctuation theorem
P(∆stot/t = s) P(∆stot/t = −s) = ets
NO (Uniform) Large Deviation Principle
P(∆stot/t = s) ∼ e−tI(s)+o(t)
Large Deviations for entropy production rate
Scaling cumulant generating function
λ∆stot(k) = lim
t→∞
1 t lnhek∆stoti λ∆stot(k) < ∞ for k ∈ (−1, 0] λ0
∆stot(k = 0) = λ0 ∆stot(k = 1+) = mτcE2
θ = hWi θ I(s) = max
k∈R {sk − λ∆stot(k)}
LDP rate function
I(s) well defined only for s ∈ −mτcE2 θ , mτcE2 θ
θ X(t) + B
Lack of large deviation principle for fluctuations on the right of average velocity
λ∆stot(k) < ∞ for k ∈ (−1, 0] I(s) well defined only for s ∈ −mτcE2 θ , mτcE2 θ
∆stot(k = 0) = λ0 ∆stot(k = 1+) = mτcE2
θ = hWi θ
hWi θ = mE θ v
Large Deviations for entropy production rate
Violation of the LDP Exponentially rare long ballistic (no collisions) trajectories
∆stot = ln P(v(0)) P(−v(t)) p(∆stot/t = s) ∼ exp ⇣ −κ √ t √s ⌘
«fat » tail
∆stot(t) = 1 θ ⇣m 2 [v2(0) − v2(t)] + m EX(t) ⌘ + ln P(v(0)) P(−v(t))
Pnc(t) = e−t/τc
∆stot ∼ m 2θE2t2 ∼ xE(t)
A constructive role of Fluctuation Theorem
P(∆stot/t = s) ∼ e−
√ t φR(s)
RIGHT tail of the distribution
A constructive role of Fluctuation Theorem
P(∆stot/t = s) ∼ e−
√ t φR(s)
P(∆stot/t = −s) = e−tsP(∆stot/t = −s) = e−ts−
√ t φR(s)
RIGHT tail of the distribution Fluctuation Relation
A constructive role of Fluctuation Theorem
P(∆stot/t = s) ∼ e−
√ t φR(s)
P(∆stot/t = −s) = e−tsP(∆stot/t = −s) = e−ts−
√ t φR(s)
RIGHT tail of the distribution Fluctuation Relation LEFT tail of the distribution
P(∆stot/t = −s) = e−tφL(s)
Two speeds for the Large Deviation Principle
s∗ P(s, t) ∼ e−
√ t φR(s)
P(s, t) ∼ e−t φL(s) s∗ = mτcE2 θ s > s∗ s < s∗
Two speeds for the Large Deviation Principle
s∗ s∗ = mE θ v v < mE θ v v > mE θ v P(v, t) ∼ e−t φL(v) P(v, t) ∼ e−
√ t φR(v)
1/2 1/2 1/2 − ε 1/2 + ε
Probability to move forward/backward Free Driven
Persistence time before jumps
1/2 1/2 1/2 − ε 1/2 + ε
Probability to move forward/backward
p(τ) = e−τ/τ0
Pε(x, t) = 1 t1/2 G ✓x − vεt t1/2 ◆
P(x, t) = 1 t1/2 G ⇣ x t1/2 ⌘
hx2(t)i ⇠ t hx(t)iε ⇠ t
Persistence time before jumps
1/2 1/2 1/2 − ε 1/2 + ε
Probability to move forward/backward
P(x, t) = 1 t1/2 G ⇣ x t1/2 ⌘
Pε(x, t) = ?
1 < α < 2 p(τ) = 1 τ 1+α
hx2(t)i ⇠ t hx(t)iε ⇠ t
Persistence time before jumps
h[δx(t)]2iε = ?
1 < α < 2 hx2(t)iε hx(t)i2
ε ⇠ t3−α
FIELD INDUCED SUPERDIFFUSION
FIELD INDUCED SUPERDIFFUSION
Pε(x, t) = 1 t1/α F ✓x − vεt t1/α ◆ Θ(x)
1 < α < 2 hx2(t)iε hx(t)i2
ε ⇠ t3−α
N(t) i the average number of jumps up to time t
The probability of being at a distance ξ from the average displacement is provided by a single long persistence event
(1) (2)
p(ξ) ∼ 1 ξ1+α Pε(ξ, t) = N(t) ξ1+α = t hτi 1 ξ1+α
Rest ¡4me ¡ Average ¡displacement ¡
τ ξ = vτ
N(t) i the average number of jumps up to time t (1) (2)
Pε(ξ, t) = N(t) ξ1+α = t hτi 1 ξ1+α Pε(ξ, t) = t−γF(ξ/tγ) γ = 1/α p(ξ) ∼ 1 ξ1+α
Rest ¡rime ¡ Average ¡displacement ¡
τ ξ = vτ
The probability of being at a distance ξ from the average displacement is provided by a single long persistence event
Pε(ξ, t) = N(t) ξ1+α = t hτi 1 ξ1+α Pε(ξ, t) = t−γF(ξ/tγ) γ = 1/α Pε(x, t) = 1 t1/α F ✓x − vεt t1/α ◆ Θ(x)
Power law tail (left) Cutoff
p(ξ) ∼ 1 ξ1+α
Rest ¡rime ¡ Average ¡displacement ¡
τ ξ = vτ
Pε(x, t) = 1 t1/α F ✓x − vεt t1/α ◆ Θ(x)
Power law tail (left) Cutoff
h[x(t)]niε ⇠ `n(t) = tn/α h[δx(t)]niε ⇠ tn+1−α
FALSE !! ‘’STRONG’’ ANOMALOUS SUPER-DIFFUSION
∆sm(t) = − X
i
δEcoll(v0
j, vj)
θ θ = Tζ 1 + α 1 + 2ζ − α W(t) + ∆Ecoll(t) = ∆E(t) ∆sm(t) = 1 θ [W(t) − ∆E(t)] ∆stot(t) = W(t) θ − m(v2
t − v2 0)
2θ + log P(v0) P(−vt)
P[{v(t)}|v0] = Pnc(t − tn)
Nc
Y
j=1
Pnc(ti − ti1) w(v0
i|vi)
P[{˜ v(t)}|˜ v0] = Pnc(t − tn)
Nc
Y
j=1
Pnc(ti − ti1) w(−vi| − v0
i)
W(t) + ∆Ecoll(t) = ∆E(t) ∆sm(t) = 1 θ [W(t) − ∆E(t)] ∆stot(t) = W(t) θ − m(v2
t − v2 0)
2θ + log P(v0) P(−vt)
v t