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 = µ E t Brownian motion of a colloidal particle in an equilibrium fluid

Simple diffusion may hidden surprises ! h x 2 ( t ) i = 2 Dt ? h x ( t ) i E = µ E t

Simple diffusion may hidden surprises ! Drift induced by external field No Large Deviation Principle Average velocity v h x 2 ( t ) i = 2 Dt Displacement X E h x ( t ) i E = µ E t P ( X E /t = v ) 6 = e − t φ ( v ) ‘’Non-­‑equilibrium ¡fluctua4ons ¡in ¡a ¡driven ¡stochas4c ¡Lorentz ¡gas’’, ¡ ¡ G. ¡Gradenigo, ¡U. ¡B. ¡M. ¡Marconi, ¡Sarracino, ¡A. ¡Puglisi, ¡PRE ¡(2012) ¡ ‘’Fluctua4on ¡rela4ons ¡without ¡uniform ¡large ¡devia4ons’’, ¡ ¡ G. ¡Gradenigo, ¡A. ¡Sarracino, ¡A. ¡Puglisi, ¡H. ¡ToucheQe, ¡J. ¡Phys. ¡A ¡(2013) ¡

Simple diffusion may hidden surprises ! Anomalous dynamics heterogeneous substrate h x 2 ( t ) i = 2 Dt h x 2 ( t ) i E � h x ( t ) i 2 E ⇠ t γ γ > 1 h x ( t ) i E = µ E t Superdiffusion ‘’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’’, ¡ ¡ R. ¡Burioni, ¡G. ¡Gradenigo, ¡A. ¡Sarracino, ¡A. ¡Vezzani, ¡A. ¡Vulpiani, ¡JSTAT ¡(2013) ¡ ‘’Scaling ¡proper4es ¡of ¡field-­‑induced ¡superdiffusion ¡in ¡CTRW’’, ¡ ¡ R. ¡Burioni, ¡G. ¡Gradenigo, ¡A. ¡Sarracino, ¡A. ¡Vezzani, ¡A. ¡Vulpiani, ¡Commun. ¡Theor. ¡Phys. ¡(2014) ¡

Plan of the talk Part I The driven stochastic Lorentz gas Entropy production Non-uniform Large Deviations (a constructive role of the Fluctuation-Relation) Part II Continuous Time Random Walk with traps Driven Anomalous (Super-) diffusion Driven Anomalous (Super-) diffusion in a simple glass model

PART I

Stochastic Lorentz gas ( m, v ) tracer ¡ scaQerer ¡ ( M, V ) Ÿ Inelastic collisions with random v i +1 = γ v i + (1 − γ ) V scatterers (independet on |v-V| !!!!) P nc ( t ) = e − t/ τ c Ÿ Free fligths at constant velocity σ 2 = T/M γ = ( ζ − α ) / (1 + ζ ) ζ = m/M P S ( V ) = Gauss v α < 1 Res4tu4on ¡coefficient ¡ τ v i +1 v i t

Driven stochastic Lorentz gas ( m, v ) probe ¡ scaQerer ¡ ( M, V ) Ÿ Inelastic collisions with random v i +1 = γ v i + (1 − γ ) V scatterers (independet on |v-V| !!!!) P nc ( t ) = e − t/ τ c Ÿ ‘’Free’’ fligths at uniform acceleration v t i t t i +1 Ÿ ¡ Ÿ ¡ Ÿ ¡ t v ( t ) = v i + ( t − t i ) E

Driven stochastic Lorentz gas Linear Boltzmann equation 1 ✓ v − γ u ◆ Z τ c ∂ t P ( v, t ) + τ c E ∂ v P ( v, t ) = − P ( v, t ) + duP ( u, t ) P S 1 − γ 1 − γ Z 1 Z 1 dv 0 w ( v | v 0 ) P ( v 0 , t ) − dv 0 w ( v 0 | v ) P ( v, t ) τ c ∂ t P ( v, t ) + τ c E ∂ v P ( v, t ) = �1 �1 Special case exactly solvable (Renewal process) v i +1 = V γ = 0 α = ζ = m/M Z ∞ r 1 ✓ ◆ ✓ ◆ M − M − u 2 T ( v − u ) 2 P ( v ) = du exp × exp 2 π T τ c E τ c E 0

Driven stochastic Lorentz gas Special case exactly solvable γ = 0 α = ζ = m/M Z ∞ r 1 ✓ ◆ ✓ ◆ M − M − u 2 T ( v − u ) 2 P ( v ) = du exp × exp 2 π T τ c E τ c E 0

Driven stochastic Lorentz gas Drift h x ( t ) i ε ⇠ t Diffusion h x 2 ( t ) i ⇠ t Z ∞ r ✓ ◆ ✓ ◆ 1 M − M − u 2 T ( v − u ) 2 P ( v ) = du exp × exp 2 π T τ c E τ c E 0

Entropy production v ∆ s tot ( t ) = log P [ { v ( t ) }| v 0 ] P ( v 0 ) P [ { ˜ v ( t ) }| ˜ v 0 ] P (˜ v 0 ) t Forw { v ( t ) } = ( v 0 , t 1 , v 1 , v 0 1 , t 2 − t 1 , v 2 , v 0 2 , . . . , v n , v 0 n , t − t n ) Back v ( t ) } = ( − v t , t − t n , − v 0 n , − v n , t n − t n � 1 , . . . , − v 0 { ˜ 1 , − v 1 , t 1 , − v 0 ) Probability of forward and backward trajectories N c Y P nc ( t i − t i � 1 ) w ( v 0 P [ { v ( t ) }| v 0 ] = P nc ( t − t n ) i | v i ) j =1 N c Y P nc ( t i − t i � 1 ) w ( − v i | − v 0 P [ { ˜ v ( t ) }| ˜ v 0 ] = P nc ( t − t n ) i ) j =1

Entropy production Total entropy production N c ( t ) w ( v 0 j | v j ) ∆ s tot ( t ) = log P [ { v ( t ) }| v 0 ] P ( v 0 ) j ) + log P ( v 0 ) X v 0 ) = log P [ { ˜ v ( t ) }| ˜ v 0 ] P (˜ w ( − v j | − v 0 P ( − v t ) j =1 Probability of forward and backward trajectories N c Y P nc ( t i − t i � 1 ) w ( v 0 P [ { v ( t ) }| v 0 ] = P nc ( t − t n ) i | v i ) j =1 N c Y P nc ( t i − t i � 1 ) w ( − v i | − v 0 P [ { ˜ v ( t ) }| ˜ v 0 ] = P nc ( t − t n ) i ) j =1

Entropy production Total entropy production N c ( t ) w ( v 0 j | v j ) ∆ s tot ( t ) = log P [ { v ( t ) }| v 0 ] P ( v 0 ) j ) + log P ( v 0 ) X v 0 ) = log P [ { ˜ v ( t ) }| ˜ v 0 ] P (˜ w ( − v j | − v 0 P ( − v t ) j =1 ∆ s tot ( t ) = − ∆ E coll ( t ) + log P ( v 0 ) P ( − v t ) θ W ( t ) + ∆ E coll ( t ) = ∆ E ( t ) = m Energy = work + collisions 2 ( v 2 t − v 2 0 ) ∆ s tot ( t ) = W ( t ) 0 ] + ln P ( v (0)) − m 2 θ [ v 2 t − v 2 P ( − v ( t )) θ ‘’temperature of the probe when E = 0 θ = θ ( α , m, M, T ) 6 = T

Entropy production rate & average velocity W ( t ) = m E X ( t ) X ( t ) = O ( t ) ∆ s tot ( t ) = W ( t ) 0 ] + ln P ( v (0)) − m 2 θ [ v 2 t − v 2 P ( − v ( t )) θ B = O (1) ∆ s tot ( t ) = m E θ X ( t ) + B P ( X ( t ) /t = s ) ∼ e − tI ( v ) P ( ∆ s tot /t = s ) ∼ e − tI ( s )+ o ( t )

Large Deviations and Fluctuation Theorem P ( ∆ s tot /t = s ) ∼ e − tI ( s )+ o ( t ) Large Deviation Principle Symmetry of the rate function I ( s ) − I ( − s ) = s P ( ∆ s tot /t = s ) Fluctuation theorem P ( ∆ s tot /t = − s ) = e ts But not today for the driven Lorenz gas!!

Large Deviations and Fluctuation Theorem P ( ∆ s tot /t = s ) ∼ e − tI ( s )+ o ( t ) NO (Uniform) Large Deviation Principle P ( ∆ s tot /t = s ) Fluctuation theorem P ( ∆ s tot /t = − s ) = e ts

Large Deviations for entropy production rate ∆ s tot ( t ) = m E θ X ( t ) + B Scaling cumulant 1 t ln h e k ∆ s tot i λ ∆ s tot ( k ) = lim generating function t →∞ LDP rate function I ( s ) = max k ∈ R { sk − λ ∆ s tot ( k ) } λ ∆ s tot ( k ) < ∞ for k ∈ ( − 1 , 0] ∆ s tot ( k = � 1 + ) = m τ c E 2 = h W i λ 0 ∆ s tot ( k = 0 � ) = � λ 0 θ θ − m τ c E 2 , m τ c E 2  � I ( s ) well defined only for s ∈ θ θ

Large Deviations for entropy production rate Lack of large deviation principle for fluctuations on the right of average velocity h W i = m E v θ θ λ ∆ s tot ( k ) < ∞ for k ∈ ( − 1 , 0] ∆ s tot ( k = � 1 + ) = m τ c E 2 = h W i λ 0 ∆ s tot ( k = 0 � ) = � λ 0 θ θ − m τ c E 2 , m τ c E 2  � I ( s ) well defined only for s ∈ θ θ

Rare events induce violation of the LDP Exponentially rare long ballistic Violation of the LDP (no collisions) trajectories ∆ s tot ( t ) = 1 + ln P ( v (0)) ⇣ m ⌘ 2 [ v 2 (0) − v 2 ( t )] + m E X ( t ) θ P ( − v ( t )) ∆ s tot ∼ m ∆ s tot = ln P ( v (0)) 2 θ E 2 t 2 ∼ x E ( t ) P ( − v ( t )) √ ⇣ t √ s ⌘ P nc ( t ) = e − t/ τ c p ( ∆ s tot /t = s ) ∼ exp − κ «fat » tail

A constructive role of Fluctuation Theorem RIGHT tail of the distribution √ t φ R ( s ) P ( ∆ s tot /t = s ) ∼ e −

A constructive role of Fluctuation Theorem RIGHT tail of the distribution √ t φ R ( s ) P ( ∆ s tot /t = s ) ∼ e − Fluctuation Relation √ t φ R ( s ) P ( ∆ s tot /t = − s ) = e − ts P ( ∆ s tot /t = − s ) = e − ts −

A constructive role of Fluctuation Theorem RIGHT tail of the distribution √ t φ R ( s ) P ( ∆ s tot /t = s ) ∼ e − Fluctuation Relation √ t φ R ( s ) P ( ∆ s tot /t = − s ) = e − ts P ( ∆ s tot /t = − s ) = e − ts − LEFT tail of the distribution P ( ∆ s tot /t = − s ) = e − t φ L ( s )

Two speeds for the Large Deviation Principle s < s ∗ s > s ∗ √ P ( s, t ) ∼ e − t φ L ( s ) t φ R ( s ) P ( s, t ) ∼ e − s ∗ = m τ c E 2 θ s ∗

Two speeds for the Large Deviation Principle v < m E v > m E θ v θ v √ P ( v, t ) ∼ e − t φ L ( v ) t φ R ( v ) P ( v, t ) ∼ e − s ∗ = m E θ v s ∗

PART II

Continuous time random walk (with traps) Probability to move forward/backward 1 / 2 1 / 2 Free 1 / 2 − ε 1 / 2 + ε Driven p ( τ ) Persistence time before jumps

Continuous time random walk (with traps) Probability to move forward/backward 1 / 2 1 / 2 ⇣ x 1 h x 2 ( t ) i ⇠ t ⌘ P ( x, t ) = t 1 / 2 G t 1 / 2 1 / 2 − ε 1 ✓ x − v ε t ◆ 1 / 2 + ε P ε ( x, t ) = t 1 / 2 G t 1 / 2 h x ( t ) i ε ⇠ t p ( τ ) = e − τ / τ 0 Persistence time before jumps