Weak-noise limit of systems driven by non-Gaussian fluctuations Adrian Baule with P. Sollich (King’s College) GGI Florence, June 2014 A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 1 / 31
Stochastic model for non-equilibrium systems Equation of motion: √ q ( t ) = F ( q ( t )) + ˙ D ξ ( t ) A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 2 / 31
Stochastic model for non-equilibrium systems Equation of motion: √ q ( t ) = F ( q ( t )) + ˙ D ξ ( t ) + z ( t ) − a Poissonian shot noise (PSN) z(t) N t � z ( t ) = A i δ ( t − t i ) i =1 t ◮ N t Poisson distribution ◮ Times t i uniform in [0 , t ] ◮ A i are i.i.d. with density ρ ( A ) A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 2 / 31
Stochastic model for non-equilibrium systems Equation of motion: √ q ( t ) = F ( q ( t )) + ˙ D ξ ( t ) + z ( t ) − a Poissonian shot noise (PSN) z(t) N t � z ( t ) = A i δ ( t − t i ) i =1 t ◮ N t Poisson distribution ◮ Times t i uniform in [0 , t ] ◮ A i are i.i.d. with density ρ ( A ) √ L´ evy noise: Γ( t ) = D ξ ( t ) + z ( t ) − a , a = � z ( t ) � A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 2 / 31
Poissonian shot noise Average number of shots: � N ( t ) � = λ t � z ( t ) � = λ � A � A 2 � Cov ( z ( t ) , z ( t ′ )) � δ ( t − t ′ ) = λ Infinite hierarchy of cumulants A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 3 / 31
Poissonian shot noise Average number of shots: � N ( t ) � = λ t � z ( t ) � = λ � A � A 2 � Cov ( z ( t ) , z ( t ′ )) � δ ( t − t ′ ) = λ Infinite hierarchy of cumulants Non-local diffusion ∂ 2 ∂ − ∂ ∂ q ( F ( q ) − a ) p ( q , t ) + D ∂ t p ( q , t ) = ∂ q 2 p ( q , t ) 2 � ∞ + λ d A p ( q − A , t ) ρ ( A ) − λ p ( q , t ) −∞ Weak-noise limit? A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 3 / 31
Characteristic functional of PSN Poissonian shot noise (PSN) z(t) N t � z ( t ) = A i δ ( t − t i ) i =1 t N t Poisson distribution Times t i uniform in [0 , t ] � � A i are i.i.d. with density ρ ( A ) Calculate noise functional � � � t � t � � �� � � G z [ g ] = exp g ( s ) z ( s ) d s = exp λ ( φ ( g ( s )) − 1) d s i 0 0 � e iAk � where φ ( k ) = A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 4 / 31
Path-integral formalism Propagator given as path-integral over path weight P [ q ] � ( q , t ) f ( q , t | q 0 ) = D q P [ q ] ( q 0 , 0) � ( q , t ) � t � � � = D q D g exp − L ( q , g ) d s ( q 0 , 0) 0 Write P [ q ] as inverse functional FT � � � � P [ q ] = D g exp − i g ( s )(˙ q − F a ( q )) d s G ξ [ g ] G z [ g ] Lagrangian: q − F a ( q )) + 1 2 Dg 2 − λ ( φ ( g ) − 1) L ( q , g ) = ig (˙ Conjugate momentum: ∂ L /∂ ˙ q = ig A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 5 / 31
Path-integral formalism Lagrangian q − F a ( q )) + 1 2 Dg 2 − λ ( φ ( g ) − 1) L ( q , g ) = ig (˙ Want: L → ˜ L / D . Introduce the scaling: → g / D ˜ g A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 6 / 31
Path-integral formalism Lagrangian A 2 � g 2 A 3 � ig 3 q − F ( q )) + 1 �� � 2 Dg 2 + λ � L ( q , g ) = ig (˙ 2! + 3! + ... Want: L → ˜ L / D . Introduce the scaling: → g / D ˜ g ˜ λ/ D µ λ → A 0 D ν ˜ A 0 → A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 7 / 31
Path-integral formalism Lagrangian A 2 � g 2 A 3 � ig 3 q − F ( q )) + 1 �� � 2 Dg 2 + λ � L ( q , g ) = ig (˙ 2! + 3! + ... Want: L → ˜ L / D . Introduce the scaling: 3 I 2 → g / D ˜ g III Ν ˜ λ/ D µ λ → 1 II A 0 D ν ˜ A 0 → 0 1 2 3 Μ Gaussian weak-noise limit: ν = 1 2( µ + 1) , µ > 1 PSN weak-noise limit: µ = ν = 1 A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 7 / 31
Euler-Lagrange equations Saddle-point approximation for D → 0 � t � � − 1 f ( q , t | q 0 ) = ψ ( q ∗ , g ∗ ) exp L ( q ∗ , g ∗ ) d s (1 + O ( D )) D 0 Optimal paths determined by coupled EL equations F a ( q ) + ig − i λφ ′ ( g ) q ˙ = − F ′ g ˙ = a ( q ) g with boundary conditions q (0) = q 0 and q ( t ) = q t Prefactor ψ ( q ∗ , g ∗ ) can be calculated by recursion relation A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 8 / 31
Euler-Lagrange equations Saddle-point approximation for D → 0 � t � � − 1 f ( q , t | q 0 ) = ψ ( q ∗ , g ∗ ) exp L ( q ∗ , g ∗ ) d s (1 + O ( D )) D 0 Optimal paths determined by coupled EL equations F a ( q ) + ig − i λφ ′ ( g ) q ˙ = − F ′ g ˙ = a ( q ) g with boundary conditions q (0) = q 0 and q ( t ) = q t Prefactor ψ ( q ∗ , g ∗ ) can be calculated by recursion relation Gaussian case ( λ = 0) q − F ′ ( q ) F ( q ) = 0 g = − i (˙ q − F ( q )) → ¨ L = 1 q − F ( q )) 2 → 2(˙ A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 8 / 31
Weak-noise limit of non-equilibrium systems 1 Escape from metastable potential → asymptotic scaling of � τ ex � 2 Large deviations of non-equilibrium observables � t Ω[ q ] = U (˙ q , q ) d s 0 I ( ω ) = D → 0 D log P Ω ( ω ) lim 3 Piecewise linear transport model ◮ Simple model for noise induced transport ◮ Stationary properties ◮ Weak-noise approximation of finite time propagator A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 9 / 31
Escape from metastable potential q m Kramer’s rate � V � q � 1 � τ ex � ∝ e − β ∆ V r = � q 0 q A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 10 / 31
Escape from metastable potential q m Kramer’s rate � V � q � 1 � τ ex � ∝ e − β ∆ V r = � q 0 q Exact asymptotics of � τ ex � ( Freidlin & Wentzell ): D → 0 D log � τ ex � = inf lim t ≥ 0 S ( q m , t ; q 0 ) A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 10 / 31
Escape from metastable potential q m Kramer’s rate � V � q � 1 � τ ex � ∝ e − β ∆ V r = � q 0 q Exact asymptotics of � τ ex � ( Freidlin & Wentzell ): D → 0 D log � τ ex � = inf lim t ≥ 0 S ( q m , t ; q 0 ) Action for PSN: � t L ( q ∗ , g ∗ ) d s S ( q m , t ; q 0 ) = 0 A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 10 / 31
Escape path Gaussian case ( λ = 0) d 1 q 2 − F ( q ) 2 ) = 0 q − F ′ ( q ) F ( q ) = 0 ¨ → 2(˙ d t A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 11 / 31
Escape path Gaussian case ( λ = 0) d 1 q 2 − F ( q ) 2 ) = 0 q − F ′ ( q ) F ( q ) = 0 ¨ → 2(˙ d t Optimal paths: q = F ( q ) ˙ q = − F ( q ) ˙ Relaxation: zero action Excitation: non-zero action A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 11 / 31
Escape path Gaussian case ( λ = 0) d 1 q 2 − F ( q ) 2 ) = 0 q − F ′ ( q ) F ( q ) = 0 ¨ → 2(˙ d t Optimal paths: q = F ( q ) ˙ q = − F ( q ) ˙ Relaxation: zero action Excitation: non-zero action Escape path is the time-reverse of a deterministic relaxation path. Action: � t S = 1 q − F ( q )) 2 d s = 2∆ V (˙ 2 0 A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 11 / 31
Escape path q m � V � q � � q 0 q Gaussian case ( λ = 0) q m � V q � s � S q 0 s A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 12 / 31
Escape path PSN case ( λ � = 0) F a ( q ) + ig − i λφ ′ ( g ) ˙ = q − F ′ g ˙ = a ( q ) g with boundary conditions q (0) = q 0 and q ( t ) = q m Action: � t L ( q ∗ , g ∗ ) d s S ( q m , t ; q 0 ) = 0 Noise-free deterministic relaxation: g = 0 → S = 0 A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 13 / 31
Escape path Gaussian case ( λ = 0) q m � V q � s � S q 0 s PSN case ( λ � = 0) q m � V q � s � S q 0 s A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 14 / 31
Time-reversal symmetry Optimal paths break time-reversal symmetry Gaussian noise PSN q � s � q � s � s s Relation with fluctuation theorems Ratio of path probabilities β ∆ E thermal noise log p [ q ( s ) | q 0 ] q t ] = β ∆ S driving p [˜ q ( s ) | ˜ ? PSN A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 15 / 31
Large deviations of non-equilibrium observables Consider functionals of q ( s ) � t Ω[ q ] = U (˙ q , q ) d s 0 We are interested in large deviations I ( ω ) = lim D → 0 − D log P Ω ( ω ) Consider scaled cumulant generating function R t � q , q ) d s � e α 0 U (˙ Λ( α ) = lim D → 0 D log Legendre transform I ( ω ) = sup α ( αω − Λ( α )) A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 16 / 31
Large deviations of non-equilibrium observables Obtain from path-integral ˜ Λ( α ) = − inf S ( q t , t ; q 0 ) q t Modified Lagrangian ˜ L ( q ∗ , g ∗ ) = L ( q ∗ , g ∗ ) − α U (˙ q ∗ , q ∗ ) A. Baule (QMUL) Weak-noise limit GGI Florence, June 2014 17 / 31
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