Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Diffusions conditioned on occupation measures Florian Angeletti Work in collaboration with Hugo Touchette 28 January 2016
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Out-of-equilibrium systems How to describe out-of-equilibrium systems?
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Out-of-equilibrium systems How to describe out-of-equilibrium systems? Idea Start with an equilibrium system Study fluctuations of this system far away from equilibrium
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Out-of-equilibrium systems How to describe out-of-equilibrium systems? Idea Start with an equilibrium system Study fluctuations of this system far away from equilibrium Application Start simple Markov process Diffusion process
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Diffusion process X T diffusion process Stochastic Differential Equation dX t = F ( X t ) dt + σ dW t Master equation: ∂ t p ( x , t ) = L † p ( x , t ) Generator : L = F · ∇ + 1 2 ∇ · D ∇ Stationary state Fluctuation around stationary state Fluctuations far away from equilibrium?
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Occupation observable Set S S = [ a , b ] S = [ a , ∞ ] S = { a } ? Occupation rate � T R T = 1 1 1 S ( X t ) dt T 0 What happens during a rare fluctuation of R ( t )?
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Occupation observable Set S S = [ a , b ] S = [ a , ∞ ] S = { a } ? Occupation rate � T R T = 1 1 1 S ( X t ) dt T 0 What happens during a rare fluctuation of R ( t )? Fluctuations far away from equilibrium Large deviation function? Conditioning?
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion x t ! ! t R T T
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Large deviation principle R additive observable Exponentially decreasing probability P ( R T = r ) = e − TI ( r )+ o ( T ) I ( r ) decay rate: rate function Concentration point r 0 : I ( r 0 ) = 0
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Cramer theorem Scaled cumulant generating function λ ( k ) = 1 � e kTR ( X T ) � T ln E Rate function: I ( r ) = inf k { rk − λ ( k ) } Tilted operator: L k = L + k 1 1 S λ ( k ) maximal eigenvalue of L k
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Conditioned process X t | R T = r R T : Non-local observable Non homogeneous
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Conditioned process X t | R T = r R T : Non-local observable Non homogeneous Strong constraint Microcanonical process
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Canonical process Canonical process C t “generator”: L k = L + k 1 1 S , i.e. E [ f ( C t 1 ) . . . f ( C t n )] = 1 � p 0 ( x 0 ) e t 1 L f ( x 1 ) ... e ( t n − t n − 1 ) L f ( x n ) dx 0 . . . dx n Z ( t n ) Non-homogeneous k = I ( r ) : same average as the conditioned process
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Driven process Y T Generalized Doobbs transform of C t : L k = r − 1 k L k r k − r − 1 k ( L k r k ) r k maximal eigenvalue of L k homogeneous
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Driven process Y t diffusion process with effective force: dY t = F k ( Y t ) dt + σ dW t Effective force: F k ( x ) = F ( x ) + D ∇ ln r k ( x )
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Process equivalence X T | R T ≍ C T ≍ Y T T ln dP X 1 a.e X ≍ Y ⇐ ⇒ lim = 0 dP Y T
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Spectral problem How to find λ ( k ) and r ( k ) L k r k = λ k r k Case 1 L k self-adjoint: reversible process with respect to Lebesgue measure Quantum mechanics results
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Spectral problem How to find λ ( k ) and r ( k ) L k r k = λ k r k Case 1 L k self-adjoint: reversible process with respect to Lebesgue measure Quantum mechanics results Case 2 L k is not self-adjoint, real eigenvalues if F = ∇ U reversible with respect to the stationary measure p = e − U symmetrisation: quantum eigenvalue problems H Ψ = λ Ψ
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Spectral problem How to find λ ( k ) and r ( k ) L k r k = λ k r k Case 1 L k self-adjoint: reversible process with respect to Lebesgue measure Quantum mechanics results Case 2 L k is not self-adjoint, real eigenvalues if F = ∇ U reversible with respect to the stationary measure p = e − U symmetrisation: quantum eigenvalue problems H Ψ = λ Ψ Case 3 True out-of-equilibrium system
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Ornstein-Uhlenbeck process dX t = − γ X t dt + σ dW t , Case 2: U ( x ) = α x 2 2 with α = 2 γ/ D and D = σ 2 . � T R t = 1 0 1 1 S ( X t ) dt T L k = L + k 1 1 S = ⇒ H k = H + k 1 1 S
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Quantum problem � x 2 � Ψ ′′ ( x ) − 4 + ν ( x ) Ψ( x ) = 0 Weber equation with piecewise constants where 2 � λ − α D � 1 √ α S ( x ) ν ( x ) = − k 1 α D 4
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Perturbed potential k < 0 k = 0 k > 0 12 12 12 10 10 10 8 8 8 6 y y y 6 6 4 4 4 2 2 2 0 0 0 � 2 � 4 � 2 0 2 4 � 4 � 2 0 2 4 � 4 � 2 0 2 4 x x x S = [ − 1 , 1]
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Exact solutions Solution space spanned by: 2; x 2 � ν 2 + 1 4; 1 � s 1 ( ν, x ) = e − x 2 4 1 F 1 2 2; x 2 � ν 2 + 3 4; 3 � s 2 ( ν, x ) = xe − x 2 4 1 F 1 , 2 where 1 F 1 ( a ; b ; x ) is the confluent hypergeometric function of the first kind. Exact solution vanishing at + ∞ : 1 � 1 � ν � � � W ( ν, x ) = cos 2 π + π Γ 4 − ν s 1 ( ν, x ) 4 √ π 2 + 1 4 2 ν 2 √ � 3 � ν � � � − 2 sin 2 π + π Γ 4 − ν s 2 ( ν, x ) . 4 2
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Piecewise solution on interval S = [ a , b ] K 1 W ( ν ′ , − x ) x < a Ψ( x ) = K 2 s 1 ( ν, x ) + K 3 s 2 ( ν, x ) a < x < b K 4 W ( ν ′ , x ) b < x , with ν ′ = ν ( S c ) and ν = ν ( S )
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Continuity condition − W ( ν ′ , − a ) s 1 ( ν, a ) s 2 ( ν, a ) 0 ∂ x W ( ν ′ , − a ) ∂ x s 1 ( ν, a ) ∂ x s 2 ( ν, a ) 0 det = 0 − W ( ν ′ , b ) 0 s 1 ( ν, b ) s 2 ( ν, b ) − ∂ x W ( ν ′ , b ) 0 ∂ x s 1 ( ν, b ) ∂ x s 2 ( ν, b )
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Rate functions λ ( k ) Rate function I ( k ) 1.5 1.0 λ ( k ) I ( r ) 0.5 0.0 0.0 0.2 0.4 0.6 0.8 r k S = [0 , 1] Parameters: α = σ = 1
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Effective potential Effective potential U k ( x ) for S = [0 , 1] 30 14 25 12 20 10 U k ( x ) U k ( x ) 8 15 6 10 4 5 2 0 0 � 4 � 2 0 2 4 � 4 � 2 0 2 4 x x k = { 0; 2; 4; .. ; 10 } k = { 0; − 2; − 4; .. ; − 10 }
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Asymptotic effective potential 14 12 10 U k ( x ) 8 6 4 2 0 6 4 2 0 2 4 6 x Black: Effective potential U −∞ ( x ) preventing any occupation in S = [0 , 1] Blue: Effective potential U ∞ ( x ) forcing a total occupation in S = [0 , 1] Parameters: α = σ = 1
Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion Half-line S = [ a , + ∞ ) � K 1 W ( ν ′ , − x ) x < a Ψ( x ) = K 2 W ( ν, x ) x > a
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