Diffusions conditioned on occupation measures Florian Angeletti - - PowerPoint PPT Presentation

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

XT diffusion process Stochastic Differential Equation dXt = F(Xt)dt + σdWt Master equation: ∂tp(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 RT = 1 T T 1 1S(Xt) dt 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 RT = 1 T T 1 1S(Xt) dt 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

! !

t xt T RT

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Large deviation principle

R additive observable Exponentially decreasing probability P(RT = r) = e−TI(r)+o(T) I(r) decay rate: rate function Concentration point r0: I(r0) = 0

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Cramer theorem

Scaled cumulant generating function λ(k) = 1 T ln E

• ekTR(XT )

Rate function: I(r) = inf

k {rk − λ(k)}

Tilted operator: Lk = L + k1 1S λ(k) maximal eigenvalue of Lk

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Conditioned process

Xt|RT = r RT: Non-local observable Non homogeneous

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Conditioned process

Xt|RT = r RT: 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 Ct “generator”: Lk = L + k1 1S, i.e. E [f (Ct1) . . . f (Ctn)] = 1 Z(tn)

• p0(x0)et1Lf (x1)...e(tn−tn−1)Lf (xn)dx0 . . . dxn

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 YT Generalized Doobbs transform of Ct: Lk = r−1

k Lkrk − r−1 k (Lkrk)

rk maximal eigenvalue of Lk homogeneous

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Driven process

Yt diffusion process with effective force: dYt = Fk(Yt)dt + σdWt Effective force: Fk(x) = F(x) + D∇ ln rk(x)

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Process equivalence

XT|RT ≍ CT ≍ YT X ≍ Y ⇐ ⇒ lim

T

1 T ln dPX dPY

a.e

= 0

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Spectral problem

How to find λ(k) and r(k) Lkrk = λkrk Case 1 Lk 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) Lkrk = λkrk Case 1 Lk self-adjoint: reversible process with respect to Lebesgue measure Quantum mechanics results Case 2 Lk is not self-adjoint, real eigenvalues if F = ∇U reversible with respect to the stationary measure p = e−U symmetrisation: quantum eigenvalue problems HΨ = λΨ

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Spectral problem

How to find λ(k) and r(k) Lkrk = λkrk Case 1 Lk self-adjoint: reversible process with respect to Lebesgue measure Quantum mechanics results Case 2 Lk is not self-adjoint, real eigenvalues if F = ∇U reversible with respect to the stationary measure p = e−U symmetrisation: quantum eigenvalue problems HΨ = λΨ Case 3 True out-of-equilibrium system

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Ornstein-Uhlenbeck process

dXt = −γXtdt + σdWt, Case 2: U(x) = αx2 2 with α = 2γ/D and D = σ2. Rt = 1

T

T

0 1

1S(Xt) dt Lk = L + k1 1S = ⇒ Hk = H + k1 1S

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Quantum problem

Ψ′′(x) − x2 4 + ν(x)

• Ψ(x) = 0

Weber equation with piecewise constants where ν(x) = 2 αD

• λ − αD

4 − k1 1√αS(x)

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Perturbed potential

k < 0

4 2 2 4 2 4 6 8 10 12 x y

k = 0

4 2 2 4 2 4 6 8 10 12 x y

k > 0

4 2 2 4 2 2 4 6 8 10 12 x y

S = [−1, 1]

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Exact solutions

Solution space spanned by: s1(ν, x) = e− x2

4 1F1

ν 2 + 1 4; 1 2; x2 2

• s2(ν, x) = xe− x2

4 1F1

ν 2 + 3 4; 3 2; x2 2

• ,

where 1F1 (a; b; x) is the confluent hypergeometric function of the first kind. Exact solution vanishing at +∞ : W (ν, x) = 1 2

ν 2 + 1 4 √π

• cos

ν

2π + π 4

• Γ

1

4 − ν 2

• s1(ν, x)

− √ 2 sin ν

2π + π 4

• Γ

3

4 − ν 2

• s2(ν, x)
• .

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Piecewise solution on interval

S = [a, b] Ψ(x) =      K1W (ν′, −x) x < a K2s1(ν, x) + K3s2(ν, x) a < x < b K4W (ν′, x) b < x, with ν′ = ν(Sc) and ν = ν(S)

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Continuity condition

det     −W (ν′, −a) s1(ν, a) s2(ν, a) ∂xW (ν′, −a) ∂xs1(ν, a) ∂xs2(ν, a) s1(ν, b) s2(ν, b) −W (ν′, b) ∂xs1(ν, b) ∂xs2(ν, b) −∂xW (ν′, b)     = 0

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Rate functions

λ(k)

k λ(k)

Rate function I(k)

0.0 0.2 0.4 0.6 0.8 0.0 0.5 1.0 1.5

r I(r)

S = [0, 1] Parameters: α = σ = 1

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Effective potential

Effective potential Uk(x) for S = [0, 1]

4 2 2 4 5 10 15 20 25 30

x Uk(x)

k = {0; 2; 4; ..; 10}

4 2 2 4 2 4 6 8 10 12 14

x Uk(x)

k = {0; −2; −4; ..; −10}

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Asymptotic effective potential

6 4 2 2 4 6 2 4 6 8 10 12 14

x Uk(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, +∞) Ψ(x) =

• K1W (ν′, −x)

x < a K2W (ν, x) x > a

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Rate function

λ(k)

10 5 5 10 2 4 6 8

k λ(k)

Rate function I(k)

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

r I(r)

S = [1, ∞) α = σ = 1.

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Effective potential

4 2 2 4 10 20 30 40

x Uk(x)

k = 0 : 2 : 10

4 2 2 4 5 10 15 20

x Uk(x)

k = 0 : 2 : 10 Black : asymptotic effective potential U±∞(x) S = [1, ∞) α = σ = 1.

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Punctual occupation

S = {a} = lim

ǫ→0{a + ǫ, a − ǫ}

Ψ(x) =

• K1W (ν′, −x)

x < a K2W (ν′, x) x > a Continuity conditions K1W (ν, −a) − K2W (ν, a) = 0 K1∂xW (ν, −a) + K2∂xW (ν, a) = kΨ(a).

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Effective potential

4 2 2 4 5 10 15 20 25 30

x Uk(x)

k = 0, 1.01, 2.02, 4.04, 6.06

4 2 2 4 2 4 6 8 10

x Uk(x)

k = −1.01, −2.02, −4.04, −10.1 α = σ = 1

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Perturbative methods

Rewrite the hamiltonian Hk as a perturbed hamiltonian Hk+∆k = Hk + ∆k 1 1S First order approximation for eigenvalues: ∂kλn(k) = Ψn(k)|1 1S|Ψn(k) Eigenvectors: ∂kΨn(k) =

• m=n

Ψm(k)|1 1S|Ψn(k) λm(k) − λn(k) Ψm(k).

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Orthogonality matrix

Ni,j(k) = Ψi(k)|1 1S|Ψj(k) ∂kλn(k) = Nn,n ∂kNi,j(k) =

• m=i

Ni,m(k)Nm,j(k) λi(k) − λm(k) +

• m=j

Nj,m(k)Nm,i(k) λj(k) − λm(k) , Methods: Truncate higher eigenvalues terms and use a numerical solvers

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Rate function

S = [0, 1] λ(k)

10 5 5 10 1 1 2 3 4 5 6

k λ(k)

RatefunctionI(k)

0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

r I(r)

Black: exact function number of modes M = 2, 5, 10, 20 α = σ = 1

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Orthogonality matrix evolution

2 4 6 8 10 2 4 6 8 10

1.0 0.5 0.5 1.0

k = −10, −5, 0, 5, 10 Top: S = [1, ∞) Bottom: S = [0, 1] M = 10, α = σ = 1

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Failures

λ(k) for point occupation at x = 0

10 5 5 10 2 4 6 8 10

k λ(k)

Black: exact λ(k) M = 20, 40, 60, 80, 100, 120 α = σ = 1

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Perspectives

Beyond dimension 1 Adaptative numerical method for the spectral problem Perturbing out-of-equilibrium well-known statistical physics system

Introduction Large deviation function Driven process Ornstein-Uhlenbeck process Perturbative methods Conclusion

Questions?