Post on 01-Jan-2016
description
Lecture 6: Langevin equations
Outline:• linear/nonlinear, additive and multiplicative noise• soluble linear example w/ additive noise: Ornstein-Uhlenbeck process• general 1-d nonlinear equation with multiplicative noise• relation to Fokker-Planck equation• Ito formulation, relation between Ito & Stratonovich approaches
Stochastic differential equationsDifferential equations which contain (“are driven by”) random functions
Stochastic differential equationsDifferential equations which contain (“are driven by”) random functions
Langevin equations: the random function is Gaussian white noise ξ(t):
Stochastic differential equationsDifferential equations which contain (“are driven by”) random functions
Langevin equations: the random function is Gaussian white noise ξ(t):
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ξ(t)ξ ( ′ t ) = 2σ 2δ(t − ′ t )
Stochastic differential equationsDifferential equations which contain (“are driven by”) random functions
Langevin equations: the random function is Gaussian white noise ξ(t):
€
ξ(t)ξ ( ′ t ) = 2σ 2δ(t − ′ t )
ξ (t1)ξ (t2)ξ (t3)ξ (t4 ) = ξ (t1)ξ (t2) ξ (t3)ξ (t4 )
+ ξ (t1)ξ (t3) ξ (t2)ξ (t4 )
+ ξ (t1)ξ (t4 ) ξ (t2)ξ (t3) etc.
Stochastic differential equationsDifferential equations which contain (“are driven by”) random functions
Langevin equations: the random function is Gaussian white noise ξ(t):
€
ξ(t)ξ ( ′ t ) = 2σ 2δ(t − ′ t )
ξ (t1)ξ (t2)ξ (t3)ξ (t4 ) = ξ (t1)ξ (t2) ξ (t3)ξ (t4 )
+ ξ (t1)ξ (t3) ξ (t2)ξ (t4 )
+ ξ (t1)ξ (t4 ) ξ (t2)ξ (t3) etc.
Simple example (Brownian motion/ Ornstein-Uhlenbeck process):
€
mdv
dt= −γv + ξ (t)
Stochastic differential equationsDifferential equations which contain (“are driven by”) random functions
Langevin equations: the random function is Gaussian white noise ξ(t):
€
ξ(t)ξ ( ′ t ) = 2σ 2δ(t − ′ t )
ξ (t1)ξ (t2)ξ (t3)ξ (t4 ) = ξ (t1)ξ (t2) ξ (t3)ξ (t4 )
+ ξ (t1)ξ (t3) ξ (t2)ξ (t4 )
+ ξ (t1)ξ (t4 ) ξ (t2)ξ (t3) etc.
Simple example (Brownian motion/ Ornstein-Uhlenbeck process):
Solution v(t) is random (because it depends on ξ(t))
€
mdv
dt= −γv + ξ (t)
Stochastic differential equationsDifferential equations which contain (“are driven by”) random functions
Langevin equations: the random function is Gaussian white noise ξ(t):
€
ξ(t)ξ ( ′ t ) = 2σ 2δ(t − ′ t )
ξ (t1)ξ (t2)ξ (t3)ξ (t4 ) = ξ (t1)ξ (t2) ξ (t3)ξ (t4 )
+ ξ (t1)ξ (t3) ξ (t2)ξ (t4 )
+ ξ (t1)ξ (t4 ) ξ (t2)ξ (t3) etc.
Simple example (Brownian motion/ Ornstein-Uhlenbeck process):
Solution v(t) is random (because it depends on ξ(t)) Want to know P[v], averages over distribution of ξ(t)
€
mdv
dt= −γv + ξ (t)
€
v(t) , v(t)v( ′ t ) , K
More generally,
multivariate:
€
dx i
dt= − γ ij x j + ξ i(t), ξ i(t)ξ j ( ′ t ) = Δ ijδ(t − ′ t )
j
∑
More generally,
multivariate:
€
dx i
dt= − γ ij x j + ξ i(t), ξ i(t)ξ j ( ′ t ) = Δ ijδ(t − ′ t )
j
∑
mi
d2x i
dt 2+ η ij
dx j
dtj
∑ = − κ ij x j + ξ i(t)j
∑higher-order:
More generally,
multivariate:
€
dx i
dt= − γ ij x j + ξ i(t), ξ i(t)ξ j ( ′ t ) = Δ ijδ(t − ′ t )
j
∑
mi
d2x i
dt 2+ η ij
dx j
dtj
∑ = − κ ij x j + ξ i(t)j
∑
dx
dt= f (x, t) + ξ (t)nonlinear:
higher-order:
More generally,
multivariate:
€
dx i
dt= − γ ij x j + ξ i(t), ξ i(t)ξ j ( ′ t ) = Δ ijδ(t − ′ t )
j
∑
mi
d2x i
dt 2+ η ij
dx j
dtj
∑ = − κ ij x j + ξ i(t)j
∑
dx
dt= f (x, t) + ξ (t)
dx
dt= f (x, t) + g(x, t)ξ (t)
nonlinear:
higher-order:
multiplicativenoise:
Brownian motion
€
mdv
dt= −γv + ξ (t); ξ (t)ξ ( ′ t ) = σ 2δ(t − ′ t )
Brownian motion
€
mdv
dt= −γv + ξ (t); ξ (t)ξ ( ′ t ) = σ 2δ(t − ′ t )
solution (with m = 1):
€
v(t) = v(0)e−γt + d ′ t e−γ (t− ′ t )ξ ( ′ t )0
t
∫
Brownian motion
€
mdv
dt= −γv + ξ (t); ξ (t)ξ ( ′ t ) = σ 2δ(t − ′ t )
solution (with m = 1):
€
v(t) = v(0)e−γt + d ′ t e−γ (t− ′ t )ξ ( ′ t )0
t
∫
v(t) = v(0)e−γtt →∞
⏐ → ⏐ ⏐ 0averages:
Brownian motion
€
mdv
dt= −γv + ξ (t); ξ (t)ξ ( ′ t ) = σ 2δ(t − ′ t )
solution (with m = 1):
€
v(t) = v(0)e−γt + d ′ t e−γ (t− ′ t )ξ ( ′ t )0
t
∫
v(t) = v(0)e−γtt →∞
⏐ → ⏐ ⏐ 0
v(t1)v(t2) = v 2(0)e−γ ( t1 +t2 ) + d ′ t 0
t1∫ e−γ ( t1 − ′ t ) d ′ ′ t 0
t 2∫ e−γ (t2 − ′ ′ t ) ξ ( ′ t )ξ ( ′ ′ t )
averages:
Brownian motion
€
mdv
dt= −γv + ξ (t); ξ (t)ξ ( ′ t ) = σ 2δ(t − ′ t )
solution (with m = 1):
€
v(t) = v(0)e−γt + d ′ t e−γ (t− ′ t )ξ ( ′ t )0
t
∫
v(t) = v(0)e−γtt →∞
⏐ → ⏐ ⏐ 0
v(t1)v(t2) = v 2(0)e−γ ( t1 +t2 ) + d ′ t 0
t1∫ e−γ ( t1 − ′ t ) d ′ ′ t 0
t 2∫ e−γ (t2 − ′ ′ t ) ξ ( ′ t )ξ ( ′ ′ t )
averages:
€
t1 < t2 : σ 2 e−γ ( t1 +t2 −2 ′ t )d ′ t =σ 2
2γe−γ ( t1 +t2 ) e2γt1 −1( ) =
0
t1∫ σ 2
2γe−γ ( t2 −t1 ) − e−γ (t1 +t2 )
( )
Brownian motion
€
mdv
dt= −γv + ξ (t); ξ (t)ξ ( ′ t ) = σ 2δ(t − ′ t )
solution (with m = 1):
€
v(t) = v(0)e−γt + d ′ t e−γ (t− ′ t )ξ ( ′ t )0
t
∫
v(t) = v(0)e−γtt →∞
⏐ → ⏐ ⏐ 0
v(t1)v(t2) = v 2(0)e−γ ( t1 +t2 ) + d ′ t 0
t1∫ e−γ ( t1 − ′ t ) d ′ ′ t 0
t 2∫ e−γ (t2 − ′ ′ t ) ξ ( ′ t )ξ ( ′ ′ t )
averages:
€
t1 < t2 : σ 2 e−γ ( t1 +t2 −2 ′ t )d ′ t =σ 2
2γe−γ ( t1 +t2 ) e2γt1 −1( ) =
0
t1∫ σ 2
2γe−γ ( t2 −t1 ) − e−γ (t1 +t2 )
( )
t1 > t2: : L =σ 2
2γe−γ (t1 −t2 ) − e−γ ( t1 +t2 )
( )
Brownian motion
€
mdv
dt= −γv + ξ (t); ξ (t)ξ ( ′ t ) = σ 2δ(t − ′ t )
solution (with m = 1):
€
v(t) = v(0)e−γt + d ′ t e−γ (t− ′ t )ξ ( ′ t )0
t
∫
v(t) = v(0)e−γtt →∞
⏐ → ⏐ ⏐ 0
v(t1)v(t2) = v 2(0)e−γ ( t1 +t2 ) + d ′ t 0
t1∫ e−γ ( t1 − ′ t ) d ′ ′ t 0
t 2∫ e−γ (t2 − ′ ′ t ) ξ ( ′ t )ξ ( ′ ′ t )
averages:
€
t1 < t2 : σ 2 e−γ ( t1 +t2 −2 ′ t )d ′ t =σ 2
2γe−γ ( t1 +t2 ) e2γt1 −1( ) =
0
t1∫ σ 2
2γe−γ ( t2 −t1 ) − e−γ (t1 +t2 )
( )
t1 > t2: : L =σ 2
2γe−γ (t1 −t2 ) − e−γ ( t1 +t2 )
( )
⇒ v(t1)v(t2) = v 2(0)e−2γ (t1 +t2 ) +σ 2
2γe−γ t2 −t1 − e−γ ( t1 +t2 )
( )
Brownian motion
€
mdv
dt= −γv + ξ (t); ξ (t)ξ ( ′ t ) = σ 2δ(t − ′ t )
solution (with m = 1):
€
v(t) = v(0)e−γt + d ′ t e−γ (t− ′ t )ξ ( ′ t )0
t
∫
v(t) = v(0)e−γtt →∞
⏐ → ⏐ ⏐ 0
v(t1)v(t2) = v 2(0)e−γ ( t1 +t2 ) + d ′ t 0
t1∫ e−γ ( t1 − ′ t ) d ′ ′ t 0
t 2∫ e−γ (t2 − ′ ′ t ) ξ ( ′ t )ξ ( ′ ′ t )
averages:
€
t1 < t2 : σ 2 e−γ ( t1 +t2 −2 ′ t )d ′ t =σ 2
2γe−γ ( t1 +t2 ) e2γt1 −1( ) =
0
t1∫ σ 2
2γe−γ ( t2 −t1 ) − e−γ (t1 +t2 )
( )
t1 > t2: : L =σ 2
2γe−γ (t1 −t2 ) − e−γ ( t1 +t2 )
( )
⇒ v(t1)v(t2) = v 2(0)e−2γ (t1 +t2 ) +σ 2
2γe−γ t2 −t1 − e−γ ( t1 +t2 )
( ) t1 ,t2 →∞ ⏐ → ⏐ ⏐ σ 2
2γe−γ t1 −t2
Brown (2)
equal-time correlation:
€
v 2(t) =σ 2
2γ
Brown (2)
equal-time correlation:
but from equilibrium stat mech:
€
v 2(t) =σ 2
2γ
v 2(t) = T (m =1)
Brown (2)
equal-time correlation:
but from equilibrium stat mech:
€
v 2(t) =σ 2
2γ
v 2(t) = T (m =1)
⇒ σ 2 = 2γT
Brown (2)
equal-time correlation:
but from equilibrium stat mech:
€
v 2(t) =σ 2
2γ
v 2(t) = T (m =1)
⇒ σ 2 = 2γT
(another Einstein relation)
Brown (2)
equal-time correlation:
but from equilibrium stat mech:
€
v 2(t) =σ 2
2γ
v 2(t) = T (m =1)
⇒ σ 2 = 2γT
(another Einstein relation)
Note: OU model also applies with v -> x (position) to overdamped motion in a parabolic potential
Solution using Fourier transform
€
−iωx(ω) = −γx(ω) + ξ (ω)
Solution using Fourier transform
€
−iωx(ω) = −γx(ω) + ξ (ω)
ξ (ω)ξ ( ′ ω ) = σ 2 ⋅2πδ(ω + ′ ω ) = 2γT ⋅2πδ(ω + ′ ω )
Solution using Fourier transform
€
−iωx(ω) = −γx(ω) + ξ (ω)
ξ (ω)ξ ( ′ ω ) = σ 2 ⋅2πδ(ω + ′ ω ) = 2γT ⋅2πδ(ω + ′ ω )
i.e., ξ (ω)ξ (−ω) = ξ (ω)2
= σ 2 = 2γT
Solution using Fourier transform
€
−iωx(ω) = −γx(ω) + ξ (ω)
ξ (ω)ξ ( ′ ω ) = σ 2 ⋅2πδ(ω + ′ ω ) = 2γT ⋅2πδ(ω + ′ ω )
i.e., ξ (ω)ξ (−ω) = ξ (ω)2
= σ 2 = 2γT
x(ω) =ξ (ω)
−iω + γsolution:
Solution using Fourier transform
€
−iωx(ω) = −γx(ω) + ξ (ω)
ξ (ω)ξ ( ′ ω ) = σ 2 ⋅2πδ(ω + ′ ω ) = 2γT ⋅2πδ(ω + ′ ω )
i.e., ξ (ω)ξ (−ω) = ξ (ω)2
= σ 2 = 2γT
x(ω) =ξ (ω)
−iω + γ
x(ω)2
=ξ (ω)
2
ω2 + γ 2
solution:
Solution using Fourier transform
€
−iωx(ω) = −γx(ω) + ξ (ω)
ξ (ω)ξ ( ′ ω ) = σ 2 ⋅2πδ(ω + ′ ω ) = 2γT ⋅2πδ(ω + ′ ω )
i.e., ξ (ω)ξ (−ω) = ξ (ω)2
= σ 2 = 2γT
x(ω) =ξ (ω)
−iω + γ
x(ω)2
=ξ (ω)
2
ω2 + γ 2=
σ 2
ω2 + γ 2
solution:
Solution using Fourier transform
€
−iωx(ω) = −γx(ω) + ξ (ω)
ξ (ω)ξ ( ′ ω ) = σ 2 ⋅2πδ(ω + ′ ω ) = 2γT ⋅2πδ(ω + ′ ω )
i.e., ξ (ω)ξ (−ω) = ξ (ω)2
= σ 2 = 2γT
x(ω) =ξ (ω)
−iω + γ
x(ω)2
=ξ (ω)
2
ω2 + γ 2=
σ 2
ω2 + γ 2=
2Tγ
ω2 + γ 2
solution:
Solution using Fourier transform
€
−iωx(ω) = −γx(ω) + ξ (ω)
ξ (ω)ξ ( ′ ω ) = σ 2 ⋅2πδ(ω + ′ ω ) = 2γT ⋅2πδ(ω + ′ ω )
i.e., ξ (ω)ξ (−ω) = ξ (ω)2
= σ 2 = 2γT
x(ω) =ξ (ω)
−iω + γ
x(ω)2
=ξ (ω)
2
ω2 + γ 2=
σ 2
ω2 + γ 2=
2Tγ
ω2 + γ 2
x(0)x(t) = σ 2 dω
2π∫ e iωt
ω2 + γ 2=
σ 2
2π2πi( )
e−γt
2iγ=
σ 2
2γe−γt , t > 0
solution:
inverse FT:
Solution using Fourier transform
€
−iωx(ω) = −γx(ω) + ξ (ω)
ξ (ω)ξ ( ′ ω ) = σ 2 ⋅2πδ(ω + ′ ω ) = 2γT ⋅2πδ(ω + ′ ω )
i.e., ξ (ω)ξ (−ω) = ξ (ω)2
= σ 2 = 2γT
x(ω) =ξ (ω)
−iω + γ
x(ω)2
=ξ (ω)
2
ω2 + γ 2=
σ 2
ω2 + γ 2=
2Tγ
ω2 + γ 2
x(0)x(t) = σ 2 dω
2π∫ e iωt
ω2 + γ 2=
σ 2
2π2πi( )
e−γt
2iγ=
σ 2
2γe−γt , t > 0
=σ 2
2π2πi( )(−1)
e+γt
−2iγ=
σ 2
2γe+γt , t < 0
solution:
inverse FT:
Solution using Fourier transform
€
−iωx(ω) = −γx(ω) + ξ (ω)
ξ (ω)ξ ( ′ ω ) = σ 2 ⋅2πδ(ω + ′ ω ) = 2γT ⋅2πδ(ω + ′ ω )
i.e., ξ (ω)ξ (−ω) = ξ (ω)2
= σ 2 = 2γT
x(ω) =ξ (ω)
−iω + γ
x(ω)2
=ξ (ω)
2
ω2 + γ 2=
σ 2
ω2 + γ 2=
2Tγ
ω2 + γ 2
x(0)x(t) = σ 2 dω
2π∫ e iωt
ω2 + γ 2=
σ 2
2π2πi( )
e−γt
2iγ=
σ 2
2γe−γt , t > 0
=σ 2
2π2πi( )(−1)
e+γt
−2iγ=
σ 2
2γe+γt , t < 0
=σ 2
2γe−γ t
solution:
inverse FT:
Solution using Fourier transform
€
−iωx(ω) = −γx(ω) + ξ (ω)
ξ (ω)ξ ( ′ ω ) = σ 2 ⋅2πδ(ω + ′ ω ) = 2γT ⋅2πδ(ω + ′ ω )
i.e., ξ (ω)ξ (−ω) = ξ (ω)2
= σ 2 = 2γT
x(ω) =ξ (ω)
−iω + γ
x(ω)2
=ξ (ω)
2
ω2 + γ 2=
σ 2
ω2 + γ 2=
2Tγ
ω2 + γ 2
x(0)x(t) = σ 2 dω
2π∫ e iωt
ω2 + γ 2=
σ 2
2π2πi( )
e−γt
2iγ=
σ 2
2γe−γt , t > 0
=σ 2
2π2πi( )(−1)
e+γt
−2iγ=
σ 2
2γe+γt , t < 0
=σ 2
2γe−γ t = Te−γ t
solution:
inverse FT:
Solution using Fourier transform
€
−iωx(ω) = −γx(ω) + ξ (ω)
ξ (ω)ξ ( ′ ω ) = σ 2 ⋅2πδ(ω + ′ ω ) = 2γT ⋅2πδ(ω + ′ ω )
i.e., ξ (ω)ξ (−ω) = ξ (ω)2
= σ 2 = 2γT
x(ω) =ξ (ω)
−iω + γ
x(ω)2
=ξ (ω)
2
ω2 + γ 2=
σ 2
ω2 + γ 2=
2Tγ
ω2 + γ 2
x(0)x(t) = σ 2 dω
2π∫ e iωt
ω2 + γ 2=
σ 2
2π2πi( )
e−γt
2iγ=
σ 2
2γe−γt , t > 0
=σ 2
2π2πi( )(−1)
e+γt
−2iγ=
σ 2
2γe+γt , t < 0
=σ 2
2γe−γ t = Te−γ t
solution:
inverse FT:
(as in direct calculation)
Damped oscillator
€
md2x
dt 2+ η
dx
dt+ κx = ξ (t)
Damped oscillator
FT:
€
−ω 2 − iωγ + ω02
( )x(ω) = ξ (ω) ω02 = κ /m, γ = η /m( )€
md2x
dt 2+ η
dx
dt+ κx = ξ (t)
Damped oscillator
FT:
€
−ω 2 − iωγ + ω02
( )x(ω) = ξ (ω) ω02 = κ /m, γ = η /m( )
⇒ x(ω)2
=ξ (ω)
2m2
ω2 −ω02
( )2
+ γ 2ω2=
σ 2 m2
ω2 −ω02
( )2
+ γ 2ω2
€
md2x
dt 2+ η
dx
dt+ κx = ξ (t)
Damped oscillator
FT:
€
−ω 2 − iωγ + ω02
( )x(ω) = ξ (ω) ω02 = κ /m, γ = η /m( )
⇒ x(ω)2
=ξ (ω)
2m2
ω2 −ω02
( )2
+ γ 2ω2=
σ 2 m2
ω2 −ω02
( )2
+ γ 2ω2
⇒ x(t)x(0) =σ 2
2ηκe−γ t / 2 cos ω0
2 − 14 γ 2 t( ) +
γ
2 ω02 − 1
4 γ 2sin ω0
2 − 14 γ 2 t( )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
inverse FT:
€
md2x
dt 2+ η
dx
dt+ κx = ξ (t)
General OU process
€
dx i
dt= − Γij x j + ξ i(t), ξ i(t)ξ j ( ′ t ) =
j
∑ Δ ijδ ijδ(t − ′ t )
General OU process
€
dx i
dt= − Γij x j + ξ i(t), ξ i(t)ξ j ( ′ t ) =
j
∑ Δ ijδ ijδ(t − ′ t )
md2x
dt 2+ η
dx
dt+ κx = ξ (t)
damped oscillator:
General OU process
€
dx i
dt= − Γij x j + ξ i(t), ξ i(t)ξ j ( ′ t ) =
j
∑ Δ ijδ ijδ(t − ′ t )
md2x
dt 2+ η
dx
dt+ κx = ξ (t)
˙ x =p
m˙ p = −κx − γp + ξ (t)
damped oscillator:
General OU process
€
dx i
dt= − Γij x j + ξ i(t), ξ i(t)ξ j ( ′ t ) =
j
∑ Δ ijδ ijδ(t − ′ t )
md2x
dt 2+ η
dx
dt+ κx = ξ (t)
˙ x =p
m˙ p = −κx − γp + ξ (t)
damped oscillator:
Is a 2-d OU process with
€
Γ=0 −1 m
κ γ
⎛
⎝ ⎜
⎞
⎠ ⎟, Δ =
0 0
0 σ 2
⎛
⎝ ⎜
⎞
⎠ ⎟
General OU process
€
dx i
dt= − Γij x j + ξ i(t), ξ i(t)ξ j ( ′ t ) =
j
∑ Δ ijδ ijδ(t − ′ t )
md2x
dt 2+ η
dx
dt+ κx = ξ (t)
˙ x =p
m˙ p = −κx − γp + ξ (t)
damped oscillator:
Is a 2-d OU process with
€
Γ=0 −1 m
κ γ
⎛
⎝ ⎜
⎞
⎠ ⎟, Δ =
0 0
0 σ 2
⎛
⎝ ⎜
⎞
⎠ ⎟
x(t) is not a Markov process (2nd order equation), but (x(t),p(t)) is (1st order equation).
Formal solution by FT
€
−iωI+ Γ( )x(ω) = ξ (ω)
Formal solution by FT
€
−iωI+ Γ( )x(ω) = ξ (ω)
x(ω)xT (−ω) = −iωI+ Γ( )−1
ξ (ω)ξ T (−ω) iωI+ ΓT( )
−1
Formal solution by FT
€
−iωI+ Γ( )x(ω) = ξ (ω)
x(ω)xT (−ω) = −iωI+ Γ( )−1
ξ (ω)ξ T (−ω) iωI+ ΓT( )
−1
= −iωI+ Γ( )−1
Δ iωI+ ΓT( )
−1
Formal solution by FT
€
−iωI+ Γ( )x(ω) = ξ (ω)
x(ω)xT (−ω) = −iωI+ Γ( )−1
ξ (ω)ξ T (−ω) iωI+ ΓT( )
−1
= −iωI+ Γ( )−1
Δ iωI+ ΓT( )
−1
damped oscillator case:
€
x(ω)xT (−ω) =−iω −1 m
κ −iω + γ
⎛
⎝ ⎜
⎞
⎠ ⎟
−10 0
0 σ 2
⎛
⎝ ⎜
⎞
⎠ ⎟
iω κ
−1 m iω + γ
⎛
⎝ ⎜
⎞
⎠ ⎟
−1
Formal solution by FT
€
−iωI+ Γ( )x(ω) = ξ (ω)
x(ω)xT (−ω) = −iωI+ Γ( )−1
ξ (ω)ξ T (−ω) iωI+ ΓT( )
−1
= −iωI+ Γ( )−1
Δ iωI+ ΓT( )
−1
damped oscillator case:
€
x(ω)xT (−ω) =−iω −1 m
κ −iω + γ
⎛
⎝ ⎜
⎞
⎠ ⎟
−10 0
0 σ 2
⎛
⎝ ⎜
⎞
⎠ ⎟
iω κ
−1 m iω + γ
⎛
⎝ ⎜
⎞
⎠ ⎟
−1
=1
−iω(−iω + γ) + ω02 2
−iω + γ 1 m
−κ iω
⎛
⎝ ⎜
⎞
⎠ ⎟0 0
0 σ 2
⎛
⎝ ⎜
⎞
⎠ ⎟iω + γ −κ
1 m iω
⎛
⎝ ⎜
⎞
⎠ ⎟
Formal solution by FT
€
−iωI+ Γ( )x(ω) = ξ (ω)
x(ω)xT (−ω) = −iωI+ Γ( )−1
ξ (ω)ξ T (−ω) iωI+ ΓT( )
−1
= −iωI+ Γ( )−1
Δ iωI+ ΓT( )
−1
damped oscillator case:
€
x(ω)xT (−ω) =−iω −1 m
κ −iω + γ
⎛
⎝ ⎜
⎞
⎠ ⎟
−10 0
0 σ 2
⎛
⎝ ⎜
⎞
⎠ ⎟
iω κ
−1 m iω + γ
⎛
⎝ ⎜
⎞
⎠ ⎟
−1
=1
−iω(−iω + γ) + ω02 2
−iω + γ 1 m
−κ iω
⎛
⎝ ⎜
⎞
⎠ ⎟0 0
0 σ 2
⎛
⎝ ⎜
⎞
⎠ ⎟iω + γ −κ
1 m iω
⎛
⎝ ⎜
⎞
⎠ ⎟
=σ 2 m2
−iω(−iω + γ) + ω02 2
1 iωm
−iωm ω2m2
⎛
⎝ ⎜
⎞
⎠ ⎟
General 1-d Langevin equation
€
dx
dt= F(x) + ξ (t)nonlinear:
General 1-d Langevin equation
€
dx
dt= F(x) + ξ (t)
dx
dt= −γ sin(x) + ξ (t)
nonlinear:
ex: overdamped pendulum
General 1-d Langevin equation
€
dx
dt= F(x) + ξ (t)
dx
dt= −γ sin(x) + ξ (t)
dx
dt= F(x) + G(x)ξ (t)
nonlinear:
ex: overdamped pendulum
with multiplicative noise
General 1-d Langevin equation
€
dx
dt= F(x) + ξ (t)
dx
dt= −γ sin(x) + ξ (t)
dx
dt= F(x) + G(x)ξ (t)
dx
dt= rx + xξ (t)
nonlinear:
ex: overdamped pendulum
with multiplicative noise
ex: geometric Brownian motion
Fokker-Planck for nonlinear case
€
x(t + Δt) = x(t) + F(x(t))Δt + ξ (t)Δt
Fokker-Planck for nonlinear case
By definition of Gaussian noise ξ,
€
x(t + Δt) = x(t) + F(x(t))Δt + ξ (t)Δt
P x(t + Δt) | x(t)( ) =1
2πσ 2Δtexp −
x(t + Δt) − F x(t)( )Δt − x(t)( )2
2σ 2Δt
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
Fokker-Planck for nonlinear case
By definition of Gaussian noise ξ,
€
x(t + Δt) = x(t) + F(x(t))Δt + ξ (t)Δt
P x(t + Δt) | x(t)( ) =1
2πσ 2Δtexp −
x(t + Δt) − F x(t)( )Δt − x(t)( )2
2σ 2Δt
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
Δx = F x(t)( )Δt, (Δx)2 = σ 2Δt
Fokker-Planck for nonlinear case
By definition of Gaussian noise ξ,
€
x(t + Δt) = x(t) + F(x(t))Δt + ξ (t)Δt
P x(t + Δt) | x(t)( ) =1
2πσ 2Δtexp −
x(t + Δt) − F x(t)( )Δt − x(t)( )2
2σ 2Δt
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
Δx = F x(t)( )Δt, (Δx)2 = σ 2Δt
r1(x) = F(x) = u(x), r2(x) = σ 2 = 2D
in terms of Kramers-Moyal expansion coefficients,
Fokker-Planck for nonlinear case
By definition of Gaussian noise ξ,
€
x(t + Δt) = x(t) + F(x(t))Δt + ξ (t)Δt
P x(t + Δt) | x(t)( ) =1
2πσ 2Δtexp −
x(t + Δt) − F x(t)( )Δt − x(t)( )2
2σ 2Δt
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
Δx = F x(t)( )Δt, (Δx)2 = σ 2Δt
r1(x) = F(x) = u(x), r2(x) = σ 2 = 2D
∂P
∂t= −
∂
∂xF(x)P(x, t)( ) + 1
2 σ 2 ∂ 2P(x, t)
∂x 2
in terms of Kramers-Moyal expansion coefficients,
=> FP equation
Fokker-Planck for nonlinear case
By definition of Gaussian noise ξ,
€
x(t + Δt) = x(t) + F(x(t))Δt + ξ (t)Δt
P x(t + Δt) | x(t)( ) =1
2πσ 2Δtexp −
x(t + Δt) − F x(t)( )Δt − x(t)( )2
2σ 2Δt
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
Δx = F x(t)( )Δt, (Δx)2 = σ 2Δt
r1(x) = F(x) = u(x), r2(x) = σ 2 = 2D
∂P
∂t= −
∂
∂xF(x)P(x, t)( ) + 1
2 σ 2 ∂ 2P(x, t)
∂x 2
in terms of Kramers-Moyal expansion coefficients,
=> FP equation
(FP equation is still linear, though Langevin equation is nonlinear)
with multiplicative noise
€
x(t + Δt) = x(t) + u(x(t))Δt + G x(t)( )ξ (t)Δt
with multiplicative noise
€
x(t + Δt) = x(t) + u(x(t))Δt + G x(t)( )ξ (t)Δt
But ξ(t) is composed of δ-functionseach δ-function causes a jump in x
with multiplicative noise
€
x(t + Δt) = x(t) + u(x(t))Δt + G x(t)( )ξ (t)Δt
But ξ(t) is composed of δ-functionseach δ-function causes a jump in xis the G(x(t)) to be evaluated before or after the jump, or in between?
with multiplicative noise
€
x(t + Δt) = x(t) + u(x(t))Δt + G x(t)( )ξ (t)Δt
But ξ(t) is composed of δ-functionseach δ-function causes a jump in xis the G(x(t)) to be evaluated before or after the jump, or in between?
If you implement it numerically exactly as written, you are adopting
with multiplicative noise
€
x(t + Δt) = x(t) + u(x(t))Δt + G x(t)( )ξ (t)Δt
But ξ(t) is composed of δ-functionseach δ-function causes a jump in xis the G(x(t)) to be evaluated before or after the jump, or in between?
If you implement it numerically exactly as written, you are adoptingthe Ito convention
with multiplicative noise
€
x(t + Δt) = x(t) + u(x(t))Δt + G x(t)( )ξ (t)Δt
But ξ(t) is composed of δ-functionseach δ-function causes a jump in xis the G(x(t)) to be evaluated before or after the jump, or in between?
If you implement it numerically exactly as written, you are adoptingthe Ito convention
An alternative approach is to takeand take the limit τ -> 0 at the end.
€
ξ(t)ξ ( ′ t ) =σ 2
τf
t − ′ t
τ
⎛
⎝ ⎜
⎞
⎠ ⎟, f (x)dx =1
−∞
∞
∫
with multiplicative noise
€
x(t + Δt) = x(t) + u(x(t))Δt + G x(t)( )ξ (t)Δt
But ξ(t) is composed of δ-functionseach δ-function causes a jump in xis the G(x(t)) to be evaluated before or after the jump, or in between?
If you implement it numerically exactly as written, you are adoptingthe Ito convention
An alternative approach is to takeand take the limit τ -> 0 at the end. This is
the Stratonovich convention
€
ξ(t)ξ ( ′ t ) =σ 2
τf
t − ′ t
τ
⎛
⎝ ⎜
⎞
⎠ ⎟, f (x)dx =1
−∞
∞
∫
with multiplicative noise
€
x(t + Δt) = x(t) + u(x(t))Δt + G x(t)( )ξ (t)Δt
But ξ(t) is composed of δ-functionseach δ-function causes a jump in xis the G(x(t)) to be evaluated before or after the jump, or in between?
If you implement it numerically exactly as written, you are adoptingthe Ito convention
An alternative approach is to takeand take the limit τ -> 0 at the end. This is
the Stratonovich conventionIt is equivalent to evaluating G at the midpoint of the jump.
€
ξ(t)ξ ( ′ t ) =σ 2
τf
t − ′ t
τ
⎛
⎝ ⎜
⎞
⎠ ⎟, f (x)dx =1
−∞
∞
∫
with multiplicative noise
€
x(t + Δt) = x(t) + u(x(t))Δt + G x(t)( )ξ (t)Δt
But ξ(t) is composed of δ-functionseach δ-function causes a jump in xis the G(x(t)) to be evaluated before or after the jump, or in between?
If you implement it numerically exactly as written, you are adoptingthe Ito convention
An alternative approach is to takeand take the limit τ -> 0 at the end. This is
the Stratonovich conventionIt is equivalent to evaluating G at the midpoint of the jump.With multiplicative noise, these 2 conventions lead to different FP equations
€
ξ(t)ξ ( ′ t ) =σ 2
τf
t − ′ t
τ
⎛
⎝ ⎜
⎞
⎠ ⎟, f (x)dx =1
−∞
∞
∫
with multiplicative noise
€
x(t + Δt) = x(t) + u(x(t))Δt + G x(t)( )ξ (t)Δt
But ξ(t) is composed of δ-functionseach δ-function causes a jump in xis the G(x(t)) to be evaluated before or after the jump, or in between?
If you implement it numerically exactly as written, you are adoptingthe Ito convention
An alternative approach is to takeand take the limit τ -> 0 at the end. This is
the Stratonovich conventionIt is equivalent to evaluating G at the midpoint of the jump.With multiplicative noise, these 2 conventions lead to different FP equations. (For additive noise, they are 2 different ways to do the problem but must give the same answer.)
€
ξ(t)ξ ( ′ t ) =σ 2
τf
t − ′ t
τ
⎛
⎝ ⎜
⎞
⎠ ⎟, f (x)dx =1
−∞
∞
∫
FP equations from Ito and Stratonovich
Ito (same argument as for additive noise case):
FP equations from Ito and Stratonovich
Ito (same argument as for additive noise case):
€
∂P
∂t= −
∂
∂xu(x)P(x, t)( ) + 1
2 σ 2 ∂ 2
∂x 2G2(x)P(x, t)( )
FP equations from Ito and Stratonovich
Ito (same argument as for additive noise case):
€
∂P
∂t= −
∂
∂xu(x)P(x, t)( ) + 1
2 σ 2 ∂ 2
∂x 2G2(x)P(x, t)( )
∂P
∂t= −
∂
∂xu(x)P(x, t)( ) + 1
2 σ 2 ∂
∂xG(x)
∂
∂xG(x)P(x, t)( )
⎛
⎝ ⎜
⎞
⎠ ⎟
Stratonovich (it can be shown that):
FP equations from Ito and Stratonovich
Ito (same argument as for additive noise case):
€
∂P
∂t= −
∂
∂xu(x)P(x, t)( ) + 1
2 σ 2 ∂ 2
∂x 2G2(x)P(x, t)( )
∂P
∂t= −
∂
∂xu(x)P(x, t)( ) + 1
2 σ 2 ∂
∂xG(x)
∂
∂xG(x)P(x, t)( )
⎛
⎝ ⎜
⎞
⎠ ⎟
∂P
∂t= −
∂
∂xu(x) + 1
2 σ 2G(x) ′ G (x)( )P(x, t)[ ] + 12 σ 2 ∂ 2
∂x 2G2(x)P(x, t)( )
Stratonovich (it can be shown that):
or,equivalently,
FP equations from Ito and Stratonovich
Ito (same argument as for additive noise case):
€
∂P
∂t= −
∂
∂xu(x)P(x, t)( ) + 1
2 σ 2 ∂ 2
∂x 2G2(x)P(x, t)( )
∂P
∂t= −
∂
∂xu(x)P(x, t)( ) + 1
2 σ 2 ∂
∂xG(x)
∂
∂xG(x)P(x, t)( )
⎛
⎝ ⎜
⎞
⎠ ⎟
∂P
∂t= −
∂
∂xu(x) + 1
2 σ 2G(x) ′ G (x)( )P(x, t)[ ] + 12 σ 2 ∂ 2
∂x 2G2(x)P(x, t)( )
Stratonovich (it can be shown that):
“anomalous drift”
or,equivalently,
Where does the anomalous drift come from?
€
dx = u(x)dt + G(x) + 12
′ G (x)dx[ ]ξ (t)dt
From Stratonovich midpoint prescription:
Where does the anomalous drift come from?
€
dx = u(x)dt + G(x) + 12
′ G (x)dx[ ]ξ (t)dt
= u(x)dt + G(x)ξ (t)dt + ′ G (x) u(x)dt + G(x)ξ (t)dt( )ξ (t)dt
From Stratonovich midpoint prescription:
Where does the anomalous drift come from?
€
dx = u(x)dt + G(x) + 12
′ G (x)dx[ ]ξ (t)dt
= u(x)dt + G(x)ξ (t)dt + ′ G (x) u(x)dt + G(x)ξ (t)dt( )ξ (t)dt
= u(x)dt + G(x)ξ (t)dt + 12
′ G (x)G(x)ξ 2(t)dt 2
From Stratonovich midpoint prescription:
Where does the anomalous drift come from?
€
dx = u(x)dt + G(x) + 12
′ G (x)dx[ ]ξ (t)dt
= u(x)dt + G(x)ξ (t)dt + ′ G (x) u(x)dt + G(x)ξ (t)dt( )ξ (t)dt
= u(x)dt + G(x)ξ (t)dt + 12
′ G (x)G(x)ξ 2(t)dt 2
= u(x)dt + σG(x)ξ (t)dt + 12
′ G (x)G(x) ξ 2(t) dt 2
From Stratonovich midpoint prescription:
Where does the anomalous drift come from?
€
dx = u(x)dt + G(x) + 12
′ G (x)dx[ ]ξ (t)dt
= u(x)dt + G(x)ξ (t)dt + ′ G (x) u(x)dt + G(x)ξ (t)dt( )ξ (t)dt
= u(x)dt + G(x)ξ (t)dt + 12
′ G (x)G(x)ξ 2(t)dt 2
= u(x)dt + σG(x)ξ (t)dt + 12
′ G (x)G(x) ξ 2(t) dt 2
= u(x)dt + G(x)ξ (t)dt + 12
′ G (x)G(x)σ 2
dtdt 2
From Stratonovich midpoint prescription:
Where does the anomalous drift come from?
€
dx = u(x)dt + G(x) + 12
′ G (x)dx[ ]ξ (t)dt
= u(x)dt + G(x)ξ (t)dt + ′ G (x) u(x)dt + G(x)ξ (t)dt( )ξ (t)dt
= u(x)dt + G(x)ξ (t)dt + 12
′ G (x)G(x)ξ 2(t)dt 2
= u(x)dt + σG(x)ξ (t)dt + 12
′ G (x)G(x) ξ 2(t) dt 2
= u(x)dt + G(x)ξ (t)dt + 12 ′ G (x)G(x)
σ 2
dtdt 2
= u(x) + 12 σ 2 ′ G (x)G(x)( )dt + G(x)ξ (t)dt
From Stratonovich midpoint prescription:
formalism using differentialsSDE is written
€
dx = u(x)dt + σG(x)dW (t)
formalism using differentialsSDE is written
€
dx = u(x)dt + σG(x)dW (t)
dW
dt= ξ (t), W (t) = ξ (t ')d ′ t
0
t
∫ ξ (t)ξ ( ′ t ) = δ(t − ′ t )
formalism using differentialsSDE is written
€
dx = u(x)dt + σG(x)dW (t)
dW
dt= ξ (t), W (t) = ξ (t ')d ′ t
0
t
∫ ξ (t)ξ ( ′ t ) = δ(t − ′ t )
dW (t) = ξ (t)dt
formalism using differentialsSDE is written
€
dx = u(x)dt + σG(x)dW (t)
dW
dt= ξ (t), W (t) = ξ (t ')d ′ t
0
t
∫ ξ (t)ξ ( ′ t ) = δ(t − ′ t )
dW (t) = ξ (t)dt
dW = 0, dW 2 = dt
formalism using differentialsSDE is written
€
dx = u(x)dt + σG(x)dW (t)
dW
dt= ξ (t), W (t) = ξ (t ')d ′ t
0
t
∫ ξ (t)ξ ( ′ t ) = δ(t − ′ t )
dW (t) = ξ (t)dt
dW = 0, dW 2 = dt
Ito’s lemma: Consider some function F(x,t). What is dF(x,t)? Expand:
€
dF = F(x + dx, t + dt) − F(x, t)
formalism using differentialsSDE is written
€
dx = u(x)dt + σG(x)dW (t)
dW
dt= ξ (t), W (t) = ξ (t ')d ′ t
0
t
∫ ξ (t)ξ ( ′ t ) = δ(t − ′ t )
dW (t) = ξ (t)dt
dW = 0, dW 2 = dt
Ito’s lemma: Consider some function F(x,t). What is dF(x,t)? Expand:
€
dF = F(x + dx, t + dt) − F(x, t)
= F x + u(x)dt + σG(x)dW (t), t + Δt( ) − F(x, t)
formalism using differentialsSDE is written
€
dx = u(x)dt + σG(x)dW (t)
dW
dt= ξ (t), W (t) = ξ (t ')d ′ t
0
t
∫ ξ (t)ξ ( ′ t ) = δ(t − ′ t )
dW (t) = ξ (t)dt
dW = 0, dW 2 = dt
Ito’s lemma: Consider some function F(x,t). What is dF(x,t)? Expand:
€
dF = F(x + dx, t + dt) − F(x, t)
= F x + u(x)dt + σG(x)dW (t), t + Δt( ) − F(x, t)
=∂F
∂xu(x) +
∂F
∂t+ 1
2 σ 2 ∂ 2F
∂x 2G2(x)
⎛
⎝ ⎜
⎞
⎠ ⎟dt + σ
∂F
∂xG(x)dW
formalism using differentialsSDE is written
€
dx = u(x)dt + σG(x)dW (t)
dW
dt= ξ (t), W (t) = ξ (t ')d ′ t
0
t
∫ ξ (t)ξ ( ′ t ) = δ(t − ′ t )
dW (t) = ξ (t)dt
dW = 0, dW 2 = dt
Ito’s lemma: Consider some function F(x,t). What is dF(x,t)? Expand:
€
dF = F(x + dx, t + dt) − F(x, t)
= F x + u(x)dt + σG(x)dW (t), t + Δt( ) − F(x, t)
=∂F
∂xu(x) +
∂F
∂t+ 1
2 σ 2 ∂ 2F
∂x 2G2(x)
⎛
⎝ ⎜
⎞
⎠ ⎟dt + σ
∂F
∂xG(x)dW
______________
“because dW = O(Δt)”