lecture 6: langevin equations

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


Langevin equations: the random function is Gaussian white noise ξ(t):



€

ξ(t)ξ ( ′ t ) = 2σ 2δ(t − ′ 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.



€

ξ(t)ξ ( ′ t ) = 2σ 2δ(t − ′ t )


+ 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)



€

ξ(t)ξ ( ′ t ) = 2σ 2δ(t − ′ t )


+ ξ (t1)ξ (t3) ξ (t2)ξ (t4 )

+ ξ (t1)ξ (t4 ) ξ (t2)ξ (t3) etc.


Solution v(t) is random (because it depends on ξ(t))

€

mdv

dt= −γv + ξ (t)



€

ξ(t)ξ ( ′ t ) = 2σ 2δ(t − ′ t )


+ ξ (t1)ξ (t3) ξ (t2)ξ (t4 )

+ ξ (t1)ξ (t4 ) ξ (t2)ξ (t3) etc.


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


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


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


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 )


€

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 )


€

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 )


€

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 )


€

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 )


€

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 )


€

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)


but from equilibrium stat mech:

€

v 2(t) =σ 2

2γ

v 2(t) = T (m =1)

Brown (2)



€

v 2(t) =σ 2

2γ

v 2(t) = T (m =1)

⇒ σ 2 = 2γT

Brown (2)



€

v 2(t) =σ 2

2γ

v 2(t) = T (m =1)

⇒ σ 2 = 2γT

(another Einstein relation)

Brown (2)



€

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)


€

−iωx(ω) = −γx(ω) + ξ (ω)

ξ (ω)ξ ( ′ ω ) = σ 2 ⋅2πδ(ω + ′ ω ) = 2γT ⋅2πδ(ω + ′ ω )


€

−iωx(ω) = −γx(ω) + ξ (ω)


i.e., ξ (ω)ξ (−ω) = ξ (ω)2

= σ 2 = 2γT


€

−iωx(ω) = −γx(ω) + ξ (ω)


i.e., ξ (ω)ξ (−ω) = ξ (ω)2

= σ 2 = 2γT

x(ω) =ξ (ω)

−iω + γsolution:


€

−iωx(ω) = −γx(ω) + ξ (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:


€

−iωx(ω) = −γx(ω) + ξ (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:


€

−iωx(ω) = −γx(ω) + ξ (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:


€

−iωx(ω) = −γx(ω) + ξ (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:


€

−iωx(ω) = −γx(ω) + ξ (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:


€

−iωx(ω) = −γx(ω) + ξ (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:


€

−iωx(ω) = −γx(ω) + ξ (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:


€

−iωx(ω) = −γx(ω) + ξ (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


j


md2x

dt 2+ η

dx

dt+ κx = ξ (t)

damped oscillator:

General OU process

€

dx i


j


md2x

dt 2+ η

dx

dt+ κx = ξ (t)

˙ x =p

m˙ p = −κx − γp + ξ (t)

damped oscillator:

General OU process

€

dx i


j


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


j


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)


€

−iωI+ Γ( )x(ω) = ξ (ω)

x(ω)xT (−ω) = −iωI+ Γ( )−1

ξ (ω)ξ T (−ω) iωI+ ΓT( )

−1


€

−iωI+ Γ( )x(ω) = ξ (ω)

x(ω)xT (−ω) = −iωI+ Γ( )−1


−1

= −iωI+ Γ( )−1

Δ iωI+ ΓT( )

−1


€

−iωI+ Γ( )x(ω) = ξ (ω)

x(ω)xT (−ω) = −iωI+ Γ( )−1


−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


€

−iωI+ Γ( )x(ω) = ξ (ω)

x(ω)xT (−ω) = −iωI+ Γ( )−1


−1

= −iωI+ Γ( )−1

Δ iωI+ ΓT( )

−1


€

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

⎛

⎝ ⎜

⎞

⎠ ⎟


€

−iωI+ Γ( )x(ω) = ξ (ω)

x(ω)xT (−ω) = −iωI+ Γ( )−1


−1

= −iωI+ Γ( )−1

Δ iωI+ ΓT( )

−1


€

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:


€

dx

dt= F(x) + ξ (t)

dx

dt= −γ sin(x) + ξ (t)

nonlinear:

ex: overdamped pendulum


€

dx

dt= F(x) + ξ (t)

dx


dx

dt= F(x) + G(x)ξ (t)

nonlinear:


with multiplicative noise


€

dx

dt= F(x) + ξ (t)

dx


dx

dt= F(x) + G(x)ξ (t)

dx

dt= rx + xξ (t)

nonlinear:



ex: geometric Brownian motion

Fokker-Planck for nonlinear case

€

x(t + Δt) = x(t) + F(x(t))Δt + ξ (t)Δt


By definition of Gaussian noise ξ,

€


P x(t + Δt) | x(t)( ) =1

2πσ 2Δtexp −

x(t + Δt) − F x(t)( )Δt − x(t)( )2

2σ 2Δt

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥



€


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



€


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,



€


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


=> FP equation



€


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


=> FP equation

(FP equation is still linear, though Langevin equation is nonlinear)


€

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


€


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


€



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

−∞

∞

∫


€




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

−∞

∞

∫


€





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

−∞

∞

∫


€





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

−∞

∞

∫


€





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):



€

∂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 ∂ 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):



€

∂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)( )


or,equivalently,



€

∂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)( )


“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:


€

dx = u(x)dt + G(x) + 12


= u(x)dt + G(x)ξ (t)dt + ′ G (x) u(x)dt + G(x)ξ (t)dt( )ξ (t)dt



€

dx = u(x)dt + G(x) + 12



= u(x)dt + G(x)ξ (t)dt + 12

′ G (x)G(x)ξ 2(t)dt 2



€

dx = u(x)dt + G(x) + 12



= 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



€

dx = u(x)dt + G(x) + 12



= 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



€

dx = u(x)dt + G(x) + 12



= 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


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

dt= ξ (t), W (t) = ξ (t ')d ′ t

0

t

∫ ξ (t)ξ ( ′ t ) = δ(t − ′ t )

dW (t) = ξ (t)dt


€


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


€


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)


€


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


€

dF = F(x + dx, t + dt) − F(x, t)

= F x + u(x)dt + σG(x)dW (t), t + Δt( ) − F(x, 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


€

dF = F(x + dx, t + dt) − F(x, t)


=∂F

∂xu(x) +

∂F

∂t+ 1

2 σ 2 ∂ 2F

∂x 2G2(x)

⎛

⎝ ⎜

⎞

⎠ ⎟dt + σ

∂F

∂xG(x)dW


€


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


€

dF = F(x + dx, t + dt) − 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)”

lecture 6: langevin equations

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