Lecture 7: Langevin equations II: geometric Brownian motion and perturbation theory Outline: Geometric Brownian motion, Stratonovich and Ito models Ito

Lecture 7: Langevin equations II:geometric Brownian motion and

perturbation theory

Outline:• Geometric Brownian motion, Stratonovich and Ito models

• Ito calculus method • small noise correlation time method (Stratonovich only)• solution using Fokker-Planck equation

• Perturbation theory for nonlinear Langevin equations• diagrammatic expansion• self-consistent approximations

“Geometric Brownian motion” (GBM) with Ito calculus and Ito convention


model of share prices


Start with equation in differential form:


€

dx = rxdt + σxdW (t)



Apply Ito’s lemma with


€


€

G(x) = x, F(x) = log x





€


G(x) = x, F(x) = log x, ′ F (x) =1

x, ′ ′ F (x) = −

1

x 2





€


G(x) = x, F(x) = log x, ′ F (x) =1

x, ′ ′ F (x) = −

1

x 2

dF = ′ F (x)u(x) + + 12 σ 2 ′ ′ F (x)G2(x)( )dt + σ ′ F (x)G(x)dW





€


G(x) = x, F(x) = log x, ′ F (x) =1

x, ′ ′ F (x) = −

1

x 2


=1

x

⎛

⎝ ⎜

⎞

⎠ ⎟rx + 1

2 σ 2 −1

x 2

⎛

⎝ ⎜

⎞

⎠ ⎟x 2

⎡

⎣ ⎢

⎤

⎦ ⎥dt + σ

1

x

⎛

⎝ ⎜

⎞

⎠ ⎟xdW





€


G(x) = x, F(x) = log x, ′ F (x) =1

x, ′ ′ F (x) = −

1

x 2


=1

x

⎛

⎝ ⎜

⎞

⎠ ⎟rx + 1

2 σ 2 −1

x 2

⎛

⎝ ⎜

⎞

⎠ ⎟x 2

⎡

⎣ ⎢

⎤

⎦ ⎥dt + σ

1

x

⎛

⎝ ⎜

⎞

⎠ ⎟xdW

= r − 12 σ 2

( )dt + σdW


F = log x(t) is normally distributed with mean (r – ½σ2)t and variance σ2t




€


G(x) = x, F(x) = log x, ′ F (x) =1

x, ′ ′ F (x) = −

1

x 2


=1

x

⎛

⎝ ⎜

⎞

⎠ ⎟rx + 1

2 σ 2 −1

x 2

⎛

⎝ ⎜

⎞

⎠ ⎟x 2

⎡

⎣ ⎢

⎤

⎦ ⎥dt + σ

1

x

⎛

⎝ ⎜

⎞

⎠ ⎟xdW

= r − 12 σ 2

( )dt + σdW


F = log x(t) is normally distributed with mean (r – ½σ2)t and variance σ2t => x(t) is log-normally distributed:




€


G(x) = x, F(x) = log x, ′ F (x) =1

x, ′ ′ F (x) = −

1

x 2


=1

x

⎛

⎝ ⎜

⎞

⎠ ⎟rx + 1

2 σ 2 −1

x 2

⎛

⎝ ⎜

⎞

⎠ ⎟x 2

⎡

⎣ ⎢

⎤

⎦ ⎥dt + σ

1

x

⎛

⎝ ⎜

⎞

⎠ ⎟xdW

= r − 12 σ 2

( )dt + σdW


F = log x(t) is normally distributed with mean (r – ½σ2)t and variance σ2t => x(t) is log-normally distributed:

€

P(x) =1

x 2πσ 2texp −

log x − r −σ 2 /2( )t( )2

2σ 2t

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥




€

x(0) =1( )

€


G(x) = x, F(x) = log x, ′ F (x) =1

x, ′ ′ F (x) = −

1

x 2


=1

x

⎛

⎝ ⎜

⎞

⎠ ⎟rx + 1

2 σ 2 −1

x 2

⎛

⎝ ⎜

⎞

⎠ ⎟x 2

⎡

⎣ ⎢

⎤

⎦ ⎥dt + σ

1

x

⎛

⎝ ⎜

⎞

⎠ ⎟xdW

= r − 12 σ 2

( )dt + σdW

moments of x(t)can get moments of x(t) from this or directly from the SDE and Ito’s lemma:


€

d(x n ) = nx n−1( )u(x)dt + 1

2 σ 2 n(n −1)x n−2( )G

2(x)dt + σ nx n−1( )G(x)dW


€

d(x n ) = nx n−1( )u(x)dt + 1

2 σ 2 n(n −1)x n−2( )G


= nx n−1( )rxdt + 1

2 σ 2 n(n −1)x n−2( )x 2dt + σ nx n−1

( )xdW


€

d(x n ) = nx n−1( )u(x)dt + 1

2 σ 2 n(n −1)x n−2( )G


= nx n−1( )rxdt + 1

2 σ 2 n(n −1)x n−2( )x 2dt + σ nx n−1

( )xdW

= n r + 12 σ 2(n −1)( )x ndt + nσx ndW


€

d(x n ) = nx n−1( )u(x)dt + 1

2 σ 2 n(n −1)x n−2( )G


= nx n−1( )rxdt + 1

2 σ 2 n(n −1)x n−2( )x 2dt + σ nx n−1

( )xdW

= n r + 12 σ 2(n −1)( )x ndt + nσx ndW

d x n

dt= n r + 1

2 σ 2(n −1)( ) x n dt


€

d(x n ) = nx n−1( )u(x)dt + 1

2 σ 2 n(n −1)x n−2( )G


= nx n−1( )rxdt + 1

2 σ 2 n(n −1)x n−2( )x 2dt + σ nx n−1

( )xdW

= n r + 12 σ 2(n −1)( )x ndt + nσx ndW

d x n

dt= n r + 1

2 σ 2(n −1)( ) x n dt

x n (t) = exp n r + 12 σ 2(n −1)( )t[ ] x(0) =1( )in particular,


€

d(x n ) = nx n−1( )u(x)dt + 1

2 σ 2 n(n −1)x n−2( )G


= nx n−1( )rxdt + 1

2 σ 2 n(n −1)x n−2( )x 2dt + σ nx n−1

( )xdW

= n r + 12 σ 2(n −1)( )x ndt + nσx ndW

d x n

dt= n r + 1

2 σ 2(n −1)( ) x n dt

x n (t) = exp n r + 12 σ 2(n −1)( )t[ ] x(0) =1( )

x(t) = exp rt( )

in particular,


€

d(x n ) = nx n−1( )u(x)dt + 1

2 σ 2 n(n −1)x n−2( )G


= nx n−1( )rxdt + 1

2 σ 2 n(n −1)x n−2( )x 2dt + σ nx n−1

( )xdW

= n r + 12 σ 2(n −1)( )x ndt + nσx ndW

d x n

dt= n r + 1

2 σ 2(n −1)( ) x n dt

x n (t) = exp n r + 12 σ 2(n −1)( )t[ ] x(0) =1( )

x(t) = exp rt( )

x 2(t) = exp 2 r + 12 σ 2

( )t[ ]

in particular,


€

d(x n ) = nx n−1( )u(x)dt + 1

2 σ 2 n(n −1)x n−2( )G


= nx n−1( )rxdt + 1

2 σ 2 n(n −1)x n−2( )x 2dt + σ nx n−1

( )xdW

= n r + 12 σ 2(n −1)( )x ndt + nσx ndW

d x n

dt= n r + 1

2 σ 2(n −1)( ) x n dt

x n (t) = exp n r + 12 σ 2(n −1)( )t[ ] x(0) =1( )

x(t) = exp rt( )

x 2(t) = exp 2 r + 12 σ 2

( )t[ ]

CV 2 =x 2(t) − x(t)

2

x(t)2 = exp σ 2t( ) −1

in particular,

geometric Brownian motion, Ito calculus with Stratonovich convention

Recall extra drift

€

dx = u(x) + 12 σ 2 ′ G (x)G(x)( )dt + G(x)ξ (t)dt


Recall extra drift

€


= u(x) + 12 σ 2 ′ G (x)G(x)( )dt + σG(x)dW (t)

in our current notation


Recall extra drift

€



= rx + 12 σ 2x( )dt + σxdW (t)



Recall extra drift

€



= rx + 12 σ 2x( )dt + σxdW (t)


same as for Ito convention except r -> r + ½σ2


Recall extra drift

€



= rx + 12 σ 2x( )dt + σxdW (t)

d log x = rdt + σdW (t)




Recall extra drift

€



= rx + 12 σ 2x( )dt + σxdW (t)




€

⇒ P(x) =1

x 2πσ 2texp −

log x − rt( )2

2σ 2t

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥


Recall extra drift

€



= rx + 12 σ 2x( )dt + σxdW (t)




€

⇒ P(x) =1

x 2πσ 2texp −

log x − rt( )2

2σ 2t

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

€

d(x n ) = n r + 12 σ 2 + 1

2 σ 2(n −1)( )x ndt + nσx ndWmoments:


Recall extra drift

€



= rx + 12 σ 2x( )dt + σxdW (t)




€

⇒ P(x) =1

x 2πσ 2texp −

log x − rt( )2

2σ 2t

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

€

d(x n ) = n r + 12 σ 2 + 1

2 σ 2(n −1)( )x ndt + nσx ndW

x n (t) = exp n r + 12 σ 2n( )t[ ] x(0) =1( )

moments:

GBM with Stratonovich, finite-τ noiseThis was based on the “midpoint prescription”. I claimed that thisprescription was equivalent to giving the noise a finite correlation time τ and letting τ -> 0 in the end. Here I show this for this model.


€

dx

dt= rx + yx;

dy

dt= −γy + ξ (t), ξ (t)ξ ( ′ t ) = σ 2γ 2give y a small

correlation time:


€

dx

dt= rx + yx;

dy

dt= −γy + ξ (t), ξ (t)ξ ( ′ t ) = σ 2γ 2

y(t1)y(t2) = 12 σ 2γe−γ t1 −t2

give y a smallcorrelation time:


€

dx

dt= rx + yx;

dy

dt= −γy + ξ (t), ξ (t)ξ ( ′ t ) = σ 2γ 2

y(t1)y(t2) = 12 σ 2γe−γ t1 −t2

x(t) = exp rt + y(t1)dt10

t

∫[ ], (x(0) =1)


solve for x(t):


€

dx

dt= rx + yx;

dy

dt= −γy + ξ (t), ξ (t)ξ ( ′ t ) = σ 2γ 2

y(t1)y(t2) = 12 σ 2γe−γ t1 −t2


t

∫[ ], (x(0) =1)

eaz = exp − 12 a2 z2

( )⇒


solve for x(t):use identity forGaussian variables:


€

dx

dt= rx + yx;

dy

dt= −γy + ξ (t), ξ (t)ξ ( ′ t ) = σ 2γ 2

y(t1)y(t2) = 12 σ 2γe−γ t1 −t2


t

∫[ ], (x(0) =1)


( )⇒

x 2(t) = exp 2rt + 2 y(t1)y(t2)0

t

∫ dt1dt20

t

∫[ ]




€

dx

dt= rx + yx;

dy

dt= −γy + ξ (t), ξ (t)ξ ( ′ t ) = σ 2γ 2

y(t1)y(t2) = 12 σ 2γe−γ t1 −t2


t

∫[ ], (x(0) =1)


( )⇒

x 2(t) = exp 2rt + 2 y(t1)y(t2)0

t

∫ dt1dt20

t

∫[ ]

= exp 2rt + σ 2γ e−γ t1 −t2

0

t

∫ dt1dt20

t

∫[ ]




€

dx

dt= rx + yx;

dy

dt= −γy + ξ (t), ξ (t)ξ ( ′ t ) = σ 2γ 2

y(t1)y(t2) = 12 σ 2γe−γ t1 −t2


t

∫[ ], (x(0) =1)


( )⇒

x 2(t) = exp 2rt + 2 y(t1)y(t2)0

t

∫ dt1dt20

t

∫[ ]


0

t

∫ dt1dt20

t

∫[ ]

γ →0 ⏐ → ⏐ ⏐ exp 2rt + 2σ 2γt( )




€

dx

dt= rx + yx;

dy

dt= −γy + ξ (t), ξ (t)ξ ( ′ t ) = σ 2γ 2

y(t1)y(t2) = 12 σ 2γe−γ t1 −t2


t

∫[ ], (x(0) =1)


( )⇒

x 2(t) = exp 2rt + 2 y(t1)y(t2)0

t

∫ dt1dt20

t

∫[ ]


0

t

∫ dt1dt20

t

∫[ ]

γ →0 ⏐ → ⏐ ⏐ exp 2rt + 2σ 2γt( )



as we got before

GBM, Stratonovich, with Fokker-Planck

recall the FP equation with Stratonovich convention can be written

€

∂P

∂t= −

∂

∂xu(x)P(x, t)( ) + 1

2 σ 2 ∂

∂xG(x)

∂

∂xG(x)P(x, t)( )

⎛

⎝ ⎜

⎞

⎠ ⎟



€

∂P

∂t= −

∂

∂xu(x)P(x, t)( ) + 1

2 σ 2 ∂

∂xG(x)

∂

∂xG(x)P(x, t)( )

⎛

⎝ ⎜

⎞

⎠ ⎟

= −∂

∂xrxP(x, t)( ) + 1

2 σ 2 ∂

∂xx

∂

∂xxP(x, t)( )

⎛

⎝ ⎜

⎞

⎠ ⎟



€

∂P

∂t= −

∂

∂xu(x)P(x, t)( ) + 1

2 σ 2 ∂

∂xG(x)

∂

∂xG(x)P(x, t)( )

⎛

⎝ ⎜

⎞

⎠ ⎟

= −∂

∂xrxP(x, t)( ) + 1

2 σ 2 ∂

∂xx

∂

∂xxP(x, t)( )

⎛

⎝ ⎜

⎞

⎠ ⎟

y = log x ⇒∂P (y, t)

∂t= −r

∂P

∂y+ 1

2 σ 2 ∂ 2P

∂y 2⇒change variables:



€

∂P

∂t= −

∂

∂xu(x)P(x, t)( ) + 1

2 σ 2 ∂

∂xG(x)

∂

∂xG(x)P(x, t)( )

⎛

⎝ ⎜

⎞

⎠ ⎟

= −∂

∂xrxP(x, t)( ) + 1

2 σ 2 ∂

∂xx

∂

∂xxP(x, t)( )

⎛

⎝ ⎜

⎞

⎠ ⎟


∂t= −r

∂P

∂y+ 1

2 σ 2 ∂ 2P

∂y 2⇒

y(t) = rt, y(t) − y(t)( )2

= σ 2t

change variables:

y(t) is Gaussian with



€

∂P

∂t= −

∂

∂xu(x)P(x, t)( ) + 1

2 σ 2 ∂

∂xG(x)

∂

∂xG(x)P(x, t)( )

⎛

⎝ ⎜

⎞

⎠ ⎟

= −∂

∂xrxP(x, t)( ) + 1

2 σ 2 ∂

∂xx

∂

∂xxP(x, t)( )

⎛

⎝ ⎜

⎞

⎠ ⎟


∂t= −r

∂P

∂y+ 1

2 σ 2 ∂ 2P

∂y 2⇒

y(t) = rt, y(t) − y(t)( )2

= σ 2t

change variables:

y(t) is Gaussian with

as obtained from working with differentials and using Ito’s lemma

GBM, Ito convention, using Fokker-Planck(Finally), the FP equation for the Ito convention

€

∂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) − 1

2 σ 2G(x) ′ G (x)( )P(x, t)[ ]12 σ 2 ∂

∂xG(x)

∂

∂xG(x)P(x, t)( )

⎛

⎝ ⎜

⎞

⎠ ⎟

can be written


€

∂P

∂t= −

∂

∂xu(x)P(x, t)( ) + 1

2 σ 2 ∂ 2

∂x 2G2(x)P(x, t)( )

∂P

∂t= −

∂

∂xu(x) − 1

2 σ 2G(x) ′ G (x)( )P(x, t)[ ]12 σ 2 ∂

∂xG(x)

∂

∂xG(x)P(x, t)( )

⎛

⎝ ⎜

⎞

⎠ ⎟

= −∂

∂x(rx − 1

2 σ 2x)P(x, t)( ) + 12 σ 2 ∂

∂xx

∂

∂xxP(x, t)( )

⎛

⎝ ⎜

⎞

⎠ ⎟

can be written

Here:


€

∂P

∂t= −

∂

∂xu(x)P(x, t)( ) + 1

2 σ 2 ∂ 2

∂x 2G2(x)P(x, t)( )

∂P

∂t= −

∂

∂xu(x) − 1

2 σ 2G(x) ′ G (x)( )P(x, t)[ ]12 σ 2 ∂

∂xG(x)

∂

∂xG(x)P(x, t)( )

⎛

⎝ ⎜

⎞

⎠ ⎟

= −∂

∂x(rx − 1

2 σ 2x)P(x, t)( ) + 12 σ 2 ∂

∂xx

∂

∂xxP(x, t)( )

⎛

⎝ ⎜

⎞

⎠ ⎟

can be written

Here:

But this is just the same equation as in the Stratonovich case, excpt for a reduced drift r - ½σ2, in agreement with what wefound using differentials and the Ito lemma.

Summary:

Both ways of treating the problem with the Ito convention (differentials + Ito’s lemma, FP) agree with each other.

Summary:

Both ways of treating the problem with the Ito convention (differentials + Ito’s lemma, FP) agree with each other.All ways of treating the problem with the Stratonovichconvention (differentials + midpoint correction + Ito’s lemma,finite-τ noise, FP) agree with each other.

Summary:

Both ways of treating the problem with the Ito convention (differentials + Ito’s lemma, FP) agree with each other.All ways of treating the problem with the Stratonovichconvention (differentials + midpoint correction + Ito’s lemma,finite-τ noise, FP) agree with each other.Ito and Stratonovich problems are different (Stratonovich hasa drift rate larger by ½σ2).

Summary:

Both ways of treating the problem with the Ito convention (differentials + Ito’s lemma, FP) agree with each other.All ways of treating the problem with the Stratonovichconvention (differentials + midpoint correction + Ito’s lemma,finite-τ noise, FP) agree with each other.Ito and Stratonovich problems are different (Stratonovich hasa drift rate larger by ½σ2).I have shown this here for GBM, but it is true in general (except for constant G(x), in which case Stratonovich and Ito are equivalent).

Perturbation theory for nonlinear Langevin equations

(now back to additive noise)


(now back to additive noise)Consider equations with a steady state, nonlinear F:



€

dx

dt= F(x) + ξ (t); ξ (t)ξ ( ′ t ) = 2Tδ(t − ′ t )



Here I will concentrate on the example F(x) = -γx – gx3

overdamped motion in a quartic potential, double-wellpotential for γ < 0 :€

dx

dt= F(x) + ξ (t); ξ (t)ξ ( ′ t ) = 2Tδ(t − ′ t )




overdamped motion in a quartic potential, double-wellpotential for γ < 0 :€

dx

dt= F(x) + ξ (t); ξ (t)ξ ( ′ t ) = 2Tδ(t − ′ t )

€

dx

dt= −γx − gx 3 + ξ (t)




overdamped motion in a quartic potential, double-wellpotential for γ < 0 :

add an external driving force:

€

dx

dt= F(x) + ξ (t); ξ (t)ξ ( ′ t ) = 2Tδ(t − ′ t )

€

dx

dt= −γx − gx 3 + ξ (t)

€

dx

dt= −γx − gx 3 + ξ (t) + h(t)

some definitions and notation

Write this as

€

G0−1x = −gx 3 + ξ + h;


Write this as

€

G0−1x = −gx 3 + ξ + h;

G0 =d

dt+ γ

⎛

⎝ ⎜

⎞

⎠ ⎟−1where


Write this as

€

G0−1x = −gx 3 + ξ + h;

G0 =d

dt+ γ

⎛

⎝ ⎜

⎞

⎠ ⎟−1

G0(ω) =1

−iω + γ

where


Write this as

€

G0−1x = −gx 3 + ξ + h;

G0 =d

dt+ γ

⎛

⎝ ⎜

⎞

⎠ ⎟−1

G0(ω) =1

−iω + γG0(t, ′ t ) = e−γ (t− ′ t )Θ(t − ′ t )

where


Write this as

€

G0−1x = −gx 3 + ξ + h;

G0 =d

dt+ γ

⎛

⎝ ⎜

⎞

⎠ ⎟−1

G0(ω) =1

−iω + γG0(t, ′ t ) = e−γ (t− ′ t )Θ(t − ′ t )

x = G0 ⋅ −gx 3 + ξ + h[ ]

x(t) = d ′ t G0(t − ′ t ) −gx 3( ′ t ) + ξ ( ′ t ) + h( ′ t )[ ]∫

where

multiply by G0:

in time domain:


Write this as

€

G0−1x = −gx 3 + ξ + h;

G0 =d

dt+ γ

⎛

⎝ ⎜

⎞

⎠ ⎟−1

G0(ω) =1

−iω + γG0(t, ′ t ) = e−γ (t− ′ t )Θ(t − ′ t )

x = G0 ⋅ −gx 3 + ξ + h[ ]

x(t) = d ′ t G0(t − ′ t ) −gx 3( ′ t ) + ξ ( ′ t ) + h( ′ t )[ ]∫

x(ω) = G0(ω) −gd ′ ω

2π∫ d ′ ′ ω

2πx( ′ ω )x( ′ ′ ω )x(ω − ′ ω − ′ ′ ω ) + ξ (ω) + h(ω)

⎡ ⎣ ⎢

⎤ ⎦ ⎥

where

multiply by G0:

in time domain:

in frequency domain:


Write this as

€

G0−1x = −gx 3 + ξ + h;

G0 =d

dt+ γ

⎛

⎝ ⎜

⎞

⎠ ⎟−1

G0(ω) =1

−iω + γG0(t, ′ t ) = e−γ (t− ′ t )Θ(t − ′ t )

x = G0 ⋅ −gx 3 + ξ + h[ ]

x(t) = d ′ t G0(t − ′ t ) −gx 3( ′ t ) + ξ ( ′ t ) + h( ′ t )[ ]∫

x(ω) = G0(ω) −gd ′ ω

2π∫ d ′ ′ ω

2πx( ′ ω )x( ′ ′ ω )x(ω − ′ ω − ′ ′ ω ) + ξ (ω) + h(ω)

⎡ ⎣ ⎢

⎤ ⎦ ⎥

( )t

∫ ≡ dt( )∫ ; ( )ω

∫ ≡dω

2π∫ ( )

where

multiply by G0:

in time domain:

in frequency domain:

notation:

iteration of equation of motion

Define

€

x0 = G0(ξ + h)


Define

equation of motion:

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x0 = G0(ξ + h)

x = x0 − G0gx 3


Define

equation of motion:

€

x0 = G0(ξ + h)

x = x0 − G0gx 3 = x0 − gG0 x0 − G0gx 3( )

3


Define

equation of motion:

€

x0 = G0(ξ + h)

x = x0 − G0gx 3 = x0 − gG0 x0 − G0gx 3( )

3

= x0 − G0g x0 − gG0 x0 − gG0x 3( )[ ]

3L


Define

equation of motion:

diagrammatic representation: key:

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x0 = G0(ξ + h)

x = x0 − G0gx 3 = x0 − gG0 x0 − G0gx 3( )

3

= x0 − G0g x0 − gG0 x0 − gG0x 3( )[ ]

3L

= +

: x0

: x

: G0

: -g

iterate diagrams:

= + 3

h

iterate diagrams:

= + 3

h: ξ

iterate diagrams:

= + 3

+9h

: ξ

h+ …

averaging over noise:Recall: To average products of arbitrary numbers of factors of noise,pair in all ways (Wick’s theorem)


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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 (t1)ξ (t3) ξ (t2)ξ (t4 )

+ ξ (t1)ξ (t4 ) ξ (t2)ξ (t3) etc.

€

G ≡δx

δh: G(t, ′ t ) =

δx(t)

δh( ′ t ), G(ω) =

dx(ω)

dh(ω)

Define the Green’s function


€


+ ξ (t1)ξ (t3) ξ (t2)ξ (t4 )

+ ξ (t1)ξ (t4 ) ξ (t2)ξ (t3) etc.

€

G ≡δx

δh: G(t, ′ t ) =

δx(t)

δh( ′ t ), G(ω) =

dx(ω)

dh(ω)

=

Define the Green’s function

averaging the diagrams:

= + 3

oo:

€

ξξ =2T


= + 3

+9

o

oo

o:

€

ξξ =2T


= + 3

+9

+18

o

oo

o

o+ …

o:

€

ξξ =2T

correlation function

o

€

C(ω) = G(ω) ⋅2T ⋅G(−ω)

correlation function

o

€

C(ω) = G(ω) ⋅2T ⋅G(−ω)

o

€

C0(ω) = G0(ω) ⋅2T ⋅G0(−ω)

=2T

ω2 + γ 2

in algebra,

= + 3

+9

+18

o

oo

o

o

+ …

€

G(ω) = G0(ω) + 3G0(ω) −g C0( ′ ω )′ ω

∫[ ]G0(ω)

+9G0(ω) −g C0( ′ ω )′ ω

∫[ ]G0(ω) −g C0( ′ ω )′ ω

∫[ ]G0(ω)

+18G0(ω) g2

′ ω ∫ C0( ′ ω )C0( ′ ′ ω )G0(ω − ′ ω − ′ ′ ω )

′ ′ ω ∫[ ]G0(ω) +L

“self-energy” (“mass operator”)

= +

+ + …


= +

+ + …

= +


= +

+ + …

= +

Dyson equation

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G = G0 + G0ΣG


= +

+ + …

= +

Dyson equation or

€

G = G0 + G0ΣG G = G0−1 − Σ( )

−1


= +

+ + …

= +

Dyson equation or

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G = G0 + G0ΣG G = G0−1 − Σ( )

−1

= 3 o

+ 6 oo

+ … Σ =

1st-order approximation

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Σ(ω) = −3g C0( ′ ω ) + 6g2 C0( ′ ω )C0( ′ ′ ω )G (ω − ′ ω − ′ ′ ω )′ ′ ω

∫′ ω

∫′ ω

∫ +L


€


∫′ ω

∫′ ω

∫ +L

Σ(t − ′ t ) = −3gC0(t − ′ t = 0) + 6g2C02(t − ′ t )G0(t − ′ t ) +L

or, in time domain,


€


∫′ ω

∫′ ω

∫ +L


or, in time domain,

A simple, low-order approximation for Σ sums an infinitenumber of terms in the series for G.


€


∫′ ω

∫′ ω

∫ +L


or, in time domain,


lowest-order approximation for Σ:

€

Σ=−3gC0(t = 0) = −3gT


€


∫′ ω

∫′ ω

∫ +L


or, in time domain,



€

Σ=−3gC0(t = 0) = −3gT

G(ω) =1

−iω + γ + 3gT


€


∫′ ω

∫′ ω

∫ +L


or, in time domain,



increase in damping constant:

€

Σ=−3gC0(t = 0) = −3gT

G(ω) =1

−iω + γ + 3gT

γ eff = γ + 3gT

Hartree approximation

Replace C0 and G0 in the expression for Σ by C and G.This sums up all self-energy insertions of this form on theinternal lines in the self-energy diagrams.


€

Σ=−3gC(t = 0)


lowest-order approximation (Hartree):


€

Σ=−3gC(t = 0)


lowest-order approximation (Hartree):

Σ = + + +

+ …

oo

o ooo

oo

oo

self-consistent solution

€

Σ=−3gdω

2πC(ω) = −3g

dω

2πG(ω) ⋅2T ⋅G(−ω)∫∫


€

Σ=−3gdω

2πC(ω) = −3g

dω

2πG(ω) ⋅2T ⋅G(−ω)∫∫

= −3gdω

2π∫ 1

−iω + γ − Σ⋅2T ⋅

1

iω + γ − Σ


€

Σ=−3gdω

2πC(ω) = −3g

dω

2πG(ω) ⋅2T ⋅G(−ω)∫∫

= −3gdω

2π∫ 1

−iω + γ − Σ⋅2T ⋅

1

iω + γ − Σ

Σ = −3gT

γ − Σ

self-consistent equation


€

Σ=−3gdω

2πC(ω) = −3g

dω

2πG(ω) ⋅2T ⋅G(−ω)∫∫

= −3gdω

2π∫ 1

−iω + γ − Σ⋅2T ⋅

1

iω + γ − Σ

Σ = −3gT

γ − Σ

Σ =γ − γ 2 +12gT

2= −3gT + O(g2)


solution:


€

Σ=−3gdω

2πC(ω) = −3g

dω

2πG(ω) ⋅2T ⋅G(−ω)∫∫

= −3gdω

2π∫ 1

−iω + γ − Σ⋅2T ⋅

1

iω + γ − Σ

Σ = −3gT

γ − Σ

Σ =γ − γ 2 +12gT

2= −3gT + O(g2)


solution:

This solution is approximate. But it is exact if x is a vector with ncomponents, with

in the limit n -> ∞.

€

dx

dt= −γx −

g

nx

2x + ξ (t)

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Lecture 7: Langevin equations II: geometric Brownian motion and perturbation theory Outline: Geometric Brownian motion, Stratonovich and Ito models Ito