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![Page 1: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/1.jpg)
Lecture 13: Field-theoretic formulation of Langevin models
Outline:• Functional (path) integral formulation• Stratonovich and Ito, again• the Martin-Siggia-Rose formalism• free fields• perturbation theory• Stratonovich and supersymmetry
![Page 2: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/2.jpg)
generating functionals
In the equilibrium case, the partition function
![Page 3: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/3.jpg)
generating functionals
In the equilibrium case, the partition function
€
Z[β,h] = Dφexp∫ −βE[φ]( )
![Page 4: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/4.jpg)
generating functionals
In the equilibrium case, the partition function
€
Z[β,h] = Dφexp∫ −βE[φ]( )
€
= Dφexp − 12 dd x∫ r0φ
2(x) + 12 u0φ
4 (x) + (∇φ(x))2 − 2h(x)φ(x)[ ]{ }∫
![Page 5: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/5.jpg)
generating functionals
In the equilibrium case, the partition function
€
Z[β,h] = Dφexp∫ −βE[φ]( )
€
= Dφexp − 12 dd x∫ r0φ
2(x) + 12 u0φ
4 (x) + (∇φ(x))2 − 2h(x)φ(x)[ ]{ }∫is a generating functional:
![Page 6: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/6.jpg)
generating functionals
In the equilibrium case, the partition function
€
Z[β,h] = Dφexp∫ −βE[φ]( )
€
= Dφexp − 12 dd x∫ r0φ
2(x) + 12 u0φ
4 (x) + (∇φ(x))2 − 2h(x)φ(x)[ ]{ }∫is a generating functional:
€
∂ logZ
∂β= E
![Page 7: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/7.jpg)
generating functionals
In the equilibrium case, the partition function
€
Z[β,h] = Dφexp∫ −βE[φ]( )
€
= Dφexp − 12 dd x∫ r0φ
2(x) + 12 u0φ
4 (x) + (∇φ(x))2 − 2h(x)φ(x)[ ]{ }∫is a generating functional:
€
∂ logZ
∂β= E
€
∂2 logZ
∂β 2= E 2 − E
2
![Page 8: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/8.jpg)
generating functionals
In the equilibrium case, the partition function
€
Z[β,h] = Dφexp∫ −βE[φ]( )
€
= Dφexp − 12 dd x∫ r0φ
2(x) + 12 u0φ
4 (x) + (∇φ(x))2 − 2h(x)φ(x)[ ]{ }∫is a generating functional:
€
δ logZ
δh(x)= φ(x)
€
∂ logZ
∂β= E
€
∂2 logZ
∂β 2= E 2 − E
2
![Page 9: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/9.jpg)
generating functionals
In the equilibrium case, the partition function
€
Z[β,h] = Dφexp∫ −βE[φ]( )
€
= Dφexp − 12 dd x∫ r0φ
2(x) + 12 u0φ
4 (x) + (∇φ(x))2 − 2h(x)φ(x)[ ]{ }∫is a generating functional:
€
δ logZ
δh(x)= φ(x)
€
δ 2 logZ
δh(x)δh( ′ x )= φ(x)φ( ′ x ) − φ(x) φ( ′ x )
€
∂ logZ
∂β= E
€
∂2 logZ
∂β 2= E 2 − E
2
![Page 10: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/10.jpg)
generating functionals
In the equilibrium case, the partition function
€
Z[β,h] = Dφexp∫ −βE[φ]( )
€
= Dφexp − 12 dd x∫ r0φ
2(x) + 12 u0φ
4 (x) + (∇φ(x))2 − 2h(x)φ(x)[ ]{ }∫is a generating functional:
€
δ logZ
δh(x)= φ(x)
€
δ 2 logZ
δh(x)δh( ′ x )= φ(x)φ( ′ x ) − φ(x) φ( ′ x )
€
∂ logZ
∂β= E
€
∂2 logZ
∂β 2= E 2 − E
2
Here we will construct a generating functional for time-dependentcorrelation functions in the Langevin-Landau-Ginzburg model
![Page 11: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/11.jpg)
Dynamics (single variable)
Start from the equation of motion
€
η(t)η ( ′ t ) = 2Tδ(t − ′ t )
€
dφ
dt= −γφ + h(t) + η (t)
![Page 12: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/12.jpg)
Dynamics (single variable)
Start from the equation of motion
€
dφ
dt= f (φ) + h(t) + η (t)
€
η(t)η ( ′ t ) = 2Tδ(t − ′ t )
€
dφ
dt= −γφ + h(t) + η (t)
or, more generally,
![Page 13: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/13.jpg)
Dynamics (single variable)
Start from the equation of motion
€
dφ
dt= f (φ) + h(t) + η (t)
€
η(t)η ( ′ t ) = 2Tδ(t − ′ t )
€
dφ
dt= −γφ + h(t) + η (t)
or, more generally,
Discretize time:
€
t = Δ, 2Δ,L MΔ; η m = η (t)dtmΔ
(m +1)Δ
∫
![Page 14: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/14.jpg)
Dynamics (single variable)
Start from the equation of motion
€
dφ
dt= f (φ) + h(t) + η (t)
€
η(t)η ( ′ t ) = 2Tδ(t − ′ t )
€
dφ
dt= −γφ + h(t) + η (t)
or, more generally,
Discretize time:
€
t = Δ, 2Δ,L MΔ; η m = η (t)dtmΔ
(m +1)Δ
∫
€
P[η ] =1
(4πTΔ)M / 2exp −
1
4TΔη n
2
n= 0
M −1
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
η mη n = 2TΔδmn
Gaussian noise:
![Page 15: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/15.jpg)
equations of motion
Ito:
![Page 16: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/16.jpg)
equations of motion
€
φn +1 − φn = Δ f (φn ) + hn( ) + η nIto:
![Page 17: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/17.jpg)
equations of motion
€
φn +1 − φn = Δ f (φn ) + hn( ) + η nIto:
Stratonovich:
![Page 18: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/18.jpg)
equations of motion
€
φn +1 − φn = Δ f (φn ) + hn( ) + η nIto:
€
φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1
2 Δ f (φn +1) + hn +1( ) + η n
Stratonovich:
![Page 19: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/19.jpg)
equations of motion
€
φn +1 − φn = Δ f (φn ) + hn( ) + η nIto:
€
φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1
2 Δ f (φn +1) + hn +1( ) + η n
Stratonovich:
Change variables from η (0 < n < M-1) to ϕ (1 < n < M):
![Page 20: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/20.jpg)
equations of motion
€
φn +1 − φn = Δ f (φn ) + hn( ) + η nIto:
€
φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1
2 Δ f (φn +1) + hn +1( ) + η n
Stratonovich:
Change variables from η (0 < n < M-1) to ϕ (1 < n < M):
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
![Page 21: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/21.jpg)
equations of motion
€
φn +1 − φn = Δ f (φn ) + hn( ) + η nIto:
€
φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1
2 Δ f (φn +1) + hn +1( ) + η n
Stratonovich:
Change variables from η (0 < n < M-1) to ϕ (1 < n < M):
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥ (Ito)
![Page 22: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/22.jpg)
equations of motion
€
φn +1 − φn = Δ f (φn ) + hn( ) + η nIto:
€
φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1
2 Δ f (φn +1) + hn +1( ) + η n
Stratonovich:
Change variables from η (0 < n < M-1) to ϕ (1 < n < M):
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥ (Ito)
€
J[φ] = det∂η
∂φ
⎛
⎝ ⎜
⎞
⎠ ⎟= det δm,n +1 − 1+ Δ ′ f (φn )( )δm,n[ ] =1Jacobian:
![Page 23: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/23.jpg)
equations of motion
€
φn +1 − φn = Δ f (φn ) + hn( ) + η nIto:
€
φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1
2 Δ f (φn +1) + hn +1( ) + η n
Stratonovich:
Change variables from η (0 < n < M-1) to ϕ (1 < n < M):
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥ (Ito)
€
J[φ] = det∂η
∂φ
⎛
⎝ ⎜
⎞
⎠ ⎟= det δm,n +1 − 1+ Δ ′ f (φn )( )δm,n[ ] =1Jacobian:
______diagonal
![Page 24: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/24.jpg)
equations of motion
€
φn +1 − φn = Δ f (φn ) + hn( ) + η nIto:
€
φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1
2 Δ f (φn +1) + hn +1( ) + η n
Stratonovich:
Change variables from η (0 < n < M-1) to ϕ (1 < n < M):
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥ (Ito)
€
J[φ] = det∂η
∂φ
⎛
⎝ ⎜
⎞
⎠ ⎟= det δm,n +1 − 1+ Δ ′ f (φn )( )δm,n[ ] =1Jacobian:
______diagonal
______________1 belowdiagonal
![Page 25: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/25.jpg)
equations of motion
€
φn +1 − φn = Δ f (φn ) + hn( ) + η nIto:
€
φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1
2 Δ f (φn +1) + hn +1( ) + η n
Stratonovich:
Change variables from η (0 < n < M-1) to ϕ (1 < n < M):
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥ (Ito)
€
J[φ] = det∂η
∂φ
⎛
⎝ ⎜
⎞
⎠ ⎟= det δm,n +1 − 1+ Δ ′ f (φn )( )δm,n[ ] =1Jacobian: (Ito)
______diagonal
______________1 belowdiagonal
![Page 26: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/26.jpg)
equations of motion
€
φn +1 − φn = Δ f (φn ) + hn( ) + η nIto:
€
φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1
2 Δ f (φn +1) + hn +1( ) + η n
Stratonovich:
Change variables from η (0 < n < M-1) to ϕ (1 < n < M):
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥ (Ito)
€
J[φ] = det∂η
∂φ
⎛
⎝ ⎜
⎞
⎠ ⎟= det δm,n +1 − 1+ Δ ′ f (φn )( )δm,n[ ] =1Jacobian: (Ito)
______diagonal
______________1 belowdiagonal
(all elements above the diagonal vanish, so det = product of diagonal elements)
![Page 27: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/27.jpg)
Stratonovich
€
φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1
2 Δ f (φn +1) + hn +1( ) + η n ⇒
![Page 28: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/28.jpg)
Stratonovich
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − 1
2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
€
φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1
2 Δ f (φn +1) + hn +1( ) + η n ⇒
![Page 29: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/29.jpg)
Stratonovich
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − 1
2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
€
φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1
2 Δ f (φn +1) + hn +1( ) + η n ⇒
Jacobian:
![Page 30: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/30.jpg)
Stratonovich
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − 1
2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
€
φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1
2 Δ f (φn +1) + hn +1( ) + η n ⇒
€
J[φ] = det (1− 12 Δ ′ f n +1)δm,n +1 − 1+ 1
2 Δ ′ f n( )δm,n[ ]Jacobian:
![Page 31: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/31.jpg)
Stratonovich
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − 1
2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
€
φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1
2 Δ f (φn +1) + hn +1( ) + η n ⇒
€
J[φ] = det (1− 12 Δ ′ f n +1)δm,n +1 − 1+ 1
2 Δ ′ f n( )δm,n[ ]
= (1− 12 Δ ′ f n )
n=1
M
∏
Jacobian:
![Page 32: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/32.jpg)
Stratonovich
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − 1
2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
€
φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1
2 Δ f (φn +1) + hn +1( ) + η n ⇒
€
J[φ] = det (1− 12 Δ ′ f n +1)δm,n +1 − 1+ 1
2 Δ ′ f n( )δm,n[ ]
= (1− 12 Δ ′ f n ) = exp log(1− 1
2 Δ ′ f n )n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
n=1
M
∏
Jacobian:
![Page 33: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/33.jpg)
Stratonovich
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − 1
2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
€
φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1
2 Δ f (φn +1) + hn +1( ) + η n ⇒
€
J[φ] = det (1− 12 Δ ′ f n +1)δm,n +1 − 1+ 1
2 Δ ′ f n( )δm,n[ ]
= (1− 12 Δ ′ f n ) = exp log(1− 1
2 Δ ′ f n )n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
n=1
M
∏ = exp − 12 Δ ′ f n
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
Jacobian:
![Page 34: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/34.jpg)
Stratonovich
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − 1
2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
€
φn +1 − φn = 12 Δ f (φn ) + hn( ) + 1
2 Δ f (φn +1) + hn +1( ) + η n ⇒
€
J[φ] = det (1− 12 Δ ′ f n +1)δm,n +1 − 1+ 1
2 Δ ′ f n( )δm,n[ ]
= (1− 12 Δ ′ f n ) = exp log(1− 1
2 Δ ′ f n )n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
n=1
M
∏ = exp − 12 Δ ′ f n
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
Jacobian:
(back to this later)
![Page 35: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/35.jpg)
Martin-Siggia-Rose
back to Ito:
![Page 36: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/36.jpg)
Martin-Siggia-Rose
€
P[φ | h] =1
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥back to Ito:
![Page 37: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/37.jpg)
Martin-Siggia-Rose
€
P[φ | h] =1
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥back to Ito:
€
exp − 14 a2
( ) =dy
2πexp −y 2 + iya( )
−∞
∞
∫ ⇒use
![Page 38: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/38.jpg)
Martin-Siggia-Rose
€
P[φ | h] =1
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥back to Ito:
€
exp − 14 a2
( ) =dy
2πexp −y 2 + iya( )
−∞
∞
∫ ⇒use
€
P[φ | h] =1
(2π )Md ˆ φ n
n= 0
M −1
∏ exp∫ −TΔ ˆ φ n2 + i ˆ φ n −φn +1 + φn + Δ( fn + hn )[ ]( )
n= 0
M −1
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
![Page 39: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/39.jpg)
Martin-Siggia-Rose
€
P[φ | h] =1
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥back to Ito:
€
exp − 14 a2
( ) =dy
2πexp −y 2 + iya( )
−∞
∞
∫ ⇒use
€
P[φ | h] =1
(2π )Md ˆ φ n
n= 0
M −1
∏ exp∫ −TΔ ˆ φ n2 + i ˆ φ n −φn +1 + φn + Δ( fn + hn )[ ]( )
n= 0
M −1
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
generating function (multivariate characteristic function)
![Page 40: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/40.jpg)
Martin-Siggia-Rose
€
P[φ | h] =1
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥back to Ito:
€
exp − 14 a2
( ) =dy
2πexp −y 2 + iya( )
−∞
∞
∫ ⇒use
€
P[φ | h] =1
(2π )Md ˆ φ n
n= 0
M −1
∏ exp∫ −TΔ ˆ φ n2 + i ˆ φ n −φn +1 + φn + Δ( fn + hn )[ ]( )
n= 0
M −1
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
generating function (multivariate characteristic function)
€
Z[h,θ] = dφn
n=1
M
∏∫ e iθ nφn
n=1
M
∏ ⎛
⎝ ⎜
⎞
⎠ ⎟P[φ]
![Page 41: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/41.jpg)
Martin-Siggia-Rose
€
P[φ | h] =1
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − f (φn ) + hn[ ]Δ( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥back to Ito:
€
exp − 14 a2
( ) =dy
2πexp −y 2 + iya( )
−∞
∞
∫ ⇒use
€
P[φ | h] =1
(2π )Md ˆ φ n
n= 0
M −1
∏ exp∫ −TΔ ˆ φ n2 + i ˆ φ n −φn +1 + φn + Δ( fn + hn )[ ]( )
n= 0
M −1
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
generating function (multivariate characteristic function)
€
Z[h,θ] = dφn
n=1
M
∏∫ e iθ nφn
n=1
M
∏ ⎛
⎝ ⎜
⎞
⎠ ⎟P[φ]
=1
(2π )Mdφn d ˆ φ n
n= 0
M −1
∏n=1
M
∏ exp∫ i θnφn
n=1
M
∑ + −TΔ ˆ φ n2 + i ˆ φ n −φn +1 + φn + Δ( fn + hn )[ ]( )
n= 0
M −1
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
![Page 42: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/42.jpg)
a field theory:
Δ -> 0:
€
Z[h,θ] = DφD ˆ φ exp −S[φ, ˆ φ ,h,θ]( )∫−S[φ, ˆ φ ,h,θ] = d∫ t −T ˆ φ 2(t) + i ˆ φ (t) − ˙ φ (t) + f (φ) + h(t)( ) + iθ(t)φ(t)[ ]“action”
putting space back in, using
€
S[φ, ˆ φ ,h,θ] = d∫ t ddx L φ(x, t), ˙ φ (x, t), ˆ φ (x, t),h(x, t),θ(x, t)( )
L(φ, ˙ φ , ˆ φ ,h,θ) = T ˆ φ 2 + i ˆ φ ∂φ
∂t+ ir0
ˆ φ φ − i ˆ φ ∇ 2φ + iu0ˆ φ φ3 − i ˆ φ h − iθφ€
f (x, t) = − r0 −∇ 2( )φ(x, t) − u0φ
3(x, t)
______________________ quadratic: L0
![Page 43: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/43.jpg)
a field theory:
Δ -> 0:
€
Z[h,θ] = DφD ˆ φ exp −S[φ, ˆ φ ,h,θ]( )∫−S[φ, ˆ φ ,h,θ] = d∫ t −T ˆ φ 2(t) + i ˆ φ (t) − ˙ φ (t) + f (φ) + h(t)( ) + iθ(t)φ(t)[ ]“action”
![Page 44: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/44.jpg)
a field theory:
Δ -> 0:
€
Z[h,θ] = DφD ˆ φ exp −S[φ, ˆ φ ,h,θ]( )∫−S[φ, ˆ φ ,h,θ] = d∫ t −T ˆ φ 2(t) + i ˆ φ (t) − ˙ φ (t) + f (φ) + h(t)( ) + iθ(t)φ(t)[ ]“action”
putting space back in, using
€
f (x, t) = − r0 −∇ 2( )φ(x, t) − u0φ
3(x, t)
![Page 45: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/45.jpg)
a field theory:
Δ -> 0:
€
Z[h,θ] = DφD ˆ φ exp −S[φ, ˆ φ ,h,θ]( )∫−S[φ, ˆ φ ,h,θ] = d∫ t −T ˆ φ 2(t) + i ˆ φ (t) − ˙ φ (t) + f (φ) + h(t)( ) + iθ(t)φ(t)[ ]“action”
putting space back in, using
€
S[φ, ˆ φ ,h,θ] = d∫ t ddx L φ(x, t), ˙ φ (x, t), ˆ φ (x, t),h(x, t),θ(x, t)( )
L(φ, ˙ φ , ˆ φ ,h,θ) = T ˆ φ 2 + i ˆ φ ∂φ
∂t+ ir0
ˆ φ φ − i ˆ φ ∇ 2φ + iu0ˆ φ φ3 − i ˆ φ h − iθφ€
f (x, t) = − r0 −∇ 2( )φ(x, t) − u0φ
3(x, t)
![Page 46: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/46.jpg)
a field theory:
Δ -> 0:
€
Z[h,θ] = DφD ˆ φ exp −S[φ, ˆ φ ,h,θ]( )∫−S[φ, ˆ φ ,h,θ] = d∫ t −T ˆ φ 2(t) + i ˆ φ (t) − ˙ φ (t) + f (φ) + h(t)( ) + iθ(t)φ(t)[ ]“action”
putting space back in, using
€
S[φ, ˆ φ ,h,θ] = d∫ t ddx L φ(x, t), ˙ φ (x, t), ˆ φ (x, t),h(x, t),θ(x, t)( )
L(φ, ˙ φ , ˆ φ ,h,θ) = T ˆ φ 2 + i ˆ φ ∂φ
∂t+ ir0
ˆ φ φ − i ˆ φ ∇ 2φ + iu0ˆ φ φ3 − i ˆ φ h − iθφ€
f (x, t) = − r0 −∇ 2( )φ(x, t) − u0φ
3(x, t)
______________________ quadratic: L0
![Page 47: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/47.jpg)
a field theory:
Δ -> 0:
€
Z[h,θ] = DφD ˆ φ exp −S[φ, ˆ φ ,h,θ]( )∫−S[φ, ˆ φ ,h,θ] = d∫ t −T ˆ φ 2(t) + i ˆ φ (t) − ˙ φ (t) + f (φ) + h(t)( ) + iθ(t)φ(t)[ ]“action”
putting space back in, using
€
S[φ, ˆ φ ,h,θ] = d∫ t ddx L φ(x, t), ˙ φ (x, t), ˆ φ (x, t),h(x, t),θ(x, t)( )
L(φ, ˙ φ , ˆ φ ,h,θ) = T ˆ φ 2 + i ˆ φ ∂φ
∂t+ ir0
ˆ φ φ − i ˆ φ ∇ 2φ + iu0ˆ φ φ3 − i ˆ φ h − iθφ€
f (x, t) = − r0 −∇ 2( )φ(x, t) − u0φ
3(x, t)
______________________ quadratic: L0
_____inter-actionterm L1
![Page 48: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/48.jpg)
a field theory:
Δ -> 0:
€
Z[h,θ] = DφD ˆ φ exp −S[φ, ˆ φ ,h,θ]( )∫−S[φ, ˆ φ ,h,θ] = d∫ t −T ˆ φ 2(t) + i ˆ φ (t) − ˙ φ (t) + f (φ) + h(t)( ) + iθ(t)φ(t)[ ]“action”
putting space back in, using
€
S[φ, ˆ φ ,h,θ] = d∫ t ddx L φ(x, t), ˙ φ (x, t), ˆ φ (x, t),h(x, t),θ(x, t)( )
L(φ, ˙ φ , ˆ φ ,h,θ) = T ˆ φ 2 + i ˆ φ ∂φ
∂t+ ir0
ˆ φ φ − i ˆ φ ∇ 2φ + iu0ˆ φ φ3 − i ˆ φ h − iθφ€
f (x, t) = − r0 −∇ 2( )φ(x, t) − u0φ
3(x, t)
______________________ quadratic: L0
_____inter-actionterm L1
________“source” terms
![Page 49: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/49.jpg)
a field theory:
Δ -> 0:
€
Z[h,θ] = DφD ˆ φ exp −S[φ, ˆ φ ,h,θ]( )∫−S[φ, ˆ φ ,h,θ] = d∫ t −T ˆ φ 2(t) + i ˆ φ (t) − ˙ φ (t) + f (φ) + h(t)( ) + iθ(t)φ(t)[ ]“action”
putting space back in, using
€
S[φ, ˆ φ ,h,θ] = d∫ t ddx L φ(x, t), ˙ φ (x, t), ˆ φ (x, t),h(x, t),θ(x, t)( )
L(φ, ˙ φ , ˆ φ ,h,θ) = T ˆ φ 2 + i ˆ φ ∂φ
∂t+ ir0
ˆ φ φ − i ˆ φ ∇ 2φ + iu0ˆ φ φ3 − i ˆ φ h − iθφ€
f (x, t) = − r0 −∇ 2( )φ(x, t) − u0φ
3(x, t)
______________________ quadratic: L0
_____inter-actionterm L1
________“source” terms
note:
€
Z[h,0] = DφD ˆ φ exp −S[φ, ˆ φ ,h,θ]( )∫ =1
![Page 50: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/50.jpg)
a field theory:
Δ -> 0:
€
Z[h,θ] = DφD ˆ φ exp −S[φ, ˆ φ ,h,θ]( )∫−S[φ, ˆ φ ,h,θ] = d∫ t −T ˆ φ 2(t) + i ˆ φ (t) − ˙ φ (t) + f (φ) + h(t)( ) + iθ(t)φ(t)[ ]“action”
putting space back in, using
€
S[φ, ˆ φ ,h,θ] = d∫ t ddx L φ(x, t), ˙ φ (x, t), ˆ φ (x, t),h(x, t),θ(x, t)( )
L(φ, ˙ φ , ˆ φ ,h,θ) = T ˆ φ 2 + i ˆ φ ∂φ
∂t+ ir0
ˆ φ φ − i ˆ φ ∇ 2φ + iu0ˆ φ φ3 − i ˆ φ h − iθφ€
f (x, t) = − r0 −∇ 2( )φ(x, t) − u0φ
3(x, t)
______________________ quadratic: L0
_____inter-actionterm L1
________“source” terms
note:
€
Z[h,0] = DφD ˆ φ exp −S[φ, ˆ φ ,h,θ]( )∫ =1 (normalization of P(ϕ|h))
![Page 51: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/51.jpg)
correlation functions
![Page 52: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/52.jpg)
correlation functions
magnetization
![Page 53: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/53.jpg)
correlation functions
€
M(x, t) = φ(x, t) = −i limθ →0
δZ
δθ(x, t)magnetization
![Page 54: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/54.jpg)
correlation functions
€
M(x, t) = φ(x, t) = −i limθ →0
δZ
δθ(x, t)magnetization
correlation functions:
![Page 55: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/55.jpg)
correlation functions
€
C(x, t; ′ x , ′ t ) = φ(x, t)φ( ′ x , ′ t ) − φ(x, t) φ( ′ x , ′ t ) = −limθ →0
δ 2Z
δθ(x, t)δθ( ′ x , ′ t )
€
M(x, t) = φ(x, t) = −i limθ →0
δZ
δθ(x, t)magnetization
correlation functions:
![Page 56: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/56.jpg)
correlation functions
€
C(x, t; ′ x , ′ t ) = φ(x, t)φ( ′ x , ′ t ) − φ(x, t) φ( ′ x , ′ t ) = −limθ →0
δ 2Z
δθ(x, t)δθ( ′ x , ′ t )
€
M(x, t) = φ(x, t) = −i limθ →0
δZ
δθ(x, t)
€
G(x, t; ′ x , ′ t ) = i ˆ φ (x, t)φ( ′ x , ′ t ) = −i limθ →0
δ 2Z
δθ(x, t)δh( ′ x , ′ t )
magnetization
correlation functions:
![Page 57: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/57.jpg)
correlation functions
€
C(x, t; ′ x , ′ t ) = φ(x, t)φ( ′ x , ′ t ) − φ(x, t) φ( ′ x , ′ t ) = −limθ →0
δ 2Z
δθ(x, t)δθ( ′ x , ′ t )
€
M(x, t) = φ(x, t) = −i limθ →0
δZ
δθ(x, t)
€
G(x, t; ′ x , ′ t ) = i ˆ φ (x, t)φ( ′ x , ′ t ) = −i limθ →0
δ 2Z
δθ(x, t)δh( ′ x , ′ t )
magnetization
correlation functions:
€
=δM(x, t)
δh( ′ x , ′ t )
![Page 58: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/58.jpg)
correlation functions
€
C(x, t; ′ x , ′ t ) = φ(x, t)φ( ′ x , ′ t ) − φ(x, t) φ( ′ x , ′ t ) = −limθ →0
δ 2Z
δθ(x, t)δθ( ′ x , ′ t )
€
M(x, t) = φ(x, t) = −i limθ →0
δZ
δθ(x, t)
€
G(x, t; ′ x , ′ t ) = i ˆ φ (x, t)φ( ′ x , ′ t ) = −i limθ →0
δ 2Z
δθ(x, t)δh( ′ x , ′ t )
magnetization
correlation functions:
€
=δM(x, t)
δh( ′ x , ′ t )= susceptibility / response function
![Page 59: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/59.jpg)
correlation functions
€
C(x, t; ′ x , ′ t ) = φ(x, t)φ( ′ x , ′ t ) − φ(x, t) φ( ′ x , ′ t ) = −limθ →0
δ 2Z
δθ(x, t)δθ( ′ x , ′ t )
€
M(x, t) = φ(x, t) = −i limθ →0
δZ
δθ(x, t)
€
G(x, t; ′ x , ′ t ) = i ˆ φ (x, t)φ( ′ x , ′ t ) = −i limθ →0
δ 2Z
δθ(x, t)δh( ′ x , ′ t )
magnetization
correlation functions:
€
=δM(x, t)
δh( ′ x , ′ t )= susceptibility / response function
€
ˆ φ (x, t) = −limθ →0
δZ
δh(x, t)= 0
![Page 60: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/60.jpg)
correlation functions
€
C(x, t; ′ x , ′ t ) = φ(x, t)φ( ′ x , ′ t ) − φ(x, t) φ( ′ x , ′ t ) = −limθ →0
δ 2Z
δθ(x, t)δθ( ′ x , ′ t )
€
M(x, t) = φ(x, t) = −i limθ →0
δZ
δθ(x, t)
€
G(x, t; ′ x , ′ t ) = i ˆ φ (x, t)φ( ′ x , ′ t ) = −i limθ →0
δ 2Z
δθ(x, t)δh( ′ x , ′ t )
magnetization
correlation functions:
€
=δM(x, t)
δh( ′ x , ′ t )= susceptibility / response function
€
ˆ φ (x, t) = −limθ →0
δZ
δh(x, t)= 0
€
ˆ φ (x, t) ˆ φ ( ′ x , ′ t ) = −limθ →0
δ 2Z
δh(x, t)δh( ′ x , ′ t )= 0
![Page 61: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/61.jpg)
free action
€
L0 = T ˆ φ 2 + i ˆ φ ∂φ
∂t+ ir0
ˆ φ φ − i ˆ φ ∇ 2φ
![Page 62: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/62.jpg)
free action
€
L0 = T ˆ φ 2 + i ˆ φ ∂φ
∂t+ ir0
ˆ φ φ − i ˆ φ ∇ 2φ
free action:
![Page 63: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/63.jpg)
free action
€
L0 = T ˆ φ 2 + i ˆ φ ∂φ
∂t+ ir0
ˆ φ φ − i ˆ φ ∇ 2φ
€
S0 = L0dt =∫ 12 dt∫ φ ˆ φ ( )
0 ∂t + r0 −∇ 2
−∂t + r0 −∇ 2 2T
⎛
⎝ ⎜
⎞
⎠ ⎟φˆ φ
⎛
⎝ ⎜
⎞
⎠ ⎟free action:
![Page 64: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/64.jpg)
free action
€
L0 = T ˆ φ 2 + i ˆ φ ∂φ
∂t+ ir0
ˆ φ φ − i ˆ φ ∇ 2φ
€
S0 = L0dt =∫ 12 dt∫ φ ˆ φ ( )
0 ∂t + r0 −∇ 2
−∂t + r0 −∇ 2 2T
⎛
⎝ ⎜
⎞
⎠ ⎟φˆ φ
⎛
⎝ ⎜
⎞
⎠ ⎟free action:
€
Z0(h,θ) = DφD ˆ φ exp − 12 dt∫ φ ˆ φ ( )
0 ∂t + r0 −∇ 2
−∂t + r0 −∇ 2 2T
⎛
⎝ ⎜
⎞
⎠ ⎟φˆ φ
⎛
⎝ ⎜
⎞
⎠ ⎟+ ih ˆ φ + iθφ
⎡
⎣ ⎢
⎤
⎦ ⎥∫
generating functional:
![Page 65: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/65.jpg)
free action
€
L0 = T ˆ φ 2 + i ˆ φ ∂φ
∂t+ ir0
ˆ φ φ − i ˆ φ ∇ 2φ
€
S0 = L0dt =∫ 12 dt∫ φ ˆ φ ( )
0 ∂t + r0 −∇ 2
−∂t + r0 −∇ 2 2T
⎛
⎝ ⎜
⎞
⎠ ⎟φˆ φ
⎛
⎝ ⎜
⎞
⎠ ⎟free action:
€
Z0(h,θ) = DφD ˆ φ exp − 12 dt∫ φ ˆ φ ( )
0 ∂t + r0 −∇ 2
−∂t + r0 −∇ 2 2T
⎛
⎝ ⎜
⎞
⎠ ⎟φˆ φ
⎛
⎝ ⎜
⎞
⎠ ⎟+ ih ˆ φ + iθφ
⎡
⎣ ⎢
⎤
⎦ ⎥∫
generating functional:
in Fourier components:
€
Z0(h,θ) = DφD ˆ φ exp − 12 φ− p,−ω
ˆ φ − p,−ω( )0 −iω + r0 + p2
iω + r0 + p2 2T
⎛
⎝ ⎜
⎞
⎠ ⎟φpω
ˆ φ pω
⎛
⎝ ⎜
⎞
⎠ ⎟
p,ω
∫ ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
∫
⋅exp i h− p,−ωˆ φ pω + i θ− p,−ωφpω
p,ω
∫p,ω
∫ ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
![Page 66: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/66.jpg)
free fields(invert the matrix in the exponent in S0)
![Page 67: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/67.jpg)
free fields
€
C0( p,ω) =φ− p,−ω
ˆ φ − p,−ω
⎛
⎝ ⎜
⎞
⎠ ⎟ φp,ω
ˆ φ p,ω( ) =
2T
ω2 + (r0 + p2)2
1
−iω + r0 + p2
1
+iω + r0 + p20
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
(invert the matrix in the exponent in S0)
![Page 68: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/68.jpg)
free fields
€
C0( p,ω) =φ− p,−ω
ˆ φ − p,−ω
⎛
⎝ ⎜
⎞
⎠ ⎟ φp,ω
ˆ φ p,ω( ) =
2T
ω2 + (r0 + p2)2
1
−iω + r0 + p2
1
+iω + r0 + p20
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
(invert the matrix in the exponent in S0)
back to time domain:
![Page 69: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/69.jpg)
free fields
€
C0( p,ω) =φ− p,−ω
ˆ φ − p,−ω
⎛
⎝ ⎜
⎞
⎠ ⎟ φp,ω
ˆ φ p,ω( ) =
2T
ω2 + (r0 + p2)2
1
−iω + r0 + p2
1
+iω + r0 + p20
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
(invert the matrix in the exponent in S0)
back to time domain:
€
C0( p, t − ′ t ) =φ− p (t)ˆ φ − p (t)
⎛
⎝ ⎜
⎞
⎠ ⎟ φp ( ′ t ) ˆ φ p ( ′ t )( )
=T exp −(r0 + p2) t − ′ t [ ] Θ(t − ′ t )exp −(r0 + p2)(t − ′ t )[ ]
Θ( ′ t − t)exp −(r0 + p2)( ′ t − t)[ ] 0
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
![Page 70: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/70.jpg)
free fields
€
C0( p,ω) =φ− p,−ω
ˆ φ − p,−ω
⎛
⎝ ⎜
⎞
⎠ ⎟ φp,ω
ˆ φ p,ω( ) =
2T
ω2 + (r0 + p2)2
1
−iω + r0 + p2
1
+iω + r0 + p20
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
(invert the matrix in the exponent in S0)
back to time domain:
€
C0( p, t − ′ t ) =φ− p (t)ˆ φ − p (t)
⎛
⎝ ⎜
⎞
⎠ ⎟ φp ( ′ t ) ˆ φ p ( ′ t )( )
=T exp −(r0 + p2) t − ′ t [ ] Θ(t − ′ t )exp −(r0 + p2)(t − ′ t )[ ]
Θ( ′ t − t)exp −(r0 + p2)( ′ t − t)[ ] 0
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
in agreement with what we found using the direct approach in Lect. 11
![Page 71: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/71.jpg)
perturbation theory
![Page 72: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/72.jpg)
perturbation theory
want to evaluate quantities like
€
φ(1) ˆ φ (2) = DφD ˆ φ ∫ φ(1) ˆ φ (2)exp −S( )
= DφD ˆ φ ∫ φ(1) ˆ φ (2)exp −S1( )exp −S0( )
= φ(1) ˆ φ (2)exp −S1( )0
(1) = (x1, t1), etc.
![Page 73: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/73.jpg)
perturbation theory
expand
€
exp −S1( ) = exp − dt dd x L1(x, t)∫( )
= exp −iu0 d∫ 1 ˆ φ (1)φ3(1)( )
want to evaluate quantities like
€
φ(1) ˆ φ (2) = DφD ˆ φ ∫ φ(1) ˆ φ (2)exp −S( )
= DφD ˆ φ ∫ φ(1) ˆ φ (2)exp −S1( )exp −S0( )
= φ(1) ˆ φ (2)exp −S1( )0
(1) = (x1, t1), etc.
![Page 74: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/74.jpg)
perturbation theory
expand
€
exp −S1( ) = exp − dt dd x L1(x, t)∫( )
= exp −iu0 d∫ 1 ˆ φ (1)φ3(1)( )
€
=1− iu0 d∫ 1 ˆ φ (1)φ3(1) − 12 u0
2 d∫ 1d2 ˆ φ (1)φ3(1) ˆ φ (2)φ3(2) +L
want to evaluate quantities like
€
φ(1) ˆ φ (2) = DφD ˆ φ ∫ φ(1) ˆ φ (2)exp −S( )
= DφD ˆ φ ∫ φ(1) ˆ φ (2)exp −S1( )exp −S0( )
= φ(1) ˆ φ (2)exp −S1( )0
(1) = (x1, t1), etc.
![Page 75: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/75.jpg)
1st order:
![Page 76: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/76.jpg)
1st order:
€
φ(1) ˆ φ (2) = φ(1) ˆ φ (2) 1− S1 +L( )0
= C0(1,2) + iu0 d∫ 3 φ(1) ˆ φ (2) ˆ φ (3)φ3(3)0
+L
![Page 77: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/77.jpg)
1st order:
€
φ(1) ˆ φ (2) = φ(1) ˆ φ (2) 1− S1 +L( )0
= C0(1,2) + iu0 d∫ 3 φ(1) ˆ φ (2) ˆ φ (3)φ3(3)0
+L
average of product of 6 Gaussian variables: Use Wick’s theoremsum of products of all pairwise averages)
![Page 78: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/78.jpg)
1st order:
€
φ(1) ˆ φ (2) = φ(1) ˆ φ (2) 1− S1 +L( )0
= C0(1,2) + iu0 d∫ 3 φ(1) ˆ φ (2) ˆ φ (3)φ3(3)0
+L
average of product of 6 Gaussian variables: Use Wick’s theoremsum of products of all pairwise averages)
€
3 φ(1) ˆ φ (2)0
d3∫ ˆ φ (3)φ(3)0
φ2(3)0
+3 d3∫ φ(1) ˆ φ (3)0
φ2(3)0
φ(3) ˆ φ (2)0
+6 d3 φ(1)φ(3)0
φ(3) ˆ φ (3)∫0
φ(3) ˆ φ (2)0
(graphs onblackboard)
![Page 79: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/79.jpg)
1st order:
€
φ(1) ˆ φ (2) = φ(1) ˆ φ (2) 1− S1 +L( )0
= C0(1,2) + iu0 d∫ 3 φ(1) ˆ φ (2) ˆ φ (3)φ3(3)0
+L
average of product of 6 Gaussian variables: Use Wick’s theoremsum of products of all pairwise averages)
€
3 φ(1) ˆ φ (2)0
d3∫ ˆ φ (3)φ(3)0
φ2(3)0
+3 d3∫ φ(1) ˆ φ (3)0
φ2(3)0
φ(3) ˆ φ (2)0
+6 d3 φ(1)φ(3)0
φ(3) ˆ φ (3)∫0
φ(3) ˆ φ (2)0
(graphs onblackboard)
but most of these vanish:
![Page 80: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/80.jpg)
1st order:
€
φ(1) ˆ φ (2) = φ(1) ˆ φ (2) 1− S1 +L( )0
= C0(1,2) + iu0 d∫ 3 φ(1) ˆ φ (2) ˆ φ (3)φ3(3)0
+L
average of product of 6 Gaussian variables: Use Wick’s theoremsum of products of all pairwise averages)
€
3 φ(1) ˆ φ (2)0
d3∫ ˆ φ (3)φ(3)0
φ2(3)0
+3 d3∫ φ(1) ˆ φ (3)0
φ2(3)0
φ(3) ˆ φ (2)0
+6 d3 φ(1)φ(3)0
φ(3) ˆ φ (3)∫0
φ(3) ˆ φ (2)0
(graphs onblackboard)
but most of these vanish:
€
φ(3) ˆ φ (3) = G( p, t = 0)p
∫ = 0
![Page 81: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/81.jpg)
1st order:
€
φ(1) ˆ φ (2) = φ(1) ˆ φ (2) 1− S1 +L( )0
= C0(1,2) + iu0 d∫ 3 φ(1) ˆ φ (2) ˆ φ (3)φ3(3)0
+L
average of product of 6 Gaussian variables: Use Wick’s theoremsum of products of all pairwise averages)
€
3 φ(1) ˆ φ (2)0
d3∫ ˆ φ (3)φ(3)0
φ2(3)0
+3 d3∫ φ(1) ˆ φ (3)0
φ2(3)0
φ(3) ˆ φ (2)0
+6 d3 φ(1)φ(3)0
φ(3) ˆ φ (3)∫0
φ(3) ˆ φ (2)0
(graphs onblackboard)
but most of these vanish:
€
φ(3) ˆ φ (3) = G( p, t = 0)p
∫ = 0 (Ito)
![Page 82: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/82.jpg)
Feynman graphs
The surviving term:
€
3 d3∫ φ(1) ˆ φ (3)0
φ2(3)0
φ(3) ˆ φ (2)0
![Page 83: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/83.jpg)
Feynman graphs
1
2
3
G(1,3)
G(3,2)
C(3,3)The surviving term:
€
3 d3∫ φ(1) ˆ φ (3)0
φ2(3)0
φ(3) ˆ φ (2)0
![Page 84: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/84.jpg)
Feynman graphs
1
2
3
G(1,3)
G(3,2)
C(3,3)The surviving term:
…
€
3 d3∫ φ(1) ˆ φ (3)0
φ2(3)0
φ(3) ˆ φ (2)0
![Page 85: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/85.jpg)
Feynman graphs
Can generate a diagrammatic expansion like that in Lect 7
1
2
3
G(1,3)
G(3,2)
C(3,3)The surviving term:
…
€
3 d3∫ φ(1) ˆ φ (3)0
φ2(3)0
φ(3) ˆ φ (2)0
![Page 86: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/86.jpg)
Feynman graphs
Can generate a diagrammatic expansion like that in Lect 7In fact, it is exactly the same diagrammatic expansion
1
2
3
G(1,3)
G(3,2)
C(3,3)The surviving term:
…
€
3 d3∫ φ(1) ˆ φ (3)0
φ2(3)0
φ(3) ˆ φ (2)0
![Page 87: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/87.jpg)
Feynman graphs
Can generate a diagrammatic expansion like that in Lect 7In fact, it is exactly the same diagrammatic expansion(except that ϕ and the correlation and response functionsnow depend on space as well as time)
1
2
3
G(1,3)
G(3,2)
C(3,3)The surviving term:
…
€
3 d3∫ φ(1) ˆ φ (3)0
φ2(3)0
φ(3) ˆ φ (2)0
![Page 88: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/88.jpg)
Feynman graphs
Can generate a diagrammatic expansion like that in Lect 7In fact, it is exactly the same diagrammatic expansion(except that ϕ and the correlation and response functionsnow depend on space as well as time)
1
2
3
G(1,3)
G(3,2)
C(3,3)The surviving term:
…
€
3 d3∫ φ(1) ˆ φ (3)0
φ2(3)0
φ(3) ˆ φ (2)0
all closed loops of response functions (including alldisconnected diagrams) vanish because for Ito G(t=0) = 0.
![Page 89: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/89.jpg)
Stratonovich, again
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − 1
2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
![Page 90: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/90.jpg)
Stratonovich, again
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − 1
2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
Can take the continuous-time limit of the exponent as in Ito, but now J ≠ 1.
![Page 91: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/91.jpg)
Stratonovich, again
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − 1
2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
Can take the continuous-time limit of the exponent as in Ito, but now J ≠ 1.
Use Grassman variables
![Page 92: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/92.jpg)
Stratonovich, again
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − 1
2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
Can take the continuous-time limit of the exponent as in Ito, but now J ≠ 1.
Use Grassman variables
€
ψ,ψ :
ψψ = −ψ ψ , ψ 2 =ψ 2 = 0,
ψ1ψ 2 = −ψ 2ψ1, etc.
![Page 93: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/93.jpg)
Stratonovich, again
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − 1
2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
Can take the continuous-time limit of the exponent as in Ito, but now J ≠ 1.
Use Grassman variables
€
ψ,ψ :
ψψ = −ψ ψ , ψ 2 =ψ 2 = 0,
ψ1ψ 2 = −ψ 2ψ1, etc.
Taylor series expansionsterminate:
![Page 94: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/94.jpg)
Stratonovich, again
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − 1
2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
Can take the continuous-time limit of the exponent as in Ito, but now J ≠ 1.
Use Grassman variables
€
ψ,ψ :
ψψ = −ψ ψ , ψ 2 =ψ 2 = 0,
ψ1ψ 2 = −ψ 2ψ1, etc.
€
exp(aψ ψ ) =1+ aψ ψ
Taylor series expansionsterminate:
![Page 95: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/95.jpg)
Stratonovich, again
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − 1
2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
Can take the continuous-time limit of the exponent as in Ito, but now J ≠ 1.
Use Grassman variables
€
ψ,ψ :
ψψ = −ψ ψ , ψ 2 =ψ 2 = 0,
ψ1ψ 2 = −ψ 2ψ1, etc.
“integrals”:
€
exp(aψ ψ ) =1+ aψ ψ
Taylor series expansionsterminate:
![Page 96: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/96.jpg)
Stratonovich, again
€
P[φ] =J[φ]
(4πTΔ)M / 2exp −
1
4TΔφn +1 − φn − 1
2 Δ fn + hn( ) − 12 Δ fn +1 + hn +1( )( )
2
n=1
M
∑ ⎡
⎣ ⎢
⎤
⎦ ⎥
Can take the continuous-time limit of the exponent as in Ito, but now J ≠ 1.
Use Grassman variables
€
ψ,ψ :
ψψ = −ψ ψ , ψ 2 =ψ 2 = 0,
ψ1ψ 2 = −ψ 2ψ1, etc.
€
dψdψ
1
ψ
ψ
ψ ψ
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
∫ =
0
0
0
1
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
“integrals”:
€
exp(aψ ψ ) =1+ aψ ψ
Taylor series expansionsterminate:
![Page 97: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/97.jpg)
determinants
€
dψdψ ∫ exp ψ iij
∑ Aijψ j
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟= detA
![Page 98: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/98.jpg)
determinants
€
dψdψ ∫ exp ψ iij
∑ Aijψ j
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟= detA
dx
2π∫ exp − 1
2 x i
ij
∑ Aij x j
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟= detA( )
−1/ 2cf for real x
![Page 99: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/99.jpg)
determinants
€
dψdψ ∫ exp ψ iij
∑ Aijψ j
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟= detA
dx
2π∫ exp − 1
2 x i
ij
∑ Aij x j
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟= detA( )
−1/ 2
dzdz*
2π∫ exp − 1
2 zi*
ij
∑ Aijz j
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟= detA( )
−1
cf for real x
and for complex z
![Page 100: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/100.jpg)
determinants
€
dψdψ ∫ exp ψ iij
∑ Aijψ j
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟= detA
dx
2π∫ exp − 1
2 x i
ij
∑ Aij x j
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟= detA( )
−1/ 2
dzdz*
2π∫ exp − 1
2 zi*
ij
∑ Aijz j
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟= detA( )
−1
cf for real x
and for complex z
€
J[φ] = det∂η
∂φ
⎛
⎝ ⎜
⎞
⎠ ⎟= det
∂
∂t− ′ f (φ)
⎛
⎝ ⎜
⎞
⎠ ⎟
= DψDψ exp dt dd xψ (t)∂
∂t− ′ f (φ)
⎛
⎝ ⎜
⎞
⎠ ⎟ψ (t)∫
⎡
⎣ ⎢
⎤
⎦ ⎥∫
so representJ as
![Page 101: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/101.jpg)
determinants
€
dψdψ ∫ exp ψ iij
∑ Aijψ j
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟= detA
dx
2π∫ exp − 1
2 x i
ij
∑ Aij x j
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟= detA( )
−1/ 2
dzdz*
2π∫ exp − 1
2 zi*
ij
∑ Aijz j
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟= detA( )
−1
cf for real x
and for complex z
€
J[φ] = det∂η
∂φ
⎛
⎝ ⎜
⎞
⎠ ⎟= det
∂
∂t− ′ f (φ)
⎛
⎝ ⎜
⎞
⎠ ⎟
= DψDψ exp dt dd xψ (t)∂
∂t− ′ f (φ)
⎛
⎝ ⎜
⎞
⎠ ⎟ψ (t)∫
⎡
⎣ ⎢
⎤
⎦ ⎥∫
so representJ as
“ghost” variables
![Page 102: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/102.jpg)
Stratonovich generating functional
€
Z[h,θ] = DφD ˆ φ DψDψ exp −S[φ, ˆ φ ,ψ ,ψ ,h,θ]( )∫(one variable)
![Page 103: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/103.jpg)
Stratonovich generating functional
€
Z[h,θ] = DφD ˆ φ DψDψ exp −S[φ, ˆ φ ,ψ ,ψ ,h,θ]( )∫
€
−S[φ, ˆ φ ,ψ ,ψ ,h,θ] = d∫ t −T ˆ φ 2(t) + i ˆ φ (t) − ˙ φ (t) + f (φ) + h(t)( ) + iθ(t)φ(t)[ ]
+ dt ψ (t) ∂t − ′ f (φ)( )ψ (t)∫
(one variable)
![Page 104: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/104.jpg)
Stratonovich generating functional
€
Z[h,θ] = DφD ˆ φ DψDψ exp −S[φ, ˆ φ ,ψ ,ψ ,h,θ]( )∫
€
−S[φ, ˆ φ ,ψ ,ψ ,h,θ] = d∫ t −T ˆ φ 2(t) + i ˆ φ (t) − ˙ φ (t) + f (φ) + h(t)( ) + iθ(t)φ(t)[ ]
+ dt ψ (t) ∂t − ′ f (φ)( )ψ (t)∫
(one variable)
(field)
€
−S[φ, ˆ φ ,ψ ,ψ ,h,θ] =
d∫ t dd x −T ˆ φ 2 + i ˆ φ − ˙ φ − r0φ − u0φ3 +∇ 2φ + h( ) +ψ ∂t + r0 + 3u0φ
2 −∇ 2( )ψ + iθφ[ ]
![Page 105: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/105.jpg)
Stratonovich generating functional
€
Z[h,θ] = DφD ˆ φ DψDψ exp −S[φ, ˆ φ ,ψ ,ψ ,h,θ]( )∫
€
−S[φ, ˆ φ ,ψ ,ψ ,h,θ] = d∫ t −T ˆ φ 2(t) + i ˆ φ (t) − ˙ φ (t) + f (φ) + h(t)( ) + iθ(t)φ(t)[ ]
+ dt ψ (t) ∂t − ′ f (φ)( )ψ (t)∫
(one variable)
(field)
€
−S[φ, ˆ φ ,ψ ,ψ ,h,θ] =
d∫ t dd x −T ˆ φ 2 + i ˆ φ − ˙ φ − r0φ − u0φ3 +∇ 2φ + h( ) +ψ ∂t + r0 + 3u0φ
2 −∇ 2( )ψ + iθφ[ ]
free action:
€
−S0 = d∫ t dd x −T ˆ φ 2 − i ˆ φ ∂t + r0 −∇ 2( )φ +ψ ∂t + r0 −∇ 2
( )ψ[ ]
![Page 106: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/106.jpg)
Stratonovich generating functional
€
Z[h,θ] = DφD ˆ φ DψDψ exp −S[φ, ˆ φ ,ψ ,ψ ,h,θ]( )∫
€
−S[φ, ˆ φ ,ψ ,ψ ,h,θ] = d∫ t −T ˆ φ 2(t) + i ˆ φ (t) − ˙ φ (t) + f (φ) + h(t)( ) + iθ(t)φ(t)[ ]
+ dt ψ (t) ∂t − ′ f (φ)( )ψ (t)∫
(one variable)
(field)
€
−S[φ, ˆ φ ,ψ ,ψ ,h,θ] =
d∫ t dd x −T ˆ φ 2 + i ˆ φ − ˙ φ − r0φ − u0φ3 +∇ 2φ + h( ) +ψ ∂t + r0 + 3u0φ
2 −∇ 2( )ψ + iθφ[ ]
free action:
€
−S0 = d∫ t dd x −T ˆ φ 2 − i ˆ φ ∂t + r0 −∇ 2( )φ +ψ ∂t + r0 −∇ 2
( )ψ[ ]
interactions:
€
−S1 = u0 d∫ t dd x −i ˆ φ φ3 + 3ψ φ2ψ( )
![Page 107: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/107.jpg)
ghost correlations:
€
ψ −p,−ωψ pω 0=
−1
−iω + r0 + p2
![Page 108: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/108.jpg)
ghost correlations:
€
ψ −p,−ωψ pω 0=
−1
−iω + r0 + p2
Now when we expand
€
φ(1) ˆ φ (2) = DφD ˆ φ ∫ φ(1) ˆ φ (2)exp −S( )
= DφD ˆ φ ∫ φ(1) ˆ φ (2)exp −S1( )exp −S0( ) = φ(1) ˆ φ (2)exp −S1( )0
![Page 109: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/109.jpg)
ghost correlations:
€
ψ −p,−ωψ pω 0=
−1
−iω + r0 + p2
Now when we expand
€
φ(1) ˆ φ (2) = DφD ˆ φ ∫ φ(1) ˆ φ (2)exp −S( )
= DφD ˆ φ ∫ φ(1) ˆ φ (2)exp −S1( )exp −S0( ) = φ(1) ˆ φ (2)exp −S1( )0
we get
€
φ(1) ˆ φ (2) 1− S1 +L( )0
= C0(1,2)
+iu0 d∫ 3 φ(1) ˆ φ (2) ˆ φ (3)φ3(3)0
− 3u0 d3∫ φ(1) ˆ φ (2)ψ (3)φ2(3)ψ (3)0
![Page 110: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/110.jpg)
ghost correlations:
€
ψ −p,−ωψ pω 0=
−1
−iω + r0 + p2
Now when we expand
€
φ(1) ˆ φ (2) = DφD ˆ φ ∫ φ(1) ˆ φ (2)exp −S( )
= DφD ˆ φ ∫ φ(1) ˆ φ (2)exp −S1( )exp −S0( ) = φ(1) ˆ φ (2)exp −S1( )0
we get
€
φ(1) ˆ φ (2) 1− S1 +L( )0
= C0(1,2)
+iu0 d∫ 3 φ(1) ˆ φ (2) ˆ φ (3)φ3(3)0
− 3u0 d3∫ φ(1) ˆ φ (2)ψ (3)φ2(3)ψ (3)0
€
3 φ(1) ˆ φ (2)0
d3∫ ψ (3)ψ (3)0
φ2(3)0
+6 d3 φ(1)φ(3)0
ψ (3)ψ (3)∫0
φ(3) ˆ φ (2)0
new terms
![Page 111: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/111.jpg)
cancellation of closed loopsBecause of the -1 in the ghost correlation function,these just cancel the terms
![Page 112: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/112.jpg)
cancellation of closed loops
€
3 φ(1) ˆ φ (2)0
d3∫ ˆ φ (3)φ(3)0
φ2(3)0
+6 d3 φ(1)φ(3)0
φ(3) ˆ φ (3)∫0
φ(3) ˆ φ (2)0
Because of the -1 in the ghost correlation function,these just cancel the terms
![Page 113: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/113.jpg)
cancellation of closed loops
€
3 φ(1) ˆ φ (2)0
d3∫ ˆ φ (3)φ(3)0
φ2(3)0
+6 d3 φ(1)φ(3)0
φ(3) ˆ φ (3)∫0
φ(3) ˆ φ (2)0
Because of the -1 in the ghost correlation function,these just cancel the terms
that were zero with Ito convention but not Stratonovich
![Page 114: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/114.jpg)
cancellation of closed loops
€
3 φ(1) ˆ φ (2)0
d3∫ ˆ φ (3)φ(3)0
φ2(3)0
+6 d3 φ(1)φ(3)0
φ(3) ˆ φ (3)∫0
φ(3) ˆ φ (2)0
Because of the -1 in the ghost correlation function,these just cancel the terms
that were zero with Ito convention but not Stratonovich
This theory has a supersymmetry
![Page 115: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/115.jpg)
the superfield
Define a combination of the real and Grassman fields
![Page 116: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/116.jpg)
the superfield
Define a combination of the real and Grassman fields
€
Φ=φ+ξ ψ +ψ ξ −iξ ξ ˆ φ
€
ξ,ξ Grassman numbers
![Page 117: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/117.jpg)
the superfield
€
f (φ) = −∂V (φ)
∂φ
Define a combination of the real and Grassman fields
Then if
€
Φ=φ+ξ ψ +ψ ξ −iξ ξ ˆ φ
€
ξ,ξ Grassman numbers
![Page 118: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/118.jpg)
the superfield
€
f (φ) = −∂V (φ)
∂φ
Define a combination of the real and Grassman fields
Then if
€
Z = DΦexp −S Φ[ ]( )∫
S = dt dξ dξ ∫ ∂Φ
∂ξT
∂Φ
∂ξ −ξ
∂Φ
∂t
⎛
⎝ ⎜
⎞
⎠ ⎟+ V Φ(t,ξ ,ξ )( )
⎡
⎣ ⎢
⎤
⎦ ⎥
≡ dτ∫ ∂Φ
∂ξT
∂Φ
∂ξ −ξ
∂Φ
∂t
⎛
⎝ ⎜
⎞
⎠ ⎟+ V Φ(τ )( )
⎡
⎣ ⎢
⎤
⎦ ⎥
€
Φ=φ+ξ ψ +ψ ξ −iξ ξ ˆ φ
€
ξ,ξ Grassman numbers
the generating functional can be written
![Page 119: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/119.jpg)
How does this happen?
Expand the potential term:
![Page 120: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/120.jpg)
How does this happen?
Expand the potential term:
€
Φ=φ+ξ ψ +ψ ξ + iξ ξ ˆ φ ⇒
V (Φ) = V (φ) + (ξ ψ +ψ ξ + iξ ξ ˆ φ ) ′ V (φ) + 12 (ξ ψ +ψ ξ + iξ ξ ˆ φ )2 ′ ′ V (φ)
![Page 121: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/121.jpg)
How does this happen?
Expand the potential term:
€
Φ=φ+ξ ψ +ψ ξ + iξ ξ ˆ φ ⇒
V (Φ) = V (φ) + (ξ ψ +ψ ξ + iξ ξ ˆ φ ) ′ V (φ) + 12 (ξ ψ +ψ ξ + iξ ξ ˆ φ )2 ′ ′ V (φ)
Integrate over the “Grassman time”
![Page 122: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/122.jpg)
How does this happen?
Expand the potential term:
€
Φ=φ+ξ ψ +ψ ξ + iξ ξ ˆ φ ⇒
V (Φ) = V (φ) + (ξ ψ +ψ ξ + iξ ξ ˆ φ ) ′ V (φ) + 12 (ξ ψ +ψ ξ + iξ ξ ˆ φ )2 ′ ′ V (φ)
€
dξ dξ ∫ V (Φ) = i ˆ φ ′ V (φ) +ψψ ′ ′ V (φ) = −i ˆ φ f (φ) +ψ ′ f (φ)ψ
Integrate over the “Grassman time”
![Page 123: Lecture 13: Field-theoretic formulation of Langevin models Outline: Functional (path) integral formulation Stratonovich and Ito, again the Martin-Siggia-Rose.](https://reader036.fdocuments.in/reader036/viewer/2022062423/56649ec45503460f94bcf32f/html5/thumbnails/123.jpg)
How does this happen?
Expand the potential term:
€
Φ=φ+ξ ψ +ψ ξ + iξ ξ ˆ φ ⇒
V (Φ) = V (φ) + (ξ ψ +ψ ξ + iξ ξ ˆ φ ) ′ V (φ) + 12 (ξ ψ +ψ ξ + iξ ξ ˆ φ )2 ′ ′ V (φ)
€
dξ dξ ∫ V (Φ) = i ˆ φ ′ V (φ) +ψψ ′ ′ V (φ) = −i ˆ φ f (φ) +ψ ′ f (φ)ψ
Integrate over the “Grassman time”
which are the terms in the action involving f.