renormalization group in stochastic theory of...

Renormalization group in stochastictheory of developed turbulence 2

Renormalization and the renormalization group

Juha Honkonen

National Defence University, Helsinki, Finland

Renormalization group in stochastic theory of developed turbulence 2 – p. 1/18

Outline

Canonical dimensions of fields and parametersCanonical dimensions in hydrodynamicsCanonical dimensions of Green functionsOne-irreducible correlation functionsPower counting for one-irreducible functionsRenormalization and countertermsRenormalized correlation functionsHomogeneous renormalization-group equationInvariant (running) parametersCritical dimensions at the fixed point of RGScaling in terms of physical variables


Canonical dimensions in field theory

Both UV and IR divergences occur in field theories.




In particle physics, the UV divergences must be eliminated,in hydrodynamics it is useful to extract IR asymptotics.





IR and UV connected in logarithmic models withdimensionless couplings. Introduce canonical dimensions.






Separate spatial and temporal dimensions convenient.Conventions: dk

k = −dkx = 1, dω

ω = −dωt = 1, dω

k = −dkω = 0.






Separate spatial and temporal dimensions convenient.Conventions: dk

k = −dkx = 1, dω

ω = −dωt = 1, dω

k = −dkω = 0.

Total dimension of a quantity Q from the homogeneity of thefree-field action under scaling k → λk, ω → λω; here:

dQ = dkQ + 2dω

Q .


Canonical dimensions in hydrodynamics

In stochastic hydrodynamics dimensions of fields from thedimensionless substantial derivative: v′ [∂tv + (v∇)v].




dkv = −1 , dω

v = 1 , dv = 1 ;

dkv′ = d + 1 , dω

v′ = −1 , dv′ = d − 1 .




dkv = −1 , dω

v = 1 , dv = 1 ;

dkv′ = d + 1 , dω

v′ = −1 , dv′ = d − 1 .

Dimensions of the viscosity and coupling constant fordf (k) = g10ν

30 k4−d−2ε (don’t forget the Fourier transform!):

dkν0

= −2 , dων0

= 1 , dν0= 0 ;

dkg0

= 2ε , dωg0

= 0 , dg0= 2ε .




dkv = −1 , dω

v = 1 , dv = 1 ;

dkv′ = d + 1 , dω

v′ = −1 , dv′ = d − 1 .

Dimensions of the viscosity and coupling constant fordf (k) = g10ν

30 k4−d−2ε (don’t forget the Fourier transform!):

dkν0

= −2 , dων0

= 1 , dν0= 0 ;

dkg0

= 2ε , dωg0

= 0 , dg0= 2ε .

The theory is logarithmic at ε = 0 in any space dimension d.


Canonical dimensions of generating functions

The basic generating function is rendered dimensionless

G(J,J′) =

∫

Dv

∫

Dv′ eSNS(v,v′)+vJ+v′J′

= e1

2

δδv

∆11δ

δv+ δ

δv∆12

δ

δv′ ev′(v∇)v+vJ+v

′J′∣

∣

v=v′=0

when dkv + dk

J = d, dωv + dω

J = 1, hence dv + dJ = d + 2.


Canonical dimensions of generating functions

The basic generating function is rendered dimensionless

G(J,J′) =

∫

Dv

∫

Dv′ eSNS(v,v′)+vJ+v′J′

= e1

2

δδv

∆11δ

δv+ δ

δv∆12

δ

δv′ ev′(v∇)v+vJ+v

′J′∣

∣

v=v′=0

when dkv + dk

J = d, dωv + dω

J = 1, hence dv + dJ = d + 2.

Generating functions W (J,J′) = ln G(J,J′) and

Γ(v,v′) = W (J,J′) − vJ − v′J′ , v =δW

δJ, v′ =

δW

δJ′

have zero canonical dimensions by definition.


Canonical dimensions of Green functions

Green functions are derivatives of generating functionals;hence, in the coordinate space

dWn,n′({ti},{xi}) = dGn,n′({ti},{xi}) = ndv + n′dv′ ,

dΓn,n′({ti},{xi}) = ndJ + n′d′J = (n + n′)(d + 2) − ndv − n′dv′ .






In Fourier transforms frequency and wave-vector δ functionsof conservation laws factorize. Thus, in the Fourier space

dWn,n′= dGn,n′

= d + 2 + ndv + n′dv′ − (n + n′)(d + 2) ,

dΓn,n′= d + 2 − ndv − n′dv′ .






In Fourier transforms frequency and wave-vector δ functionsof conservation laws factorize. Thus, in the Fourier space

dWn,n′= dGn,n′

= d + 2 + ndv + n′dv′ − (n + n′)(d + 2) ,

dΓn,n′= d + 2 − ndv − n′dv′ .

For instance

dΓ22= d+2− 2(d− 1) = 4− d , dW12

= d+2+d− 2(d+2) = −2 .


Power counting for one-irreducible functions

The superficial degree of divergence δ of a 1PI graphγ({ωi}, {ki}) is defined by the homogeneity relation

γ({λ2ωi}, {λki}) = λδγ({ωi}, {ki})

in which the same stretching is implied for the integrationvariables.


Power counting for one-irreducible functions

The superficial degree of divergence δ of a 1PI graphγ({ωi}, {ki}) is defined by the homogeneity relation

γ({λ2ωi}, {λki}) = λδγ({ωi}, {ki})

in which the same stretching is implied for the integrationvariables.

The real degree of divergence δ′ of a graph γ is obtained bysubtracting minus the number of factorizing vertex wavevectors. In our case

δ′γ = d + 2 − V dg0− Ndv − N ′dv′ − N ′ .


Classification of models

Finite number of graphs with δ′ ≥ 0: superrenormalizablemodel;




Finite number of 1PI functions with graphs δ′ ≥ 0:renormalizable model;




Finite number of 1PI functions with graphs δ′ ≥ 0:renormalizable model;

Infinite number of 1PI functions with graphs δ′ ≥ 0:nonrenormalizable model.


Divergent 1PI functions of the NS problem

UV divergences in the one-irreducible graphs Γ. Let dg0= 0.

Then for the graphs γ of a 1PI function Γ

δγ = dΓ .





δγ = dΓ .

Graphs with N ′ = 0 and N ′ = 1, N = 0 vanish: δ ≥ 0 for theone-irreducible correlation functions 〈vv′〉1−ir, 〈vvv′〉1−ir:

δ〈vv′〉1−ir= 1 , δ〈vvv′〉1−ir

= 0 , δ〈v′v′〉1−ir= 2 − d .





δγ = dΓ .

Graphs with N ′ = 0 and N ′ = 1, N = 0 vanish: δ ≥ 0 for theone-irreducible correlation functions 〈vv′〉1−ir, 〈vvv′〉1−ir:

δ〈vv′〉1−ir= 1 , δ〈vvv′〉1−ir

= 0 , δ〈v′v′〉1−ir= 2 − d .

The UV-divergent part of a graph is subtracted to produce aUV-finite renormalized function and a counterterm.


One-irreducible correlation functions

One-loop graph for the one-irreducible function 〈vv′〉1−ir


One-irreducible correlation functions

One-loop graph for the one-irreducible function 〈vv′〉1−ir

There are, however, 8 graphs in the two-loop approximation


Renormalization and counterterms

Subtraction of divergences effected by a local counterterm.




For instance, the divergent part of the graph

has been calculated as [with the sharp IR cutoff θ(m − k)]






Γ(1)mn(ω,p) ∼ −ν0p

2 (d − 1)2(m)−2ε g0Sd

8(d + 2) ε.






Γ(1)mn(ω,p) ∼ −ν0p

2 (d − 1)2(m)−2ε g0Sd

8(d + 2) ε.

To subtract this term, add to the action the counterterm

v′ν0(d − 1)2(m)−2ε g0Sd

8(d + 2) ε∇2v .


Renormalization of the Navier-Stokes problem

Counterterms produced by the graphs of the Navier-Stokesproblem contain at least one factorized ∇, so there are nocounterterms of structure v′∂tv.




Galilei invariance preserves the covariant derivative∂tv + (v∇)v: counterterm v′(v∇)v is not generated either.





The only generic counterterm is v′∇2v (for d > 2).





The only generic counterterm is v′∇2v (for d > 2).

Separate analysis for d = 2, let d > 2 for the time being.


Renormalized correlation functions

A model with a finite number of one-irreducible correlationfunctions with δ ≥ 0 is renormalizable.




Basic statement of renormalization theory: UV divergencesof a renormalizable model may be absorbed in a redifinitionof parameters such that the renormalized action

SNSR(v,v′) =1

2v′Dv′ − v′

[

∂tv + (v∇)v − νZν∇2v

]

generates UV finite correlation functions such that∫

DvDv′v(1) . . . v(n) eSNS(v,v′) =

∫

DvDv′v(1) . . . v(n) eSNSR(v,v′) .




Basic statement of renormalization theory: UV divergencesof a renormalizable model may be absorbed in a redifinitionof parameters such that the renormalized action

SNSR(v,v′) =1

2v′Dv′ − v′

[

∂tv + (v∇)v − νZν∇2v

]

generates UV finite correlation functions such that∫

DvDv′v(1) . . . v(n) eSNS(v,v′) =

∫

DvDv′v(1) . . . v(n) eSNSR(v,v′) .

Most straightforwardly the renormalization constant Zν isfound just from this condition.


Homogeneous renormalization-group equation

Introduce a scaling parameter µ in the connection betweenrenormalized and unrenormalized (bare) parameters:

ν0 = νZν , g01 = D01ν−30 = g1µ

2ǫZ−3ν .




ν0 = νZν , g01 = D01ν−30 = g1µ

2ǫZ−3ν .

Bare quantities independent of µ: the homogeneous RGequation; for the pair correlation function

[µ∂µ + β1∂g1− γνν∂ν ] G = 0 , γν = µ∂µ

∣

∣

0ln Zν , β1 = µ∂µ

∣

∣

0g1




ν0 = νZν , g01 = D01ν−30 = g1µ

2ǫZ−3ν .

Bare quantities independent of µ: the homogeneous RGequation; for the pair correlation function

[µ∂µ + β1∂g1− γνν∂ν ] G = 0 , γν = µ∂µ

∣

∣

0ln Zν , β1 = µ∂µ

∣

∣

0g1

where derivatives taken with bare parameters fixed and∫

dr exp [i(k · r)] 〈vn(t,x + r)vm(t,x)〉 = Pnm(k)G(k)


Invariant (running) parameters

RG solution for the velocity correlation function:

G(k) = ν2k2−dR

(

k

µ, g1,

m

µ

)

= ν2k2−dR(

1, g1,m

k

)

.




G(k) = ν2k2−dR

(

k

µ, g1,

m

µ

)

= ν2k2−dR(

1, g1,m

k

)

.

Invariant (running) parameters g1(µ/k, g1), ν(µ/k, g1): firstintegrals of the RG equation, e.g.

[µ∂µ + β1∂g1− γνν∂ν ] g1 = 0 , g1(1, g1) = g1 .




G(k) = ν2k2−dR

(

k

µ, g1,

m

µ

)

= ν2k2−dR(

1, g1,m

k

)

.

Invariant (running) parameters g1(µ/k, g1), ν(µ/k, g1): firstintegrals of the RG equation, e.g.

[µ∂µ + β1∂g1− γνν∂ν ] g1 = 0 , g1(1, g1) = g1 .

Connection between bare and invariant parameters:

g10 = g1k2εZ−3

ν

(

g1,m

k

)

, ν =

(

D10k−2ε

g1

)1/3

.


Critical dimensions at the fixed point of RG

For ε > 0 ∃ an IR-stable fixed point: g1 → g1∗ ∝ ε.Asymptotics of correlation and response functions W aregeneralized homogeneous functions

W∣

∣

IR({λ−∆ωti}, {λ

−1xi}) = λ∑

Φ∆ΦW

∣

∣

IR({ti}, {xi}) .

Here, ∆ω and ∆Φ are critical dimensions of ω and Φ = {v, v′}.


Critical dimensions at the fixed point of RG

For ε > 0 ∃ an IR-stable fixed point: g1 → g1∗ ∝ ε.Asymptotics of correlation and response functions W aregeneralized homogeneous functions

W∣

∣

IR({λ−∆ωti}, {λ

−1xi}) = λ∑

Φ∆ΦW

∣

∣

IR({ti}, {xi}) .

Here, ∆ω and ∆Φ are critical dimensions of ω and Φ = {v, v′}.They are all expressed through γ∗

ν ≡ γν(g∗):

∆v = 1 − γ∗ν , ∆v′ = d − ∆ϕ , ∆ω = 2 − γ∗

ν .

Due to Galilei invariance, basic critical dimensions are exact:

∆v = 1 − 2ε/3 , ∆ω = 2 − 2ε/3 .


Scaling in terms of physical variables



IR fixed point yields large-scale limit (k → 0, u = m/k =const)

G(k) ∼ (D10/g1∗)2/3 k 2−d−4ε/3R(1, g1∗, u) , R(1, g1∗, u) =

∞∑

n=1

εnRn(u) .



IR fixed point yields large-scale limit (k → 0, u = m/k =const)

G(k) ∼ (D10/g1∗)2/3 k 2−d−4ε/3R(1, g1∗, u) , R(1, g1∗, u) =

∞∑

n=1

εnRn(u) .

Translate in traditional variables; trade D10 for the meanenergy injection rate E (2 > ε > 0):

E =(d − 1)

2(2π)d

∫

dk df (k) ⇒ D10 =4(2 − ε) Λ2ε−4E

Sd(d − 1), Λ = (E/ν3

0)1/4 .


Freezing of dimensions in the inertial range

Large-scale scaling in terms of E and ν0 for 2 > ε > 0:

G(k) ∼[

4(2 − ε)/Sd(d − 1)g1∗

]2/3ν2−ε0 E

ε/3k 2−d−4ε/3R(1, g1∗, u) .




G(k) ∼[

4(2 − ε)/Sd(d − 1)g1∗

]2/3ν2−ε0 E

ε/3k 2−d−4ε/3R(1, g1∗, u) .

The desired Kolmogorov scaling, when ε → 2 (IR pumping).




G(k) ∼[

4(2 − ε)/Sd(d − 1)g1∗

]2/3ν2−ε0 E

ε/3k 2−d−4ε/3R(1, g1∗, u) .


Freezing of scaling dimensions for ε > 2 [Adzhemyan, Antonov& Vasil’ev (1989)]: D10 acquires scale dependence through

D10 = 4(ε − 2) m4−2εE/Sd(d − 1) , m = 1/L .




G(k) ∼[

4(2 − ε)/Sd(d − 1)g1∗

]2/3ν2−ε0 E

ε/3k 2−d−4ε/3R(1, g1∗, u) .



D10 = 4(ε − 2) m4−2εE/Sd(d − 1) , m = 1/L .

Yields independence of ν0, Kolmogorov exponents ∀ ε > 2:

G(k) ∼[

4(ε − 2)/Sd(d − 1)g1∗

]2/3E

2/3k−d−2/3u4(2−ε)/3R(1, g1∗, u) .




G(k) ∼[

4(2 − ε)/Sd(d − 1)g1∗

]2/3ν2−ε0 E

ε/3k 2−d−4ε/3R(1, g1∗, u) .



D10 = 4(ε − 2) m4−2εE/Sd(d − 1) , m = 1/L .

Yields independence of ν0, Kolmogorov exponents ∀ ε > 2:

G(k) ∼[

4(ε − 2)/Sd(d − 1)g1∗

]2/3E

2/3k−d−2/3u4(2−ε)/3R(1, g1∗, u) .

Analysis of the inertial-range limit u = m/k → 0 beyond RG.Renormalization group in stochastic theory of developed turbulence 2 – p. 18/18

renormalization group in stochastic theory of...

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