6. renormalized perturbation theory - ruhr-uni-bochum.de

What is a plasma ?

Kinetic description

Fluid description

Navier-Stokes turbulence

Closure Theories

Renormalized . . .

Renormalization Group

Passive Scalar

Dynamic Functional

Magnetic Reconnection

Mean Field . . .

Magneto-Rotational . . .

Interstellar Medium (ISM)

Turbulence in Fusion

Literature

6. Renormalized Perturbation Theory

6.1. Perturbation expansion of the Navier-Stokes equation

Kraichnan (1959): direct-interaction approximationWyld (1961): partial summation for a simplified scalar modelLee (1965): partial summation for Navier-Stokes and MHD

6.1.1. The zero-order isotropic propagators

λ: bookkeeping parameter, Dαβ: projector on divergence free part

zero-order Green tensor:

with

so that

What is a plasma ?

Kinetic description

Fluid description


Closure Theories

Renormalized . . .


Passive Scalar

Dynamic Functional


Mean Field . . .




Literature

zero-order velocity:

Fourier transform in time:

=⇒ Navier-Stokes

zero-order propagator is given by

What is a plasma ?

Kinetic description

Fluid description


Closure Theories

Renormalized . . .


Passive Scalar

Dynamic Functional


Mean Field . . .




Literature

6.1.2. The primitive perturbation expansion

Stirring force

after Fourier-trafo

zero-order velocity field after in Fourier-space

which is the zero-order term in expansion

correlation Qαβ(k;ω, ω′)

What is a plasma ?

Kinetic description

Fluid description


Closure Theories

Renormalized . . .


Passive Scalar

Dynamic Functional


Mean Field . . .




Literature

homogeneous isotropic turbulence

and

Perturbation expansion for Q

Take Fourier transformed NS-equation, invert linear part and substituteforcing by u(0)

What is a plasma ?

Kinetic description

Fluid description


Closure Theories

Renormalized . . .


Passive Scalar

Dynamic Functional


Mean Field . . .




Literature

Now substitute perturbation expansion for u

equating coefficients:

u(2) can be expressed by u(0)

What is a plasma ?

Kinetic description

Fluid description


Closure Theories

Renormalized . . .


Passive Scalar

Dynamic Functional


Mean Field . . .




Literature

stirring Gaussian =⇒ u(0) Gaussian=⇒

⟨u(0)u(0)u(0)u(0)

⟩can be factored as in Quasi-Normal approximation

second-order correlation tensor:

What is a plasma ?

Kinetic description

Fluid description


Closure Theories

Renormalized . . .


Passive Scalar

Dynamic Functional


Mean Field . . .




Literature

6.1.3. Graphical representation of the perturbation series

all orders can be expressed by zero-order terms, but divergent series

three main constituents: u(0), G0 and M

zero-order:

first-order: wavenumber conservation

What is a plasma ?

Kinetic description

Fluid description


Closure Theories

Renormalized . . .


Passive Scalar

Dynamic Functional


Mean Field . . .




Literature

second-order: two M factors:

third-order: three M factors:

What is a plasma ?

Kinetic description

Fluid description


Closure Theories

Renormalized . . .


Passive Scalar

Dynamic Functional


Mean Field . . .




Literature

graphical expansion for correlation tensor

zero-order:

second-order:

this is middle second-order term:

What is a plasma ?

Kinetic description

Fluid description


Closure Theories

Renormalized . . .


Passive Scalar

Dynamic Functional


Mean Field . . .




Literature

this is the last second-order term:

The third is a mirrow image of this one.

What is a plasma ?

Kinetic description

Fluid description


Closure Theories

Renormalized . . .


Passive Scalar

Dynamic Functional


Mean Field . . .




Literature

fourth-order showing four of the 29 fourth-order diagrams:

Now resummation (renormalisation): new diagram elements

Write correlation tensor as:

What is a plasma ?

Kinetic description

Fluid description


Closure Theories

Renormalized . . .


Passive Scalar

Dynamic Functional


Mean Field . . .




Literature

6.1.4. Class A diagram: the renormalized propagator

Wyld (1961): Class A diagrams are those diagrams which can be split intotwo pieces by cutting a single Q0 line.

zero-order: Q0 can be expressed in terms of two zero-order propagatorsacting on the spectrum of the stirring forces w(k;ω, ω′)

This looks graphically like

Now second-order:

Let’s summarize: at zero order, we have w with a G0 on each side. Atsecond order, w has a G0 on one side and a diagram which connects like aG0 on the other. This holds for all orders. Thus we have a generalization

What is a plasma ?

Kinetic description

Fluid description


Closure Theories

Renormalized . . .


Passive Scalar

Dynamic Functional


Mean Field . . .




Literature

of

which reads

where G(k, ω) is the renormalized propagator.

Graphically, this corresponds to

What is a plasma ?

Kinetic description

Fluid description


Closure Theories

Renormalized . . .


Passive Scalar

Dynamic Functional


Mean Field . . .




Literature

6.1.5. Class B diagrams: renormalized perturbation series

Class B diagrams can’t be split into two by cutting a single Q0 line.

In the class A diagrams, certain diagram parts were propagator like, that is,they connected like G0: renormalize G0 by adding up all diagrams whichconnect like G0.

Renormalize vertex: add up all diagrams which connect like a vertex

Example: consider fourth-order diagram

The part

connects like a point vertex =⇒ renormalized vertex

What is a plasma ?

Kinetic description

Fluid description


Closure Theories

Renormalized . . .


Passive Scalar

Dynamic Functional


Mean Field . . .




Literature

replace vertex by renormalized vertex:

Therefore the key to the class B diagrams is as follows:

1. Find those diagrams which cannot be reduced to a lower order byreplacing diagram parts.

2. Call these the irreducible diagrams.

3. Replace all elements in the irreducible diagrams by their renormalizedforms.

4. Write down all these modified diagrams in order, thus generating arenormalized perturbation expansion.

What is a plasma ?

Kinetic description

Fluid description


Closure Theories

Renormalized . . .


Passive Scalar

Dynamic Functional


Mean Field . . .




Literature

Result for Q(k;ω, ω′)

This is an integral equation for Q(k;ω, ω)

Combine vertex and propagator expansions:

Integral equation for the renormalized vertex

What is a plasma ?

Kinetic description

Fluid description


Closure Theories

Renormalized . . .


Passive Scalar

Dynamic Functional


Mean Field . . .




Literature

Integral equation for the renormalized propagator G(k, ω)

Pecularity of this diagram:

unrenormalized propagator emerging from the left !!!

Reason for this: symbolic form of Navier-Stokes

L0u(k) = λM(k)u(j)u(k − j) , L0 = ∂t + νk2

and renormalize r.h.s., then invert L0 which results in G0

What is a plasma ?

Kinetic description

Fluid description


Closure Theories

Renormalized . . .


Passive Scalar

Dynamic Functional


Mean Field . . .




Literature

6.1.6. Second-order closures

What have we done:

We replaced a wildly divergent series with one of unknown properties !

We have hope that it might be assymptotic, but we simple don’t know !

Well known examples recovered from this Wyld (1961) formulation:

Example 1:

correlation tensor: truncate at second order (in number vertices)vertex: truncate at first order (unrenormalized vertex)propagator: truncate at zero order (unrenormalized propagator)

This is Chandrasekhar’s theory (1955) which is the two-time analog ofquasi-normality.

What is a plasma ?

Kinetic description

Fluid description


Closure Theories

Renormalized . . .


Passive Scalar

Dynamic Functional


Mean Field . . .




Literature

Example 2:

correlation tensor: truncate at second ordervertex: truncate at first orderpropagator: truncate at second order

This is the pioneering direct-interaction approximation (DIA) by Kraichnan(1959): second-order closure with line and with no vertex renormalization.

6. renormalized perturbation theory - ruhr-uni-bochum.de

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