We find the exact N-point generating function in Polyakov's approach to
Burgers turbulence.Comment: 7 pages,Latex,no figure

B. Davoudi

Borue

DeDominicis

Fisher

Janssen

Kardar

L'vov

M.R. Rahimi Tabar

Medina

Polyakov

S. Rouhani

Yakhot

Zakharov

Zalesky

Zeldovich

English

arXiv

Crossref

Physics Letters A

The exact N-point generating function in Polyakov-Burgers turbulence

We find the exact N-point generating function in Polyakov's approach to Burgers turbulence

Rouhani, S

Davoudi, B

CERN Document Server

The Exact N-Point Generating Function in Polyakov -
Burgers Turbulence
M.R.Rahimi Tabar, S.Rouhani and B.Davoudi
Department of Physics, Sharif University of Technology
Tehran P.O.Box: 11365-9161, Iran.
Institue for Studies in Theoretical Physics and Mathematics
Tehran P.O.Box: 19395-5746, Iran.
Abstract
We nd the exact N-point generating function in Polyakov’s approach to Burgers
turbulence.
1
1- Introduction
A theoretical understanding of turbulence has eluded physicists for a long time. Recently
Polyakov [1] has oerd a eld theoretic method for deriving of the probability distribution
or density of states in (1+1)-dimensional turbulent systems. Polyakov formulates a new
method for analyzing the inertial range correlation functions based on the two important
ingredients in eld theory and statistical physics namely the operator product expansion
(OPE) and anomalies. Despite existence of many eld theoretic approaches to turbulence
[2,3,4], it appears that this new approach is more promising. Polyakov argues that in the
limit of high Reynold’s number because of existence of singularities at coinciding point, dis-
sipation remains nite and all sublleading terms give vanishing contributions in the inertial
range. By develop the OPE one nds the leading singularities and can show that this ap-
proach is self-consistent. Here we consider Polyakov‘s approach [1] to the Burger turbulence
when the gradient of pressure is negligible and solve the N-point master equation, calculating
the N-point generating function. Our result also apply to the Kardar-Parisi-Zhang (KPZ)
equation in (1+1)-dimensions, crystal growth [5], the nonlinear dynamics of a moving line
[6], galaxy formations [7], dissipative transport [9], dynamics of a Sine-Gordan chain [10],
behavior of magnetic flux line in superconductor [11], and spin glasses [12].
2- N-Point Generating Functions
The Burgers equation has following form
ut + uux = uxx + f(x; t) (1)
where u is the velocity eld, and  is the viscosity and f(x; t) is the Gaussian random force
2
with the following correlation:
< f(x; t)f(x
0
; t
0
) >= k(x− x
0
)(t− t
0
) (2)
The transformation, u(x; t) = −@xh(x; t) maps eq.(1) to the well-known KPZ equation [5],
@th = @xxh +

2
[@xh]
2 + f(x; t) (3)
It is noted that the parameter  that appears in the above transformation is not renormalized
under any renormalization procedure [8]. Following Polyakov [1] consider the following
generating functional
ZN (1; 2; : : : N ; x1; : : : xN) =< exp(
NX
j=1
ju(xj; t)) > (4)
Noting that the random force f(x; t) has a Gaussian distribution ZN satises a closed dif-
ferential equation provided that the viscosity  the tends to zero:
_ZN +
X
j
@
@j
(
1
j
@ZN
@xj
) =
X
k(xi − xj)ijZN +DN (5)
where DN is :
DN = 
X
j < u
00
(xj; t) exp
X
ku(xk; t) > (6)
To reach the inertial range we must, how ever, keep  innitesimal but non-zero. Polyakov
argues that the anomaly mechanism implies that innitesimal viscosity produces a nite
eect. To compute this eect Polakov makes the F-conjecture, which is the existence of an
operator product expansion or the fusion rules. The fusion rule is the statement concerning
the behaviour of correlation functions, when some subset of points are put close together.
Let us use the following notation;
Z(1; 2; : : : ; x1; : : : xN ) =< e1(x1) : : : eN (xN) > (7)
3
then Polyakov‘s F-conjecture is that in this case the OPE has the following form,
e1(x+ y=2)e2(x− y=2) = A(1; 2; y)e1+2(x) +B(1; 2; y)
@
@x
e1+2 + o(y
2) (8)
This implies that ZN fuses into functions ZN−1 as we fuse a couple of points together. The
F-conjecture allows us to evaluate the following anomaly operator (i.e. the DN -term in
eq.(5)),
a(x) = lim
!0
(u\(x) exp(u(x)) (9)
which can be written as:
a(x) = lim
;y;!0

@3
@@y2
e(x+ y)e(x) (10)
As discussed in [1] the only possible Galilean invarant expression is:
a(x) = a()e(x) + ~()
@
@x
e(x) (11)
Therefore in steady state the master equation takes the following form,
X
(
@
@xj
− (j))
@
@xj
ZN −
X
k(xi − xj)ijZN =
X
a(j)ZN
() = ~() +
1

(12)
Polyakov has found the explicit form of Z2 in the case that k(xi − xj) = K(0)(1− (xi −
xj)2=l2), and the desity of states as;
W (u; y) =
Z c+i1
c−i1
d
2i
e−u(y) (13)
where
(y) = e2=3(y)
3=2
4
and
 = 2(1 − 2); y = x1 − x2:
It can be easly shown that with the following denition of variables Polyakov‘s master
equation with the scaling conjecture[1] is:
f
@2
@2@y2
+
@2
@3@y3
− (y22 + y33)
2gf3 = 0 (14)
where
f3 = (123)
−bZ3
y1 =
x1 + x2 + x3
3
y2 = x1 −
x2 + x3
2
y3 = x2 − x3
1 =
1 + 2 + 3
3
2 =
3
2
(1 −
2 + 3
2
) 3 = 2(2 − 3) (15)
Now we set f as;
f3 = 
S2
2 
S3
3 g3(2y2; 3y3) (16)
inserting in eq.(15) results in
g3(2y2; 3; y3) = e
2=3(2y2+3y3)
3=2
(17)
and
S2 = S3 = −5=4
Now if we use the following transformation;
y1 =
x1 + x2 + x3 + : : : xN
N
y2 = x1 −
x2 + x3 + : : : xN
N − 1
y3 = x2 −
x3 + x4 + : : : xN
N − 2
5
and yN = xN−1 − xN (18)
and
1 =
1 + 2 + : : :+ N
N
2 =
N
N − 1
[1 −
2 + 3 + : : :+ N
N − 1
]
3 =
N − 1
N − 2
[2 −
3 + 4 + : : :+ N
N − 2
]
and N = 2(N−1 − N ) (19)
we get the following partial dierential equation for fN ;
f
@2
@y2@2
+ : : :+
@2
@yN@N
gfN − (y22 + : : :+ yNN )
2fN = 0 (20)
which is solved by:
fN = (23 : : : N )
−[ 2N−1
2(N−1) ] exp2=3(2y2+:::+NyN )
3=2
(21)
In principle the parameter b in eq.(15) can be evaluated by means of the exponents of
(2 : : : N ) term in eq.(21) which for Z2 it turns out that b = 3=4. Our exact result for ZN
allows us to determaine the OPE coecients in eq.(8), and calculate ZN in the case where
the random force is conservative, which is important for the KPZ-equation, work in this
direction is on the way.
Acknowledgements: We wish to thank Vahid Karimipour for valuable discussions.
6
References
[1] A.M.Polyakov, "Turbulence Without Pressure" PUPT-1546, hep-th/9506189
[2] A.M.Polyakov, Nucl. Phys. B396,367 (1993)
[3] V.S.L‘Vov Phys. Reports 207, 2 (1991)
[4] V.Zakharov, V.L‘Vov and G.Falkovich," Kolmogorov spectra of turbulence I" ,Springer-
Verlag, Berlin Heidelberg, 1992.
[5] M.Kardar, G.Parisi and Y.Zhang, Phys. Rev. Lett. 56 , 889 (1986)
[6] T.Hwa and M.Kardar, Phys. Rev. A45, 7002 (1992)
[7] Ya.Zeldovich Astron and Astroph. 5,84 (1972)
[8] E.Medina, T.Hwa, M.Kardar and Y.Zhang, Phys. Rev. A39, 3053 (1989)
[9] H.K.Janssen and B.Schmittmann, Z.Phys. B63, 517 (1986)
[10] S.Zalesky, Physica D34, 417 (1989)
[11] T.Hwa, Phys. Rev. Lett. 69, 1552 (1992)
[12] D.S.Fisher and D.A.Huse, Phys. Rev. B43, 10728 (1991)
7


The Exact N-point Generating Function in Polyakov-Burgers Turbulence

The Exact N-point Generating Function in Polyakov-Burgers Turbulence

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