VOLUME 78, NUMBER 10
PHYSICAL REVIEW LETTERS
10 MARCH 1997
Probability Distribution Functions for the Random Forced Burgers Equation
Weinan E,1 Konstantin Khanin,2 Alexandre Mazel,3 and Yakov Sinai4
1
Courant Institute of Mathematical Sciences, New York University, New York , New York 10012
2
Heriot-Watt University, Edinburgh and Landau Institute, Moscow, Russia
3
Princeton University, Princeton, New Jersey 08544
4
Princeton University, Princeton and Landau Institute, Moscow, Russia
(Received 8 November 1996)
We propose a new approach for the analysis of stationary correlation functions of the 1D Burgers
equation driven by a random force. We use this to study the asymptotic behavior of the probability
distribution of velocity gradients and velocity increments. [S0031-9007(97)02681-1]
PACS numbers: 47.27.Gs, 03.40.Gc
Statistical properties of solutions of the random forced
Burgers equation have been a subject of intensive studies recently (see Refs. [1–6]). Of particular interest
are the asymptotic properties of probability distribution
functions associated with velocity gradients and velocity
increments. Aside from the fact that such issues are of
direct interest to a large number of problems such as
the growth of random surfaces [1], it is also hoped that
the field-theoretic techniques developed for the Burgers
equation will eventually be useful for understanding more
complex phenomena such as turbulence.
In this paper, we propose a new and direct approach for
analyzing the scaling properties of the various distribution
functions for the random forced Burgers equation. We
will consider the problem
1
ut 1 2 su2 dx ­ nuxx 2 Vx sx, td .
(1)
Most of our discussion will be limited to the inviscid case
when n ­ 0. But we will summarize, at the end, the
necessary changes for the case when 0 , n ø 1. The
potential V of the force is a random function
X Ω s1d
2pkx s1d
Ck cos
Bk std
V sx, td ­
L
jkj#k0
æ
2pkx s2d
s2d
1 Ck sin
(2)
Bk std .
L
s1d
s2d
Here Bk and Bk are identically distributed independent
white noises and L is the size of the system. We will
consider only periodic solutions of (1) with period L.
Asshxstdjdd ­
Z
0
2`
µ
It is clear that the statistical behavior of the solutions
s1d
depends on the decay properties of the coefficients hCk j
s2d
and hCk j. In this paper we deal with the simplest
case, when the forcing is limited to a finite number
of modes.
Our study of stationary correlation functions is based
on an idea that appeared earlier in Ref. [7]. We will
construct a statistically stationary functional of the forces
s1d
s2d
usx, td ­ CsshBk std, Bk std, t # tjdd such that u is a
solution of (1). The stationary distribution of (1) is then
s1d
s2d
given by the distribution of C0 ­ CsshBk std, Bk std,
t # 0jdd.
Our construction of C0 (and hence C by time translation) is based on the following idea. Given an initial data,
(1) as a hyperbolic differential equation can be solved
using the method of characteristics. The characteristics
satisfy Newton’s equation,
dx
­ y,
dt
dy
­ 2Vx sx, td .
dt
(3)
The solution u at sx, td is given by the velocity of the
characteristic which reaches x at time t. Of course the solution as well as the field of characteristics depends heavily on the initial data. To get the stationary distribution
of solutions, we need to select special initial data which
amounts to selecting a special field of characteristics.
This special class of characteristics is given by what we
call minimizers [8]. A curve hfxstd, tg, t # 0j is a minimizer if it minimizes the action,
(
æ∂
µ ∂
X
2pkx s1d
2pkx s2d
1 dx 2
s1d
s2d
Ck cos
1
Bk std 1 Ck sin
Bk std dt ,
2 dt
L
L
jkj#k0
(4)
with respect to arbitrary perturbations on finite time intervals. It is easy to see that minimizers satisfy Newton’s
equation for characteristics.
Using a limiting procedure for action-minimizing characteristics on increasingly large time intervals, we can
show that with probability 1, minimizers exist, i.e., for
each x, there exists y ­ ysxd such that the solution of
(3) with an initial value fx, ysxdg gives rise to a minimizer. Furthermore, two different minimizers never intersect in the past, i.e., if x1 std and x2 std are two different
minimizers, then x1 std fi x2 std for all t , 0. This remarkable property is a consequence of the randomness,
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© 1997 The American Physical Society
0031-9007y97y78(10)y1904(4)$10.00
VOLUME 78, NUMBER 10
PHYSICAL REVIEW LETTERS
and endows minimizers with an intrinsic meaning: The
minimizers are unique, except for a set E of countably
many points. This is because the minimizers can only
intersect at t ­ 0. If x0 is a point where two minimizers x1 std and x2 std intersect, then x1 and x2 enclose an
interval at t ­ 21. Intervals corresponding to different
x0 ’s do not intersect. Hence there can only be countably many points of intersection. Now the functional
C0 can be defined by the initial velocity of the minimizs1d
s2d
ers: usx, 0d ­ C0 s hBk std, Bk std, t # 0jdd ­ ysxd. This
is well defined everywhere, except on E which corresponds to the locations of shocks where y is discontinuous. At each location of shocks, there are at least two
minimizers corresponding to limits of velocities from the
left and right.
Another basic property of the minimizers is the hyperbolicity well known in the theory of dynamical systems.
Assume that usx, 0d given by C0 is continuous on an interval fx1 , x2 g. Hyperbolicity means that the minimizers
emanating from x1 and x2 converge exponentially fast to
each other in the past, i.e.,
jx1 std 2 x2 stdj # Cjx1 2 x2 je 2mjtj ,
where C is a random constant and m is the Lyapunov
exponent of the field of minimizers. Using the terminology
of the dynamical systems theory, one can say that each
continuous component of the solution usx, 0d is an unstable
manifold of any point on the graph s x, usx, 0ddd, x1 # x #
x2 . This implies that there are only finitely many shocks
at each time.
We now show how this construction can be used to study
the stationary probability distribution of velocity gradients
and velocity increments. The velocity gradient ≠uy≠x can
be represented as a sum of a continuous component ≠u1 y≠x
and a sum of delta functions
P representing contribution from
the shocks: ≠u2 y≠x ­ i wi dssx 2 xi stddd, where s xi std, tdd
is in D — the set of shocks in the sx, td plane. Since shocks
can only be created and can merge but never disappear, the
shock set D should basically have a spine structure with
a skeleton shock running from t ­ 2` to t ­ 1` and
newly created shocks forming finite ribs that eventually
merge with each other and with the skeleton shock.
Since we are concerned with stationary probabilities,
we can restrict ourselves to t ­ 0. First, consider the
probability Ph≠u1 y≠x . zj for large z. These are associated with steep ramps and are due to large fluctuations
of the force. To estimate this probability, let x1 , x2 be
close. For t , 0, the minimizers passing through sx1 , 0d
and sx2 , 0d satisfy
Z 0
Vx s x1 ssd, sddds ,
y1 std ­ y1 s0d 2
y2 std ­ y2 s0d 2
with y1 s0d , y2 s0d, and
Z
t
0
t
Vx s x2 ssd, sddds ,
x1 std ­ x1 1 y1 s0dt 2
x2 std ­ x2 1 y2 s0dt 2
10 MARCH 1997
Z
Z
0
dt
t
0
dt
Z
Z
t
0
t
0
t
Vx s x1 ssd, sdd ds ,
Vx s x2 ssd, sdd ds .
If fy2 s0d 2 y1 s0dgysx2 2 x1 d is large, in the absence of
forces these characteristics would intersect very quickly in
the past at time 2sx1 2 x2 dyfy1 s0d 2 y2 s0dg. However,
since they are minimizers, they cannot intersect. Therefore the action of the forces results in the inequality,
0 , x2 std 2 x1 std ­ x2 2 x1 1 s y2 s0d 2 y1 s0dddt
Z 0 Z 0
2
dt
fVx s x2 ssd, sdd
t
t
2 Vx s x1 ssd, sdd gds ,
i.e.,
y2 s0d 2 y1 s0d
t
x2 2 x 1
Z 0 Z 0 V s x ssd, sdd 2 V s x ssd, sdd
x 2
x 1
.
dt
ds
x2 2 x 1
t
t
Z 0 Z 0
x2 ssd 2 x1 ssd
­
dt
Vxx s x p ssd, sdd
ds .
x2 2 x 1
t
t
11
(5)
Take t , 21yz (which is the time that the curves would
intersect in the absence of forces), e.g., t ­ 23yz in
(5), then fy2 s0d 2 y1 s0dgysx2 2 x1 d is close to ≠uy≠x.
Therefore the left-hand side is negative with an absolute
value greater than 1. Thus
Ç
ÇZ 0
Z 0
x2 ssd 2 x1 ssd
p
dt
Vxx s x ssd, sdd
ds . 1 . (6)
x2 2 x 1
23yz
t
The ratio fx2 ssd 2 x1 ssdgysx2 2 x1 d is continuous and
bounded, and the distribution of the random variable
inside the absolute value sign in (6) is roughly the same as
the distribution of the integral of Brownian motion over an
1
interval of length t , z which is Gaussian with variance
proportional to z 23 . This gives the estimate
ΩZ 0
æ
Z 0
x2 ssd 2 x1 ssd
p
P
dt
Vxx s x ssd, sdd
ds . 1
x2 2 x1
23yz
t
, exph2const 3 z 3 j . (7)
The constant in the last expression is not universal and
depends on the details of the distribution of the force.
A similar result was obtained in [3,5] using an entirely
different argument.
Next, we consider the case when ≠uy≠x , z ø 21.
These are associated with preshock events, i.e., places on
the sx, td plane where shocks are generated. (See also
[9], where the concept of preshock events are discussed.)
Take such a point sx0 , t0 d; at sx0 , t0 d, ≠uy≠x ­ 2`. We
will describe in more detail the set of sx, td near sx0 , t0 d,
where ≠uy≠x , z.
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VOLUME 78, NUMBER 10
PHYSICAL REVIEW LETTERS
Generically, we have Du ­ usx, t0 d 2 usx0 , t0 d ­
Csx 2 x0 d1y3 near x0 , where C is a random constant. For
small jt 2 t0 j, the action of forces is negligible. Therefore the solution near sx0 , t0 d satisfies
usx, td ­ u0 s yd,
y ­ x 2 usx, td st 2 t0 d .
From (8) we have
≠u
du0 ydy
­
.
≠x
1 1 sdu0 ydyd st 2 t0 d
(8)
(9)
This shows that near sx0 , t0 d the set hsx, td, ≠uy≠x , zj
is a curvilinear region centered at sx0 , t0 d and bounded
by the curves x ­ x0 1 u0 sx0 d st 2 t0 d 6 f23zyCs1 1
zjt 2 t0 jdg23y2 7 Cf23zyCs1 1 zjt 2 t0 jdg21y2 st 2 t0 d.
The width of this region behaves as Osjzj23y2 d and
the height is roughly Osjzj21 d. Therefore its area is
approximately Ojzj25y2 .
The set of preshocks forms a stationary random point
field in the sx, td plane with density qdxdt. Stationarity means that q is independent of t, i.e., q ­ qsxd.
Hence we conclude
RL that Ph≠uy≠x , zj is proportional to
Qjzj25y2 , Q ­ 0 qsxddx. The density of this distribution decays as Qjzj27y2 .
We turn now to the probability distribution of w ­
DuyDx ­ fusx2 , td 2 usx1 , tdgyx2 2 x1 . For large positive values of w, the analysis remains essentially the same
and gives the superexponential behavior exph2const w 3 j.
For negative values the distribution of DuyDx remains
close to that of ≠uy≠x until some threshold value w p ,
after which it deviates due to the contribution from
existing small shocks. We can split this contribution
into two parts: one part due to shocks generated recently, and one part due to shocks generated in the
distant past. This means that, for Du ø 1, the density
of the distribution of the size of the shock inside an
interval sx, x 1 Dxd can be written as fsDu, Dxd ­
f1 sDu, Dxd 1 f2 sDu, Dxd, where f1 sDu, Dxd is the
probability density that a shock of size Du has appeared
at approximately t ­ DxyDu , 2sDud3 yDu ­ 2sDud2
in sx, x 1 Dxd. Since the size of the shock after its
first appearance grows as sDtd1y2 [10], the probability
of having shocks of size Du in an interval of length
Dx should be qDxDt ­ qDxsDud2 . Taking the derivative with respect to Du, we get the probability density
f1 ­ qDxDu.
f2 comes from solutions having shocks (of size Du)
which originated in the distant past and become weaker
due to fluctuations of the forces. For small Du, f2 should
have a powerlike behavior: f2 sDu, Dxd , sDxd sDudb .
So far, our analysis has yielded little specific information about the value of b, but there are indications
that b should depend on details of the distribution of
the force and hence is nonuniversal. Numerical work
is now going on to validate this assumption. Assuming this, we have psDuyDx ­ wddw ­ C1 sDxd3 wdw 1
C2 sDxdb12 w b dw.
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10 MARCH 1997
Case 1. b . 1. In this case, the first term dominates.
The crossover (the value of w p ) can be found from
w 27y2 , sDxd3 w which gives w p , sDxd22y3 .
Case 2. b # 1. In this case, the second term dominates and w p , sDxd2s2b14dys2b17d .
The overall picture for the probability distribution of
≠uy≠x and DuyDx is summarized in Fig. 1. We end this
paper with several remarks.
(1) The asymptotic behavior of psDuyDxd found above
implies intermittency, i.e.,
Ω
sDxdn , for n # 1 ,
n
ksDud l ,
Dx,
for n . 1 ,
i.e., the behavior is bifractal.
(2) Our calculation of the probability distribution of w
is valid only for the regime jDuj ø 1. When jDuj ø
1, the behavior is again superexponential and can be
estimated with the same exponent as was used earlier.
(3) In the presence of a finite viscosity we can show
that the stationary probability distributions converge to
that of the inviscid ones as viscosity goes to zero [8].
Therefore, for n ø 1, the picture presented in Fig. 1
remains valid for z ¿ 1 and 2n 21 ø z ø 21. For
z ­ Osn 21 d, viscous corrections to shock profiles have
to be taken into account.
(4) Our results imply seemingly erroneous result that
ks≠uy≠xd2 l , `. This is because we avoid contributions
from the neighborhood of shocks. In other words, we
used ks≠uy≠xd2 l ­ limR!1` limn!0 ks≠uy≠xd2 ln,R , where
ks≠uy≠xd2 ln,R is computed for nonzero viscosity n by averaging s≠uy≠xd2 over realizations such that j≠uy≠xj , R.
We thank a. Polyakov, V. Yakhot, V. Lebedev,
A. Cheklov, S. Chen, and U. Frisch for helpful discussions. The work of E was supported in part by the
Research Grant Council of Hong Kong. The work of
K. K. and A. M. was supported by RFFR under Grant
FIG. 1. Probability density function for velocity gradients and
velocity increments.
VOLUME 78, NUMBER 10
PHYSICAL REVIEW LETTERS
No. 96-01-00377. the work of Y. S. was supported in
part by NSF through Grant No. DMS-9404437.
[1] J. Krug and H. Spohn in Solids Far from Equilibrium,
edited by C. Godreche (Cambridge University Press,
Cambridge, England, 1992).
[2] A. Cheklov and V. Yakhot, Phys. Rev. E 51, R2739
(1995); 52, 5681 (1995).
[3] A. Polyakov, Phys. Rev. E 52, 6183 (1995).
10 MARCH 1997
[4] J. P. Bouchaud, M. Mezard, and G. Parisi, Phys. Rev. E
52, 3656 (1995).
[5] V. Gurairie and A. Migdal, Phys. Rev. E (to be published).
[6] E. Balkovsky, G. Falkovich, I. Kolokolov, and
V. Lebedev, Phys. Rev. Lett. (to be published).
[7] Ya. Sinai, J. Stat. Phys. 64, 1–10 (1994).
[8] Weinan E, Konstantin Khanin, Alexandre Mazel, and
Yakov Sinai (to be published).
[9] J. D. Fournier and U. Frisch, J. Mec. Theor. Appl. 2, 699 –
750 (1983).
[10] This follows from pure kinematic calculations. Du ,
sDxd1y3 , Dx ­ DuDt. This implies that Du , sDtd1y2 .
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