Table Of ContentIntegrable Structure of Interface Dynamics
Mark Mineev-Weinstein
Theoretical Division, MS-B213, LANL, Los Alamos, NM 87545, USA
Paul B. Wiegmann
James Franck Institute and Enrico Fermi Institute of the University of Chicago
5640 S.Ellis Avenue, Chicago, IL 60637, USA and Landau Institute for Theoretical Physics
Anton Zabrodin
Joint Institute of Chemical Physics, Kosygina str. 4, 117334, Moscow, Russia
and ITEP, 117259, Moscow, Russia
0
0
0 erty: The harmonic moments of the oil domain Ck =
2 We establish the equivalence of 2D contour dynamics to z−kdxdy (k = 1,2,..., and z = x+iy)1 do not
n thedispersionless limit of theintegrable Toda hierarchy con- RchDa−n(tg)e in time, while the area of water domain, grows
strained by a string equation. Remarkably, the same hierar-
a linearly in time [3]. The proof:
chy underlies 2D quantumgravity.
J
0 PACS number(s):02.30.Em, 02.30,.Dk, 68.10.-m, 68.70.+w, dCk = VndC = (p∂ z−k z−k∂ p)d ,
1 04.60.-m dt IC(t) zk IC(t) n − n C
]
because V = ∂ p andp=0 along the (t) . By virtue
I n n
S of the Gauss’ t−heorem, it equals C
1. Laplacian growth. Contour dynamics takes place
.
n in many physical processes, where an interface moves
nli binettewrfeaecne tdwyonaimmimcsistcoibliellupsthraastees.theThmeaikneyreesuxlatmopflethoisf =ZD−(t)∇(p∇z−k−z−k∇p)dxdy =−qδk,0.
[
work is the Laplacian growth (LG) [1]. This process This property may be used as the definition of the ideal-
2 is dissipative, unstable, ubiquitous (applications range ized LG problem:
v
from oil/gas recovery to tumor growth), and universal:
7
a steadyself-similarpatternappearsgovernedbyscaling – Tofindtheformofthedomainwhoseareaincreases
0
0 laws, most of which still yet to be derived [1,2]. while all harmonic moments remain fixed.
1 In this paper we show that an arbitrary interface dy-
0 namics has an integrable structure which is the same Thisproblemisknowntobeill-defined[1]. Foralmost
0 all sets of harmonic moments, the boundary develops
as the one that underlies models of 2D quantum grav-
0 cusp-likesingularitiesinfinitetime(area)[1]. Onceasin-
ity. This structure links the interface dynamics, and es-
/
n pecially LG, with other branches of theoretical physics, gularityoccurs,theidealizedLGmodelisnolongervalid.
i Surfacetension,omittedabove,stabilizesthegrowthand
l where scaling laws are also expected [17].
n simultaneously ruins the conservation of harmonic mo-
2. To be specific, we will speak about Hele-Shaw flow
:
v [1]: a viscous fluid (oil) and a non-viscous fluid (water) ments. Simulations and experiments show that differ-
i entmechanismsofregularizationofsingularities(surface
X areconfinedinanarrowgapbetweentwoparallelplates.
tension,latticeetc.) exhibitthesameself-similarpattern
Theinteriorwaterdomain,D ,issurroundedbyanexte-
r +
[1,2]. This suggestsa fixedpoint (or points)in the space
a rioroildomain,D ,occupyingtherestoftheplane. Wa-
−
ofharmonicmoments,whichcorrespondtoobservedsta-
ter is supplied from the origin and pushes the oil/water
blepatterns. Toidentifythefixedpointsandtheirscaling
interface, (t). Both liquids are incompressible, so oil
C properties is the challenge of the growth phenomena.
is extracted at infinity at the same rate q as water is
Approachto a fixed point requires a change of all mo-
supplied.
ments. This is the question we address in this paper.
– the normal velocity of the interface is V = ∂ p We present the set of differential equations which de-
n n
−
(the D’Arcy law); the pressure p is kept constant scribetheevolutionofthedomainunderavariationofall
(p=0) inside the water domain D (t) and on the harmonic moments. This prompts to a connection with
+
interface(surfacetensionandviscosityofthewater the inversepotential problem [4]: to restorethe shape of
areneglected);andpressureisaharmonicfunction,
2p = 0, inside the oil domain D (t), while p
−
∇ →
(q/2π)log x2+y2 at infinity.
− p 1C1 andC2 arefinite: thedivergenceas|z|→∞cancelsby
This (idealized) LG problem has an important prop- integration over argz.
1
a body from a given Newtonian potential of a uniform resolve the unitary condition for the Schwarz function
mass distribution inside the body. andgiveitthefollowinginterpretation. Ifw istheimage
It remains to be seen whether these equations help of a point z, then S(z) is the complex conjugate pre-
to describe the pattern of growth; however, they reveal image of w−1: S(z)=z¯(w−1(z)).
the integrable structure of the growth problems. We 4. The idealized LG problem has an instructive form
will show that the equations describing the evolution of in terms of the Schwarz function:
a domain form an integrable hierarchy. Moreover, the q
very same hierarchy emerges in c = 1 string theory and ∂tS = ∂zlogw. (4)
π
topological gravity [5], and in 2-matrix models [6]. It
is the dispersionless limit of the 2D Toda hierarchy [7] Toderive(4)(following[10]),wedifferentiatez¯(t,w−1)=
constrained by the so-called string equation [8]. S(t,z(t,w)). We get ∂tz¯(t,w−1) = ∂tS(t,z) +
3. Toproceedfurtherweneedsomeknownfactsabout ∂zS(t,z)∂tz(t,w) and, by virtue of (b), Vn =Im(z¯tzl)=
the Schwarz function. (See e.g., [9]). An equation for a St/(2i√Sz). From logw(z) = 2πp(x,y)/q +iφ(x,y)
−
curve, FC(x,y) = FC(z+2z¯,z2−iz¯) = 0, can be resolved (at and p = const along C(t), we conclude that −∂np =
least locally) with respect to one of the complex vari- qwz/(2πiw√Sz). Since Vn = ∂np, we obtain(4). From
−
ables, say z¯=x iy. The result, z¯=S(z), is called the now on we set q =π by making a proper time rescaling.
−
Schwarz function of the curve (see e.g., [9]): The equation (4) written in terms of the conformal
(a) S(z) is a unitary operationC: S¯(S(z))=z; maps (2), (3) has the form [1]:
(b) The unit vector tangential to the curve is dz/dl =
z(t,w), z¯(t,w−1) =1, (5)
dz/√dzdz¯=(dz¯(z)/dz)−1/2 =1/√Sz = S¯z¯; { }
(c) For simple analytic curves, the Schwaprz function can
where we define the Poisson bracket by f, g
be analytically continued to some strip-like domain con- { } ≡
w(∂ f∂ g ∂ f∂ g). Oncomparingpowersofw forboth
w t t w
taining the curve. −
sides of (5), we get a set of equations for the coefficients
The function S(z) can be decomposed into a sum of
u ,u¯ of the Laurent series, (2) and (3), with fixed C .
two functions S(±)(z) that are regular in D : S(z) = k k k
± Eq.(5) suggests the Hamiltonian structure of the prob-
S(+)(z)+ S(−)(z). Under the condition S(−)( ) = 0 lem: z(t,w), z¯(t,w−1) and logw, t are canonical pairs.
∞
this decomposition is unique. The functions S(±)(z) can
5. Consider the function Ω(z) defined on the curve by
be represented by a Taylor series convergent near the
origin (which is assumed to be in D ) and near infinity z 2
+
in D−: S(+)(z)= ∞k=0Skzk, S(−)(z) = ∞k=1S−kz−k, Ω(z)= |2| +2iA(z), z ∈C, (6)
The coefficients S Pare nothing but harmPonic moments
±k
where we have separated the real and imaginary parts.
of the exterior, D , and the interior, D , domains:
− +
Here A(z) is the area of the sector enclosed by and
C
S(z)dz bounded by the ray argz and some fixed reference ray.
C = z∓kdxdy= =πS . (1)
±k ∓Z I 2iz±k ±k−1 Just like the Schwarz function, this function can be an-
D∓ C(t) alytically continued within a strip containing the curve.
In other words, S(±)(z) is the gradient of the Newto- Indeed, writing (6) in terms of S(z), we have
nian potential created by matter uniformly distributed
z z
in the interior (exterior) of . In these terms the ideal-
C Ω(z)=zS(z)/2+2i [S(z′)dz′ z′dS(z′)]/4i= S(z′)dz′
izedLGproblemimplies thatS(+) does notvaryintime Z − Z
and πS−1 =(Area ofD+) grows linearly in time. ∞ ∞
The Schwarz function is closely related to conformal = C zk/π+tlogz v /2 C z−k/π, (7)
k 0 −k
− −
maps. Letφ(x,y)bethefunctionharmonicallyconjugate kX=1 Xk=1
to 2πp(x,y)/q. Then w = e−2πp/q+iφ univalently maps
where v does not depend on z. As is seen from the
the oil domain to the exterior of the unit circle. This 0
above definition, Ω is defined up to a purely imaginary
map sends w = to z = . Let us write
∞ ∞ z-independentterm. We fix it by requiringv0 to be real.
∞ ThefunctionΩisthegeneratingfunctionofthecanon-
z(t,w)=r(t)w+ uk(t)w−k (2) icaltransformation(logw, t) (z, z¯). Indeed,from(7)
X →
k=0 wehaveS(z)=∂zΩ(z)and,byvirtueof(4),logw =∂tΩ.
Therefore,thedifferentialdΩ=S(z)dz+logwdtencodes
for the inverse map, where r is chosen to be real, so
the LG equations (4) or (5).
the map z(w) is unique. This map and the map to the
complex conjugate domain D¯ , 6. Now we extend the differential dΩ to include varia-
−
tions of all higher moments. For k 1, let us denote
≥
∞
z¯(t,w−1)=r(t)w−1+ u¯k(t)wk, (3) t = Ck, H = ∂Ω(z), H¯ = ∂Ω(z). (8)
Xk=0 k πk k ∂tk k − ∂t¯k
2
Then the multi-time Hamiltonian system is defined as where z belongs to the curve. Eq.(11) follows by virtue
of (8). Now we see that H and H¯ (defined as in (8))
∞ k k
dΩ=S(z)dz+logwdt+ (H dt H¯ dt¯ ). are indeed complex conjugates on the curve.
k k k k
− Eqs.(9) or alternatively (10) together with (14), (2),
X
k=1
and(5)provideanalgorithmgeneratingequationsforthe
Thus, the flows with respect to the “times” tk are coefficients of the conformal map (2). Two first Hamil-
tonians are H =rw+u /2, and H =r2w2+2ru w+
∂kz ={z,Hk}, ∂k¯z ={z,H¯k}, (9) ru1+u20/2. T1he first equ0ation of th2e hierarchy is 0
where ∂k = ∂/∂tk, ∂k¯ = ∂/∂t¯k, and {,} is the canoni- ∂12¯1ϕ=∂texp(∂tϕ), (17)
cal Poisson bracket introduced above. These equations
are consistent due the symmetry relations ∂H (z) = wherer2 =exp(∂ ϕ). Onecanseefrom∂ Ω=logw that
l k t t
∂ H (z) whichfollow from (8). In terms of w, these con- ϕ is the constant term in the expansion (7): ϕ=v .
k l 0
ditions have the form of the zero-curvature equations: 7. The unitarity condition (a) for the Schwarz func-
tion and the properties (6), (16) of the generating func-
∂kHl(w) ∂lHk(w)= Hk(w),Hl(w) . (10) tion Ω (which actually follow from the unitarity) impose
− { }
importantrelationsamongtheharmonicmomentsofany
We now proceed to calculate the Hamiltonians. Below
smooth simply connected domain and the harmonic mo-
we will prove that in addition to (8), for z on the curve,
ments of its complement. First, from (6) one can derive
the Hamiltonians can be equivalently defined as
the following sum rules:
H = ∂ Ω¯(z¯). (11)
k − k ktkvk = kt¯kv¯k, v¯1=tt1+ ktkvk−1. (18)
X X X
(The derivative is taken at fixed z¯.) Then (7) gives us k≥1 k≥1 k≥2
∞ Second, there are symmetry relations for derivatives of
Hk =zk ∂kv0/2 ∂kvlz−l/l = (12) the harmonic moments vk = C−k/π of the interior do-
− −
Xl=1 main D+ with respect to the (rescaled) harmonic mo-
∞ ments tj of the exterior domain: ∂jvk = ∂kvj, ∂¯jvk =
=∂kv0/2+ ∂kv¯lz¯−l/l, (13) ∂kv¯j, ∂tvk = ∂kv0. The proof: it follows from (13) that
X
l=1 H dH =0 for all j,k, then it is easy to see that
C j k
H
where we setv =C /π. Eq.(12)implies thatthe Lau-
k −k
rent expansion of the Hk at w = 0 does not contain ∂jvk =I zkdHj/2πi=I zjdHk/2πi=∂kvj.
powers of w higher than wk. Moreover, all non-negative C C
powersofw comefromthe firsttwoterms ofEq.(12). In This implies that for each analytic curve (t,t ) there
j
C
turn, Eqs.(13) and (3) imply that Hk does not contain exists a real function (prepotential) F(t,tj,t¯j) such that
negative powers of w. Altogether they mean that H is
k
a polynomial in w of degree k. It reads vj =∂jF, v¯j =∂¯jF , v0 =∂tF . (19)
1 This function determines H (z) via (12).
H (w)=(zk(w)) + (zk(w)) . (14) k
k + 2 0 8. Equations (9), (14) are familiar in the soliton lit-
erature as the dispersionless limit [7] of the 2D Toda hi-
The symbol (f(w)) means a truncated Laurent series,
+ erarchy. Dispersionless hierarchies of this kind [11] are
where only terms with positive powers of w are kept;
extensions of the integrable equations of hydrodynamic
(f(w)) is the constant (w0) part of the series.
0 type [12]to the multidimensional case. Many special so-
It remains to prove Eq.(11). We first notice that
lutions were found in [13]. Eq.(5) is knownas the string
∂ ReΩ(z) = 0 if z belongs to the curve. This prop-
j equation[8]. (See also[14]for morerecentdevelopments
erty is proved by differentiating the real part of (6),
in the 2D Toda hierarchy.) To make the contact we now
Ω(z)+Ω¯(z¯)= z 2. Theanalyticcontinuationawayfrom
the curvegives|Ω|(z)+Ω¯(S(z))=zS(z). Takingthe par- review, following [7], the standard setup of the 2D Toda
hierarchy and its dispersionless limit.
tialderivativewithrespecttot andrestrictingtheresult
j The2DTodahierarchyisusuallyintroducedbymeans
to the curve again, we get:
of two difference Lax operators:
∂kΩ(z)+∂kΩ¯(z¯)+∂kS(z)∂z¯Ω¯(z¯)=z∂kS(z). (15) ∞
L=r(t)eh¯∂t + u (t)e−kh¯∂t,
k
But the r.h.s. and the last term in the l.h.s. of (15) are X
k=1
equal since z =S¯(z¯)=∂z¯Ω¯(z¯). Thus (15) reads ∞
L¯ =r(t ¯h)e−h¯∂t + u¯ (t)e−kh¯∂t, (20)
∂k[Ω(z)+Ω¯(z¯)]=0, (16) − Xk=1 k
3
where r, u and u¯ are functions of t and of two sets of 10. To summarize, comparing the semiclassical limit
k k
independent parameters t , t¯ , k > 0. These functions ofthe TodaEqs.(20),(21),andthe stringequation,with
k k
obey the Lax-Sato equations: Eqs.(2), (3), (9), (14), (5) of an arbitrary interface dy-
namics,wefindanexactequivalencebetweenthem. This
¯h∂tkL=[L,Hk], where Hk =(Lk)++(Lk)0/2, isthemainresultofthiswork. Thesumrules(18)forthe
¯h∂t¯kL¯ =[H¯k,L], where H¯k =(L¯k)−+(L¯k)0/2. (21) harmonic moments are nothing else but (a part of) the
W-constraints for the τ-function. It also follows from
The symbol (Lk)± means the part of the operator that above that the interface dynamics is equivalent to the
consistsofpositive(negative)powersoftheshiftoperator N planar limit of the matrix model (24): the (log-
eh¯∂t, and (Lk)0 is the part that does not contain the arit→hm∞of) the partition function of the latter is the pre-
shift operator. The first equation of the hierarchy is the potential function F (19).
familiar 2D Toda equation: Itistemptingtounderstandwhatdotherobustscaling
behaviorobservedinvarietyofgrowthproblems[2]anda
∂2 ϕ(t)=eϕ(t+h¯)−ϕ(t) eϕ(t)−ϕ(t−h¯), (22)
1¯1 − scalingbehaviorof2Dgravity[17],[18],haveincommon.
We acknowledge useful discussions with B. Dubrovin,
where r2(t) = eϕ(t+h¯)−ϕ(t). It is also customary to con-
G. Doolen, J. Gibbons, L. Kadanoff, V. Kazakov,I. Kri-
sider the Orlov-Shulmanoperators [15]
chever, L. Levitov, S. Novikov, A. Orlov, B. Shraiman
∞ ∞ andT.Takebe. P.W.wouldliketothanktheLadyDavis
M = kt Lk+t+ v L−k, foundation for the hospitality in Hebrew University in
k k
X X Jerusalem, where this work has been completed. P. W.
k=1 k=2
∞ ∞ wassupportedbygrantsNSFDMR9971332andMRSEC
M¯ = kt¯kL¯k+t+ v¯kL¯−k, (23) NSF DMR 9808595. The work of A. Z. was partially
X X
k=1 k=2 supported RFBR grant 98-01-00344.
where v = v (t,t ,t¯), v¯ = v¯ (t,t ,t¯) are functions
k k j j k k j j
such that the operators obey the conditions [L,M] =
¯hL, [L¯,M¯] = ¯hL¯. These operators satisfy the follow-
−
ing linear equations: LΨ = zΨ, ∂ Ψ = H Ψ, h¯z∂ Ψ =
k k z
MΨ, and similarly for the bar-operators acting on Ψ¯.
9. One particular solution of the 2D Toda hierarchy [1] D.Bensimonetal,Rev.Mod.Phys58,977(1986);“Dy-
describes 2-matrix models. Consider the integral over namics of Curved Fronts” ed. by P. Pelce (Academic
two hermitian N N matrices Press, Inc., Boston, 1988); for recent advances in this
×
problem related to this work see M. Mineev-Weinstein,
τ = eNtr(MM¯+ k>0(tkMk+t¯kM¯k))dMdM¯ (24) Phys.Rev.Lett. 80, 2113 (1998); M. Mineev-Weinstein
Z P and O. Kupervasser, patt-sol/9902007, M. Feigenbaum,
I. Procaccia, and B. Davidovich, J. Stat. Physics to ap-
(thepartitionfunction). Ithasbeenshownthatthisinte- pear, chao-dyn/9908007
gralistheτ-functionforaspecialsolutionofthe2DToda [2] L. M. Sander et al., Phys. Rev. A 32, 3160 (1985),
hierarchy with h¯ =1/N [16]. The solution is selected by T. W. Witten and L. M. Sander, Phys. Rev. Lett. 47,
the string equation [L,L¯]=h¯. The coefficients v of the 1400 (1981)
k
M-operator are given by v =∂ logτ =<Mk >, where [3] S. Richardson, J. Fluid Mech. 56 (1972) 609
k k [4] P. S. Novikov, Soviet Math. Dokl. 18, 165 (1938);
< > means an average over matrices with the weight
··· M. Sakai, Proc. Amer. Math. Soc. 70, 35 (1978); on
(24). A scaling behavior of a proper largeN limit of the
the links with the LG problem see e.g. P. Etingof and
matrix model is expected to describe 2D gravity [17].
A.Varchenko,UniversityLectureSeries, 3, AMS,Provi-
The dispersionless hierarchy is obtained in the limit
dence, RI,1992
¯h 0. Inthislimit,theshiftoperatoreh¯∂t isreplacedby [5] R. Dijkgraaf et al., Nucl.Phys. B394, 356 (1993);
→
a classical variable w, the Lax operators are substituted A. Hanany et al., ibid. B425, 150 (1994); K. Takasaki,
by their eigenvalues L z(w) and L¯ z¯(w−1), and Commun. Math. Phys. 170, 101 (1995); T. Eguchi and
→ →
the operator, L−1M, becomes a function S(z). At the H. Kanno, Phys.Lett. 331B, 330 (1994)
same time all commutators are replaced by the Poisson [6] J. M. Daul, V.A.Kazakov and I.K.Kostov, Nucl. Phys.
brackets with the symplectic structure w,t = w, so B409, 311 (1993); L. Bonora and C. S. Xiong, Phys.
[L,L¯] = h¯ turns into (5). Eq.(17) is the{h¯ }0 limit of Lett. B347, 41 (1995)
→ [7] For a comprehensive review, see K. Takasaki and
Toda’sequation(22). ThewavefunctionΨisreplacedby
eΩ/h¯,whereΩisthe generatingfunctionofthe canonical T. Takebe, Rev.Math. Phys. 7, 743 (1995)
[8] M. R.Douglas, Phys. Lett. B238, 176 (1990)
transformation(logw,t) (z,z¯). Atlast,theτ-function
is τ = eF/h¯2 as ¯h 0→, where the function F is the [9] P.Davis,TheSchwarzfunctionanditsapplications.The
Carus Math. Monographs, No.17, MAA, Buffalo (1994)
→
prepotential introduced in (19).
4
[10] S.D. Howison, Euro. J. Appl. Math. 3, 209 (1992)
[11] I. M. Krichever, Comm. Pure. Appl. Math. 47, 437
(1994)
[12] B. A. Dubrovin and S. P. Novikov, Soviet Math. Dokl.
27, 665 (1983); S. P.Tsarev, ibid..31, 488 (1985)
[13] Y. Codama and J. Gibbons, Phys. Lett. 135 A, 167
(1989)
[14] M. Adler and P. van Moerbeke, Commun. Pure Appl.
Math. 50, 241 (1997)
[15] A. Orlov and E. Shulman, Lett. Math. Phys. 12, 171
(1986)
[16] A.Gerasimov et al, Nucl. Phys. B357, 565 (1991).
[17] Forareview,seee.g.P.GinspargandG.Moore,Lectures
on 2D gravity and 2D string theory, Lectures given at
TASIsummer school, 1992, hep-th/9304011
[18] E.Brezin et al., Phys.Lett. B237,43 (1990); D.J. Gross
and A. A.Migdal Phys. Rev.Lett. 64, 717 (1990)
5
