Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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at a detailed leve l with respect to radio wave s, the

surprises from gravitational waves used as tools to

view the universe are potentially even greater.

See also: Asymptotic Structure and Conformal

Infinity; Computational Methods in General Relativity:

The Theory; General Relativity: Experimental Tests;

General Relativity: Overview.

Further Reading

Bartusiak M (2003) Einstein’s Unfinished Symphony: Listening to

the Sounds of Space-Time. New York: Berkley Publishing Group.

Cutler C and Thorne K (2002) An overview of gravitational wave

sources, Arxiv.org:gr-qc/0204090.

Douglass D and Braginsky V (1979) Gravitational radiation

experiments. In: Hawking S and Israel W (eds.) General

Relativity: An Einstein Centenary Survey. Cambridge:

Cambridge University Press.

Kennefick D (1997) Controversies in the history of the radiation

reaction problem in general relativity, Arxiv.org:gr-qc/

9704002.

Saulson P (1994) Fundamentals of Interferometric Gravitational

Wave Detectors.Singapore: World Scientific.

Schutz B (1984) Gravitational waves on the back of the envelope.

American Journal of Physics 52: 412.

Taylor J (1994) Nobel lecture: binary pulsars and relativistic

gravity. Reviews of Modern Physics 66: 711.

Thorne K (1987) Gravitational radiation. In: Hawking S and

Israel W (eds.) 300 Years of Gravitation. Cambridge:

Cambridge University Press.

Thorne K (1995a) Gravitational waves, Arxiv.org:gr-qc/9506086.

Thorne K (1995b) Black Holes and Time Warps: Einstein’s

Outrageous Legacy. New York: W W Norton.

Throne K (1997) Gravitational radiation: a new window on the

universe, Arxiv.org:gr-qc/9704042.

The LIGO project web page, which includes links to all the other

experimental efforts: http://www.ligo.caltech.edu.

Growth Processes in Random Matrix Theory

K Johansson, Kungl Tekniska Ho

gskolan, Stockholm,

Sweden

Introduction

Probability distributions coming from random matrix

theory (RMT), RMT laws, occur in different contexts,

notably in quantum physics and in number theory.

RMT laws are also seen in certain local random growth

models and related problems in discrete probability,

random permutations, exclusion processes, and ran-

dom tilings (dimer models). In these models limit laws

for height/shape fluctuations are given by limit laws

from RMT, in particular the largest eigenvalue or

Tracy–Widom distributions. These models belong to

the Kardar–Parisi–Zhang (KPZ) universality class.

Models in this class have two universal exponents, 1/

3 describing the interface fluctuations and 2/3 describ-

ing the correlations in the transversal direction. By a

local random growth model, we mean a model where

the random growth mechanism is local in that it does

not depend on the global geometry as in diffusion

limited aggregation (DLA). Typically there is also some

smoothing mechanism. The connection with RMT can

only be established for special exactly solvable models.

Below we discuss a basic model based on a last-passage

percolation problem, which translates into a poly-

nuclear growth (PNG) process. Other models that can

be treated are in a sense variations of this model. Point

processes with determinantal correlation functions

play a central role in RMT and in the analysis of the

basic model and we start by discussing these. The basic

model has several different interpretations that will be

outlined. Another basic tool, which can be formulated

in different ways, is the Robinson–Schensted–Knuth

(RSK) correspondence well known in combinatorics.

One approach is in terms of nonintersecting paths

which translates into a multilayer PNG process. Limit

theorems can be formulated for the height above a

fixed location, and also for the whole height function in

terms of the Airy process which extends the Hermitian

Tracy–Widom distribution F

. It is expected that

several results should generalize to a broader class of

models. There is a natural universality problem of

extending the validity of the RMT laws.

Determinantal Processes

Point processes with determinantal correlation func-

tions play an important role in the exactly solvable

models. We consider probability measures on



,   R, of the form

detð

ðx

ÞÞ

i; j¼1

det ð

ðx

ÞÞ

i; j¼1

ðxÞ½1

which can be thought of as describing random points

in  at positions x

, ..., x

.Here, is a reference

measure on , for example, Lebesgue or counting

measure, Z

a normalization constant, and 

given

functions. A measure of this form has determinantal

correlation functions in the sense that the density, with

respect to d

(y), of particles at y

, ..., y

ðy

; ...; y

Þ¼detðK

ðy

; y

ÞÞ

i; j¼1

½2

There is an explicit formula for the correlation

kernel K

in terms of the functions 

586 Growth Processes in Random Matrix Theory

The eigenvalue measures in the basic random

matrix ensembles have the form

j

ðxÞj



j¼1

wðx

Þd

ðxÞ½3

where 

(x) = det(x

i1

)

i, j= 1

is Vandermonde’s

determinant, x 2 

and x

, ..., x

are the eigen-

values. For the Gaussian unitary ensemble (GUE

1

exp(tr M

)dM of n  n Hermitian matrices M,

we have  = 2, =R, w(x) = exp(x

) and  the

Lebesgue measure. For the Laguerre unitary ensem-

ble (LUE

n, 

) of complex covariance matrices, M



where M is an (n þ )  n-matrix with standar d

complex Gaussian elements, we have w(x) =



x

,   0,  = 2, =[0, 1) and  the Lebesgue

measure. The  = 2 case of [3] can be put into the

form [1] and hence has determinantal correlation

functions. In this case the correlation kernel can be

expressed in terms of the normalized orthogonal

polynomials p

(x) with respect to w(x)d(x)on.

Because of this when  = 2 the ensemble [3] is

referred to as an orthogonal polynomial ensemble

(OPE). The kernel is given by

ðx; yÞ¼

n1

k¼0

ðxÞp

ðyÞðwðxÞwðyÞÞ

1=2

½4

A consequence of [2] is that the probability of

finding no particle in a set J   is given by a

Fredholm determinant,

P½no particle in J¼detðI  K

ðJ;Þ

½5

In particular the distribution function F()ofthelargest

eigenvalue or rightmost particle x

max

= max

1jn

in an OPE is given by [5] with K

as in [4] and

J = (, 1).

A Basic Model

Let (w(i, j))

(i, j)2Z

be independent geometric random

variables with parameter a

P½wði; jÞ¼k¼ð1  a

Þða

½6

k  0 and 0  a

< 1. As a limiting case we can

obtain exponential random variables. Consider the

last-passage time

GðM; NÞ¼max



ði;jÞ2

wði; j Þ½7

where the maximum is over all up/right paths 

from (1, 1) to (M, N), that is,  = {(i

, j

), ...,(i

, j

)}

with (i

kþ1

, j

kþ1

)  (i

, j

) = (1, 0) or (0, 1), (i

, j

) =

(1, 1) and (i

, j

) = (M, N), m = M þ N  1. We can

also think of this as a zero- temperature directed

polymer, by thinking of the w(i, j) : s as (minus)

energies and  as random walk paths.

As will be explained in some more detail below,

if the w(i, j) : s are exponential with mean 1 and

M  N,thenG(M, N) = 

max

in distribution, M  N,

where 

max

is the largest eigenvalue in LUE

N, MN

Hence, in this case G(M, N) behaves exactly like a

largest eigenvalue. If the w(i, j):s are geometric with

parameter q,thenG(M, N) has the same distribution

as the rightmost particle in an OPE, namely [3],

with  = 2, w(x) =



MNþx



and  the counting

measure on =N, called the Meixner ensemble.

Since in this case the relevant orthogonal polynomials

are discrete the ensemble is referred to as a discrete

OPE.

The random variables {G(M, N)}

(M, N)2Z

have two

interpretations related to random growth. It follows

from [7] that

GðM; NÞ

¼maxðGðM  1; NÞ; GðM; N  1ÞÞ

þ wðM; NÞ½8

This can be thought of as a growth rule. We change

variables by letting G(M, N) = h(M  N, M þ N  1)

and w(M,N) = !(M N,M þN 1) with w(M,N)=0

if (M,N) 62Z

. Then

hðx; t þ 1Þ

¼ maxðhðx  1; tÞ; hðx; tÞ; hðx þ 1; tÞÞ

þ !ðx; tÞ½9

x  Z, t  N, h(x,0) 0, and !(x, t) = 0ifjxjt or if

x  t is even. We can extend it to the whole real line

by letting h(x, t ) = h([x], t). The growth rule [9] is a

discrete polynuclear growth (PNG) model. Up-steps

in the inte rface, x ! h(x, t

) (see the top curve in

Figure 1), move at unit speed to the left and down-

steps move at unit speed to the right and they merge

at collision. On top of this smoothing mechanism,

we have random deposition given by !(x, t). Look-

ing at the definition of !, we see that all deposition

up to time t is on top of a basic layer (t, t). The

asymptotic shape will look like a droplet and this

setting of PNG is called the droplet geometry. We

see that height fluctuations are directly related to

fluctuations of G(M, N).

We get another growth model, the corner growth

model, somewhat similar to the classical Eden

growth model, by considering the random shape

(see Figure 2),

ðnÞ¼fðM; NÞ2Z

; GðM; NÞþM þ N  1  ng

þ½1; 0

½10

Growth Processes in Random Matrix Theory 587

The complement of this set in Z

has a boundary

B(n) which we can think of as an interface. By [8]

and the lack of memory property of the geometric/

exponential distribution, the region (n) grows by

adding new squares independently at each corner of

B(n) with geometric/exponential waiting times (see

Figure 2). If we look at B(n) in a coordinate system

with M = N as vertical axis and write a 1 for every

unit down-step on B(n) and a 0 for every unit up-

step (see Figure 2), the corner growth dynamics

translates into the totally asymmetric simple exclu-

sion process (TASEP), in discrete or continuous

time, with initial configuration ...1111000 ... .As

shown by Jockush, Propp, and Shor, (n) also

occurs in a uniform random domino tiling of a

region called the Aztec diamond (see Figure 3). The

shape (n), when q = 1=2, has the same law as the

completely regular (frozen) North Polar Region

(NPR) in the tiling and hence the boundary

fluctuations of the NPR are related to the fluctua-

tions of G(M, N). The NPR in Figure 3 has the same

shape as (5) in Figure 2. This connects the models

considered here with dimer or tiling problems in

two-dimensional equilibrium statistical mechanics.

Consider a (Poissonized) random permutation 

from S

, where N is a Poisson() random variable.

Let L() denote the length of the longes t increasing

subsequence in , for example,  = 316452 has

L = 3. By thinking of the representation of a

permutation by its permutation matrix, we see that

G(N, N) with w(i, j) geometric with parameter

q = =N

converges to L() in distribution as

N !1. We call this limit the Poisson limit. Taking

this limit in the PNG process yields the Pra¨ hofer–

Spohn continuous time PNG (cont-PNG) model,

which is similar to the discrete PNG defined above

but where all steps have unit size and we have

continuous time dynamics with deposition events

according to a two-dimensional spacetime Poisson

process. The study of L(), and its de-Poissonization

when N is nonrandom, is known as Ulam’s problem

in combinatorial probability.

The RSK Correspondence

The mapping of the last-passage problem [7] into a

determinantal proces s is based on the RSK corre-

spondence. This correspondence maps the integer

matrix (w(i, j))

1i, jM

bijectively to a pair of semi-

standard Young tableaux (P, Q) with common

shape , which is a partition  = (

, 

, ...)of

1i, jM

w(i, j). This map has the property that

G(M, N) = 

, the length of the first row in the

Young diagram. From the combinatorial definition

of the Schur polynomials s



it follows that the

measure [6] on the integer matrix is mapped to a

–1

–2

–3

Figure 1 Multilayer PNG model.

01 01

Ω(5)

Figure 2 Corner growth model at time n = 5. The crosses are

the possible growth sites.

Figure 3 Domino tiling of an Aztec diamond of size n = 5.

Dominos marked by dots form the NPR.

588 Growth Processes in Random Matrix Theory

probability measure on partitions, the Schur mea-

sure, given by

Schur

½¼



ða

; ...; a

Þs



ðb

; ...; b

Þ½11

This measure has determinantal correlation functions

if we think of x

= 

 i as the positions of particles

in Z.Ifweusex

= 

þ N  i as variables and

specialize to a

=  = a

ﬃﬃﬃ

, b

=  = b

ﬃﬃﬃ

and b

= 0forj > N we get the Meixner ensemble.

The case of exponential random variables, for

example the relation to LUE discussed above, is

obtained from the Meixner ensemble by taking an

appropriate limit. In the Poisson limit we get the

Poissonized Plancherel measure,



Plan

½¼

N ¼0

=

½12

where P

Plan, N

[] = (dim )

=N! if  is a partition of

N and 0 otherwise. Here dim  is the dimension

of the irreducible representation of S

labeled by .

In the work of Borodin and Olshanski in representa-

tion theory various measures on partitions with

determinantal correlation functions occur naturally.

Also Okounkov and co-workers have used the

Plancherel and Schur measures in Gromov–Witten

theory. The correlation kernel for the Plancherel

measure repre sented as the point process (x

)

i1

in Z

with x

= 

 i has the correlation kernel, called the

discrete Bessel kernel,



ðx; yÞ¼

ﬃﬃﬃﬃ



ðx  yÞ

1

ðJ

ð2

ﬃﬃﬃﬃ



ÞJ

yþ1

ð2

ﬃﬃﬃﬃ



J

xþ1

ð2

ﬃﬃﬃﬃ



ÞJ

ð2

ﬃﬃﬃﬃ



Þ½13

where J

is the ordinary Bessel function. The

random variable L() has the same distribution as

max x

þ 1. Henc e, by [5],

P½LðÞn¼detðI  B



‘

ðfn; nþ1;...gÞ

½14

The random variable L() also gives the height

above the origin in cont-PNG.

There is a geometric interpretation of RSK going

back to Viennot. The pair (P, Q) is represented as a

family of nonintersecting paths in a directed graph.

These paths can be obtained by running a multilayer

version of the PNG process where the size of

collisions are deposited as growth in lower layers

which evolve according to the same PNG dynamics.

The information lost in the collisions is recorded in

the lower layers. This can be done also for

{w(i, j)}

iþj1t

and leads to a multilayer version of

[9],{h

j

(x,t)}

j = 1

, where h

j

(x,0) j, h

j

(t, t) =

j, and h(x, t) = h

(x, t) is the top path (see Figure 1).

The Karlin–McGregor theorem or the Gessel–

Viennot method say that the weight (probability)

of a family of nonintersecting paths with fixed initial

and final positions on a weighted directed acyclic

graph is given by a determinant. It follows that the

probability of a certain configuration { h

j

(0, t)}

j1

given by a product of two determinants and hence

has the form [1].InFigure 1, the weights of the

horizontal line segments will be 1, whereas each unit

vertical step has weight a

or b

as indicated in the

figure. This leads to [9] using the Jacobi–Trudi

formula for the Schur polynomial.

Limit Theorems

The existenc e of a limit shape in a model like [6]

with w(i, j) independent random variables, and in

related problems follows by a subadditivity argu-

ment, although explicit shapes are only known in a

few cases . The formalism described above makes it

possible to get more detailed results about the

fluctuations around the limit shape, like a central-

limit theorem, but with a non-normal limit law. We

know that G(M, N) has the same distribution as

max

 N þ 1, where x

max

is the rightmost particle in

the Meixner ensemble. This, together with [4], [5],

and an asymptotic analysis of the Meixner poly-

nomials, gives



GðM; NÞ!ð;qÞN þð;qÞN

1=3



ðÞ½15

as N !1, M !1, M=N !   1, where

!ð; qÞ¼

ð1 þ

ﬃﬃﬃﬃﬃﬃ

q

1  q

 1 ½16

and

ð; qÞ¼

ðq=Þ

1=6

1  q

ﬃﬃﬃ



ﬃﬃﬃ

2=3

ð1 þ

ﬃﬃﬃﬃﬃﬃ

q

2=3

½17

The limiting distribution function F

is the Tracy–

Widom distribution given by

ðÞ¼detðI  AÞ

ð;1Þ

½18

where

Aðx; yÞ¼

AiðxÞAi

ðyÞAi

ðxÞAiðyÞ

x  y

½19

is the Airy kernel. It is also the limiting largest

eigenvalue distribution for GUE

lim

N!1

ﬃﬃﬃﬃﬃﬃ



ðnÞ

max

 2n

1=3

 

¼F

ðÞ½20

The function F

can also be expressed in terms of a

Painleve´ II function. The limit theorem [15]

Growth Processes in Random Matrix Theory 589

translates into a fluctuation result for the height

function in the corner growth and the PNG models,

saying that the height fluctuations above a fixed

location at time t are of order t

1=3

and given by the

-distribution. Here we see the KPZ exponent 1/3.

For the length L() of the length of a longest

increasing subsequence in a random permutation or

the height above the origin in the cont-PNG, [14]

and asymptotics for Bessel funct ions yield

P½ðLðÞ2

ﬃﬃﬃﬃ



Þ=

1=6

 !F

ðÞ½21

as  !1. This result was first proved by Baik,

Deift, and Johansson using a Toeplitz determinant

formula (Gessel’s formula) for the left-hand side of

[14] and the Deift–Zhou nonlinear steepest descent

method for oscil latory Riemann–Hilbert problems.

The above limit theorems can be extended to limit

theorems for the whole point process rescaled

around the rightmost point. This results in a

limiting determinantal point process given by the

Airy kernel [19].

The Airy Process

From the point of view of the growth processes, for

example, the PNG process [7], it is natural to consider

a scaling limit of the whole height function x ! h(x, t)

as t !1. Looking at the height configuration in the

multilayer growth process x ! {h

j

(x, t)}

j1

at differ-

ent locations x

, ..., x

, x

rþ1

, ..., x

M1

leads, via the

Karlin–McGregor or Gessel–Viennot method, to

probability measures of the form

M1

r¼0

det 

r;rþ1

; y

rþ1



i;j¼1

½22

with y

and y

fixed configurations. Here, in the

discrete PNG model, 

r, rþ1

(x, y) is the transition

probability (weight) to go from height x to height y

between positions x

and x

rþ1

. This measure

generalizes [1] and it also has determinantal correla-

tion functions. Measures of this form also arise in

multimatrix models and in Dyson’s Brownian

motion model, t ! M(t), t 2 R, for Hermitian

matrices, which is a Gaussian multimatrix model.

The elements of the time-dependent Hermitian

matrix M(t) evolve according to independent

Ornstein–Uhlenbeck processes and we have the

transition kernel

1

exp½trðMðtÞqMð0ÞÞ

=ð1  q

Þ ½23

where q = exp(t). This process has GUE as its

stationary distribution. The Harish–Chandra/Itzykson–

Zuber integral can be used to show that the joint

eigenvalue measure for M(t

), ..., M(t

M1

)hasthe

form [22] and hence has determinantal correlation

functions. The correlation kernel is the extended

Hermite kernel,

ð;x; ; yÞ

k¼1

kðÞ

nk

ðxÞp

nk

ðyÞe



ðx

þy

if   



k¼1

kðÞ

nk

ðxÞp

nk

ðyÞe



ðx

þy

if <

½24

with p

the normalized Hermite polynomials, p

 0

if k < 0. Notice that this reduces to the Hermite

kernel [4] when  = . This machinery can be used

to show that the largest eigenvalue process

t !

(n)

max

(t) induced by M( t) converges in the sense

of finite-dimensional distributions to a limiting

process, the Airy process,

ﬃﬃﬃﬃﬃﬃ



max

ðn

1=3

tÞ2n



1=3

!AðtÞ½25

as n !1. The Airy process A(t), which is a

stationary process, can be viewed as the top curve

of a multilayer process t ! (A

j

(t))

j1

, A(t) = A

(t)

such that the point process {A

j

)}

1kM, j0

has

determinantal correlation functions with correlation

kernel

Að;; 

;

ð

Ai ð þ ÞAi ð

þ Þd

if   



1

ð

Ai ð þ Þ Ai ð

þ Þd

if <

½26

the extended Airy kernel, which reduces to the

ordinary Airy kernel [19] when  = 

. The Airy

process can be viewed as an extension of the Tracy–

Widom distribution F

. For the PNG model above,

the multilayer process is described by an extended

kernel, which in the cont-PNG is an extended

version of the discrete Bessel kernel [13].Ina

suitable scaling limit, this extended kernel converges

to the extended Airy kernel. For the PNG process

[7], this leads to the limit law

ðdN

1=3

1

h 2d

1

1 þ

ﬃﬃﬃ

1 

ﬃﬃﬃ

2=3

;2N  1





ﬃﬃﬃ

1  q



!AðÞ½27

590 Growth Processes in Random Matrix Theory

as N !1, where d = (1  q)

1

(

ﬃﬃﬃ

)

1=3

(1 þ

ﬃﬃﬃ

)

1=3

Notice the exponent 2/3 which is the second KPZ

exponent. This exponent can also be seen in the

transversal fluctuations of the maximal paths in [6]

for G(N, N). These are superdiffusive, they have

fluctuations of order N

2=3

around the diagonal,

compared to N

1=2

for random walk paths between

the same points. A fluctuation result like [27] can

also be proved for the corner growth model and

hence also for the Aztec diamond. The boundary of

the NPR suitably rescaled converges to the Airy

process.

Variations

Above we discussed one possible geometry, the

droplet, for the PNG process. If we start with

h(x,0)  0 and allow random depositions along the

whole line, we get an interface that is macroscopi-

cally flat, and not curved as in the droplet case. In

this case, the height fluctuations above a fixed

location at time t are again of size t

1=3

and described

by the Gaussian Orthogonal Ensemble largest

eigenvalue distribution. This law comes from the

scaling limit of the rightmost particle in [3] with

 = 1, w(x) = exp(x

), =R,and the Lebesgue

measure. In this case, the correlation functions are

not determinants but rather pfaffians. The result for

flat PNG follows from the Baik–Rains analysis of

symmetrized last -passage or permutation problems.

In the PNG model we can also consider an interface

in equilibrium. This can be put into the last-passage

percolation picture by suit able boundary conditions,

different parameters for w(i, j) when i or j equals 1

or extra Poisson points on the axes in the Poisson

limit. Results by Baik and Rains show that in the

cont-PNG in equilibrium the height fluctuations are

given by a relative of the Tracy–W idom distribution,

. In these last two cases, the scaling limit of the

whole height profile is not known.

The types of results discussed above can only be

obtained for very special models. However, it is

expected that many of the results (in particular the

KPZ exponents 1/3 and 2/3, and also the fluctuation

laws, including the Airy process) should generalize

to many other models. The different interpretations

of [6] mentioned above suggest different general-

izations, various local growth models, directed

polymers, asymmetric exclusion processes, and

dimer/tiling problems. RMT laws are natural limit

laws for which the domain of attraction is not

understood.

See also: Combinatorics: Overview; Determinantal

Random Fields; Dimer Problems; Integrable Systems in

Random Matrix Theory; Random Walks in Random

Environments; Random Matrix Theory in Physics;

Random Partitions.

Further Reading

Baik J and Rains E (2001) Symmetrized random permutations. In:

Ruck M (ed.) Random Matrix Models and Their Applications,

Math. Sci. Res. Publ., vol. 4, pp. 1–19. Cambridge: Cambridge

University Press. (arXiv/math.CO/9910019).

Baik J, Deift PA, and Johansson K (1999) On the distribution of

the length of the longest increasing subsequence in a random

permutation. Journal of American Mathematical Society 12:

1119–1178.

Borodin A and Olshanski G (2001) In: z-Measures on Partitions,

Robinson–Schensted–Knuth Correspondence, and  = 2

Rand om Matrix Ensembles, Ma th. sci. Res. Publ., vol. 4,

pp. 71–94. Cambridge: Cambridge University Press. (arXiv/

math.CO/9905189).

Ferrari PL, Pra¨ hofer M, and Spohn H (2003) Stochastic growth in

one dimension and Gaussian multi-matrix models. Proceed-

ings of the International Congress of Mathematical Physics,

Lissabon 2003 (arXiv/mathph/0310053).

Johansson K (2002) Toeplitz determinants, random growth and

determinantal processes. Proceedings of the ICM Beijing

2002, vol. 3, pp. 53–62 (arXiv/math.PR/0304368).

Krug J and Spohn H (1992) Kinetic roughening of growing

interfaces. In: Godre` che C (ed.) Solids Far from Equilibrium:

Growth, Morphology and Defects, pp. 479–582. Cambridge:

Cambridge University Press.

Mehta ML (1991) Random Matrices, 2nd edn. San Diego:

Academic Press.

Okounkov A (2003) The uses of random partitions, arXiv/math-ph/

0309015 Proceedings of the International Congress of

Mathematical Physics, Lissabon 2003 (arXiv/mathph/

0309015).

Stanley RP (1999) Enumerative Combinatorics, vol. 2. Cambridge:

Cambridge University Press.

Tracy C and Widom H (2002) Distribution functions for largest

eigenvalues and their applications. Proceedings of the ICM

Beijing 2002, vol. 1, pp. 587–596 (arXiv/math.PR/0210034).

Growth Processes in Random Matrix Theory 591

Hamiltonian Fluid Dynamics

P J Morrison, University of Texas at Austin, Austin,

TX, USA

Introduction

The ideal fluid description is one in which viscosity or

other phenomenological terms are neglected. Thus, as is

the case for systems governed by Newton’s second law

without dissipation, such fluid descriptions possess

Lagrangian and Hamiltonian descriptions. In fact, in

the eighteenth century, Lagrange himself discussed

what is in essence the action principle for the

incompressible fluid. The subsequent history of action

functional and Hamiltonian formulations of the ideal

fluid is long and convoluted with contributions from

Clebsch in the nineteenth century, and the likes of L

Landau and V Arnol’d in the mid-twentieth century.

In the early 1980s, there was a flurry of activity on

the noncanonical Poisson bracket formulation, and

this formulation is the focus of the present treatment,

which is motivated by the wo rk of the author, D

Holm,JMarsden,TRatiu,AWeinstein,andothers.

Noncanonical Hamiltonian Structure

The traditional arena for Hamiltonian dynamics is the

cotangent bundle M:= T



Q, the phase space, which is

naturally a symplectic manifold with a closed non-

degenerate 2-form. In coordinates, the 2-form is given

by !

= dq ^ dp,whereq denotes the configuration

coordinate for the base space manifold Q and p

denotes the corresponding canonical momenta that

arise from Legendre (convex) transformation. The

2-form !

provides a natural identification at a point

z = (q, p) 2Mof T

M with T



M, and because of

nondegeneracy its inverse, the cosymplectic form,

provides the map J

: T



M!T

M.Thus,fora

Hamiltonian H : M!R we have the Hamiltonian

system of ordinary differential equations

z = J

dH,

which in canonical coordinates has the familiar form

¼ @H=@p

;

¼@H=@q

½1

with i = 1, 2, ..., N, where N is the number of

degrees of freedom.

Hamilton’s equations can also be written in terms

of the Poisson bracket [f , g]:= !

df , J

dg), where

f , g : M!R are smooth phase-space functions. In

terms of z = (q, p), Hamilton’s equations a re



¼ J





¼½z



; H½2

where the Poisson bracket is

½f ; g¼







½3

with



I



½4

Note, repeated indices are to be summed with

,  = 1, 2, ...,2N.In[4],0

is an N  N matrix

of zeros and I

is the N  N unit matrix.

Noncanonical Poisson Brackets

The canonical Poisson bracket description of [2]–[4]

suggests a generalization, with antecedents to S Lie

and others, that was termed noncanonical Hamilto-

nian form in the fluid mechanics context by

P Morrison and J Greene (1980):

A system has noncanonical Hamiltonian form if it

can be written as

z = [z, H], where the noncanonical

Poisson bracket [,]is a Lie product for a realization of

a Lie enveloping algebra on phase-space functions.

Recall a Lie enveloping algebra a is a Lie algebra,

with the usual product [ , ] that is bilinear, anti-

symmetric, and satisfies the Jacobi identity, which in

addition has a product a  a !a that satisfies

the Leibniz identity [fg, h] = f [g, h] þ [f , h]g for all

f , g, h 2 a .

The geometric description of noncanonical Hamil-

tonian form has evolved into a structure called the

Poisson manifold, a differential manifold Z

endowed with the binary bracket operation [ , ]

defined on smooth functions, say, f , g : Z!R.

Poisson manifolds differ from symplectic manifolds

because the nondegeneracy condition is removed. In

coordinates, [ , ] is given by

½f ; g¼







;

;  ¼1; 2; ...; M

½5

where M = dim Z. Note that J need not have the

form of [3], may depend upon the coordinate

z, and may have vanishing determinant. Bilinearity,

[f , g] = [g, f ]forallf , g, and the Jacobi

identity, [f,[g, h]] þ [g,[h, f ]] þ [h,[f , g]]  0, for

all f , g, h, imply that t he cosym plectic matr ix

satisfies J



= J



and







þ J







þ J







 0 ½6

respectively, for , , ,  = 1, 2, ..., M.

ThelocalstructureofZ is elucidated by the Darboux–

Lie theorem, which states that in a neighborhood of a

point z 2Z,forwhichrankJ = M, there exist coordi-

nates in which J has the following form:

ðJÞ¼

I

000

M2N

½7

From [7] it is clear that in the right coordinates, the

system looks like a canonical N-degree-of-freedom

Hamiltonian system with some extraneous coordi-

nates, M  2N in fact. Through any point of the

M-dimensional phase space Z, there exists a local

foliation by symplectic leaves of dimension 2N.

A consequence of the degeneracy is that there

exists a special class of invariants called Casimir

invariants that is built into the phase space. Since

the rank of J is 2N, there exist possibly M  2N

independent null eigenvectors. A consequence of the

Darboux–Lie theorem is that the independent null

eigenvectors exist and, moreover, the null space can

in fact be spanned by the gradients of the Casimir

invariants, which satisfy J



(a)

=@z



= 0, where

a = 1, 2, 3, ..., M  2N. That the Casimir invariants

are constants of motion follows from

ðaÞ







¼0 ½8

Thus, Casimir invariants are constants of motion for any

Hamiltonian. The symplectic leaves of dimension 2N

are the intersections of the M  2N surfaces defined by

(a)

= constant. Dynamics generated by any H that

begins on a particular symplectic leaf remains there. The

structure of Poisson manifolds has now been widely

studied, but we will not pursue this further here.

Let us turn to infinite-dimensional systems, field

theories such as those that govern ideal fluids, where

the governing equations are partial differential

equations. Although the level of rigor does not

match that achieved for the finite systems described

above, formally one can parody most of the steps

and, consequently, the finite theory provides cogent

imagery and serves as a beacon for shedding light. In

infinite dimensions, an analog of [5] is given by

fF; Gg¼



d

F



G



¼:

F



; J

G





½9

where F and G are functionals of the functions

(, t),

which are functions of  = (

, ..., 

), independent

variables of some kind, F=

denotes the functional

(variational) derivative, and ,

is a pairing between a

vector (function) space and its dual. The

, i = 1, ..., n,

are n field components, and now J is a cosymplectic

operator. To be noncanonically Hamiltonian requires

antisymmetry, {F, G} = {G, F}, and the Jacobi iden-

tity, {F,{G, H}} þ {G,{H, F}} þ {H,{F, G}}  0, for all

functionals F, G,andH. Antisymmetry requires J to be

skew-symmetric, that is, hf ,Jgi= J

f ,g



=  g,Jf

The Jacobi identity for infinite-dimensional systems has

a condition analogous to [6]; it can be shown that one

need only consider variations of J when calculating, for

example, {F,{G,H}}.

Lie–Poisson Brackets

As noted in the Introduction, the usual variables of

fluid mechanics are not a set of canonical variables,

and, consequently, the Hamiltonian description in

terms of these variables is noncanonical. There is a

special general form that the Poisson bracket takes

for equations that describe media in terms of

Eulerian-like variables, the so-called Lie–Poisson

brackets, a special form of noncanonical Poisson

bracket. Lie–Poisson brackets describe essentially

every fundamental equation that describes classical

media. In addit ion to the equations for the ideal

fluid, they describe Liouville’s equation for the

dynamics of the phase-space densi ty of a collection

of particles, the various hierarchy of kinetic theory,

the Vlasov equation of plasma physics, and various

approximations thereof, and magnetized and other

more complicated fluids.

Both finite- and infinite-dimensional Lie–Poisson

brackets are intimately associated with a Lie group G .

We use the pairing between a vector space and its

dual, ,

, where the second slot is reserved for

elements of the Lie algebra g of G and the first slot

for elements of its dual g



. Thus, ,

: g



 g !R.In

terms of the pairing, noncanonical Lie–Poisson

brackets have the following compact form:

fF; Gg¼h; ½F



; G



i ½10

594 Hamiltonian Fluid Dynamics

where we suppose the dynamical variable  2 g



,[,]is

the Lie algebra product, which takes g g !g ,andwe

have introduced the shorthand F



:= F=.The

quantities F



and G



are, of course, in g . We refer to

{ , } as the ‘‘outer’’ bracket of the realization enveloping

algebra and [ , ] as the ‘‘inner’’ bracket of the Lie algebra

g. The binary operator [ , ]

is defined as follows:

h; ½f ; gi ¼: ½; g

; f

½11

where evidently  2 g



, g, f 2 g ,and[,]

: g



 g !g



The operator [ , ]

, which defines the coadjoint orbit,

is necessary for obtaining the equations of motion

from a Lie–Poisson bracket.

For finite-dimensional systems, the group G must be

a finite-parameter Lie group, the variable corre-

sponds to w, and the cosymplectic form in coordinates

is given by J

= c

, where the c

are the structure

constants for the Lie algebra g , which satisfy

¼c

þ c

¼ 0

½12

relations that imply [10] satisfies the antisymmetry

condition and the Jacobi identity.

For infinite-dimensional systems, the group G

must be an infinite-parameter Lie group and the

cosymplectic operator has the form J

= C



where C

are structure operators. The meaning of

these structure operators will be clarified when we

consider brackets for fluid mechanics.

The Fluid State

Fluid mechanics has a long history, and thus it

comes as no surprise that the fluid state has been

described in many ways. Because the Hamiltonian

structure depend s on the state variab les, some of

these ways are described below, beginning with

Lagrangian variable description.

Lagrangian Variables

The description of a fluid that is most like that of

particle mechanics occurs in terms of variables usually

referred to as Lagrangian variables. This description

dates to the eighteenth century. The idea behind the use

of these variables is a simple one: if a fluid is described

as a continuum collection of fluid particles, also called

fluid parcels or elements, then its motion is governed by

an equation that is an infinite-dimensional version of

Newton’s second law and, consequently, as we will see,

both the Hamiltonian and the Lagrangian descriptions

are infinite degree-of-freedom generalizations of those

of ordinary particle mechanics.

The position of a fluid element, referred to a fixed

rectangular coordinate systems, is given by q = q(a, t),

where q = (q

, q

)anda = (a

, a

)isacon-

tinuum label that replaces the index i of [1].In

practice, the label can be any quantity that identifies

a fluid particle, but it is often taken to be the position

of the fluid particle at time t = 0inrectangular

coordinates. The quantities q

(a, t) are coordinates

for the configuration space Q, which is in fact a

function space because in addition to the three indices

‘‘i’’ there is the continuum label a.Weassumethata

varies over a fixed domain,   R

, which is

completely filled with fluid, and that the function

q :  ! is one-to-one and onto. We will assume that

as many derivatives of q with respect to a as needed

exist, but we will not say more about Q; in fact, not

much is known about the solution function space for

the 3D fluid equations in Lagrangian variables. Often

in the Hamiltonian context the functions q = q(a, t)are

assumed to be diffeomorphisms and their collection is

referred to as the diffeomorphism group.

In the sequel several manipulations are needed and

so we record here some identities for later use. Viewing

the map a 7!q at fixed t as a coordinate change, the

Jacobian matrix @q

=@a

=: q

has an inverse given by

¼ 

½13

where A

is the cofac tor of q

and J is its

determinant. A convenient expression for A

given by



kjl



imn

½14

where 

ijk

( = 

ijk

) is the skew-symmetric tensor

(density). Evidently, @J =@q

= A

follows from [13].

Eulerian Variables

In the Lagrangian variable description, one picks out

a particular particle, labeled by a, and keeps track in

time t of where it goes. How ever, in the Eulerian

variable description, one stays at a spatial observa -

tion point r = (x

, x

) 2  and monitors the

nature of the fluid at r at time t.

The most important Eulerian variable is the Eulerian

velocity field v(r, t). This quantity is the velocity of the

particular fluid element that is located at the spatial

point r at time t. The label of that particular fluid

element is given by a = q

1

(r, t), and so

vðr; tÞ¼

qða; tÞj

a¼q

1

ðr; tÞ

:¼

q  q

1

ðr; tÞ½15

where  denotes differentiation with respect to time

at fixed label a. Attached to a fluid element is a

certain amount of mass descr ibed by a density

Hamiltonian Fluid Dynamics 595