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

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suitable bound, for the finite-volume Gibbs state





({



}j

) under the boundary conditions 

,on

the probability that a fixed point in  is encircled

by a contour from @. If this is the case, we say that

the phase m is s table. It turns out that such a bound

is actually an integral part of the construction of

metastable free energies f

(t) yielding the home-

omorphism t 7!a (t). In this way, we get the main

claim formulated as follows:

Theorem 1 Consider a parametric set of Hamilto-

nians H

= H

(0)

r1

‘ = 1

‘

(‘)

with periodic finite-

range interactions satisfying the condition of

removal of degeneracy as well as the Peierls

condition with respect to the reference set

G = g(H

(0)

). Let d  2 and let  be sufficiently

large. Then there exists a homeomorphism t 7!a(t)

of a neighborhood V



of the origin of the parameter

space R

r1

onto a neighborhood U



of the origin of

such that, for any t 2 V



, the set of all stable

phases is {m 2 {1, ..., r}ja

(t) = 0}.

The Peierls condition can be actually assumed

only for the Hamiltonian H

(0)

inferring its validity

for H

on a sufficiently small neighborhood V



Notice also that the result can be actually stated

not as a claim about phase diagram in a space of

parameters, but as a statement about stable phases

of a fixed Hamiltonian H. Namely, for a Hamilto-

nian H satisfying Peierls condition with respect to a

reference set G, one can assure the existence of

parameters a

labeled by elements from G such that

the set of extremal periodic Gibbs states of H

consists of all those m-phases for which a

= 0.

Construction of Metastable Free Energies

An important part of the Pirogov–Sinai theory is

an actual construction of the metastable free

energies – a set of functions f

(t), m = 1, ..., r,

that provide the homeomorphism a(t)bytaking

(t) = f

(t)  min



(t).

We start with a contour representation of

partition function Z(j

). Considering, for each

contributing configuration , the collection @()of

its contours, we notice that, in addition to the fact

that different contours , 

2 @() have disjoint

supports,  \ 

= ;, the contours from @()have

to satisfy the matching conditions:ifC is a

connected component of Z

2@

, then the

restrictions of the spin configurations 



to C

are the same for all contours  2 @()with

dist(, C) = 1. In other words, the contours touch-

ing C inducethesamelabelonC. Let us observe

that there is actually one-to-one correspondence

between configurations  that coincide with 



and collections M(, q) of contours @ in 

satisfying the matching condition, and such that the

external among them are q-contours. Here, a contour

 2 @ is called an external contour in @ if   Ext 

for all 

2 @ different from .

With this observation and using 

(@) to denote

the union of all components of  n

2@

 with label

m, we get

Zðj

Þ¼

@2Mð;qÞ

e

ðHÞj

ðÞj

2@

 ðÞ

½9

Usefulness of such contour representations stems

from an expectation that, for a stable phase q,

contours should constitute a suppressed excitation

and one should be able to use cluster expansions to

evaluate the behavior of the Gibbs state 

However, the direct use of the cluster expansion on

[9] is trammeled by the presence of the energy terms

e

(H)j

(@)j

and, more seriously, by the require-

ment that the contour labels match.

Nevertheless, one can rewrite the partition func-

tion in a form that does not involve any matching

condition. Namely, considering first a sum over

mutually external contours @

ext

and resumming over

collections of contours which are contained in their

interiors without touching the boundary (being thus

prevented to ‘‘glue’’ with external contours), we get

Zðj

Þ¼

ext

e

ðHÞjExtj



2@

ext



 ðÞ

dil

ðInt

j



½10

Here the sum goes over all collections of

compatible external q-contours in , Ext =

Ext



ext

) =

2@

ext

(Ext  \ ), and the partition

function Z

dil

ðj

Þ is defined by [9] with

Mð, qÞ replaced by M

dil

ð, qÞMð, qÞ,the

set of all those collections whose external coun-

tours  are such that dist ð, 

Þ > 1: Multiplying

now each term by

1 ¼

2@

ext

dil

ðInt

j

dil

ðInt

j

½11

we get

Zðj

Þ¼

ext

e

ðHÞjExtj



2@

ext

e

ðHÞjj

ðÞZ

dil

ðInt j



½12

where w

() is given by

ðÞ¼e

 ðÞ

e

ðHÞjj

dil

ðInt

j

dil

ðInt

j

½13

62 Pirogov–Sinai Theory

Observing that a similar expression is valid for

dil

ðj

Þ (with an appropriate restriction on the

sum over external contours @

ext

) and proceeding by

induction, we eventually get the representation

Zðj

Þ¼e

e

ðHÞjj

@2Cð;qÞ

2@

ðÞ½14

where C(, q) denotes the set of all collections of

nonoverlapping q-contours in . Clearly, the sum on

the right-hand side is exactly of the form needed to

apply cluster expansion, provided the contour weights

satisfy the necessary convergence assumptions.

Even though this is not necessarily the case, there

is a way to use this representation. Namely, one can

artificially change the weights to satisfy the needed

bound, for example, by modifying them to the form

ðÞ¼min w

ðÞ; e

jj



½15

with a suitable constant . The modified partition

function

ðj

Þ¼e

e

ðHÞjj

@2Cð; q Þ

2@

ðÞ½16

can then be controlled by cluster expansion allowing

to define

ðHÞ¼



lim

jj!1

jj

log Z

ðj

Þ½17

This is the metastable free energy corresponding to the

phase q. Applying the cluster expansion to the

logarithm of the sum in [16],wegetjf

(H)  e

(H)j

=2

. The metastable free energy corresponds to

taking the ground state 

and its excitations as long

as they are sufficiently suppressed. Once w

() exceeds

the weight e

jj

(and the contour would have been

actually preferred), we suppress it ‘‘by hand.’’ The

pointisthatifthephaseq is stable, this never happens

and w

() = w

()forall q-contours .Thisistheidea

behind the use of the function f

(H) as an indicator of

the stability of the phase q by taking

ðtÞ¼f

ðH

Þmin

ðH

Þ½18

Of course, the difficult point is to actually prove that

the stability of phase q (i.e., the fact that a

(t) = 0)

indeed implies w

() = w

() for all . The crucial step

is to prove, by induction on the diameter of  and ,

the following three claims (with  = 2e

=2

1. If  is a q-contour with a

(t) diam   =4, then

() = w

().

2. If a

(t) diam   =4, then Z(j

) = Z

(j

) 6¼ 0

and

Zðj



 e

f

ðH

Þjjj@j

½19

3. If m 2 G, then

Zðj

 e

min

ðH

Þjj

j@j

½20

A standard example illuminating the perturbative

construction of the metastable free energies and

showing the role of entropic contributions is the

Blume–capel model. It is defined by the Hamiltonian



ðÞ¼J

hx;yi

ð





x2



h

x2



½21

with spins 

2 { 1, 0, 1}. Taking into account only

the lowest-order excitations, we get:



ð; hÞ¼  h 



ð2dhÞ

(sea of pluses or minuses with a single spin flip !0)

and

ð; hÞ¼



ð2dþÞ

h

þ e

h



(sea of zeros with a single spin flip either 0 !þor

0 !)

Since these functions differ from full metastable free

energies f



(, h), f

(, h) by terms of higher order

(e

(4d2)

), the real phase diagram differs in this

order from the one constructed by equating the

functions



(, h)and

(, h). It is particularly

interesting to inspect the origin,  = h = 0. It is only

the phase 0 that is stable there at all small

temperatures since

ð0; 0Þ



2d

< f



ð0; 0Þ



2d

½22

The only reason why the phase 0 is favored at this

point with respect to phases þ and  is that there

are two excitations of order e

2d

for the phase 0,

while there is only one such excitation for þ or .

The entropy of the lowest-order contribution to

(0, 0) is overweighting the entropy of the contribu-

tion to f



(0, 0) of the same order.

Applications

Several applications, stemming from the Pirogov–

Sinai theory, are based on the fact that, due to the

cluster expansion, we have quite accurate descrip-

tion of the model in finite volume.

One class of applications concerns various

problems featuring interfaces between coexisting

phases. To be able to transform the problem into a

study of the random boundary line separating the

two phases, one needs a precise cluster expansion

formula for partition functions in volumes occupied

by those phases. In the situation with no symmetry

Pirogov–Sinai Theory 63

between the phases, the use of the Pirogov–Sinai

theory is indispensable.

Another interesting class of applications concerns

the behavior of the system with periodic boundary

conditions. It is based on the fact that the partition

function Z

on a torus T

consisting of N

sites

can be, again with the help of the cluster expan-

sions, explicitly and very accurately evaluated in

terms of metastable free energies,



q¼1

f

ðHÞN



 expf min

ðHÞN

 bNg½23

with a fixed constant b. This formula (and its

generalization to the case of complex parameters)

allows us to obtain various results concerning the

behavior of the model in finite volumes.

Finite-Size Effects

Considering as an illustration a perturbation of the

Ising model, so that it does not have the  symmetry

any more (and the value h

() of external field

at which the phase transition between plus and

minus phase occurs is not known), we can pose a

natural question that has an importance for correct

interpretation of simulation data. Namely, what is

the asymptotic behavior of the magnetization

per

(, h) = 

(1/

x2



) on a torus? In the

thermodynamic limit, the magnetization m

per

(, h)

displays, as a function of h, a discontinuity at

h = h

(). For finite N, we get a rounding of the

discontinuity – the jump is smoothed. What is the

shift of a naturally chosen finite-volume transition

point h

(N) with respect to the limiting value h

The answer can be obtained with the help of [23]

once sufficient care is taken to use the freedom in

the definition of the metastable free energies f

(h)

and f



(h) to replace them with a sufficiently smooth

version allowing an approximation of the functions



(h) around limiting point h

in terms of their

Taylor expansion.

As a result, in spite of the asymmetry of the model,

the finite-volume magnetization m

per

(, h)hasauni-

versal behavior in the neighborhood of the transition

point h

. With suitable constants m and m

,wehave

per

ð; hÞm

þ m tanhfN

mðh  h

Þg ½24

Choosing the inflection point h

max

(N)ofm

per

(, h)

as a natural finite-volume indicator of the occurence

of the transition, one can show that

max

ðNÞ¼h

3

2

2d

þ OðN

3d

Þ½25

Zeros of Partition Functions

The full strength of the formula [23] is revealed

when studying the zeros of the partition function

(z) as a polynomial in a complex parameter z

entering the Hamiltonian of the model. To be able

to use the theory in this case, one has to extend the

definitions of the metastable free energies to com-

plex values of z. Indeed, the construction still goes

through, now yielding genuinely complex, contour

models w



with the help of an inductive procedure.

Notice that no analytic continuation is involved. An

analog of [23] is still valid,

ðzÞ

m¼1

f

ðzÞN



 expf min

<ef

ðzÞN

 bNg½26

Using [26], it is not difficult to convince oneself

that the loci of zeros can be traced down to the

phase coexistence lines. Indeed, on the line of

the coexistence of two phases <ef

= <ef

,the

partition function Z

(z) is approximated by

fN

=mf

þ e

=mf

). The zeros of this

approximation are thus given by the equations

<ef

¼<ef

< < ef

‘

for all ‘ 6¼ m; n

N

ð=mf

=mf

Þ¼ mod 2

½27

The zeros of the full partition function Z

(z) can

be proved to be exponentially close, up to a shift

of order O(e

bN

), to those of the discussed

approximation.

Briefly, the zeros of Z

(z) asymptotically con-

centrate on the phase coexistence curves with the

density (1=2)N

j(d=dz)(f

 f

)j.

Bibliographical Remarks

and Generalizations

The original works Pirogov and Sinai (1975, 1976)

and Sinai (1982) introduced an analog of the weights

()andparametersa

(H) as a fixed point of a

suitable mapping on a Banach space. The inductive

definition used here was introduced in Kotecky´and

Preiss (1983) and Zahradnı´k (1984).Thecompleteness

of phase diagram – the fact that the stable phases

exhaust the set of all periodic extremal Gibbs states

was first proved in Zahradnı´k (1984). Extension to

complex parameters was first considered in Gawe¸dzki

et al. (1987) and Borgs and Imbrie (1989).Forareview

of the standard Pirogov–Sinai theory, see Sinai (1982)

and Slawny (1987).

Application of Pirogov–Sinai theory for finite-size

effects was studied in Borgs and Kotecky´ (1990) and

64 Pirogov–Sinai Theory

general theory of zeros of partition functions is

presented in Biskup et al. (2004).

The basic statement of the Pirogov–Sinai theory

yielding the construction of the full phase diagram

has been extended to a large class of models. Let us

mention just few of them (with rather incomplete

references):

1. Continuous spins. The main difficulty in these

models is that one has to deal with contours

immersed in a sea of fluctuating spins (Dobrushin

and Zahradnı´k 1986, Borgs and Waxler 1989).

2. Potts model. An example of a system a transi-

tion in temperature with the coexistence of the

low-temperature ordered and the high-tempera-

ture disordered phases. Contour reformulation is

employing contours between ordered and dis-

ordered regions (Bricmont et al. 1985, Kotecky´

et al. 1990). The treatment is simplified with help

of Fortuin–Kasteleyn representation (Laanait

et al. 1991).

3. Models with competing interactions. ANNNI

model, microemulsions. Systems with a rich

phase structure (Dinaburg and Sinai 1985).

4. Disordered systems . An example is a proof of

the existence of the phase transition for the three-

dimensional random field Ising model (Bricmont

and Kupiainen 1987, 1988) using a renormaliza-

tion group version of the Pirogov–Sinai theory

first formulated in Gaw¸edzki et al. (1987).

5. Quantum lattice models. A class of quantum

models that can be viewed as a quantum perturba-

tion of a classical model. With the help of Feyn-

man–Kac formula these are rewritten as a (d þ 1)-

dimensional classical model that is, in its turn,

treated by the standard Pirogov–Sinai theory (Datta

et al. 1996, Borgs et al. 1996).

6. Continuous systems. Gas of particles in con-

tinuum interacting with a particular potential of

Kac type. Pirogov–Sinai theory is used for a proof

of the existence of the phase transitions after a

suitable discretisation (Lebowitz et al. 1999).

See also: Cluster Expansion; Falicov–Kimball Model;

Phase Transiti ons in Continuous Syst ems; Quantum

Spin Systems.

Further Reading

Biskup M, Borgs C, Chayes JT, and Kotecky´ R (2004) Partition

function zeros at first-order phase transitions: Pirogov–Sinai

theory. Journal of Statistical Physics 116: 97–155.

Borgs C and Imbrie JZ (1989) A unified approach to phase

diagrams in field theory and statistical mechanics. Commu-

nications in Mathematical Physics 123: 305–328.

Borgs C and Kotecky´ R (1990) A rigorous theory of finite-size

scaling at first-order phase transitions. Journal of Statistical

Physics 61: 79–119.

Borgs C, Kotecky´ R, and Ueltschi D (1996) Low temperature phase

diagrams for quantum perturbations of classical spin systems.

Communications in Mathematical Physics 181: 409–446.

Borgs C and Waxler R (1989) First order phase transitions in

unbounded spin systems: construction of the phase diagram.

Communications in Mathematical Physics 126: 291–324.

Bricmont J, Kuroda T, and Lebowitz J (1985) First order phase

transitions in lattice and continuum systems: extension of

Pirogov–Sinai theory. Communications in Mathematical Phy-

sics 101: 501–538.

Bricmont J and Kupiainen A (1987) Lower critical dimensions for

the random field Ising model. Physical Review Letters 59:

1829–1832.

Bricmont J and Kupiainen A (1988) Phase transition in the 3D

random field Ising model. Communications in Mathematical

Physics 116: 539–572.

Datta N, Ferna´ndez R, and Fro¨ hlich J (1996) Low-temperature

phase diagrams of quantum lattice systems. I. Stability for

quantum perturbations of classical systems with finitely-many

ground states. Helv. Phys. Acta 69: 752–820.

Dinaburg EL and Sinaı¨ YaG (1985) An analysis of ANNNI model

by Peierls contour method. Communications in Mathematical

Physics 98: 119–144.

Dobrushin RL and Zahradnı´k M (1986) Phase diagrams of

continuous lattice systems. In: Dobrushin RL (ed.) Math.

Problems of Stat. Physics and Dynamics, pp. 1–123. Dordrecht:

Reidel.

Gawe¸dzki K, Kotecky´ R, and Kupiainen A (1987) Coarse-graining

approach to first-order phase transitions. Journal of Statistical

Physics 47: 701–724.

Kotecky´ R, Laanait L, Messager A, and Ruiz J (1990) The q-state

Potts model in the standard Pirogov–Sinai theory: surface

tensions and Wilson loops. Journal of Statistical Physics 58:

199–248.

Kotecky´ R and Preiss D (1983) An inductive approach to PS

theory, Proc. Winter School on Abstract Analysis, Suppl. ai

Rend. del Mat. di Palermo.

Lebowitz JL, Mazel A, and Presutti E (1999) Liquid–vapor phase

transitions for systems with finite range interactions. Journal

of Statistical Physics 94: 955–1025.

Laanait L, Messager A, Miracle-Sole´ S, Ruiz J, and Shlosman SB

(1991) Interfaces in the Potts model I: Pirogov–Sinai theory of

the Fortuin–Kasteleyn representation. Communications in

Mathematical Physics 140: 81–91.

Pirogov SA and Sinai YaG (1975) Phase diagrams of classical

lattice systems (Russian). Theoretical and Mathematical

Physics 25(3): 358–369.

Pirogov SA and Sinai YaG (1976) Phase diagrams of classical

lattice systems. Continuation (Russian). Theoretical and

Mathematical Physics

26(1): 61–76.

Sinai YaG (1982) Theory of Phase Transitions: Rigorous Results.

New York: Pergamon.

Slawny J (1987) Low temperature properties of classical lattice

systems: phase transitions and phase diagrams. In: Domb C and

Lebowitz JL (eds.) Phase Transitions and Critical Phenomena,

vol. 11, pp. 127–205. New York: Academic Press.

Zahradnı´k M (1984) An alternate version of Pirogov–Sinai theory.

Communications in Mathematical Physics 93: 559–581.

Pirogov–Sinai Theory 65

Point-Vortex Dynamics

S Boatto, IMPA, Rio de Janeiro, Brazil

D Crowdy, Imperial College, London, UK

Introduction

Vortices have a long fascinating history. Descartes

wrote in his Le Monde:

...que tous les mouvements qui se font au Monde sont

en quelque fac¸on circulaire: c’est a` dire que, quand un

corps quitte sa place, il entre toujours en celle d’un

autre, et celui-ci en celle d’un autre, et ainsi de suite

jusques au dernier, qui occupe au meˆme instant le lieu

de´laisse´ par le premier.

In particular, Descartes thought of vortices to

model the dynamics of the solar system, as reported

by W W R Ball (1940):

Descartes’ physical theory of the universe, embodying

most of the results contained in his earlier and

unpublished Le Monde, is given in his Principia,

1644, ... He assumes that the matter of the universe

must be in motion, and that the motion must result in a

number of vortices. He stated that the sun is the center

of an immense whirlpool of this matter, in which the

planets float and are swept round like straws in a

whirlpool of water.

Descartes’ theory was later on recused by Newton

in his Principia in 1687. Few centuries later,

W Thomson (1867) the later Lord Kelvin, made use

of vortices to formulate his atomic theory: each atom

was assumed to be made up of vortices in a sort of

ideal fluid. In 1878–79 the American physicist A M

Mayer conducted a few experiments with needle

magnets placed on floating pieces of cork in an

applied magnetic field, as toy models for studying

atomic interactions and forms (Mayer 1878, Aref

et al. 2003). In 1883 inspired by Mayer experiments,

J J Thomson combined W Thomson’s atomic theory

with H von Helmholtz’s point-vortex theory

(Helmholtz 1858): he thought as the electrons were

point vortices inside a positively charged shell (see

Figure 1), the vortices being located at the vertices of

regular parallelograms and investigated about the

stability of such structures (see Thomson (1883,

section 2.1)). The vortex-atomic theory survived for

quite a few years up to Rutherford’s experiments

proved that atoms have quite a different structure!

Before continuing this historical/modeling overview,

let’s address the following question:

what is a vortex and, more specifically, what is a point-

vortex?

Roughly speaking, following Descart es, a vortex

is an entity which makes particles move along

circular-like orbits. Examples are the cyclones and

anticyclones in the atmosphere (see Figure 3).

Mathematically speaking, let u = (u, v, w) 2 R

be a

velocity field, the associated vorticity field ! is

defined to be

! ¼r^u ½1

In this article we are considering exclusively inviscid

flows which are also incompressible, that is,

ru ¼ 0 ½2

and have constant density , which we normalize to

be equal to 1 ( = 1). In two dimensions, a point-

vortex field is the simplest of all vorticity fields: it

can be thought as an entity where the vorticity field

is concentrated into a point. In other words, point

vortices are singularities of the vorticity field! Then,

in the plane the vorticity field associated to a system

of N point vortices is

!ðrÞ¼

¼1





ðr  r



Þ½3

Figure 2 Hurricane Jeanne. Reproduced with permission from

the National Oceanic and Atmospheric Administration (NOAA)

(www.noaanews.noaa.gov).

(a) (b)

Figure 1 Thomson atomic model: (a) atom with three

electrons and (b) atom with four electrons. From Thomson JJ

(1883) A Treatise on the Motion of Vortex Rings. New York:

Macmillan and Thomson JJ (1904) Electricity and Matter.

Westmister: Archibald Constable.

66 Point-Vortex Dynamics

where 



,  = 1, ..., N, is a constant and corre-

sponds to the vorticity (or circulation) of the

-vortex, situated at r



. In fact by definition,

the circulation around a curve C de limiting a region

 with boundary C,



u  ds ¼



ðr ^ uÞn dA ¼



! ½4

where we have used Stokes’ theorem to bring in the

vorticity. Then if the region contains only the th

point vortex, we obtain





!  dA ¼ 



½5

by eqn [3]. A positive (resp. negative) sign of 



indicates that the corresponding point vortex

induces an anticlockwise (resp. clockwise) particle

motion, see Figure 4a)). Is there an analog of a

point-vortex system for a three-dimensional flow?

Yes, and this brings in the analogy between vortex

lines and magnetic field lines that Mayer used in his

experiments with floating magnets. In fact, in three

dimensions, the notion of a point vortex can be

extended to that one of a straight vortex line (see

Figure 4b), where, by definition, a vortex line is a curve

that is tangent to the vorticity vector ! at each of its

point. In this context we would like to mention the

beautiful experiments of Yarmchuck–Gordon–Packard

on vortices in superfluid helium. They observed the

formation of stable polygonal configurations of iden-

tical vortices, quite similar to the ones observed by

Mayer with his magnets (see Figures 5 and 1).

One would like to understand how such configura-

tions form and to give a theoretical account about their

stability. In order to answer these questions we have to

first be able to describe the dynamics of a system of

point vortices from a mathematical point of view.

Evolution Equations

Can point vortices be viewed as ‘‘discrete’’ (or

localized) solutions of Euler equ ation in two dimen-

sions? Let us consider the Euler equ ation

þ u ru ¼rp þ f ½ 6

where p is the pressure, f = rU is a conservative

force, and restrict our attention to the two-dimensional

setting, for example, vortex dynamics on the plane (or a

sphere). Then it is immediate that by taking the curl of

eqn [6] we obtain the evolution equation of the

vorticity, that is,

þ u r! ¼ 0; or

¼ 0 ½7

where the operator D=Dt = @=@t þ u ris called the

material derivative and describes the evolution along

the flow lines. It follows from eqn [7] that in two

dimensions the vorticity is conserved as it is trans-

ported along the flow lines. Then a natural question

arises: supposing the vorticity field ! is known, is it

possible to deduce the velocity field u generating !?Or

in other words, is it possible to solve the system of eqns

[1]–[2]? It is immediate to see that in general the

solution is not unique, if some boundary conditions

are not specified (see Marchioro and Pulvirenti

(1993)). Furthermore, as already observed by Kirchh-

off in 1876 (Boatto and Cabral 2003), in two

dimensions we can recast the fluid equations [1]–[2]

into a Hamiltonian formalism. In fact, notice that on

the plane u = (

y)andeqn [2] is still satisfied if we

represent the velocity components as

Figure 3 Cyclones and anticyclones in the atmosphere. Repro-

duced from Boatto S and Cabrel HE, SIAM Journal of Applied

Mathematics 64:216–230 (2003). With the permission of SIAM.

Γ > 0 Γ > 0

c|u|

particle

(a) (b)

Figure 4 (a) Advected by the velocity field of one point vortex, a test particle follows a circular orbit, with a speed proportional to the

absolute value of the vortex circulation and inversely proportional to the square of its distance from the vortex. (b) Straight vortex lines.

Point-Vortex Dynamics 67

x ¼

@

;

y ¼

@

½8

that is, by means of , called the stream function.

Formally,  plays the roˆ le of a Hamiltonian for the pair

of conjugate variables (x, y) and it is used to describe the

dynamics of a test particle, located at (x, y) and advected

by the flow. By substituting [8] into [1], we obtain

ðrÞ¼!ðrÞ½9

that is, a Poisson equation with ! as a source term.

Then, once we specify the vorticity field, by

inverting [9] we obtain the stream function  to be

ðrÞ¼

Gðr; r

Þ!ðr

Þdr

½10

where G(r, r

) is the Green’s function, solution of

the equation G(x, y) = (x, y). The Green’s func-

tion both for the plane and the sphere is (Marchioro

and Pulvirenti 1993)

Gðr; r

Þ¼

4

log kr  r

½11

where kr  r

= (x  x

)

þ (y  y

)

.By[10],once

we specify the vorticity field !(r) we can compute ,

and by replacing it into [8] the velocity field becomes

uðrÞ¼

Kðr; r

Þ!ðr

Þdr

½12

where K(r, r

) = ( r  r

)

=½2kr  r

 and it

represents the velocity field generated by a point

vortex of intensity one, located at r

. Then by

considering the vorticity field generated by point

vortices, eqn [3], together with eqn [11], eqn [10]

becomes

ðrÞ¼

4

log kr  r

¼1





ðr

 r



¼

4

¼1





log kr  r



½13

Equation [13] describes together with [8], the

dynamics of a test particle at a point r = ( x, y)in

the plane. Analogously, it can be shown that the

dynamics of a systems of point vortices in the plane

is given by the equations





; 



¼



½14

where (q



, p



) = (x



, 



),  = 1, ..., N, is a pair of

conjugate variables and H

is the generalization of

the stream function  (eqn [13]):

¼

4

;¼1

6¼









log kr



 r



½15

Notice that the vortex Hamiltonian H

(eqn [15])is

an autonomous Hamiltonian and, as we will discuss

in the first subsection, it provides a good Lyapunov-

like function to study stability properties of some

vortex configurations. Moreover, H

is invariant

with respect to rotations and translations, then by

the Noether theorem there are other first integrals of

motion, that is,

L ¼

k¼1



k x

; M

k¼1



;

k¼1



Figure 5 Photographs of vortex configurations in a rotated

sample of superfluid helium with 1, ...,11 vortices. Reprinted

figure with permission from Yarmchuk EJ, Gordon MJV, and

Packard RE (1979) Observation of stationary vortices arrays in

rotating superfluid Helium. Physical Review Letters 43(3): 214–

217. Copyright (1979) by the American Physical Society.

68 Point-Vortex Dynamics

expressing, respectively, the conservation of angular

momentum, L, and linear momentum, M =

, M

), on the plane. We shall denote with M

the magnitude of M (i.e., M = kMk). Furthermore,

by introducing the Poisson bracket

½f ; g¼

¼1









¼1











we can construct three integrals in involution out of

the four conserved quantities L, M

, M

, and H

These are L, M

þ M

and H

: in fact,

½H

; L¼0; H

; M

þ M

¼0;

L; M

þ M

¼0

It is then possible to reduce the system of equations

from N to N  2 degrees of freedom. A Hamiltonian

system with N degrees of freedom is integrable

whenever there are N independent integrals of

motion in involution. It follows that a vortex system

with N  3 is integrable, wher eas the system of

equations of four identical vortices has been shown

by Ziglin to be nonintegrable in the sense that there

are no other first integrals analytically depending on

the coordinates and circulations, and functionally

independent of L, H

, M

(see Ziglin (1982)).

The following, however, has been shown:

1. Let K =

 = 1



be the total vorticity,

M = (M

, M

) the total momentum and M = kMk .

Then, as shown by Aref and Stremler (1999),ifK = 0

and M = 0, N-vortex problem [16] is integrable.

2. A system of four identical vortices (i.e., k



= k

for  = 1, ..., 4) can undergo periodic or quasi-

periodic motion for special initial conditions (see

Khanin (1981) Russian Math. Surveys 36: 231;

Aref and Pomphrey (1982) Proc. R. Soc. Lond. A

380: 359–387). More specifically, the motion of a

system of four identical vortices can be periodic,

quasiperiodic, or chaotic depending on the symme-

try of the initial configuration. In fact, every vortex

configuration that belongs to the subspace of

symmetric configurations – x



= x

þ2

and y



þ2

,  = 1, 2 – gives rise to an integrable vortex

motion.

We have that up to two vortices, the motion is

almost always periodic and the orbits are circles; the

only exception being the case for which k

= k

when the circles degenerate into straight lines. Thus,

a configuration of two point vortices is always a

relative equilibrium configuration, that is, there exists

a specific reference frame in which the two vortices

are at rest. If the vortices are identical (

=

=),

the motion is synchronous with frequency ==

and the vortices share the same circular orbit (see

Figure 6a). If the vortices are not identical and have

vorticities of different magnitudes (say j

j > j

j),

their motion is still synchronous and periodic, with

frequency =(

þ 

)=(2), and the vortices move

on different circular orbits (with r

< r

) both

centered at the center of vorticity. Note that for

both cases, identical and nonidentical vortices, we

can view the vortex dynamics in a co-rotating frame

where the vortices are simply at rest.

For three vortices we can have periodic and

quasiperiodic motion, depending on the initial

conditions, and for four vortices we can have

periodic, quasiperiodic, or weakly chaotic motion.

Remarks

(i) The nonintegrability of the 4-vortex system was

also proved for configurations of nonidentical vortices.

Koiller and Carvalho (1989) gave an analytical proof

for 

= 

and 

=

= ,0   1. Moreover,

Castilla et al. (1993) considered the case:



=

= 1and

= .

(ii) Due to the translational and rotational

symmetries of H

, there are some analogies between

the N-vortex problem and the N-body problem,

especially for what concerns configurations of

relative equilibria (see Albouy (1996) and Glass

(2000)). A relative equilibrium is a vortex (or mass)

configuration that moves without change of shape

or form, that is, a configuration which is steadily

Γ = Γ

⏐⏐⏐⏐

–Γ

0 < Γ

< Γ

(a) (b)

Figure 6(a–d) For N = 2 the vortex dipole exhibits a synchro-

nous and the orbits are in general circular orbits, with the

exception of the case (d) for which 

= 

and the circular

orbit degenerates into a line (or a circle of infinite radius).

Point-Vortex Dynamics 69

rotating or translating. A few examples are vortex

polygons (see Figure 7) like the ones studied by

Thomson, Mayer, Yarmchuk–Gordon–Packard,

Boatto–Cabral (2003), Cabral–Schmidt (1999/

2000), Dritschel–Polvani (1993), Lim–Montaldi–

Roberts (2001), Sakajo (2004). For an exhaust ive

review on relative equilibria of vortices, see the

article by Aref et al. (2003). We shall discuss

stability of polygonal vortex configu ration in the

following subsection.

(iii) As shown by Kimura (1999) in a beautiful

geometrical formalism, on the unit sphere (S

) and

on the Hyperbolic plane (H

), the vortex Hamilto-

nians [15] are

¼

4

6¼









logð1  cos 



Þ on S

¼

4

6¼









log

cosh 



 1

cosh 



þ 1

on H

where

cos 



¼ cos 



cos 



þ sin 



sin 



cosð



 



Þ on S

cosh 



¼ cosh 



cosh 



þ sinh 



sinh 



cosð



 



Þ on H

On S

, 



and 



are, respectively, the co-latitude

and the longitude of the -vortex,  = 1, ..., N.We

can define cano nical varia bles q



and p



on S

and

, respectively, as



¼ 



cos 



; p



¼ 



on S



¼ 



cosh 



; p



¼ 



on H

Montaldi et al. (2002) studied vortex dyna mics on

a cylindrical surface, and Soulie` re and Tokieda

(2002) considered vortex dynamics on surfaces

with symmetries.

(iv) As we shall see in the section on point

vortex motion, it is sometimes useful to employ

the complex analysis formalism. Then the vari-

ables of interest are z



= x



þ iy



,  = 1, ..., N,and

its conjugate





, the Hami ltonian [15] takes the

form

¼

2

6¼









log jz



 z



and the equations of motions become



2

6¼;¼1







 z





 z



;¼ 1; ...; N ½16

(v) Equation [14] can we rewritten in a more

compact form as

¼ Jr

½17

where

X ¼ðq

; ...; q

; p

; ...; p

; ...;

;

; ...;



J ¼

O I

I O



I being the N  N identit y matrix.

(vi) How clos e is the point-vortex model to the

original Euler equation? Point-vortex systems repre-

sent discrete solutions of the Euler equation in a

‘‘weak’’ sense – see both the book and the article by

Marchioro and Pulvirenti (1993, 1994). These

authors proved that the Euler dynamics is ‘‘similar’’

to the vortex dynamics in which the vortices are

localized in very small regions, and the vorte x

intensities are the total vorticities associated to

such small regions. In particular, let us consider a

vorticity field with compact support on a family of

-balls, that is,



i¼1



with support of !



contained in the ball of center x

(independent of ) and radius . Furthermore let us

assume that

jrr

j



dr ¼ 

(a) (b)

Figure 7 Polygonal configuration of vortices: (a) planar

configurations and (b) configurations of vortex rings on a sphere,

with and without polar vortices.

70 Point-Vortex Dynamics

with the 

independent of . Then in the limit  !0

the dynamics of the center of vorticity



(t) =



(r, t)dr, of a given -ball, ‘‘converges’’

to the motion of a single point vortex (see Figure 8).

This result is important to illustrate as vortex

systems provide both a useful heuristic tool in the

analysis of the general properties of the solutions of

Euler’s equations (Poupaud 2002, Schochet 1995),

and a useful starting point for the construction of

practical algorithms for solving equations in specific

situations. In particular, it provides a theoretical

justification to the vortex method previously intro-

duced by Carnevale et al. (1992). These authors

constructed a numerical algorithm to study turbu-

lence decaying in two dimensions. Their vortex

method greatly simplifies fluid simulations as basi-

cally it relies on a discretization of the fluid into

circular patches. The dynamics of patches is given

by the centers of vorticity, which interact as a point-

vortex system, endowed with a rule dictating how

patches merge (see Figure 9).

Stability of a Vortex Ring

As mentioned in the Introduction section, the study

of vortex relative equilibria has a long history.

Kelvin showed that steadily rotating patterns of

identical vortices arise as solutions of a variational

problem in which the interaction energy (vortex

Hamiltonian) is minimized subject to the constraint

that the angular impulse be maintain ed (see Aref

(2003). In 1883, while studying and modeling the

atomic structure, J J Thomson investigated the linear

stability of co-rotating point vortices in the plane. In

particular, his interest was in configurations of

identical vortices equally spaced along the circum-

ference of a circle, that is, located at the vertices of a

regular polygon (see Figure 7). He proved that for

six or fewer vortices the polygonal configurations

are stable, while for seven vortices – the Thomson

heptagon – he erroneously concluded that the

configuration is slightly unstable. It took more

than a century to make some progresses on this

problem. D G Dritschel (1985) succeeded in solving

the heptagon mystery for what concerns its linear

stability analysis, leaving open the nonlinear stabi-

lity question: he proved that the Thomson heptagon

is neutrally stable and that for eight or more vortices

the corresponding polygonal configurations are

linearly unstable. Later on in 1993, Polvani and

Dritschel (1993) gen eralized the techniques used in

Dritschel (1985) to study the linear stability of a

‘‘latitudinal’’ ring of point vortices on the sphere, as

a function of the number N of vortices in the ring,

and of the ring’s co-latitude  (see Figure 10). They

proved that polygonal configurations are more

unstable on the sphere than in the plane. In

particular, they showed that at the pole, for N < 7

the configuration is stable, for N = 7 it is neutrally

stable and for N > 7itisunstable.Bymeansofthe

energy momentum method (Marsden–Meyer–Weistein

reduction), J E Marsden and S Pekarsky (1998)

studied the nonlinear stability analysis for the

integrable case of polygonal configurations of

three vortices of arbitrary vorticities (

, 

and



) on the sphere, leaving open the stability

analysis for nonintegrable vortex systems (N > 3).

In 1999 H E Cabral and D S Schmidt completed

the linear an d nonlinear stability analysis at once

for polygonal configurations in the plane. In 2003

Boatto and Cabral studied the nonlinear stability of

a ring of vortices on the sphere, as a function of the

number of vortices N and the ring colatitude .

Figure 8 In the limit  !0, the dynamics of the center of

vorticity of a vortex -ball is approximated by the dynamics of a

point vortex.

Figure 9 In Carnevale et al. (1992) the fluid is modeled by a

dilute vortex gas with density  and typical radius a . The

dynamics is governed by the point-vortex dynamics of the disk

centers, each disk corresponding to a point vortex of intensity

=

ext

, where 

ext

plays the role of a vorticity density. Two

vortices or radius a

and a

merge when their center-to-center

distance is less or equal to the sum of their radii, a

þ a

. Then a

new vortex is created and its radius a

is given by

= (a

þ a

)

1=4

Figure 10 Latitudinal ring of vortices. Reproduced with

permission from Boatto S and Cabral HE SIAM Journal of

Applied Mathematics 64: 216–230 (2003).

Point-Vortex Dynamics 71