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

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which is uniformly attractive. It would be pleasing if

, m

could be deduced easily from q, m. One

might guess q

= jnjq, m

= nm, in analogy with

monopoles. Unfortunately, this is false: q

, m

grow approximately exponentially with jnj.

Vortex Scattering

The AHM being Lorentz invariant, one can obtain

time-dependent solutions wherein a single n-vortex

travels at constant velocity, with speed 0 < v < 1

and E

tot

= (1  v

)

1=2

, by Lorentz boosting the

static solutions described above. Of more dynamical

interest are solutions in which two or more vortices

undergo relative motion. The simplest problem is

vortex scattering. Two vortices, initially well sepa-

rated, are propelled towards one another. In the

center-of-mass (COM) frame they have, as t !1,

equal speed v, and approach one another along

parallel lines distance b (the impact param eter)

apart, see Figure 4.Ifb = 0, they approach head-

on. Assuming they do not capture one another, they

interact and, as t !1, recede along parallel straight

lines having been deflected through an angle  (the

scattering angle). If scattering is elastic, the exit lines

also lie b apart and each vortex travels at speed v as

t !1. The dependence of  on v, b, and  has

been studied through lattice simulations by several

authors, perhaps most comprehensively by Myers,

Rebbi, and Strilka (1992). We shall now describe

their results.

Note first that vortex scattering is actually

inelastic: vortices recede with speed < v because

some of their initial kinetic energy is dispersed by

the collision as small-amplitud e traveling waves

(‘‘radiation’’). This energy loss can be as high as

80% in very fast collisions at small b. At small v the

energy loss is tiny, but can still have important

consequences for type I vortices: if v is very small,

they start with only just enough energy to escape

their mutual attraction. In undergoing a small b

collision they can lose enough of this energy to

become trapped in an oscillating bound state. In this

case they do not truly scatter and  is ill-defined.

Myers et al. find that v  0.2 suffices to avoid

capture when  = 1=2. Since type I vortices attract,

one might expect  to be always negative, indicating

that the vortices deflect towards one another. In

fact, as Figure 5a shows, this happens only for small

v and large b. Anot her naive expectation is that

=0or=180



when b = 0 (either vortices pass

through one another or ricochet backwards in a

head-on collision). In fact =90



, the only other

possibility allowed by reflexion symmetry of the

initial data. Figure 6 depicts snapshots of such a

scattering process at modest v. The vortices deform

each other as they get close until, at the moment of

coincidence, they are close to the static 2-vortex

ring. They then break apart along a line perpendi-

cular to their line of approach. One may consider

them to have exchanged half-vortices, so that each

emergent vortex is a mixture of the incoming

vortices. This rather surprising phenomenon was

actually predicted by Ruback in advance of any

numerical simulations and turns out to be a generic

feature of planar topological solitons.

Consider now the type II case ( = 2, Figure 5b).

Here,  > 0 for all v, b as one expects of particles

that repel each other. Head-on scattering is more

interesting now since two regimes emerge: for v >

crit

 0.3, one has the surprising 90



scattering

already described, while for v < v

crit

the vortices

bounce backwards, =180



. This is easily

explained. In order to undergo 90



head-on scatter-

ing, the vortices must become coincident (otherwise

reflexion symmetry is violated), hence must have

initial energy at least E

. For v < v

crit

, where

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1  v

crit

¼ E

½17

they have too little energy, so come to a halt before

coincidence, then recede from one another. The

solution v

crit

of [17] depends on  and is plotted in

Figure 7. For v slightly above v

crit

, we see that, in

contrast to the type I case, (b) is not monotonic:

maximum deflection occurs at nonzero b.

The point vortex formalism yields a simple model

of type II vortex scattering which is remarkably

successful at small v. One writes down the Lagrangian

for two identical (nonrelativistic) point particles of

mass E

moving along trajectories x

(t), x

(t) under

the influence of the repulsive potential E

int

L ¼

ðj

þj

ÞE

int

ðjx

 x

jÞ ½18

Energy and angular momentum conservation reduce

(v, b) to an integral over one variable (s = jx

 x

which is easily computed numerically. To illustrate,

Figure 5b shows the result for  = 2, v = 0.1

in comparison with the lattice simulations of

Figure 4 The geometry of vortex scattering.

Abelian Higgs Vortices 155

Myers et al. The agreement is almost perfect. For

large v the approximation breaks down not only

because relativistic corrections become significant,

but also because small b collisions then probe the small

 x

j region where vortex core overlap effects

become important. For the same reason, the point

vortex model is less useful for type I scattering.

Here there is no repulsion to keep the vortices well

separated, so its validity is restricted to the small v,

large b regime.

Critical coupling is theoretically the most inter-

esting regime, wher e most analytic progress has been

made. Since E

int

 E

int

 0, one might expect vortex

scattering to be trivial ((v, b)  0), but this is quite

wrong, as shown in Figure 5c. In particular,

(v,0)= 90



for all v, just as in the large v type I

and type II cases. The point is that scalar attraction

and magnetic repulsion of vortices are mediated by

fields with different Lorentz transformation proper-

ties. While they cancel for static vortices, there is no

reason to expect them to cancel for vortices in

relative motion.

Critical Coupling

The AHM with  = 1 has many remarkable prop er-

ties, at which we have so far only hinted. These all

stem from Bogomol’nyi’s crucial observa tion

(Manton and Sutcliffe 2004, pp. 197–202) that the

potential energy in this case can be rewritten as

0 1 2 3 4 5 6

–50

100

(a)

1 2 3 4 5 6

(c)

0 1 2 3 4 5 6

100

120

140

160

180

(b)

Figure 5 The 2-vortex scattering angle  as a function of impact parameter b for v = 0.1 (5), v = 0.2 (4),

v = 0.3 (}), v = 0.4 (&), v = 0.5 (), and v = 0.9 (þ), as computed by Myers et al. (1992): (a)  = 1=2; (b)  = 2; (c)  = 1. The

dotted curves are merely guides to the eye. The solid curves in (b), (c) were computed using the point vortex model. Note that Myers

et al. use different normalizations, so b =

ﬃﬃﬃ

MRS

and  = 

MRS

=2.

156 Abelian Higgs Vortices

E ¼

B 

ð1 jj



(

þjD

 þiD

j

þ B



x  i

dð



DÞ½19

The last integral vanishes by Stokes’s theorem, so

E  n by flux quantization [6], and E = n if and

only if

ðD

þ iD

Þ ¼ 0 ½20

ð1 jj

Þ¼B ½21

Note that system [20], [21] is first order, in contrast

to the second-order field equations [3]. No explicit

solutions of [20], [21] are known. However, Taubes

has proved that for each unordered list

, z

, ..., z

]ofn points in C, not necessarily

distinct, there exists a solution of [20], [21], unique

up to gauge transformations, with (z

) = (z

) =

 = (z

) = 0 and  nonvanishing elsewhere, the

zero at z

having the same multiplicity as z

has in

the list. Note that the list is unordered: a solution is

uniquely determined by the positions and multi-

plicities of the zeroes of , but the order in which we

label these is irrelevant. The solution minimizes E

within the class C

of winding n configurations, so is

automatically a stable static solution of the model.

Equation [20] applied to the symmetric n-vortex,

 = (r)e

in

, A = a( r)d implies a(r) = n  r

(r)=(r).

Comparing with [8], [9], it follows that q

= m

when  = 1 as previously claimed, since K

= K

Tong has conjectured, based on a string duality

argument, that q

= 28

1=4

. This is consistent with

current numerics but has no direct derivation so far.

Figure 6 Snapshots of the energy density during a head-on

collision of vortices. This 90



scattering phenomenon is a

generic feature of planar topological soliton dynamics.

1 2 3 4 5 6 7 8

0.1

0.2

0.3

0.4

0.5

crit

Figure 7 The critical velocity for 90



head-on scattering of type

II vortices v

crit

as a function of , as predicted by equation [17]

(solid curve), in comparison with the results of Myers et al.

(1992), (crosses).

Abelian Higgs Vortices 157

Taubes’s theorem shows that this n-vortex is just

one point, correspon ding to the list [0, 0, ..., 0], in a

2n-dimensional space of static multivortex solutions

called the moduli space M

. This space may be

visualized as the flat, finite-dimensional valley

bottom in C

on which E attains its minimum

value, n. Points in M

are in one-to-one correspon-

dence with distinct unordered lists [z

, z

, ..., z

which are themselves in one-to-one correspondence

with points in C

, as follows. To each list, we assign

the unique monic polynomial whose roots are z

pðzÞ¼ðz  z

Þðz  z

Þðz  z

¼ a

þ a

z þþa

n1

þ z

½22

This polynomial is uniquely determined by its

coefficients (a

, a

, ..., a

n1

) 2 C

, which give good

global coordinates on M

ﬃ C

. The zeros z

of 

may be used as local coordinates on M

, away from

, the subset of M

on which two or more of the

zeros z

coincide, but are not good global

coordinates.

Let ( , A)

denote the static solution correspond-

ing to a 2 C

. If the zeros z

are all at least s apart,

Taubes showed the solution is just a linear super-

position of 1-vortices located at z

, up to corrections

exponentially small in s. Imagine these constituent

vortices are pushed with small initial velocities.

Then ((t), A(t)) must remain close to the valley

bottom M

, since departing from it costs kinetic

energy, of which there is little. Manton has

suggested, therefore, that the dynamics is well

approximated by the constrained variational problem

wherein ((t), A(t)) = (, A)

a(t)

for all t.Since

the action S =

x =

kin

 E)dt,andE = n,

constant, on M

, this constrained problem amounts

to Lagrangian mechanics on configuration space M

with Lagrangian L = E

kin

.NowE

kin

is real,

positive, and quadratic in time derivatives of , A,so

L ¼



ðaÞ



½23



forming the entries of a positive-definite n  n

Hermitian matrix (

 

). Since ( , A)

is not

known explicitly, neither are 

(a). Observe, how-

ever, that L is the Lagrangian for geodesic motion in

with respect to the Riemannian metric

 ¼ 

ðaÞda



½24

Manton originally proposed this geodesic approx-

imation for monopoles, but it is now standard for all

topological solitons of Bogomol’nyi type (where one

has a moduli space of static multisolitons saturating

a topological lower bound on E). Note that

geodesics are independent of initial speed, which

agrees with Myers et al: Figure 5c shows that (v, b)

is approximately independent of v for v  0.5.

Further, Stuart (1994) has proved that, for initial

speeds of order , small, the fields stay (point wise) 

close to their geodesic approximant for times of

order 

1

On symmetry grou nds, two vortex dynami cs in

the COM frame reduces to geodesic motion in M

ﬃ

C, the subspace of centered 2-vortices (a

= 0, so

= z

), with induced metric



¼ Gðja

jÞda



½25

G being some positive function. Note that a

= z

so the intervortex distance jz

 z

j= 2jz

j= 2ja

1=2

The line a

=  2 R, traversed with  increasing, say,

is geodesic in M

. The vortex positions (roots of

þ a

)are

ﬃﬃﬃﬃﬃﬃ

jj

for   0andi

ﬃﬃﬃ



for >0.

This describes perfectly the 90



scattering phenom-

enon: two vortices approach head-on along the x

axis, coincide to form a 2-vortex ring, then break

apart along the x

axis, as in Figure 6. This behavior

occurs because a

= z

, rather than z

 z

,isthe

correct global coordinate on M

, since vortices are

classically indistinguishable.

Samols found a useful formula (Manton and

Sutcliffe 2004, pp. 205–215) for  in terms of the

behavior of j

j close to its zeros, using which he

devised an efficient numerical scheme to evaluate

G(ja

j), and computed (b) in detail, finding

excellent agreement with lattice simulations at low

speeds. He also studied the quantum scattering of

vortices, approximating the quantum state by a

wave function  on M

evolving according to the

natural Schro¨ dinger equation for quantum geodesic

motion,

ih

@

¼

h





 ½26

where 



is the Laplace–Beltrami operator on

, ). This technique, introduced for monopoles

by Gibbons and Manton, is now standard for

solitons of Bogomol’nyi type.

By analyzing the forces between moving point

vortices at  = 1, Manton and Speight (2003)

showed that, as the vortex separations become

uniformly large, the metric on M

approaches



¼







4

s6¼r

ðjz

 z

jÞ

ðdz

 dz

Þðd



 d





½27

This formula can also be obtained by a method of

matched asymptotic expansions. We can use [27] to

study 2-vortex scattering for large b, when the

158 Abelian Higgs Vortices

vortices remain well separated. (Note that 

is not

positive definite if any jz

 z

j becomes too small.)

The results are good, provided v  0.5 and b  3

(see Figure 5c).

Other Developments

The (critically coupled) AHM on a compact physical

space  is of considerable theoretical and physical

interest. Bradlow showed that M

() is empt y unless

V = Area()  4n, so there is a limit to how many

vortices a space of finite area can accommodate

(Manton and Sutcliffe 2004, pp. 227–230). Manton

has analyzed the thermodynamics of a gas of

vortices by studying the statistical mechanics of

geodesic flow on M

(). In this context, spatial

compactness is a technical device to allow nonzero

vortex density n=V for finite n, without confining

the fields to a finite box, which would destroy the

Bogomol’nyi properties. In the limit of interest,

n, V !1 with n/V fixed, the thermodynamical

properties turn out to depend on  only through

V,so=S

and =T

give equivalent results, for

example. The equation of state of the gas is

(P = pressure, T = temperature)

P ¼

V  4n

½28

which is similar, at low density n/V, to that of a gas

of hard disks of area 2. The crucial step in deriving

[28] is to find the volume of M

() which, despite

there being no formula for , may be computed

exactly by remarkable indirect arguments (Manton

and Sutcliffe 2004, pp. 231–234).

The static AHM coincides with the Ginzburg–

Landau model of superconductivity, which has

precisely the same type I/II clas sification. Here the

‘‘Higgs’’ field represents the wave function of a

condensate of Cooper pairs, usually (but not always)

electrons. There has been a parallel development of

the static model by condensed matter theorists,

therefore; see Fossheim and Sudbo (2004), for

example. In fact the vortex was actually first

discovered by Abrikosov in the condensed matter

context. One important difference is that type I

superconductors do not support vortex solutions in

an external magnetic field B

ext

because the critical

ext

j required to create a single vortex is greater

than the critical jB

ext

j required to destroy the

condensate completely (  0). Type II supercon-

ductors do suppo rt vortices, and there are such

superconductors with   1, but the vortex

dynamics we have described is not relevant to these

systems. In this context there is an obvious preferre d

reference frame (the rest frame of the superconduc-

tor) so it is unsurprising that the Lorentz-invariant

AHM is inappropriate. Insofar as vortices move at

all, they seem to obey a first-order (in time)

dynamical system, in contrast to the second-order

AHM. Manton has devised a first-order system

which may have relevance to superconductivity, by

replacing E

kin

with a Chern–Simons–Schro¨ dinger func-

tional (Manton and Sutcliffe 2004, pp. 193–197).

Rather than attracting or repelling, vortices now

tend to orbit one another at constant separation.

There is again a moduli space approximation to

slow vortex dynamics for   1, but it has a

Hamiltonian-mechanical rather than Riemannian-

geometric flavor.

Finally, an interesting simplification of the AHM,

which arises, for example, as a phenomenological

model of liquid helium-4, is obtained if we discard the

gauge field A



, or equivalently set the electric charge of

 to e = 0. There is now no type I/II classification, since

 may be absorbed by rescaling. The resulting model,

which has only global U(1) phase symmetry, supports

n-vortices  = (r)e

in

for all n, but these are not

exponentially spatially localized,

ðrÞ¼1 

r



ð8 þ n

2

þ Oðr

6

Þ½29

and cannot have finite E by Derrick ’s theorem. They

are unstable for jnj > 1, and 1-vortices uniformly

repel one another. They can be given an interesting

first-order dynamics (the Gross–Pitaevski equation).

Abbreviations



electromagnetic gauge potential

b impact parameter



gauge-covariant derivative

E potential energy

kin

kinetic energy



electromagnetic field strength tensor

L Lagrangian

L Lagrangian density

S action

 Higgs field

 scattering angle

See also: Fractional Quantum Hall Effect;

Ginzburg–Landau Equation; High T

Superconductor

Theory; Integrable Systems: Overview; Nonperturbative

and Topological Aspects of Gauge Theory; Quantum

Fields with Topological Defects; Solitons and Other

Extended Field Configurations; Symmetry Breaking in

Field Theory; Topological Defects and Their Homotopy

Classification; Variational Techniques for

Ginzburg–Landau Energies.

Abelian Higgs Vortices 159

Further Reading

Atiyah M and Hitchin N (1988) The Geometry and Dynamics of

Magnetic Monopoles. Princeton: Princeton University Press.

Fossheim K and Sudbo A (2004) Superconductivity: Physics and

Applications. Hoboken NJ: Wiley.

Jaffe A and Taubes C (1980) Vortices and Monopoles: Structure

of Static Gauge Theories. Boston: Birkha¨ user.

Nakahara M (1990) Geometry, Topology and Physics. Bristol:

Adam-Hilger.

Manton NS and Speight JM (2003) Asymptotic interactions of

critically coupled vortices. Communications in Mathematical

Physics 236: 535–555.

Manton NS and Sutcliffe PM (2004) Topological Solitons.

Cambridge: Cambridge University Press.

Myers E, Rebbi C, and Strilka R (1992) Study of the interaction

and scattering of vortices in the abelian Higgs (or Ginzburg-

Landau) model. Physical Review 45: 1355–1364.

Rajaraman R (1989) Solitons and Instantons. Amsterdam: North-

Holland.

Stuart D (1994) Dynamics of abelian Higgs vortices in the near

Bogomolny regime. Communications in Mathematical Physics

159: 51–91.

Vilenkin A and Shellard EPS (1994) Cosmic Strings and Other

Topological Defects. Cambridge: Cambridge University Press.

Adiabatic Piston

Ch Gruber, Ecole Polytechnique Fe

rale de

Lausanne, Lausanne, Switzerland

A Lesne, Universite

P.-M. Curie, Paris VI, Paris,

France

Introduction

Macroscopic Problem

The ‘‘adiabatic piston’’ is an old problem of

thermodynamics which has had a long and con-

troversial history. It is the simplest example con-

cerning the time evolution of an adiabatic wall, that

is, a wall which does not conduct heat. The system

consists of a gas in a cylin der divided by an

adiabatic wall (the piston). Initially, the piston is

held fixed by a clamp and the two gases are in

thermal equilibrium character ized by (p



, T



, N



where the index /þ refers to the gas on the left/right

side of the piston and (p, T, N) denote the pressure,

the temperature, and the number of particles

(Figure 1). Since the piston is adiabatic, the whole

system remains in equilibr ium even if T



6¼ T

.At

time t = 0, the clamp is removed and the piston is let

free to move without any friction in the cylinder. The

question is to find the final state, that is, the final

position X

of the piston and the parameters (p



, T



)

of the gases .

In the late 1950s , using the two laws of

equilibrium thermodynamics (i.e., thermostatics),

Landau and Lifshitz concluded that the adiabatic

piston will evolve toward a final state where



= p

. Later, Callen (1963) and others

realized that the maximum entropy condition

implies that the system will reach mechanical

equilibrium where the pressures are equal p



= p

;

however, nothing could be said concerning the final

position X

or the final temperatures T



which

should depend explicitly on the viscosity of the

fluids. It thus became a controversial problem since

one was forced to accept that the two laws of

thermostatics are not sufficient to predict the final

state as soon as adiabatic movable walls are

involved (see early referenc es in Gruber (1999)).

Experimentally, the adiabatic piston was used

already before 1924 to measure the ratio c

the specific heats of gases. In 2000, new measure-

ments have shown that one ha s to distinguish

between two regimes, corresponding to weak damp-

ing or strong damping, with very different proper-

ties, for example, for weak damping the frequency

of oscillations corresponds to adiabatic oscillations,

whereas for strong damping it corresponds to

isothermal oscillations.

Microscopic Problem

The ‘‘adiabatic piston’’ was first considered from a

microscopic point of view by Lebowitz who intro-

duced in 1959 a simple model to study heat

conduction. In this model, the gas consists of point

particles of mass m making purely elastic collisions

on the wall of the cylinder and on the piston.

Furthermore, the gas is very dilute so that the

–

Figure 1 The adiabatic piston problem.

160 Adiabatic Piston

equation of state p = nk

T is satisfied at equili-

brium, where n is the density of particles in the gas

and k

the Boltzmann constant. The adiabatic piston

is taken as a heavy particle of mass M  m without

any internal degree of freedom. Using this same

model Feynman (1965) gave a qualitative analysis in

Lectures in Physics. He argued intuitively but

correctly that the system should converge first

toward a state of mechanical equilibrium where



= p

and then very slowly toward thermal

equilibrium. This approach toward thermal equili-

brium is associated with the ‘‘wiggles’’ of the piston

induced by the random collisions with the atoms of

the gas. Of course, this stochastic behavior is not

part of thermodynamics and the evolution beyond

the mechanica l equilibrium cannot appear in the

macroscopical framework assuming that the piston

does not conduct heat.

From a microscopical point of view, one is

confronted with two different problems: the

approach toward mechanical equilibrium in the

absence of any a priori friction (where the entropy

of both gases should increase) and, on a different

timescale, the approach toward thermal equilibrium

(where the entropy of one gas should decrease but

the total entropy increase).

The conceptual difficulties of the problem beyond

mechanical equilibrium come from the following

intuitive reasoning. When the piston moves toward

the hotter gas, the atoms of the hotter gas gain

energy, whereas those of the cooler gas lose energy.

When the piston moves toward the cooler side, it is

the opposite. Since on an average the hotter side

should cool down and the cold side should warm

up, we are led to conclude that on an average the

piston should move toward the colder side. On the

other hand, from p = nk

T, the piston should move

toward the warmer side to maintain pressure

balance.

In 1996, Crosignani, Di Porto, and Segev intro-

duced a kinetic model to obtain equations descr ibing

the adiabatic approach toward mechanical eq uili-

brium. Starting with the microsc opical model

introduced by Lebowitz, Gruber, Piasecki, and

Frachebourg, later joined by Lesne and Pache,

initiated in 1998 a systematic investigation of the

adiabatic piston within the framework of statistical

mechanics, together with a large number of numer-

ical simulations. This analysis was based on the fact

that m=M is a very small parameter to investigate

expansions in powers of m=M (see Gruber and

Piasecki (1999) and Gruber et al. (2003) and

reference therein). An approach using dynamical

system methods was then developed by Lebowitz

et al. (2000) and Chernov et al. (2002).An

extension to hard-disk particles was analyzed at

the same time by Kestemont et al. (2000). Recently,

several other authors have contributed to this

subject.

The genera l picture which emerges from all the

investigations is the following. For an infinite

cylinder, starting with mechanical equilibrium



= p

= p, the piston evolves to a stationary

stochastic state with nonzero velocity toward the

warmer side

hVi¼

ﬃﬃﬃﬃﬃﬃﬃﬃ

k

ﬃﬃﬃﬃﬃﬃﬃ



ﬃﬃﬃﬃﬃﬃﬃ



Þþo



½1

with relaxation time

 ¼

ﬃﬃﬃﬃﬃﬃﬃﬃ

k

ﬃﬃﬃﬃﬃﬃﬃ



ﬃﬃﬃﬃﬃﬃﬃ



1

½2

where M=A is the mass per unit area of the piston.

In this state the piston has a temperature

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ



and there is a heat flux

¼ð

ﬃﬃﬃﬃﬃﬃﬃ



ﬃﬃﬃﬃﬃﬃﬃ

ﬃﬃﬃﬃﬃﬃﬃﬃ

m

p þ o



ðp



¼ p

¼ pÞ½3

For a finite cylinder and p

6¼ p



, the evolution

proceeds in four different stages. The first two are

deterministic and adiabatic. They correspond to the

thermodynamic evolution of the (ma croscopic)

adiabatic piston. The last two stages, which go

beyond thermodynamics, are stochastic with heat

transfer across the piston. More precisely:

1. In the first stage whose duration is the time

needed for the shock wave to bounce back on the

piston, the evolution corresponds to the case of

the infinite cylinder (with p



6¼ p

). If

R = Nm=M > 10, the piston will be able to

reach and maintain a constant velocity

V ¼ðp



p

ﬃﬃﬃﬃﬃﬃﬃﬃ

k

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ



ﬃﬃﬃﬃﬃﬃﬃ



þp



ﬃﬃﬃﬃﬃﬃﬃ

þO



for jp



p

j1 ½4

2. In the second stage the evolution toward

mechanical equilibrium is either weakly or

strongly damped depending on R.IfR < 1, the

evolution is very weakly damped, the dynamics

takes place on a timescale t

ﬃﬃﬃﬃ

t, and the effect

of the collisions on the piston is to introduce an

external potential (X) = c

þ c

=(L  X)

On the other hand, if R > 4, the evolution is

strongly damped (with two oscillations only) and

depends neither on M nor on R.

Adiabatic Piston 161

3. After mechanical equilibrium has been reached,

the third stage is a stochastic approach tow ard

thermal equilibrium associated with heat transfer

across the piston. This evolution is very slow and

exhibits a scaling property with respect to

= mt=M.

4. After thermal equilibrium has been reached



= T

, p



= p

), in a fourth stage the gas

will evolve very slowly toward a state with

Maxwellian distribution of velocities, induced

by the collision with the stochastic piston.

The general conclusion is thus that a wall which is

adiabatic when fixed will become a heat conductor

under a stochastic motion. However, it should be

stressed that the time required to reach thermal

equilibrium will be several orders of magnitude larger

than the age of the universe for a macroscopical piston

and such a wall could not reasonably be called a heat

conductor. However, for mesoscopic systems, the effect

of stochasticity may lead to very interesting properties,

as shown by Van den Broeck et al. (2004) in their

investigations of Brownian (or biological) motors.

Microscopical Model

The system consists of two fluids separated by an

‘‘adiabatic’’ piston inside a cylinder with x-axis,

length L, and area A. The fluids are made of N



identical light particles of mass m. The piston is a

heavy flat disk, without any internal degree of

freedom, of mass M  m, orthogonal to the

x-axis, and velocity parallel to this x-axis. If the

piston is fixed at some position X

, and if the two

fluids are in thermal equilibrium characterized by



, T



, N



), then they will remain in equilibrium

forever even if T

6¼T



: it is thus an ‘‘adiabatic

piston’’ in the sense of thermodynamics. At a certain

time t = 0, the piston is let free to move and the

problem is to study the time evolution. To define the

dynamics, we consider that the system is purely

Hamiltonian, that is, the particles and the piston

move without any friction according to the laws of

mechanics. In particular, the collisions between the

particles and the walls of the cylinder, or the piston,

are purely elastic and the total energy of the system

is conserved. In most studies, one considers that the

particles are point particles making purely elastic

collisions. Since the piston is bound to move only in

the x-direction, the velocity components of the

particles in the transverse directions play no role in

this problem. Moreover, since there is no coupling

between the components in the x- and transverse

directions, one can simplify the model further by

assuming that all probability distributions are

independent of the transverse coordinates. We are

thus led to a formally one-dimensional problem

(except for normalizations). Therefore, in this

review, we con sider that the particles are noninter-

acting and all velocities are parallel to the x-axis.

From the collision law, if v and V denote the

velocities of a particle and the piston before a

collision, then under the collis ion on the piston:

v ! v

¼ 2V  v þ ðv  VÞ

V ! V

¼ V þ ðv  VÞ

½5

where

 ¼

M þ m

½6

Similarly, under a collision of a pa rticle with the

boundary at x = 0orx = L:

v ! v

¼v ½7

Let us mention that more general models have also

been considered, for example, the case where the

two fluids are made of point particles with different

masses m



, or two-dimensional models where the

particles are hard disks. However, no significant

differences appear in these more general models and

we restrict this article to the simplest case.

One can study different situations: L = 1, L

finite, and L !1. Furthermore, taking first M and

A finite, one can investigate several limits.

1. Thermodynamic limit for the piston only. In

this limit, L is fixed ( finite or infinite) and

A !1, M !1, keeping constant the initial

densities n



of the fluid and the parameter

 ¼

2mA

M þ m

¼ A  2m

½8

If L is finite, this means that N



!1 while

keeping constant the parameters



gas

½9

2. Thermodynamic limit for the whole system,

where L !1 and A  L

, N



 L

. In this

limit, space and time variables are rescaled

according to x

= x=L and t

= t=L. This limit

can be considered as a limiting case of (1) where





ﬃﬃﬃﬃ

!1 (and time is scaled).

3. Continuum limit where L and M are fixed and



!1, m !0 keeping M



gas

constant, that is,



= cte .

The case L infinite and the limit (1) have been

investigated using statistical mechanics (Liouville or

162 Adiabatic Piston

Boltzmann’s equations). On the other hand, the

limit (2) has been studied using dynamical system

methods, reducing first the system to a billiard in an

þ N



þ 1)-dimensional polyhedron. The limit

(3) has been introduced to derive hydrodynamical

equations for the fluids.

In this article, we present the approach based on

statistical mechanics. Although not as rigorous as (2)

on a mathematical level, it yields more informations

on the approach toward mechanical and thermal

equilibrium. Moreover, it indicates what are the

open problems which should be mathematically

solved. In all investigations, advantage is taken of

the fact that m/M is very small and one introduces

the small parameter

 ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

m=M

 1 ½10

Let us note that  measures the ratio of thermal

velocities for the piston and a fluid particle, whereas

  

measures the ratio of velocity changes during

a collision.

Starting Point: Exact Equations

Using the statistical point of view, the time evolution

is given by Liouville’s equation for the probability

distribution on the whole phase space for (N



þ 1) particles, with L, A, N



, and M finite.

Initially (t  0), the piston is fixed at (X

, V

= 0)

and the fluids are in thermal equilibrium with

homogeneous densities n



, velocity distributions

’



(v) = ’



(v), and temperatures



¼ m

1

dvn



’



ðvÞv

½11

Integrating out the irrelevant degrees of freedom,

the Liouville’s equation yields the equations for

the distribution 



(x, v; t) of the right and left

particles:





ðx; v; tÞþv@





ðx; v ; tÞ¼I



ðx; v ; tÞ½12

The collision term I



(x, v; t) is a functional of



, P

(X, v; X, V; t), the two-point correlation func-

tion for a right (resp. left) particle at (x = X, v) and

the piston at (X, V). Similarly, one obtains for the

velocity distribution of the pisto n:

ðV; tÞ¼A

1

ðV  vÞ ðV  v Þ



surf

ðv

; V

; tÞ



þðv  VÞ



surf

ðv ; V; tÞ



 A

1

ðV  vÞ ðv  VÞ

surf

ðv

; V

; tÞ



þðV  vÞ

surf

ðv ; V ; tÞ



dv ½13

where (v

, V

) are given by eqn [5] and





surf

ðv; V; tÞ¼

1

dX

; P

ðX; v ; X; V; tÞ½14

We thus have to solve eqns [12]–[13] with initial

conditions





ðx; v; t ¼ 0Þ¼n



’



ðvÞðxÞðX

 xÞ



ðx; v; t ¼ 0Þ¼n

’

ðvÞðL  xÞðx  X

ðV; t ¼ 0Þ¼ðVÞ

½15

Using the fact that  = 2m=(M þ m)  1, we can

rewrite eqn [13] as a form al s eries in powers of :

ðV; tÞ¼

k¼1

ð1Þ



k1



kþ1

ðV; tÞ½16

ðV; tÞ¼

ðv  VÞ





surf

ðv ; V; tÞdv



1

ðv  VÞ



surf

ðv ; V; tÞdv ½17

from which one obtains the equations for the

moments of the piston velocity:



dhV

k¼1



k1

k!ðn kÞ!

1

dVV

nk

kþ1

ðV; tÞ½18

However, we do not know the two-point correlation

functions.

If the length of the cylinder is infinite, the

condition M  m implies that the probability for

a particle to make more than one collision on the

piston is negligible. Alternatively, one could choose

initial distributions ’



(v) which are zero for jv j <

min

,wherev

min

is taken such that the probability

of a r ecollision is strictly zero. Therefore, if L = 1,

one can consider that before a collision on the

piston the particles are d istributed with ’



(v)for

all t, and the two-point correlation functions

factorize, that is,





surf

ðv; V; tÞ¼



surf

ðv; tÞðV; tÞ; if v > V



surf

ðv; V; tÞ¼

surf

ðv; tÞðV; tÞ; if v < V

½19

where for L = 1, 



surf

(v; t) = n



’



(v) and thus the

conditions to obtain eqn [18] are satisfied.

If L is finite, one can show that the factorization

property (eqn [19]) is an exact relation in the

thermodynamic limit for the piston (A !1,

M=A = cte). For finite L and finite A, we introduce

Adiabatic Piston 163

Assumption 1 (Factorization condition). Before a

collision the two-point correlation functions have the

factorization property (eqn [19]) to first order in .

Under the factorization condition, we have

ðV; tÞ¼F

ðV; tÞðV; tÞ½20

with

ðV; tÞ¼

dvðv  VÞ





surf

ðv; tÞ



1

dvðv  VÞ



surf

ðv; tÞ

¼ F



ðV; tÞF

ðV; tÞ½21

and from eqn [18]



hVi¼MhF

ðV; tÞi



½22



i¼M hVF

ðV;tÞi



þhF

ðV;tÞi



½½23

Introducing



V =hV i



then from eqns [12] and [20],

it follows that the (kinetic) energies satisfy





¼M



ðV; tÞi





þhðV 



VÞF



ðV; tÞi







ðV; tÞi



½24

which implies conservation of energy.

From the first law of thermodynamics,





P !

þ P

P !

½25

where P

P !

and P

P !

denote the work- and

heat-power transmitted by the piston to the fluid,

we conclude from eqns [22] and [ 25] that the heat

flux is

P!

¼M

hðV 



VÞF



ðV; tÞi







ðV; tÞi



½26

Since   1, it is interesting to introduce the

irreducible moments



¼hðV 



VÞ



½27

and the expansion around



V = hVi



ðV; tÞ¼

r¼0

ðr; Þ



VÞðV 



VÞ

½28

from which one obtains equations for d

=dt.In

particular, using the identities

ðrþ1; Þ

¼3F

ðr; Þ

; F

ðrþ2; Þ

¼ 2F

ðr; Þ

½29

in [22] and [24], we have



ðV; tÞ





¼F





V; tÞ

r0

ð2 þ rÞ!

ðr; Þ



2þr

½30





¼M





ðV; tÞ













V; tÞþ

r2

ð2r  3Þ

 F

ðr1; Þ



V; tÞ



½31

Depending on the questions or approximations one

wants to study, either the distribution (V; t) or the

moments hV

will be the interesting objects.

Finally, with the condition [19], one can take

eqn [12] for x 6¼ X

and impose the boundary

conditions at x = X





ðX

; v; tÞ¼



ðX

; v

; tÞ; if v < V



ðX

; v; tÞ¼

ðX

; v

; tÞ; if v > V

½32

and similarly for x = 0 and x = L with v

= v.

Let us note that this factorization condition is of

the same nature as the molecular chaos assumption

introduced in kinetic theory, and with this condition

eqn [13] yields the Boltzmann equation for this

model.

In the following, to obtain explicit results as a

function of the initial temperatures T



, we take

Maxwellian distributions ’



(v) and initial condi-

tions (p



, T



, n



) such that the velocity of the piston

remains small (i.e., jhVi

jjhv



j).

Distribution (V ; t) for the Infinite

Cylinder (L = 1)

To lowest order in  =

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

m=M

, and assuming

j1  p



j is of order , one obtains from eqn [16]

the usual Fokker–Planck equation whose solution

gives



ðV; tÞ¼

ﬃﬃﬃﬃﬃﬃ

2

ðtÞ

exp 

ðV 



ðtÞÞ

2

ðtÞ

½33

164 Adiabatic Piston