Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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supersymmetric extension of [14], one needs to
add a number of endomorphisms X
I
2End
T
E
that play the role of bosonic scalar fields in the
adjoint representation of the gauge group and a
number of odd Grassmann parity endomorphisms
i
2End
T
E endowed with an SO(d)-spinor
index . The latter ones are analogs of the usual
fermionic fields.
In string theory, one considers a maximally
supersymmetric extension of the Yang–Mills theory
[14]. In this case, the supersymmetric action depends
on 10 d bosonic scalars X
I
, I = d, ..., 9, and the
fermionic fields can be collected into an SO(9, 1)
Majorana–Weyl spinor multiplet
, = 1, ..., 16.
The maximally supersymmetr ic Yang–Mills action
takes the form
S
SYM
¼
V
4g
2
tr
F
F
þ½r
; X
I
½r
; X
I
þ½X
I
; X
J
½X
I
; X
J
2
½r
;
2
I
½X
I
;
½17
Here the curvature indices F
, , = 0, ..., d 1,
are assumed to be contracted with a Minkowski
signature metric, and
A
are blocks of the ten-
dimensional 32 32 gamma-matrices
A
¼
0
A
ð
A
Þ
0
!
; A ¼ 0; ...; 9
This action is invariant under two kinds of super-
symmetry transformations denoted by
,
~
and
defined as
¼
1
2
ð
jk
F
jk
þ
jI
½r
j
; X
I
þ
IJ
½X
I
; X
J
Þ
r
j
¼
j
;
X
J
¼
J
~
¼ ;
~
r
j
¼ 0;
~
X
J
¼ 0
½18
where is a co nstant 16-component Majorana–Weyl
spinor. Of particular interest for string theory
applications are solutions to the equations of motion
corresponding to [17] that are invariant under some
of the above supersymmetry transformations.
Further discussion can be found in Konechny and
Schwarz (2002).
Morita Equivalence
The role of Morita equivalence as a duality
transformation in noncommutative Yang–Mills
theory was elucidated by Schwarz (1998). We will
adopt a definition of Morita equivalence for
noncommutative tori which can be shown to be
essentially equivalent to the standard definition of
strong Morita equivalence. We will say that two
noncommutative tori T
d
and T
d
ˆ
are Morita equiva-
lent if there exists a (T
d
, T
d
ˆ
)-bimodule Q and a
(T
d
ˆ
, T
d
)-bimodule P such that
Q
T
^
P ffiT
; P
T
Q ffiT
^
½19
where T
on the right-hand side is considered as a
(T
, T
)-bimodule and analogously for T
ˆ
. (It is
assumed that the isomorphisms are canonical.)
Given a T
-module E one obtains a T
ˆ
-module
^
E as
^
E ¼P
T
E ½20
One can show that this mapping is functorial.
Moreover, the bimodule Q provides us with an
inverse mapping Q
T
ˆ
^
E ffiE.
We further introduce a notion of gauge Morita
equivalence (originally called ‘‘complete Morita
equivalence’’) that allows one to transport
connections along with the mapping of modules
[20].LetL be a d-dimensional commutative Lie
algebra. We say that the (T
d
ˆ
, T
d
) Morita equiva-
lence bimodule P establishes a gauge Morita
equivalence if it is endowed with operators
r
P
X
, X 2L that determine a constant curvature
connection simultaneously with respect to T
d
and
T
d
ˆ
, that is, s atisfy
r
P
X
ðeaÞ¼ r
P
X
e
a þeð
X
aÞ
r
P
X
ð
^
aeÞ¼
^
a r
P
X
e
þð
^
X
^
aÞe
r
P
X
; r
P
Y
¼2i
XY
1
½21
Here
X
and
^
X
are standard derivations on T
and
T
ˆ
, respectively. In other words, we have two Lie
algebra homomorphisms
: L !L
;
^
: L !L
^
½22
If a pair (P, r
P
X
) specifies a gauge (T
, T
ˆ
)-
equivalence bimodule, then there exists a correspon-
dence between connections on E and connections on
^
E. The connection
^
r
X
on
^
E corresponding to a
given connection r
X
on E is defined as
r
X
7!
^
r
X
¼1 r
X
þr
P
X
1 ½23
More precisely, an operator 1 r
X
þr
P
X
1on
P
C
E descends to a connection
^
r
X
on
^
E = P
T
E.
It is straightforward to check that under this
mapping gauge equivalent connections go to gauge
equivalent ones,
Z
y
r
X
Z ¼
^
Z
y
^
r
X
^
Z
where
^
Z = 1 Z is the endomorphism of
^
E = P
T
E
corresponding to Z 2End
T
d
E.
Noncommutative Tori, Yang–Mills, and String Theory 527
The curvatures of
^
r
X
and r
X
are connected by
the formula
F
r
XY
¼
^
F
r
XY
þ 1
XY
½24
which in particular shows that constant curvature
connections go to constant curvature ones.
Since noncommutative tori are labeled by an
antisymmetric d d matrix , gauge Morita equiva-
lence establishes an equivalence relation on the set
of such matrices. To describe this equivalence
relation, consider the action 7!h =
ˆ
of
SO(d, djZ) on the space of antisymmetric d d
matrices by the formula
^
¼ðM þNÞðR þSÞ
1
½25
where the d d matrices M, N, R, S are such that
the matrix
h ¼
MN
RS
½26
belongs to the group SO(d, djZ). The above action is
defined whenever the matrix A R þ S is inverti-
ble. One can prove that two noncommutative tori
T
d
and T
ˆ
are gauge Morita equivalent if and only if
the matrices and
ˆ
belong to the same orbit of the
SO(d, djZ) action [25].
The duality group SO(d, djZ) also acts on the
topological numbers of moduli 2
even
(Z
d
). This
action can be shown to be given by a spinor
representation constructed as follows. First note
that the operators a
i
=
i
, b
i
= @=@
i
act on (R
d
)
and give a representation of the Clifford algebra
specified by the metric with signature (d, d). The
group O(d, djC) can thus be regarded as a group of
automorphisms acting on the Clifford algebra
generated by a
i
, b
j
. Denote the latter action by W
h
for h 2O(d, djC). One defines a projective action V
h
of O(d, djC)on(R
d
) according to
V
h
a
i
V
1
h
¼W
h
1
ða
i
Þ; V
h
b
j
V
1
h
¼W
h
1
ðb
j
Þ
This projective action can be restricted to yield a
double-valued spinor representation of SO(d, djC)
on (R
d
) by choosing a suitable bilinear form on
(R
d
). The restrictio n of this representation to the
subgroup SO(d, djZ ) acting on
even
(Z
d
) gives the
action of Morita equivalence on the topological
numbers of moduli.
The mapping [23] preserves the Yang–Mills
equations of motion [15]. Moreover, one can define
a modification of the Yang–Mills action functional
[14] in such a way that the values of the functionals
on r
X
and
^
r
X
coincide up to an appropriate
rescaling of coupling constants. The modified action
functional has the form
S
YM
¼
V
4g
2
trðF
jk
þ
jk
1ÞðF
jk
þ
jk
1Þ½27
where
jk
is a scalar-valued tensor that can be
thought of as some background field. Adding this
term will allow us to compensate for the curvature
shift by adopting the transformatio n rule
XY
7!
XY
XY
Note that the new action functional [27] has the
same equations of motion [15] as the original one.
To show that the functional [27] is invariant
under gauge Morita equivalence, one has to take
into account two more effects. Firstly, the values of
trace change by a factor c = dim (
^
E)( dim(E))
1
as
^
tr
^
X = ctr X. Secondly, the identification of L
and
L
ˆ
is established by means of some linear transfor-
mation A
k
j
, the deter minant of which will rescale the
volume V. Both effects can be absorbed into an
appropriate rescaling of the coupling constant.
One can show that the curvature tensor, the
metric tensor , the background field
ij
, and the
volume element V transform according to
F
^
r
ij
¼A
k
i
F
r
kl
A
l
j
þ
ij
^
g
ij
¼A
k
i
g
kl
A
l
j
^
ij
¼A
k
i
kl
A
l
j
ij
^
V ¼Vjdet Aj
½28
where A = R þ S and = RA
t
. The action func-
tional [27] is invariant under the gauge Morita
equivalence if the coupling constant transforms
according to
^
g
2
YM
¼g
2
YM
jdet Aj
1=2
½29
Supersymmetric extensions of Yang–Mills theory
on noncomm utative tori were shown to arise within
string theory essentially in two situations. In the first
case, one considers compactifications of the (BFSS
or IKKT) matrix model of M-theory (Connes et al.
1998). A discussion regarding the connection
between T-duality and Morita equivalence in this
case can be found in Seiberg and Witten (1999,
section 7). Noncommutative gauge theories on tori
can also be obtained by taking the so-called Seiberg–
Witten zero slope limit in the presence of a Neveu–
Schwarz B-field background (Seiberg and Witten
1999). The emergence of noncommutative geometry
in this limit is discussed in this article. Below we give
some details on the relation between T-duality and
Morita equivalence in this approach. Consider a
number of Dp-branes wrapped on T
p
parametrized by
528 Noncommutative Tori, Yang–Mills, and String Theory
coordinates x
i
x
i
þ 2r with a closed-string metric
G
ij
and a B-field B
ij
. The SO(p, pjZ) T-duality group
is represented by the matrices
T ¼
ab
cd
½30
that act on the matrix
E ¼
r
2
0
ðG þ 2
0
BÞ
by a fractional transformation
T : E 7!E
0
¼ðaE þbÞð cE þdÞ
1
½31
The transformed metric and B-field are obtained by
taking, respectively, the symmetric and antisym-
metric parts of E
0
. The string coupling constant is
transformed as
T : g
s
7!g
0
s
¼
g
s
ðdetðcE þ dÞÞ
1=2
½32
The zero slope limit of Seiberg and Witten is
obtained by taking
0
ffiffi
p
!0; G
ij
!0 ½ 33
Sending the closed-string metric to zero implies that
the B-field dominates in the open-string boundary
conditions. In the limit [33], the compactification is
parametrized in terms of open-string moduli
g
ij
¼ð2
0
Þ
2
ðBG
1
BÞ
ij
ij
¼
1
2r
2
ðB
1
Þ
ij
½34
which remain finite. One can demonstrate that
ij
is
a noncommutativity parameter for the torus and the
low-energy effective theory living on the Dp-brane is
a noncommutative maximally supersymmetric gauge
theory with a coupling constant
G
s
¼ g
s
det g
det G
1=4
½35
From the transformation law [31], it is not hard to
derive the transformation rules for the moduli [34]
in the limit [33],
T : g 7!g
0
¼ða þ bÞgða þ bÞ
t
T : 7!
0
¼ðc þ dÞða þ bÞ
t
½36
Furthermore, the effective gauge theory becomes a
noncommutative Yang–Mills theory [17] with a
coupling constant
ðg
YM
Þ
2
¼
ð
0
Þ
ð3pÞ=2
ð2Þ
p2
G
s
which goes to a finite limit under [33] provided one
simultaneously scales g
s
with as
g
s
ð3pþkÞ=4
where k is the rank of B
ij
. The limiting coupling constant
g
YM
transforms under the T-duality [31], [32] as
T : g
YM
7!g
0
YM
¼ g
YM
ðdetða þ bÞÞ
1=4
½37
We see that the transformation laws [31] and [37]
have the same form as the corresponding transfor-
mations in [25], [28], [29] provided one identifies
matrix [26] with matrix [30] conjugated by
T ¼
01
10
The need for conjugation reflects the fact that in the
BFSS M(atrix) model in the framework of which
the Morita equivalence was originally considered, the
natural degrees of freedom are D0 branes versus Dp
branes considered in the above discussion of T-duality.
One can further check that the gauge field transfor-
mations following from gauge Morita equivalence
match with those induced by the T-duality. It is worth
stressing that in the absence of a B-field background
the effective action based on the square of the gauge
field curvature is not invariant under T-duality.
Instantons on Noncommutative T
4
Consider a Yang–Mills field r
X
on a projective
module E over a noncommutative 4-torus T
4
.
Assume that the Lie algebra of shifts L
is equipped
with the standard Euclidean metric such that the
metric tensor in the basis [7] is given by the identity
matrix. The Yang–Mills field r
i
is called an instanton
if the self-dual part of the corresponding curvature
tensor is proportional to the identity operator,
F
þ
jk
1
2
F
jk
þ
1
2
jkmn
F
mn
¼ i!
jk
1 ½38
where !
jk
is a constant matrix with real entries. An
anti-instanton is defined the same way by replacing
the self-dual part with the anti-self-dual one.
One can define a noncommutative analog of
Nahm transform for instantons (Astashkevich et al.
2000) that has properties very similar to those of the
ordinary (commutative) one. To that end, consider a
triple (P, r
i
,
^
r
i
) consisting of a (finit e projective)
(T
4
, T
4
ˆ
)-bimodule P, T
4
-connection r
i
and T
4
ˆ
-
connection
^
r
i
that satisfy the following properties.
The connection r
i
commutes with the T
ˆ
-action on
P and the connection
^
r
i
with that of T
. The
commutators [r
i
, r
j
], [
^
r
i
,
^
r
j
], [r
i
,
^
r
j
] are propor-
tional to the identity operator
Noncommutative Tori, Yang–Mills, and String Theory 529
½r
i
; r
j
¼!
ij
1
½
^
r
i
;
^
r
j
¼
^
!
ij
1
½r
i
;
^
r
j
¼
ij
1
½39
The above conditions mean that P is a T
8
(
ˆ
)
-
module and r
i
^
r
i
is a constant curvature connec-
tion on it. In addition, we assume that the tensor
ij
is nondegenerate.
For a connection r
E
on a right T
4
-module E,we
define a Dirac operator D =
i
(r
E
i
þr
i
) acting on
the tensor product
ðE
T
PÞS
where S is the SO(4) spinor representation space and
i
are four-dimensional Dirac gamma-matrices. The
space S is Z
2
-graded: S = S
þ
S
and D is an odd
operator so that we can consider
D
þ
: ðE
T
PÞS
þ
!ðE
T
PÞS
D
: ðE
T
PÞS
!ðE
T
PÞS
þ
A connection r
E
i
on a T
4
-module E is called
P-irreducible if there exists a bounded inverse to the
Laplacian
¼
X
i
r
E
i
þr
i
r
E
i
þr
i
One can show that if r
E
is a P-irreducible instanton,
then ker D
þ
= 0andD
D
þ
=.Denoteby
^
E the
closure of the kernel of D
.SinceD
commutes wit h
the T
4
ˆ
-actionon(E
T
P) S
the space
^
E is a right
T
4
ˆ
-module. One can prove that t his module is finite
projective. Let P :(E
T
P) S
!
^
E be a Hermitian
projector. Denote by r
^
E
the composition P
^
r.One
can show that r
^
E
is a Yang–Mills field on
^
E.
The noncommutative Nahm transform of a
P-irreducible instanton connection r
E
on E is
defined to be the pair (
^
E, r
^
E
). One can further
show that r
^
E
is an instanton.
See also: Electroweak Theory; Hopf Algebras and
q-Deformation Quantum Groups; Noncommutative
Geometry from Strings; Quantum Group Differentials,
Bundles and Gauge Theory; Quantum Hall Effect; String
Field Theory; von Neumann Algebras: Introduction,
Modular Theory, and Classification Theory.
Further Reading
Astashkevich A, Nekrasov N, and Schwarz A (2000) On
noncommutative Nahm transform. Communications in Math-
ematical Physics 211: 167–182.
Connes A (1980) C
alge` bres et ge´ome´trie differentielle. Comptes
Rendus Hebdomaclaires des Seances l’Academie des Sciences,
Paris Ser. A-B 290.
Connes A (1994) Noncommutative Geometry. Academic Press.
Connes A, Douglas MR, and Schwarz A (1998) Noncommutative
geometry and Matrix theory: compactification on tori. Journal
of High Energy Physics 02: 003.
Douglas MR and Nekrasov N (2001) Noncommutative field
theory. Reviews of Modern Physics 73: 977–1029.
Konechny A and Schwarz A (2002) Introduction to M(atrix) theory
and noncommutative geometry. Physics Reports 360: 353–465.
Li H (2004) Strong Morita equivalence of higher-dimensional
noncommutative tori. Journal fu¨ r die Reineund Angewandte
Mathematik 576: 167–180.
Polishchuk A and Schwarz A (2003) Categories of holomorphic
vector bundles on noncommutative two-tori. Communications
in Mathematical Physics 236: 135–159.
Rieffel MA (1982) Morita equivalence for operator algebras. In:
Kadison RV (ed.) Operator Algebras and Applications,vol.38,
Proc. Symp. Pure Math. Providence, RI: American Mathematical
Society.
Rieffel MA (1988) Projective modules over higher-dimensional
non-commutative tori. Canadian Journal of Mathematics
40(2): 257–338.
Rieffel MA (1990) Noncommutative tori – a case study of non-
commutative differentiable manifolds. Contemporary Mathe-
matics 105: 191–211.
Rieffel MA and Schwarz A (1999) Morita equivalence of
multidimensional noncommutative tori. International Journal
of Mathematics 10(2): 289.
Schwarz A (1998) Morita equivalence and duality. Nuclear
Physics B 534: 720–738.
Seiberg N and Witten E (1999) String theory and noncommuta-
tive geometry. Journal of High Energy Physics 9909: 032.
Szabo RJ (2003) Quantum field theory on noncommutative
spaces. Physics Reports 378: 207.
Nonequilibrium Statistical Mechanics (Stationary): Overview
G Gallavotti, Universita
`
di Roma ‘‘La Sapienza,’’
Rome, Italy
ª 2006 G Gallavotti. Published by Elsevier Ltd.
All rights reserved.
Nonequilibrium
Systems in stationary nonequilibrium are mechanical
systems subject to nonconservative external forces
and to thermostat forces which forbid indefinite
increase of the energy and allow reaching statisti-
cally stationary states. A system is described by
the positions and velocities of its n particles X,
_
X,
with the particle positions confined to a finite
volume container C
0
.
If X = (x
1
, ..., x
n
) are the particl e pos itions in
a Car tesian inertial system of coordinates, the
equations of motion are determined by their masses
m
i
> 0, i = 1, ..., n, by the potential energy of
530 Nonequilibrium Statistical Mechanics (Stationary): Overview
interaction V(x
1
, ..., x
n
) V(X), by the external
nonconservative forces F
i
(X, F), and by the thermo-
stat forces J
i
as
m
i
€
x
i
¼@
x
i
VðXÞþF
i
ðX; FÞJ
i
; i ¼1;...;n ½1
where F = (’
1
,...,’
q
) are strength parameters on
which the external forces depend. All forces and
potentials will be supposed smooth, that is, analytic,
in their variables aside from possible impulsive elastic
forces describing shocks, and with the property
F(X;0)=0. The impulsive forces are allowed here to
model pos sible shocks with the walls of the container
C
0
or between hard core particles.
A thermostat is a ‘‘reservoir’’ which may consist
of one or more infinite systems which are asympto-
tically in thermal equilibrium and are separated by
boundary surfaces from each other as well as from
the system: with the latter, they interact through
short-range conservative forces, see Figure 1.
The reservoirs occupy infinite regions of the space
outside C
0
, for example, sectors C
a
R
3
, a = 1, 2 ...,
in space and their particles are in a configuration
which is typical of an equilibrium state at temperature
T
a
. This means that the empirical probability of
configurations in each C
a
is Gibbsian with some
temperature T
a
. In other words, the frequency with
which a configuration (
_
Y, Y þ r) occurs in a region
þ r C
a
while a configuration (
_
W, W þ r) occurs
outside þ r (with Y , W \ =;) averaged over
the translations þ r of by r (with the restriction
that þ r C
a
)is
average
rþC
a
ðf
þr
½ð
_
Y; Y þ rÞ;
_
W; W þ rÞ
¼
e
a
ð1=2m
a
Þj
_
Yj
2
þV
a
ðYjWÞ
ðÞ
normalization
½2
Here m
a
is the mass of the particles in the ath
reservoir and V
a
(YjW) is the energy of the short-
range potential between pairs of particles in Y C
a
or with one point in Y and one in W. Since the
configurations in the system and in the thermostats
are not random, [2] should be considered as an
‘‘empirical’’ probability in the sense that it is the
frequency density of the events {(
_
Y, Y þ r); W þ r}:
in other words, the configurations w
a
in the
reservoirs should be ‘‘typical’’ in the sense of
probability theory of distributions which are asymp-
totically Gibbsian.
The property of being ‘‘thermostats’’ means that
[2] remains true for all times, if initially satisfied.
Mathematically, there is a problem at this point:
the latter property is either true or false, but a
proof of its validity seems out of reach of the
present techniques except in very simple cases.
Therefore, here we follow an intuitive approach
and assume that such thermostats exist and,
actually, that any configuration which is typical of
a stationary state of an infinite size system of
interacting particles in the C
a
’s, with physically
reasonable microscopic interactions, satisfies the
property [2].
The above thermostats are examples of ‘‘determi-
nistic thermostats’’ because, together with the
system, they form a deterministic dynamical system.
They are called ‘‘Hamiltonian thermostats’’ and are
often considered as the most appropriate models of
‘‘physical thermostats.’’
A closely related thermostat model is obtained by
assuming that the particles outside the system are
not in a given configuration but they have a
probability distribution whose conditional distribu-
tions satisfy [2] initially. Also in this case, it is
necessary to assume that [2] remains true for all
times, if initially satisfied. Such thermostats are
examples of ‘‘stochastic thermostats’’ because their
action on the system depends on random variables
w
a
which are the initial configurations of the
particles belonging to the thermostats.
Other kinds of stochastic thermostats are ‘‘colli-
sion rules’’ with the container boundary @C
0
of :
every time a particle collide s with @C
0
it is reflected
with a momentum p in d
3
p that has a probability
distribution proportional to e
a
(1=2m)p
2
d
3
p where
a
, a = 1, 2 , ... depends on which boundary portion
(labeled by a = 1, 2, ...) the collision takes place and
T
a
= (k
B
a
)
1
and its ‘‘temperature’’ if k
B
is Boltz-
mann’s constant. Which p is actually chosen after
each collision is determined by a random variable
w = (w
1
, w
2
, ...).
The distinction between stochastic and deter-
ministic thermostats ultimately rests on what we
call ‘‘system.’’ If reservoirs or the randomness
generators are included in the system, then the
system becomes deter ministic (possibl y infinite);
and finite deterministic thermostats can also be
regarded as simplified models for infinite reservoirs,
see the section ‘‘Heat, temperat ure, an d entropy
produ ction.’’
T
1
T
2
T
3
Σ
Figure 1 A symbolic drawing of the container C
0
for the
system and of the surrounding regions containing the particles
acting as thermostats at temperatures T
1
, T
2
, ....
Nonequilibrium Statistical Mechanics (Stationary): Overview 531
It is also possible, and convenient, to consider
‘‘finite deterministic thermostats.’’ In the latter case,
J is a force only depending upon the configuration
of the n particles v of in their finite container C
0
.
Examples of finite deterministic reservoirs are forces
obtained by imposing a nonholonomic constraint via
some ad hoc principle like the Gauss principle.For
instance, if a system of particles driven by a force
G
i
=
def
@
x
i
V(X) þ F
i
(X)isenclosedinaboxC
0
and J
is a thermostat enforcing an anholonomic constraint
(
_
X, X) 0 via Gauss’ principle, then
J
i
ð
_
X; XÞ
¼
"
P
j
_
x
j
@
x
j
ð
_
X; XÞþð1=mÞG
j
@
_
x
j
ð
_
X; XÞ
P
j
1
m
ð@
_
x
j
ð
_
X; XÞÞ
2
#
@
_
x
i
ð
_
X; XÞ½3
Gauss’ principle says that the force which needs to
be added to the other forces G
i
acting on the system
minimizes
X
i
ðG
i
m
i
a
i
Þ
2
m
i
given
_
X, X, among all accelerations a
i
which are
compatible with the constraint .
It should be kept in mind that the only known
examples of mathematically treatable thermostats
modeled by infinite reservoirs are cases in which the
thermostat particles are either noninteracting parti-
cles or linear (i.e., noninteracting) oscillators. For
simplicity stochastic or infinite thermostats will not
be considered here and we restrict attention to finite
deterministic systems.
In general, in order that a force J can be
considered a deterministic ‘‘thermostat force’’ a
further property is necessary: namely that the system
evolves according to [1] towards a stationary state.
This means that for all initial particle configurations
(
_
X, X), except possibly for a set of zero phase-space
volume, any smooth function f (
_
X, X) evolves in time
so that, if S
t
(
_
X, X) denotes the configuration into
which the initial data evolve in time t according to
[1], then the limit
lim
T!1
1
T
Z
T
0
f ðS
t
ð
_
X; XÞÞdt ¼
Z
f ðzÞðdzÞ½4
exists and is independent of (
_
X, X). The probability
distribution is then called the SRB distribution for
the system. The maps S
t
will have the group
property S
t
S
t
0
= S
tþt
0
and the SRB distribution
will be invariant under time evolution.
It is important to stress that the requirement that
the exceptional configurations form just a set of zero
phase volume (rather than a set of zero probability
with respect to another distribution, singular with
respect to the phase volume) is a strong assumpti on
and it should be considered an axiom of the theory:
it corresponds to the assumption that the initial
configuration is prepared as a typical configuration
of an equilibrium state, which, by the classical
equidistribution axiom of equilibrium statistical
mechanics, is a typical configuration with respect
to the phase volume.
For this reason, the SRB distribution is said to
describe a ‘‘stationary nonequilibrium state’’ of
the system. The SRB distribution depends on the
parameters on which the forces acting on the
system depend, for example, jC
0
j (volume), F
(strength of the forcings), {
1
a
} (temperatures), etc.
The collection of SRB distributions obtained by
letting the parameters vary defines a ‘‘nonequilibrium
ensemble.’’
In the stochastic case, the distribution is
required to be invariant in the sense that it can be
regarded as a marginal distribution of an invariant
distribution for the larger (deterministic) system
formed by the thermostats and the system itself.
For more details, the reader is referred to Evans
and Morriss (1990), Ruelle (1999), and Eckmann
et al. (1999).
Nonequilibrium Thermodynamics
The key problem of nonequilibrium statistical
mechanics is to derive a macroscopic ‘‘nonequili-
brium thermodynamics’’ in a way similar to the
derivation of equilibrium thermodynamics from
equilibrium statistical mechanics.
The first difficulty is that nonequilibrium thermo-
dynamics is not well understood. For instance, there
is no (agreed upon) definition of entropy of a
nonequilibrium stationary state, whi le it should be
kept in mind that the effort to find the microscopic
interpretation of equilibrium entropy, as defined by
Clausius, was a driving factor in the foundations of
equilibrium statistical mechanics.
The importance of entropy in classical equilibrium
thermodynamics rests on the implication of univer-
sal, parameter-free relations which follow from its
existence (e.g., @
V
(1=T) @
U
(p=T)ifU is the
internal energy, T the absolute temperature, and p
the pressure of a simple homogeneous material).
Are there universal relations among averages of
observables with respect to SRB distributions?
The question has to be posed for systems ‘‘really’’
out of equilibrium, that is, for F 6¼ 0 (see [1]): in
fact, there is a well-developed theory of the
derivatives with respe ct to F of averages of
532 Nonequilibrium Statistical Mechanics (Stationary): Overview
observables evaluated at F = 0. The latter theory is
often called, and here we shall do so as well,
‘‘classical nonequilibrium thermodynamics’’ or
‘‘near-equilibrium thermodynamics’’ and it has
been quite successfully developed on the basis of
the notions of equilibrium thermodynamics, paying
particular attention to the macroscopic evolution of
systems described by macro scopic continuum equa-
tions of motion.
‘‘Stationary nonequilibrium statistical mechanics’’
will indicate a theory of the relations between
averages of observables with respect to SRB dis-
tributions. Systems so large that their volume
elements can be regarded as being in locally
stationary nonequilibrium states could also be
considered. This would extend the fam iliar ‘‘local
equilibrium states’’ of classical nonequilibrium ther-
modynamics: howev er, they are not considered here.
This means that we shall not attempt to find the
macroscopic equations regulating the time evolution
of continua locally in nonequilibrium stationary
states but we shall only try to determine the
properties of their ‘‘volume elements’’ assuming
that the timescale for the evolution of large
assemblies of volume elements is slow compared to
the timescales necessary to reach local stationarity.
For more details, the reader is referred to
de Groot and Mazur (1984), Lebowitz (1993),
Ruelle (1999, 2000), Gallavotti (1998, 2004), and
Goldstein and Lebowitz (2004).
Chaotic Hypothesis
In equilibrium statistical mechanics, the ergodic
hypothesis plays an important conceptual role as it
implies that the motions of ergodic systems have an
SRB statistics and that the latter coincides with the
Liouville distribution on the energy surface.
An analogous role has been proposed for the
‘‘chaotic hypothesis,’’ which states that the
motion of a chaotic system, developing on its attracting
set, can be regarded as an Anosov flow.
This means that the attracting sets of chaotic
systems, physically defined as systems with at least
one positive Lyapunov exponent, can be regarded as
smooth surfaces on which motion is highly unstabl e:
1. Around every point, a curvilinear coordinate
system can be established which has three planes,
varying continuously with x, which are covariant
(i.e., they are coordinate planes at a point x
which are mapped, by the evolution S
t
, into the
corresponding coordinate planes around S
t
x).
2. The planes are of three types, ‘‘stable,’’ ‘‘unstable,’’
and ‘‘marginal,’’ with respective positive dimen-
sions d
s
, d
u
, and 1: infinitesimal lengths on the
stable plane and on the unstable plane of any
point contract at exponential rate as time
proceeds towards the future or towards the past.
The length along the marginal direction neither
contracts nor expands (i.e., it varies around the
initial value staying bounded away from 0 and
1): its tangent vector is parallel to the flow. In
cases in which time evolution is discrete, and
determined by a map S, the marginal direction is
missing.
3. The contraction over a time t, positive for lines
on the stable plane and negative for those on the
unstable plane, is exponential, i.e. lengths are
contracted by a factor uniformly bounded by
Ce
jtj
with C, >0.
4. There is a dense trajectory.
It has to be stressed that the chaotic hypothesis
concerns physical systems: mathematically, it is
very easy to find dynamical systems for which it
doesnothold,atleastaseasyasitistofind
systems in which the ergodic hypothesis does not
hold (e.g., harmonic lattices or blackbody radia-
tion). However, if suitably i nterpreted, the ergodic
hypothesis leads, even for these systems, to physi-
cally correct results (the specific heats at high
temperature, the Raleigh–Jeans distribution at low
frequencies). Moreover, the failures of the ergodic
hypothesis in physically important systems have led
to new scientific paradigms (like quantum
mechanics from the specific heats at low tempera-
ture and Planck’s law).
Since phy sical systems are almost always not
Anosov systems, it is very likely that probing
motions in extreme regimes will make visible the
features that distinguish Anosov systems from non-
Anosov systems, much as it happens with the
ergodic hypothesis.
The interest of the hypothesis is to provide a
framework in which properties like the existence of
an SRB distribution is a priori guaranteed, together
with an expression for it which can be used to work
with formal expressions of the averages of the
observables: the role of Anosov systems in chaotic
dynamics is similar to the role of harmonic oscillators
in the theory of regular motions. They are the
paradigm of chaotic systems, as the harmonic
oscillators are the paradigm of order. Of course, the
hypothesis is only a beginning and one has to learn
how to extract information from it, as it was the case
with the use of the Liouville distribution, once the
ergodic hypothesis guaranteed that it was the
Nonequilibrium Statistical Mechanics (Stationary): Overview 533
appropriate distribution for the study of the statistics
of motions in equilibrium situations.
For more details, the reader is referred to Ruelle
(1976), Galla votti and Cohen (1995), Ruelle (1999),
Gallavotti (1998), and Gallavotti et al. (2004).
Heat, Temperature, and Entropy
Production
The amount of heat
_
Q that a system produces while in
a stationary state is naturally identified with the work
that the thermostat forces J perform per unit time
_
Q ¼
X
i
J
i
_
x
i
½5
A system may be in contact with several reservoirs:
in models, this will be reflected by a decomposition
J ¼
X
J
ðaÞ
ð
_
X; XÞ½6
where J
(a)
is the force due to the ath thermostat and
depends on the coordinates of the particles which
are in a region
a
C
0
of a decomposition
[
m
a = 1
a
= C
0
of the container C
0
occupied by the
system (
a
\
a
0
= ; if a 6¼ a
0
).
From several studies based on simulations of finite
thermostatted systems of particles arose the proposal
to consider the average of the phase-space contrac-
tion
(a)
(
_
X, X) due to the ath thermostat
ðaÞ
ð
_
X; XÞ¼
def
X
j
@
_
x
j
J
ðaÞ
j
ð
_
X; XÞ½7
and to identify it with the rate of entropy creation in
the ath thermostat.
Another key notion in thermodynamics is the
temperature of a reservoir; in the infinite determi-
nistic thermost at case, of the sect ion ‘‘Noneq uili-
brium ,’’ it is defi ned as ( k
B
a
)
1
but in the finite
deterministic thermostats considered here it needs to
be defined. If there are m reservoirs with which the
system is in contact, one sets
ðaÞ
þ
¼
def
h
ðaÞ
ð
_
X; XÞi
Z
ðaÞ
ð
_
X; XÞðd
_
X dXÞ
_
Q
a
¼
def
X
i
J
ðaÞ
i
_
x
i
½8
where is the SRB distribution describing the
stationary state. It is natural to define the absolute
temperature of the ath thermostat to be
T
a
¼
h
_
Q
a
i
k
B
ðaÞ
þ
½9
It is not clear that T
a
> 0: this happens in a rather
general class of models and it would be desirable, for
the interpretation that is proposed here, that it could
be considered a property to be added to the require-
ments that the forces J
ðaÞ
be thermostat models.
An im portant class of thermostats for which the
property T
a
> 0 holds can be described as follows.
Imagine N particles in a container C
0
interacting via
a potential V
0
=
P
i<j
’(q
i
q
j
) þ
P
j
V
0
(q
j
) (where
V
0
models external conservative forces like obsta-
cles, walls, gravity, ...) and, furthermore, interacting
with M other systems
a
,ofN
a
particles of mass
m
a
, in containers C
a
contiguous to C
0
. The latter
will model M parts of the system in contact with
thermostats at temperatures T
a
, a = 1, ..., M.
The coordinates of the particles in the ath system
a
will be denoted x
a
j
, j = 1, ..., N
a
, and they will
interact with each other via a potential V
a
=
P
N
a
i, j
’
a
(x
a
i
x
a
j
). Furthermore, there will be an
interaction between the particles of each thermostat
and those of the system via potentials W
a
=
P
N
i = 1
P
N
a
j = 1
w
a
(q
i
x
a
j
), a = 1, ..., M.
The potentials will be assumed to be either hard
core or nonsingular potentials and the external V
0
is
supposed to be at least such that it forbids the
existence of obvious constants of motion.
The temperature of each
a
will be defined by
the total kinetic energy of its particles, that
is, by K
a
=
P
N
a
j = 1
(1=2)m
a
(
_
x
a
j
)
2
¼
def
(3=2)N
a
k
B
T
a
:the
particles of the ath thermostat will be kept at
constant temperature by further forces J
a
j
. The latter
are defined by imposing via a Gaussian constraint
that K
a
is a constant of motion (see [3] with K
a
).
This means that the equations of motion are
m
€
q
j
¼@
q
j
V
0
ðQÞþ
X
N
a
a¼1
W
a
ðQ; x
a
Þ
!
m
a
€
x
a
j
¼@
x
a
j
V
a
ðx
a
ÞþW
a
ðQ; x
a
ÞðÞJ
a
j
½10
and an application of Gauss’ principle yields
J
a
j
¼
L
a
_
V
a
3N
a
k
B
T
a
_
x
a
j
¼
def
a
_
x
a
j
where L
a
is the work per unit time done by the
particles in C
0
on the particles of
a
and V
a
is their
potential energy.
In this case, the partial divergence
a
ð3N
a
1Þ
a
is, up to a constant factor ð1ð1=3N
a
ÞÞ;
a
¼
L
a
k
B
T
a
_
V
a
k
B
T
a
and it will make [9] identically satisfied with T
a
> 0
because L
a
can be naturally interpreted as heat Q
a
ceded, per unit time, by the particles in C
0
to the
subsystem
a
(hence to the ath thermostat because
the temperature of
a
is constant), while the
534 Nonequilibrium Statistical Mechanics (Stationary): Overview
derivative of V
a
will not contribute to the value of
a
þ
. The phase-space contraction rate is, neglecting
the total derivative terms (and OðN
1
a
ÞÞ,
true
ð
_
X; XÞ¼
X
N
a
a¼1
_
Q
a
k
B
T
a
½11
where the subscript ‘‘true’’ is to remind that an
additive total derivative term distinguishes it from
the complete phase-space contraction.
Remarks
(i) The above formula provides the motivation of the
name ‘‘entropy creation rate’’ attributed to the
phase-space contraction . Note that in this way
the definition of entropy creation is ‘‘reduced’’ to
the equilibrium notion because what is being
defined is the entropy increase of the thermostats
which have to be considered in equilibrium. No
attempt is made here to define neither the entropy
of the stationary state nor the notion of tempera-
ture of the nonequilibrium system in C
0
(the T
a
are temperatures of the
a
, not of the particles in
C
0
). This is an important point as it leaves open
the possibility of envisaging the notion of ‘‘local
equilibrium’’ which becomes necessary in the
approximation (not considered here) in which
the system is regarded as a continuum.
(ii) In the above model, another viewpoint is
possible: that is, to consider the system to
consist of only the N particles in C
0
and the M
systems
a
to be thermostats. From this point of
view, it can be considered a model of a system
subject to thermostats. The Gibbs distribution
characterizing the infinite thermostats of the
sect ion ‘‘N onequilibr ium’’ becomes in this case
the constraint that the kinetic energies K
a
are
constants, enforced by the Gaussian forces. In
the new viewpoint, the appropriate definition
should be simply the right-hand side (RHS) of
[11], i.e. the work per unit time done by the
forces of the system on the thermostats divided
by the temperature of the thermostats. This
suggests a different and general definition of
entropy creation rate, applying also to thermo-
stats that are often considered ‘‘more physical’’
and that needs to be further investigated. In the
example [10] the new definition differs from the
phase space contraction rate by a total time
derivative, i.e. rather trivially for the purposes of
the following.
For more details, the reader is referred to Evans
and Morriss (1990), Gallavotti and Cohen (1995),
Ruelle (1996, 1997), and Gallavotti (2004).
Thermodynamic Fluxes and Forces
Nonequilibrium stationary states depend upon
external parameters ’
j
like the temperatures T
a
of
the thermostats or the size of the force parameters
F = (’
1
, ..., ’
q
), see [1]. Nonequilibrium thermo-
dynamics is well developed at ‘‘low forcing’’: strictly
speaking, this means that it is widely believed that
we understand the properties of the derivatives of
the averages of observables with respect to the
external parameters if evaluated at ’
j
= 0. Important
notions are the notions of thermodynamic fluxes J
i
and of thermodynamic forces ’
i
; hence, it seems
important to extend such notions to nonequilibrium
systems (i.e., F 6¼ 0).
A possible extension could be to define the
thermodynamic flux J
i
associated with a force ’
i
as
J
i
= h@
’
i
i
SRB
where (X,
_
X; F) is the volume
contraction per unit time. This definition seems
appropriate in several concrete cases that have been
studied and it is appealing for its generality.
An interesting example is provided by the model
of thermostatted system in [10]: if the container of
the system is a box with periodic boundary condi-
tions, one can imagine to add an extra consta nt
force E acting on the particles in the container.
Imagining the particles to be charged by a charge e
and regarding such force as an electric field, the first
equation in [10] is modified by the addition of a
term eE.
The constraints on the thermostat temperatures imply
that depends also on E:infact,ifJ = e
P
j
_
q
j
is the
electric current, energy balance implies
_
U
tot
= E J
P
a
(L
a
_
V
a
)ifU
tot
is the sum of all kinetic and
potential energies. Then, the phase-space contraction
X
a
L
a
_
V
a
T
a
can be written, to first order in the temperature
variations T
a
with respect to a common value
T
a
= T,as
X
a
L
a
_
V
a
T
T
a
T
þ
E J
_
U
tot
T
hence
true
, see [11],is
true
¼
E J
k
B
T
X
a
_
Q
a
k
B
T
T
a
T
½12
The definition and extension of the conjugacy
between thermodynamic forces and fluxes is com-
patible with the key results of classical nonequili-
brium thermodynami cs, at least as far as Onsager
Nonequilibrium Statistical Mechanics (Stationary): Overview 535
reciprocity and Green–Kubo’s formulas are con-
cerned. It can be checked that if the equilibrium
system is reversible, that is, if there is an isometry I
on phase space which anticommutes with the
evolution (IS
t
= S
t
I in the case of continuous-time
dynamics t !S
t
or IS = S
1
I in the case of discrete-
time dynamics S), then, shortening (
_
X, X) into x,
L
ij
¼
def
@
i
J
j
j
F¼0
¼@
i
h@
j
ðx;FÞi
SRB
j
F¼0
¼@
j
J
i
j
F¼0
¼L
ji
¼
1
2
Z
1
1
h@
j
ðS
t
x;FÞ@
i
ðx;FÞi
SRB
j
F¼0
dt ½13
The (x;F) plays the role of ‘‘Lagrangian’’ generat-
ing the duality between forces and fluxes. The
extension of the duality just considered might be of
interest in situations in which F 6¼0.
For more details, the reader is referred to de Groot
and Mazur (1984), Gallavotti (1996),andGallavotti
and Ruelle (1997).
Fluctuations
As in equilibrium, large statistical fluctuations of
observables are of great interest and already there is,
at the moment, a rather large set of experiments
dedicated to the analysis of large fluctuations in
stationary states out of equilibrium.
If one defines the dimensionless phase-space
contraction
pðxÞ¼
1
Z
0
ðS
t
xÞ
þ
dt ½14
(see also [11]), then there exists p
1 such that the
probability P
of the event p 2 [a, b] with [a, b]
(p
, p
) has the form
P
ðp 2½a; bÞ ¼ const: e
max
p2½a;b
ðpÞþOð1Þ
½15
with (p) analytic in (p
, p
). The function (p ) can
be conveniently normalized to have value 0 at p = 1
(i.e., at the average value of p).
Then, in Anosov systems which are rever sible and
dissipative (see the previous section), a general
symmetry property, called the ‘‘fluctuation theorem’’
and reflecting the reversibility symmetry, yields the
parameterless relation
ðpÞ¼ðpÞp
þ
p 2ðp
; p
Þ½16
This relation is interesting because it has no free
parameters; in other words, it is universal for
reversible dissipative Anosov systems. In connection
with the flux–force duality in the previous section, it
can be checked to reduce to the Green–Kubo
formula and to Onsager reciprocity, see [13], in the
case in which the evolution depends on several fields
F and F !0 (of course the relation becomes trivial
as F !0 because
þ
!0 and to obtai n the result
one has first to divide both sides by suitable powers
of the fields F).
A more informal (but imprecise) way of writing
[15] and [16] is
P
ðpÞ
P
ðpÞ
¼ e
p
þ
þOð1Þ
; for all p 2ðp
; p
Þ½17
where P
(p) is the probability density of p.An
obvious but interesting consequence of [17] is that
he
p
þ
i
SRB
= 1
in the sense that (1=) loghe
p
þ
i
SRB
!
!1
0.
Occasionally, systems with singularities have to be
considered. In such cases, the relation [16] may
change in the sense that the function (p) may not be
analytic: in such cases, one expects that the relati on
holds in the largest analyticity interval symmetric
around the origin. In Anasov systems and also
various cases considered in the literature, such
interval appears to contain the interval (1, 1).
Note that in the theory of fluctuations of the time
averages p we can replace by any other bounded
quantity which is a total time derivative: hence, in the
example discussed above, it can be replaced by
true
,
see [12], which has a natural physical meaning.
It is important to remark that the above fluctua-
tion relation is the first representative of several
consequences of the reversibility and chaotic
hypotheses. For instance, given F
1
, ..., F
n
arbitrary
observables which are (say) odd under time reversal
I (i.e., F(Ix) = F(x)) and given n functions t 2
[=2, =2] !’
j
(t), j = 1, ..., n, one can ask which
is the probability that F
j
(S
t
x) ‘‘closely follows’’ the
‘‘pattern’’ ’
j
(t) and at the same time
1
Z
0
ðS
xÞ
þ
d
has value p. Then calling P
(F
1
’
1
, ..., F
n
’
n
, p)
the probability of this event, which we write in the
imprecise form corresponding to [17] for simplicity,
and defining I’
j
(t) =
def
’
j
(t), it is
P
ðF
1
’
1
; ...; F
n
’
n
; pÞ
P
ðF
1
I’
1
; ...; F
n
I’
n
; pÞ
¼ e
þ
p
p 2ðp
; p
Þ½18
which is remarkabl e because it is parameterless and
at the same time surprisi ngly independent of the
choice of the observables F
j
. The relation [18] has
far-reaching consequences: for instan ce, if n = 1 and
F
1
= @
i
(x; F) the relation [18] has been used to
derive the mentioned Onsager reciprocity and
Green–Kubo’s formulas at F = 0.
536 Nonequilibrium Statistical Mechanics (Stationary): Overview