Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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Here ^
a
is the unit normal to H,
ab
the shear of ‘
a
(i.e., the tracefree part of q
am
q
bm
r
m
‘
n
), and
a
=
q
ab
^
r
c
r
c
‘
b
, where q
ab
is the projector onto the
tangent space of the cuts S. The first integral on
the right-hand side can be directly interpreted as the
flux across H of matter–energy (relative to the
vector field N‘
a
). The second term is pur ely
geometric and is interpreted as the flux of energy
carried by gravitational waves across H. It has
several properties which support this interpretation.
Thus, not only does the second law of black hole
mechanics hold for a dynamical horizon H, but the
‘‘cause’’ of the increase in the area can be directly
traced to physical processes happening near H.
Another natural question is whether the first law
[8] can be generalized to fully dynamical situations,
where is replaced by a finite transition. Again, the
answer is in the affirmative. We will outline the idea
for the case when there are no gauge fields on H.As
with isolated horizons, to have a well-defined notion
of angular momentum, let us suppose that the
intrinsic 3-metric on H admits a rotational Killing
field ’. Then, the angular momentum associated
with any cut S is given by
J
ð’Þ
S
¼
1
8G
I
S
K
ab
’
a
^
r
b
d
2
V
1
8G
I
S
j
ð’Þ
d
2
V ½11
where K
ab
is the extrinsic curvature of Hin (M, g
ab
)and
j
(’)
is interpreted as ‘ ‘the angular momentum density.’ ’
Now, in the Kerr family, the mass, surface gravity, and
the angular velocity can be unambiguously expressed as
well-defined functions
m(a, J),
(a, J), and
(a, J)ofthe
horizon area a and angular momentum J. The idea is to
use these expressions to associate mass, surface gravity,
and angular velocity with each cut of H. Then, a
surprising result is that the difference between the
horizon masses associated with cuts S
1
and S
2
can be
expressed as the integral of a locally defined flux across
the portion H of H bounded by H
1
and H
2
:
m
2
m
1
¼
1
8G
Z
H
da þ
1
8G
I
S
2
j
’
d
2
V
I
S
1
j
’
d
2
V
Z
2
1
d
I
S
j
’
d
2
V
½12
If the cuts S
2
and S
1
are only infinitesimally separated,
this expression reduces precisely to the standard first
law involving infinitesimal variations. Therefore, [12] is
an integral generalization of the first law.
Let us con clude with a general perspective. On the
whole, in the passage from event horizons in
stationary spacetimes to isolated horizons and then
to dynamical horizons, one considers increasingly
more realistic situations. In all the three cases, the
analysis has been extended to allow the presence of
a cosmological constant .(Theonlysignificant
change is that the topology of cuts S of dynamical
horizons is restricted to be S
2
if > 0andis
completely unrestricted if < 0.) In the first two
frameworks, results have also been extended to higher
dimensions. Since the notions of isolated and dynami-
cal horizons make no reference to infinity, these
frameworks can be used also in spatially compact
spacetimes. The notion of an event horizon, by
contrast, does not naturally extend to these space-
times. On the other hand, the generalization [4] of the
first law [3] is applicable to event horizons of
stationary spacetimes in a wide class of theories while
so far the isolated and dynamical horizon frameworks
are tied to general relativity (coupled to matter
satisfying rather weak energy conditions). From a
mathematical physics perspective, extension to more
general theories is an important open problem.
See also: Asymptotic Structure and Conformal Infinity;
Branes and Black Hole Statistical Mechanics; Dirac
Fields in Gravitation and Nonabelian Gauge Theory;
Geometric Flows and the Penrose Inequality; Loop
Quantum Gravity; Minimal Submanifolds; Quantum Field
Theory in Curved Spacetime; Quantum Geometry and its
Applications; Random Algebraic Geometry, Attractors
and Flux Vacua; Shock Wave Refinement of the
Friedman–Robertson–Walker Metric; Stationary Black
Holes.
Further Reading
Ashtekar A, Beetle C, and Lewandowski J (2001) Mechanics
of rotating black holes. Physical Review 64: 044016 (gr-qc/
0103026).
Ashtekar A, Fairhurst S, and Krishnan B (2000) Isolated horizons:
Hamiltonian evolution and the first law. Physical Review D
62: 104025 (gr-qc/0005083).
Ashtekar A and Krishnan B (2003) Dynamical horizons and their
properties. Physical Review D 68: 104030 (gr-qc/0308033).
Ashtekar A and Krishnan B (2004) Isolated and dynamical
horizons and their applications. Living Reviews in Relativity
10: 1–78 (gr-qc/0407042).
Bardeen JW, Carter B, and Hawking SW (1973) The four laws of
black hole mechanics. Communications in Mathematical
Physics 31: 161.
DeWitt BS and DeWitt CM (eds.) (1972) Black Holes.
Amsterdam: North-Holland.
Frolov VP and Novikov ID (1998) Black Hole Physics.
Dordrecht: Kluwer.
Hawking SW and Ellis GFR (1973) Large Scale Structure of
Space-Time. Cambridge: Cambridge University Press.
Hayward S (1994) General laws of black hole dynamics. Physical
Review D 49: 6467–6474.
Iyer V and Wald RM (1994) Some properties of noether charge
and a proposal for dynamical black hole entropy. Physical
Review D 50: 846–864.
Wald RM (1994) Quantum Field Theory in Curved Spacetime and
Black Hole Thermodynamics. Chicago: University of Chicago
Press.
Black Hole Mechanics 305
Boltzmann Equation (Classical and Quantum)
M Pulvirenti, Universita
`
di Roma ‘‘La Sapienza,’’
Rome, Italy
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
Ludwig Boltzmann (1872) established an evolution
equation to describe the behavior of a rarefied gas,
starting from the mathematical model of elastic balls
and using mechanical and statistical considerations.
The importance of this equation is twofold. First, it
provides a reduced description (as well as the
hydrodynamical equations) of the microscopic
world. Second, it is also an important tool for the
applications, especially for dilute fluids when the
hydrodynamical equations fai l to hold.
The starting point of the Boltzmann analysis is to
abandon the study of the gas in terms of the detailed
motion of molecules which constitute it because of
their large number. Instead, it is better to investigate
a function f (x, v), which is the probability density of
a given particle, where x and v denote its position
and velocity. Actually, f (x, v)dx dv is often confused
with the fraction of molecules falling in the cell of
the phase space of size dx dv around x, v. The two
concepts are not exactly the same, but they are
asymptotically equivalent (when the number of
particles is diverging) if a law of large numbers holds.
The Boltzmann equation is the following:
ð@
t
þ v r
x
Þf ¼ Qðf ; f Þ½1
where Q, the collision operator, is defin ed by eqn [2]:
Qðf ; f Þ¼
Z
R
3
dv
1
Z
S
2
þ
dnðv v
1
Þn
½f ðx; v
0
Þf ðx; v
0
1
Þf ðx; vÞf ðx; v
1
Þ ½2
and
v
0
¼ v n½n ðv v
1
Þ
v
0
1
¼ v
1
þ n½n ðv v
1
Þ
½3
Moreover, n (the impact parameter) is a unitary
vector and S
2
þ
= {njn (v v
1
) 0}.
Note that v
0
, v
0
1
are the outgoing velocities after a
collision of two elastic balls with incoming velocities
v and v
1
and centers x and x þ rn, r being the
diameter of the spheres. Obviously, the collision
takes place if n (v v
1
) 0. Equations [3] are a
consequence of the conservation of total energy,
momentum, and angular momentum. Note also that
r does not enter in eqn [1] as a parameter.
As fundamental features of eqn [1], we have the
conservation in time of the following five quantities
Z
dx
Z
dvfðx; v; tÞ v
½4
with = 0, 1, 2, expressing conservation of the
probability, momentum, and energy.
From now on we shall set
R
=
R
R
3
for notational
simplicity.
Moreover, Boltzmann introduced the (kinetic)
entropy defined as
Hðf Þ¼
Z
dx
Z
dvf log f ðx; vÞ½5
and proved the famous H-theorem asserting the
decreasing of H(f (t)) along the solutions to eqn [1].
Finally, in the case of bounde d domains or
homogeneous solut ions (f = f (v; t) is independent of
x), the distribution defined for some >0, >0,
and u 2 R
3
by
Mðx; vÞ¼
ð2=Þ
3=2
e
ð=2Þjvuj
2
½6
called Maxwellian distribution, is stationary for the
evolution given by eqn [1]. In addition, M minimizes
H among all distributions with given total mass ,
given mean velocity u, and mean energy. The
parameter is interpreted as the inverse
temperature.
In conclusion, Boltzmann was able to introduce
not only an evolutionary equation with the remark-
able properties expressing mass, momentum, and
energy conservation, but also the trend to the
thermal equilibrium. In other words, he tried to
conciliate the Newton’s laws with the second
principle of thermodynamics.
The Boltzmann Heuristic Argument
Thus, we want to find an evolution equation for the
quantity f (x, v; t). The molecular system we are
considering consists of N identical particles of
diameter r in the whole space R
3
. We denote by
x
1
, v
1
, ..., x
N
, v
N
a state of the system, where x
i
and
v
i
indicate the position and the velocity of the
particle i. The particles cannot overlap (i.e., the
centers of two particles cann ot be at a distance
smaller than the particle diameter r).
The particles are moving freely up to the first
instance of contact, that is, the first time when two
particles (say particles i and j) arrive at a distance r.
Then the pair interacts when an elastic collision
occurs. This means that they change instantaneously
306 Boltzmann Equation (Classical and Quantum)
their velocities, according to the conservation of
the energy and linear and angular momentum.
More precisely, the velocities after a collision
with incoming velocities v and v
1
are t hose g iven
by formula [3]. After the first collision, the
system evolves by iterating the procedure. Here
we neglect triple collisions because they are
unlikely. The evolution equation for a tagged
particle is then of the form
ð@
t
þ v r
x
Þf ¼ Coll ½7
where Coll denotes the variation of f due to the
collisions.
We have
Coll ¼ G L ½8
where L and G (the loss and gain terms, respe ctively)
are the negative and pos itive contributions to the
variation of f due to the collisions. More precisely,
L dx dv dt is the probability of the test particle to
disappear from the cell dx dv of the phase space
because of a collision in the time interval (t, t þ dt)
and Gdx dv dt is the probability to appear in the
same time interval for the same reason. Let us
consider the sphere of center x with radius r and a
point x þ rn over the surface, where n denotes the
generic unit vector. Consider also the cylinder with
base area dS = r
2
dn and height jVjdt along the
direction of V = v
2
v.
Then a given particle (say particle 2) with velocity
v
2
can contribute to L because it can collide with the
test particle in the time dt, provided it is localized in
the cylinder and if V n 0. Therefore, the contri-
bution to L due to the particle 2 is the probability of
finding such a particle in the cylinder (conditioned to
the presence of the first particle in x). This quantity is
f
2
(x, v, x þ nr, v
2
) j(v
2
v) njr
2
dn dv
2
dt ,wheref
2
is the joint distribution of two particles. Integrating in
dn and dv
2
, we obtain that the total contribution to
L due to any predetermined particle is
r
2
Z
dv
2
Z
S
2
dnf
2
ðx; v; x þ nr; v
2
Þjðv
2
vÞnj½9
where S
2
is the unit hemisphere (v
2
v) n < 0.
Finally, we obtain the total contribution multiplying
by the total number of particles:
L ¼ðN 1Þr
2
Z
dv
2
Z
S
dnf
2
ðx; v; x þ nr; v
2
Þjðv
2
vÞnj½10
The gain term can be derived analogously by
considering that we are looking at particles which
have velocities v and v
2
after the collisions so
that we have to integrate over the hemisphere
S
2
þ
= {(v
2
v) n > 0}:
G ¼ðN 1Þr
2
Z
dv
2
Z
S
þ
dnf
2
ðx; v; x þ nr; v
2
Þjðv
2
vÞnj½11
Summing G and L, we get
Coll ¼ðN 1Þr
2
Z
dv
2
Z
dnf
2
ðx; v; x þ nr; v
2
Þðv
2
vÞn ½12
which, however, is not a very useful expression
because the time derivative of f is expressed in terms
of another object, namely f
2
. An evolution equation
for f
2
will imply f
3
, the joint distribution of three
particles, and so on, up to we include the total
particle number N. Here the basic main assumption
of Boltzmann enters, namely that two given particles
are uncorrelated if the gas is rarefied, namely
f ðx; v; x
2
; v
2
Þ¼f ðx; vÞf ðx
2
; v
2
Þ½13
Condition [13], referred to as the propagation of
chaos, seems contradictory at first sight: if two
particles collide, correlations are created. Even though
we could assume eqn [13] at some time, if the test
particle collides with particle 2, such an equation
cannot be satisfied anymore after the collision.
Before discussing the propagation of chaos
hypothesis, we first analyze the size of the collision
operator. We remark that, in practical situations
for a rarefied gas, the combination Nr
3
10
4
cm
3
(i.e., the volume occupied by the particles) is very
small, while Nr
2
= O(1). This implies that G = O(1).
Therefore, since we are dealing with a very large
number of particles, we are tempted to perform the
limit N !1 and r ! 0 in such a way that
r
2
= O(N
1
). As a consequence, the probability that
two tagged particles collide (which is of the order of
the surface of a ball, i.e., O(r
2
)) is negligible.
However, the probability that a given particle
performs a collision with any one of the remaining
N 1 particles (which is O(Nr
2
) = O(1)) is not
negligible. Therefore, condition [13] is referring to
two preselected particles (say particles 1 and 2), so
that it is not unreasonable to conceive that it holds
in the limiting situation in which we are working.
However, we cannot insert [13] in [12] because
this latter equation refers to instants before and after
the collision and, if we know that a collision took
place, we certainly cannot invoke eqn [13]. Hence, it
is more convenient to assume eqn [13] in the loss
term and work over the gain term to keep advantage
Boltzmann Equation (Classical and Quantum) 307
of the factorization property which will be assumed
only before the collision.
Coming back to eqn [11] for the outgoing pair
velocities v, v
2
(satisfying the condition (v
2
v) n > 0),
we make use of the continuity property
f
2
ðx; v; x þ nr; v
2
Þ¼f
2
x; v
0
; x þ nr; v
0
2
½14
where the pair v
0
, v
0
2
is pre-collisional. On f
2
expressed before the collision, we can reasonably
apply condition [13] and obtain
G L ¼ðN 1Þr
2
Z
dv
2
Z
S
2
þ
dnðv v
2
Þn
½f ðx; v
0
Þfx nr; v
0
2
f ðx; vÞf ðx þ nr; v
2
Þ ½15
after a change n !n in the gain term, using the
notation S
2
þ
for the hemisphere {nj= (v
2
v) n 0}.
This transforms the pair v
0
, v
0
2
from a pre-collisional
to a post-collisional pair.
Finally, in the limit N !1, r !0, Nr
2
=
1
,we
find
ð@
t
þ v r
x
Þf
¼
1
Z
dv
2
Z
S
þ
dnðv v
2
Þn
½f ðx; v
0
Þfx; v
0
2
f ðx; vÞf ðx; v
2
Þ ½16
The parameter , called mean free path, represents,
roughly speaking, the typical length a particle can
cover without undergoing any collision. In eqns [1]
and [2], we just chose = 1.
Equation [16] (or, equivalently, eqns [1] and [2])is
the Boltzmann equation for hard spheres. Such an
equation has a statistical nature, and it is not
equivalent to the Hamiltonian dynamics from which
it has been derived. Indeed, the H-theorem shows that
such an equation is not reversible in time as expected
of any law of mechanics.
This concludes the heuristic preliminary analysis of
the Boltzmann equation. We certainly know that the
above arguments are delicate and require a more
rigorous and deeper analysis. If we want the Boltzmann
equation not to be a phenomenological model, derived
by ad hoc assumptions and justified only by its
practical relevance, but rather that it is a consequence
of a mechanical model, we must derive it rigorously. In
particular, the propagation of chaos should be not a
hypothesis but the statement of a theorem.
Beyond the Hard Spheres
The heuristic arguments we ha ve developed so far
can be extended to different potentials than that of
the hard-sphere systems. If the particles interact via
a two-body interaction V = V(r), the resulting
Boltzmann equation is eqn [1], with
Qðf ; f Þ¼
Z
dv
1
Z
S
2
þ
dnBðv v
1
; nÞ f
0
f
0
1
ff
1
½17
where we are using the usual shorthand notation:
f
0
¼ f ðx; v
0
Þ; f
0
1
¼ fx; v
0
1
; f ¼ f ðx; vÞ;
f
1
¼ f ðx; v
1
Þ
½18
and B = B(v v
1
; n) is a suitable function of the
relative velocity v v
1
and the impact parameter n,
which is proportional to the cross section relative to
the potential V. Another equivalent, sometimes
more convenient, way, to expres s eqn [17] is
Qðf ; f Þ¼
Z
dv
1
Z
dv
0
Z
dv
0
1
Wv; v
1
jv
0
; v
0
1
f
0
f
0
1
ff
1
½19
with
Wv; v
1
jv
0
; v
0
1
¼ wv; v
1
jv
0
; v
0
1
v þ v
1
v
0
v
0
1
1
2
v
2
þ v
2
1
v
0
ðÞ
2
v
0
1
2
½20
where w is a suitable kernel. All the qualitative
properties, such as the conservation laws and the
H-theorem, are obviously still valid.
Consequences
The Boltzmann equation provoked a debate involving
Loschmidt, Zermelo, and Poincare´, who outlined
inconsistencies between the irreversibility of the equa-
tion and the reversible character of the Hamiltonian
dynamics. Boltzmann argued the statistical nature of
his equation and his answer to the irreversibility
paradox was that ‘ ‘most’’ of the configurations behave
as expected by the thermodynamical laws. However,
he did not have the probabilistic tools for formulating
in a precise way the statements of which he had a
precise intuition.
Grad (1949) stated clearly the limit N !1,
r ! 0, Nr
2
! const :, where N is the number of
particles and r is the diameter of the molecul es, in
which the Boltzmann equation is expected to hold.
This limit is usually called the Boltzmann–Grad limit
(B–G limit in the sequel).
The problem of a rigorous derivation of the
Boltzmann equation was an open and challenging
problem for a long time. Lanford (1975) showed that,
although for a very short time, the Boltzmann equation
can be derived starting from the mechanical model of the
hard-sphere system. The proof has a deep content but is
relatively simple from a technical viewpoint.
308 Boltzmann Equation (Classical and Quantum)
Existence
The mathematical study of the Boltzmann equation
starts with the problem of proving the existence of
the solutions. One would like to be able to show that,
for all (or at least for a physically significant family
of) initial distributions (which are positive and
summable functions) with finite momentum, energy,
and entropy, there exists a unique solution to eqn [1]
with the same mass, momentum, and energy as of the
initial distribution. Moreover, the entropy should
decrease and the solution should approach the right
Maxwellian as t !1. The problem, in such a
generality, is still unsolved, but several results in this
direction have been achieved since the pioneering
works due to Carleman (1933) for the homogeneous
equation. Actually, there are satisfactory results for
some special situations, such as the homogeneous
solutions (independent of x) close to the equilibrium,
to the vacuum, or to homogeneous data. The most
general result we have up to now is, unfortunately,
not constructive. This is due to Di Perna and Lions
(1989), who showed the existence of suitable weak
solutions to eqn [1]. However, we still do not know
whether such solutions, which preserve mass and
momentum, and satisfy the H-theorem, are unique
and also preserve the energy.
Hydrodynamics
The derivation of hydrodynamical equations from
the Boltzmann equation is a problem as old as the
equation itself and, in fact, it goes back to Maxwell
and Hilbert. Preliminary to the discussion of the
hydrodynamic limit, we establish a few properties of
the collision kernel.
It is a well-known fact that the only solution to
the equation
Qðf ; f Þ¼0 ½21
is a local Maxwellian, namely
f ðx; vÞ:¼ Mðx; vÞ
¼
ðxÞ
ð2TðxÞÞ
3=2
e
jvuðxÞj
2
=2TðxÞ
½22
where the local parameters , u, and T satisfy the
relations
Z
M dv ¼ ½23
Z
vM ¼ u ½24
1
2
Z
v
2
M dv ¼
3
2
T þ
1
2
u
2
½25
Moreover, the only solution to the equ ation
Z
hðvÞQðf ; f Þdv ¼ 0 ½26
is any linear combination of the quantities (1, v, v
2
),
called collision invariants. The last property
obviously corresponds to the mass, momentum,
and energy conservation.
With this in mind, consider a change of
variables in the Boltzmann equation [1],passing
from microscopic to macroscopic variables,
x !"x, t !"t.Here" is a s mall scale parameter
expressing the ratio between the typical inter-
particle distances and the typical distances over
which the macroscopic equations are varying.
Such a change yields
ð@
t
þ v r
x
Þf
"
¼
1
"
Qðf
"
; f
"
Þ½27
We need to allow the small parameter " (mean free
path or the Knudsen number) to tend to zero. In
order to eliminate the singularity on the right-hand
side of [27], we multiply both sides by the collision
invariants v
with = 0, 1, 2; and obtain the five
equations:
Z
dvv
ð@
t
þ v r
x
Þf
"
¼ 0 ½28
On the other hand, if f
"
converges to f,as" !0,
necessarily Q(f , f ) = 0 and hence f = M. Therefore,
we expect that in the limit " ! 0,
Z
dvv
ð@
t
þ v r
x
ÞM ¼ 0 ½29
Equation [29] fixes a relation among the fields , u, T
as functions of x and t. A standard computation gives
us the Euler equations for compressible gas
@
t
þ divðuÞ¼0 ½30
@
t
u þðu rÞu þ
1
rp ¼ 0 ½31
@
t
T þðu rÞT þ
2
3
Tru ¼ 0 ½32
where the pressure p is related to the density and
the temperature T by the perfect gas law
p ¼ T ½33
In order to make the above arguments rigorous,
Hilbert (1916) developed a useful tool, called the
Hilbert expansion, to control the limiting procedure.
Boltzmann Equation (Classical and Quantum) 309
Namely, he expressed a formal solution to eqn [27]
in the form of a power series expansion:
f
"
¼
X
j0
f
j
"
j
½34
where f
0
is the local Maxwellian, with the para-
meters , u, T satisfying the Euler equations. All the
other coefficients f
j
of the developments can be
determined by recurrence, inverting suitable opera-
tors. However, the series is not expected to be
convergent, so that the way to show the validity of
the hydrodynamical limit rigorously is to truncate
the expansion and to control the remainder. The
first result in this direction was obtained by Caflisch
(1980). However, this approach is based on the
regularity of the solutions to the Euler equations,
which is known to hold only for short times since
shocks can be formed. How to approximate the
shocks in terms of a kinetic description is still a
difficult and open problem.
Note that the hydrodynamical picture of the
Boltzmann equation just means that we are looking
at the solutions of this equation at a suitable
macroscopic scale. The rarefaction hypothesis
underlying the Boltzmann description is reflected in
the law of perfect gas, which states that the
particles, in the local thermal equilibrium, are free.
Stationary Problems
Stationary non-Maxwellian solutions to the
Boltzmann equation should describe stationary
nonequilibrium states exhibiting nontrivial flows.
In spite of the physical relevance of these problems,
not many complete mathematical results are, at the
moment, available. Among them, there is the
traveling-wave problem, which can be formulated
in the following way. We look for a solution
f = f ( x ct, v), f : R R
3
!R
þ
, constant in form
but traveling with a constant velocity c > 0, to
ðv
1
cÞf
0
¼ Qðf ; f Þ½35
where v
1
is the first component of v and f
0
denotes
the spatial derivative of f. Equation [35] must be
complemented by the boundary conditions which
are f !M
,asx !1, where M
are the right
and left Maxwellians, namely two prescribed equili-
brium situations at infinity. The parameters (density,
mean velocity, and temperature) of the Maxwel-
lians, however, cannot be chosen arbitrarily. Indeed,
the conservations of the mass, momentum, and
energy (which are properties of Q) imply the
conservations (in x) of the fluxes of these quantities.
Hence, we have to impose five equations that relate
the upstream and the downstream values of the
densities, mean velocities, and temperatures. Such
relations are known in gas dynamics as the
Rankine–Hugoniot conditions. A solution of this
problem has been found by Caflisch and Nikolaenko
(1983) in case of a weak shock (namely, when M
þ
and M
are close) by using Hilbert expansion
techniques. More recently, Liu and Yu (2004)
established also stability and positivity of this
solution.
Quantum Kinetic Theory
Uehling and Uhlembeck (1933) introduced the
following kinetic equation for describing a large
system of weakly interacting bosons or fermions:
ð@
t
þ v r
x
Þf ¼
Z
dv
1
Z
dv
0
Z
dv
0
1
Wv; v
1
jv
0
; v
0
1
fð1 f Þð1 f
1
Þf
0
f
0
1
ð1 f
0
Þ 1 f
0
1
ff
1
g½36
Here the þ/ sign, stand for bosons/fermions,
respectively, and
Wv; v
1
jv
0
; v
0
1
¼ð
^
Vðv
0
vÞ
^
Vðv
0
v
1
ÞÞ
2
v þ v
1
v
0
v
0
1
1
2
v
2
þ v
2
1
ðv
0
Þ
2
v
0
1
2
½37
Moreover,
^
VðpÞ¼4
Z
dx e
ipx
½38
where V is the interaction potential. Note that eqn
[37] is the expression of the cross section of a
quantum scattering in the Born approximation.
The unknown f = f (x, v; t)ineqn [37] is the expected
number of molecules falling in the unit (quantum) cell
of the phase space. This function is proportional to the
one-particle Wigner function, introduced by Wigner
(1932) to handle kinetic problems in quantum
mechanics, and defined as (setting h = 1):
1
ð2Þ
3
Z
dy e
iyv
x þ
1
2
y; x
1
2
y
where (x; z) is the kernel of a one-particle density
matrix. Basically, the Wigner function is an equiva-
lent way to describe a state of a quantum system.
For instance, eqn [40] below expres ses the equili-
brium distributions for bosons and fermions in
terms of Wigner funct ions. In general, the Wigner
functions, due to the uncertainty principle, are real
but not necessarily positive; however, the inte gral
with respect to x and v gives the probability
310 Boltzmann Equation (Classical and Quantum)
distributions of the velocity and the position,
respectively. In the kinetic regime, in which we are
interested, the scales are mesoscopic, namely the
typical quantum osci llations are on a scale much
smaller than the characteristic scal es of the problem,
so that we expect that f should be a genuine
probability distribution, since the Heisenberg
principle does not play an essential role. However,
the inte raction oc curs on a microscopic scale, so that
we expect that the statistics play a role in addition
to the quantum rules for the scattering.
In this framework, the entropy functional is
Hðf Þ¼
Z
dx
Z
dv ½f ðx; vÞlog f ðx; vÞ
ð1 f ðx; vÞÞlogð1 f ðx; vÞÞ ½39
It is decreasing along the solutions to eqn [35] and it is
also minimized (among the distributions with give n
mass, momentum, and energy) by the equilibria
MðvÞ¼
z
e
ð=2Þjvuj
2
z
½40
namely the Bose–Einstein and the Fermi–Dirac
distributions, respectively. Here >1andz > 0
are the inverse temperature and the activity, respec-
tively. Note that, for the Bose–Einstein distribution,
z < 1. This creates, in a sense, an inconsistency with
eqn [36]. Indeed, assuming u = 0 and an initial
distribution f = f
0
(v) with the density larger than the
maximal density allowed by eqn [40], namely
c
:¼
Z
dv
1
e
ð=2Þv
2
1
½41
it cannot converge to any equilibrium. In order to
overcome this difficulty related to the Bose con-
densation, one can enlarge the definition of the
equilibria family by setting
MðvÞ¼
1
e
ð=2Þv
2
1
þ ðvÞ½42
to take care of excess of mass by means of a condensate
component. However, it is not clear whether eqn
[36] can actually describe the Bose condensation
since its derivation from the Schro¨ dinger equation
requires, just from the very beginning, the existence of
bosonic quasifree states which can be constructed only
if the density is moderate. Further analyses are certainly
needed to clarify the situation. A rigorous derivation of
the Uehling and Uhlembeck equation is, up to now, far
from being obtained even for short times; nevertheless,
such an equation is extensively used in the applications.
Equation [36] concerns a weakly interacting gas of
quantum particles. From a mathematical viewpoint, it
is expected to be valid in the so-called weak-coupling
limit, which consists in scaling space and time and the
interaction potential as
x ! "x; t ! "t;!
ffiffiffi
"
p
½43
where "
1
= N
1=3
is a parameter diverging when the
number of particles N tends to infinity.
We mention, incidentally, that under such a
scaling, a classical system is described by a transport
equation, called Fokker–Planck–Landau equation,
with a diffusion operator in the velocity space.
The B–G limit considered for classical particle
systems is different from that considered here
for weakly interacting qua ntum systems. It is actually
equivalent to rescaling space an d time according to
x ! "x; t ! "t ½44
leaving the interaction unscaled but, in order to
control the total interaction, we make the density
diverging gently as "
1
= N
1=2
.
A quantum system under such a scaling is expected to
be described by a Boltzmann equation [1] with the
collision operator Q computed with the full quantum
cross section. Now we do not have any effect of the
statistics because in this rarefaction limit these correc-
tions disappear. On the other hand, the cross section is
that arising from the analysis of the quantum scattering.
Since we do not rescale the interaction, all the other
terms in the Born expansion of the cross section play a
role. This kind of Boltzmann equation is a good
description of a rarefied gas in which quantum effects
are not negligible.
See also: Adiabatic Piston; Evolution Equations: Linear
and Nonlinear; Gravitational N-Body Problem (Classical);
Interacting Particle Systems and Hydrodynamic
Equations; Kinetic Equations; Multiscale Approaches;
Nonequilibrium Statistical Mechanics: Dynamical
Systems Approach; Quantum Dynamical Semigroups.
Further Reading
Balesku R (1978) Equilibrium and Nonequilibrium Statistical
Mechanics. Moscow: Mir (distributed by Imported Publica-
tions, Chicago, Ill).
Caflisch RE (1980) The fluid dynamical limit of the nonlinear
Boltzmann equation. Communications of Pure and Applied
Mathematics 33: 651–666.
Caflisch RE and Nicolaenko B (1983) Shock waves and the
Boltzmann equation. Nonlinear partial differential equations.
Contemporary Mathematics 17: 35–44.
Carleman T (1933) Sur la the´orie de l’e´quation inte´gro-differentielle
de Boltzmann. Acta Mathematica 60: 91–146.
Cercignani C (1998) Ludwig Boltzmann. The Man Who Trusted
Atoms. Oxford: Oxford University Press.
Cercignani C, Illner R, and Pulvirenti M (1994) The Mathema-
tical Theory of Dilute Gases. Springer Series in Applied
Mathematics, vol. 106. New York: Springer.
Boltzmann Equation (Classical and Quantum) 311
Di Perna RJ and Lions P-L (1989) On the Cauchy problem for the
Boltzmann equations: Global existence and weak stability.
Annals of Mathematics 130: 321–366.
Grad H (1949) On the kinetic theory of rarified gases.
Communications in Pure and Applied Mathematics
2: 331–407.
Hilbert D (1916) Begru¨ ndung der Kinetischen Gastheorie.
Mathematische Annalen 72: 331–407.
Lanford OE III (1975) Time evolution of large classical systems.
In: Ehlers J, Hepp K, and Weidenmu¨ ller HA (eds.) Lecture
Notes in Physics, vol. 38, pp. 1–111. Berlin: Springer.
Liu T-P and Yu S-H (2004) Boltzmann equation: micro–macro
decompositions and positivity of shock profiles. Communica-
tions in Mathematical Physics 246(1): 133–179.
Spohn H (1994) Quantum kinetic equations. In: Fannes M, Maes C,
and Verbeure A (eds.) On Three Levels: Micro, Meso and Macro
Approaches in Physics. New York: Plenum.
Uehling EA and Uhlembeck GE (1933) Transport phenomena in
Einstein–Bose and Fermi–Dirac gases. I. Physical Reviews
43: 552–561.
Wigner EP (1932) On the quantum correction for thermodynamic
equilibrium. Physical Reviews 40: 749–759.
Bose–Einstein Condensates
F Dalfovo, L P Pitaevskii, and S Stringari,
Universita
`
di Trento, Povo, Italy
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
In 1924 the Indian physicist S N Bose introduced a new
statistical method to derive the blackbody radiation law
in terms of a gas of light quanta (photons). His work,
together with the contemporary de Broglie’s idea of
matter–wave duality, led A Einstein to apply the same
statistical approach to a gas of N indistinguishable
particles of mass m. An amazing result of his theory was
the prediction that below some critical temperature a
finite fraction of all the particles condense into the
lowest-energy single-particle state. This phenomenon,
named Bose–Einstein condensation (BEC), is a conse-
quence of purely statistical effects. For several years,
such a prediction received little attention, until 1938,
when F London argued that BEC could be at the basis of
the superfluid properties observed in liquid
4
He below
2.17 K. A strong boost to the investigation of Bose–
Einstein condensates was given in 1995 by the observa-
tion of BEC in dilute gases confined in magnetic traps
and cooled down to temperatures of the order of a few
nK. Differently from superfluid helium, these gases
allow one to tune the relevant parameters (confining
potential, particle density, interactions, etc.), so to make
them an ideal test-ground for concepts and theories on
BEC.
What Is BEC?
In nature, particles have either integer or half-
integer spin. Those having half-integer spin, like
electrons, are called f ermions and obey the Fermi–
Dirac statistics; those having integer spin are
called bosons and obey the Bose–Einstein statis-
tics. Let us consider a system of N bosons. In
order to introduce the concept of BEC on a
general ground, one can start with the definition
of the one-body density matrix
n
ð1Þ
ðr; r
0
Þ¼
^
y
ðrÞ
^
ðr
0
Þ
½1
The quantities
^
y
(r) and
ˆ
(r) are the field operators
which create and annihilate a particle at point r,
respectively; they satisfy the bosonic commutation
relations
½
^
ðrÞ;
^
y
ðr
0
Þ ¼ ðr r
0
Þ; ½
^
ðrÞ;
^
ðr
0
Þ ¼ 0 ½2
If the system is in a pure state described by the
N-body wave function (r
1
,...,r
N
), then the
average [1] is taken following the standard rules of
quantum mech anics and the one-body density
matrix can be written as
n
ð1Þ
ðr;r
0
Þ
¼N
Z
dr
2
dr
N
ðr;r
2
;...;r
N
Þðr
0
;r
2
;...;r
N
Þ½3
involving the integration over the N 1 variables
r
2
,...,r
N
. In the more general case of a statistical
mixture of pure states, expression [3] must be
averaged according to the probability for a system
to occupy the different states.
Since n
(1)
(r, r
0
) = (n
(1)
(r
0
, r))
the quantity n
(1)
,
when regarded as a matrix function of its indices
r and r
0
, is Her mitian. It is therefore always possible
to find a complete orthonormal basis of single-
particle eigenfunctions, ’
i
(r), in terms of which the
density matrix takes the diagonal form
n
ð1Þ
ðr; r
0
Þ¼
X
i
n
i
’
i
ðrÞ’
i
ðr
0
Þ½4
The real eigenvalues n
i
are subject to the normal-
ization condition
P
i
n
i
= N and have the meaning of
occupation numbers of the single-particle states ’
i
.
BEC occurs when one of these numbers (say, n
0
)
becomes macroscopic, that is, when n
0
N
0
is a
number of order N, all the others remaining of order 1.
312 Bose–Einstein Condensates
In this case eqn [4] can be conveniently rewritten in
the form
n
ð1Þ
ðr; r
0
Þ¼N
0
’
0
ðrÞ’
0
ðr
0
Þþ
X
i6¼0
n
i
’
i
ðrÞ’
i
ðr
0
Þ½5
and the state represented by ’
0
(r) is called
Bose–Einstein condensate. This definition is rather
general, since it applies to any macroscopic (N 1)
system of indistinguishable bosons independently of
mutual interactions and external fields.
The one-body density matrix [1] contains informa-
tion on important physical observables. By setting
r = r
0
one finds the diagonal density of the system
nðrÞn
ð1Þ
ðr; rÞ¼h
^
y
ðrÞ
^
ðrÞi ½6
with N =
R
dr n(r). The off-diagonal components
can instead be used to calculate the momentum
distribution
nðpÞ¼h
^
y
ðpÞ
^
ðpÞi ½7
where
^
(p) = (2h)
3=2
R
dr
^
(r) exp [ip r=h] is the
field operator in momentum representation. By
inserting this expression for
^
(p) into eqn [7] one
finds
nðpÞ¼
1
ð2hÞ
3
Z
dR ds n
ð1Þ
R þ
s
2
; R
s
2
e
ips=h
½8
where s = r r
0
and R = (r þ r
0
)=2.
Let us consider a uniform system of N particles in
a volume V and take the thermodynamic limit
N, V !1 with density N/V kep t fixed. The eigen-
functions of the density matrix are plane waves and
the lowest-energy state has zero momentum, p = 0,
and constant wave function ’
0
(r) = V
1=2
. BEC in
this state implies a macroscopic number of particles
having zero momentum and constant density N
0
=V.
The density matrix only depends on s = r r
0
and
can be written as
n
ð1Þ
ðsÞ¼
N
0
V
þ
1
V
X
p6¼0
n
p
e
ips=h
½9
In the s !1limit, the sum on the right vanishes due
to destructive interference between different plane
waves, but the first term survives. One thus finds that,
in the presence of BEC, the one-body density matrix
tends to a constant finite value at large distances. This
behavior is named off-diagonal long-range order,
since it involves the off-diagonal components of the
density matrix. Its counterpart in momentum space is
the appearance of a singular term at p = 0:
nðpÞ¼N
0
ð pÞþ
X
p
0
6¼0
n
p
0
ð p p
0
Þ½10
The sum on the right is the number of noncondensed
particles (N N
0
), and the quan tity N
0
=N is called
condensate fraction.
If the system is not uniform, the eigenfunctions of
the density matrix are no longer plane waves but,
provided N is sufficiently large, the concept of BEC
is still well defined, being associated with the
occurrence of a macroscopic occupation of a
single-particle eigenfunction ’
0
(r) of the density
matrix. Thus, the condensed bosons can be
described by means of the function (r) =
ffiffiffiffiffiffiffi
N
0
p
’
0
(r), which is a classical compl ex field playing
the role of an order parameter. This is the analog of
the classi cal limit of quantum electrodynamics,
where the electromagnetic field replaces the micro-
scopic description of photons. The function may
also depend on time and can be written as
ðr; tÞ¼jðr; tÞje
iSðr;tÞ
½11
Its modulus determines the contribution of the
condensate to the diagonal density [6], while the
phase S is crucial in characterizing the coherence
and superfluid propertie s of the system. The order
parameter [11], also named macroscopic wave
function or condensate wave function, is defined
only up to a constant phase factor. One can always
multiply this function by the numerical factor e
i
without changing any physical property. This
reflects the gauge symmet ry exhibited by all the
physical equations of the problem. Making an
explicit choice for the value of the order parameter,
and hence for the phase, corresponds to a formal
breaking of gauge symmetry.
BEC in Ideal Gases
Once we have defined what is a Bose–Einstein
condensate, the next question is when such a
condensation occurs in a given system. The ideal
Bose gas provides the simplest example. So, let us
consider a gas of noninteracting bosons described
by the Hamiltonian
^
H =
P
i
^
H
(1)
i
, where the Schro¨-
dinger equation
^
H
(1)
’
i
(r) =
i
’
i
(r)givesthespec-
trum of single-particle wave functions and
energies. One can define an occupation number
n
i
as the num ber of particles in the state with
energy
i
. Thus, any given state of the many-body
system is specified by a set {n
i
}. The mean
occupation numbers,
n
i
, c an be calculated by
using the standard rules of statistical mechanics.
For instance, by considering a grand canonical
ensemble at temperature T, one finds
n
i
¼fexp½ð
i
Þ 1g
1
½12
Bose–Einstein Condensates 313
with = 1=(k
B
T). The chemical potential is fixed
by the normalization condition
P
i
n
i
= N, where N
is the averag e number of particles in the gas. For
T !1the chemical potential is negative and large.
It increases monotonically when T is lowered. Let us
call
0
the lowest single-particle level in the
spectrum. If at some critical temperature T
c
the
normalization condition can be satisfied with
!
0
, then the occupation of the lowest state,
n
0
= N
0
, becomes of order N and BEC is realized.
Below T
c
the normalization condition must be
replaced with N = N
0
þ N
T
, where N
T
=
P
i6¼0
n
i
is
the num ber of particles out of the condensate, that
is, the thermal component of the gas. Whether BEC
occurs or not, and what is the value of T
c
depends
on the dimensionality of the system and the type of
single-particle spectrum .
The simplest case is that of a gas confined in a
cubic box of volume V = L
3
with periodic boundary
conditions, where
^
H
(1)
= (h
2
=2m)r
2
. The eigen-
functions are plane waves ’
p
(r) = V
1=2
exp [ip
r=h], with energy
p
= p
2
=2m and momentum
p = 2hn=L. Here n is a vector whose components
n
x
, n
y
, n
z
are 0 or integers. The lowest eigenvalue
has zero energy (
0
= 0) and zero momentum. The
mean occupation numbers are given by
n
p
= {exp [(p
2
=2m )] 1}
1
. In the thermo-
dynamic limit (N, V !1with N/V kept constant),
one can replace the sum
P
p
with the integral
R
d(), where () = (2)
2
V(2m=h
2
)
3=2
ffiffi
p
is the
density of states. In this way, one can calculate the
thermal component of the gas as a function of T,
finding the critical temperature
k
B
T
c
¼
2h
2
m
N
Vð3=2Þ
2=3
½13
where is the Riemann zeta function and (3=2) ’
2.612. For T > T
c
, one has <0 and N
T
= N. For
T < T
c
one instead has = 0, N
T
= N N
0
and
N
0
ðTÞ¼N½1 ðT=T
c
Þ
3=2
½14
The critical temperature turns out to be fully
determined by the density N/V and by the mass of
the constituents. These results were first obtained
by A Einstein in his seminal paper and used by
F London in the context of superfluid helium. We
notice that the replacement of the sum with an
integral in the above deri vation is justified only if
the thermal energy k
B
T is much larger than the
energy spacing between single-particle levels, that is,
if k
B
T h
2
=2mV
2=3
. Is is also worth noticing that
the above expression for T
c
can be written as
3
T
N=V ’ 2.612, where
T
= [2h
2
=(mk
B
T)]
1=2
is
the thermal de Broglie wavelength. This is
equivalent to saying that BEC occurs when the
mean distance between bosons is of the order of
their de Broglie wavelength.
Another interesting case, which is relevant for the
recent experiments with BEC in dilute gases con-
fined in magnetic and/or optical traps, is that of an
ideal gas subje ct to ha rmonic potentials. Let us
consider, for simplicity, an isotropic exter nal poten-
tial V
ext
(r) = (1=2)m!
2
ho
r
2
. The single-particle Hamil-
tonian is
^
H
(1)
= (h
2
=2m)r
2
þ V
ext
(r) and its
eigenvalues are
n
x
, n
y
, n
z
= (n
x
þ n
y
þ n
z
þ 3=2)h!
ho
.
The corresponding density of states is () =
(1=2)(h!
ho
)
3
2
. A natural therm odynamic limit for
this syst em is obtained by letting N !1 and
!
ho
!0, while keeping the product N!
3
ho
constant.
The condition for BEC to occur is that approaches
the value
000
= (3=2)h!
ho
from below by cooling the
gas down to T
c
. Following the same procedure as
for the uniform gas, one finds
k
B
T
c
¼ h!
ho
½N=ð3Þ
1=3
¼ 0:94h!
ho
N
1=3
½15
and
N
0
ðTÞ¼N ½1 ðT=T
c
Þ
3
½16
Notice that the condensate is not uniform in this case,
since it corresponds to the lowest eigenfunction of the
harmonic oscillator, which is a Gaussian of width
a
ho
= [h=(m!
ho
)]
1=2
. Correspondingly, the condensate
in the momentum space is also a Gaussian, of width
a
1
ho
. This implies that, differently from the gas in a box,
here the condensate can be seen both in coordinate and
momentum space in the form of a narrow distribution
emerging from a wider thermal component. Finally,
results [15] and [16] remain valid even for anisotropic
harmonic potentials, with trapping frequencies !
x
, !
y
,
and !
z
, provided the frequency !
ho
is replaced by the
geometric average (!
x
!
y
!
z
)
1=3
.
BEC in Interacting Gases
Actual condensates are made of interacting particles.
The full many-body Hamiltonian is
^
H ¼
Z
dr
^
y
ðrÞ
^
H
0
^
ðrÞ
þ
1
2
Z
dr
0
dr
^
y
ðrÞ
^
y
ðr
0
ÞVðr r
0
Þ
^
ðr
0
Þ
^
ðrÞ½17
where V(r r
0
) is the particle–particle interaction and
^
H
0
= (h
2
=2m)r
2
þ V
ext
(r). Differently from the
case of ideal gases,
^
H is no longer a sum of single-
particle Hamiltonians. However, the general defini-
tions given in the section ‘‘What is BEC?’’ a re still
valid. In particular, the one-body density matrix, in the
presence of BEC, can be separated as in eqn [5].One
314 Bose–Einstein Condensates