Spohn H. Dynamics of Charged Particles and their Radiation Field
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18.2 Equilibrium states and their perturbations, KMS condition 283
If H is invariant under time-reversal, we may choose C = T , with time-reversal
T , and CHC = THT = H . Then the Liouvillean is given by
L = H ⊗ 1 − 1 ⊗ H (18.24)
as an operator on
H ⊗ H. Clearly the spectrum of L consists of the energy
differences {E
i
− E
j
| i, j = 0, 1,...}, where E
i
, i = 0, 1,..., are the eigenval-
ues of H .
18.2 Equilibrium states and their perturbations, KMS condition
Of the many possible quantum states thermal equilibrium plays a special role. It is
defined by the density matrix
ρ
β
= Z
−1
e
−β H
, Z = tr[e
−β H
]. (18.25)
As an element of
T
2
(H) we set
κ
β
= Z
−1/2
e
−β H/2
. (18.26)
Then ρ
β
= κ
β
κ
∗
β
. Since ρ
β
is strictly positive, a
∗
a
β
= tr[ρ
β
a
∗
a] = 0 for a ∈ A
implies a = 0. Equivalently,
(a)κ
β
= 0 implies a = 0, (18.27)
which means that κ
β
is separating for the algebra (A).Inprinciple, one should
allow for additional conservation laws like total charge or total number of particles.
However, this is ignored here since the Hamiltonian (18.1) does not have such a
structure.
For the photon field in infinite space the spectrum of H is continuous and
Z
−1
e
−β H
as such makes no sense. On the other hand, the atom is a small per-
turbation. Thus the equilibrium state of the coupled system relative to that of the
uncoupled system remains meaningful even at infinite volume and is the object of
thermal perturbation theory.
We consider
H = H
0
+ I. (18.28)
H
0
is the unperturbed reference system and I is the perturbation, assumed to be
bounded, I < ∞.Bythe Golden–Thompson inequality
Z
β
= tr[e
−β H
] = tr[e
−β(H
0
+I )
] ≤ tr[e
−β H
0
e
−β I
]
≤ e
βI
tr[e
−β H
0
] = e
βI
Z
0
β
. (18.29)
Thus if Z
0
β
< ∞,asassumed, then ρ
β
= Z
−1
β
e
−β H
∈ T
1
(H).
284 Relaxation at finite temperatures
The Liouvillean of the reference system is given by
L
0
= (H
0
) −r (H
0
) (18.30)
and the Liouvillean of the perturbed system by
L = (H ) −r (H ). (18.31)
Under the isomorphism I
C
the Liouvilleans become
L = I
C
LI
−1
C
= (H
0
+ I ) ⊗ 1 − 1 ⊗ C(H
0
+ I )C = L
0
+ W,
L
0
= H
0
⊗ 1 − 1 ⊗CH
0
C, W = I ⊗ 1 −1 ⊗CIC. (18.32)
We also define the Radon–Nikodym operators,
L
and
L
r
, through
L
= L
0
+ (I ), L
r
= L
0
−r (I ). (18.33)
Then, with κ
β
= (Z
β
)
−1/2
e
−β H/2
,κ
0
β
= (Z
0
β
)
−1/2
e
−β H
0
/2
,wehave
e
−βL
/2
κ
0
β
= e
−β((H
0
)+(I )−r(H
0
))/2
κ
0
β
= e
−β(H
0
+I )/2
κ
0
β
e
β H
0
/2
= (Z
β
/Z
0
β
)
1/2
(Z
β
)
−1/2
e
−β H/2
= (Z
β
/Z
0
β
)
1/2
κ
β
(18.34)
and by a similar calculation
e
βL
r
/2
κ
0
β
= (Z
β
/Z
0
β
)
1/2
κ
β
. (18.35)
κ
0
β
is in the domain of the operators e
−βL
/2
and e
βL
r
/2
and their action maps
unperturbed to perturbed equilibrium,
κ
β
= (Z
0
β
/Z
β
)
1/2
e
−βL
/2
κ
0
β
,κ
β
= (Z
0
β
/Z
β
)
1/2
e
βL
r
/2
κ
0
β
. (18.36)
The thermal state e
−β H
is related to the unitary time evolution e
−itH
through an-
alytic continuation to β = it, which gives rise to a very powerful analytic structure
of equilibrium time correlations known as the Kubo–Martin–Schwinger (KMS)
boundary condition. We define the time correlations as
aα
t
(b)
β
= F
ab
(t), α
t
(b)a
β
= G
ab
(t). (18.37)
They are linked through
aα
t
(b)
β
= Z
−1
β
tr[e
−β H
ae
itH
be
−itH
] = Z
−1
β
tr[e
−β H
e
(β+it)H
be
−(β+it)H
a]
=α
−iβ+t
(b)a
β
, (18.38)
which is the KMS condition.Itstates that F
ab
(t) is the boundary value of a function
G
ab
(z) which is analytic in the strip S
−β
={z |−β<Imz < 0} such that
lim
η↑β
G
ab
(t − iη) = F
ab
(t). (18.39)
18.3 Spectrum of the Liouvillean and relaxation 285
Equivalently, G
ab
(t) is the boundary value of a function F
ab
(z) analytic in the strip
S
β
such that
lim
η↑β
F
ab
(t + iη) = G
ab
(t). (18.40)
A state which satisfies either of these boundary conditions is called a KMS state
with respect to the time evolution α
t
.Inour set-up, the only KMS state is ρ
β
. The
KMS condition is used as a defining property for equilibrium states in infinitely ex-
tended systems. In general, for the same group of automorphisms there could then
be several KMS states. Physically they represent distinct thermodynamic phases.
18.3 Spectrum of the Liouvillean and relaxation
As discussed in section 17.5, at zero temperature the relaxation to the ground state
can be reduced to a scattering problem, see Proposition 17.5. As a simplification,
at finite temperature it suffices to have sufficiently strong spectral properties of
the Liouvillean. We have in mind now a situation where the size of the black-body
cavity is huge on the atomic scale. Therefore the relevant mathematical idealization
is to have the photon field infinitely extended. The algebra B(
H) must be replaced
then by a suitable algebra
A of quasi-local observables. Its construction will be
explained in the following. At the moment we focus on the abstract structure. Thus
we have given the C
∗
-algebra A and a one-parameter group α
t
of ∗-automorphisms
as the dynamics. The distinguished state on
A is the KMS state ω
β
at inverse
temperature β. Its time correlations are defined by
F
ab
(t) = ω
β
(aα
t
(b)), G
ab
(t) = ω
β
(α
t
(b)a) (18.41)
and they satisfy the KMS boundary condition
G
ab
(t − iβ) = F
ab
(t), F
ab
(t + iβ) = G
ab
(t) ; (18.42)
compare with (18.39), (18.40). Note that ω
β
is necessarily time-invariant, since
ω
β
(1α
t
(b)) = ω
β
(α
t
(b)1) and by the KMS condition
F
1b
(t) = F
1b
(t + iβ). (18.43)
Let us define the ∗-algebra
A
0
through smoothing in time with a test function of
compact support in Fourier space,
A
0
=
a
f
=
dtf(t)α
t
(a) |a ∈ A,
f ∈ C
00
(R)
. (18.44)
For b ∈
A
0
, z → F
1b
(z) is an entire function bounded as |F
1b
(z)|≤α
z
(b)≤
α
iImz
(b).By(18.43) F
1b
is periodic with period iβ. Hence F
1b
is bounded and
286 Relaxation at finite temperatures
thus constant by Liouville’s theorem, which implies
ω
β
(α
t
(b)) = ω
β
(b) (18.45)
for all t ∈
R.
We assume that
A is a simple C
∗
-algebra, which means that the only two-sided
∗-ideals of
A are either {0} or A itself. The KMS condition then ensures that for
every a ∈
A
ω
β
(a
∗
a) = 0 implies a = 0. (18.46)
To prove (18.46) we define
N ={a ∈ A|ω
β
(a
∗
a) = 0} with the goal of establish-
ing that
N is a two-sided ∗-ideal. Clearly, if ω
β
(a
∗
a) = 0 and b ∈ A,then
ω
β
(a
∗
b
∗
ba) ≤ ω
β
(a
∗
b
∗
bb
∗
ba)
1/2
ω
β
(a
∗
a)
1/2
= 0 (18.47)
by the Schwarz inequality. Hence
AN ⊂ N .Toshow the converse one chooses
b ∈
A.Bythe KMS condition
ω
β
(b
∗
a
∗
ab) = ω
β
((b
∗
a
∗
a)b) = ω
β
(α
−iβ
(b)b
∗
a
∗
a) = 0 (18.48)
by the Schwarz inequality as before. Thus
NA ⊂ N and N is a two-sided ∗-ideal.
Since
A is simple, (18.46) follows.
Next we need the analog of
T
2
(H) and of the Liouvillean, which is the content
of the Gelfand–Naimark–Segal (GNS) construction. The GNS Hilbert space
H
β
is
defined as the completion of
A equipped with the scalar product
a|b=ω
β
(a
∗
b). (18.49)
By the argument above a|a=0 implies a = 0, as it should. In our context
H
β
is
aseparable Hilbert space. We set
β
= 1 and define the left representation of A
through
(a)b = ab. (18.50)
Thereby (
A) ⊂ B(H
β
).Inaddition we define
e
−itL
a = α
t
a (18.51)
on
H
β
. Since b|e
−itL
a=ω(b
∗
α
t
a) = ω(α
−t
b
∗
a) =e
itL
b|a and since
e
−itL
b|e
−itL
a=ω(α
t
(b
∗
a)) =b|a by time-invariance of ω
β
,e
−itL
is a
strongly continuous unitary group on
H
β
.ByStone’s theorem it has a self-adjoint
generator, which by definition is the Liouvillean
L.
The initial state of interest is a local perturbation of the equilibrium state ω
β
.It
can be written as
ρ(a) = ω
β
(b
∗
ab), b ∈ A ,ω
β
(b
∗
b) = 1. (18.52)
18.3 Spectrum of the Liouvillean and relaxation 287
In the GNS representation ρ corresponds to the state given by the vector b ∈ H
β
.
More generally, a perturbed state can be written as
ρ(a) =
∞
n=1
p
n
ω
β
(b
∗
n
ab
n
) (18.53)
with b
n
∈ A, ω(b
∗
n
b
n
) = 1, p
n
≥ 0,
∞
n=1
p
n
= 1. States of the form (18.53) are
called normal. A state not covered by this class would be a two-temperature state
of the photon gas, for example, where the temperature to the far right differs from
that to the far left. In fact, its long-time behavior would be rather different from
that of the initial states discussed here.
Let ρ be a normal state with time evolved ρ
t
(a) = ρ(α
t
(a)).Byrelaxation to
equilibrium we mean
lim
t→∞
ρ
t
(a) = ω
β
(a) (18.54)
for all a ∈
A.
Proposition 18.1 (Relaxation to equilibrium as a spectral property). Suppose the
Liouvillean
L has a purely absolutely continuous spectrum except for a nondegen-
erate eigenvalue at 0. Then for all a ∈
A
lim
t→±∞
ρ
t
(a) = ω
β
(a). (18.55)
Proof: Since ω
β
is time invariant, the (unique) zero eigenvector of
L is
β
.Byas-
sumption the spectral measure of ψ|e
−itL
ϕ has the point mass ψ|
β
β
|ϕ at
zero and is otherwise absolutely continuous. Therefore by the Riemann–Lebesgue
lemma
lim
t→∞
ψ|e
−itL
ϕ=ψ|
β
β
|ϕ (18.56)
for all ψ, ϕ ∈
H
β
.
In view of the structure of normal states it suffices to study
ω
β
(b
∗
α
t
(a)c) = ω
β
α
−iβ
(c)b
∗
α
t
(a)
=(b)(α
iβ
(c
∗
))
β
|(α
t
(a))
β
=(b)(α
iβ
(c
∗
))
β
|e
−itL
(a)
β
. (18.57)
We assume that a, b, c ∈
A
0
, see (18.44). Then (b)(α
iβ
(c
∗
))
β
, (a)
β
∈ H
β
.
Therefore from (18.56)
lim
t→∞
ω
β
(b
∗
α
t
(a)c) =(b)(α
iβ
(c
∗
))
β
|
β
β
|(a)
β
= ω
β
(α
−iβ
(c)b
∗
)ω
β
(a)
= ω
β
(b
∗
c)ω
β
(a), (18.58)
288 Relaxation at finite temperatures
which, upon inserting in (18.53), implies the limit (18.55). Note that the KMS
condition is used twice, in the first identity of (18.57) and in the last identity of
(18.58).
2
Proposition 18.1 suggests that relaxation to equilibrium can be established in
two steps: (i) One has to find for the equilibrium state a sufficiently concrete repre-
sentation of the algebra of local observables and of the Liouvillean. (ii) The spec-
tral properties of the Liouvillean must be studied. For (i) the natural representation
is the Araki–Woods representation of the free photon gas in infinite volume. It will
be taken up in the following section. The coupled system is constructed through
perturbation series. For the dynamics the time-dependent Dyson series is used and
for the thermal state the thermal perturbation theory of section 18.2. Of course, the
convergence of both series relies on the atom being modeled as an N-level system
and on the explicit control of the free photon gas. Only through the convergence
of the perturbation series are we assured of the correct representation spaces for
the interacting system. Nevertheless, we skip this important point completely and
jump to the spectral analysis of the interacting Liouvillean.
18.4 The Araki–Woods representation of the free photon field
For photons in a cavity , the spectrum of allowed momenta is discrete,
tr[exp[−β H
f,
]] < ∞, and the rules of thermal equilibrium for bounded quantum
systems are applicable, through which the time-correlations of local observables
in the form ω
β
(aα
t
(b)) are defined. A macroscopic cavity with its surface
kept at a uniform temperature is extremely well approximated by the infinite-
volume limit ↑
R
3
.For the Hamiltonian (18.1) the infinite-volume limit of time-
correlations can be established. Rather than going through the construction, we
merely state the final answer, which will serve as a basis for the study of relaxa-
tion.
We work in the momentum space representation. Without risk of confusion we
set k = (k,λ) ∈
R
3
×{1, 2} and
λ=1,2
d
3
k =
dk, δ(k −k
) = δ
λ,λ
δ(k −
k
). The bosonic field operators are
a( f ) =
dkf(k)a(k) =
λ=1,2
d
3
kf(k,λ)a(k, λ), a
∗
( f ) =
dkf(k)a
∗
(k)
(18.59)
with f ∈
S
0
(R
3
×{1, 2}), the Schwartz space of functions that decrease rapidly
and vanish at k = 0. Observe that our convention for the complex conjugation
of the test function f differs from that in (13.59), (13.60). Let us also intro-
duce the complex conjugation τ f (k) = f (k)
∗
=
f (k, 1)
∗
, f (k, 2)
∗
. Its second
18.4 The Araki–Woods representation of the free photon field 289
quantization is the anti-unitary time-reversal operator T on F with the properties
T = , Ta
( f )T = a
(τ f ), T = T
∗
= T
−1
. (18.60)
Note that (a
∗
( f ))
∗
= a(τ f ) and f, g
h
=
dk(τ f )g. The boson fields satisfy the
canonical commutation relations (CCR),
[a
∗
( f ), a
∗
(g)] = 0 = [a( f ), a(g)], (18.61)
[a(τ f ), a
∗
(g)] =f, g
h
. (18.62)
Let
P denote the polynomial ∗-algebra generated by
{a
∗
( f ), a(g) | f, g ∈ S
0
(R
3
)
2
}. (18.63)
On
P the time evolution α
f
t
is defined through
α
f
t
(a
∗
(k)) = e
itω(k)
a
∗
(k), α
f
t
(a(k)) = e
−itω(k)
a(k). (18.64)
The equilibrium state ω
f
β
of the photon field at inverse temperature β is a quasi-free
state on
P. Set
ρ
β
(k) =
1
e
βω(k)
− 1
. (18.65)
Then the two-point function is given by
ω
f
β
a
∗
(τ f )a(g)
=f,ρ
β
g
h
(18.66)
and all other moments by
ω
f
β
n
i=1
a
∗
(τ f
i
)
m
j=1
a(g
j
)
= δ
mn
det{ f
i
,ρ
β
g
j
h
}
i, j=1,...,n
. (18.67)
ω
f
β
satisfies the KMS condition as can be seen directly from
ω
f
β
a(k)a
∗
(k
)
= δ(k − k
) + ω
f
β
a
∗
(k
)a(k)
= e
βω(k)
ρ
β
(k)δ(k − k
)
= ω
f
β
α
f
−iβ
(a
∗
(k
))a(k)
. (18.68)
Through the GNS construction the data (
P,α
f
t
,ω
f
β
) determine a separable
Hilbert space
H
f
β
,aleft representation of P on H
f
β
,avector
f
β
∈ H
f
β
cyclic
for (
P), and a unitary one-parameter group e
−itL
f
, t ∈ R, such that
ω
f
β
(a) =
f
β
|(a)
f
β
(α
t
(a)) = e
itL
f
(a)e
−itL
f
, a ∈ P. (18.69)
290 Relaxation at finite temperatures
We follow Araki and Woods to construct, as for a bounded quantum system, an
isomorphism I
T
between H
f
β
and F ⊗F .OnF ⊗ F we introduce the Bose fields
a
( f ) = a
( f ) ⊗ 1, a
r
( f ) = 1 ⊗ a
(τ f ). (18.70)
Note that a
r
is an antilinear representation of the CCR. The isomorphism I
T
is
then defined through the following relations,
I
T
f
β
= ⊗ , (18.71)
I
T
(a( f ))I
−1
T
= a
(
1 + ρ
β
f ) + a
∗
r
(
√
ρ
β
f ), (18.72)
I
T
r(a( f ))I
−1
T
= a
∗
(
√
ρ
β
τ f ) + a
r
(
1 + ρ
β
τ f ). (18.73)
As it should be, (18.72) is linear and (18.73) is antilinear in f .
I
T
(a
( f ))I
−1
T
and I
T
r(a
( f ))I
−1
T
satisfy the CCR and one only has to check
that the two-point function is properly transported,
⊗ |I
T
(a
∗
( f ))(a(g))I
−1
T
⊗
= ⊗ |a
r
(
√
ρ
β
f )a
∗
r
(
√
ρ
β
g) ⊗ =τ f,ρ
β
g
h
= ω
f
β
(a
∗
( f )a(g)) =
f
β
|(a
∗
( f ))(a(g))
f
β
(18.74)
and likewise for the right representation. We conclude that, indeed, I
T
: H
f
β
→
F ⊗ F is an isometry.
The Liouvillean is transported as L
f
= I
T
L
f
I
−1
T
. From the bounded systems
one would expect that
L
f
=
dkω(k)
a
∗
(k)a
(k) − a
∗
r
(k)a
r
(k)
. (18.75)
Then indeed, as required,
e
itL
f
a
(k)e
−itL
f
= e
−itω(k)
a
(k), e
itL
f
a
r
(k)e
−itL
f
= e
itω(k)
a
r
(k) (18.76)
and
e
itL
f
I
T
(a(k))I
−1
T
e
−itL
f
= e
−itω(k)
I
T
(a(k))I
−1
T
= I
T
(α
t
(a(k)))I
−1
T
; (18.77)
similarly for the right representation.
18.5 Atom in interaction with the photon gas
The atomic Hamiltonian H
at
has N , possibly degenerate, eigenvalues, ε
1
≤ ε
2
≤
···≤ε
N
, and the atomic Hilbert space is H
at
= C
N
.Wefixacorresponding
eigenbasis, H
at
ϕ
j
= ε
j
ϕ
j
, j = 1,...,N . The algebra of observables is the N × N
complex matrices
M
N
and it carries the thermal state ω
at
β
= Z
−1
e
−β H
at
.Aslong
18.5 Atom in interaction with the photon gas 291
as there is no interaction we merely tensor the atom with the photon field. The
algebra of observables is
M
N
⊗ P, the thermal state is
ω
0
β
= ω
at
β
⊗ ω
f
β
, (18.78)
and the dynamics is generated by α
0
t
= α
at
t
⊗ α
f
t
.Asbefore, the GNS construction
determines a separable Hilbert space
H
0
β
with cyclic vector
0
β
and a unitary time
evolution e
−itL
0
.With C denoting complex conjugation in the given basis of H
at
,
the map I
0
= I
C
⊗ I
T
: H
0
β
→ H
at
⊗ H
at
⊗ F ⊗ F =
H
β
is an isomorphism. In
particular, I
0
0
β
=
0
β
with
0
β
=
N
j=1
e
−βε
j
ϕ
j
⊗ ϕ
j
⊗ ⊗ . (18.79)
The Liouvillean is mapped as
I
0
L
0
I
−1
0
= L
0
= L
at
⊗ 1 + 1 ⊗ L
f
, L
at
= H
at
⊗ 1 − 1 ⊗ H
at
. (18.80)
The real task is to find out how the interaction is mapped to the Araki–Woods
representation space. According to (18.1) one has
H
int
= Q · E
ϕ
=
λ=1,2
d
3
k ϕ(k)
ω/2Q · e
λ
ia(k,λ)− ia
∗
(k,λ)
. (18.81)
It is more convenient to slightly generalize from (18.81) as
H
int
=
dk
G(k) ⊗ a
∗
(k) + G(k)
∗
⊗ a(k)
, (18.82)
where G :
R
3
×{1, 2}→M
N
as a matrix-valued function, with the memo that
some specific features of the coupling in (18.81) will be used in the spectral ana-
lysis below.
If G ∈
S
0
(R
3
×{1, 2}, M
N
) as matrix-valued function, one has H
int
∈
M
N
⊗ P. Thus the Liouvillean in the GNS space necessarily takes the form
L
λ
= L
0
+ λ(H
int
) − λr(H
int
) = L
0
+ λL
int
(18.83)
and only the transformations (18.72), (18.73) have to be applied, resulting in
L
int
= I
0
L
int
I
−1
0
(18.84)
=
dk
1 + ρ
β
G
(k) −
√
ρ
β
G
∗
r
(k)
a
∗
(k)
+
1 + ρ
β
G
∗
(k) −
√
ρ
β
G
r
(k)
a
(k)
+
√
ρ
β
G
∗
(k) −
1 + ρ
β
G
r
(k)
a
∗
r
(k)
+
√
ρ
β
G
(k) −
1 + ρ
β
G
∗
r
(k)
a
r
(k)
.
292 Relaxation at finite temperatures
Here G
= G
⊗ 1, G
r
= 1 ⊗ G
, G
= I
C
G
I
−1
C
. Extending the test function
notation to matrix-valued test functions, L
int
may be written more concisely as
L
int
= a
∗
1 + ρ
β
G
−
√
ρ
β
G
∗
r
+ a
1 + ρ
β
G
∗
−
√
ρ
β
G
r
+a
∗
r
√
ρ
β
G
∗
−
1 + ρ
β
G
r
+ a
r
√
ρ
β
G
−
1 + ρ
β
G
∗
r
. (18.85)
With some effort we thus achieved our goal of writing the Liouvillean for the exci-
tations away from equilibrium. Note that through ρ
β
the interaction is temperature-
dependent and becomes singular as β →∞, which only reflects the fact that the
ground state does not fall into the scheme explained before.
Two problems remain to be sorted out. First, L
λ
= L
0
+ λL
int
must generate a
unitary time evolution on
H
β
.If
dk
ω(k) + ω(k)
−3
G(k)
2
< ∞, (18.86)
then the self-adjointness of L
λ
follows from the Nelson commutator theorem.
Note that for the physical case (18.81) the condition (18.86) translates to
d
3
k|ϕ|
2
(ω
2
+ ω
−2
)<∞,which is satisfied.
Secondly, the equilibrium state of the interacting system must be represented by
avector in
H
β
. The thermal perturbation theory of section 18.2 tells us that this
new vector is formally given by
λ
β
= (Z
β
)
−1/2
e
−β L
/2
0
β
= (Z
β
)
−1/2
e
β L
r
/2
0
β
, (18.87)
where, according to (18.33), (18.85)
L
= L
0
+ λL
int
, L
r
= L
0
− λL
intr
(18.88)
and
L
int
= a
∗
(
1 + ρ
β
G
) + a
(
1 + ρ
β
G
∗
) + a
∗
r
(
√
ρ
β
G
∗
) + a
r
(
√
ρ
β
G
),
L
intr
= a
∗
(
√
ρ
β
G
∗
r
) + a
(
√
ρ
β
G
r
) + a
∗
r
(
1 + ρ
β
G
r
) + a
r
(
1 + ρ
β
G
∗
r
).
(18.89)
(Z
β
)
−1/2
normalizes the vector to one. It can be shown that Z
β
< ∞ provided
dk(1 +ω
−1
)G(k)
2
< ∞. (18.90)
Therefore under the condition (18.86),
λ
β
∈
H
β
.
By construction L
λ
λ
β
= 0. Thus L
λ
has a zero eigenvector, which does not
change under the dynamics and represents the state of global equilibrium. Accord-
ing to Proposition 18.1, we have to make sure that
λ
β
is the only eigenvector