J OINT DISTRIBUTION OF TWO LINEAR STATISTICS
IN THE L AGUERRE ENSEMBLE .
C OMPLEX IMPEDANCE OF CHAOTIC CAVITIES
A URÉLIEN G RABSCH & C HRISTOPHE T EXIER
Laboratoire de Physique Théorique et Modèles Statistiques, Orsay, France
T HE PROBLEM
C OULOMB GAS METHOD
P
Density of eigenvalues ρ(x) = (1/N ) i δ(x − γi /N )
β 2
has weight P({γi }) ∝ exp − 2 N E [ρ] where
Wigner-Smith time-delay matrix :
VG
Q = −i~S
† ∂S
(4)
∞
Z
E [ρ] =
∂ε
In a chaotic cavity with perfect coupling (hSi = 0) :
dx ρ(x) (x − ln x)
0
Z
−1
Q
VG
∞
dxdx0 ρ(x) ρ(x0 ) ln |x − x0 |
−
belongs to the Laguerre ensemble
0
0.5
Brouwer, Frahm & Beenakker, PRL (1997)
=C
−1
+
−1
Cq
2
with Cq = (e /h) Tr {Q} (2)
2
h Tr Q
Rq = 2
2 (3)
2e Tr {Q}
C OUL . GAS UNDER CONSTRAINTS
Introduce
H = (1/N )
2 s =2 Tr {Q} /τP
N Tr Q /τH = (1/N ) i x−2
i
e2
s
Cq =
∆
P
−1
x
and r =
i i
r
h
.
Rq =
2
2
N e 2s
and
(5)
“Equilibrium” condition (δF /δρ(x) = 0) :
−1
γ
i i
P
2 ?
(8)
2
exp −(β/2)N Ψ(s, r) ,
(9)
Ψ(s, r) = E [ρ? (x; s, r)] − E [ρMP (x)]
(ρMP (x) : solution for µ1 = µ2 = 0)
YHs,rL=E @Ρ*D-E @ΡMPD
2.5
Ψ(s, r) ' δs
r=2
(15)
δRq
(16)
4
x
(17)
bulk+charge
?
splitted
C
r
Μ2=0
MP
2
compact
(18)
forbidden HJensenL
1
(19)
0
0.0
0.5
1.0
1.5
2
s
Remark : case of uncorrelated variables:
Szavits-Nossan, Evans & Majumdar, J.Phys.A (2014)
J Szavits-Nossan et al
ρ(x) = (1/N ) δ(x − x1 ) + ρ̃(x) for µ2 = 0
“condensate”
1.0
0.5
0.0
0.0
0.5
1.0
1.5
2.0
2.5
1
r=
+
2
N x1
3.0
s2
2
Z
dx
ρ̃(x)
2
x |{z}
⇒ x1 ' p
1
N (r − r(s))
ρ∗ (x;s)
1
Ψ(s, r) ' − ln(r − s2 ) for r − s2 → 0+
2
βN 2 /4
PN (Rq ) ∼ [Rq −h/(2N e )]
E [ρ∗ (x;s)] ,
(20)
2
for Rq → h/(2N e ).
(14)
2
J. Phys. A: Math. Theor. 47 (2014) 455004
1.5
4
24
5
Veff (x) = x−ln x+µ1 /x+µ2 /x2 ⇒ µ2 < 0: instability
⇓
Phase transition
2.0
−1 24
δs
.
160
δr
P HASE DIAGRAM
P HASE TRANSITION
r=3
δr
4
hδCq δRq i
1
q
= +√
2 '
2 ' βN 2 ,
2
hCq i
2 hRq i
hδCq2 ihδRq2 i
a/b and v = ab. Then the Lagrange multipliers are given by
h
i
2
3
4
2 2
2
2 2
µ1 = v 2u (−9 + 4u − 6u + 4u − 9u ) + 3v (1 − u ) (1 + u ) / 2u(1 − u ) ,
h
i
µ2 = −v 2 2u (−3 + 2u − 3u2 ) + v (1 − u2 )2 /(1 − u2 )2 .
r=1
(12)
Expansion :
2
E [ρ? ] ' E [ρ
]−(1/2)(s−1)
∂µ
MP
1 /∂s MP −(1/2)(r −
2)2 ∂µ2 /∂rMP − (s − 1)(r − 2)∂µ1 /∂rMP
√
F-HsL
3.0
(11)
R
s∗ = ρMP (x)/x = 1 → hCq i = e2 /∆
R
(13)
2
2
r∗ = ρMP (x)/x = 2 → hRq i ' h/(N e )
3
E NERGY OF THE GAS
6
x
2
δCq
x2 + c x + d p
Solution of (8) :
ρ? (x; s, r) =
(b − x)(x − a) ,
3
2πx
√
√
where c = µ1 / ab + µ2 (a + b)/(ab)3/2 and d = 2µ2 / ab. Given s and r, we obtain [a, b] from
h
i
2
3
4
4
2
3
s = 2u (3 − 4u + 18u − 4u + 3u ) − v (1 − u) (1 + 4u + u ) / 32u v ,
h
i
2
3
4
5
6
2 4
4 2
r = 2u (9 − 10u + 39u − 12u + 39u − 10u + 9u ) − 3v (1 − u ) / 128u v ,
where u =
5
T YPICAL FLUCTUATIONS
(10)
C OMPACT PHASE
p
4
∂E [ρ? (x; s, r)]
= −µ?1 (s, r) ,
∂s
∂E [ρ? (x; s, r)]
?
= −µ2 (s, r) .
∂r
(
where
for x ∈ [a, b]
3
Marčenko-Pastur law : (equilibrium density)
p
√
(x+ − x)(x − x− )
ρMP (x) =
,
x± = 3 ± 2 2
2πx
Joint distribution:
N →∞
2
T HERMODYNAMIC IDENTITIES
→ get ρ? (x; s, r) minimizing F [ρ]
∼
1
−2
γ
i i
P
Solution of (8)
function of µ1 & µ
2
( : ρ(x)
(
R
s = dx ρ(x)/x
µ1 = µ?1 (s, r)
R
constraints
−→
r = dx ρ(x)/x2
µ2 = µ?2 (s, r)
PN (s, r)
0
x
L ARGE DEVIATION FUNCTION Ψ
(7)
The joint distribution PN (s, r) is dominated by ρ(x)
minimizing E [ρ] under three constraints
R
dx
ρ(x)
−
1
+
⇒ consider
F
[ρ]
=
E
[ρ]
+
µ
0
R
R
2
µ1
dx ρ(x)/x − s + µ2
dx ρ(x)/x − r .
Z b
0
1 µ1
2µ2
ρ(x
)
0
1 − − 2 − 3 = 2 r dx
0
x x
x
x
−
x
a
Question :
h
2e2
0.2
0.0
x’
k=1
Statistical properties of Rq =
0.3
0.1
βN/2 −βγk /2
γk
e
Eigenvalue of Q: proper time τi = τH /γi (τH = h/∆)
Büttiker, Prêtre & Thomas (1993) :
−1
Cµ
i<j
⇒
frozen
(1)
|γi − γj |β
N
Y
Vconf (x)
Ρ* HxL
1
Z(ω) =
+ Rq + O(ω)
−iω Cµ
P(γ1 , · · · , γN ) ∝
Y
0.4
Ρ*Hx;1L
Coherent AC transport :
(6)
cf. [1]
PN (s, r) ∼ (r − r(s))
exponential function (case (iii)); the critical line was calculated numerically for the
Pareto distribution, f (m ) = (γ − 1) mγ for m > 1 and γ = 7 2 . Shaded area σ < μ2 is
forbidden owing to Jensenʼs inequality.
R EFERENCES
}|
{
z
Energy : E [ρ? ] ' E [ρMP ] + Φ− (s) −(1/N ) ln x1
−βN/4
Figure 2. Phase diagram in μ–σ plane for p = 1 2 and f (m ) that falls off slower than
2
where Z L, r (M , V ) is a constant that depends on r , M and V and is given in (54). The bottom
expression in (21), which corresponds to the condensate bump, is described by a bivariate
Gaussian distribution, where the vector x(m ), mean vector e and covariance matrix Σ are
given in (56) and (57). Interestingly, as the non-diagonal elements of the covariance matrix Σ
are non-zero, the position of the bump is generally shifted away from the expected occupation
number of (Lσ − Lσc ) p .
Case (iii). This is the case where f (m ) is heavy-tailed, i.e.
exp − (β/2)N Φ− (s) .
(21)
[1] Christophe Texier and Satya N. Majumdar,
Phys. Rev. Lett. 110, 250602 (2013).
[2] Aurélien Grabsch and Christophe Texier,
preprint cond-mat arXiv:1407.3302.
f (m )
decays more slowly than exp (−km)
for any k > 0.
(22)
Recall that the standard condensation applies here when only ML is fixed provided μ > 〈m〉,