Recent developments in experimental Gravitational Waves (GW) detection have led to the emergence of concerns regarding the inconsistency of treating the appa-rati as equilibrium systems. Nonequilibrium considerations surface, for example, in examining the electronic feedback machinery that is used to damp and to sta-bilize GW oscillator-bar detectors, or when considering temperature gradients produced by high-powered lasers used in interferometric GWdetectors. Either way, the experimental apparatus often consists of an oscillator with a very high quality factor, subjected to some power supply, and capableof extraordinary sensitivity in length displacement. Two theoretical paradigms (with their advan-tages and disadvantages) are presented: one in which the oscillator is described by an ad hoc Langevin equation with memory, and one in which it is modelled by a deterministic one-dimensional chain of anharmonic oscillators. Within both, it is seen that GW experiments suggest entirely novel problems to theorist

Bonaldi, Michele

Conti, Livia

De Gregorio, Paolo

Rondoni, Lamberto

PUblication MAnagement

GW detectors
Modelling nonequilibrium macroscopic
oscillators of interest to experimentalists.
P. De Gregorio
“RareNoise”: L. Conti, M. Bonaldi, L. Rondoni
ERC-IDEAS: www.rarenoise.lnl.infn.it
Nonequilibrium Processes:
The Last 40 Years and the Future
Obergurgl, Tirol, Austria, 29 August - 2 September 2011
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors
Modelling nonequilibrium macroscopic
oscillators of interest to experimentalists.
P. De Gregorio
“RareNoise”: L. Conti, M. Bonaldi, L. Rondoni
ERC-IDEAS: www.rarenoise.lnl.infn.it
In celebration of Prof. Denis J. Evans’ 60th birthday
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
GW detectors and fluctuations
Ground-based Detectors
Often, can ‘hear’ their own thermal fluctuations.
Expected to approach the quantum limit in the future.
Nonequilibrium stationary states and noise
Past studies had assumed the noise be Gaussian.
However the experimentalists’ interest is in the tails of the
distributions. There, they may be not.
Then the question
We detect a rare burst. Is it of an external source? Or false
positive due to rare nonequilibrium (and non-Gaussian)
fluctuations? Knowledge correct statistics is indispensable.
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
The idea behind the RareNoise project
The experiment
A heated mass is appended to a high Q rod, generating a
current. Few modes of oscillation are monitored.
Experiment performed under tunable and controllable
thermodynamic conditions.
As has been described in L. Conti’s presentation and M. Bonaldi’s poster.
For a nice outline of the experiment see: Conti, Bonaldi, Rondoni, Class. Quantum Grav. 27, 084032 (2010)
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Thermal gradients are present in actual GW detectors
LCGT Project. From: T. Tomaru et al., Phys. Lett. A 301, 215 (2002)
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Another experimental context - AURIGA
The AURIGA experiment
INFN Padova - Italy
2 tons - 3 meters long -
Aluminium-alloy resonating
bar cooled to an effective
temperature of O(mK ), via
feedback currents (extra
damping). The feedback
employs a delay-line,
generating effectively a
nonequilibrium-state
current.
Also in M. Bonaldi’s poster A. Vinante et al., PRL 101,
033601 (2008)
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Feedback cooling - the two approaches
L I˙(t) + R I(t) + 1C q(t) = VT (t) ← feedback off
γ = R/L; ω2o = 1/LC; 〈VT (t)VT (t′)〉 = 2RkBTδ(t − t′)
(L− Lin) I˙(t) + R I(t) + 1C q(t) = VT (t)− Lin I˙s(t)
Is(t) = I(t) + Id (t) ← feedback on ↑
quasi harmonic appr.: R → R′ = (R + Rd )
L I˙(t) + R′I(t) + 1C q(t) = VT (t)
general case:
I˙(t) +
R t
−∞ f (t − t′)I(t′)dt′ +
R t
−∞ g(t − t′)q(t′)dt′ = N(t)
Effective ‘cooling’
-1.5
-1
-0.5
0
0.5
1
1.5
2
0.99 0.995 1 1.005 1.01
L
o
g
[ ω
0
S
I
s
/
〈 I
2
〉 e
q
]
ω/ω0
G = 0.06, 0.15, 0.4, 0.75
2tdω0 = pi
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Work and heat distribution, from experiment to
quasi-harmonic approximation
Qτ = ∆Uτ + Wτ
Pτ = Qτ + Q(→bath)τ ' Qτ
ρ(²τ ) =
1
τ ln
PDF(²τ )
PDF(−²τ )
²τ = PτL/(kBTR)
D2(²τ ) = ∂
2
∂²2τ
h ln PDF(²τ )
τ
i
M. Bonaldi, L. Conti, PDG, L. Rondoni, G. Vedovato, A. Vinante et al (AURIGA team) PRL 103, 010601 (2009)
J. Farago J Stat Phys 107, 781 (2002)
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Feedback cooling - the two approaches
L I˙(t) + R I(t) + 1C q(t) = VT (t) ← feedback off
γ = R/L; ω2o = 1/LC; 〈VT (t)VT (t′)〉 = 2RkBTδ(t − t′)
(L− Lin) I˙(t) + R I(t) + 1C q(t) = VT (t)− Lin I˙s(t)
Is(t) = I(t) + Id (t) ← feedback on ↑
quasi harmonic appr.: R → R′ = (R + Rd )
L I˙(t) + R′I(t) + 1C q(t) = VT (t)
general case:
I˙(t) +
R t
−∞ f (t − t′)I(t′)dt′ +
R t
−∞ g(t − t′)q(t′)dt′ = N(t)
Effective ‘cooling’
-1.5
-1
-0.5
0
0.5
1
1.5
2
0.99 0.995 1 1.005 1.01
L
o
g
[ ω
0
S
I
s
/
〈 I
2
〉 e
q
]
ω/ω0
G = 0.06, 0.15, 0.4, 0.75
2tdω0 = pi
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Feedback cooling via the full equation
L I˙(t) + R I(t) + 1C q(t) = VT (t) ← feedback off
γ = R/L; ω2o = 1/LC; 〈VT (t)VT (t′)〉 = 2RkBTδ(t − t′)
(L− Lin) I˙(t) + R I(t) + 1C q(t) = VT (t)− Lin I˙s(t)
Is(t) = I(t) + Id (t) ← feedback on ↑
quasi harmonic appr.: R → R′ = (R + Rd )
L I˙(t) + R′I(t) + 1C q(t) = VT (t)
general case:
I˙(t) +
R t
−∞ f (t − t′)I(t′)dt′ +
R t
−∞ g(t − t′)q(t′)dt′ = N(t)
N(t) =
R t
−∞ h(t − t′)η(t′)dt′
〈q(0)N(t)〉 6= 0; 〈I(0)N(t)〉 6= 0 Kubo’s causality broken
An interesting problem in statistical mechanics in ots own right
Effective ‘cooling’
-1.5
-1
-0.5
0
0.5
1
1.5
2
0.99 0.995 1 1.005 1.01
L
o
g
[ ω
0
S
I
s
/
〈 I
2
〉 e
q
]
ω/ω0
G = 0.06, 0.15, 0.4, 0.75
2tdω0 = pi
Shift of the resonance frequency.
From the full equation
PDG, Rondoni, Bonaldi, Conti, J.Stat.Mech., P10016 (2009)
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
The digital protocol
Assume Id (t) = GIs(t − td ); x = Lin/L, y = 1− x , |Gy | < 1
q¨s(t) + γq˙s(t) + ω20qs(t)−
x
y
∞∑
k=1
(Gy)k [γq˙s(t − ktd ) + ω20qs(t − ktd )] = N(t)
Generalized FDT (Kubo) would entail causality @ t > t ′ 〈N(t)q(t
′)〉 = 0
〈N(t)q˙(t ′)〉 = 0
⇒ 〈N(t)N(t ′)〉 = 〈q˙2〉f (t − t ′)
But here
N(t) =
∞∑
k=0
(Gy)kη(t − ktd ) = 1L
∞∑
k=0
(Gy)k VT (t − ktd )
⇒ ∃ k s.t. (t − ktd ) < t ′
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
The digital protocol
Assume Id (t) = GIs(t − td ); x = Lin/L, y = 1− x , |Gy | < 1
q¨s(t) + γq˙s(t) + ω20qs(t)−
x
y
∞∑
k=1
(Gy)k [γq˙s(t − ktd ) + ω20qs(t − ktd )] = N(t)
Generalized FDT (Kubo) would entail causality @ t > t ′ 〈N(t)q(t
′)〉 = 0
〈N(t)q˙(t ′)〉 = 0
⇒ 〈N(t)N(t ′)〉 = 〈q˙2〉f (t − t ′)
But here
N(t) =
∞∑
k=0
(Gy)kη(t − ktd ) = 1L
∞∑
k=0
(Gy)k VT (t − ktd )
⇒ ∃ k s.t. (t − ktd ) < t ′
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
The digital protocol
Assume Id (t) = GIs(t − td ); x = Lin/L, y = 1− x , |Gy | < 1
q¨s(t) + γq˙s(t) + ω20qs(t)−
x
y
∞∑
k=1
(Gy)k [γq˙s(t − ktd ) + ω20qs(t − ktd )] = N(t)
Generalized FDT (Kubo) would entail causality @ t > t ′ 〈N(t)q(t
′)〉 = 0
〈N(t)q˙(t ′)〉 = 0
⇒ 〈N(t)N(t ′)〉 = 〈q˙2〉f (t − t ′)
But here
N(t) =
∞∑
k=0
(Gy)kη(t − ktd ) = 1L
∞∑
k=0
(Gy)k VT (t − ktd )
⇒ ∃ k s.t. (t − ktd ) < t ′
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Solving in Fourier space → The power spectrum
Hyp. : 〈η(t)η(t ′)〉 = 2kBTγL δ(t − t
′)
⇒ SIs(ω) =
∫ +∞
−∞
eiωt〈Is(0)Is(t)〉dt =
(2kBTγ
L
) ω2
D(ω)
D(ω) = α2 + G2β2 + (1 + G2)γ2ω2 Def. : α = ω2 − ω20
+2xGγω3 sin (ωtd )− 2G(αβ + γ2ω2) cos (ωtd ) β = y ω2 − ω20
For very high quality factors
ω0
γ
≡ Q À 1
xG À 1 ⇒ SIs(ω)
∼= 2 z ω
2
(ω2 − ω2r )2 + µ2ω2
and if 2tdω0 = pi, then : ω
4
r =
1 + G2
1 + y2G2
ω
4
0 ; z =
kBTγ
L (1 + y2G2)
µ
2 = 2
„ p1 + G2q
1 + y2G2
− 1 + yG
2
1 + y2G2
«
ω
2
0 −→G→0 x
2G2ω20
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Solving in Fourier space → The power spectrum
Hyp. : 〈η(t)η(t ′)〉 = 2kBTγL δ(t − t
′)
⇒ SIs(ω) =
∫ +∞
−∞
eiωt〈Is(0)Is(t)〉dt =
(2kBTγ
L
) ω2
D(ω)
D(ω) = α2 + G2β2 + (1 + G2)γ2ω2 Def. : α = ω2 − ω20
+2xGγω3 sin (ωtd )− 2G(αβ + γ2ω2) cos (ωtd ) β = y ω2 − ω20
For very high quality factors
ω0
γ
≡ Q À 1
xG À 1 ⇒ SIs(ω)
∼= 2 z ω
2
(ω2 − ω2r )2 + µ2ω2
and if 2tdω0 = pi, then : ω
4
r =
1 + G2
1 + y2G2
ω
4
0 ; z =
kBTγ
L (1 + y2G2)
µ
2 = 2
„ p1 + G2q
1 + y2G2
− 1 + yG
2
1 + y2G2
«
ω
2
0 −→G→0 x
2G2ω20
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Q = 105 - shift of the resonance frequency
-1.5
-1
-0.5
0
0.5
1
1.5
2
0.99 0.995 1 1.005 1.01
L
o
g
[ ω
0
S
I
s
/
〈 I
2
〉 e
q
]
ω/ω0
G = 0.06, 0.15, 0.4, 0.75
2tdω0 = pi
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Q = 1 - "transition" of the resonance frequency
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
ω
0
S
I
s
/
〈 I
2
〉 e
q
ω/ω0
G = 0 (Equil.)
G = 0.5
2tdω0 = pi
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Q = 1 - changing td
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
ω
0
S
I
s
/
〈 I
2
〉 e
q
ω/ω0
2tdω0 = pi
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Q = 1 - "transition" of the resonance frequency as td ↑
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
ω
0
S
I
s
/
〈 I
2
〉 e
q
ω/ω0
2tdω0 = 1.15 pi
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Q = 1 - "transition" of the resonance frequency as td ↑
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
ω
0
S
I
s
/
〈 I
2
〉 e
q
ω/ω0
2tdω0 = 1.4 pi
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Q = 1 - "transition" of the resonance frequency as td ↑
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
ω
0
S
I
s
/
〈 I
2
〉 e
q
ω/ω0
2tdω0 = 1.5 pi
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Q = 1 - "transition" of the resonance frequency as td ↑
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
ω
0
S
I
s
/
〈 I
2
〉 e
q
ω/ω0
2tdω0 = 1.7 pi
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
The analog protocol
The feedback is implemented via a low-pass filter
I˜d(ω) =
AΩ
Ω− iω I˜s(ω)
⇓
SIs(ω) =
(2kBTγ
L
) ω2 (ω2 +Ω2)
B(ω)
B(ω) = ω2(ω2 − F 2)2 + (aω2 − b3)2
F 2 = ω20+γΩ(1−A); a = (1−yA)Ω+γ; b3 = Ωω20(1−A)
For consistency with digital protocol we set AΩ = Gω0
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Analog vs digital @ Q = 105
-1
-0.5
0
0.5
1
1.5
2
0.995 0.9975 1 1.0025 1.005
L
o
g
[ ω
0
S
I
s
/
〈 I
2
〉 e
q
]
ω/ω0
Ω = 0.2 ω0
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Analog vs digital decreasing Ω/ω0
-1
-0.5
0
0.5
1
1.5
2
0.995 0.9975 1 1.0025 1.005
L
o
g
[ ω
0
S
I
s
/
〈 I
2
〉 e
q
]
ω/ω0
Ω = 0.1 ω0
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Analog vs digital decreasing Ω/ω0
-1
-0.5
0
0.5
1
1.5
2
0.995 0.9975 1 1.0025 1.005
L
o
g
[ ω
0
S
I
s
/
〈 I
2
〉 e
q
]
ω/ω0
Ω = 0.001 ω0
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Modeling the rod with one-dimensional chains
L as the observable
The idea is to define a variation of the celebrated FPU chain:
Vt(x) = h
[
(x − 1)2−λ(x − 1)3 + µ(x − 1)4]
to then compare it to a more ‘phenomelogical’ interaction potential, to
overcome certain inconsistencies
VLJ(x) = ²
(
x−12 − 2 x−6 ) x = r/r0
-1
-0.6
-0.2
0.2
0.6
0.8 1 1.2 1.4
V
/
ǫ
x
VLJ
Vt2
Vt1
λ = µ = 1 → Vt1
λ = 7, µ = 53λ/12 → Vt2(as if we expanded LJ)
0
0.005
0.01
0.015
0.02
0.025
0.03
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
l o
g
( L
/
L
0
)
kbT [ ǫ ]
VLJ
Vt1
Vt2











△△
△△
△△
△△
△
△
△
△
△
△
△
♦♦♦ ♦ ♦
♦ ♦
♦ ♦
♦ ♦
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Comparing thermo-elastic properties
E , the elastic modulus: F ' E (∆L/L), if ∆L¿ L. PDG, L.Rondoni, M.Bonaldi, L.Conti, to be published
40
60
80
100
120
140
160
180
200
220
0 0.02 0.04 0.06 0.08 0.1 0.12
E
[
ǫ
r
0
]
kBT [ ǫ ]
(a)
(b)
(c)




△
△
△
△ △
△
△
△
♦♦ ♦ ♦ ♦
-0.1
0
0.1
-1 0 1
F
[
ǫ
r
0
]
(∆ L/L) × 103
⋆
⋆
⋆
⋆
⋆
⋆
⋆
⋆
⋆
⋆
⋆
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Comparing thermo-elastic properties
E , the elastic modulus: F ' E (∆L/L), if ∆L¿ L.
40
60
80
100
120
140
160
180
200
220
0 0.02 0.04 0.06 0.08 0.1 0.12
E
[
ǫ
r
0
]
kBT [ ǫ ]
(a)
(b)
(c)




△
△
△
△ △
△
△
△
♦♦ ♦ ♦ ♦
-0.1
0
0.1
-1 0 1
F
[
ǫ
r
0
]
(∆ L/L) × 103
⋆
⋆
⋆
⋆
⋆
⋆
⋆
⋆
⋆
⋆
⋆
100
150
200
250
300
350
400
450
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
E
[
ǫ
r
0
]
kBT [ ǫ ]
(a)



△△
△
△
△
△
△
♦ ♦ ♦ ♦
♦
♦
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Comparing thermo-elastic properties
E , the elastic modulus: F ' E (∆L/L), if ∆L¿ L. ω0 resonance frequency of 1st mode.
40
60
80
100
120
140
160
180
200
220
0 0.02 0.04 0.06 0.08 0.1 0.12
E
[
ǫ
r
0
]
kBT [ ǫ ]
(a)
(b)
(c)




△
△
△
△ △
△
△
△
♦♦ ♦ ♦ ♦
-0.1
0
0.1
-1 0 1
F
[
ǫ
r
0
]
(∆ L/L) × 103
⋆
⋆
⋆
⋆
⋆
⋆
⋆
⋆
⋆
⋆
⋆
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
1.01
1.02
0 0.002 0.004 0.006 0.008 0.01 0.012
( ω
r
ω
0
) 2 L
J
E [ ǫ ]







0.01
1
100
0.85 0.95 1.05
S
( ω
)
(
a
.
u
.
)
ω/ω0
(a)(b) (c)
100
150
200
250
300
350
400
450
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
E
[
ǫ
r
0
]
kBT [ ǫ ]
(a)



△△
△
△
△
△
△
♦ ♦ ♦ ♦
♦
♦
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Comparing thermo-elastic properties
E , the elastic modulus: F ' E (∆L/L), if ∆L¿ L. ω0 resonance frequency of 1st mode. Q = ω0/∆ω1/2
40
60
80
100
120
140
160
180
200
220
0 0.02 0.04 0.06 0.08 0.1 0.12
E
[
ǫ
r
0
]
kBT [ ǫ ]
(a)
(b)
(c)




△
△
△
△ △
△
△
△
♦♦ ♦ ♦ ♦
-0.1
0
0.1
-1 0 1
F
[
ǫ
r
0
]
(∆ L/L) × 103
⋆
⋆
⋆
⋆
⋆
⋆
⋆
⋆
⋆
⋆
⋆
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
1.01
1.02
0 0.002 0.004 0.006 0.008 0.01 0.012
( ω
r
ω
0
) 2 L
J
E [ ǫ ]







0.01
1
100
0.85 0.95 1.05
S
( ω
)
(
a
.
u
.
)
ω/ω0
(a)(b) (c)
100
150
200
250
300
350
400
450
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
E
[
ǫ
r
0
]
kBT [ ǫ ]
(a)



△
△
△
△
△
△
△
△
♦ ♦ ♦ ♦
♦
♦
0
200
400
600
800
1000
0 0.002 0.004 0.006 0.008 0.01
Q
E [ ǫ ]







500
1000
1500
2000
0.972 0.976 0.98
S
( ω
)
(
a
.
u
.
)
ω/ω0
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Effects of gradients (under investigation) - ∇T ≡ 0
T T
1
10
100
1000
10000
100000
0.5 1 1.5 2 2.5 3 3.5
S
( ω
)
[ a
u
]
ω/ω0
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Effects of gradients on Q: ∇T 6= 0 -vs- ∇T ≡ 0
T− T+
1
10
100
1000
10000
100000
0.5 1 1.5 2 2.5 3 3.5
S
( ω
)
[ a
u
]
ω/ω0
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Effects of gradients on Q: ∇T 6= 0 -vs- ∇T ≡ 0
T− T+
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
0.94 0.96 0.98 1 1.02 1.04 1.06
S
( ω
)
[ a
u
]
ω/ω0
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.
GW detectors And nonequilibrium considerations
Acknowledgments
FP7 - IDEAS - ERC grant agreement n. 202680
INFN Padova - INFN Torino
The whole RareNoise team:
L. Conti, L. Rondoni, M. Bonaldi and
P. Adamo, R. Hajj, G. Karapetyan, R.-K. Takhur, C. Lazzaro
Former: C. Poli, A.B. Gounda, S. Longo, M. Saraceni
Thanks to the AURIGA collaboration - INFN.
G. Vedovato, A. Vinante, M. Cerdonio, G.A. Prodi, J.-P- Zendri,
. . .
De Gregorio, Conti, Bonaldi, Rondoni Modelling nonequilibrium macroscopic oscillators.


Modelling nonequilibrium macroscopic oscillators of interest to experimentalists

http://www.rarenoise.lnl.infn.it/sources/Obergurgl_DeGregorio.pdf

Modelling nonequilibrium macroscopic oscillators of interest to experimentalists

Abstract

Similar works

Full text

Available Versions

PUblication MAnagement