Modelling nonequilibrium macroscopic oscillators of interest to experimentalists. · 2016-06-02 ·...

GW detectors

Modelling nonequilibrium macroscopicoscillators 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 macroscopicoscillators 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


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 thedistributions. There, they may be not.

Then the question

We detect a rare burst. Is it of an external source? Or falsepositive due to rare nonequilibrium (and non-Gaussian)fluctuations? Knowledge correct statistics is indispensable.



The idea behind the RareNoise project

The experiment

A heated mass is appended to a high Q rod, generating acurrent. Few modes of oscillation are monitored.

Experiment performed under tunable and controllablethermodynamic 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)



Thermal gradients are present in actual GW detectors

LCGT Project. From: T. Tomaru et al., Phys. Lett. A 301, 215 (2002)



Another experimental context - AURIGA

The AURIGA experiment

INFN Padova - Italy

2 tons - 3 meters long -Aluminium-alloy resonatingbar cooled to an effectivetemperature of O(mK ), viafeedback currents (extradamping). The feedbackemploys a delay-line,generating effectively anonequilibrium-statecurrent.Also in M. Bonaldi’s poster A. Vinante et al., PRL 101,

033601 (2008)



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 Is(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

Log[ω

0S

Is/〈I

2〉 e

q]

ω/ω0

G = 0.06, 0.15, 0.4, 0.752tdω0 = π



Work and heat distribution, from experiment toquasi-harmonic approximation

Qτ = ∆Uτ + Wτ

Pτ = Qτ + Q(→bath)τ ' Qτ

ρ(ετ ) = 1τ ln PDF(ετ )

PDF(−ετ )

ετ = PτL/(kBTR)

D2(ετ ) = ∂2

∂ε2τ

hln PDF(ετ )

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)



Feedback cooling - the two approaches






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)


-1.5

-1

-0.5

0

0.5

1

1.5

2

0.99 0.995 1 1.005 1.01

Log[ω

0S

Is/〈I

2〉 e

q]

ω/ω0

G = 0.06, 0.15, 0.4, 0.752tdω0 = π



Feedback cooling via the full equation






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


-1.5

-1

-0.5

0

0.5

1

1.5

2

0.99 0.995 1 1.005 1.01

Log[ω

0S

Is/〈I

2〉 e

q]

ω/ω0

G = 0.06, 0.15, 0.4, 0.752tdω0 = π

Shift of the resonance frequency.From the full equation

PDG, Rondoni, Bonaldi, Conti, J.Stat.Mech., P10016 (2009)



The digital protocol

Assume Id (t) = GIs(t − td ); x = Lin/L, y = 1− x , |Gy | < 1

qs(t) + γqs(t) + ω20qs(t)−

xy

∞∑k=1

(Gy)k [γqs(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 ′)〉 = 〈q2〉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 ′



Solving in Fourier space → The power spectrum

Hyp. : 〈η(t)η(t ′)〉 =2kBTγ

Lδ(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 1

xG 1 ⇒ SIs(ω) ∼=

2 z ω2

(ω2 − ω2r )2 + µ2ω2

and if 2tdω0 = π, then : ω4r =

1 + G2

1 + y2G2ω

40 ; z =

kBTγ

L (1 + y2G2)

µ2 = 2

„ p1 + G2q

1 + y2G2−

1 + yG2

1 + y2G2

«ω

20 −→

G→0x2G2

ω20



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

Log[ω

0S

Is/〈I

2〉 e

q]

ω/ω0

G = 0.06, 0.15, 0.4, 0.752tdω0 = π



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

ω0S

Is/〈I

2〉 e

q

ω/ω0

G = 0 (Equil.)

G = 0.5

2tdω0 = π



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

ω0S

Is/〈I

2〉 e

q

ω/ω0

2tdω0 = π



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

ω0S

Is/〈I

2〉 e

q

ω/ω0

2tdω0 = 1.15 π




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

ω0S

Is/〈I

2〉 e

q

ω/ω0

2tdω0 = 1.4 π




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

ω0S

Is/〈I

2〉 e

q

ω/ω0

2tdω0 = 1.5 π




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

ω0S

Is/〈I

2〉 e

q

ω/ω0

2tdω0 = 1.7 π



The analog protocol

The feedback is implemented via a low-pass filter

Id(ω) =AΩ

Ω− iωIs(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 = Ωω2

0(1−A)

For consistency with digital protocol we set AΩ = Gω0



Analog vs digital @ Q = 105

-1

-0.5

0

0.5

1

1.5

2

0.995 0.9975 1 1.0025 1.005

Log[ω

0S

Is/〈I

2〉 e

q]

ω/ω0

Ω = 0.2 ω0



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

Log[ω

0S

Is/〈I

2〉 e

q]

ω/ω0

Ω = 0.1 ω0



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

Log[ω

0S

Is/〈I

2〉 e

q]

ω/ω0

Ω = 0.001 ω0



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, toovercome certain inconsistencies

VLJ(x) = ε(

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 → Vt1λ = 7, µ = 53 λ/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

log

(L/L

0)

kbT [ ǫ ]

VLJ

Vt1

Vt2

♦♦♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦



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

r0

]

kBT [ ǫ ]

(a)

(b)

(c)

♦♦ ♦ ♦ ♦

-0.1

0

0.1

-1 0 1

F[

ǫ

r0

]

(∆ L/L) × 103

⋆⋆

⋆⋆

⋆⋆

⋆⋆

⋆⋆

⋆




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

r0

]

kBT [ ǫ ]

(a)

(b)

(c)

♦♦ ♦ ♦ ♦

-0.1

0

0.1

-1 0 1

F[

ǫ

r0

]

(∆ 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

r0

]

kBT [ ǫ ]

(a)

♦♦♦♦ ♦ ♦ ♦♦




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

r0

]

kBT [ ǫ ]

(a)

(b)

(c)

♦♦ ♦ ♦ ♦

-0.1

0

0.1

-1 0 1

F[

ǫ

r0

]

(∆ 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

(

ωr

ω0

)

2 LJ

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

r0

]

kBT [ ǫ ]

(a)

♦♦♦♦ ♦ ♦ ♦♦




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

r0

]

kBT [ ǫ ]

(a)

(b)

(c)

♦♦ ♦ ♦ ♦

-0.1

0

0.1

-1 0 1

F[

ǫ

r0

]

(∆ 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

(

ωr

ω0

)

2 LJ

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

r0

]

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



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



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



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



Acknowledgments

FP7 - IDEAS - ERC grant agreement n. 202680INFN Padova - INFN Torino

The whole RareNoise team:L. Conti, L. Rondoni, M. Bonaldi andP. Adamo, R. Hajj, G. Karapetyan, R.-K. Takhur, C. LazzaroFormer: 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,. . .


Modelling nonequilibrium macroscopic oscillators of interest to experimentalists. · 2016-06-02 ·...

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