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Detection of gravitational waves On Effective Temperatures Discussion and open questions Effects of breaking vibrational energy equipartition on measurements of temperature in macroscopic oscillators L. Rondoni, Politecnico Torino R.

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions

Effects of breaking vibrational energy equipartition on measurements of temperature in macroscopic oscillators

  • L. Rondoni, Politecnico Torino
  • R. Belousov, M. Bonaldi, L. Conti, P. De Gregorio, C. Giberti

Firenze – 29 May 2014 PRL 2009; J. Stat. Mech. 2009 and 2013;

  • Class. Quant. Grav. 2010; PRB 2011; PRE 2011 and 2012

http://www.rarenoise.lnl.infn.it/

  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions

Outline

1

Detection of gravitational waves Langevin equation Fluctuations

2

On “Effective Temperatures” 1-dimensional models: comparison with experiment Simulations and “theory” Results

3

Discussion and open questions

  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions Langevin equation Fluctuations

Thermal fluctuations unobservable in macroscopic objects?

General Relativity predicts gravitational waves (GW): e.g. accelerating binary systems of neutron stars or black holes; vibrations of black holes or neutron stars. Hulse-Taylor measurement of orbits of two neutron stars, spiralling as if losing energy by GW emission; in excellent agreement with predictions, were awarded Nobel prize in 1993. GW: kind of space-time ripples, in two fundamental states of polarization, cross and plus. Effect of GW on matter: squeezing and stretching, depending on phase. plus cross

  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Weakest assumptions approach News from RareNoise

The idea which was behind the RareNoise project

Ground-based Detectors Can detect thermal fluctuations intrinsic to the test mass. 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? Knowing correct statistics is mandatory.

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions Langevin equation Fluctuations

Gravitational Wave detector

M otivation: GWs will provide new and unique information about astrophysical processes GW amplitude: A detection rate of few events/year requires sensitivity of

  • ver timescales as short as 1msec

small signal noise  noise sources must be reduced to very low levels

  • L. Conti - Bologna 17.03.2014
  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions Langevin equation Fluctuations

GW detectors (interferometers)

LIGO @Livingston (USA) VIRGO @Cascina (Italy) GEO600 @Hannover (Germany) TAM A300 @Tokyo (Japan)

  • L. Conti - Bologna 17.03.2014
  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions Langevin equation Fluctuations

GW detector noise budget

Dominant Sources of Noise:

  • seismic noise
  • thermal noise
  • photon shot noise

Thermal Shot Seismic sensitivity increases

  • L. Conti - Bologna 17.03.2014
  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions Langevin equation Fluctuations

Thermal compensation

666 m 687 m

Applied thermal gradient deforms the mirror and corrects the ROC

to correct mismatch of the mirror Radius Of Curvature (ROC) due to: fabrication thermal lensing thermo-elastic deformation What is the ‘thermal noise’ of such a non-equilibrium body?

  • L. Conti - Bologna 17.03.2014
  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions Langevin equation Fluctuations

Resonant-bar GW detectors: feedback cooling down to mK: viscous force reduces thermal noise on length of resonant-bar detector AURIGA (PRL top ten stories, 2008). Steady state modelled by 3 electro-mechanical oscillators with stochastic driving.

  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions Langevin equation Fluctuations

LdIs(t) dt + Is(t) [R + Rd] + qs(t) C =

  • 2kBT0R Γ(t)

Id(t) = GIs(t − td) td = π 2ωr G ≪ 1 Rd = GωrLin expresses viscous damping due to feedback; No time reversal invariance (q′

s = qs, I ′ s = −Is, t′ = −t),

violates Einstein relation, but formally identical to equilibrium

  • scillator at fictitious temperature Teff = T0/(1 + g)

with feedback efficiency g = Rd/R, so that: I 2

s = 2kBTeff/L

Hence, usually treated as equilibrium system!

  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions Langevin equation Fluctuations

PDF and fluctuation relation of injected power Pτ: Farago, ’02 ρ(˜ ǫτ) = lim

τ→∞

1 τ ln PDF(˜ ǫτ) PDF(−˜ ǫτ) =

  • 4γ˜

ǫτ, ˜ ǫτ < 1

3;

γ˜ ǫτ

  • 7

4 + 3 2˜ ǫτ − 1 4˜ ǫ2

τ

  • ,

˜ ǫτ ≥ 1

3.

˜ ǫτ = PτL/(kBT0R) =; γ = (R + Rd)/L, Teff = (22 ± 1) mK second derivative

  • f PDF;

b) ρ(˜ ǫτ). Data from May 2005 to May 2008. Shades = experimental uncertainty on τeff, Teff, T0.

  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Weakest assumptions approach News from RareNoise

RN aluminum exp. - longitudinal and flexural oscillations

longitudinal flexural

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions 1-dimensional models: comparison with experiment Simulations and “theory” Results

For macroscopic systems in local thermodynamic equilibrium (LTE) “the properties of a ‘long’ metal bar should not depend on whether its ends are in contact with water or with wine ‘heat reservoirs’ at temperature T1 and T2” (Rieder, Lebowitz, Lieb, JMP 1967) But modelling by 1-dimensional systems incurs in violations of conditions of LTE, hence strong dependence on details of microscopic dynamics: care must be taken in tuning parameters to

  • btain

“proper thermo-mechanical” behaviour. Wanted “realistic” equilibrium properties: thermal expansion, and temperature dependent elasticity, resonance frequencies and quality factor. and non-equilibrium: linear “temperature” profile.

  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions 1-dimensional models: comparison with experiment Simulations and “theory” Results

Deterministic reversible semi-open Nos´ e-Hoover at T1 and T1 + ∆T Nearest- and next-nearest-neighbors L-J. N = 128, 256, 512 V (ri, ri±ℓ) = ǫ

  • ℓr0

|ri − ri±ℓ| 12 − 2

  • ℓr0

|ri − ri±ℓ| 6 ; ℓ = 1, 2 m¨ ri = F int

i

(ri, ri±1, ri±2) − χi ˙ ri ˙ χi = m τ 2 K kBTi − 1

  • ; Ki = m˙

r2

i

for i = 1, 2 and N − 1, N χi = 0 for i = 1, 2, N − 1, N Looks more like 3D

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 0.5 1 1.5 2 2.5 3 V (x) [ ǫ ] x = r/r0

  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions 1-dimensional models: comparison with experiment Simulations and “theory” Results

Canonical and local canonical appear consistent with observed results from simulations (elasticity etc.) Kinetic temperature profile straight apart from thermostatted borders, i = 1, 2, N − 1, N Maybe better mixing?

  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions 1-dimensional models: comparison with experiment Simulations and “theory” Results

Spectral density - Experiment and Simulations

For given z = z(t) real, Sz(ω) = +∞

−∞ eiωtz(t)z(0)dt

e.g. z → x(t) = L(t) − L,

  • r

z → v(t) = ˙ x(t); z → V (t)

10−4 10−3 10−2 10−1 100 101 102 5 10 15 20 25 ω1 Sx(ω) / r 2 ω/ω1

✲

10−3 10−1 101 0.6 1 1.4

  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions 1-dimensional models: comparison with experiment Simulations and “theory” Results

Equilibrium & Nonequilibrium thermo-elasticity - Exp+Sim

T1 T1 + ∆T

0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 0.004 0.008 0.012 0.016 0.02 0.024 ωr ω0 2

LJ

E [ ǫ ]

  • 116

118 120 122 124 126 128 130 0.06 0.07 0.08 0.09 0.1 0.11 E [ ǫ

R0 ]

kBT [ ǫ ]

  • △

△

  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions 1-dimensional models: comparison with experiment Simulations and “theory” Results

Equilibrium & Nonequilibrium thermo-elasticity

1) 1D model reproduces thermo-elastic properties at equilibrium, e.g. linearity of elastic modulus E or of ωres with T; 2) It works out of equilibrium as well: e.g. ωr = ωr(T), with average temperature T = (T1 + T2)/2, and ωr(T) the equilibrium resonance frequency. 3) Non trivial: for larger ∆T, theory does not apply. Explanation in terms of local canonical, ψi = exp (−Ei/kBTi) i.e. under local equilibrium.

  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions 1-dimensional models: comparison with experiment Simulations and “theory” Results

Experiment: low-loss (high quality factor) bar. ⇒ dynamics: independent damped oscillators forced by thermal noise (PSD sum of Lorentzian curves). Equilibrium is canonical and independent of damping. ⇒ normal modes of reduced mass µi, resonating at ωi: H(x, v) = 1 2

  • i

µi(ω2

i x2 i + v2 i ) ;

P(x, v) = e−H(x,v)/kBT/Z Experiment: one end fixed and nearly all mass at other end. Hence numerical simulations with µ1 ≈ M. At equilibrium, averaging over P: x2

1 = kB T

Mω2

1

i.e. x1 yields a measurement of temperature. On previous grounds, could one just use T in place of T, in general, if moderately out of equilibrium?

  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions 1-dimensional models: comparison with experiment Simulations and “theory” Results

Experiment says NO

x2 =

kB Teff mω2

For growing gradients T separates from Teff given by spectrum!

  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions 1-dimensional models: comparison with experiment Simulations and “theory” Results

Simulations

T T

10−2 10−1 100 101 0.6 0.8 1 1.2 1.4 ωr Sx(ω) / r 2 ω/ωr

  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions 1-dimensional models: comparison with experiment Simulations and “theory” Results

Effects of growing gradients: ∇T ↑ at same T

T T +

10−2 10−1 100 101 0.6 0.8 1 1.2 1.4 ωr Sx(ω) / r 2 ω/ωr

  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions 1-dimensional models: comparison with experiment Simulations and “theory” Results

Effects of growing gradients: ∇T ↑ at same T

T − T +

10−2 10−1 100 101 0.6 0.8 1 1.2 1.4 ωr Sx(ω) / r 2 ω/ωr

  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions 1-dimensional models: comparison with experiment Simulations and “theory” Results

Effects of growing gradients: ∇T ↑ at same T

T − T +

10−2 10−1 100 101 0.6 0.8 1 1.2 1.4 ωr Sx(ω) / r 2 ω/ωr

  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions 1-dimensional models: comparison with experiment Simulations and “theory” Results

Effects of growing gradients: ∇T ↑ at same T

T − T +

10−2 10−1 100 101 0.6 0.8 1 1.2 1.4 ωr Sx(ω) / r 2 ω/ωr

  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions 1-dimensional models: comparison with experiment Simulations and “theory” Results

Effects of growing gradients: ∇T ↑ at same T

T − T +

10−2 10−1 100 101 0.6 0.8 1 1.2 1.4 ωr Sx(ω) / r 2 ω/ωr

To be published

  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions 1-dimensional models: comparison with experiment Simulations and “theory” Results

Mode-mode correlations

In 1D models, a current J = 0 means xivj = 0 for some i, j Hyp.: For steady state, in canonical ensemble (under harmonic approximation), Jou et al. βH ⇒ βH + γJ with e−βH(x,v) ⇒ e−βH(x,v)−γJ(x,v); with J = − 1 N

1,N

  • i=k

jik xivk γ = Lagrange multiplier of heat flux. J ∝ ∇T for small ∇T. Guess β and make even simpler, more general, assumption on xivk: if w is one velocity correlated with x1, consider: PNEQ(x1, w) = exp (−Mω2

1 x2 1/2kBT − µ w2/2kBT + λMω2 1 x1w)/κ

where κ = 2π

  • Mω2

1

  • µ/(kBT)2 − λ2Mω2

1

; and T = (T1 + T2)/2

  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions 1-dimensional models: comparison with experiment Simulations and “theory” Results

x1w = λ µ/(kBT)2 − λ2Mω2

1

; x2

1 =

µx1w λMω2

1kBT

Introduce φ = −Mω2

1x1w ;

η = µ Mω2

1(kBT)2 ;

λ(φ) = 1 −

  • 1 + 4ηφ2

2φ then

  • x2

1

  • =

η η − λ(φ)2

  • x2

1

(eq) (T) with limit cases

  • x2

1

  • ≃
  • 1 + ηφ2

x2

1

(eq) (T) , |φ| ≪ 1/√η

  • x2

1

  • ≃ √η|φ|
  • x2

1

(eq) (T) , |φ| ≫ 1/√η i.e.

  • x2

1

  • x2

1

  • eq

− 1 ∝ (∆T)2 , ∆T ≪ T

  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions 1-dimensional models: comparison with experiment Simulations and “theory” Results

Simulations and experiment

  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators

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Detection of gravitational waves On “Effective Temperatures” Discussion and open questions

Discussion and open questions

In experiment, normal-mode analysis justified by high Q; Fourier law by small gradients; experimental data agree with numerical results for such simple model, for thermo-mechanical properties and as well as for vibrational energy of solids, as functions of T, at small ∇T; temperature immediately ceases to be the sole parameter characterizing fluctuations of long-wavelength modes: indeed strong dependence of “Teff ”, i.e. of

  • x2

, on ∇T; Experiment constitutes protocol to measure value of Lagrange multiplier λ, the “heatability” of the mode; dependence on initial conditions? theory and range of applicability?

  • L. Rondoni, Politecnico Torino

breaking energy equipartition in macroscopic oscillators