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
Detection of gravitational waves On “Effective Temperatures” Discussion and open questions Outline Detection of gravitational waves 1 Langevin equation Fluctuations On “Effective Temperatures” 2 1-dimensional models: comparison with experiment Simulations and “theory” Results Discussion and open questions 3 L. Rondoni, Politecnico Torino breaking energy equipartition in macroscopic oscillators
Detection of gravitational waves Langevin equation On “Effective Temperatures” Fluctuations Discussion and open questions 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
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.
Detection of gravitational waves Langevin equation On “Effective Temperatures” Fluctuations Discussion and open questions 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 over 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
Detection of gravitational waves Langevin equation On “Effective Temperatures” Fluctuations Discussion and open questions GW detectors (interferometers) TAM A300 @Tokyo (Japan) GEO600 @Hannover (Germany) LIGO @Livingston (USA) VIRGO @Cascina (Italy) L. Conti - Bologna 17.03.2014 L. Rondoni, Politecnico Torino breaking energy equipartition in macroscopic oscillators
Detection of gravitational waves Langevin equation On “Effective Temperatures” Fluctuations Discussion and open questions GW detector noise budget Dominant Sources of Noise: seismic noise • thermal noise • photon shot noise • Seismic sensitivity increases Shot Thermal L. Conti - Bologna 17.03.2014 L. Rondoni, Politecnico Torino breaking energy equipartition in macroscopic oscillators
Detection of gravitational waves Langevin equation On “Effective Temperatures” Fluctuations Discussion and open questions Thermal compensation to correct mismatch of the mirror fabrication Radius Of Curvature (ROC) due to: thermal lensing thermo-elastic deformation Applied thermal gradient deforms the mirror and corrects the ROC 687 m 666 m 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
Detection of gravitational waves Langevin equation On “Effective Temperatures” Fluctuations Discussion and open questions 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
Detection of gravitational waves Langevin equation On “Effective Temperatures” Fluctuations Discussion and open questions LdI s ( t ) + I s ( t ) [ R + R d ] + q s ( t ) � = 2 k B T 0 R Γ( t ) dt C I d ( t ) = GI s ( t − t d ) π t d = 2 ω r G ≪ 1 R d = G ω r L in expresses viscous damping due to feedback; s = − I s , t ′ = − t ), No time reversal invariance ( q ′ s = q s , I ′ violates Einstein relation, but formally identical to equilibrium oscillator at fictitious temperature T eff = T 0 / (1 + g ) � I 2 with feedback efficiency g = R d / R , so that: s � = 2 k B T eff / L Hence, usually treated as equilibrium system! L. Rondoni, Politecnico Torino breaking energy equipartition in macroscopic oscillators
Detection of gravitational waves Langevin equation On “Effective Temperatures” Fluctuations Discussion and open questions PDF and fluctuation relation of injected power P τ : Farago, ’02 ǫ τ < 1 � 4 γ ˜ ǫ τ , ˜ 3 ; 1 τ ln PDF(˜ ǫ τ ) ǫ τ ) = lim ρ (˜ ǫ τ ) = � � τ →∞ 7 3 1 ǫ τ ≥ 1 PDF( − ˜ γ ˜ ǫ τ 4 + ǫ τ − , ˜ 3 . ǫ 2 2˜ 4˜ τ ˜ ǫ τ = P τ L / ( k B T 0 R ) =; γ = ( R + R d ) / L , T eff = (22 ± 1) mK second derivative of PDF; b) ρ (˜ ǫ τ ). Data from May 2005 to May 2008. Shades = experimental uncertainty on τ eff , T eff , T 0 . L. Rondoni, Politecnico Torino breaking energy equipartition in macroscopic oscillators
Weakest assumptions approach News from RareNoise RN aluminum exp. - longitudinal and flexural oscillations longitudinal flexural
Detection of gravitational waves 1-dimensional models: comparison with experiment On “Effective Temperatures” Simulations and “theory” Discussion and open questions 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 T 1 and T 2 ” (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 obtain “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
Detection of gravitational waves 1-dimensional models: comparison with experiment On “Effective Temperatures” Simulations and “theory” Discussion and open questions Results Deterministic reversible semi-open Nos´ e-Hoover at T 1 and T 1 + ∆ T Nearest- and next-nearest-neighbors L-J. N = 128 , 256 , 512 �� � 6 � � 12 ℓ r 0 � ℓ r 0 V ( r i , r i ± ℓ ) = ǫ − 2 ; ℓ = 1 , 2 | r i − r i ± ℓ | | r i − r i ± ℓ | r i = F int 0.6 m ¨ ( r i , r i ± 1 , r i ± 2 ) − χ i ˙ r i i 0.4 � K 0.2 � χ i = m r 2 ˙ − 1 ; K i = m ˙ 0 i τ 2 V ( x ) [ ǫ ] k B T i -0.2 for i = 1 , 2 and N − 1 , N -0.4 -0.6 -0.8 χ i = 0 for i � = 1 , 2 , N − 1 , N -1 Looks more like 3D 0 0.5 1 1.5 2 2.5 3 x = r /r 0 L. Rondoni, Politecnico Torino breaking energy equipartition in macroscopic oscillators
Detection of gravitational waves 1-dimensional models: comparison with experiment On “Effective Temperatures” Simulations and “theory” Discussion and open questions 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
Detection of gravitational waves 1-dimensional models: comparison with experiment On “Effective Temperatures” Simulations and “theory” Discussion and open questions Results Spectral density - Experiment and Simulations For given z = z ( t ) real, � + ∞ −∞ e i ω t � z ( t ) z (0) � d t S z ( ω ) = 10 2 e.g. z → x ( t ) = L ( t ) − � L � , 10 1 10 1 ✲ or z → v ( t ) = ˙ x ( t ); z → V ( t ) ω 1 S x ( ω ) / r 2 10 0 0 10 − 1 10 − 1 10 − 3 0.6 1 1.4 10 − 2 10 − 3 10 − 4 0 5 10 15 20 25 ω / ω 1 L. Rondoni, Politecnico Torino breaking energy equipartition in macroscopic oscillators
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