Modelling nonequilibrium macroscopic oscillators. oscillators of interest to experimentalists.

Transcription

1 Modelling nonequilibrium macroscopic oscillators of interest to experimentalists. P. De Gregorio RareNoise : L. Conti, M. Bonaldi, L. Rondoni ERC-IDEAS: Nonequilibrium Processes: The Last 4 Years and the Future Obergurgl, Tirol, Austria, 29 August - 2 September 2

2 Modelling nonequilibrium macroscopic oscillators of interest to experimentalists. P. De Gregorio RareNoise : L. Conti, M. Bonaldi, L. Rondoni ERC-IDEAS: In celebration of Prof. Denis J. Evans 6th birthday

3 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.

4 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, 8432 (2)

5 Thermal gradients are present in actual GW detectors LCGT Project. From: T. Tomaru et al., Phys. Lett. A 3, 25 (22)

6 GW detectors 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, 336 (28)

7 Feedback cooling - the two approaches L İ(t) + R I(t) + C q(t) = V T (t) feedback off γ = R/L; ω 2 o = /LC; V T (t)v T (t ) = 2Rk B T δ(t t ) (L L in ) İ(t) + R I(t) + C q(t) = V T (t) L in İ s(t) I s(t) = I(t) + I d (t) feedback on Effective cooling quasi harmonic appr.: R R = (R + R d ) L İ(t) + R I(t) + C q(t) = V T (t) general case: İ(t) + R t f (t t )I(t )dt + R t g(t t )q(t )dt = N(t) Log[ωSIs/ I 2 eq] G =.6,.5,.4,.75 2tdω = π ω/ω

8 Work and heat distribution, from experiment to quasi-harmonic approximation Q τ = U τ + W τ P τ = Q τ + Q τ ( bath) Q τ ρ(ɛ τ ) = τ ln PDF(ɛτ ) PDF( ɛ τ ) ɛ τ = P τ L/(k B TR) D 2 (ɛ τ ) = 2 ɛ 2 τ h i ln PDF(ɛτ ) τ M. Bonaldi, L. Conti, PDG, L. Rondoni, G. Vedovato, A. Vinante et al (AURIGA team) PRL 3, 6 (29) J. Farago J Stat Phys 7, 78 (22)

9 Feedback cooling - the two approaches L İ(t) + R I(t) + C q(t) = V T (t) feedback off γ = R/L; ω 2 o = /LC; V T (t)v T (t ) = 2Rk B T δ(t t ) (L L in ) İ(t) + R I(t) + C q(t) = V T (t) L in İ s(t) I s(t) = I(t) + I d (t) feedback on quasi harmonic appr.: R R = (R + R d ) L İ(t) + R I(t) + C q(t) = V T (t) general case: İ(t) + R t f (t t )I(t )dt + R t g(t t )q(t )dt = N(t) Log[ωSIs/ I 2 eq] G =.6,.5,.4,.75 Effective cooling 2tdω = π ω/ω

10 Feedback cooling via the full equation L İ(t) + R I(t) + C q(t) = V T (t) feedback off γ = R/L; ω 2 o = /LC; V T (t)v T (t ) = 2Rk B T δ(t t ) (L L in ) İ(t) + R I(t) + C q(t) = V T (t) L in İ s(t) I s(t) = I(t) + I d (t) feedback on quasi harmonic appr.: R R = (R + R d ) general case: L İ(t) + R I(t) + C q(t) = V T (t) İ(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()n(t) ; I()N(t) = Kubo s causality broken Log[ωSIs/ I 2 eq] G =.6,.5,.4,.75 Effective cooling 2tdω = π An interesting problem in statistical mechanics in ots own right ω/ω Shift of the resonance frequency. From the full equation PDG, Rondoni, Bonaldi, Conti, J.Stat.Mech., P6 (29)

11 The digital protocol GW detectors Assume I d (t) = GI s (t t d ); x = L in /L, y = x, Gy < q s (t) + γ q s (t) + ω 2 q s (t) x y (Gy) k [γ q s (t kt d ) + ωq 2 s (t kt d )] = N(t) k= Generalized FDT (Kubo) would entail t > t N(t)q(t ) = N(t)N(t ) = q 2 f (t t ) N(t) q(t ) = But here N(t) = (Gy) k η(t kt d ) = (Gy) k V T (t kt d ) L k= k s.t. (t kt d ) < t k=

12 The digital protocol GW detectors Assume I d (t) = GI s (t t d ); x = L in /L, y = x, Gy < q s (t) + γ q s (t) + ω 2 q s (t) x y (Gy) k [γ q s (t kt d ) + ωq 2 s (t kt d )] = N(t) k= Generalized FDT (Kubo) would entail t > t N(t)q(t ) = N(t)N(t ) = q 2 f (t t ) N(t) q(t ) = But here N(t) = (Gy) k η(t kt d ) = (Gy) k V T (t kt d ) L k= k s.t. (t kt d ) < t k=

13 The digital protocol GW detectors Assume I d (t) = GI s (t t d ); x = L in /L, y = x, Gy < q s (t) + γ q s (t) + ω 2 q s (t) x y (Gy) k [γ q s (t kt d ) + ωq 2 s (t kt d )] = N(t) k= Generalized FDT (Kubo) would entail t > t N(t)q(t ) = N(t)N(t ) = q 2 f (t t ) N(t) q(t ) = But here N(t) = (Gy) k η(t kt d ) = (Gy) k V T (t kt d ) L k= k s.t. (t kt d ) < t k=

14 Solving in Fourier space The power spectrum Hyp. : S Is (ω) = + η(t)η(t ) = 2k BT γ δ(t t ) L ( e iωt 2kB T γ ) ω 2 I s ()I s (t) dt = L D(ω) D(ω) = α 2 + G 2 β 2 + ( + G 2 )γ 2 ω 2 Def. : α = ω 2 ω 2 +2xGγω 3 sin (ωt d ) 2G(αβ + γ 2 ω 2 ) cos (ωt d ) β = y ω 2 ω 2 For very high quality factors ω γ Q xg S I s (ω) 2 z ω 2 = (ω 2 ωr 2 ) 2 + µ 2 ω 2 and if 2t d ω = π, then : ω 4 r = + G2 + y 2 G 2 ω4 ; z = k B T γ L ( + y 2 G 2 ) p µ 2 + G 2 = 2 q + y 2 G 2 + yg 2 + y 2 G 2 «ω 2 G x 2 G 2 ω 2

15 Solving in Fourier space The power spectrum Hyp. : S Is (ω) = + η(t)η(t ) = 2k BT γ δ(t t ) L ( e iωt 2kB T γ ) ω 2 I s ()I s (t) dt = L D(ω) D(ω) = α 2 + G 2 β 2 + ( + G 2 )γ 2 ω 2 Def. : α = ω 2 ω 2 +2xGγω 3 sin (ωt d ) 2G(αβ + γ 2 ω 2 ) cos (ωt d ) β = y ω 2 ω 2 For very high quality factors ω γ Q xg S I s (ω) 2 z ω 2 = (ω 2 ωr 2 ) 2 + µ 2 ω 2 and if 2t d ω = π, then : ω 4 r = + G2 + y 2 G 2 ω4 ; z = k B T γ L ( + y 2 G 2 ) p µ 2 + G 2 = 2 q + y 2 G 2 + yg 2 + y 2 G 2 «ω 2 G x 2 G 2 ω 2

16 Q = 5 - shift of the resonance frequency 2.5 G =.6,.5,.4,.75 2t d ω = π Log[ωSIs/ I 2 eq] ω/ω

17 Q = - "transition" of the resonance frequency ωsis/ I 2 eq G = (Equil.) 2t d ω = π G = ω/ω

18 Q = - changing t d ωsis/ I 2 eq t d ω = π ω/ω

19 Q = - "transition" of the resonance frequency as t d ωsis/ I 2 eq t d ω =.5 π ω/ω

20 Q = - "transition" of the resonance frequency as t d ωsis/ I 2 eq t d ω =.4 π ω/ω

21 Q = - "transition" of the resonance frequency as t d ωsis/ I 2 eq t d ω =.5 π ω/ω

22 Q = - "transition" of the resonance frequency as t d ωsis/ I 2 eq t d ω =.7 π ω/ω

23 The analog protocol GW detectors The feedback is implemented via a low-pass filter Ĩ d (ω) = AΩ Ω iω Ĩs(ω) ( 2kB T γ ) ω 2 (ω 2 + Ω 2 ) S Is (ω) = L B(ω) B(ω) = ω 2 (ω 2 F 2 ) 2 + (aω 2 b 3 ) 2 F 2 = ω 2 +γω( A); a = ( ya)ω+γ; b3 = Ωω 2 ( A) For consistency with digital protocol we set AΩ = Gω

24 Analog vs Q = Ω =.2 ω Log[ωSIs/ I 2 eq] ω/ω

25 Analog vs digital decreasing Ω/ω 2.5 Ω =. ω Log[ωSIs/ I 2 eq] ω/ω

26 Analog vs digital decreasing Ω/ω 2.5 Ω =. ω Log[ωSIs/ I 2 eq] ω/ω

27 Modeling the rod with one-dimensional chains L as the observable The idea is to define a variation of the celebrated FPU chain: V t (x) = h [ (x ) 2 λ(x ) 3 + µ(x ) 4] to then compare it to a more phenomelogical interaction potential, to overcome certain inconsistencies V LJ (x) = ɛ ( x 2 2 x 6 ) x = r/r V /ǫ Vt λ = µ = V t λ = 7, µ = 53 λ/2 V t2 (as if we expanded LJ) x Vt2 VLJ log (L/L).2 VLJ.5 Vt2..5 Vt kbt [ ǫ ]

28 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 E [ ǫ r ] 22 2 (c) (a) 8 6 (b) ( L/L) kbt [ ǫ ] F [ ǫ r ]

29 Comparing thermo-elastic properties E, the elastic modulus: F E ( L/L), if L L. E [ ǫ r ] 22 2 F [ ǫ r ] (c) (a) 8 6 (b) ( L/L) 3 E [ ǫ r ] (a) kbt [ ǫ ] kbt [ ǫ ]

30 Comparing thermo-elastic properties E, the elastic modulus: F E ( L/L), if L L. ω resonance frequency of st mode. E [ ǫ r ] 22 2 F [ ǫ r ] (c) (a) 8 6 (b) ( L/L) 3 E [ ǫ r ] (a) kbt [ ǫ ] kbt [ ǫ ].2 ( ωr ) 2 LJ ω S(ω) (a.u.) (b) (a) (c) ω/ω E [ ǫ ]

31 Comparing thermo-elastic properties E, the elastic modulus: F E ( L/L), if L L. ω resonance frequency of st mode. Q = ω / ω /2 E [ ǫ r ] 22 2 F [ ǫ r ] (c) (a) 8 6 (b) ( L/L) 3 E [ ǫ r ] (a) kbt [ ǫ ] kbt [ ǫ ] ( ωr ) 2 LJ ω S(ω) (a.u.) (b) (a) (c) ω/ω E [ ǫ ] Q S(ω) (a.u.) E [ ǫ ] ω/ω

32 Effects of gradients (under investigation) - T T T S(ω) [au] ω/ω

33 Effects of gradients on Q: T -vs- T T T + S(ω) [au] ω/ω

34 Effects of gradients on Q: T -vs- T T T S(ω) [au] ω/ω

35 Acknowledgments GW detectors FP7 - IDEAS - ERC grant agreement n 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,...