Transient effects in hydronization

Transcription

1 Transient effects in hydronization Michał Spaliński University of Białystok & National Centre for Nuclear Research Foundational Aspects of Relativistic Hydrodynamics, ECT* Trento, 9th May 2018

2 Introduction Hydronization: the process of reaching a stage of evolution where hydrodynamics works. AdS/CFT Gradient expansion, order 1 Proceeds by the decay of transients, leaving long-lived excitations. Very convenient to use special variables, which behave universally at late times, leading to attractor behaviour. For SYM, transients exhibit damped-oscillatory behaviour, which can be seen in numerical solutions at intermediate times. Out there, where most non-hydro modes have decayed

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4 Standard approach: MIS, BRSSS, ( D + 1) µ = µ +... can be matched to the least damped QNM of N=4 SYM HJSW: an alternative designed to have oscillating transient modes ( 1 1 T D)2 +2! I T D +! 2 µ =! 2 µ c 1 T D ( µ )+... The choice determines the non-hydrodynamic sector (transient behaviour) The gradient expansion: matching hydrodynamics to a microscopic theory µ = µ + D( µ )+... Bonus: divergent gradient expansions carry information about the non-hydro sector.

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sha1_base64="w3ypaer1wo2c3fzhd4sds7opimw=">aaacxnicbzbbaxqxfmezo9a6wru1f8gdwux0tm4uor6lrebxbbctbjbltezng5pkhurnyxlm6kfxwj+mt34um9sv7nyhgx/+v/d4yt+vtfkupr97yyohjzyebz7pp3229xx7sppi2fenkzirla7caq4etbi4iuuat2uhyhknj/nfycdpltf5vdnvtkhxzudmqljjogjnb68fkv1gugq58mpwutqq8ryeenzw7xwwtefpsvh9ka3ekk1qpn/pvrnfjrudlqqg76dzwtmsgcmlnbz90xisqv7agu6jtgdqz8lyjy1/g52cl5wlxxjfuv9obddel0weow3quv9nnfk/nm2o/dqlytynozw3i8pgc6p4fwsvlenjehefskfiw7k8h5gfxfd6fwhxslbggc2cogpden0kczofte1dikwho/rlhdx1xzsokeabrqd5xxzvjbj0lh37odz4vap5k71ib9h7lrf9dsc+sjgbmml+sj/smv3q3sqbyvayfdua9fyzu+xojs//afx1udo=</latexit> <latexit sha1_base64="w3ypaer1wo2c3fzhd4sds7opimw=">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</latexit> Bjorken flow w T / As a function of the clock variable the pressure anisotropy A P T P P L is very special: it shows universal behaviour at late times: A(w) =8( /s) 1 w +... independently of initial conditions. Because of the simple scaling form this takes, one can further rescale w w /s so the leading behaviour is even independent of the model. For this observable a special attractor solution exists.

6 The attractor in BRSSS hydrodynamics Evolution equation C 1+ A 12 A 0 + C 3w + C 1 8C A 2 = 3 2 8C w A in terms of dimensionless transport coefficients C = T, C = /s, C 1 = T 1 / Gradient expansion solution Hydro 1 Numerics Attractor A = 8C w + 16C (C C 1 ) 3w Note: the attractor depends on the values of the transport coefficients. Question: how closely can one model the attractor of a microscopic theory?

7 Asymptotic behaviour in BRSSS Singularities of the analytic continuation of the Borel transform Borel transform method: A B ( ) = 1X n=1 a n n! n Branch point location determined by Cannot integrate over the real line Z A resummed (w) =w C d e w à B ( ) Complex ambiguity

8 The ambiguity is resolved by incorporating trans-series sectors 1X 1 X A = a n w n + ce 3 2C w w C 2C 1 C a (1) n w n +... n=0 n=0 The exponentially-suppressed corrections: transient, non-hydrodynamic modes. The coefficients in the trans-series sectors are related by resurgence relations. The complex trans-series parameter c is partially fixed by consistency. The remaining freedom is a single real integration constant. The perturbative sector has no memory of the initial conditions The exponential decay shows how the solution forgets the initial conditions.

9 Borel summation with oscillating transients N=4 SYM HJSW hydro The pattern of divergence in SYM ang HJSW hydro is almost identical No singularities on the real axis; ignore instanton sectors Idea: use HJSW hydro as a testbed for the Borel summation of the hydro series

10 Borel sum in HJSW We adopt HJSW as a testing ground: Use 240 terms of the series Attractor Borel The result can be compared to the numerically determined attractor The summation breaks down for w < 0.3, but gets better for larger values of w. Could be improved by including trans-series sectors, but this would require determining appropriate values of trans series parameters. Next: proceed in the same way to sum the series for N=4 SYM

11 Attractor in N=4 SYM Hydro 1 Numerics Attractor Hydro 1 Numerics Attractor Very early hydronization Pressure anisotropy at hydrodynamization high A 0 = 2530w w 2 570w The difference between the attractor and hydro-1 is parameter dependent Behaviour for small w hard to get this way

12 MIS attractor SYM attractor Gradient expansion, order1 Comparing attractors: the BRSSS attractor tracks the SYM attractor very closely: is reaches the MIS attractor as soon as the calculation makes sense.

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sha1_base64="8+4uvga57b7u4xphdgoyqrfxp7w=">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</latexit> <latexit sha1_base64="y8cbfqfvp4gwy1nyxumhmbg56hm=">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</latexit> Seeing the transients in SYM plasma The QNM can be seen in the Borel-Pade plot. They can also be seen directly in numerical solutions of Bjorken flow obtained using AdS/CFT. The form of the trans-series correction is apple 3 3 A(w) A H (w)+e 3 2 I w w +(w) cos 2 Rw I log(w) + (w)sin 2 Rw I log(w) where ±(w) =C ± 1+ X n>0 a (±) n w n! is the 1-instanton sector series, which we can approximate by a constant. But we do not really know the universal part.

14 <latexit sha1_base64="oyj0/wi+gxvb2yf/wvprojrfrpm=">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</latexit> <latexit sha1_base64="yovlshaplpivjsxgfs2qur4/shw=">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</latexit> <latexit sha1_base64="yovlshaplpivjsxgfs2qur4/shw=">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</latexit> <latexit sha1_base64="0ncqqkk0inwrdltlgn/oivnpxmu=">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</latexit> To see that the transient, damped oscillations can be resolved with the existing numerical methods we can consider pairs of solutions; their difference will not involve the universal, hydrodynamic part: A 1 (w) A 2 (w) e 3 2 I w w apple C (+) 3 12 cos 2 Rw I log(w) + C ( ) 3 12 sin 2 Rw I log(w) Here all the parameters are fixed apart from the two amplitudes, which reflect the initial conditions and differ from one pair of solution to another. The two amplitudes appearing in the formula above can then be fitted to the numerical solution.

15 <latexit sha1_base64="slblz64dmrhlrqfynmiqjgh4/1i=">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</latexit> <latexit sha1_base64="car3x/jw0ji/bqwzemh9ojjbrqy=">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</latexit> <latexit sha1_base64="car3x/jw0ji/bqwzemh9ojjbrqy=">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</latexit> <latexit sha1_base64="3s0tma+6hmloid/epa3pluejyj0=">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</latexit> Alternatively, instead of looking at differences of solutions, we can try to subtract the universal part ( hydro to all orders ) by estimating it using the gradient expansion. Can we just remove the universal part by subtracting first or second order of the gradient expansion? First order of the gradient expansion A (1) H (w) =8( /s) 1 w

16 <latexit sha1_base64="ui4azheqv6uopoof/c3sdukx/pg=">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</latexit> <latexit sha1_base64="kkt+f+jjji1mjzwspgdu45v4kim=">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</latexit> <latexit sha1_base64="kkt+f+jjji1mjzwspgdu45v4kim=">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</latexit> <latexit sha1_base64="tzbwnlg6rytgciy3byobvuajm5k=">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</latexit> Instead, we can try to use the estimate of all-order hydro provided by the attractor calculated as the Borel sum of the truncated gradient expansion: 3 3 A(w) A? (w) e 3 2 I w w R applec (+) cos 2 Rw I log(w) + C ( ) sin 2 Rw I log(w)

17 Summary Hydrodynamics can be formulated based on very general arguments and matched to microscopic theories by comparing gradient expansions. The emergence of hydrodynamic behaviour is governed by the decay of non-hydrodynamic modes rather than local equilibration. Universal observables which exhibit attractor behaviour provide a clean way to study the decay of transients. The naive Borel sum provides a reasonable approximation to the attractor. The least-damped QNM can be seen directly in the late time behaviour of numerical solutions of Bjorken flow.