Thermal processes generated in quark-gluon plasma by yoctosecond laser pulses
TThermal processes generated in QGP by yoctosecond (10 -24 s) laser pulsesJ. Marciak-Kozłowska M. Kozłowski
Institute of Electron Technology, Warsaw Physics Department, Warsaw University, Warsaw * corresponding author, e-mail: [email protected] Abstract
In this paper the thermal processes generated by yoctosecond (10 -24 s) laserpulses in QGP are investigated. Considering that the relaxation time in QGP is of theorder of 1 ys it is shown that in QGP the yoctosecond laser pulses can generate thethermal waves with velocity v = c (0.3 fm/ys). Key words : QGP, thermal waves, yoctosecond pulses
1. Introduction
The QGP (Quark Gluon Plasma) is formed in collisions of hadrons.
In this paper we develop the model for the thermal energy transport in QGP generatedby yoctosecond laser pulses (1ys = 10 -24 s = 0.3 fm/ c ). As was shown by A. Ipp et al.,Phys. Rev. [1] the laser with yoctosecond photon pulses can be arranged in RHIC andLHC hadron colliders. In QGP the thermal relaxation time is of the order of ys. It meansthat for ys photons the duration time of the laser pulse and relaxation time is the sameorder. For those thermal processes the Fourier approximation – parabolic thermaldiffusion is not valid. In this paper we compare both models – hyperbolic and parabolicand conclude that in the case of the hyperbolic model the new mode of QGP excitation– thermal wave can be generated.
2. Heaviside and Klein – Gordon thermal equation
First of all let us consider, for the moment, the parabolic heat transportequation [2] . TmtT (cid:61649)(cid:61501)(cid:61622)(cid:61622) (cid:61544) (2.1)When the real time (cid:61529)(cid:61614)(cid:61614)
Titt ,2/ , Ψ is the quantum wave function,Eq. (2.1) has the form of a free Schrödinger equation.2 (cid:61529)(cid:61649)(cid:61485)(cid:61501)(cid:61622)(cid:61529)(cid:61622) mti (cid:61544)(cid:61544) (2.2)The complete Schrödinger equation has the form,2 (cid:61529)(cid:61483)(cid:61529)(cid:61649)(cid:61485)(cid:61501)(cid:61622)(cid:61529)(cid:61622) Vmti (cid:61544)(cid:61544) (2.3)where V denotes the potential energy. When we go back to real time Titt (cid:61614)(cid:61529)(cid:61614) ,2 , the parabolic quantum heat transport is obtained .2 TVTmtT (cid:61544)(cid:61544) (cid:61485)(cid:61649)(cid:61501)(cid:61622)(cid:61622) (2.4)Equation (2.4) describes the quantum heat transport for (cid:61556) (cid:61502)(cid:61508) t . For heattransport initiated by yoctosecond laser pulses, (cid:61556) (cid:61500)(cid:61508) t , one obtains the generalizedquantum hyperbolic heat transport equation with memory term added.2 TVTmtTtT (cid:61544)(cid:61544) (cid:61485)(cid:61649)(cid:61501)(cid:61622)(cid:61622)(cid:61483)(cid:61622)(cid:61622) (cid:61556) (2.5)
Considering that (cid:61557)(cid:61556) m (cid:61544) (cid:61501) , Eq. (2.5) can be written as follows:.21 TTVmtTmtT (cid:61649)(cid:61501)(cid:61483)(cid:61622)(cid:61622)(cid:61483)(cid:61622)(cid:61622) (cid:61544)(cid:61544) (cid:61557) (2.6)Equation (2.6) describes the heat flow when besides the temperature gradient, thepotential energy V operates.In the following, we consider one-dimensional heat transfer phenomena, i.e .21 xTTVmtTmtT (cid:61622)(cid:61622)(cid:61501)(cid:61483)(cid:61622)(cid:61622)(cid:61483)(cid:61622)(cid:61622) (cid:61544)(cid:61544) (cid:61557) (2.7)For quantum heat transfer equation (2.7) we seek the solution in the form ),(),( txuetxT τt (cid:61501) (2.8)After substitution (2.8) into Eq. (2.7), one obtains,0),(1 (cid:61501)(cid:61483)(cid:61622)(cid:61622)(cid:61485)(cid:61622)(cid:61622) txquxut uυ (2.9)where (cid:61687)(cid:61688)(cid:61686)(cid:61671)(cid:61672)(cid:61670)(cid:61485)(cid:61501) (cid:61544)(cid:61544) (cid:61557) mVmq . (2.10)In the following, we will consider a constant potential energy V = V . The generalsolution of Eq. (2.9) for the Cauchy boundary conditions [2], ),(),(),()0,( xFt txuxfxu t (cid:61501)(cid:61690)(cid:61691)(cid:61689)(cid:61674)(cid:61675)(cid:61673) (cid:61622)(cid:61622)(cid:61501) (cid:61501) (2.11)has the form [2] (cid:61682) (cid:61483)(cid:61485) (cid:61510)(cid:61483)(cid:61483)(cid:61483)(cid:61485)(cid:61501) tx tx dzzyxtxftxftxu (cid:61557)(cid:61557) (cid:61557)(cid:61557)(cid:61557) ,),,(212 )()(),( (2.12)where (cid:61480) (cid:61481) ' 2 2 202 2 20 2 2 222 22 ( )1( , , ) ( ) ( ) ( ) ,( )22 b J z x tbx t z F z J z x t btf z z x tm Vmb (cid:61557) (cid:61557)(cid:61557)(cid:61557) (cid:61557) (cid:61557)(cid:61557) (cid:61557) (cid:61485) (cid:61485)(cid:61670) (cid:61686)(cid:61510) (cid:61501) (cid:61485) (cid:61485) (cid:61483)(cid:61671) (cid:61687)(cid:61672) (cid:61688) (cid:61485) (cid:61485)(cid:61670) (cid:61686)(cid:61501) (cid:61485)(cid:61671) (cid:61687)(cid:61672) (cid:61688) (cid:61544) (cid:61544) (2.13) and J (z) denotes the Bessel function of the first kind. The function u ( x,t ) describes thepropagation of the distorted thermal quantum waves with characteristic lines tx (cid:61557) (cid:61617)(cid:61501) . We can define the distortionless thermal wave as the wave that preserves theshape in the field of the potential energy V . The condition for conserving the shape canbe formulated as Vm mq (cid:61557) (cid:61670) (cid:61686)(cid:61501) (cid:61485) (cid:61501)(cid:61671) (cid:61687)(cid:61672) (cid:61688) (cid:61544) (cid:61544) . (2.14)When Eq. (2.14) holds, Eq. (2.10) has the form.),( xut txu (cid:61622)(cid:61622)(cid:61501)(cid:61622)(cid:61622) (cid:61557) (2.15)Equation (2.15) is the quantum thermal wave equation with the solution (for Cauchyboundary conditions (2.11)) .)(212 )()(),( (cid:61682) (cid:61483)(cid:61485) (cid:61483)(cid:61483)(cid:61483)(cid:61485)(cid:61501) tx tx dzzFtxftxftxu (cid:61557)(cid:61557) (cid:61557)(cid:61557)(cid:61557) (2.16)It occurs that structure of equation (2.9) depends on the sign of the parameter q . For quantum heat transport, e.g. in nuclear matter, parameter q is the function ofpotential barrier height V .In monograph [1] the velocity of thermal waves in nuclear (protons andneutrons) and in QGP was calculated, 1, 2 i i c i (cid:61557) (cid:61537) (cid:61501) (cid:61501) (2.17)where c is the light velocity and α = 0.16 for nuclear matter and α = 1 for QGP.Respectively we obtain for thermal relaxation time , 1, 2 i i i im (cid:61556) (cid:61557) (cid:61501) (cid:61501) (cid:61544) (2.18)and (cid:61480) (cid:61481) , h m c (cid:61556) (cid:61537) (cid:61501) (cid:61544) . q m c (cid:61556) (cid:61501) (cid:61544) For Cauchy initial condition [2] )()0,(),()0,( xgtxuxfxu (cid:61501)(cid:61622)(cid:61622)(cid:61501) the solution of Eq. (2.10) for q <0 has the form (cid:61480) (cid:61481) (cid:61480) (cid:61481) (cid:61480) (cid:61481) ( ) ( )( , ) 21 ( ) ( )2 ( ) ( )2 ( ) x tx t x tx t f x t f x tu x t g I q t x dI q t xq t f dt x (cid:61557)(cid:61557) (cid:61557)(cid:61557) (cid:61557) (cid:61557)(cid:61526) (cid:61557) (cid:61526) (cid:61526)(cid:61557) (cid:61557) (cid:61526)(cid:61557) (cid:61526) (cid:61526)(cid:61557) (cid:61526) (cid:61483)(cid:61485) (cid:61483)(cid:61485) (cid:61485) (cid:61483) (cid:61483)(cid:61501) (cid:61673) (cid:61689)(cid:61483) (cid:61485) (cid:61485) (cid:61485)(cid:61674) (cid:61690)(cid:61675) (cid:61691)(cid:61673) (cid:61689)(cid:61485) (cid:61485) (cid:61485)(cid:61485) (cid:61674) (cid:61690)(cid:61675) (cid:61691)(cid:61483) (cid:61485) (cid:61485) (cid:61682) (cid:61682) (2.19)and the equation (2.10) is the modified Heaviside equation. When q >0 equation (2.10)is the modified Klein – Gordon thermal equation. The solution of M K-G equation hasthe same structure as the formula (2.19) but with q q (cid:61614) (cid:61485) and ( ), ( ) ( ), ( ) I z I z J z J z (cid:61614) [2].
3. The model thermal quantum equation
For free thermal energy transport equation (2.7) can be written as:
T m T Tt t x (cid:61557) (cid:61622) (cid:61622) (cid:61622)(cid:61483) (cid:61501)(cid:61622) (cid:61622) (cid:61622) (cid:61544) (3.1)In the subsequent we are concerned with the solution to (3.1) for a nearly delta functiontemperature generated in nuclear matter. The pulse transfered has the shape: for 0 ,0 for . T x lT x l (cid:61508) (cid:61500) (cid:61500) (cid:61508)(cid:61508) (cid:61501) (cid:61502) (cid:61508) (3.2)With t = 0 temperature profile the solution of equation (3.1) is [2]: /21 0 0 02 /20 0 1 04 ( , ) ( ) ( )1 ( ) ( ) ( ),2 t tt T l t T e t t t t ttT e I z I z t tz (cid:61556)(cid:61556)(cid:61556) (cid:61556) (cid:61485) (cid:61485)(cid:61508) (cid:61501) (cid:61508) (cid:61521) (cid:61485) (cid:61521) (cid:61483) (cid:61508) (cid:61485)(cid:61676) (cid:61692)(cid:61483) (cid:61508) (cid:61483) (cid:61521) (cid:61485)(cid:61677) (cid:61693)(cid:61678) (cid:61694) (3.3)where (cid:61480) (cid:61481) τttz (cid:61485)(cid:61501) and / S t l (cid:61557) (cid:61501) . The solution to eq.(3.3) when there arereflecting boundaries is the superposition of the temperature at l from the originaltemperature and from image heat sources at nl (cid:61617) : /20 /20 0 0 1 ( ) ( )( , ) ,1( ) ( ) ( )2 2 t i iti i i ii T e t t t t tT l t t tT e I z I z t tz (cid:61556) (cid:61556) (cid:61556) (cid:61556) (cid:61485)(cid:61605) (cid:61485)(cid:61501) (cid:61673) (cid:61689)(cid:61508) (cid:61521) (cid:61485) (cid:61521) (cid:61483) (cid:61508) (cid:61485)(cid:61674) (cid:61690)(cid:61501) (cid:61676) (cid:61692)(cid:61508)(cid:61674) (cid:61690)(cid:61483)(cid:61508) (cid:61483) (cid:61521) (cid:61485)(cid:61677) (cid:61693)(cid:61674) (cid:61690)(cid:61678) (cid:61694)(cid:61675) (cid:61691) (cid:61669) (3.4)
In Eqs (3.3) and (3.4) ( ) y (cid:61521) is the Heaviside step function.
4. Yoctosecond photon pulses from quark – gluon plasma
Among the shortest possible time scales that are available experimentallyare those obtained through high energy collisions. Particularly interesting in thiscontext are heavy ion collisions that can produce a quark – gluon plasma (QGP). Heavyion collision at the Relativistic Heavy Ion Collider (RHIC) and LHC at CERN producethis new state of matter up to the size of nucleous (~ 15 fm) for a duration of a few tensof yoctoseconds (1ys = 10 -24 s = 0.3 fm/ c ) [3]. In such a collision the QGP is producedinitially in a very anisotropic state and reaches a hydrodynamic evolution throughinternal interactions only after a few relaxation time τ . The observed particle spectraturned out to agree well with ideal hydrodynamical model predictions which led to theassumption that relaxation time is of the order of 1 ys.In this paragraph we calculate the temperature field T ( x,t ) for hadroncollisions following the model presented in paragraph 2. In monograph [2] therelaxation time for QGP was calculated: (cid:61480) (cid:61481) ~ 10 s 1ys, q q m c (cid:61556) (cid:61537) (cid:61485) (cid:61501) (cid:61627) (cid:61501) (cid:61544) (4.1)where q (cid:61537) = 1, m q = 417 MeV and (cid:61544) is the Planck constant.In Fig. 1-3 the results of calculations are presented. For all figures theduration of laser pulse is 1 ys = 0.3 fm/ c . In Fig. 1 a, b the relaxation time is 5 fm/ c and v = c . Fig. 1 a presents the result of calculation with Klein – Gordon thermal relaxationtime. The temperature field has the structure of thermal wave with velocity c . theFourier model (parabolic equation) is presented in Fig. 1 b. In Fig. 2, 3 the calculationsere presented for relaxation time 1 fm/ c and 0.5 fm/ c respectively.
5. Conclusions
In this paper the interaction of yoctosecond laser pulses with QGP isinvestigated. It is shown that for relaxation time in the range 0.5 – 5 fm/ c (1 –10 ys) inQGP the thermal waves with velocities can be created. References [1] A. Ipp, Ch. H. Keitel and J. Evers, Phys. Rev. Lett. (2009) 152301.[2] M. Kozłowski, J. Marciak-Kozłowska,
Thermal processes using attosecond laserpulses , Springer 2006.[3] The CMS Collaborations, arXiv:1005.3299.
Figure captions :Fig. 1 a. Temperature field T ( x,t ). Parameters: relaxation time τ = 5 fm/ c , v = c .Hyperbolic model calculation.Fig. 1 b. Temperature field T ( x,t ). Parameters: relaxation time τ = 5 fm/ c , v = c .Parabolic model calculation.Fig. 2 a,b The same as in Fig. 1 a,b but τ = 1 fm/ c .Fig. 3 a,b The same as in Fig. 1 a,b but τ = 0.5 fm/ c ≈ 1 ys.≈ 1 ys.