A compact formula for the quantum fluctuations of energy
Keywords:Quantum fluctuations, Energy density, Relativistic heavy-ion collisions
A formula to calculate the quantum fluctuations of energy in small subsystems of a hot and relativistic gas is derived. We find an increase in fluctuations for subsystems of small sizes, but we agrees with the energy fluctuations in the canonical ensemble if the size is large enough. Not only one can use our expression to find the limit of the concepts of energy density or fluid element in connection to relativistic heavy-ion collisions, but also in other areas of physics where one studies matter with high temperature and velocity.
K. Huang, Statistical Mechanics, (Jhon Wiley & Sons Inc., New York, 2nd ed., 1987).
M. Smoluchowski, Beitrag zur Theorie der Opaleszenz von Gasen im kritischen Zustande, Bulletin international de l’Academie des sciences de Cracovie (1911) 493.
S. Jeon and V. Koch, Charged particle ratio fluctuation as a signal for quark-gluon plasma, Phys. Rev. Lett. 85 (2000) 2076. https://doi.org/10.1103/PhysRevLett.85.2076.
D. J. Gross, R. D. Pisarski, and L. G. Yaffe, QCD and Instantons at Finite Temperature, Rev. Mod. Phys. 53 (1981) 43. https://doi.org/10.1103/RevModPhys.53.43.
S. Haussler, M. Bleicher, and H. Stocker, Susceptibilities and fluctuations in a Quark-Hadron System with Dynamical Recombination, arXiv:0803.2846 [hep-ph]. .
C. Herzog and K.-W. Huang, Boundary Fluctuations and A Reduction Entropy, Phys. Rev. D 95 (2017) 021901, https://doi.org/10.1103/PhysRevD.95.021901.
V. Vovchenko, D. V. Anchishkin, M. I. Gorenstein, R. V. Poberezhnyuk, and H. Stoecker, Critical fluctuations in models with van der Waals interactions, Acta Phys. Polon. Supp. 10 (2017) 753,
J. Steinheimer, V. Vovchenko, J. Aichelin, M. Bleicher, and H. StA¨ocker, Conserved charge fluctuations ˆ are not conserved during the hadronic phase, Phys. Lett. Supl. Rev. Mex. Fis. 3 0308115 4 RAJEEV SINGH B 776 (2018) 32, https://doi.org/10.1016/j.physletb.2017.11.012.
D. S. Lohr-Robles, E. Lopez-Moreno, and P. O. Hess, Quantum Phase Transitions within a nuclear cluster model and an effective model of QCD, Nucl. Phys. A 1016 (2021) 122335, https://doi.org/10.1016/j.nuclphysa.2021.122335.
Z. Bai, W.-j. Fu, and Y.-x. Liu, Identifying the QCD Phase Transitions via the Gravitational Wave Frequency from Supernova Explosion, ApJ 922 (2021) 266, https://doi.org/10.3847/1538-4357/ac2a31.
L. Fortunato, Quantum phase transitions in algebraic and collective models of nuclear structure, Prog. Part. Nucl. Phys. 121 (2021) 103891. https://doi.org/10.1016/j.ppnp.2021.103891.
X. Li, S. Shu, and J.-R. Li, The quantum fluctuation in an inhomogeneous background and its influence to the phase transition in a finite volume system, arXiv:2108.12325 [hep-ph].
Z.-C. Yang, Y. Li, M. P. A. Fisher, and X. Chen, Entanglement phase transitions in random stabilizer tensor networks, arXiv:2107.12376 [cond-mat.stat-mech].
M. Sami and R. Gannouji, Spontaneous symmetry breaking in the late Universe and glimpses of early Universe phase transitions a la baryogenesis, ´ Int. J. Mod. Phys. D 30 (2021) 2130005, https://doi.org/10.1142/S0218271821300056.
G. P. de Brito, O. Melichev, R. Percacci, and A. D. Pereira, Can quantum fluctuations differentiate between standard and unimodular gravity?, J. High Energ. Phys. 2021 (2021), https://doi.org/10.1007/JHEP12(2021)090.
E. M. Lifshitz and I. M. Khalatnikov, Investigations in relativistic cosmology, Adv. Phys. 12 (1963) 185. https://doi.org/10.1080/00018736300101283.
A. H. Guth and S.-Y. Pi, Fluctuations in the new inflationary universe, Phys. Rev. Lett. 49 (1982) 1110. https://doi.org/10.1103/PhysRevLett.49.1110.
S. Choudhury and A. Mazumdar, Primordial blackholes and gravitational waves for an inflection-point model of inflation, Phys. Lett. B 733 (2014) 270, https://doi.org/10.1016/j.physletb.2014.04.050.
S. Choudhury, S. Panda, and R. Singh, Bell violation in the Sky, Eur. Phys. J. C 77 (2017) 60, nurlf https://doi.org/10.1140/epjc/s10052-016-4553-3.
S. Choudhury, S. Panda, and R. Singh, Bell violation in primordial cosmology, Universe 3 (2017) 13, https://doi.org/10.3390/universe3010013.
S. Choudhury, The Cosmological OTOC: A New Proposal for Quantifying Auto-correlated Random Nonchaotic Primordial Fluctuations, Symmetry 13 (2021) 599, arXiv:2106.01305 [physics.gen-ph]. https://doi.org/10.3390/sym13040599.
L. L. Graef, Constraining the spectrum of cosmological perturbations from statistical thermal fluctuations, Phys. Lett. B 819 (2021) 136418. https://doi.org/10.1016/j. physletb.2021.136418.
R. Kubo, The Fluctuation-Dissipation Theorem, Rep. Prog. Phys. 29 (1966) 255. https://doi.org/10.1088/0034-4885/29/1/306.
J. Berges and K. Rajagopal, Color superconductivity and chiral symmetry restoration at nonzero baryon density and temperature, Nucl. Phys. B 538 (1999) 215, https://doi.org/10.1016/S0550-3213(98)00620-8.
A. M. Halasz, A. D. Jackson, R. E. Shrock, M. A. Stephanov, and J. J. M. Verbaarschot, On the phase diagram of QCD, Phys. Rev. D 58 (1998) 096007, https://doi.org/10.1103/PhysRevD.58.096007.
M. A. Stephanov, K. Rajagopal, and E. V. Shuryak, Signatures of the tricritical point in QCD, Phys. Rev. Lett. 81 (1998) 4816, https://doi.org/10.1103/PhysRevLett.81.4816.
M. A. Stephanov, K. Rajagopal, and E. V. Shuryak, Event-byevent fluctuations in heavy ion collisions and the QCD critical point, Phys. Rev. D 60 (1999) 114028, https://doi.org/10.1103/PhysRevD.60.114028.
Y. Hatta and T. Ikeda, Universality, the QCD critical / tricritical point and the quark number susceptibility, Phys. Rev. D 67 (2003) 014028, 014028, https://doi.org/10.1103/PhysRevD.67.014028.
D. T. Son and M. A. Stephanov, Dynamic universality class of the QCD critical point, Phys. Rev. D 70 (2004) 056001, https://doi.org/10.1103/PhysRevD.70.056001.
M. Stephanov, Non-Gaussian fluctuations near the QCD critical point, Phys. Rev. Lett. 102 (2009) 032301, https://doi.org/10.1103/PhysRevLett.102.032301.
B. Berdnikov and K. Rajagopal, Slowing out-of-equilibrium near the QCD critical point, Phys. Rev. D 61 (2000) 105017, https://doi.org/10.1103/PhysRevD.61.105017.
S. Caron-Huot, P. M. Chesler, and D. Teaney, Fluctuation, dissipation, and thermalization in non-equilibrium AdS5 black hole geometries, Phys. Rev. D 84 (2011) 026012, https://doi.org/10.1103/PhysRevD.84.026012.
M. Kitazawa, M. Asakawa, and H. Ono, Non-equilibrium time evolution of higher order cumulants of conserved charges and event-by-event analysis, Phys. Lett. B 728 (2014) 386, https://doi.org/10.1016/j.physletb.2013.12.008.
J. Goswami, F. Karsch, C. Schmidt, S. Mukherjee, and P. Petreczky, Comparing conserved charge fluctuations from lattice QCD to HRG model calculations, Acta Phys. Polon. Supp. 14 (2021) 251.
W.-j. Fu, X. Luo, J. M. Pawlowski, F. Rennecke, R. Wen, and S. Yin, High-order baryon number fluctuations within the fRG approach, 9, (2021).
M. Pradeep, K. Rajagopal, M. Stephanov, and Y. Yin, Freezing out critical fluctuations, in International Conference on Critical Point and Onset of Deconfinement. 9, (2021).
J. Goswami, F. Karsch, S. Mukherjee, P. Petreczky, and C. Schmidt, Conserved charge fluctuations at vanishing netbaryon density from Lattice QCD, in 19th International Conference on Strangeness in Quark Matter. 9, (2021).
D. Bollweg et al., Second order cumulants of conserved charge fluctuations revisited I. Vanishing chemical potentials, Phys. Rev, D 104 (2021) 074512, https://doi.org/10.1103/PhysRevD.104.074512.
C. Schmidt et al., Net-baryon number fluctuations, in Criticality in QCD and the Hadron Resonance Gas. 1, (2021).
X. Guo, K. A. Milton, G. Kennedy, W. P. McNulty, N. Pourtolami, and Y. Li, The energetics of quantum vacuum friction. I. Field fluctuations, Phys. Rev. D 104 (2021) 116006, https://doi.org/10.1103/PhysRevD.104.116006.
A. Das, W. Florkowski, R. Ryblewski, and R. Singh, Quantum fluctuations of energy in subsystems of a hot relativistic gas, arXiv:2012.05662 [hep-ph].
A. Das, W. Florkowski, R. Ryblewski, and R. Singh, Pseudogauge dependence of quantum fluctuations of the energy in a hot relativistic gas of fermions, Phys. Rev. D 103 (2021) L091502, https://doi.org/10.1103/PhysRevD.103.L091502.
A. Das, W. Florkowski, R. Ryblewski, and R. Singh, Quantum baryon number fluctuations in subsystems of a hot and dense relativistic gas of fermions, in 9th Large Hadron Collider Physics Conference. 9 (2021), arXiv:2105.02125 [nucl-th].
R. Singh, Mathematical expressions for quantum fluctuations of energy for different energy-momentum tensors, in 9th Large Hadron Collider Physics Conference. 9 2021. arXiv:2109.11068 [quant-ph]. .
R. Singh, Quantum fluctuations of baryon number density, J. Phys.: Conference Series 2105 (2021) 012006, https://doi.org/10.1088/1742-6596/2105/1/012006.
K. Fukushima and C. Sasaki, The phase diagram of nuclear and quark matter at high baryon density, Prog. Part. Nucl. Phys. 72 (2013) 99, https://doi.org/10.1016/j.ppnp.2013.05.003.
A. Jaiswal and V. Roy, Relativistic hydrodynamics in heavyion collisions: general aspects and recent developments, Adv. High Energy Phys. 2016 (2016) 9623034, https://doi.org/10.1155/2016/9623034.
W. Florkowski, M. P. Heller, and M. Spalinski, New theories of relativistic hydrodynamics in the LHC era, Rept. Prog. Phys. 81 (2018) 046001, https://doi.org/10.1088/1361-6633/aaa091.
K. Fukushima, Extreme matter in electromagnetic fields and rotation, Prog. Part. Nucl. Phys. 107 (2019) 167, https://doi.org/10.1016/j.ppnp.2019.04.001.
P. Romatschke and U. Romatschke, Relativistic Fluid Dynamics In and Out of Equilibrium. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 5 (2019).
S. Bhadury, J. Bhatt, A. Jaiswal, and A. Kumar, New developments in relativistic fluid dynamics with spin, Eur. Phys. J. ST 230 (2021) 655, https://doi.org/10.1140/epjs/s11734-021-00020-4.
S. Coleman, Lectures of Sidney Coleman on Quantum Field Theory. (WSP, Hackensack 2018). .
C. Itzykson and J. Zuber, Quantum Field Theory. (International Series In Pure and Applied Physics. McGraw-Hill, New York, 1980). .
T. Evans and D. A. Steer, Wick’s theorem at finite temperature, Nucl. Phys. B 474 (1996) 481, https://doi.org/ 10.1016/0550-3213(96)00286-6.
N. Phillips and B. Hu, Vacuum energy density fluctuations in Minkowski and Casimir states via smeared quantum fields and point separation, Phys. Rev. D 62 (2000) 084017, https: //doi.org/10.1103/PhysRevD.62.084017.
S. Mrowczynski, Density fluctuations in the quark - gluon plasma, Phys. Rev. C 57 (1998) 1518, https://doi.org/10.1103/PhysRevC.57.1518.
A. Kisiel, T. Taluc, W. Broniowski, and W. Florkowski, THERMINATOR: THERMal heavy-IoN generATOR, Comput. Phys. Commun. 174 (2006) 669, https://doi.org/10.1016/j.cpc.2005.11.010.
J. Kapusta, B. Muller, and M. Stephanov, Relativistic Theory of Hydrodynamic Fluctuations with Applications to Heavy Ion Collisions, Phys. Rev. C 85 (2012) 054906, https://doi.org/10.1103/PhysRevC.85.054906.
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