Extensive and Intensive Quantities in HEC
Abdel Nasser Tawfik
Egyptian Center for Theoretical Physics (ECTP),
World Laboratory for Cosmology And Particle Physics (WLCAPP),
Modern University for Technology and Information (MTI Univ.)
http://atawfik.net/; http://wlcapp.net/
THREE DAYS OF CRITICAL BEHAVIOUR IN HOT AND DENSE QCD, June 14, 2013
Agenda
• Thermodynamics/Statistics in HEC • Motivation: Particle Ratios and Freeze-Out Parameters • Intensive/Extensive Quantities in HEC • Produced Particles in HEC • Elliptic Flow in HEC • Net-Charge Fluctuations in HEC • Energy Density (Bjorken) in HEC • Conclusions and Outlook
Terminology “intensive/extensive” was introduced by Richard C. Tolman (1881 –1948), in 1927
Thermodynamics & Statistics
• Thermodynamics is • a "theory of principle“ (entropy, arrow of time, etc.) • the study of macroscopic behavior of physical systems
under exchange of work and heat with surroundings • based on thermodynamic equilibrium; no macroscopic
properties change with time
• Statistical mechanics • explains the 2nd Law as an effect of microscopic particles
that make up the universe (system of interest).
Statistical Physics & HEC: History
(1901-1954)
Heinz Koppe
Die Mesonenausbeute beim Beschuß von leichten Kernen mit α-Teilchen, Zeitschrift Naturforschung 3, 251 1948
High Energy Nuclear Events, PTP 5, 570 1950
Statistical Thermodynamics of Strong Interactions at High Energies, Nuovo Cimento Suppl. 3, 147-186, 1965
Computing HE Collisions of proton with multiple productier of particles by statistical wieghts of the various possibilities
Systematical analysis of HE phenomenna using all tools of statistical physics
Statistical Estimation of Meson Yield from the Bombardment of Light Nuclei with α-Particles
Enrico Fermi Rolf Hagedorn
(1919–2003)
Early days of statistical hadron production theory: Canonical statistical mechanics method used to treat small hadron abundances First relativistic p-p collision was conducted.
1955:
1975:
Motivation: Early History
V.B. Magalinskii and Ia.P. Terletskii, Sov. Physics. JETP 2, 143, 1955. E.V. Shuryak, Sov.J.Nuc.Phys 20, 295, 1975.
Johann Rafelski and Jean Letessier, J.Phys. G28 (2002) 1819-1832
X1+X2+…=X Extensive
X1=X2=…=X Intensive
Intensive & Extensive Quantities
Intensive & Extensive Quantities
Intensive (Bulk) properties do not depend on system size or amount of existing material. Therefore, it is scale invariant The same for all subsystems
Extensive properties are additive for independent and non-interacting subsystems. They are directly proportional to the amount of existing material. Sum of subsystems’ properties
Extensive properties are counterparts of intensive properties.
There are measured physical properties which are neither intensive nor extensive, e.g. electric resistance, invariant mass and special relativity.
Ratio of two extensive properties that scale in the same way is scale-invariant, and hence an intensive property.
Intensive properties: • chemical potential • temperature, critical temperature • density • viscosity, concentration • specific volume, energy, heat capacity, • pressure, elasticity • velocity, Acceleration …, etc. Only two independent intensive variables are needed to fully specify the entire state of a system. Other intensive properties can be derived from the two known values.
Extensive properties: • amount of critical heat, • energy • entropy • particle number • mass • momentum • volume • electrical charge …, etc. The value of an additive property is proportional to system size, or to its quantity of matter.
Intensive & Extensive Quantities
In(Ex)clusive Quantities in HEC
The charge distribution is inclusive, while isotropically resolved particle observation is exclusive
“the assignment of some properties as intensive or extensive may depend on the way in which subsystems are arranged”: Otto Redlich (1896 - 1978)
Measurements of Inclusive and exclusive HEC • Particle, • jet production, • Particle decays, • Cross sections • Hadronic process, • Diffraction, (Compton) processes, • etc.
Exclusive implies that E and p, for instance, of all the products are measured. Inclusive means that some quantities of the products are left unmeasured.
Non-Extensive Statistical Mechanics
Constantino Tsallis proposed non-extensive entropy as generalization of Boltzmann-Gibbs entropy.
Through mass spectrum for bosons/fermions or density of their states
A statistical analogy was proposed by Koppe (1948) and Fermi (1950)
BG-entropy makes systems having strong dependence on initial conditions.
Non-extensive statistical mechanics are power laws (no longer exponentials).
J. Statistical Phys. 52, 479–487 (1988)
Jean Cleymans’ Talk
Self-consistency is related to non-extensivity.
Rolf Hagedorn used it to explain the thermodynamics of fireballs in HE physics collisions.
Motivation: particle production
Nucl.Phys. A859 (2011) 63-72, Int.J.Theor.Phys. 51 (2012) 1396-1407
Motivation: particle production
Canonical vs. Grand-Canonical Ensemble
𝑍 𝑁, 𝑇, 𝑉 𝑍 , 𝑇, 𝑉
𝑍 𝑁, 𝑇, 𝑉 = 𝑇𝑟𝑁 𝑒− 𝐻 𝑍 , 𝑇, 𝑉 = 𝑇𝑟𝑁 𝑒
−(𝐻−𝑁)
𝑍 , 𝑇, 𝑉 = 𝑍 𝑁, 𝑇, 𝑉 𝑒− 𝑁
𝑍 𝑁, 𝑇, 𝑉 =1
2 𝑑 𝑍 𝑖𝑇, 𝑇, 𝑉 𝑒−𝑖𝑁2 𝑝𝑖
0
Extensive Intensive
𝑃 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 =𝑍 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒, 𝑇, 𝑉
𝑍 , 𝑇, 𝑉𝑒− 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒
Dirac Delta Function and is Wick rotated
N is not constant N is constant
Motivation: chemical freeze-out
1306.1025 [hep-ph]
Nucl.Phys. A764 (2006) 387-392, Europhys.Lett. 75 (2006) 420
Chemical freeze-out: Centrality
dependence STAR-ESP
Lokesh Kumar [STAR Collaboration], Central Eur.J.Phys. 10 (2012) 1274-1277
Orpheus I. Mall [STAR Collaboration] Acta Phys.Polon.Supp. 5 (2012) 491-496
Motivation: chemical freeze-out
Freeze-out diagram (2CFO): empty circles (squares): strange (non-strange). Points with equal energies are connected by lines. Filled square (circle): QCD crossover at vanishing chemical potential (critical endpoint).
Strange freeze-out
1306.2006 [nucl-th]
2CFO: All strange particles and decouple together at one time, and all other non-strange particles together at another time.
Helmut Satz, Int.J.Mod.Phys. E21 (2012) 1230006
Motivation: hadronization temperature
Hadronization T calculated in HRG for different initial collisions configuration and energies
Freeze-out: In(Ex)tensive
Universal Description of Freeze-Out: (T-) (intensive-intensive) • <E>/<n> 1GeV Extensive/Extensive Intensive • nb+nab 0.14fm3 Extensive/Extensive Intensive • s/T3 = 7 Extensive/Intensive
Normalized Extensive or Intensive • Vanishing 2 Intensive
To understand the latter, we recall that susceptibility (2nd order moments or 2) is the derivative of extensive n wrt nonconjugate variable fluctuations
Motivation: Produced particles (Extensive)
NSD=Non-Single-Diffractive INEL=INELastic (INEL=NSD+SD)
1304.2969 [nucl-ex]: Comparison of dNch/dη per participating nucleon at midrapidity in central heavy-ion collisions to corresponding results from p+p(¯p) and p(d)+A collisions
Motivation: normalized produced particles
(Extensive as well)
Elliptic Flow: Center-of-mass energy
1304.2969 [nucl-ex]: transverse momentum integrated v2 close to midrapidity for z=1 particles from central collisions (20-30%)
Elliptic Flow: Center-of-mass energy
(Extensive?)
Elliptic Flow: charged produced particles
Elliptic Flow: transverse momentum
Elliptic Flow: Preliminary pPb at 5.02 TeV
[ALICE Collaboration] Phys. Rev. Lett. 110, 152301 (2013)
Net-Charge Fluctuations
Net-Charge Fluctuations
1304.2969 [nucl-ex]:
Net-Charge Fluctuations
1304.2969 [nucl-ex]:
Energy Density: Bjorken
is formation time
Energy density: Bjorken
Thomas A. Trainor, arXiv:1303.4774 [hep-ph]
arXiv:13045.0387
Intensive Extensive
Intensive Extensive
Conclusions and Outlook
• Systematic Estimation for Intensive and Extensive
Measured Quantities, especially for collective flow, fluctuations and correlations.
• Thermodynamical Properties in Canonical and Grand-Canonical Ensemble in Thermal in Dense Medium.
• Dependences on System-Size, Energies, etc.
Thanks for your Attention!
