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
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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
Heinz Koppe
Die Mesonenausbeute beim
Beschuß von leichten Kernen
mit α-Teilchen,
Zeitschrift Naturforschung 3,
251 1948
Statistical Estimation of
Meson Yield from the
Bombardment of Light
Nuclei with α-Particles
Enrico Fermi
Rolf Hagedorn
(1901-1954)
(1919–2003)
High Energy Nuclear Events,
PTP 5, 570 1950
Computing HE Collisions
of proton with multiple
productier of particles by
statistical wieghts of the
various possibilities
Statistical Thermodynamics of
Strong Interactions at High
Energies,
Nuovo Cimento Suppl. 3, 147186, 1965
Systematical analysis of HE
phenomenna using all
tools of statistical physics
Motivation: Early History
1955:
1975:
Early days of statistical hadron
production theory:
Canonical statistical mechanics method
used to treat small hadron abundances
First relativistic p-p collision was
conducted.
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
Intensive & Extensive Quantities
X1=X2=…=X
X1+X2+…=X
Intensive
Extensive
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.
Ratio of two extensive properties that scale in the same way is
scale-invariant, and hence an intensive property.
There are measured physical properties which are neither intensive
nor extensive, e.g. electric resistance, invariant mass and special
relativity.
Intensive & Extensive Quantities
Intensive properties:
Extensive properties:
• chemical potential
• temperature, critical temperature
• density
• viscosity, concentration
• specific volume, energy, heat capacity,
• pressure, elasticity
• velocity, Acceleration
…, etc.
• amount of critical heat,
• energy
• entropy
• particle number
• mass
• momentum
• volume
• electrical charge
…, 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.
The value of an additive property is
proportional to system size, or to its
quantity of matter.
In(Ex)clusive Quantities in HEC
“the assignment of some properties as intensive or extensive may depend on
the way in which subsystems are arranged”: Otto Redlich (1896 - 1978)
The charge distribution is inclusive,
while isotropically resolved particle observation is exclusive
Measurements of
Inclusive and exclusive HEC
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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.
J. Statistical Phys. 52, 479–487 (1988)
BG-entropy makes systems having strong dependence on initial conditions.
Non-extensive statistical mechanics are power laws (no longer exponentials).
Jean Cleymans’ Talk
Self-consistency is related to non-extensivity.
Rolf Hagedorn used it to explain the thermodynamics
of fireballs in HE physics collisions.
Through mass spectrum for bosons/fermions or density of their states
A statistical analogy was proposed by Koppe (1948) and Fermi (1950)
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
Intensive
N is constant
Extensive
N is not constant
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 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒, 𝑇, 𝑉 −  𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒
𝑃 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 =
𝑒
𝑍 , 𝑇, 𝑉
1
𝑍 𝑁, 𝑇, 𝑉 =
2
2 𝑝𝑖
𝑑 𝑍 𝑖𝑇, 𝑇, 𝑉
0
𝑒

−𝑖𝑁
Dirac Delta Function and  is Wick rotated
Motivation: chemical freeze-out
Nucl.Phys. A764 (2006) 387-392,
Europhys.Lett. 75 (2006) 420
1306.1025 [hep-ph]
Chemical freeze-out: Centrality
dependence STAR-ESP
Lokesh Kumar [STAR Collaboration], Central Eur.J.Phys. 10 (2012) 1274-1277
Motivation: chemical freeze-out
Orpheus I. Mall [STAR Collaboration] Acta Phys.Polon.Supp. 5 (2012) 491-496
Strange freeze-out
1306.2006 [nucl-th]
2CFO:
All strange particles
and  decouple
together at one time,
and all other nonstrange particles
together at another
time.
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).
Motivation: hadronization temperature
Hadronization T calculated in HRG for different initial collisions configuration and energies
Helmut Satz, Int.J.Mod.Phys. E21 (2012) 1230006
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
Intensive
• Vanishing  2
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)
Motivation: normalized produced particles
(Extensive as well)
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
Elliptic Flow: Center-of-mass energy
Elliptic Flow: Center-of-mass energy
(Extensive?)
1304.2969 [nucl-ex]: transverse momentum integrated v2 close to midrapidity for
z=1 particles from central collisions (20-30%)
Elliptic Flow: charged produced particles
Elliptic Flow: transverse momentum
Elliptic Flow: Preliminary pPb at 5.02 TeV
Net-Charge Fluctuations
[ALICE Collaboration] Phys. Rev. Lett. 110, 152301 (2013)
Net-Charge Fluctuations
1304.2969 [nucl-ex]:
Energy Density: Bjorken
1304.2969 [nucl-ex]:
 is formation time
Energy density: Bjorken
Thomas A. Trainor, arXiv:1303.4774 [hep-ph]
Intensive  Extensive
arXiv:13045.0387
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 GrandCanonical Ensemble in Thermal in Dense Medium.
• Dependences on System-Size, Energies, etc.
Thanks for your Attention!