Lectures on Hydrodynamics Jean-Yve µν T µ ( x) 0 We can always diagonalize point by point by a Lorentz transformation T Λ µν x ( u ) µ and space-rotation R ( Ω) Then, in general, λ0 0 0 0 0 λx 0 0 Λ Λ 0 0 λy 0 0 0 0 λ z µν * T T x T R TR
Lectures on Hydrodynamics Jean-Yve If isotropic: T µν ( x) λ0 0 0 0 0 λ 0 0 0 0 λ 0 0 0 0 λ We assume local thermal equilibrium and use the thermodynamical relations 0 ns,, λ p( n s) λ ε ε and also the conservation of baryon number, ( nu µ ) 0 µ,,
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Courtesy: François
Equation of State of Hadronic Matter Preliminaries: Relativistic Ideal Gas Hadron Gas EoS and Resonances Excluded volume effect QGP Equation of State (Bag Model, Lattice) Maxwell Construction for the mixed phase Exercise!!
Volume V Average Number of Particles N Average Total Energy E
ZZ(β,µ,V): Partition function for Grand Canonical Ensemble µ β µβ ν µ β β N E V e E V Z,,, >, < E Z p E Z µ β β µ β µ β N E V e N V Z,,, > <, N Z p N Z
ln Z ln Boson i ( β ( εi µ ) e ), ln Z ln Fermion i ( β ( εi µ ) + e ), Average values of energy, number of particles and entropy i < E > ln Z β ( ε µ ) < β ν βµ ε i e + N > β µ ln Z β ( εi µ ) β i e + < S > β < E > µ < N > + ln i Z and in theromodynamical limit, P V lnz T
Relativistic Ideal Gas Free single particle energy state p {i} { }, i Plane wave 4 ε p c + m c gv 3 3 d p i π : Large V (thermodynamical) limit g: statistical factor < E > < N > gv 3 4 + π β p c m c µ gv d 3 d + 3 p e p 3 4 + π β p c m c µ e p c m c 4
Pressure: P P( β, µ ) g ( π ) 3 g ( π ) 3 3 T V π 3 gv ( π ) 3 d 3 p d p / dp dp p β ( ε p µ ) e p p v β ( ε µ ) e p 4 3 3 d Thermodynamical relation: n ( β, µ ) µ P β s ( β, µ ) β P ε βp( β, µ ) β, µ, ν βµ number density entropy density P + s µ n, β + ε ( ) β ε p µ ln e energy density
For bosons, always e β ( ε p µ ) > but for fermions, also possible e β ( ε µ ) p : degenerate gas Non degenerate case: e β ( ε µ ) p > ( β ( ε p µ ) ln e ) n n ( ) nβ ( ε p µ ) e, n P g m β π c n with Φ 3 0 n νn ( ) e ( Φ nβmc ), n z x + ( z) x dxe. ν βµ,
Integral representation of K-Bessel functions:. ) / ( 0 + + + Γ dx e x x z z K x z λ λ λ λ π. ) 3/ ( 0 + + Γ dx e x x z z K x z π Φ + 0 ) ( dx x e z x z Γ z K z z ) (3/ π. K z z Finally,, 3 n mc n z n n z K z n e c m g P β βµ β π, 3 n mc n z n n z K z e c m g n β βµ π. 3 n mc n z n n z K z dz d e mc c m g β βµ π ε
>>, µ ε β p e Boltzmann Limit:, 3 mc K e mc T c m g P β π βµ, 3 mc K mc T e c m g n β π βµ. 3 mc z z K z dz d e mc c m g β βµ π ε Note: PnT, and. ln ln 3 mc K mc T c m g n T β π µ
Contributions from Boltzmann-type expansion Boson Fermion
Ultra-relativistic limit : mc <<T, ex. quark gas In equilibrium mixture of particles + anti-particles ZZ particle Z anti-particle Z(β,µ) Ζ(β, µ) and P gt 3 β ( ε p µ ) d p ln e + ln π ( ) 3 [ ( )] β ε + µ e 4 gt F( a, b), mc µ 3 a, b, 6π ( c) T T p F a e b b 3 / ( a, b ) dx ( x a ) +. x e x +
nt (, µ ) g + 3 d p ( π ) exp p m µ / T θ Particles 3 + + exp p m µ / T θ Anti-parricles In general, µ0 implies n0.. (but see later)
For ultra-relativistic case, a<<,. 3, 0 3 a O e e x a x dx b a F b x b x + + +. 60 7 4 3 4 ), ( 4 4 π π + + a b b a b b a F Fermion F Boson (a,b)?
Degenerate Case (Fermions) : b>>. + + + + a b x b x e e a x dx b a F, 3 / + a b x e a x dx 3 / ), ( b a F [ ] 4 4 ln 3 5 8 ξ ξ ξ ξ + + b ξ π + b + + 3 4 3 60 7 ξ ξ π b. a ξ where
Hadronic Zoo
Mixture of different particles : index of particle species P E S a a P E S,,, P P T, µ ; m, m : mass of th particle
How to determine µ? Remember: Maximize S, fixing the total Baryon number, total charge, total strangeness etc. Answer: µ B µ B + S µ S + Q µ Q +... B : baryon number of the particle, µ B : chemical potential for the baryon number of the system S : strangeness quantum number of the particle, µ S : chemical potential for the strangeness of the system Q : electric charge of the particle, µ Q : chemical potential for the charge of the system
Resonance Gas The ideal gas of a set of h hadrons with m conserved number PT (, µ, t,.. t ) P( T, µ ) t h m { } 3 d p µ θ + θ 3 + µ P ( T, ) g T ln exp p m / T ( π ) θ + (Fermions) θ (Bosons) t ( t) µ Q µ () t Q
400 µ B 0 Hadron Gas Massless Pion Temperature MeV 300 00 00 0 0 4 6 8 Entropy Density fm -3
Hagedorn Temperature In early 960, R. Hagedorn proposed the possible existence of limiting temperature in hadronic gas.
Energy Density MeV/fm 3 0000 000 00 0 µ B 0 Hadron Gas Massless Pion Pressure MeV/fm 3 000 00 0 0. µ B 0 Hadron Gas Massless Pion 0. 0.0 0. 0 Entropy Density fm -3 0.0 0.0 0. 0 Entropy Density fm -3
Particle ratio in RHIC (, µ ) B, µ S, µ Q (, µ,, ) B µ S µ Q N n T + decay from resonances N n T + decay from resonances β
CERN
The above picture shows that when the baryon chemical potential is not zero (n B >0) then even for n S 0, µ S 0 ) How can we say so?? ) Why this happens??
Effect of Unstable Particles (resonance width)
Hadronic resonance width (particle data group) Γ (GeV) 0.5 0.4 0.3 0. CB0 With C With B 0. 0 0 4 8 M (GeV)
Can large width resonances be considered as usual particles in the EoS? How the temperature influences?
Presence of unstable particle (resonance) in themal equlibrium m m M, Γ m m m + m M
Virial Expansion 0 + N PT (, µ ) P( T, µ ) T b ( T) e βµ N N P0( T, µ ) ideal gas ( π) 4πi W 3 dp β P + W bn( T) dw e Tr 3 N AS S T dw W K ( βw ) TrN AS S π 4πi W W Connected ( N ) 0 Connected R.Dashen, S.Ma, H.Bernstein, Phys.Rev. 87 (969) 345
Second virial coefficient N S L (W) e iδ L (W ) Tr N AS W S C 4i L (L δl + ) W b (T) T π W0 dw W K ( βw) π L (L δl + ) W Relativistic Beth-Uhlenbeck formula
When the width is narrow (Γ 0) dδ L dw πδ (W M) TM (T) (L + ) K ( βm) π b P(T, µ ) βµ P0 (T, µ ) + T b(t) e P0 (T, µ ) + PR (T, µ R ) T M PR (T, µ ) gr K( βm)exp( βµ R ) Ideal Boltzmann gas! π µ R µ, gr L +
finite width / tan M W δ Γ Breit-Wigner: 4 / W M / dw d + Γ Γ δ βµ + Γ β π Γ µ 0 R W 3 R R R 4 / W M W) ( K W dw e 4 T g ) (T, P
Correction factor due to a finite width F Γ / π W K( βw) dw β W +Γ M K ( M) M W /4 0 T M βµ R PR (T, µ R ) gr K( βm)e F π nr nr( Γ 0) F correction to the number density Effective mass: [ βm (T)] M K ( βm) F M (T) K
Example of the integrant for F ρ Meson M 770 MeV, Γ 30 MeV Meson ρ T 50 MeV
Example of the integrant for F ρ Meson M 770 MeV, Γ 30 MeV T 50 MeV T 00 MeV
ρ Meson F vs. Temperature M vs. Temperature
f (70) Meson M 70 MeV, Γ 85 MeV T 50 MeV T 00 MeV
f (70) Meson F vs. Temperatura M vs. Temperatura
Remark on the effects of resonance width While Γ << T, resonances can be treated as a set of particles with mass distribution corresponding to the mass width. This effectively reduces the mass depending on the temperature and its abundance increase with T. In practice, replace the Breit-Wigner by a Gaussian to kill the effect of tail.
Effect of Excluded Volume The ideal hadronic gas under-estimate the interaction volume to fit the total multiplicity It has long been known that there exist a strong repulsion force between nucleons to get the saturation prpoerty of nuclei. Not easy (actually extremely difficult) to include the interaction among hadrons in a EoS However, it is an fundamental importance to include such effects for hydrodynamical applications.
Consider the Boltzmann gas Z GC ( T, µ, V ) e ( V V ) βe + βµ N N Z,, (, ) Can T N V Φ β m V N! N Φ β, m g π m 3 K β, m βm Z (,, ) (,, ) GC T µ V ZCan T N V e N! N 0 βµ N
We may introduce the effect of excluded volume effect by reducing the free space for particles as Z T, µ, V; v GC N 0 0 Z Can T, N, V Nv0 θ V v0n e N! 0 Not easy to deal with. Z (,, ) (,, ) IB T µ P dv ZGC T µ V e β PV βµ N Laplace transform (Legendre transformation for V-P) Isobaric Ensemble (pressure constant)
Z T P dv e m V v N V v N e IB PV N (, µ, ) ( β, ) θ N N! Φ 0 β βµ N 0 0 N N! Φ β, m e dv e V v N θ V v N βµ N β PV βp Φ β, m e β µ Pv0 0 N 0 0 β β µ (, ) P P T v P 0 0 P ( T, µ ) 0 : Expression of Grand Canonical Ensemble Pressure
IB (, µ, ) Z T P (, ) P P T v P β β µ 0 0 E V N Z IB β Z IB Z IB β Z µ IB Z IB, β Z P IB,,
N V + β vp T, µ vp 0 0 0 βp β P T, µ v P 0 0 β vp T, µ vp 0 0 0 βp β P T, µ v P 0 0,, n P T, µ v P / µ 0 0 + β vp 0 0 T, µ 0 vp n T, µ v P 0 0 + β vp 0 0 T, µ 0 vp
Thermodynamical limit V β vp T, µ vp 0 0 0 βp β P T, µ v P 0 0 Search for the root of Then calculate n ε n T, µ v P 0 0 + β vp T, µ vp 0 0 0 ( T, v P) ε µ 0 0 + β vp 0 0 T, µ 0 vp βp β P T, µ v P 0 0 0
QGP EoS Can we go over the Hagedorn temperature? Answer: Yes. Existence of phase transition.
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Presence of boundary Courtesy of François
Vacuum pressure B
P P + P B QGP quarks Gluons ε ε ε + + QGP quarks Gluons B So that P + + ε P ε QGP quarks+ gluons dε Tds+ µ dn QGP dp ndµ + sdt QGP If m q m g 0, 7 π hc 4 90 4 PQGP g 3 F + gg T B
800 n B 0 Hadron Gas QGP Bag Pressure MeV/fm 3 400 0 Phase transition -400 0 50 00 50 00 50 T MeV
Maxwell construction of mixed phase (first order phase transition) T P QGP QGP T P hadron hadron µ µ QGP hadron
Phase boundary and Adiabatic path (s/n B const) 300 8. 64. 3. Temperature 00 00 5. 6. 8. 4... 0 0 400 800 00 600 000 µ B
Bag Model Courtesy of François