Spinodal Instabili.es at the Deconfinement Phase Transi.on

Transcription

1 Spinodal Instabili.es at the Deconfinement Phase Transi.on Jørgen Randrup (Berkeley Lecture I: Phase coexistence (equilibrium WED 1:- 13: Lecture II: Phase separa.on (non- equilibrium THU 16:3-17:3 Discussion THU 18:- 19: Lecture III: Effects on collision dynamics (clumping SAT 11:- 1: Jørgen Randrup Dubna: 4 July 15 1

2 Basic thermodynamics 1 X 1 = {E 1, N 1, V 1, } => S 1 (X 1 X = {E, N, V, } => S (X The combined system is in equilibrium provided S has a local maximum - which requires δs = and δ S < : X = {E, N, V, } = X 1 +X + E = E 1 +E + N = N 1 +N + V = V 1 +V + S = S 1 +S + δs: δx l δ S: = i. = δs = i δx l i. = > δ S = i δs i = i δ S i ( l δe = i δn = i δv = i = i S i X l i ( l 1 l δx l i δe i. = δn i. = δv i. = = l S i X l 1 i X l i ( λ l iδxi l i => δx l 1 i δx l i λ l 1. = λ l λ l i S i X l i. =... λ E i λ N i λ V i = S i E i = β i = 1 T i = S i N i = α i = µ i T i = S i V i = π i = p i T i T 1 = T = µ 1 = µ = p 1 = p = S i => The entropy curvature matrices have only negative eigenvalues X l 1 i X l i Jørgen Randrup Dubna: 4 July 15

3 Microcanonical thermodynamics: E and N are specified: entropy density temperature chemical poten.al ε σ ρ ε σ ε ρ σ ρ σ < stability pressure enthalpy density Canonical thermodynamics: <E> and N are specified: Same: Then replace S by S = S βe and require δs = & δ S < - or consider F = -TS = E-TS and require δf= & δ F> free energy density ρ f T (ρ > stability µ T (ρ = ρ f T (ρ σ T (ρ = T f T (ρ Phase coexistence ó f T (ρ has common tangent Free energy density 1 1 f T (ρ Density Jørgen Randrup Dubna: 4 July 15 3

4 Thermodynamics of non- uniform maper: microcanonical Non-uniform charge density Non-uniform energy density Non-uniform entropy density ρ(r ε(r σ[ ρ(r, ε(r](r δs = [ β(rδ ε(r+ α(rδ ρ(r]dr δ ε(r, δ ρ(r : =. δs β δe α δn = N = ρ(rdr E = ε(rdr S = σ(rdr T (r =1/ β(r µ(r = α(r T (r [( β(r β δ ε(r+( α(r α δ ρ(r]dr Constant temperature: Constant chemical potential: Constant pressure: r :. β(r = β β =. r : α(r =. α α =. δπ = εδβ ρδα π = ε β ρ α => π p/t = σ βε αρ p(r =p Gradient corrections: p(r p (ε(r, ρ(r Cρ ρ(r Jørgen Randrup Dubna: 4 July 15 4

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6 Applica.on to rela.vis.c nuclear collisions Equations of state Two-phase EoS => Interface tension One-phase EoS (Maxwell partner Fluid dynamics Gradient term => Spinodal amplification Collisions Density moments: Enhancement Jørgen Randrup Dubna: 4 July 15 6

7 Hadron Gas versus Quark-Gluon Plasma p H = p π + p N + p N + p w Free pions, nucleons, and antinucleons: p π (T = g π T p N (T = g N T p N (T = g N T + compressional energy density: [ ( α ( ] β ρ ρ w(ρ = A + B ρ ρ s m π m π m π p w (ρ =ρ ρ (w(ρ/ρ pεdε π ln[1 e βε ] pεdε π ln[1 + e β(ε µ ] pεdε π ln[1 + e β(ε+µ ] ρ s p Q = p g + p q + p q B Free gluons, quarks, and antiquarks: Temperature T (MeV π p g = g g 9 T 4 [ 7π p q + p q = g q 36 T µ qt + 1 ] 4π µ4 q Phase crossing: QGP Hadron gas µ = µ + ρ w =3µ q Compression!/! s Jørgen Randrup Dubna: 4 July 15 7

8 Equation of state: Spline between HG and QGP QGP Free energy density f T (ρ spline ρ f T (ρ > match f,, f, f 16 HG ρ f T (ρ > match f,, f, f Temperature T (MeV QGP Hadron gas Compression!/! s Density ρ Jørgen Randrup Dubna: 4 July 15 8

9 Isothermal speed of sound Microcanonical representaqon: entropy density σ(ε, ρ v T = ρ h ( p ρ β = ε σ(ε, ρ =σ ε =1/T α = ρ σ(ε, ρ =σ ρ = µ/t = ρ ρt [σ εε σ ρρ σ h σ ερ] εε T π = σ βε αρ = p/t What happens at T=? Canonical representaqon: free- energy density µ T (ρ = ρ f T (ρ f = ε T σ σ T (ρ = T f T (ρ p T (ρ =ρ ρ f T (ρ f T (ρ ε T (ρ =f T (ρ T T f T (ρ Use ρ p T (ρ = ρ f T (ρ + ρ ρ f T (ρ ρ f T (ρ = ρ ρ µ T (ρ to get Jørgen Randrup Dubna: 4 July 15 9

10 Example: Free gluons, quarks, and antiquarks Gluons: (g g = 16 Quarks: (g q = 1 = ⅓ for T = Jørgen Randrup Dubna: 4 July 15 1

11 Two-Phase Equation of State Free energy density f T (ρ T < T c ρ f T (ρ > QGP Maxwell line (tangent ρ f T (ρ < critical point 15 Coexistence v T (!,T = HG ρ f T (ρ > Temperature T (MeV 1 5 v s (!,T = 1 A B Compression!/! s 1 A B Density ρ Jørgen Randrup Dubna: 4 July 15 11

12 Phase diagrams Temperature T (MeV c.p. (ρ,t Coexistence v T (!,T = v s (!,T = Compression!/! s Energy density " (in units of " s Coexistence boundary Isothermal spinodal Isentropic spinodal c.p. (ρ,ε compressional energy " T= (! forbidden region Net baryon density! (in units of! s The spline construction yields f T (ρ Fluid dynamics needs p(ε,ρ First tabulate p T (ρ and ε T (ρ, then get p(ρ,ε by interpolation p T (ρ =ρ ρ f T (ρ f T (ρ ε T (ρ =f T (ρ T T f T (ρ Jørgen Randrup Dubna: 4 July 15 1

13 Finite-range (ideal fluid dynamics Gradient term in free energy density: γ T = ρ ρ 1 { C[fT (ρ f M T (ρ] } 1/ dρ => gradient term in the pressure: p(r =p ( ρ(r, ε(r C ρ(r ρ(r Laplace operator Interface tension! T (MeV/fm Interface tension γ(t a =.1 fm a =.8 fm a =.1 fm Temperature T (MeV Spinodal dispersion relation ρ(r, t =ρ + δρ(x, t. = ρ + ρ k e ikx iωt p k p k + Cρ k ρ k γk = vs k C ρ k 4 ε + p - easy to insert into a fluid dynamic transport code maximum! Jørgen Randrup Dubna: 4 July 15 13

14 Unstable matter in a box Energy density " (in units of " s Interface tension! T (MeV/fm Net baryon density! (in units of! s Phase diagram in (ρ,ε plane Coexistence boundary Isothermal spinodal Isentropic spinodal c.p. compressional energy " T= (! forbidden region Interface tension a =.1 fm a =.8 fm a =.1 fm Temperature T (MeV C = ε s ρ s Amplification of irregularities? => p(r =p ( ρ(r, ε(r C ρ(r ρ(r 4 fm 4 fm J. Steinheimer & JR, PRL 19, 131 (1 J. Steinheimer & JR, PRC 87, 5493 (13 a Jørgen Randrup Dubna: 4 July 15 14

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16 Unstable matter in a box Energy density " (in units of " s Phase diagram in (ρ,ε plane Coexistence boundary Isothermal spinodal Isentropic spinodal c.p. compressional energy " T= (! forbidden region Net baryon density! (in units of! s => p(r =p ( ρ(r, ε(r C ρ(r ρ(r 4 fm 4 fm C = ε s ρ s a Interface tension! T (MeV/fm Interface tension a =.1 fm a =.8 fm a =.1 fm Temperature T (MeV Growth rate! k (c/fm Spinodal growth rate! = v s k a = a =.1 a =.8 a = Wave number k (fm -1 numerical viscosity Jørgen Randrup Dubna: 4 July 15 16

17 Maximum compression in Pb+Pb collisions Pb E lab b = Pb Energy density " (in units of " s Coexistence boundary Isothermal spinodal Isentropic spinodal c.p compressional energy " T= (! forbidden region 5 events E lab = 5 A GeV E lab = A GeV Net baryon density! (in units of! s => Optimal energy range: -4 A GeV? Jørgen Randrup Dubna: 4 July 15 17

18 Evolu.on of the (net baryon density ρ(r,t E lab = 3 A GeV Pb E lab Pb b = Total baryon number: A = ρ(r d 3 r ρ N 1 N th density moment: A N ρ(r N ρ(r d 3 r Mean baryon density: ρ 1 ρ(r ρ(r d 3 r = ρ N=1 A Jørgen Randrup Dubna: 4 July 15 18

19 One-Phase Equation of State QGP Free energy density f T (ρ ρ f T (ρ > ρ f T (ρ < HG ρ f T (ρ > Construct a corresponding one-phase Equation of State by replacing each concave section by the Maxwell line Maxwell line: common tangent ρ f T (ρ = 1 Density ρ Jørgen Randrup Dubna: 4 July 15 19

20 Two-phase EoS: unstable 3 A GeV Pb + Pb (b= Identical initial conditions Maxwell EoS: stable Density ρ(x,y,z= Difference: unstable - stable Jørgen Randrup Dubna: 4 July 15

21 Evolu.on of the density moments Density moment: ρ N 1 A N ρ(r N ρ(r d 3 r A = ρ(r d 3 r Normalized moment: ρ N / ρ N (dimensionless Density moments <! N >/<!> N N = 7 N = 6 N = 5 N = 4 N = 3 N = E lab = 3 A GeV Elapsed time t - t (fm/c Maximum enhancement of <! N >/<!> N N = 7 N = 5 std EoS Maxw EoS std EoS Maxw EoS Beam energy E lab (A GeV Jørgen Randrup Dubna: 4 July 15 1

22 Spinodal Instabili.es at the Deconfinement Phase Transi.on SUMMARY I: Phase coexistence Thermodynamics of non- uniform systems 4 Gradient correcqon Interface tension! T (MeV/fm 3 1 a =.1 fm a =.8 fm a =.1 fm Interface tension between coexisqng phases Temperature T (MeV Energy density " (in units of " s Growth rate! k (c/fm Coexistence boundary Isothermal spinodal Isentropic spinodal c.p Net baryon density! (in units of! s! = v s k a = a =.1 a =.8 a =.1 compressional energy " T= (! forbidden region Wave number k (fm -1 p(r =p ( ρ(r, ε(r C ρ(r ρ(r Maximum enhancement of <! N >/<!> N N = 7 N = 5 II: Phase separaqon III: Collision dynamics std EoS Maxw EoS Two- phase equaqon of state Growth rates of spinodal instabiliqes Needed: theoreqcal equaqon of state Needed: finite- range dissipaqve fluid dynamics, Finite- range fluid dynamics several flavors Collisions: density enhancements, opqmal energy Big queson: Is the presence of a phase transiqon experimentally visible? Beam energy E lab (A GeV std EoS Maxw EoS Jørgen Randrup Dubna: 4 July 15