Spinodal Instabilities in Phase Transitions

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1 Helmholtz International Summer School Dense Matter in Heavy Ion Collisions and Astrophysics JINR, Dubna, Russia, August 28 September 8, 2012 Spinodal Instabilities in Phase Transitions Jørgen Randrup (LBNL) Lecture I: Phase coexistence (equilibrium) TUE 11:30-12:30 Lecture II: Phase separation (non-equilibrium) THU 10:00-11:00 Seminar: Problem Solving, Discussion FRI 17:00-18:00 Lecture III: Nuclear collisions (fresh results) SAT 11:30-12:30

2 Lecture I: Phase coexistence (equilibrium) Thermodynamics: statistical mechanics of large uniform systems Non-uniform systems: gradient effects, interface tension

Basic thermodynamics 1 2 X 1 = {E 1, N 1,

2, } => S 2 (X 2 ) The combined system is

maximum - which requires δs = 0 and δ 2 S

+E 2 + N = N 1 +N 2 + V = V 1 +V 2 + S =

3 Basic thermodynamics 1 2 X 1 = {E 1, N 1, V 1, } => S 1 (X 1 ) X 2 = {E 2, N 2, V 2, } => S 2 (X 2 ) The combined system is in equilibrium provided S has a local maximum - which requires δs = 0 and δ 2 S < 0: X = {E, N, V, } = X 1 +X 2 + E = E 1 +E 2 + N = N 1 +N 2 + V = V 1 +V 2 + S = S 1 +S 2 + δs: => T 1 = T 2 = µ 1 = µ 2 = p 1 = p 2 = δ 2 S: => The entropy curvature matrices have only negative eigenvalues

4 Statistical equilibrium in bulk matter Thermodynamics with no conserved charge Control parameter(s) {X}: Energy E = Vε Volume V ε = E/V E, V Entropy function S{X}: S(E,V) = Vσ(ε) Derivative(s) λ X = X S: β = 1/T = E S(E,V) = ε σ(ε) π = p/t = V S(E,V) = σ βε temperature pressure Thermodynamic coexistence: => T 1 = T 2 & p 1 = p Thermodynamic (local) stability: δ 2 S tot < 0 => Entropy curvature ε2 σ must be negative σ concave ε

5 Single-phase system Entropy density: ε σ(ε) > 0 ε2 σ(ε) < 0 σ concave ε Temperature: β(ε) = ε σ(ε) > 0 β T ε β(ε) = ε2 σ(ε) < 0 ε2 β(ε) > 0 ε ε Pressure: π(ε) = σ(ε) β(ε)ε = β(ε) p(ε) ε π(ε) = β β ε ε β = -ε ε2 σ(ε) > 0 ε π(ε) = ε βp = β ε p +p ε β β(ε) ε p(ε) = [ε+p(ε)] ε β(ε) = h(ε) ε β(ε) > 0 p ε

6 First order <=> Phase coexistence <=> Spinodal instability Extensive variable X Entropy function S(X) occur when S(X) is locally convex: X=E: ε σ(ε) = β: β 1 = β 2 π = σ βε: π 1 = π 2 Intensive variable: Maxwell construction: X=E => λ=β

7 Phase transformation with no conserved charge Equation of State concave convex: σ εε =0 β=σ ε : β ε =0 σ εε =0 βp ε = hσ εε : p ε =0 σ εε =0

8 Statistical equilibrium in bulk matter Thermodynamics with one conserved charge Control parameter(s) {X}: Energy E = Vε Charge N = Vρ Volume V ε = E/V ρ = N/V E, N, V Entropy function S{X}: S(E,N,V) = Vσ(ε,ρ) β = 1/T = E S(E,N,V) = ε σ(ε,ρ) Derivative(s) λ X = X S: α = -µ/t = N S(E,N,V) = ρ σ(ε,ρ) π = p/t = V S(E,N,V) = σ βε αρ Thermodynamic coexistence: δs tot = 0 => T 1 = T 2 & µ 1 = µ 2 & p 1 = p Thermodynamic (local) stability: δ 2 S tot < 0 => Curvature matrix { χ χ σ(ε,ρ) } has only negative eigenvalues: ε2 σ ρ ε σ ε ρ σ ρ2 σ

9 Microcanonical scenario: E and N are specified: entropy density temperature chemical potential pressure enthalpy density Canonical scenario: <E> and N are specified: Same: Then replace S by S = S βe and require δs =0 & δ 2 S <0 - or consider F = -TS = E-TS and require δf=0 & δ 2 F>0 free energy density f T (ρ) 1 2 Phase coexistence f T (ρ) has common tangent!

10 Example: Nuclear matter Equation of state: p T (ρ) unstable meta stable stable (ρ,t) phase diagram The spinodal boundary occurs at T p T (ρ) = 0 => v 2 T ρ h ρp T (ρ) =0

11 Nuclear phase diagram in different representations ε ρ µ T

12 Isentropic changes Entropy density: σ(ε,ρ) Energy density: ε Net baryon density: ρ Temperature: T(ε,ρ) = 1/σ ε Chemical potential: µ(ε,ρ) = Τσ ρ Pressure: p(ε,ρ) = Tσ ε + µρ Enthalpy density: h(ε,ρ) = p + ε Entropy per (net) baryon: s(ε,ρ) = σ/ρ Changes: (δε,δρ) => δs : ρ 2 Tδs = ρ 2 Tδ(σ/ρ) = ρtδσ Tσδρ = ρδε µρδρ [h µρ]δρ = ρδε hδρ δs = 0 => ρδε = hδρ <=> δε/δρ = h/ρ

13 Isentropic phase trajectories in different representations δε/δρ = h/ρ ε ρ µ T

14 Example: Nambu Jona-Lasino model C. Sasaki, B. Friman, K. Redlich, Phys. Rev. Lett. 99, (2007): Density fluctuations in the presence of spinodal instabilities v T =0 v s =0

15 Canonical description: T specified Free energy density Phase coexistence common tangent: f T (ρ) common tangent = Maxwell line

16 uniform matter common tangent = Maxwell line JRandrup: Dubna School, 2010

17 Liquid-gas phase coexistence uniform gas phase uniform liquid phase droplets can coexist in mutual equilibrium phase mixture

18 common tangent = Maxwell line uniform matter JRandrup: Dubna School, 2010

Thermodynamics of non-uniform matter Consider two coexisting phases of bulk matter: Same temperature T 0 Same chemical potential µ 0 Same pressure p 0 Place them in contact with a planar interface: T

19 Thermodynamics of non-uniform matter Consider two coexisting phases of bulk matter: Same temperature T 0 Same chemical potential µ 0 Same pressure p 0 Place them in contact with a planar interface: T 0 µ 0 p 0 T 0 µ 0 p 0 ρ 1 ε 1 σ 1 ρ 2 ε 2 σ 2 PHASE 1 PHASE 2 A diffuse interface will then develop: The density profiles change smoothly through the interface region from one coexistence value to the other coexistence value: Matter density ρ(x) ρ 1 ρ 2 Depth x Energy density ε(x) ε 1 ε 2 Depth x

20 Thermodynamics of non-uniform matter: microcanonical Non-uniform charge density Non-uniform energy density Non-uniform entropy density Constant temperature: Constant chemical potential: Constant pressure: =>

21 Thermodynamics of non-uniform matter: canonical Constant temperature T Non-uniform charge density Non-uniform free-energy density Constant chemical potential: Constant pressure: => J. Randrup, Phys. Rev. C 79, (2009) H. Heiselberg et al., Phys. Rev. Lett. 70, 1355 (1993)

22 Finite range: gradient term Free energy density for uniform matter at temperature T: But we need to treat non-uniform systems: Local density approximation: implies: F T ( ) = F T ( ) No good! => Finite range must be taken into account =>

23 Interface profile => Matter density ρ(x) ρ 1 ρ 2 Depth x -Δf T (ρ) ρ 1 ρ 2 potential energy conservation

24 Interface tension Density profile ρ(x) ρ 1 ρ 2 Depth x The diffuse interface adds free energy (relative to a sharp interface); The additional free energy (per unit area) is the interface tension: 2 1 -Δf T (ρ) ρ 1 ρ 2 The interface profile is not needed, only the EoS for uniform matter! dx = dρ/ x ρ

25 Two recent examples: Marcus B. Pinto, V. Koch, and JR: Phys. Rev. C 86, (2012): Surface tension of quark matter in a geometric approach Linear σ model: σ Surface profiles ρ Nambu-Jona-Lasino model: Free energy density Excess free energy Δf Interface tension

26 Interface tension between hadron gas and QGP * Lecture II Construct* Equation of State (spline between HG & QGP) Phase diagram Add gradient term in free energy density: Interface tension JR: PRC 82 (2010) Jan Steinheimer & JR (2012)

Spinodal Instabili.es at the Deconﬁnement Phase Transi.on

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