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Thermal Properties of Dense Matter The Homogeneous Phase C. - - PowerPoint PPT Presentation

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Thermal Properties of Dense Matter The Homogeneous Phase C. Constantinou IKP, FZ J ulich 17 August 2015, Stockholm Workshop on Microphysics In Computational Relativistic Astrophysics Collaborators: M. Prakash, B. Muccioli & J.M.

slide-1
SLIDE 1

Thermal Properties of Dense Matter

The Homogeneous Phase

  • C. Constantinou

IKP, FZ J¨ ulich 17 August 2015, Stockholm Workshop on Microphysics In Computational Relativistic Astrophysics

Collaborators: M. Prakash, B. Muccioli & J.M. Lattimer

  • C. Constantinou

Thermal Properties of Dense Matter

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SLIDE 2

The Models

  • 0.4
  • 0.2

0.2 0.2 0.4 0.6 0.8

  • 0.4
  • 0.2
0.2

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 Dirac/3 Landau 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8

MDI(A)

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8

SkO'

0.2 0.4 0.6 0.8

Dirac M*/m Landau MFT dlog(m*)/dlog(n) m*/m n (fm-3) SNM PNM ◮ For n large, dm∗ dn ≃ 0 ◮ m∗ = m 1+β(x)n ◮ m∗ = E ∗ F = (p2 F + M∗2)1/2 ◮ Minimum at n s.t. pF M∗ + dM∗ dpF = 0

  • C. Constantinou

Thermal Properties of Dense Matter

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SLIDE 3

Degenerate Limit Thermodynamics

◮ Interaction switched-on adiabatically ◮ Entropy density and number density maintain their free

Fermi-gas forms: s = 1 V

  • p

[fp ln fp + (1 − fp) ln(1 − fp)] n = 1 V

  • p

fp(T)

◮

dε δs

δT

⇒ s = 2anT a = π2 2 m∗ p2

F

level density parameter

  • C. Constantinou

Thermal Properties of Dense Matter

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SLIDE 4

Degenerate Limit Thermodynamics

Rest of thermodynamics via Maxwell’s relations or other identities:

◮ Energy density dε ds = T

εth = anT 2

◮ Pressure dP dT = −n2 d(s/n) dn

Pth = 2

3nQT 2

; Q = 1 − 3

2 n m∗ dm∗ dn ◮ Chemical potential dµ dT = − ds dn

µ(n, T) = −a

  • 1 − 2Q

3

  • T 2

◮ Specific Heats

CV = T d(s/n)

dT

  • n = 2aT

CP = T d(s/n)

dT

  • P = 2aT
  • C. Constantinou

Thermal Properties of Dense Matter

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SLIDE 5

Degenerate Limit Thermodynamics Beyond Leading Order

Degenerate limit implications:

◮ η = µ−ǫ(p=0) T

≫ 1 ⇒ Sommerfeld

◮ ǫ = p2 2m + U(n, p; T) → p2 2m + U(n, p; 0)

For a general U(n, p), define an effective mass function M(n, p) = m

  • 1 + m

p ∂U(n, p) ∂p

  • n

−1 . Relation to Landau m∗ : M(n, p = pF) = m∗ Applying the Sommerfeld expansion to the integral of the entropy density gives s = 2anT − 16 5π2 a3nT 3(1 − LF) LF = 7 12p2

F

M′2

F

m∗2 + 7 12p2

F

M′′

F

m∗ + 3 4pF M′

F

m∗ ; M′

F ≡ ∂M(n, p)

∂p

  • p=pF
  • C. Constantinou

Thermal Properties of Dense Matter

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SLIDE 6

Degenerate Limit Thermodynamics Beyond Leading Order

◮ Thermal Energy:

Eth = aT 2 + 12 5π2 a3T 4(1 − LF)

◮ Thermal Pressure:

Pth = 2 3anQT 2 − 8 5π2 a3nQT 4

  • 1 − LF + n

2Q dLF dn

  • ◮ Thermal Chemical Potential:

µth = −a

  • 1 − 2Q

3

  • T 2+ 4

5π2 a3T 4

  • (1 − LF)(1 − 2Q) − ndLF

dn

  • ◮ Specific Heat at constant volume:

CV = 2aT + 48 5π2 a3T 3(1 − LF)

◮ Specific Heat at constant pressure: CP = CV + T n2 ∂Pth

∂T

  • n

2

∂P ∂n |T

  • C. Constantinou

Thermal Properties of Dense Matter

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SLIDE 7

Results: S and Eth

0.5 1 1.5 2 0.2 0.4 0.6 0.8 PNM

0.5 1 1.5 2

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.5 1 1.5 2

0.2 0.4 0.6 0.8

SNM MDI(A) T = 20 MeV

0.5 1 1.5 2 0.2 0.4 0.6 0.8

SkO'

0.2 0.4 0.6 0.8

MFT S (kB) n (fm-3) FLT FLT+NLO Exact 5 10 15 20 25 0.2 0.4 0.6 0.8 PNM

5 10 15 20 25

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 5 10 15 20 25

0.2 0.4 0.6 0.8

SNM MDI(A) T = 20 MeV

5 10 15 20 25 0.2 0.4 0.6 0.8

SkO'

0.2 0.4 0.6 0.8

MFT Eth (MeV) n (fm-3) FLT FLT+NLO Exact

◮ The three models produce quantitatively similar results. ◮ Agreement with exact results extended to n ≃ 0.1 fm−3. ◮ Better agreement for PNM than for SNM.

  • C. Constantinou

Thermal Properties of Dense Matter

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SLIDE 8

Results: Pth and µth

1 2 3 4 0.2 0.4 0.6 0.8 PNM

1 2 3 4

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1 2 3 4

0.2 0.4 0.6 0.8

SNM MDI(A) T = 20 MeV

1 2 3 4 0.2 0.4 0.6 0.8

SkO'

0.2 0.4 0.6 0.8

MFT Pth (MeV fm-3) n (fm-3) FLT FLT+NLO Exact

  • 10
  • 8
  • 6
  • 4
  • 2

0.2 0.4 0.6 0.8 PNM

  • 10
  • 8
  • 6
  • 4
  • 2

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8

  • 10
  • 8
  • 6
  • 4
  • 2
0.2 0.4 0.6 0.8

SNM MDI(A) T = 20 MeV

  • 10
  • 8
  • 6
  • 4
  • 2
0.2 0.4 0.6 0.8

SkO'

0.2 0.4 0.6 0.8

MFT µn,th (MeV) n (fm-3) FLT FLT+NLO Exact

◮ Model dependence is evident- due to dm∗ dn . ◮ Agreement with exact results extended to n ≃ 0.1 fm−3. ◮ Better agreement for PNM than for SNM.

  • C. Constantinou

Thermal Properties of Dense Matter

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SLIDE 9

Results: Specific Heats

0.5 1 1.5 0.2 0.4 0.6 0.8 PNM

0.5 1 1.5

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.5 1 1.5

0.2 0.4 0.6 0.8

SNM MDI(A) T = 20 MeV

0.5 1 1.5 0.2 0.4 0.6 0.8

SkO'

0.2 0.4 0.6 0.8

MFT CV (kB) n (fm-3) FLT FLT+NLO Exact 0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 PNM

0.5 1 1.5 2 2.5 3

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1 2 3 4 5 6 7

0.2 0.4 0.6 0.8

SNM MDI(A) T = 20 MeV

1 2 3 4 5 6 7 0.2 0.4 0.6 0.8

SkO'

0.2 0.4 0.6 0.8

MFT CP (kB) n (fm-3) FLT FLT+NLO Exact

◮ The MDI and MFT CV exceed the classical value of 1.5 in the

nondegenerate limit. In this regime the T-dependence of the spectrum becomes important.

◮ The peaks in CP are due to the proximity to the nuclear

liquid-gas phase transition.

  • C. Constantinou

Thermal Properties of Dense Matter

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SLIDE 10

Binary Mergers and the EOS

◮ The EOS is necessary for a complete description of the

dynamics of a merger.

◮ The EOS is relevant to:

◮ GW frequency ◮ Size, type, and lifetime of remnant

◮ EOSs used in simulations:

◮ Realistic ◮ Polytropic, P = κnΓS ◮ Ideal Fluid, Pth = (Γth − 1)εth

  • C. Constantinou

Thermal Properties of Dense Matter

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SLIDE 11

Γth-General Considerations

◮ Γth = 1 + Pth εth ◮ Degenerate Limit

◮ Nonrelativistic

Γth = 1 + 2

3Q − 4 5π2 a2nT 2 dLF dn n→0

− →

5 3

Q = 1 + 3

2 n m∗ dm∗ dn

◮ Relativistic

Γth = 1 + Q

3 + 8 15π2 a2T 2(1 − Q)

  • LF − 5

3 p4

F

E ∗4

F

  • n→∞

− →

4 3

Q = 1 +

  • M∗

E ∗

F

2 1 − 3n

M∗ dM∗ dn

  • ◮ CC, B. Muccioli, M. Prakash & J.M. Lattimer,

arXiv:1504.03982

  • C. Constantinou

Thermal Properties of Dense Matter

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SLIDE 12

Γth-SkO’

1.2 1.4 1.6 1.8 2 2.2

0.0001 0.001 0.01 0.1 1

T = 5 MeV 1 0.1 0.01 0.001 0.0001 1.2 1.4 1.6 1.8 2 2.2

0.0001 0.001 0.01 0.1 1

T = 10 MeV 1.2 1.4 1.6 1.8 2 2.2

0.0001 0.001 0.01 0.1 1

T = 20 MeV 1.2 1.4 1.6 1.8 2 2.2

0.0001 0.001 0.01 0.1 1

SkO' T = 50 MeV Γth No Leptons

  • Deg. on top of Exact

n (fm-3)

Yp = 0.1 = 0.3 = 0.5

◮ No T dependence:

For Skyrme models, Pth(n, T) = Pid

th(n, T; m∗)Q

εth(n, T) = εid

th(n, T; m∗) Pid

th

εid

th = 2

3

Γth = 8

3 − m∗ m ◮ Weak x dependence ◮ Sharp rise in homogeneous phase

  • C. Constantinou

Thermal Properties of Dense Matter

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SLIDE 13

Γth-MDI

1.2 1.6 2

0.0001 0.001 0.01 0.1 1

T = 5 MeV 1 Exact = Deg. 0.1 0.01 0.001 0.0001 1.2 1.6 2

0.0001 0.001 0.01 0.1 1

T = 10 MeV Exact Deg 1.2 1.6 2

0.0001 0.001 0.01 0.1 1

T = 20 MeV Exact Deg 1.2 1.6 2

0.0001 0.001 0.01 0.1 1

MDI(A) T = 50 MeV Γth Exact Deg (a)

Yp = 0.5 = 0.3 = 0.1

◮ Weak T dependence ◮ Weak x dependence ◮ Maximum around n ∼ n0: dΓth dn = 0 ⇒ dm∗ dn

  • 1 −

n m∗ dm∗ dn

  • + n d2m∗

dn2 = 0

  • C. Constantinou

Thermal Properties of Dense Matter

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SLIDE 14

Γth-MFT

0.0001 0.001 0.01 0.1 1

1.2 1.6 2 T = 5 MeV Exact = Deg. 1 0.1 0.01 0.001 0.0001

0.0001 0.001 0.01 0.1 1

1.2 1.6 2 T = 10 MeV Exact = Deg.

0.0001 0.001 0.01 0.1 1

1.2 1.6 2 T = 20 MeV Deg. Exact

0.0001 0.001 0.01 0.1 1

1.2 1.6 2 MFT T = 50 MeV Γth Deg. Exact (c) x = 0.1 = 0.3 = 0.5

◮ Weak T dependence ◮ Weak x dependence ◮ Maximum around 2n0

  • C. Constantinou

Thermal Properties of Dense Matter

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SLIDE 15

Γth-Leptons and photons included

1.2 1.6 2

0.0001 0.001 0.01 0.1 1

T = 5 MeV 1 Exact Deg 0.1 0.01 0.001 0.0001 1.2 1.6 2

0.0001 0.001 0.01 0.1 1

T = 10 MeV Exact Deg 1.2 1.6 2

0.0001 0.001 0.01 0.1 1

T = 20 MeV Exact Deg 1.2 1.6 2

0.0001 0.001 0.01 0.1 1

MDI(A) T = 50 MeV Γth Exact Deg (a)

Ye = 0.5 = 0.3 = 0.1

1.2 1.6 2 0.0001 0.001 0.01 0.1 1

T = 5 MeV 1 Exact Deg 0.1 0.01 0.001 0.0001

1.2 1.6 2 0.0001 0.001 0.01 0.1 1

T = 10 MeV Exact Deg

1.2 1.6 2 0.0001 0.001 0.01 0.1 1

T = 20 MeV Exact Deg

1.2 1.6 2 0.0001 0.001 0.01 0.1 1

SkO' T = 50 MeV Exact Deg n (fm-3) (b)

Ye = 0.1 = 0.3 = 0.5

0.0001 0.001 0.01 0.1 1

1.2 1.6 2 T = 5 MeV 1 Exact Deg 0.1 0.01 0.001 0.0001

0.0001 0.001 0.01 0.1 1

1.2 1.6 2 T = 10 MeV Exact Deg

0.0001 0.001 0.01 0.1 1

1.2 1.6 2 T = 20 MeV Exact Deg

0.0001 0.001 0.01 0.1 1

1.2 1.6 2 MFT T = 50 MeV

th

Exact Deg (c)

Ye = 0.1 = 0.3 = 0.5

◮ Γth n→0

− → 4

3 ◮ Maximum even for Skyrme

  • C. Constantinou

Thermal Properties of Dense Matter

slide-16
SLIDE 16

Γth-Beyond MFT

1.2 1.4 1.6 1.8 2 2.2 10-5 10-4 10-3 10-2 10-1 100 Γth ρB (fm-3) T=20 MeV TL (a) 0.5 0.3 0.5,e 0.3,e 0,e 10-4 10-3 10-2 10-1 100 ρB (fm-3) MFT (b)

◮ X. Zhang & M. Prakash, work in progress ◮ Finite range effects via 2-loop calculation ◮ Lower peak relative to a similarly-calibrated MFT

  • C. Constantinou

Thermal Properties of Dense Matter

slide-17
SLIDE 17

ΓS-General Comments

◮ ΓS(n, S) = ∂ ln P ∂ ln n

  • S = n

P ∂P ∂n

  • S

◮ ΓS(n, T) = CP CV n P ∂P ∂n

  • T

◮ Relation to sound speed, cs:

cs

c

2 = ΓS

P h+mn ◮ Degenerate Limit:

ΓS(n, S) =

n P0+ nQS2

6a

  • dP0

dn + QS2 6a

  • 1 + 2

3Q + Q n dQ dn

  • ΓS(n, T) =

n P0+Pth

  • K

9 + ∂Pth ∂n

  • T +

T n2CV

  • ∂Pth

∂T

  • n

2 (using CP = CV + T

n2 ( ∂P

∂T |n) 2 ∂P ∂n |T

)

  • C. Constantinou

Thermal Properties of Dense Matter

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SLIDE 18

ΓS=0

1 2 3 4 0.001 0.01 0.1 1 Yp,e = 0 (d) 1 2 3 4

0.001 0.01 0.1 1

Yp,e = 0.1 (c) 1 2 3 4

0.001 0.01 0.1 1

Yp,e = 0.3 (b) 1 2 3 4

0.001 0.01 0.1 1

MDI(A) T = 0 MeV Γ Yp,e = 0.5 n (fm-3) (a)

◮ Only nucleons ⇒ mechanical

instability

◮ With leptons, nuclear matter is stable ◮ At S = 0, Pl ∼ n4/3

⇒ ΓS=0 =

  • 4

3 + n Pl dPb dn

1 + Pb

Pl

−1 For n ≃ n0, Pb(n, α) ≃

n2(n−n0) 9n2

  • K0 + α2

3n0Lv (n−n0) + Kv

  • ◮ SNM (α = 0, Pb = 0)

ΓS=0(n = n0) = 4

3 + K0 9n0Pl ∼ 2.1,

n(sp) = 2n0

3 ◮ PNM (α = 1, Pl = 0)

ΓS=0(n = n0) = 2 + K0+Kv

3Lv

∼ 2.8 n(sp) = n0

3(K0+Kv) 2(K0+Kv)−6Lv < 0

  • C. Constantinou

Thermal Properties of Dense Matter

slide-19
SLIDE 19

ΓS=0

0.5 1 1.5 2 2.5 3 0.001 0.01 0.1 1 SkO' Ye = 0.0 0.3 0.5 0.1 (b) S = 1 S = 0 1 1.5 2 2.5 3 3.5

0.001 0.01 0.1 1

MDI(A) Ye = 0.0 0.3 0.5 0.1 (a) ΓS n (fm-3) S = 1 S = 0

◮ Low n, ΓS > Γ0

High n, ΓS < Γ0

1.5 2 2.5 0.01 0.1 1 SkO' (c) Ye = 0.5 S = 2 3 4 Exact Deg 0.01 0.1 1 1.5 2 2.5 SkO' (d) Ye = 0.5 S = 2 3 4 Exact ND 1.5 2 2.5

0.01 0.1 1

MDI(A) (a) Ye = 0.5 S = 2 3 4 Exact Deg

0.01 0.1 1

1.5 2 2.5 MDI(A) (b) Ye = 0.5 S = 2 3 4 n (fm-3) ΓS Exact ND

◮ nX such that 1 P0 dP0 dn = 1 Pth ∂Pth ∂n

  • S

(indep. of S in Deg. Limit)

  • C. Constantinou

Thermal Properties of Dense Matter

slide-20
SLIDE 20

Conclusions

◮ m∗ is crucial in the determination of thermal effects. ◮ Both Γth and ΓS depend weakly on T but their density

dependence cannot be ignored.

◮ Finite-range effects suppress the density dependence of Γth. ◮ Finite entropy restricts the range of values ΓS can attain. ◮ Leptons are essential to the stability of nuclear matter.

  • C. Constantinou

Thermal Properties of Dense Matter