Ideal Fermi Gases at Zero Temperature

lnZ =X where we have dropped the term 2 ln

1 + (2n+ 1)²π²

, because it is independent of thermodynamic quantities and will therefore vanish later anyway when taking derivatives of lnZ. By using the summation formula

∞ The integrals are now easy to do, which yields

lnZ = 2X

Finally, in the large volume limit, we get lnZ = 2V Compared to the case of spin-0 particles, Eq. (E.78), there is an extra factor 2 in front of the integral. This is due to the fact that there are two possible spin states for fermions: ”up” and ”down”. Or more precisely, it is because the spin space is two dimensional. The next thing we notice is the contribution from the vacuum energy, which we shall simply drop, as was argued for in the previous section. The remaining logarithmic terms are separate contributions from particles (with chemical potential µ) and antiparticles (with chemical potential−µ), respectively. Fortunately, we can conclude that quantum field theory manages to reproduce the well-known partition function for an ideal Fermi gas [54].

E.6 Ideal Fermi Gases at Zero Temperature

In the previous section we found the partition function for an ideal Fermi gas, Eq. (E.120). We decided to omit the contribution from the vacuum energy, after which the partition function reads

lnZ = 2V

106 Appendix E: Finite-Temperature Field Theory

In this section, we consider the zero-temperature limit of the thermodynamics quan-tities resulting from the above partition function. This will yield us a simple model of the matter inside a compact star. The following derivations are inspired by [57].

Notice that at zero temperature (β → ∞), if we use the convention that µ is positive, the term in the partition function concerning antiparticles completely vanishes. This leaves us only to consider

lnZ = 2V

Z d³p (2π)³ ln

1 +e^{−β(ωp−µ)}

. (E.122)

Let us begin by calculating the average particle number density hni, which is given by Hence, the average occupation number for a single-particle state with momentump is

hn_pi= 2

e^β(ωp−µ)+ 1, (E.124)

where the factor 2 comes from the fact that a particle with momentum p can have two independent spin states. In the limit β → ∞, this expression becomes the Heaviside step function

β→∞lim

2

e^β(ωp−µ)+ 1 = 2θ(µ−ω_p). (E.125) Consequently, at zero temperature, only single-particle states with energy less than the chemical potential are occupied. But this is exactly the definition of theFermi energy: the energy of the highest occupied single-particle energy state in a Fermi gas at zero temperature. The chemical potential therefore equals the Fermi energy E_F in the zero-temperature limit,

µ(T = 0) =E_F. (E.126)

The Fermi momentum, on the other hand, is defined implicitly by the energy-momentum relation

E_F = q

p²_F +m². (E.127)

We can now easily calculate the particle number density in terms of the Fermi momentum. Since ω_p only depends on |p| = p, integrating over angles just gives a factor 4π. We therefore get

hni= 2

Next, let us calculate the pressureP, which is given by P = 1

E.6 Ideal Fermi Gases at Zero Temperature 107 Integrating over angles followed by integration by parts yields

P = 1 and since the boundary term vanishes, this simply reduces to

P = 1 We again see the appearance of the step function in the zero-temperature limit. The integral then becomes quite easy to solve, leading to

P = 1

The internal energy density can be calculated similarly. We first note that the average total energy in the system is given by

hEi=D X Thus, in the zero-temperature limit, the energy density becomes

= hEi

The expression for the pressure, Eq. (E.132), together with the expression for the energy density, Eq. (E.134), form an equation of state for an ideal Fermi gas at zero temperature, parametrized in terms of the Fermi momentum.

108 Appendix E: Finite-Temperature Field Theory

F Wolfram Mathematica Notebook

Quark Matter Equation of State

Massive S Quark

Setup

In[1]:= L[Λ_]:=2 LogΛ  ΛMS

α[L_]:= 4 Pi β0 L * 1-2β1 Log[L] β0 ^ 2 L

ms[α_]:= hatmSα Pi^49 1+0.895062α Pi

In[4]:= ωS0[μS_, mS_]:=

-μSμS²-^{5 mS2}

2  μS²-mS² +³

2mS⁴Log ^μS2-mS2+μS

mS 

4π² ωS1[μS_, mS_, Λ_]:=1 2 Pi ^ 3

-2-mS²+μS²²+mS² 4+6 Log Λ mS

 μS -mS²+μS² -mS²LogμS+ -mS²+μS² mS

 +

3 -μS -mS²+μS² +mS²LogμS+ -mS²+μS² mS



2

n₀:

In[6]:= dωS0dmS[μS_, mS_] = D[ωS0[μS, mS], mS]; dωS1dmS[μS_, mS_, Λ_] = D[ωS1[μS, mS, Λ], mS]; dωS1dΛ[μS_, mS_, Λ_] = D[ωS1[μS, mS, Λ], Λ];

dmSdΛ[Λ_] =D[ms[α[L[Λ]]], Λ]; dαSdΛ[Λ_] =D[α[L[Λ]], Λ];

n_s* ( =n_s when the coupling and mass are not running):

In[11]:= nSstar[μS_, mS_, Λ_] =

Simplify[-D[ωS0[μS, mS], μS]] +Simplify[-D[ωS1[μS, mS, Λ], μS]α];

Output

n_s*:

In[12]:= nSstarOut=Collect[nSstar[μS, mS, Λ],α] /. Sqrt[μS ^ 2 - mS ^ 2] → uS

Out[12]=

-mS²+μS²^3/2 π²

-2 uSα 2 mS²+uSμS+3 mS²Log^Λ

mS-3 mS²Log^uS+μS

mS 

π³

Printed by Wolfram Mathematica Student Edition

109

110 Appendix F: Wolfram Mathematica Notebook

n₀(The first five expressions are outputted in order. The last two end with ; and are therefore not outputted)

:

dωS0dmSOut=Simplify[dωS0dmS[μS, mS]] /. Sqrt[μS ^ 2 - mS ^ 2] → uS dωS1dmSOut=Simplify[dωS1dmS[μS, mS,Λ]] /. Sqrt[μS ^ 2- mS ^ 2] → uS dmSdΛOut=Simplify[dmSdΛ[Λ]] /. Sqrt[μS ^ 2- mS ^ 2] → uS

dωS1dΛOut=Simplify[dωS1dΛ[μS, mS,Λ]] /. Sqrt[μS ^ 2 - mS ^ 2] → uS dαSdΛOut = Simplify[dαSdΛ[Λ]] /. Sqrt[μS ^ 2- mS ^ 2] → uS

ωS0Out= ωS0[μS, mS] /. Sqrt[μS ^ 2- mS ^ 2] → uS;

ωS1Out= ωS1[μS, mS, Λ] /. Sqrt[μS ^ 2- mS ^ 2] → uS;

Out[14]=

3 mS -mS²μS+μS³+^mS

4-mS²μS(uS+μS)Log^uS+μS mS  uS+μS

2π² -mS²+μS²

Out[15]= 1

π³uS+μS -mS²+μS²

mS mS²-μS² 4 mS²-5μSuS+μS+5 mS⁴+mS²μSuS+μS-6μS³uS+μSLoguS+μS mS + 6 mS²-mS²+μSuS+μSLoguS+μS

mS ² -6 LogΛ

mS

 μSuS+μS mS²-μS²+-2 mS⁴+2 mS²μSuS+μSLoguS+μS mS



Out[16]= hatmS -0.604796β0⁵Log Λ

ΛMS³+β0²β1 Log Λ

ΛMS 3.79817+10.5559 LogLog Λ ΛMS + β0³Log Λ

ΛMS² -3.51864β0+0.233629β1+1.20959β1 LogLog Λ ΛMS + β1² -0.942147-6.2371 LogLog Λ

ΛMS-7.03728 LogLog Λ

ΛMS² 

β0⁶ΛLog Λ ΛMS

⁵

β0²Log ^Λ

ΛMS-β1 Log2 Log ^Λ

ΛMS

β0³Log ^Λ

ΛMS²

5/9

Out[17]=

3 mS²uSμS-3 mS⁴Log^uS+μS

mS  π³Λ

Out[18]=

-2π β1+β0²Log ^Λ

ΛMS-2β1 Log2 Log ^Λ

ΛMS

β0³ΛLog ^Λ

ΛMS³

Export (Expressions to .txt files)

2 Quark Matter.nb

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G Python Code

This appendix contains all the python code used to solve the numerical problems in this thesis. Notice that the code listings have automatic line breaks generated by LaTeX to fit the page. These line breaks are marked with an arrow ,→. The code is organized as such: each problem in this thesis has its own Python module with a function that gets called from main, and the settings for each problem are contained in the beginning of those functions. The module ”running_combined.py” covers both Section 4.7 and 4.8. It combines the results of ”running_massless_SM.py” and

”running_massive_SM.py” into the same figures. Lastly, the module named starlib contains a lot of useful and shared functions and constants, e.g., an implementation of the Runge–Kutta 4 method and the structure equations.

G.1 Main

Module name: main.py

1 import incompressible_fluid

2 import linear_eos

3 import polytropic_eos

4 import full_ideal_fermi

5 import constant_massless_SM

6 import running_massless_SM

7 import running_massive_SM

8 import running_combined

9 import matplotlib.pyplot as plt

10

11 plt.rc('text', usetex=True)

12 plt.rc('font', family='serif')

13 plt.rcParams.update({'font.size': 13}) #Size guide: Three figs side by side:

16, Two figs: 11, One fig: 13

,→

14

15 KEYBOARD_INPUT = False

16

17 TASK = (

18 'incompressible_fluid',

19 'linear_eos',

20 'polytropic_eos',

21 'full_ideal_fermi',

22 'constant_massless_quarks_first_order',

23 'running_massless_quarks_first_order',

24 'running_massive_s_quark_first_order',

25 'running_combined'

26 )[int(input('Task number: ')) if KEYBOARD_INPUT else 7] # <--- CHANGE TASK MANUALLY HERE

,→

111

112 Appendix G: Python Code

27 28

29 def main():

30

31 if TASK == 'incompressible_fluid':

32 incompressible_fluid.incompressible_fluid()

33 elif TASK == 'linear_eos':

34 linear_eos.linear_eos()

35 elif TASK == 'polytropic_eos':

36 polytropic_eos.polytropic_eos()

37 elif TASK == 'full_ideal_fermi':

38 full_ideal_fermi.full_ideal_fermi()

39 elif TASK == 'constant_massless_quarks_first_order':

40 constant_massless_SM.massless_strange_matter_constant_coupling()

41 elif TASK == 'running_massless_quarks_first_order':

42 running_massless_SM.massless_strange_matter_running_coupling()

43 elif TASK == 'running_massive_s_quark_first_order':

44 running_massive_SM.massive_strange_matter_running_coupling_and_mass()

45 elif TASK == 'running_combined':

46 running_combined.running_combined()

47 48

49 if __name__ == '__main__':

50 main()

In document Neutron and Quark Stars (sider 117-124)