Note: This is a simple breakdown of research paper titled “Comparison of Equation of States for Neutron star with First Order phase transition: A Qualitative study” Published by Anshuman Verma, Asim Kumar Saha, and Ritam mallick. The content will NOT be same however only explained version of it for students who does not have prior experience to this field. Here you’ll find almost all the terms defined, graph/visual interpretation and final summary of the paper.
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Abstract: What exactly is this paper about?!
Firstly this paper talks about Equation of States (EoS) ¹ and how it’s important to understand matter at extreme conditions and to understand neutron star (or anything that’s related to extreme physical condition).
This paper intends to find solution to overcome a critical challenge that lies in modelling thermodynamic consistent,² and accurate First-order phase transition.³
Researchers have studied two different approaches for constructing EoSs with First Order phase transitions which are- The Polytropic interpolation method⁴ and the randomized speed of sound interpolation approach.⁵
Researchers has found that Polytropic interpolation method shows discontinuities in chemical potential while Speed of sound interpolation approach ensures continuity as well as effectively captures the softer segment of the mass-radius spectrum.⁶
Hence, it shows the importance of thermodynamic consistency in EoS and advantages of Randomized speed of sound method for modelling phase transition.
Introduction to these terms:
1. Equation of States: Basically equation that deals with matter and its State. EoS shows relation between physical quantities(i.e. density, pressure, temperature, energy) under extreme conditions .
How is it related ?
EoS defines how pressure in neutron star responds to density or even temprature sometimes. It also helps understanding internal composition of neutron star.
2. Thermodynamic consistent: It means that EoS or physical model describing any system must strictly follow thermodynamic laws.
3. First Order phase transition: So Phase transitions are just changes between states of matter. First order phase transition is when there’s a discontinuity in first derivative of free energy.
4. The Polytropic Interpolation method: is where pressure and density follows a certain relation, P = kp^(1+1/n), where n is Polytropic index. PIM constructs an EoS by joining multiple Polytropic segments, each representing different density regimes.
5. Randomized speed of sound interpolation approach: so this is another important thing when it comes to EoS, it’s about EoS stability and casualty, defined as C² (C being speed of sound) = dp/d€ , where € is energy density.
6. Mass radius spectrum: is relationship between mass and radius for compact objects. Different EoS predicts different curves, like stiffer EoS allows larger radius for a given mass. This Spectrum basically shows all possible mass radius combinations allowed by a particular EoS or set of it.
Introduction section
In introduction section research explains importance of EoS when it comes to understand matter under extreme conditions like densities and all as well as EoS defining relationship between thermodynamics variables of a system , defining matter properties under these conditions.
Author also mentions that EoS as of now at extreme densities remain unknown, meanwhile model calculations are used, with prediction matched against experiments and observation from Neutron stars.
One effective method of constraining EoS involves creating EoS ensemble by randomizing the speed of sound and applying thermodynamic bounds. Later it was applied to first order phase transition (FOPT) using piece wise Polytrope approach(PP) , where EoS with FOPT indicates rapid change in dense matter , even potentially leading to quark matter core.
At last, author add the need of thermodynamic consistency to avoide violation while constructing EoS.
Construction methods
Author uses following methods of construction in this section:
1. Piecewise- Polytropic (PP) method (Gorda et al. (2023)).
2. Piecewise-polytropic consistent (PP-consistent) method.
3. Speed of sound interpolation (SoSI) method (Annala et al. (2021)).
Readers are advised to read paper to understand the construction steps in detail.
Thjs graph is divided into four regions: CET (yellow), Hadronic (red), Quark (blue) , and pQCD (green). These regions shows different phases of matter , with data points marked by circles to show the speed of sound across the varying chemical potential miu*
Result and discussion Section
Now we find that: PP-consistent method uses predefined functional forms, resulting in more “constrained” EoS while the SoSI offers greater flexibility, allows a border spectrum of phase transition scenarios.
Both methods comply with causality constraints and satisfy astrophysical constraints
The SoSI method exhibits a border spread and can cover a larger area in the M-R space. The PP method produces more tightly clustered M-R curves.
The PP-consistent method is limited due to the consistency condition, while the SoSI method allows for significant variation in stiffness even after the early PT to satisfy astrophysical bounds. PP-based EoS cannot cover the softer region of the M-R spectrum due to the monotonically increasing stiffness of polytropes.
The PP method exhibits a discontinuity in the µ-P curve, while the SoSI and PP-consistent methods produce smooth, continuous curves. The speed of sound, defined in terms of pressure and energy density or using number density and chemical potential, should maintain causality for a physically consistent EoS. The SoSI method EoS maintains causality and generates the same curve with two different definitions, whereas the inconsistent PP EoS does not.
Note:
This post has only been made by undergrad student, so if there has been any mistakes or wrong interpretation of paper’s content kindly let us know. It’ll be our top priority to correct the mistakes.
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