Equation of State for Quark Gluon Plasma

Priyanka S. Mahalle

Sant Gadge Baba Amravati University, Amravati 444602.

Guided by:

Dr. Jajati K. Nayak,

Variable energy Cyclotron Centre, Sector 1, Block-AF, Bidhan Nagar, Kolkata 700064

Abstract

The early Universe is widely believed to be in the state of Quark Gluon Plasma (QGP) after few microseconds of the Big Bang. The study of QGP is one of the fascinating and frontier areas of science. Such a state has been created in the laboratory by colliding nuclei at very high energies. People since decades have been trying to understand its properties. In this work, I try to get an overview of thermodynamic properties of QGP. I learned the equation of state for QGP through the MIT Bag model. The EoS gives us the information on various thermodynamic quantities like pressure, energy density, entropy etc.

To understand QGP, I started with the elementary particles in the universe, how they interact with each other and the various forces responsible for their interactions. I also studied various conservation laws that apply to such interactions. My study covered an overview of Quantum Chromodynamics, their properties, and their behavior at extreme conditions of temperature and density. These explain under what circumstances QGP is formed. For further undserstanding, I learnt about the state of electromagnetic plasma and its different properties; about the equation of state related to ideal gas of massless particles; about the variation of pressure, energy, density etc. with temperature by considering Ideal Gas and Van der Waal equation of state; and about the thermodynamics of relativistic massless particles and Bag Model of quark gluon plasma. My goal was to calculate the thermodynamic quantities of QGP state assuming MIT Bag Model for quarks and gluons and how quarks undergo phase transition to hadron.

Keywords: Quarks, Gluons, Plasma, Chromodynmics, MIT Bag Model.

Abbreviations

 QGP Quark Gluon Plasma EoS Equation Of State QCD Quantum Chromodynamics QED Quantum Electrodynamics

A review of particles in the Universe

Till Rutherford’s scattering experiment in 1911, people believed the fundamental constituent of matter is an atom. But with the emergence of powerful particle accelerators, we have come to know that atom is not the fundamental constituent of matter. As per current knowledge, an atom is subdivided into nucleus and electron, the nucleus in turn being composed of nucleons (neutron and proton), which further can be divided into elementary particles called quarks. The fundamental constituents of matter are basically leptons, quarks and carrier bosons. Apart from fundamental constituents, there are composite particles (which are basically the bound states of fundamental particles). The classification of particles according to different properties (like spin, flavor, mass, interaction etc..) is given below:

Particles in Universe

In the above figure, the classification of particles is shown. The particles are classified as fermions and bosons; each is subclassified as fundamental (elementary) and composite. The fundamental particles are also called elementary and are assumed to have no structure; whereas composite particles are made up of fundamental or elementary particles.

Fermions

All the particles can be classified broadly into two categories, such as fermions and bosons. The fermions are particles with half integer spins and bosons are with integer spins. Similarly the elementary particles can also be classified as fermions and bosons. elementary fermions are called the matter (anti-matter) particles and they obey the Fermi-Dirac statistics. Fermions follow Pauli Exclusion Principle which states that, no two fermions can stay in the same Quantum state. Elementary Ff ermions are further classified into light fermions known as leptons and heavy fermions known as quarks. A few notes on quarks and leptons are described below:

Quarks

Quarks are color charge particles having color quantum number. Each flavor has three colors i.e. red, green, and blue. They carry fractional charge. Quarks combine to form composite particles called hadrons. Due to a phenomenon known as quark confinement quarks are never directly observed in isolation, they can be found only within the hadrons. There are six types (or flavors) of quarks i.e. up, down, charm, strange, top and bottom. Out of six flavors up, down and strange quarks are light quarks and charm, top and bottom quarks are heavy quarks. Quarks experience all four types of interactions. Color- charge particles interact via gluon exchange in the same way that charged particles interact via photon exchange. However, gluons are themselves color charged and it amplifies the strong force as Color-charge particles are separated. As baryons have three quarks content, so each quark has baryon number 1/3. The properties of each quark flavor are given below in a tabular form.

Quarks
 Flavor Charge Mass Baryon Number Hypercharge Y Isospin I I3 u 2/3 5 MeV/C2 1/3 1/3 ½ +1/2 d -1/3 7 MeV/C2 1/3 1/3 ½ -1/2 c 2/3 1.5 GeV/C2 1/3 1/3 0 0 s -1/3 135 MeV/C2 1/3 -2/3 0 0 t 2/3 175 GeV/C2 1/3 1/3 0 0 b -1/3 5GeV/C2 1/3 1/3 0 0

Leptons

Leptons carry integer charge and experience all interactions except strong force. Leptons are of six types, out of them electron, muon and tauon are charged leptons and their corresponding neutrinos are neutral leptons. Out of these electrons have least mass than the other charged leptons.The properties of leptons are given in tabular form:

Table for Leptons
 Leptons Charge Mass Lifetime Le Lµ Lτ e -1 0.511003MeV/C2 ∞ 1 0 0 νe 0 0 ∞ 1 0 0 µ -1 105.659 2.197×10-6 0 1 0 νµ 0 0 ∞ 0 1 0 τ -1 1784 3.3×10-13 0 0 1 ντ 0 0 ∞ 0 0 1

Baryons

Baryons are the fermions. Baryons are not fundamental particles rather they are composite particles made of quarks held together by the strong force in a similar way as molecules are held together by the electromagnetic force. Every Baryon is composed of three quarks and every antibaryon is composed of three antiquarks. The best known baryons are protons and neutrons. Barons are colorless i.e. they must have three colors (RGB). According to strange quark content Baryons are further classified into nucleon and hyperons. The properties carry by each baryons are given below in a tabular manner.

Table for Baryons
 Baryon Composition Strangeness MassMeV/c2 Mean Lifetime IsospinI I3 p uud 0 938.28 2×1031 ½ +1/2 n udd 0 939.6 925±11 ½ -1/2 Ʌ s((du-ud)/2) -1 1115.6 2.63×10-10 0 0 Ξ0 dss -2 1314.9 2.90×10-10 ½ ½ Ξ- uss -2 1321.3 1.6×10-10 ½ -1/2 ∑0 s((du=ud)/2) -1 1192.5 (6±1)×10-20 1 0 ∑- dds -1 1197.3 1.48×10-10 1 -1 ∑+ uus -1 1189.4 0.80×10-10 1 +1 Δ++ uuu 0 1232 0.6×10-23 Ω- sss -3 1672 0.82×10-10

Bosons

Bosons are particle with integral spin. Bosons are also known as gauge bosons. The spin zero particles are known as scalar bosons and the elementary particles having spin 1 are known as vector bosons. Bosons are the mediator of particle interactions found in our nature. They obey the Bose-Einstein statistics and don’t follow Pauli Exclusion Principle. The fundamental bosons are photon, gluon, W +, W-, Z and Higgs bosons. Out of them Gluon and photon are chargeless and massless and they are the stable vector bosons. Whereas W + and Z bosons are very heavy having mass 80.4GeV/C2 and 91.2GeV/C2 respectively. W and Z bosons are short-lived gauge bosons. The Z boson is chargeless and W+ is of charge 1 and W- is of charge -1. Gluons possesses color and is of 8 types. Gluons are the massless particles that act between quarks, antiquarks and other gluons and carries color charge. Color charges are same as electromagnetic charge, but it comes in three types (±red, ±green, ±blue). They are bicolor particles carrying one positive and one negative color charge ( according to QCD which we will discuss later ).

 Mediator Charge Mass Lifetime Spin gluon 0 0 ∞ 2 photon (ϒ) 0 0 ∞ 1 W± ±1 81800 Unknown 1 graviton 0 0 Stable 2 Higg boson 0 125090(predicted) Unknown 0

Mesons

Mesons are composite Bosons made up of two fermions (quark and antiquark). Because of their composition they have zero baryon number .Mesons are also colorless like baryons. Out of all the mesons pions have less mass i.e. 140 MeV. Pions play a role in holding atomic nuclei together via strong interaction. More about mesons is given below

 Meson Composition Charge Strangeness MassMeV/c2 Mean Lifetime IsospinI I3 ∏± ud̅, du̅ +1, -1 0 139.569 2.60×10-8 1 +1 ,-1 ∏0 (uu̅-dd̅)/2 0 0 134.964 8.7×10-17 1 0 K0, K̅0 ds̅, d̅s 0 +1 497.72 K0S= 0.83×10-10K0L= 5.18×10-8 1/2 -1/2 K± us̅, ūs +1, -1 -1 493.67 1.24×10-8 1/2 +1/2,-1/2 Ƞ ((uu̅-dd̅+ss̅)/√6) 0 0 548.8 7×10-19 0 0 ρ ud̅, du̅, (uu̅-dd̅)/√2 +1, -1, 0 0 770 0.4×10-23 1 +1,0,-1

Notes on particle interactions

In nature there are four types of interaction i.e. strong interaction, electromagnetic interaction, weak interaction and gravitational interaction that governs how object or particle interacts and how certain particle decay. All the known forces of nature can be traced to these fundamental interactions .These four interactions explained accordingly their strength is given below:

Strong interaction

Before the 1970s, physicists were uncertain as to how the atomic nucleus was bound together. To explain it the stronger attractive force were postulated. This hypothesized force was called the strong force. This is strongest force among all three fundamental forces. Despite’s its strength, the strong forces does not manifest itself in macroscopic universe because of its extremely limited range. It is confined to an operating distance of about 10-15 meter (1fm) - of the order of diameter of proton. They are responsible for the binding of the quarks inside the neutron and proton, also the binding of neutron and proton within nuclei. Strong forces interact through the “gluon”. Quantum Chromo Dynamics (QCD) is the theory of strong interaction.

Electromagnetic interaction

In 19th century James Clerk Maxwell gave scientific definition of electromagnetic forces. Electromagnetic interaction is responsible for phenomenon related to extra nuclear physics, like the interaction between electron and nuclei, intermolecular forces between the molecules etc. Electric charges are responsible for electromagnetic forces. Range of electromagnetic forces is infinity. Photons (ϒ) are the force carrier particles for electromagnetic interaction. Photons are massless and chargeless particles which move with the speed of light. It’s spin is equal to one. The coupling constant for electromagnetic interaction is given by:

$\displaystyle {∝}{}{=}{}\frac{{e}^{2}}{4πℏc}{=}{}\frac{1}{137}$

This is called the fine structure constant, because it determines the fine structure (spin- orbit splitting) of atomic spectra.

Weak interaction

The first theory of weak forces was presented by Fermi in 1933. Weak forces are responsible for the disintegration of the particles. Weak interaction involves the process like nuclear β decay, involving the emission by radioactive nuclei of electron and neutrino. The strength of weak interaction coupling is in the order of 10-7 compared to strong interaction coupling. The weak forces are mediated by intermediated vector boson W± and Z, which is very massive, lifetime, is about 10-24 seconds, and give rise to interaction of very short range. The weak interaction takes place only at very small, subatomic distances less than the diameter of proton. The coupling constant for weak interaction is given by

$\displaystyle \frac{G\left(Mc^2\right)^2}{\left(ℏc\right)^3}=1.17\times10^{-5}$

In 1970 Glashow, Weinberg, and Salam unified the weak and electromagnetic forces through GWS theory by showing them to be the two aspects of a single force now termed as electroweak force. All mesons are unstable because of weak decay.

Gravitational interaction

The gravitational force was first described systematically by Sir Isaac Newton in 17th century and least understand till now. Gravitational interaction acts between all types of particles having mass. It is the weakest interaction between all fundamental interactions. The strength of gravitational interaction is in the order of 10-39 with respect to strong interaction. Gravitational force is infinite range force. Coupling constant of Gravitational Interaction is given by

$\displaystyle \frac{{{G}_{N}}{×}{{M}^{2}}}{4πℏc}{=}{5}{×}{{10}^{-40}}$

Thus for mass scale common in high energy physics the gravitational coupling is negligibly small.

Gravitational interaction is mediated through a boson particle called “graviton”. Graviton having spin equal to 2 and is massless particle. Experiments to detect graviton are currently under way.

This is about all four fundamental interactions. Scientists are making an effort to unify the fundamental forces into single theory. In 1970 Glashow, Weinberg, and Salam unifies the weak and electromagnetic forces through GWS theory. The grand unified theory unifies the three interactions strong, electromagnetic, and weak through GUT theory. This unification occurs around 1015GeV. Now we are waiting for theory of everything TOE where gravitational interaction will unified with this three interaction. It is worth to mention that the highly successful Standard Model does not include gravity.

Conservation rules

Conservation laws are very crucial to our understanding of physical world, in that they describe which process can or cannot occur in nature. Conservation laws are the properties of interaction; they are applicable for scattering as well as decay processes. All particles will interact and decay to lighter particles unless prevented by doing so by some conservation laws. Here we list below the conservation rules required for the allowed interaction/decay of particles.

• Energy And Momentum Conservation

Energy and momentum is always conserved during all interaction we consider in particle physics according to the assumed symmetries.

• Charge Conservation

Conservation of charge is take place in all three interactions. In the case of weak interaction the lepton or quark that enter in may not have the same charge that come out but the difference is carried out by charge W boson (as is the W charged boson i.e, W±).

• Baryon Number And Lepton Number Conservation

Baryon number and lepton number is also conserved during all three interactions. We assign B=1 for Baryon, B=-1 for antibaryon, and B=0 for everything else. Conservation of lepton number is carried out by conservation of electron number, tau number, and muon number independently.

• Flavor Conservation

Flavor is conserved only during strong and electromagnetic interaction and not in weak interaction, since during weak interaction up quark may convert into down quark or strange quark

e.g. Decay of lambda

Ʌ (uds) → p+ (uud) + ∏- (u̅d) an example of weak interaction where flavor is not conserved.

• Color Conservation

During weak and electromagnetic interaction the color conservation is not much more effective. While during strong interaction color is conserved. Quark color does change but the difference is carried by gluon.

• Parity Conservation

Parity conservation is related to mirror symmetry. During strong and electromagnetic interaction parity is conserved but during weak interaction parity is not conserved.

e.g. K+ → π+ + π0 parity = +1

K+ → π+ + π+ + π0 parity = -1

Here we obtaine two different final state parity (+1,-1) for decay of same kaon. Presence of two different final parities indicate that the initial state should also have different parities, hence two distinct particles. However precise measurement indicate that they are the same particles, and it is impossible to have two parities for same particle. Hence during weak interaction parity is not conserved.

• Charge Conjugation Conservation

Charge conjugation converts each particle into its antiparticle by changing its sign. Charge conjugation changes the sign of all internal quantum number. While mass, momentum, energy, spin remain untouched. Charge Conjugation is conserved during strong and electromagnetic interaction. During weak interaction charge conjugation does not conserved. An example given below

π- → µ- + ν̅µ

Neutrinos are found to have intrinsic parities: neutrinos have left-handed parity and antineutrinos right-handed. Since charge conjugation would leave the spatial coordinates untouched, then if you operated on a neutrino with the charge conjugation operator, you would produce a left-handed antineutrino. But there is no experimental evidence for such a particle; all antineutrinos appear to be right-handed; this lead to conclusion that charge conjugation does not conserved during weak interactuion.

• Time Reversal Conservation

Time reversal reverses the order in which event occurs. Time reversal conserved during the strong and electromagnetic interaction while during weak interaction it does not conserved in fact during weak interaction the time violation is about 10 -3 sec.

• Isospin Conservation

Many particles occur in group called multiplets, in which the particle is of almost same mass but different in charge. The proton and Neutron form such a multiplet. The new quantity describes mathematically the effect of changing neutron to proton or vice versa is called Isospin. Isospin has nothing to do with the spin of the particle, but is represented by a vector that can have various orientations in an imaginary space known as isotopic spin space. Isospin is conserved only in strong interaction. During electromagnetic interaction isospin is not conserved and can be explained by using following example.

π0 → ϒ + ϒ here Isospin I goes from 1 to 0

Also in weak interaction Isospin is not conserved. Let us consider an example of lambda decay.

Ʌ0 → p+ + π - here Isospin I goes from 0 to 2

Those are all conservation rules for all three interactions. Now by using this conservation rules we predict which reaction or decay is allowed and which is forbidden. Now after this let us move towards understanding the concept of quark gluon pasma.

Quark Gluon Plasma

Getting an overview of particles in universe now I want to study about the thermodynamics state of quarks, one type of elementary particles. In the particular state of quarks, I am interested in quark gluon plasma (QGP). The theory that describes thedynamics of quarks is called Quantum Chromodynamics (QCD). Here I will briefly describe about QCD and then about QGP, I will also discuss about Normal plasma ( i.e. Quantum Electrodnamic Plasma).

Quantum Chromodynamics

In Chromodynamics, the color charge plays the same role as an electric charge in Electrodynamics.. As leptons do not carry any color charge hence they do not involve in strong interaction. Quantum Chromodynamics is the formal theory of the strong color interaction between the quarks. Quarks have three possible values of color charge red, blue, and green. Antiparticles carry anticolor. Gluons which are the mediator of strong forces are bicolored, carrying one unit of positive color and one negative. So there are 9 kinds of gluon combinations ( g1, g2,g3,...,g9 ) are as follows:

rb̅, rg̅, bg̅, br̅, gr̅, gb̅, $\frac{1}{\sqrt{2}}$(rr̅ - bb̅), $\frac{1}{\sqrt{6}}$(rr̅ + bb̅ -2gg̅), $\frac{1}{\sqrt{3}}$(rr̅ + bb̅ +gg̅)

But one is color singlet $\frac{1}{\sqrt{3}}$(rr̅ + bb̅ +gg̅) and, since it carries no net color, has to be excluded.

Quantum Chromodynamics (QCD) is very similar to Quantum Electrodynamics (QED). However there are differences between them which are as follows:

• The color charge of the strong interaction is same as the electric charge in electromagnetic interaction. The difference between these two is that there is only one kind of electric charge (positive or negative), but there are three kinds of color charge (red, blue, and green or antired, antiblue, and antigreen).
• In QED photon (ϒ) is the mediator which is electrically neutral and they do not interact directly. Since gluons itself carry the color charge they interact directly to other gluons, and hence in addition to fundamental quark-gluon vertex, we also have primitive gluon-gluon vertex. There are two kinds of gluon-gluon vertex i.e. three gluon vertices and four gluon vertices.
Quark-gluon and gluon-gluon vertices

This direct gluon-gluon coupling makes QCD more complicated than QED.

• Size of the coupling constant adds another difference between QCD and QED. Each vertex in QED introduce a factor α=1137smallness of this number means we need Feynman diagram with small number of the vertices. In QCD this factor is not constant but depends upon the separation distance between the interacting particles we call it as a running coupling constant. This will introduce the important property of QCD as Asymptotic Freedom which we discuss later.
• Another difference is QED is an Abelian Gauge Theory and QCD is a Non Abelian Gauge Theory.

Quark Confinement

When two electrically charged particles are separated from each other the force between them gradually decreases, but when color charge (quarks) separate from each other then force between them increases in accordance with the distance. As we move quarks far and far away from each other the force is become stronger and stronger and infinite amount of energy is required to separate them. Eventually a quark-antiquark pair is produce as meson or baryon instead of producing an isolated quark. Hence quarks always remains in bound state because of separation which form hadron. This property of QCD is called as Quark Confinement.

A possible scenario for quark confinement

(As we pull u quark out of the proton a pair of quarks is created, and instead of free quark, we are left with pion and neutron.)

Asymptotic Freedom

David Politzer, Frank Wilzek and David Gross found that when two quarks are brought closer through very high energy reaction then they interact very weakly and behave as free particles. This property is known as Asymptotic Freedom.

Hence at low energy there is confinement and at high energy there is asymptotic freedom in QCD. This two phenomenon are explained by using the Coupling factor αs.

Running Coupling Constant

Running coupling constant for QCD is given by

$\displaystyle {{α}_{s}}{{Q}^{2}}{=}\frac{12π}{\left({33}{-}{2}{{N}_{f}}\right){×}{l}{n}{⁡}\left(\frac{{Q}^{2}}{{Ʌ}^{2}}\right)}$

Where,

Q is related to momentum transfer

Nf is the number of participating quark ﬂavor and is determined by the available energy characterized by Q2.

The parameter Λ has to be determined by comparing QCD predictions to experimental results.

The dependence of αs on Q2 shows the behavior of quarks as:

For smaller value of Q, αs is large and hence quarks remain in confined state (as hadron) and cannot be isolated.

For large value of Q, αs is small and due to this the quarks behave as if free and this is known as Asymptotic Freedom.

These are the remarkable properties of QCD.

QCD at Extremis (At high temperature or at high baryon density)

At low energy, the interaction between quarks is so strong that they have to form the bound state hadron because of color confinement. When we provide high energy to the hadronic system ( or increase the temperature of hadronic system ) then the hadron (bound state) break up ( When the energy exceeds the thresh hold). The effective coupling become smaller due to asymptotic freedom. Similar things happens when we increase the density i.e. the hadron overlap loosing their boundaries. In such case quarks deconfine leading to a state of quark gluon plasma. Collins and Perry first showed that at extreme condition of density, hadronic matter goes to deconfined stage where they exhibit asymptotic freedom. This is called deconfinement phase transition from hadron to quarks.

What is plasma?

Irvin Langmuir first described the plasma for electrically charged particles. In Ancient Greek it is called “πλασµα” which means mouldable substance. Plasma is artificially generated by heating or subjecting neutral gas to strong electromagnetic field. Under normal condition the atom as a whole is neutral and remain confined as their binding energy is greater than the ambient thermal energy. When we increase the thermal energy near or greater than it’s binding energy by subjecting it to strong electromagnetic field then atom will ionize by removing electron and produce positively charge ions. Although these charged particles are unbound, they are not free in the sense of not experiencinf any force. These charged particles are strongly affected by each other’s electromagnetic field. This plasma may be partially or fully ionized. It is the composition of charged as well as neutral particles. Therefore electromagnetic or electrodynamics plasma it is the collection of charged and neutral particle showing the property of quasi-neutrality and collective behaviour.

Electromagnetic plasma shows following properties:

• Quasi-neutrality

Plasma contain negative (ne) as well as positively charge particles (say ni), then overall plasma is approximately neutral and shows the property of quasi-neutrality.

i.e ${{n}_{e{}}}{≅}{}{}{}{{n}_{i}}$

• rd≤L
• ωp τc ∼ 1
• ND≫1

Here L is the dimension of container in which plasma is form, and ${{r}_{d}}$is the Debye’s screening radius, ωp is plasma frequency , τc is mean collision time and ND is number of particle within the Debye’s sphere. We know that the range of electromagnetic force is infinite in vacuum, but when we think of any media this is not the case. In plasma we consider the Debye’s radius rd upto which the Coloumbian force of attraction or repulsion is experienced and after that it will not experience or negligibly experienced. In such case we say that charge is screened by 1/e value.

Plasma inside the interior of sun stars corona, neon sign ,florosant lamp, and lightning are some of the examples of electromagnetic plasma. Now let us move towards quark gluon plasma.

Quark-Gluon Plasma

Before 1975 Collins and Perry already discussed the physics of Quark gluon plasma at extreme conditions of temperature and very high density. But the term quark-gluon plasma is coined by Russian-American physicist E. V. Shuryak for the assemblage of quarks and gluons in 1980. Quark gluon plasma is state of matter in QCD which exist at extremely high temperature and density. As we know quarks are color confined to each other so it is thought that this is the state of asymptotically free strong interacting quark and gluon. Quark gluon plasma is similar to electromagnetic plasma. In Electromagnetic plasma charges are screened but in Quark-Gluon plasma color charge of quarks and gluons is screened. Here we have hadronic matter as quarks and gluons instead of charged ion. Here role of charge is played by color charge. We know that these quark and gluons remain in confined state inside the hadrons. However at extreme condition of temperature and color charge density when the ambient thermal energy is large enough to exceed the binding energy between the quarks and gluons, they undergo deconfinement and produce free color charge particles i.e. quarks and gluons. Here the state of system is completely described by the degrees of freedom of color charged particles. The temperature at which QGP is formed is nearly equal to 164-175 MeV (see Ref. [2] to [4]), and it was first predicted by Lattice Guage Theory.

It is expected that the required high temperatures and densities can be accomplished by colliding nuclei at relativistic energies where large fraction of kinetic energy of the beam will be converted to thermal energy. The Relativistic Heavy Ion Collider at Brookhaven National Laboratary and the Large Hydron Collider at Geneva can provide nuclear beam for the production of QGP, the primordial fluid which exist when universe was few microsecond old. Therefore the study of QGP is important for early universe study. Currently, there are rigorous international experimental and theoretical efforts to create and study QGP.

Equation of state

A system is composed of particles, whose average motion define its properties those properties are in turn related to one another through Equation of state. Equation of state is thermodynamic equation relating state variables which describe the state of matter such as pressure, volume, temperature or internal energy.

Equations of state are useful in describing the properties of fluids, mixture of fluids, solid and interior of stars.

Ideal Gas Equation

Ideal gas equation is the equation of state of hypothetical ideal gas. This equation is given by Emile Clapeyron in 1834 as combination of the Boyl’s law, Charl’s Law, Avogadro’s law, and Gay-Lussac’s Law. The ideal gas equation is written as :

PV = nRT

Where P,V,T is the Pressure, Volume, and absolute Temperature; n is number 0f moles of gas; and R is ideal Gas constant. This law based on the assumptions of kinetic theory of gases. They treat gas molecule as point particle that do not interact except in elastic collision.

Using statistical mechanics we are able to derive this expression;

Graphical representation for ideal gas equation

Vander Wall Equation of State:

The Van der Wall equation based on the plausible reason that real gas do not follow the ideal fas law. This equation is corrects for the volume of gas, and the attractive forces between gas molecule, which is given by:

$\displaystyle {{P+\frac{{{n}^{2}}{a}}{{V}^{2}}}}{V-nb}{=}{n}{R}{T}$

Where the first $\frac{{{n}^{2}}{a}}{{V}^{2}}$, alter the pressure in ideal gas equation, it account for intermolecular attractive forces between gas molecule the magnitude for “a” indicates the strength of intermolecular strong forces. The factor “nb” account for the volume occupied by the gas molecule. The value of a and b is experimentally determined.

Van der Waals isotherm

Here experimentally, ${a}{=}\frac{27{{R}^{2}}{{T}_{c}^{2}}}{64{{P}_{c}}}$and ${b}{=}\frac{R{{T}_{c}}}{8{{P}_{c}}}$; and Pc, Tc are critical quantities.

MIT Bag Model

Before moving towards Bag model (see Ref [7] & [8]), let us understand the concept of vacuum first. Vacuum is the space devoid of matter. Classically it is the state in which no particles are present. Hence the vacuum state has zero energy value which also is the minimum for any Hamiltonian ($E\geq0$). However, there may be Hamiltonians whose minimum energy may not always be zero. For example consider the following energy profile:

Energy profile for vacuum

Clearly $E_0\neq0$. This means the elementary particle excitations will be above this state and the produced particle will have a minimum energy E0. Hence the corrected way to define vacuum is a minimum energy state of a system whose value may or may not be zero. Of course, Quantum mechanics complicates the above picture. Due to uncertainty relation $\triangle E.\triangle t\geq\hslash$and hence even a zero energy state can have quantum fluctuation which can give rise to virtual particle-antiparticle picture. It is so happen that QCD vacuum is a minimum energy state and $E_0\neq0$. Due to which even the vacuum has finite energy density and exert a pressure. We will call this pressure as Bag pressure denoted by B. Let us suppose that we have a system in QCD vacuum. Now heat the system so that energy of the system changes, which is responsible for the production of particle and antiparticle pair. (Like neutron antineutron, proton, antiproton etc.). Let us name that particle antiparticle as “Bags” which contain quark and gluon. If we increase the temperature gradually the number of bags i.e. bag density in the state increase, and they start overlapping on each other within the system. When we continuously increase the temperature all bags overlap with each other and their boundaries disappear. Due to this, matter particle inside the bag start moving freely and quark gluon plasma is formed. The temperature at which this phase transition occurs is called Transition temperature Tc.

As we know that quarks, antiquarks, and gluons are present inside the bag. They remain confined inside the bag as pressure experienced by the quarks (say Pq) is less than the outside pressure in QCD vacuum called Bag’s pressure i. e. $B>P_q$. When we heat the bag the pressure inside the bag is increased. At certain temperature Tc the pressure Pq is greater than or equal to Bag’s pressure i.e. ${P}_{q}\ge B$then the wall of bag or boundaries of bag disappear and Quark start moving freely inside the vacuum and QGP is formed.

Formation of QGP
Quark and gluons confined inside Bag

Equation of state for Relativistic Massless Quarks and Gluons using MIT Bag Model

Grand Canonical Ensemble is given by equation

${Z}{=}{{∑}_{N=0}^{∞}}{{{e}^{βµN}}{}{{Q}_{N}}{V,T}}$ ……………… (1)

Where,

${{Q}_{N}}{=}{{∑}_{\left\{i\right\}}}{{e}^{-βE}}$………. (Here {i} indicates the sum over all configuration or possibilities)

$E=\sum_p{\in_pn_p}$…………. ( E is the energy of the system)

$⇒\mathcal{Z}=\sum _{N=0}^{\infty }{e}^{\beta µN}\sum _{\left\{{n}_{p\right\}}}{e}^{-\beta \sum _{p}{\in }_{p}{n}_{p}}$
$⇒\mathcal{Z}=\sum _{N=0}^{\infty }{\sum _{{n}_{p=N}}{e}^{\beta µN}\left(e}^{-\beta {\in }_{1{n}_{1}}}\right).\left({e}^{-\beta {\in }_{2}{n}_{2}}\right).{\left(e}^{-\beta {\in }_{3}{n}_{3}}\right)\dots$

For bosons np varies from 0 to ∞, and for fermions either np=0 or np=1

In general we can write

${Z}{=}{{∏}_{p=0}^{∞}}{{{∑}_{n}}{{{e}^{-β\left({{∈}_{p-}}{µ}\right){n}}}}}{}$ ………………………… (2)

For fermions either n=o or n=1, so we can write equation (2) as:

$\therefore{\mathcal Z}_f=\prod_{p=o}^\infty{(1+}e^{-\beta(\in_{p-}µ)})$
${\log{{\mathcal Z}_f=}}\sum_{p=o}^\infty{{\log{(1+e^{-\beta(\in_p-µ)}}})}$

Now to convert discrete summation over momentum into continuous we can write

${log⁡{{{Z}_{f}}{=}}}\frac{gV}{{{\left({2}{π}\right)}^{3}}}{{∫}_{0}^{∞}}{{log⁡{\left({1}{+}{{e}^{-β{{{{∈}_{p}}{-}{µ}}}}}\right)}}}{{d}^{3}}{p}$

Degeneracy factor g is introduce as fermions can be distinguish from each other if we consider their spin

And V is the volume of the system

${log⁡{{{Z}_{f}}{=}}}\frac{gV}{{{\left({2}{π}\right)}^{3}}}{{∫}_{0}^{∞}}{{log⁡{\left({1}{+}{{e}^{-β{{{{∈}_{p}}{-}{µ}}}}}\right)}}}{{d}^{3}}{p}$ ………………………… (3)

This is required partition function for fermions.

Similarly,

For bosons n varies from o to ∞,

${{∑}_{n=0}^{∞}}{{{e}^{-β{{{{∈}_{p}}{-}{μ}}}{n}}}}{=}{1}{+}{{e}^{-β{{{{∈}_{p}}{-}{μ}}}}}{+}{{e}^{-2β{{{{∈}_{p}}{-}{μ}}}}}{+}{{e}^{-3β{{{{∈}_{p}}{-}{μ}}}}}{+}{…}$
${⇒}{{∑}_{n=0}^{∞}}{{{e}^{-β{{{{∈}_{p}}{-}{μ}}}{n}}}}{=}\frac{1}{1-{{e}^{-β\left({{∈}_{p}}{-}{μ}\right)}}}$

So we can write equation (2) as:

${\log{{\mathcal Z}_B=}}-\sum_{p=o}^\infty{{\log{(1-e^{-\beta(\in_p-µ)}}})}$

${⇒log⁡{{{Z}_{B}}{=}}}\frac{-gV}{{{\left({2}{π}\right)}^{3}}}{{∫}_{0}^{∞}}{{log⁡{\left({1}{-}{{e}^{-β{{{{∈}_{p}}{-}{µ}}}}}\right)}}}{{d}^{3}}{p}$ ……………….. (4)

Degeneracy factor “g” is due to different color and spin of boson, and V is volume of the system. So this is partition function for bosons

In general for both fermions and bosons the partition function can be written as,

${log⁡Z=}\frac{κgV}{{{\left({2}{π}\right)}^{3}}}{{∫}_{0}^{∞}}{{log⁡{\left({1}{+}{κ}{{e}^{-β{{{{∈}_{p}}{-}{µ}}}}}\right)}}}{{d}^{3}}{p}$ ………………….. (5)

For particles µ >0, and for antiparticles µ<0

Hence for system having particles and antiparticles we can write equation (5) as:

${log⁡Z=}\frac{gVκ}{{{\left({2}{π}\right)}^{3}}}{{{{∫}_{0}^{∞}}{{log⁡{\left({1}{+}{κ}{{e}^{-β{{{{∈}_{p}}{-}{µ}}}}}\right)}}}{{d}^{3}}{p}{+}{{∫}_{0}^{∞}}{{log⁡{\left({1}{+}{κ}{{e}^{-β{{{{∈}_{p}}{+}{µ}}}}}\right)}}}{{d}^{3}}{p}}}$ …………. (6)

Now we assume that particles are massless and relativistic

$\Rightarrow\in_p=\sqrt[{}]{p^2+m^2}=p$
${∴log⁡Z=}\frac{gVκ}{{{\left({2}{π}\right)}^{3}}}{{{{∫}_{0}^{∞}}{{log⁡{{{1+κ{{e}^{-βp-µ}}}}{{d}^{3}}{p}{+}{{∫}_{0}^{∞}}{{log⁡{{{1+κ{{e}^{-βp+µ}}}}{{d}^{3}}{p}}}}}}}}}$

${log⁡Z=}\frac{gVκ}{{{\left({2}{π}\right)}^{3}}}{4}{π}{{{{∫}_{0}^{∞}}{{log⁡{{{1+κ{{e}^{-βp-µ}}}}{{p}^{2}}{d}{p}{+}{{∫}_{0}^{∞}}{{log⁡{{{1+κ{{e}^{-βp+µ}}}}{{p}^{2}}{d}{p}}}}}}}}}$……………. (7)

Using spherical polar co-ordinate.

By using Integration by parts we have

$\mathrm{log}\mathcal{Z}=\frac{gV\kappa }{{\left(2\pi \right)}^{3}}4\pi \left\{{\left[\mathrm{log}\left(1+\kappa {e}^{-\beta \left(p-µ\right)}\right).\frac{{p}^{3}}{3}\right]}_{0}^{\infty }-\left[{\int }_{0}^{\infty }\frac{-\beta \kappa {e}^{-\beta \left(p-\mu \right)}{p}^{3}}{3\left(1+\kappa {e}^{-\beta \left(p-\mu \right)}\right)}dp\right]+{\left[\mathrm{log}\left(1+\kappa {e}^{-\beta \left(p+µ\right)}\right).\frac{{p}^{3}}{3}\right]}_{0}^{\infty }-\left[{\int }_{0}^{\infty }\frac{-\beta \kappa {e}^{-\beta \left(p+\mu \right)}{p}^{3}}{3\left(1+\kappa {e}^{-\beta \left(p+\mu \right)}\right)}dp\right]\right\}$
${log⁡Z=}\frac{gVκ4πβ}{3{{2π}^{3}}}{{{{∫}_{0}^{∞}}{\frac{{{e}^{-βp-μ}}{{p}^{3}}}{{{1+κ{{e}^{-βp-μ}}}}}{d}{p}}{+}{{∫}_{0}^{∞}}{\frac{{{e}^{-βp+μ}}{{p}^{3}}}{{{1+κ{{e}^{-βp+μ}}}}}}}}$

${log⁡Z=}\frac{gVβ}{6{{π}^{2}}}{{{{∫}_{0}^{∞}}{\frac{{p}^{3}}{{{{{e}^{βp-μ}}{+}{κ}}}}{d}{p}}{+}{{∫}_{0}^{∞}}{\frac{{p}^{3}}{{{{{e}^{βp+μ}}{+}{κ}}}}{d}{p}}}}$ …………………. (8)

Consider first term,

${{∫}_{0}^{∞}}{\frac{{p}^{3}}{{{{{e}^{βp-μ}}{+}{κ}}}}{d}{p}{=}\frac{1}{{β}^{4}}}{{∫}_{-βμ}^{∞}}{\frac{{{\left({x}{+}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}$……………. (By replacing β (p-µ) =x)

Similarly,

${{∫}_{0}^{∞}}{\frac{{p}^{3}}{{{{{e}^{βp+μ}}{+}{κ}}}}{d}{p}{=}\frac{1}{{β}^{4}}}{{∫}_{βμ}^{∞}}{\frac{{{\left({x}{-}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}$……………… (By replacing β (p+µ) =x)

${∴}{log⁡Z=}\frac{gV}{{{β}^{3}}{6}{{π}^{2}}}{{{{∫}_{-βμ}^{∞}}{\frac{{{\left({x}{+}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}{+}{{∫}_{βμ}^{∞}}{\frac{{{\left({x}{-}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}}}$
${log⁡Z=}\frac{gV}{{{β}^{3}}{6}{{π}^{2}}}{{{{∫}_{-βμ}^{0}}{\frac{{{\left({x}{+}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}{+}{{∫}_{0}^{∞}}{\frac{{{\left({x}{+}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}{+}{{∫}_{0}^{∞}}{\frac{{{\left({x}{-}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}{-}{{∫}_{0}^{βμ}}{\frac{{{\left({x}{-}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}}}$
${log⁡Z=}\frac{gV}{{{β}^{3}}{6}{{π}^{2}}}{{{{∫}_{-βμ}^{0}}{\frac{{{\left({x}{+}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}{+}{{∫}_{0}^{∞}}{\frac{{{\left({x}{+}{β}{μ}\right)}^{3}}{{\left({x}{-}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}{-}{{∫}_{0}^{βμ}}{\frac{-{{\left({β}{μ}{-}{x}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}}}$

For first term replace x by –x, we get

${{∫}_{-βμ}^{0}}{\frac{{{\left({x}{+}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}{=}{-}{{∫}_{βμ}^{0}}{\frac{{βμ-x}^{3}}{{{e}^{-x}}{+}{κ}}{d}{x}}{=}{{∫}_{0}^{βμ}}{\frac{{{\left({β}{μ}{-}{x}\right)}^{3}}}{{{e}^{-x}}{+}{κ}}{d}{x}}$
${⇒log⁡Z=}\frac{gV}{{{β}^{3}}{6}{{π}^{2}}}{{{{∫}_{0}^{βμ}}{\frac{{{\left({β}{μ}{-}{x}\right)}^{3}}}{{{e}^{-x}}{+}{κ}}{d}{x}}{+}{{∫}_{0}^{∞}}{\frac{{{\left({x}{+}{β}{μ}\right)}^{3}}{{\left({x}{-}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}{-}{{∫}_{0}^{βμ}}{\frac{-{{\left({β}{μ}{-}{x}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}}}$
${⇒log⁡Z=}\frac{gV}{{{β}^{3}}{6}{{π}^{2}}}{{{{∫}_{0}^{βμ}}{{{\left({β}{μ}{-}{x}\right)}^{3}}{{\frac{1}{{{e}^{-x}}{+}{κ}}{+}\frac{1}{{{e}^{x}}{+}{κ}}}}{d}{x}}{+}{{∫}_{0}^{∞}}{\frac{{{\left({x}{+}{β}{μ}\right)}^{3}}{{\left({x}{-}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}}}$
${⇒log⁡Z=}\frac{gV}{{{β}^{3}}{6}{{π}^{2}}}{{{{∫}_{0}^{βμ}}{{{\left({β}{μ}{-}{x}\right)}^{3}}{κ}{d}{x}}{+}{2}{{∫}_{0}^{∞}}{\frac{{{x}^{3}}{+}{3}{x}{{\left({β}{μ}\right)}^{2}}}{{{e}^{x}}{+}{κ}}{d}{x}}}}$
${⇒log⁡Z=}\frac{gV}{{{β}^{3}}{6}{{π}^{2}}}{{\frac{{{κ\left({β}{μ}\right)}^{4}}}{4}{+}{2}{{∫}_{0}^{∞}}{\frac{{{x}^{3}}{+}{3}{x}{{\left({β}{μ}\right)}^{2}}}{{{e}^{x}}{+}{κ}}{d}{x}}}}$

${⇒log⁡Z=}\frac{gV{{T}^{3}}}{6{{π}^{2}}}{{\frac{{{κ\left({β}{μ}\right)}^{4}}}{4}{+}{2}{{∫}_{0}^{∞}}{\frac{{{x}^{3}}{+}{3}{x}{{\left({β}{μ}\right)}^{2}}}{{{e}^{x}}{+}{κ}}{d}{x}}}}$…….. (9) ………… (As β=1/T)

Here µ is the chemical potential; if µ is fixed then number of particles in the system is fixed. But here we consider relativistic massless particles, where number of particles in the system does not remain constant. So we look for the conserved quantity during relativistic case. Baryon number is the quantity which remains fixed in relativistic case. Hence we replace µ by µB called baryon chemical potential, which means if µB is fixed then B is fixed.

${⇒log⁡Z=}\frac{gV{{T}^{3}}}{6{{π}^{2}}}{{\frac{{{κ\left({β}{{μ}_{B}}\right)}^{4}}}{4}{+}{2}{{∫}_{0}^{∞}}{\frac{{{x}^{3}}{+}{3}{x}{{\left({β}{{μ}_{B}}\right)}^{2}}}{{{e}^{x}}{+}{κ}}{d}{x}}}}$……………. (10)

Now for Gluons: µB =0 as B=0 ${a}{n}{d}{}{κ}{=}{-}{1}$

Therefore equation………

${⇒log⁡{{{Z}_{g}}{=}}}\frac{{{g}_{g}}{V}{{T}^{3}}}{6{{π}^{2}}}{{0+2{{∫}_{0}^{∞}}{\frac{{x}^{3}}{{{e}^{x}}{-}{1}}{d}{x}}}}$
${⇒log⁡{{{Z}_{g}}{=}}}\frac{{{g}_{g}}{V}{{T}^{3}}}{6{{π}^{2}}}{{2×\frac{{π}^{4}}{30}}}$

… Using standard result from Riemann-Zeta function

${{\Rightarrow\log}{{\mathcal Z}_g=}}\frac{g_gVT^3\pi^2}{90}$………………….. (11)

This is required partition function for gluons.

For Quarks:$\kappa=+1$,

${log⁡{{{Z}_{q}}{=}}}\frac{{{g}_{q}}{V}{{T}^{3}}}{6{{π}^{2}}}{{\frac{{{\left({β}{{μ}_{q}}\right)}^{4}}}{4}{+}{2}{{∫}_{0}^{∞}}{\frac{{x}^{3}}{{{e}^{x}}{+}{1}}{+}\frac{3x{{\left({β}{{μ}_{q}}\right)}^{2}}}{{{e}^{x}}{+}{1}}{d}{x}}}}$

${log⁡{{{Z}_{q}}{=}}}\frac{{{g}_{q}}{V}{{T}^{3}}}{6{{π}^{2}}}{{\frac{{{\left({β}{{μ}_{q}}\right)}^{4}}}{4}{+}{2}{{\frac{7{{π}^{4}}}{120}{+}\frac{3{{μ}^{2}}{{β}^{2}}{{π}^{2}}}{12}}}}}$…Using standard result of integral from Riemann Zeta function

Here we consider the condition for quark as well as antiquarks.

${log⁡{{{Z}_{q}}{=}}}\frac{{{g}_{q}}{V}{{T}^{3}}}{6{{π}^{2}}}{{\frac{{{\left({β}{{μ}_{q}}\right)}^{4}}}{4}{+}\frac{7{{π}^{4}}}{60}{+}\frac{{{{μ}_{q}}^{2}}{{β}^{2}}{{π}^{2}}}{2}}}$ …………… (12)

Consider a system of massless and non-interacting quarks and gluons

${{\Rightarrow\log}{\mathcal Z=}}{\log{{\mathcal Z}_q+{\log{{\mathcal Z}_g}}}}$

${⇒log⁡Z=}\frac{{{g}_{q}}{V}{{T}^{3}}}{6{{π}^{2}}}{{\frac{{{\left({β}{{μ}_{q}}\right)}^{4}}}{4}{+}\frac{7{{π}^{4}}}{60}{+}\frac{{{{μ}_{q}}^{2}}{{β}^{2}}{{π}^{2}}}{2}}}{+}\frac{{{g}_{g}}{V}{{T}^{3}}{{π}^{2}}}{90}$ ……………… (13)

Now Pressure (P) is given by

${P}{=}\frac{∂}{∂V}{Tlog⁡Z}$
${⇒}{P}{=}\frac{∂}{∂V}{{\frac{{{g}_{q}}{V}{{T}^{3}}}{6{{π}^{2}}}{{\frac{{{\left({β}{{μ}_{q}}\right)}^{4}}}{4}{+}\frac{7{{π}^{4}}}{60}{+}\frac{{{{μ}_{q}}^{2}}{{β}^{2}}{{π}^{2}}}{2}}}{+}\frac{{{g}_{g}}{V}{{T}^{3}}{{π}^{2}}}{90}}}$
Expression for pressure
 $⇒P=\frac{{g}_{q}}{12}\left[{\frac{7{\pi }^{2}{T}^{4}}{30}+\mu }_{q}^{2}{T}^{2}+\frac{{{\mu }_{q}}^{4}}{2{\pi }^{2}}\right]+\frac{{g}_{g}{T}^{4}{\pi }^{2}}{90}$

…………………...... .................................... (14)

This is required equation of pressure for massless relativistic non-interacting Quark Gluon phase.

Now particle density nq is given by,

${n}_{q}=\frac{1}{V}\frac{\partial }{\partial {\mu }_{B}}\left(T\mathrm{log}Z\right)$
${⇒}{{n}_{q}}{=}\frac{1}{V}\frac{∂}{∂{{μ}_{q}}}{{\frac{{{g}_{q}}{V}{{T}^{4}}}{6{{π}^{2}}}{{\frac{{{\left({β}{{μ}_{q}}\right)}^{4}}}{4}{+}\frac{7{{π}^{4}}}{60}{+}\frac{{{{μ}_{q}}^{2}}{{β}^{2}}{{π}^{2}}}{2}}}{+}\frac{{{g}_{g}}{V}{{T}^{4}}{{π}^{2}}}{90}}}$
${⇒}{{n}_{q}}{=}\frac{{{g}_{q}}{{T}^{4}}}{6{{π}^{2}}}{{\frac{{{4β}^{4}}{{μ}_{q}^{3}}}{4}{+}{0}{+}\frac{2{{μ}_{q}}{{β}^{2}}{{π}^{2}}}{2}}}{+}{0}$
Expression for particle density
 $⇒{n}_{q}=\frac{{g}_{q}}{6}\left[\frac{{\mu }_{q}^{3}}{{\pi }^{2}}+{\mu }_{q}{T}^{2}\right]$

………………………. (15)

This is the expression of particle density for massless relativistic non-interacting Quark Gluon phase.

Entropy is given by the equation,

$S=\frac{1}{V}\frac{\partial }{\partial T}\left[T\mathrm{log}Z\right]$
${⇒}{S}{=}\frac{1}{V}\frac{∂}{∂T}{{\frac{{{g}_{q}}{V}{{T}^{4}}}{6{{π}^{2}}}{{\frac{{{\left({β}{{μ}_{q}}\right)}^{4}}}{4}{+}\frac{7{{π}^{4}}}{60}{+}\frac{{{{μ}_{q}}^{2}}{{β}^{2}}{{π}^{2}}}{2}}}{+}\frac{{{g}_{g}}{V}{{T}^{4}}{{π}^{2}}}{90}}}$
Expression for Entropy of system
 $⇒S=\frac{{g}_{q}}{12}\left\{\left[\frac{14{\pi }^{2}{T}^{3}}{15}+2{\mu }_{q}^{2}T\right]+\frac{2{\pi }^{2}}{45}{g}_{g}{T}^{3}\right\}$

……………………. (16)

This is required expression for entropy of system.

Now, by Thermodynamics relation we have

$TS=\in+P-\mu_Bn$
${⇒}{ϵ}{=}{T}{S}{-}{P}{+}{{μ}_{B}}{n}{}$
${⇒}{ϵ}{=}\frac{{g}_{q}}{12}{{{{\frac{14{{π}^{2}}{{T}^{4}}}{15}{+}{2}{{μ}_{q}^{2}}{{T}^{2}}}}{+}\frac{2{{π}^{2}}}{45}{{g}_{g}}{{T}^{4}}}}{-}{{\frac{{g}_{q}}{12}{{{{\frac{7{{π}^{2}}{{T}^{4}}}{30}{+}{μ}}_{q}^{2}}{{T}^{2}}{+}\frac{{{{μ}_{q}}^{4}}}{2{{π}^{2}}}}}{+}\frac{{{g}_{g}}{{T}^{4}}{{π}^{2}}}{90}}}{+}{{\frac{{g}_{q}}{6}{{\frac{{μ}_{q}^{4}}{{π}^{2}}{+}{{μ}_{q}^{2}}{{T}^{2}}}}}}$
${⇒}{ϵ}{=}{{\frac{{g}_{q}}{12}{{{{\frac{21{{π}^{2}}{{T}^{4}}}{30}{+}{3}{μ}}_{q}^{2}}{{T}^{2}}{+}\frac{3{{{μ}_{q}}^{4}}}{2{{π}^{2}}}}}{+}\frac{{{g}_{g}}{{T}^{4}}{{π}^{2}}}{30}}}$
Equation of State
 $⇒ϵ=3P$

…………………….(18)

This is required equation of state for Relativistic massless non-interacting quarks and gluons.

At LHC energy it has been found that most of the hadrons detected are the pions (π). Pions are the lightest mesons with mass approximately equals to 140MeV.

Let us consider the system of non-interacting massless pion

i.e. mπ = 0 and S=0;

As pions are the mesons they have same pressure equation like gluons,

$P_\pi=\frac{\pi^2g_\pi T^4}{90}$ …………………….(19)

As we have,

${P}_{q+\stackrel{̅}{q}+g}=\frac{{g}_{q}}{12}\left[{\frac{7{\pi }^{2}{T}^{4}}{30}+\mu }_{q}^{2}{T}^{2}+\frac{{{\mu }_{q}}^{4}}{2{\pi }^{2}}\right]+\frac{{g}_{g}{T}^{4}{\pi }^{2}}{90}\mathrm{}$…………………..(from equation 14) ${}{}{}{}{}{}{}$

Here we consider the vaccume pressure is zero. But when the vaccume pressure of QCD is equal to the Bag’s pressure then pressure then the partition function for QGP is given by:

$\mathrm{log}{\mathcal{Z}}_{QGP}=\mathrm{log}{\mathcal{Z}}_{q+\stackrel{̅}{q}+g}+\frac{BV}{T}$ ………………. (15)

Pressure of QGP is given by:

$P_{QGP}=\frac\partial{\partial V}(T{\log{{\mathcal Z}_{QGP})}}$
${⇒P}_{QGP}={P}_{q+\stackrel{̅}{q}+g}-B$

${{⇒P}_{QGP}}{=}{{\frac{{g}_{q}}{12}{{{{\frac{7{{π}^{2}}{{T}^{4}}}{30}{+}{μ}}_{q}^{2}}{{T}^{2}}{+}\frac{{{{μ}_{q}}^{4}}}{2{{π}^{2}}}}}{+}\frac{{{g}_{g}}{{T}^{4}}{{π}^{2}}}{90}}}{-}{B}$ ……………… (16)

Also, ${ϵ}_{QGP}={ϵ}_{q+\stackrel{̅}{q}+g}+B$ ……………… (17)

The equation of state for relativistic ideal gas is given by $P=\frac\in3$

${AsP}_{QGP}={P}_{q+\stackrel{̅}{q}+g}-B$
${⇒P}_{QGP}=\frac{{ϵ}_{q+\stackrel{̅}{q}+g}}{3}-B$

${{⇒P}_{QGP}}{=}\frac{1}{3}{{{{ϵ}_{QGP}}{-}{4}{B}}}$ …………. (18)

Here PQGP is the function of T and µB and Pressure of hadrons i.e. Ph I also the function of T and µB. Now if QGP is goes to hadronic phase then for phase transition it must satisfy the following conditions:

${{P}_{QGP}}{{{{T}_{q}}{,}{{μ}_{q}}}}{=}{{P}_{h}}\left({{T}_{h}}{,}{{μ}_{B}}\right)$
$T_q=T_h$
$\mu_q=\frac13\mu_B$

To know at what temperature and pressure this phase transition occurs we have to solve this equation simultaneously. Let’s calculate the transition temperature for quark-gluon system connected to a pionic system i.e. the hadronic system only contains pions (ud̅).

${{\frac{{g}_{q}}{12}{{{{\frac{7{{π}^{2}}{{T}_{c}^{4}}}{30}{+}{μ}}_{q}^{2}}{{T}_{c}^{2}}{+}\frac{{{{μ}_{q}}^{4}}}{2{{π}^{2}}}}}{+}\frac{{{g}_{g}}{{T}_{c}^{4}}{{π}^{2}}}{90}}}{-}{B}{=}\frac{{{π}^{2}}{{g}_{π}}{{T}_{c}^{4}}}{90}$ ……………… (19)

Now consider case I : when $\mu_B=0$

Equation 19 can be written as:

${{\frac{{g}_{q}}{12}{{\frac{7{{π}^{2}}{{T}_{c}^{4}}}{30}}}{+}\frac{{{g}_{g}}{{T}_{c}^{4}}{{π}^{2}}}{90}}}{-}{B}{=}\frac{{{π}^{2}}{{g}_{π}}{{T}_{c}^{4}}}{90}$
Critical temperature is given by:
 $⇒{T}_{c}={\left[\frac{90B}{{\pi }^{2}\left({g}_{eff}-{g}_{\pi }\right)}\right]}^{\frac{1}{4}}$

Where $g_{eff}=g_g+\frac74g_q$now $g_g=16$and $g_q=12;$ ${\Rightarrow g}_{eff}=37$

 $\therefore {T}_{c}={\left[\frac{90B}{{\pi }^{2}34}\right]}^{\frac{1}{4}}$

……………… (20)

Case II : when $T_c=0$;

Equation (19) can be written as:

Expression to calculate criticle Baryonic Chemical potential
 ${\mu }_{B}={\left[\frac{12×81×2{\pi }^{2}×B}{{g}_{q}}\right]}^{\frac{1}{4}}$

……………..(21)

Summary and Conclusion

In this project I learnt about the elementary particles and about the fundamental interactions, which exist among these particles. I also tried to learn the overview of Quantum Chromodynamics (QCD) which is the theory of strong interaction that exists among the fermionic particles called quarks. The quarks interact strongly via force carrier particles called gluons. The thermodynamic state of a quark-gluon system called quark-gluon plasma is studied here.

The equation of state, which is the thermodynamics relationship between the thermodynamic variable, for QGP is studied for MIT Bag model. The expressions for pressure, energy density are derived. The phase transition from QGP to pionic system has been studied usin Baf EoS. The expression for transition temperature T c has been obtained.

The equations of state derived from the MIT Bag model is one of the oldest equation of state for QGP. Even today it remains the baseline. Hence it has been widely accepted in astrophysics and cosmology. Curve obtained from the MIT Bag model for Baryon chemical potential Vs Bag pressure and Temperature Vs Bag pressure suggest that the phase transition is of first order. But experimental Lattice QCD calculation shows that there is smooth crossover between phase transition of QGP. Which suggests us to modify the Bag model to suits better with the experimental results. Uptill now many modifications are done and their results are calculated.

The energy density and pressure of QGP for Bag Model is given by

$\in=\sigma T^4-B$ $P=\frac\sigma3T^4-B$

Where $\sigma$is the Stephan-Boltzmann Constant whose value is given by

${σ}{=}\frac{{π}^{2}}{30}{{{{g}_{q}}{+}\frac{7{{g}_{g}}}{8}}}$ …( where gq and gg are the degeneracy factors for massless quarks and gluons)

$\displaystyle {{Also {}{}{P}}_{QGP}}{=}\frac{1}{3}{{{{ϵ}_{QGP}}{-}{4}{B}}}$

Some Modifications to Bag EoS are suggested as follows (see Ref. [8])

Modifications:

• A reduction in Stephan-Boltzmann constant:

Pressure and energy density is given as:

$\in=\sigma_1T^4-B_1$ , $P=\frac{\sigma_1}3T^4-AT-B_1$

• The introduction of another temperature dependent term in the pressure and also in energy density

With this modification the equation for pressure and energy density is reduced tas:

$\in=\sigma_2T^4-CT^2+B_2$ , $P=\frac{\sigma_2}3T^4-CT^2-B_2$

This modification was suggested by Pisarski (see Ref. [9]).

• A Bag constant with negative sign

References

[1] Introduction to elementary particles, David Griffith, WIELY-VCLT Publication

[2] J.C. Collins and M.J. Perry, Phys. Rev. Lett. 34, 1353 (1975)

[3] A. Bazavor etal. Phys. Rev. D, 85, 059503 (2012)

[4] S. Barsongi etal. J. Phys. G, 38, 124101 (2011)

[5] M. Cheng etal. Phys. Rev. D, 81, 054504 (2010)

[6] S. Gupta etal. Science, 332, 1525 (2011)

[7] QGP Book R. C. Hroz

[8] S. M. Sanches Jr., F.S. Navarra, D. A. Fogaca, Nucl. Phys. A, 937, 1 (2015)

[9] R. D. Pisarski, Phys. Rev. D 74 121703 (2006)

Acknowledgement

Success is where preparation and opportunities meet. I am very fortunate to get this opportunity from Indian Academy of Sciences Bangalore. I show my sincere gratitude towards them for offering me a Summer Research Fellowship for two months.

I owe my profound gratitude to my project guide Dr. Jajati K. Nayak, VECC Kolkata, for their most valuable guidance and support. The way they teach and explain physics inspire me a lot. My two months journey is really fruitful for me under his experience and knowledge.

I extend my gratitude to Dr. P. Y. Nabhiraj for providing me an opportunity to undergo summer training at VECC.

I would like to thank Mr. Sushant Singh and Mrs. Purabi Ghosh, who took keen interest and guided me all along, till the completion of project work by providing all necessary information.

I am thankful to Prof. Dr. S. K. Omanwar (HOD), SGBAU Amravati for their constant support, encouragement and also their most valuable advice and helping hand without which I could not be able to complete this project.

Last but not the least I would like to thank my parents and family for their moral support and my friends who boost me up every time.

Appendices

Natural units

The basic units in physics are mass, length, and time and the SI system express these in meters,

kilograms and seconds. Such units are not very appropriate in high energy physics, where typical length are 10-15m typical masses are 10-27Kg.The units using in high energy physics are known to be natural units.

In natural units, we take h=c=1.In high energy we express all the physical quantities in the order of energy units (order of GeV or TeV).The conversion of SI units to natural units of the physical quantities are given below:

Mass in natural units:

According to SI system unit of mass is Kg.

According Einstein mass energy relation

$E=m\times c^2$
${⟹}{m}{=}{}\frac{E}{{c}^{2}}{=}{}\frac{eV}{{c}^{2}}$

We have

${1}{e}{V}{≅}{1.602}{×}{{10}^{-19}}{}{J}$
$c\cong3\times{10}^8$
$\Longrightarrow\frac{1eV}{c^2}=\frac{1.602\times{10}^{-19}}{({3\times{10}^8)}^2}\cong1.78\times{10}^{-36}Kg$
$\Longrightarrow\frac{1GeV}{c^2}\cong1.78\times{10}^{-27}Kg$

Therefore in Natural units,

Natural unit
 Unit of mass GeV/c2.

The SI unit of length is meter, but in high energy physics the unit of length is express in terms of GeV-1

We have

${l}{e}{n}{g}{t}{h}{=}\frac{ℏc}{{{M{{c}^{2}}}}}{=}\frac{6.588×{{10}^{-25}}{{GeV{}{s}}}{×}{3}{×}{{10}^{8}}{m/s}}{GeV}{≅}{0.1975}{}{f}{m}$
${⟹}{1}{}{G}{e}{{V}^{-1}}{≅}{0.1975}{}{f}{m}$

In Natural units

Natural unit
 The unit of length is GeV-1

Similarly,

${t}{i}{m}{e}{=}\frac{ℏc}{{{M{{c}^{3}}}}}{≅}{6.59}{×}{{10}^{-25}}{}{s}$
$time=1GeV^{-1}\cong6.59\times{10}^{-25}s$
Natural unit
 The unit of time is 1GeV-1

For temperature conversion,

We have,

${E}{=}{}{{K}_{B}}{T}{}$,

Where KB is Boltzmann’s Constant and T is temperature

$⟹T=\frac{E}{{K}_{B}}\cong \frac{1eV}{1.38×{10}^{-23}×0.625×{10}^{19}\left(eV/K\right)}\cong 1.1605×{10}^{4}$
${⟹}{1}{e}{V}{≅}{11,605}{}{K}{≅}{{10}^{4}}{}{K}$
$\Longrightarrow1GeV\cong{10}^{13}K$
 Unit of Temperature is GeV

Source

• : https://en.wikipedia.org/wiki/Van_der_Waals_equation
• : inspirehep.net
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