equilibration of the quark gluon plasma - dylan van zyl · 2019-01-14 · 2 relativistic kinetic...
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HONOURS THESIS : 26/10/2018
Equilibration of the Quark Gluon Plasma
Author : Dylan van Zyl, Supervisor : Prof. Andre Peshier
Department of Physics, University of Cape Town
E-mail: [email protected], [email protected]
Abstract: We study the evolution of a gluon plasma using the relativistic Boltzmann
equation in the relaxation-time approximation under boost invariance. We first generalize
work by Baym by considering a non-zero formation time, and we calculate for CGC-like
initial conditions various bulk properties as functions of time. We extend the formalism to
the case of (approximate) gluon number conservation to explore a necessary condition for
the formation of a gluonic Bose-Einstein condensate in heavy-ion collisions. 1
1A thesis submitted in fulfillment of the requirements for the degree of Bachelor of Science (Honours).
Contents
1 Introduction 1
2 Relativistic Kinetic Theory 2
2.1 Conventions 2
2.2 Basic definitions 2
2.3 The particle four-current 2
2.4 The energy-momentum tensor 3
2.5 Energy-momentum and particle number conservation 3
2.6 The entropy four-current 3
2.7 The Boltzmann transport equation 4
2.8 The relaxation time approximation 4
3 The Baym model 4
3.1 The kinetic equation under boost invariance 5
3.2 Implications of energy-momentum conservation 6
3.2.1 The free streaming and ideal hydrodynamic limit 7
3.2.2 The general case 8
3.3 The initial distribution function 10
3.4 The g(t) function 10
3.5 Evolution of the distribution function 13
3.6 Time dependence of the particle number density 14
3.7 Time dependence of the entropy density 18
4 Extending the Baym model 18
4.1 Implications of particle number conservation 18
4.2 Formation of a Bose-Einstein condensate 20
4.3 The condensation onset - time problem 22
5 Summary 23
A Hyperbolic Space-Time Coordinates 23
B Deriving (3.27) 23
C Deriving (3.36) 24
D Deriving the Volterra equation (3.43) 25
E Derivation of (3.61) 26
F Derivation of (4.9) 27
– i –
G Derivation of (4.10) 27
References 28
1 Introduction
One of the main goals of ultra-relativistic heavy ion collisions is the observation of the
deconfinement/chiral phase transition predicted by QCD [1]. At low baryon density and/or
low temperature quarks and gluons are confined within hadrons, however at high baryon
density and/or high temperature a phase transition can occur where the quarks and gluons
become deconfined. Present experimental evidence indicates that in an ultra-relativistic
heavy ion collision a very dense and strongly interacting matter called the quark gluon
plasma (QGP) is formed. The study of the QGP plays an important role in the effort to
consolidate the Standard Model. It also has relevance in thermal quantum field theory as
well as grand unification theories. Another motivation to study the QGP is that it existed in
the early universe. The Big bang model predicts that the universe existed as a deconfined
state of quarks and gluons until about 1µs after the big bang. When the temperature
dropped below the transition temperature Tc ∼ 100MeV the deconfinement-confinement
transition occurred and hadrons were formed [2].
z
t
strong fields classical dynamics
gluons & quarks out of eq. viscous hydro
gluons & quarks in eq. ideal hydro
hadrons kinetic theory
freeze out
Figure 1: The stages of a heavy ion collision [3].
Consider the space-time picture of an ultra-relativistic heavy ion collision. Figure 1
depicts the collision of two nuclei in the (t, z) plane. The two Lorentz contracted nuclei
collide at almost the speed of light at (t = 0, z = 0). The subsequent expansion in space-
time goes through the various stages until the created particles ‘freeze out’ meaning that
the system is in the phase where the final hadrons are emitted.
Using relativistic kinetic theory we aim to describe the evolution of the QGP via the
Boltzmann transport equation using the relaxation-time approximation, which can model
the behavior of the parton system from the early pre-equilibrium stages to hadronization.
– 1 –
2 Relativistic Kinetic Theory
In this section we present the fundamentals of kinetic theory that will be used throughout
this thesis following [1] and [2].
2.1 Conventions
We work in natural units (c = ~ = kB = 1). We work with the ‘east coast’ convention for
the Minkowski metric
ηµν =
1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −1
. (2.1)
Throughout this thesis the only particles we are considering are gluons 1, which are massless
bosons, so 2
ηµνpµpν = (p0)2 − |~p|2 = m2 = 0 =⇒ p0 = |~p| = p. (2.2)
To simplify notation we also define, ∫dΓ =
∫d3p
(2π)3. (2.3)
The gluon degeneracy is dg = 16. For notational convenience we set dg to 1 unless specified
otherwise.
2.2 Basic definitions
In kinetic theory a system is studied in terms of its distribution function f(x, p) = f(~x, ~p, t)
which gives the density of particles at each point in phase space, such that
∆N = f(~x, ~p, t)∆3x∆3p is the average number of particles in a phase-space volume element
∆3x∆3p around the phase-space point (~x, ~p) at time t. The main task of kinetic theory is to
describe the time evolution of f(~x, ~p). For the purposes of this thesis, it is understood that
the distribution function depends on position, momentum and time even if for notational
convenience some of the arguments are suppressed.
2.3 The particle four-current
The particle four-current is defined as the first moment of the distribution function,
Nµ(x) =
∫dΓpµ
p0f(x, ~p). (2.4)
The particle four-current is a 4-vector since the integration measure dΓ/p0 as well as f are
Lorentz invariant. The components of the particle four-current can be identified as follows:
The particle density is
n(x) = N0(x) =
∫dΓf(x, ~p). (2.5)
1The reason for this will become clear in Section 3.2From here on the notation p and p0 will be used interchangeably without further mention.
– 2 –
The particle flow is
N i(x) =
∫dΓ
pi
p0f(x, ~p) with i = 1, 2, 3. (2.6)
2.4 The energy-momentum tensor
The energy-momentum tensor for the system is defined as a second moment of the distri-
bution function,
Tµν =
∫dΓpµpν
p0f(x, ~p). (2.7)
The components of Tµν have the following physical interpretation. The 00-component gives
the local energy density ε,
ε(x) = T 00 =
∫dΓp0f(x, ~p). (2.8)
The momentum density is given by the components,
T i0(x) =
∫dΓpif(x, ~p) with i = 1, 2, 3, (2.9)
which coincides with the energy flux. The energy flow is defined as
T 0i(x) =
∫dΓpif(x, ~p) with i = 1, 2, 3. (2.10)
Lastly, the momentum flow is the Maxwell-stress tensor
T ij(x) =
∫dΓpipj
p0f(x, ~p) with i = 1, 2, 3. (2.11)
An important property to note is that the energy-momentum tensor is traceless,
Tr(T ) =
∫dΓf(x, ~p)
(p0 − (p1)2
p0− (p2)2
p0− (p3)2
p0
)= 0. (2.12)
2.5 Energy-momentum and particle number conservation
Conservation of particle number is described by the equation
∂µNµ = 0, (2.13)
which is a continuity equation. Energy-momentum conservation is similarly given by
∂µTµν = 0, (2.14)
which are four continuity equations.
2.6 The entropy four-current
The entropy four-current is defined as
Sµ(x) =
∫dΓpµ
p0Φ[f(x, ~p)], (2.15)
where for bosons we have
Φ[f(x, ~p)] = (1 + f) ln(1 + f)− f ln f. (2.16)
In particular, the entropy density is defined by,
s(x) =
∫dΓ[(1 + f) ln(1 + f)− f ln f ]. (2.17)
– 3 –
2.7 The Boltzmann transport equation
The kinetic or transport equation describes the space-time development of the distribution
function. It is a non-linear integro-differential equation that describes the evolution of an
initial distribution towards equilibrium. The Boltzmann transport equation reads [4]
pµ∂µf(x, p) = C[f(x, p)]. (2.18)
C[f(x, p)] is known as the collision term and describes the scattering processes that causes
the evolution of the system. In 3-vector notation the Boltzmann equation has the form(∂
∂t+ ~vp · ~∇
)f(~x, ~p, t) = C∗[f(~x, ~p, t)], (2.19)
with ~vp = ~pp0
and C∗[f(x, p)] = C[f(x,p)]p0
.
In systems where the effects of collisions are negligible the collision term vanishes and the
relativistic Boltzmann equation reduces to(∂
∂t+ ~vp · ∇
)f(~x, ~p, t) = 0. (2.20)
It should be noted that we require that any collision term which describes a physical
situation to uphold energy-momentum conservation.
2.8 The relaxation time approximation
We take the following Ansatz for the collision term in the Boltzmann transport equation,
which is known as the relativistic covariant form of the relaxation time approximation [4].
C∗[f(x, p)] =pµuµp0
feq − fθ
, (2.21)
where feq is the local equilibrium distribution function which here takes the form of a
Juttner-Bose-Einstein distribution and the relaxation time θ is a parameter that controls
the rate of approach to equilibrium. The term pµuµ/p0 can be thought of as a ‘Doppler
factor’ with uµ being the collective flow velocity.
3 The Baym model
In this section, we follow Baym’s paper [5] in which he studies the approach to local thermo-
dynamic equilibrium in a central ultra-relativistic heavy ion collision under the assumption
of boost invariance. This ‘Baym model’ only considers energy-momentum conservation
(without particle number conservation). Also, the quark gluon system in heavy ion col-
lisions is simplified to a system where we only consider gluons. This assumption has the
motivation that in the early stages of a heavy ion collision high gluon densities may domi-
nate the effects of quarks.
In Sections 3.1-3.2 we discuss the basic elements of the Baym model after which we study
the model by numerically calculating the distribution function and investigating the behav-
ior of the number density and entropy density for various the relaxation times and initial
distributions.
– 4 –
3.1 The kinetic equation under boost invariance
Starting from (2.19), under the assumption that the system is homogenous in the transverse
plane we have that the initial distribution function is only dependent on z. Therefore we
have∂f(~p, z, t)
∂t+ vpz
∂f(~p, z, t)
∂z= C∗[f(~x, ~p, t)]. (3.1)
From the fact that gluons are massless we have p20 − p2z = p2⊥ with p2⊥ = p2x + p2y. One can
therefore rewrite
p0 = p⊥ cosh y and pz = p⊥ sinh y, (3.2)
where y is the so-called longitudinal rapidity. From this change of variables and by further
switching to spacetime-hyperbolic coordinates (t, z) → (τ, η), the distribution function f
can be written as
f(~p, z, t) = f(p⊥, y, τ, η). (3.3)
By making the assumption that the flow is boost invariant we then have that
f(p⊥, y, τ, η) = f(p⊥, y − η, τ). (3.4)
From this (3.1) becomes,
(cosh η
∂
∂τ+t
τsinh η
∂
∂(y − η)−tanh(y−η)(sinh η
∂
∂τ+t
τcosh η
∂
∂(y − η))
)f(p⊥, y−η, τ)
= C∗[f(p⊥, y − η, τ)]). (3.5)
See appendix A. On the central slice η = 0 the eigentime and lab time agree, τ → t,
and equation (3.5) simplifies to(∂
∂t− 1
ttanh y
∂
∂y
)f(p⊥, y, t) = C∗[f(p⊥, y, t)]. (3.6)
Switching back to momentum coordinates results in
∂f(p⊥, pz, t)
∂t− pz
t
∂f(p⊥, pz, t)
∂pz= C∗[f(~x, ~p, t)]. (3.7)
It is useful to notice that the left hand side of (3.7) is in fact the total time derivative of
f(p⊥, pz, t) at pzt = c = constant,
∂f(p⊥, pz, t)
∂t
∣∣∣∣pzt
=∂f(p⊥, pz, t)
∂t+∂f(p⊥, pz, t)
∂pz
∂
∂t(c/t)
=∂f(p⊥, pz, t)
∂t− pz
t
∂f(p⊥, pz, t)
∂pz.
(3.8)
Therefore we have that∂f(p⊥, pz, t)
∂t
∣∣∣∣pzt
= C∗[f(~x, ~p, t)]. (3.9)
– 5 –
3.2 Implications of energy-momentum conservation
As mentioned previously, for a collision term to be physical it has to satisfy energy-
momentum conservation. It can be shown that the following condition on the collision
term ∫dΓpνC∗[f ] = 0 for ν = 0, 1, 2, 3 (3.10)
implies energy-momentum conservation (2.14). Specifically, taking ν = 0 in (3.10) we have
that ∫dΓp0C∗[f ] = 0. (3.11)
For the rest of this thesis this result will be referred to as saying that the energy moment
of the collision term vanishes. Also when multiplying an equation through by p0 and inte-
grating with respect to dΓ, we will simply call this procedure taking the energy moment
of the equation.
Taking the energy moment of (3.7),∫dΓp0
∂f(p⊥, pz, t)
∂t− 1
t
∫dΓp0pz
∂f(p⊥, pz, t)
∂pz= 0, (3.12)
implies that∂ε(t)
∂t− 1
t
∫dΓp0pz
∂f(p⊥, pz, t)
∂pz= 0. (3.13)
Note that via the chain rule we have pz∂f(p⊥,pz ,t)
∂pz= −f(p⊥, pz, t) + ∂
∂pz
(pzf(p⊥, pz, t)
).
Therefore,∫dΓp0pz
∂f(p⊥, pz, t)
∂pz= −
∫dΓp0f(p⊥, pz, t) +
∫dΓp0
∂
∂pz
(pzf(p⊥, pz, t)
)= −ε(t) +
∫dΓp0
∂
∂pz
(pzf(p⊥, pz, t)
).
(3.14)
Since p0 =√p2⊥ + p2z we can write the 2nd term of the above equation as∫
dΓp0∂
∂pz
(pzf(p⊥, pz, t)
)=
∫ ∞0
dp⊥(2π)2
p⊥
∫ ∞−∞
dpz
√p2⊥ + p2z
∂
∂pz
(pzf(p⊥, pz, t)
)=
1
(2π)2
∫dp⊥p⊥
[pzf(p, t)
√p2⊥ + p2z
]pz=∞pz=−∞
−∫dΓp2zp0f(p⊥, pz, t)
= −∫dΓp2zp0f(p⊥, pz, t),
(3.15)
where we have used integration by parts and the fact that f is localized in pz, i.e.
limpz→±∞
f(p⊥, pz, t) = 0. (3.16)
– 6 –
Due to momentum isotropy in the xy plane of the system under consideration the longitu-
dinal pressure PL(t) and the transverse pressure PT (t) read
PL(t) = T 33 =
∫dΓp2zpf(p, t), (3.17)
PT (t) = T 11 = T 22 =1
2
∫dΓp2⊥pf(p, t). (3.18)
Therefore we can write ∫dΓp0
∂
∂pz
(pzf(p⊥, pz, t)
)= −PL(t). (3.19)
Therefore from (3.13) and (3.14) we have
∂ε(t)
∂t+ε(t) + PL(t)
t= 0. (3.20)
We can also rewrite this equation as
PL(t) = −(t∂ε(t)
∂t+ ε(t)
)= − ∂
∂t
(tε(t)
).
(3.21)
From the fact that the energy-momentum tensor is traceless, cf. (2.12), we can also write
ε(t)− 2PT (t)− PL(t) = 0 =⇒ PT (t) =1
2(ε(t)− PL(t))
=⇒ PT (t) =1
2(ε(t) +
∂
∂t(tε(t)))
=⇒ PT (t) =1
2t
∂
∂t
(t2ε(t)
).
(3.22)
Therefore we have that the longitudinal and transverse pressure are fully determined from
the energy density.
3.2.1 The free streaming and ideal hydrodynamic limit
We first consider the free streaming limit (θ →∞). In the absence of collisions, C[f(x, p)] =
0, (3.9) becomes∂f(p⊥, pz, t)
∂t
∣∣∣∣pzt
= 0. (3.23)
This implies that the distribution function at different times are related by the constraint
pzt = c = constant. Therefore,
f(p⊥, pz1 , t1) = f(p⊥, pz2 , t2) with pz1t1 = pz2t2. (3.24)
– 7 –
In particular,
f(p⊥, pz, t) = f(p⊥, pz0 , t0) with pzt = pz0t0
= f
(p⊥,
pzt
t0, t0
)= f0
(p⊥,
pzt
t0
),
(3.25)
with f0 being the distribution function at initial time t0.
The energy density in the free-streaming limit can be easily calculated as
ε(t) =
∫dΓpf(p⊥, pz, t)
=
∫dΓpf0
(p⊥,
pzt
t0
).
(3.26)
One can show (see appendix B) that (3.26) implies that
ε(t) =t0t
∫dΓp
√1+
[(t0t
)2
− 1
](cos θp)2f0(p⊥, pz), (3.27)
where θp is the angle between ~p and the z-axis. In this section and everything that follows,
we make the physically motivated assumption [5] that the initial distribution function
is sharply peaked around pz = 0. Therefore θp in (3.27) is taken to be π2 . Using this
simplification the energy density becomes
ε(t) =t0t
∫dΓpf0(p⊥, pz). (3.28)
With the energy density at t = t0 denoted by,
ε0 = ε(t0) =
∫dΓpf0(p⊥, pz), (3.29)
we have the following expression for the energy density in the free-streaming limit
ε(t) =t0tε0. (3.30)
In the hydrodynamic limit (θ → 0) it is easy to show that the energy density is [5]
ε(t) = ε0
(t0t
)4/3
. (3.31)
3.2.2 The general case
Now we consider the general case in which the collision term is given by the relaxation-time
approximation (2.21). Since we are considering the system at mid rapidity the Doppler
factor is unity and the relaxation time approximation is simply
C∗[f(x, p)] = −f − feqθ
. (3.32)
– 8 –
Taking the energy moment of (3.32),∫dΓpC∗[f(x, p)] = −1
θ
∫dΓp(f − feq), (3.33)
implies that
ε(t) =
∫dΓpfeq(p, t), (3.34)
where in this case feq is the Bose-Einstein distribution function,
feq =1
ep/T (t) − 1. (3.35)
Therefore (3.34) becomes,
ε(t) =
∫dΓ
p
ep/T (t) − 1
=π2
30T (t)4
(3.36)
The above integral is standard, see appendix C.
Now substituting the relaxation-time expression (3.32) into (3.9) results in
θ∂f(p⊥, pz, t)
∂t
∣∣∣∣pzt
+ f = feq. (3.37)
For constant θ the above equation can be integrated analytically as follows: To start off,
it us useful to note that∂
∂t
(etθ f
)= e
tθ
(1
θf +
∂f
∂t
). (3.38)
Therefore (3.37) can be written as
∂
∂t
(etθ f
)∣∣∣∣pzt
= etθfeqθ. (3.39)
By integrating (3.39) with respect to t, keeping pzt = c constant we have,
et′θ f(p⊥,
c
t′, t′)
∣∣∣∣tt0
=1
θ
∫ t
t0
et′θ feq(t
′)dt′, (3.40)
which becomes,
etθ f(p⊥, pz, t)− e
t0θ f
(p⊥,
pzt
t0, t0
)=
1
θ
∫ t
t0
et′θ feq
(p⊥,
pzt
t′, t′)dt′. (3.41)
Writing the components of the momentum in feq in terms of the magnitude, (3.41) becomes,
etθ f(p⊥, pz, t) = e
t0θ f0
(p⊥, pz
t
t0
)+
∫ t
t0
dt′
θet′θ feq
(√p2⊥ +
(pzt
t′
)2
, t′). (3.42)
– 9 –
It can be shown that by taking the energy moment of (3.42) and using the appropriate
change of variables that this produces the following Volterra integral equation for g(t) [5]
(see Appendix D).
g(t) = e−(t−t0)/θ +
∫ t
t0
dt′
θe−(t−t
′)/θg(t′)h(t′/t), (3.43)
with
g(t) ≡ tε(t)
t0ε(t0), (3.44)
and
h(α) =1
2
(α+
arccos(α)√1− α2
). (3.45)
Baym solves (3.43) under the additional assumption that t0 → 0. We will generalize
this result keeping both t0 and θ as relevant timescales. A non-zero t0 seems important as
it signifies the time when a transport treatment of the QGP becomes viable.
3.3 The initial distribution function
As the initial distribution, we take the following Color-Glass-Condensate (CGC) inspired
distribution
f0(p⊥, pz) =a
eβ(p⊥−Q) + 1e−(pz/σ)
2. (3.46)
This initial distribution function is dependent on the following parameters : (i) a scaling
parameter a, (ii) σ which controls the width of the Gaussian in pz, (iii) Q which sets the
energy scale and (iv) β which can be seen as a smearing parameter. We may set Q = 1,
therefore all energies are measured in units of Q. In the case that we take β to be large
this distribution simplifies to
f0(p⊥, pz) = aΘ(Q− p⊥)e−(pz/σ)2. (3.47)
3.4 The g(t) function
In section 3.3.2 we derived the integral equation (3.43), which can be solved using a nu-
merical scheme [6]. In the free streaming limit corresponding to θ → ∞ we have that
g(t) = 1 and in the ideal hydrodynamics limit θ → 0 we have that g(t) = t−1/3, which is
evident from (3.30) and (3.31) respectively. Figure 2(a) shows g(t) for various relaxation
times versus t/t0. We see that increasing the relaxation time brings the solution closer to
the free streaming limit and conversely decreasing the relaxation time brings the solution
closer to the hydrodynamic limit. Fig 2(b) shows g(t) for various initial times t0 versus
t/θ.
Since we have solved (3.43) we can then use (3.44) and (3.36) to obtain the energy
density and the temperature in terms of g(t) for various relaxation times,
ε(t) =ε0t0tg(t) T (t) =
(30
π2ε(t)
)1/4
. (3.48)
– 10 –
0 5 10 15t/ t0
0.4
0.6
0.8
1.0
g(t)
θ
θ → ∞
θ → 0
(a) g vs t/t0 for relaxation times θ = 0.5, 1, 2, 4, 8t0
0.5 1 5 10t/θ0.4
0.5
0.6
0.7
0.8
0.9
1.0g(t)
t0
(b) g vs t/θ for t0 = 0, 0.0125, 0.25, 0.5.1, 2θ
Figure 2: The solution g(t) of (3.43) [7].
Using (3.36) we find
T (t) = T0
(t0tg(t)
)1/4
. (3.49)
We now generalize Baym’s proof of equilibration. Analogous to Baym’s approach3 we
consider the integral in the Volterra equation (3.43),
I =
∫ t
t0
dt′
θe−(t−t
′)/θg(t′)h(t′/t). (3.50)
Integrating by parts
I = e−(t−t′)/θg(t′)h(t′/t)
∣∣∣∣t′=tt′=t0
−∫ t
t0
dt′e−(t−t′)/θ ∂
∂t′(g(t′/)h(t′/t))
= g(t)− e−(t−t0)/θg(t0)h(t0/t)−∫ t
t0
dt′e−(t−t′)/θ ∂
∂t′(g(t′)h(t′/t))
= g(t)− e−(t−t0)/θg(t0)h(t0/t)−∫ t
t0
dt′e−(t−t′)/θ ∂
∂t′(g(t′)h(t′/t)).
(3.51)
3See equation (23) in [5]
– 11 –
2 4 6 8 10t/ t00.05
0.10
0.50
1ϵ/ ϵ0
θ
θ → ∞θ → 0
(a) Energy density
2 4 6 8 10t/ t0
0.6
0.7
0.8
0.9
1.0
T/ T0
θ
θ → ∞θ → 0
(b) Temperature
Figure 3: The energy density and temperature for relaxation times θ = 0.5, 1, 2, 4, 8
Now substituting the original expression for g(t) from (3.43)
I = e−(t−t0)/θ(1− g(t0)h(t0/t)) + I −∫ t
t0
dt′e−(t−t′)/θ ∂
∂t′(g(t′)h(t′/t)). (3.52)
This implies that∫ t
t0
dt′e−(t−t′) ∂
∂t′
(g(t′)h(t′/t)
)= e−(t−t0)/θ(1− g(t0)h(t0/t)). (3.53)
In order for the integral on the left hand side to converge for large t we need to have
∂
∂t′
(g(t′)h(t′/t)
)∣∣∣∣t′→t→ 0 as t→∞. (3.54)
Given that h(1) = 1 and h′(1) = 1/3 (3.54) reduces to
g′(t) = −g(t)1
3tas t→∞. (3.55)
From (3.55) we have that
g(t) ∼ t−1/3 as t→∞, (3.56)
– 12 –
which corresponds to the time behaviour in the ideal hydrodynamic limit. We therefore
see that the solution approaches the hydrodynamic limit for large times. Conversely, if the
system is in a state such that g(t) ∼ t−1/3 one can say that the system has thermalized. In
effect, one can calculate the time t∗ at which the system thermalizes in that g(t∗) ∼ (t∗)−1/3.
3.5 Evolution of the distribution function
The distribution function can be obtained from (3.42). In the limit of free streaming
(θ →∞) we have,
f(p⊥, pz, t) = f0
(p⊥, pz
t
t0
). (3.57)
We can see that (3.57) corresponds to the initial distribution that shrinks and becomes
more and more peaked along the pz axis as time progresses. Therefore our assumption
that f be narrow in pz is consistent.
In the hydrodynamic limit we simply have
f(p, t) = feq(p, t), (3.58)
since for θ → 0 the system thermalizes immediately.
For the general case we can rewrite (3.42) in a way that reveals the explicit t′ dependence
in (3.35) which comes from the time dependence of the temperature T (t′),
et/θf(p⊥, pz, t) = et0/θf0
(p⊥, pz
t
t0
)+
∫ t
t0
dt′
θet′/θ
(e
√p2⊥+(pz
tt′ )
2
T (t′) − 1
)−1(3.59)
Substituting (3.49) in (3.59) allows us to compute the distribution function for finite re-
laxation times and initial distributions.
Fig’s 4-8 show plots of the initial distribution as well as the distribution function for
θ = 1, 4 at t = 2, 5. One can see qualitatively how the distribution changes towards
equilibrium distribution.
Figure 4: The initial distribution p⊥f0(p⊥, pz) with a = 1, β = 16, σ = 0.25.
– 13 –
Figure 5: The distribution function p⊥f(p⊥, pz) at t = 2 for θ = 1 from the initial
distribution in Fig 4.
Figure 6: The distribution function p⊥f(p⊥, pz) at t = 5 for θ = 1 from the initial
distribution in Fig 4.
3.6 Time dependence of the particle number density
Taking the zeroth moment of (3.42) results in
et/θn(t) = et0/θt0n0t
+
∫ t
t0
dt′
θet′/θ t′
t
∫feq(p, t)dΓ
= et0/θt0n0t
+
∫ t
t0
dt′
θet′/θ t′
tneq(t
′),
(3.60)
with
neq(t′) =
∫feq(p, t
′)dΓ =ζ(3)
π2T 3(t′). (3.61)
– 14 –
Figure 7: The distribution function p⊥f(p⊥, pz) at t = 2 for θ = 4 from the initial
distribution in Fig 4.
Figure 8: The distribution function p⊥f(p⊥, pz) at t = 5 for θ = 4 from the initial
distribution in Fig 4.
(See appendix E). Continuing from (3.60) we have
et/θn(t) = et0/θt0n0t
+
∫ t
t0
dt′
θet′/θ t′
t
ζ(3)
π2T 3(t′) (3.62)
Using (3.48) we have
et/θn(t) = et0/θt0n0t
+ζ(3)
π2
(30t0ε0π2
)3/4 ∫ t
t0
dt′
θet′/θ t′
t
(g(t′)
t′
)3/4
. (3.63)
Recognizing that T0 = (30ε0/π2)1/4 and that neq(t0) = ζ(3)
π2 T30 we can write (3.63) as
n(t)t
n0t0= e−(t−t0)/θ +
neq(t0)
n0
∫ t
t0
dt′
θe−(t
′−t0)/θ(t′
t0g3(t′)
)1/4
. (3.64)
– 15 –
2 4 6 8 10 12 14t/ t00.9
1.0
1.1
1.2
1.3nt/ n0
θ
(a) Overpopulated case, a = 0.5
2 4 6 8 10 12 14t/ t01.0
1.2
1.4
1.6
1.8
2.0nt/ n0
θ
(b) Underpopulated case, a = 0.1
2 4 6 8 10 12 14t/ t00.9
1.0
1.1
1.2
1.3nt/ n0
θ
(c) Critically-populated case, a = 0.23765
Figure 9: nt/n0 vs t/t0 according to equation (3.64) for relaxation times θ =
0.5, 1, 2, 4, 8. The arrow shows the direction of increasing θ.
We can explicitly consider the limit of ideal hydrodynamics and free streaming on the
particle number density. For ideal hydrodynamics we have g(t′) = ( t0t′ )1/3. Then we have(
n(t)t
n0t0
)hydro
= e−(t−t0)/θ +neq(t0)
n0
∫ t
t0
dt′
θe−(t
′−t0)/θ
= e−(t−t0)/θ +neq(t0)
n0(1− e−(t−t0)/θ)
=neq(t0)
n0since θ → 0.
(3.65)
– 16 –
2 4 6 8 10t/ t0
0.1
0.2
0.5
1
n/ n0
a
(a) θ = 0.5
2 4 6 8 10t/ t0
0.1
0.2
0.5
1
n/ n0
a
(b) θ = 2
2 4 6 8 10t/ t0
0.1
0.2
0.5
1
n/ n0
a
(c) θ = 4
Figure 10: n/n0 vs t/t0 for different relaxation times in the case where particle number
is not conserved, varying the scaling parameter a = 0.1, 0.2, 0.5, 1, 2 of the initial dis-
tribution and setting β = 100, σ = 0.5. The arrow indicates the direction of increasing
a.
For free streaming g(t′) = 1 so,(n(t)t
n0t0
)fs
= e−(t−t0)/θ +neq(t0)
n0
∫ t
t0
dt′
θe−(t
′−t0)/θ(t′
t0
)1/4
= 1 since θ →∞.(3.66)
– 17 –
3.7 Time dependence of the entropy density
Now that we know how the distribution function evolves with time (see section 3.5), we
can also calculate the entropy density (2.17) for finite values of the relaxation time and the
scaling parameter. It is important to note that the entropy density depends on the initial
distribution while the energy density is ‘universal’ since it was obtained from (3.43).
4 Extending the Baym model
In this section we consider extending the Baym model to the case that the gluon number is
conserved. This is an assumption which can only be approximately satisfied for a sufficiently
small time interval. If gluon number conservation holds for a sufficiently long time window
it can have interesting implications such as the formation of a Bose-Einstein condensate.
4.1 Implications of particle number conservation
By imposing particle number conservation (2.13) we have∫dΓ
1
p0pµ∂µf =
∫dΓ
1
p0C[f ]
=
∫dΓC∗[f ]
= 0.
(4.1)
Therefore we have that the zeroth moment of the collision term vanishes. In the case of
free streaming we can calculate the particle density immediately
n(t) =
∫dΓf(~p, t)
=
∫dΓf0(p⊥, pz
t
t0)
=t0t
∫dΓf0(p⊥, pz)
=t0n0t.
(4.2)
Comparing (4.2) and (3.60) we see that for the free streaming limit, the particle number
is conserved by ‘definition’. For the general case we start by taking the zeroth moment of
the relaxation time approximation (2.21)∫dΓC∗[f(x, p)] = −1
θ
∫dΓ(f − feq), (4.3)
from which we can deduce
n(t) =
∫dΓfeq(p, t). (4.4)
Following [7] we take the zeroth moment of eq (3.7)∫dΓ∂f
∂t−∫dΓpzt
∂f
∂pz= 0, (4.5)
– 18 –
0 10 20 30 40 50 60t/ t0
0.20
0.25
0.30
0.35
s t
θ
(a) Initially Overpopulated a = 2ac
0 10 20 30 40 50 60t/ t0
0.100
0.125
0.150
0.175
0.200
s t
θ
(b) Initially Critically-populated ac = 0.388039
0 10 20 30 40 50 60t/ t0
0.06
0.08
0.10
0.12
s t
θ
(c) Initially Underpopulated a = 0.5ac
Figure 11: st vs t/t0 with θ = 1, 4 for the case where particle number is not conserved.
β = 16, σ = 0.25. The arrow indicates the direction of increasing θ.
which implies that
dn(t)
dt=
1
t
∫dΓpz
∂f
∂pz
=1
t
∫dpxdpy(2π)3
(pzf
∣∣∣∣∞−∞−∫ ∞−∞
fdpz
)= −1
tn(t).
(4.6)
– 19 –
Equation (4.6) is solved by
n(t) =t0tn0. (4.7)
4.2 Formation of a Bose-Einstein condensate
When considering particle number conservation in addition to energy-momentum conser-
vation the local equilibrium distribution will take the form
feq(p, t) =1
ep−µ(t)T (t) − 1
, (4.8)
where µ is the chemical potential. We require that the µ ≤ 0 in order for the distribution
function to be well defined. We can calculate the energy density and the particle number
density from (3.34) and (4.4) to be
ε(t) =3
π2T 4(t)Li4
(eµ(t)/T (t)
), (4.9)
and
n(t) =1
π2T 3(t) Li3
(eµ(t)/T (t)
). (4.10)
See appendix F and G. Note that for a given initial condition, the parameters that define
the equilibrium distribution are fully determined by the conservation laws. The initial
number density and energy density are determined by the parameters a, σ, β from (3.46).
By fixing β and σ we can use a as a ‘population parameter’ to determine in what state the
system is. More concretely, we consider contours of constant particle number density and
energy density
n0(a) =1
π2T 3 Li3
(eµ/T
), (4.11)
and
ε0(a) =3
π2T 4Li4
(eµ/T
). (4.12)
In the underpopulated and critically populated cases the values of the equilibrium param-
eters T and µ are found where the contours intersect. In the critically populated case
the intersection occurs at the maximal possible value µ = 0. In the overpopulated case,
no solution for µ < 0 exists and a condensate is necessary to contain the excess particles
[8]. We can write down a dimensionless combination of the particle number and energy
density that distinguishes between these cases and that gives an indication whether or not
a condensate will form. We find this critical value by setting µ = 0 in (4.11) and (4.12),
neq(T, 0) =ζ(3)
π2T 3, (4.13)
and
εeq(T, 0) =π2
30T 4. (4.14)
– 20 –
2 4 6 8 10t/ t00.000
0.005
0.010
n4/ϵ3
θθ → ∞
θ → 0
(a) Initially Overpopulated a = 0.5
2 4 6 8 10t/ t0
-0.001
0.000
0.001
0.002
0.003
0.004
0.005
0.006
n4/ϵ3
θ
θ → ∞
θ → 0
(b) Initially Critically-populated a = 0.23765
2 4 6 8 10t/ t0
-0.001
0.000
0.001
0.002
0.003
0.004
0.005
0.006
n4/ϵ3
θ
θ → ∞
θ → 0
(c) Initially Underpopulated a = 0.2
Figure 12: The time-evolution of the system with respect to the condensation criteria
for the case where particle number is conserved. The critical value of n4/ε3 = 0.00619 is
indicated by the dotted line. The direction of increasing relaxation time θ is denoted by
the arrows. β = 100, σ = 0.5.
Calculating(neq)
4
(εeq)3≈ 6.17966× 10−3. (4.15)
– 21 –
Setting(neq)
4
(εeq)3=
(n0(a))4
(ε0(a))3, (4.16)
allows us to solve for the critical value ac. For the case that β = 100 and σ = 0.5 we
have that ac = 0.23765. For values of a < ac we define the system to be underpopulated
and for values a > ac we define the system to be overpopulated. When the system is
overpopulated, the local equilibrium distribution function takes the form
feq(p, t) =1
ep
T (t) − 1+ fc(t), (4.17)
with
fc = (2π)3ncδ(~p). (4.18)
The zeroth moment of fc is given by∫dΓfc =
∫d3p
(2π)3(2π)3ncδ(~p) = nc. (4.19)
Therefore the term fc provides an additional contribution to the particle number den-
sity. Note that the condensate particles make no contribution to the energy density since∫dΓpfc = 0. Therefore, for a system with a condensate, the new equations for the energy
density and the particle number density are given by
ε(t) =
∫dΓpfeq(p, t) =
∫dΓp
(1
epT − 1
+ (2π)3ncδ(~p)
)=π2
30T 4(t), (4.20)
which is the same as in section 3. And,
n(t) =
∫dΓfeq(p, t) =
∫dΓ
(1
epT − 1
+ (2π)3ncδ(~p)
)=ζ(3)
π2T 3(t) + nc(t). (4.21)
Note that when particle number is not conserved, the notion of a condensate does not
exist. We can classify the system as being overpopulated or underpopulated with the effect
that the for underpopulated systems the particle number will increase over time towards
equilibrium and for overpopulated systems the particle number will decrease over time
towards equilibrium due to the relaxation process.
4.3 The condensation onset - time problem
When considering a system that is initially overpopulated, the relaxation time model that
is used in this thesis is perhaps too simplistic to make a statement regarding the formation
of a Bose-Einstein condensate. It does however provide us with a window of time when
a condensate could form. Efforts have been made [8] to determine the onset time of the
condensate - i.e. the time period ∆t = t∗ − t0 that elapses before the condensate starts to
form (which in the relaxation-time approximation is zero). From this one can argue that
having an initial system that is overpopulated does not guarantee that a condensate will
form. If the time it takes for a condensate to form is greater than the time it takes for the
system to switch from overpopulated to underpopulated, a condensate will not form.
– 22 –
5 Summary
Following Baym’s approach we considered the Boltzmann transport equation using a re-
laxation time approximation with the assumption of longitudinal boost invariance. We
calculated the energy density and the number density at mid rapidity and studied their
time dependence. In deriving the integral equation (3.43) we also looked at the case where
instead of taking t0 → 0 we treat t0 as a relevant timescale. We saw that as the system
thermalizes it approaches a state such that g(t) ∼ t−1/3. We extended the Baym Model to
include particle number conservation and derived an analytic result for the particle number
density at mid rapidity. We also showed that in the overpopulated case there exists a time
window in which it is possible for a condensate to form.
A Hyperbolic Space-Time Coordinates
We consider the transformation from space-time coordinates to hyperbolic coordinates. In
one dimension we have,
(t, z)→ (τ, η), (A.1)
with
τ =√t2 − z2 and η = arctan(z/t) (A.2)
∂τ
∂t= cosh η
∂η
∂t= −sinh η
τ(A.3)
∂τ
∂z= − sinh η
∂η
∂z=
cosh η
τ. (A.4)
B Deriving (3.27)
Starting from
ε(t) =
∫dΓpf0
(p⊥,
pzt
t0
), (B.1)
we make the following change of variables,
p′z =pzt
t0and p′ =
√p2⊥ + p′z
2. (B.2)
So we have
dp′z =t
t0dpz. (B.3)
– 23 –
Then (B.1) becomes,
ε(t) =
∫t0t
dp⊥dp′z
(2π)3pf0(p⊥, p
′z)
=
∫t0t
dp⊥dp′z
(2π)3
√p2⊥ + p2zf0(p⊥, p
′z)
=
∫t0t
dp⊥dp′z
(2π)3
√p2⊥ +
(t0tp′z
)2
f0(p⊥, p′z)
=
∫t0t
dp⊥dp′z
(2π)3p′
√p2⊥ + ( t0t p
′z)
2
p′2f0(p⊥, p
′z)
=
∫t0t
dp⊥dp′z
(2π)3p′
√p′2 − p′z2 + ( t0t p
′z)
2
p′2f0(p⊥, p
′z)
=
∫t0t
dp⊥dp′z
(2π)3p′
√1 +
(p′zp′
)2[( t0t
)2
− 1
]f0(p⊥, p
′z).
(B.4)
Since the limits of integration remain unchanged when making the substitution we can
relabel dp′z → dpz,
ε(t) =t0t
∫dΓp
√1 +
(pzp
)2[( t0t
)2
− 1
]f0(p⊥, pz)
=t0t
∫dΓp
√1 + (cos θp)2
[(t0t
)2
− 1
]f0(p⊥, pz),
(B.5)
which follows since pzp = cos θp where θp is the angle between ~p and the z-axis.
C Deriving (3.36)
Consider
ε(t) =
∫dΓ
p
ep/T (t) − 1
=1
(2π)3
∫p
ep/T (t) − 1dpxdpydpz.
(C.1)
Switching to spherical coordinates we have,
ε(t) =1
(2π)3
∫ ∞0
p3
ep/T (t) − 1dp
∫dΩ
=4π
(2π)3
∫ ∞0
p3
ep/T (t) − 1dp.
(C.2)
After making the substitution u = p/T we have,
ε(t) =4π
(2π)3T 4
∫ ∞0
u3
eu − 1du. (C.3)
– 24 –
At this point it is useful to recall the definition of the Riemann Zeta function,
ζ(x) =1
Γ(x)
∫ ∞0
ux−1
eu − 1du. (C.4)
Therefore, ∫ ∞0
u3
eu − 1du =
∫ ∞0
u4−1
eu − 1du = Γ(4)ζ(4). (C.5)
Finally since Γ(4) = 3! = 6 and ζ(4) = π4
90 we arrive at,
ε(t) =π2
30T 4(t). (C.6)
D Deriving the Volterra equation (3.43)
Taking the energy moment of equation (3.42),
etθ ε(t) = e
t0θ
∫dΓp0f0
(p⊥, pz
t
t0
)+
∫dΓp0
∫ t
t0
dt′
θet′θ feq
(√p2⊥ +
(pzt
t′
)2
, t′), (D.1)
and interchanging the order of integration results in
etθ ε(t) = e
t0θ
∫dΓp0f0
(p⊥, pz
t
t0
)+
∫ t
t0
dt′
θet′θ
∫dΓp0feq
(√p2⊥ +
(pzt
t′
)2
, t′). (D.2)
Let us consider only the integral,
I =
∫dΓp0feq
(√p2⊥ +
(pzt
t′
)2
, t′). (D.3)
Performing a change of integration variables similar as demonstrated in appendix B we
have,
I =t′
t
∫dΓpfeq(p, t
′)
√1−
(1− t′2
t2
)cos2 θp. (D.4)
In switching to spherical coordinates (D.4) becomes,
I =t′
t
∫ 2π
0
∫ π
0
∫ ∞0
dθpdφpdp
(2π)3p3 sin θpfeq(p, t)
√1− (1− t′2
t2) cos2 θp
= 2πt′
t
∫ π
0dθp sin θp
√1− (1− t′2
t2) cos2 θp
∫ ∞0
dp
(2π)3p3feq(p, t
′).
(D.5)
Using the substitution u = cos θp,
I = 2πt′
t
∫ 1
−1du
√1− (1− t′2
t2)u2∫ ∞0
dp
(2π)3p3feq
= 4πt′
t
∫ 1
0du
√1− (1− t′2
t2)u2∫ ∞0
dp
(2π)3p3feq.
(D.6)
– 25 –
Defining the function that results from the u-integral,
h(α) =
∫ 1
0du√
1− (1− α2)u2
=1
2
(α+
arcsin(√
1− α2)√1− α2
)=
1
2
(α+
arccos(α)√1− α2
),
(D.7)
allows us to rewrite the expression I as
I = 4πt′
th(t′/t)
∫ ∞0
dp
(2π)3p3feq(t
′). (D.8)
At this point it is very useful to note that by energy-momentum conservation,
4π
∫ ∞0
dp
(2π)3p3feq(t
′) = εeq(t′) = ε(t′) (D.9)
Then,
I =t′
th(t′/t)ε(t′). (D.10)
So (D.2) becomes,
et/θε(t) =t0tet0/θε0 +
∫ t
t0
dt′
θet′θ
(t′
th(t′/t)ε(t′)
), (D.11)
which if we let g(t) = tε(t)t0ε0
can be written as,
g(t) = e−(t−t0)/θ +
∫ t
t0
dt′
θe−(t−t
′)/θg(t′)h(t′/t). (D.12)
E Derivation of (3.61)
Starting with,
neq(t′) =
∫feq(p, t
′)dΓ
=
∫1
ep/T (t′) − 1dΓ
=1
(2π)3
∫1
ep/T (t′) − 1dpxdpydpz
=1
(2π)3
∫ ∞0
1
ep/T (t′) − 1p2dp
∫dΩ
=4π
(2π)3
∫ ∞0
1
ep/T (t′) − 1p2dp.
(E.1)
After making the substitution u = p/T we have,
neq(t) =4π
(2π)3T 3(t)
∫ ∞0
u2
eu − 1du. (E.2)
– 26 –
Making use of the definition of the Riemann Zeta function(cf. appendix C) and the fact
that Γ(3) = 2! = 2 we have,
neq(t) =ζ(3)
π2T 3(t). (E.3)
F Derivation of (4.9)
We start from
εeq =
∫dΓ
p
e(p−µ)/T − 1. (F.1)
After switching to spherical coordinates we have
εeq =4π
(2π)3
∫ ∞0
p3
e(p−µ)/T − 1dp. (F.2)
After making the substitution w = p/T , (F.2) becomes
εeq =4π
(2π)3T 4
∫ ∞0
w3
ewe−µ/T − 1dw. (F.3)
Making use of an integral representation of the polylogarithm function
Lis(z) =1
Γ(s)
∫ ∞0
ws−1
ewe−z − 1dw, (F.4)
allows us to write (G.3) as
εeq =4π
(2π)3T 4Γ(4)Li4(e
µ/T )
=3
π2T 4Li3(e
µ/T ).
(F.5)
G Derivation of (4.10)
Starting from
neq =
∫dΓ
1
e(p−µ)/T − 1(G.1)
we switch to spherical coordinates to produce
neq =4π
(2π)3
∫ ∞0
p2
e(p−µ)/T − 1dp. (G.2)
After making the substitution w = p/T , (G.2) becomes
neq =4π
(2π)3T 3
∫ ∞0
w2
ewe−µ/T − 1dw. (G.3)
Making use of an integral representation of the polylogarithm function
Lis(z) =1
Γ(s)
∫ ∞0
ws−1
ewe−z − 1dw, (G.4)
– 27 –
allows us to write (G.3) as
neq =4π
(2π)3T 3Γ(3)Li3(e
µ/T )
=1
π2T 3Li3(e
µ/T )
(G.5)
References
[1] W. Florkowski, Phenomenology of Ultra-Relativistic Heavy-Ion Collisions. World Scientific,
2010.
[2] A. K. Chaudhuri, A short course on Relativistic Heavy Ion Collisions. IOPP, 2014.
[3] “https://inspirehep.net/record/930986/plots.” Accessed = 2018-11-17.
[4] R. Hakim, Introduction to Relativistic Statistical Mechanics. World Scientific, 2011.
[5] G. Baym, “Thermal Equilibration in Ultra-Relativistic Heavy-Ion Collisions,” Physics Letters,
vol. 138B, 1984.
[6] A. Peshier. Private communication.
[7] M. Steyn, “Evolution of the quark gluon plasma in the ultra-relativistic heavy ion collisions :
The boltzmann equation vs ideal hydrodynamics.” Honours Thesis, 2017.
[8] B. Harrison, “Bose-einstein condensation from a gluon transport equation.” MSc. Thesis, 2018.
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