$n_{\textrm{infl}}$ of the pressure at zero temperature. Moreover, going
beyond the approximation used in Eq. (10), we see that at zero temperature the
pressure at $n_{B}=n_{c}$ will have a very small and positive curvature, which
means that the critical density is somewhat larger than the inflection point
density, $n_{c}\gtrsim n_{\textrm{infl}}$.

The VDF model largely reproduces the behavior described above. First, due to
the fact that the pressure fits in the VDF model are “minima” fits reproducing
(among other constraints) two local extrema, a maximum at $\eta_{L}$ and a
minimum at $\eta_{R}$, the inflection point of the pressure lies roughly in
the middle between $\eta_{L}$ and $\eta_{R}$. Second, due to the thermal part
of the pressure being just like that of an ideal gas, the location of the
critical point $n_{c}$ and the location of the inflection point of the
pressure at zero temperature $n_{\textrm{infl}}$ are related by
$n_{c}=n_{\textrm{infl}}+\delta n$, where $\delta n$ is a small positive
correction. This explains why in the VDF model the critical baryon number
density $n_{c}$ lies roughly in the middle of the spinodal region
$(\eta_{L},\eta_{R})$.

##  Chapter 20 The speed of sound

The isentropic speed of sound is defined as the following total derivative

$\displaystyle
c_{\sigma}^{2}\equiv\left(\frac{d{P}}{d{\mathcal{E}}}\right)_{S/N}~{},$ (1)

where $\sigma\equiv S/N$, $S$ is the entropy, and $N$ is the number of
particles, so that $\sigma$ is the entropy per particle. The isothermal speed
of sound is defined as

$\displaystyle
c_{T}^{2}\equiv\left(\frac{d{P}}{d{\mathcal{E}}}\right)_{T}~{},$ (2)

In practice, carrying out either of these derivatives depends strongly on the
primary variables in terms of which the variations in pressure, energy
density, and entropy per particle can be obtained.

In general, the total differential of the pressure can be given in terms of
its two primary variables $A$ and $B$,

$\displaystyle
dP(A,B)=\left(\frac{d{P}}{d{A}}\right)_{B}dA+\left(\frac{d{P}}{d{B}}\right)_{A}dB~{}.$
(3)

In order to calculate the speed of sound as defined in Eq. (1) or (2), we
instead need to have

$\displaystyle
dP\big{(}\mathcal{E},Y\big{)}=\left(\frac{d{P}}{d{\mathcal{E}}}\right)_{Y}d\mathcal{E}+\left(\frac{d{P}}{d{Y}}\right)_{\mathcal{E}}dY~{},$
(4)

where $Y=\\{(S/N),T\\}$. The above equation has the obvious advantage that we
can explicitly set $dY$ to zero.

The following is based on the assumption that since $A$ and $B$ are the
primary variables in which the pressure is given (either as an analytic
formula or as a table in $A$ and $B$), we know what the derivatives
$\left(d{P}/d{A}\right)_{B}$ and $\left(d{P}/d{B}\right)_{A}$ are. Therefore,
we only need to find expressions for $dA$ and $dB$ in terms of $d\mathcal{E}$
and $dY$, as this will allow us to compute
$\left(\frac{d{P}}{d{\mathcal{E}}}\right)_{Y}$ and
$\left(\frac{d{P}}{d{Y}}\right)_{\mathcal{E}}$. To this end we express the
total differentials of $\mathcal{E}$ and $C$ in terms of $A$ and $B$,

$\displaystyle
d\mathcal{E}=\left(\frac{d{\mathcal{E}}}{d{A}}\right)_{B}dA+\left(\frac{d{\mathcal{E}}}{d{B}}\right)_{A}dB~{},$
(5) $\displaystyle
dY=\left(\frac{d{Y}}{d{A}}\right)_{B}dA+\left(\frac{d{Y}}{d{B}}\right)_{A}dB~{}.$
(6)

We can solve the above equations for the differentials $dA$ and $dB$ in terms
of $d\mathcal{E}$ and $dY$,

$\displaystyle
dA=\left[d\mathcal{E}-\frac{\left(\frac{d{\mathcal{E}}}{d{B}}\right)_{A}}{\left(\frac{d{Y}}{d{B}}\right)_{A}}dY\right]\left[\left(\frac{d{\mathcal{E}}}{d{A}}\right)_{B}-\frac{\left(\frac{d{Y}}{d{A}}\right)_{B}}{\left(\frac{d{Y}}{d{B}}\right)_{A}}\left(\frac{d{\mathcal{E}}}{d{B}}\right)_{A}\right]^{-1}~{},$
(7) $\displaystyle
dB=\left[\frac{\left(\frac{d{\mathcal{E}}}{d{A}}\right)_{B}}{\left(\frac{d{Y}}{d{B}}\right)_{A}}dY-\frac{\left(\frac{d{Y}}{d{A}}\right)_{B}}{\left(\frac{d{Y}}{d{B}}\right)_{A}}d\mathcal{E}\right]\left[\left(\frac{d{\mathcal{E}}}{d{A}}\right)_{B}-\frac{\left(\frac{d{Y}}{d{A}}\right)_{B}}{\left(\frac{d{Y}}{d{B}}\right)_{A}}\left(\frac{d{\mathcal{E}}}{d{B}}\right)_{A}\right]^{-1}~{},$
(8)

Because the derivatives in Eqs. (1) and (2) are taken at constant $Y$, we can
set $dY=0$, which allows us to write

$\displaystyle
dA\Big{|}_{dY=0}=\frac{\left(\frac{d{Y}}{d{B}}\right)_{A}}{\left[\left(\frac{d{\mathcal{E}}}{d{A}}\right)_{B}\left(\frac{d{Y}}{d{B}}\right)_{A}-\left(\frac{d{Y}}{d{A}}\right)_{B}\left(\frac{d{\mathcal{E}}}{d{B}}\right)_{A}\right]}d\mathcal{E}~{},$
(9) $\displaystyle
dB\Big{|}_{dY=0}=-\frac{\left(\frac{d{Y}}{d{A}}\right)_{B}}{\left[\left(\frac{d{\mathcal{E}}}{d{A}}\right)_{B}\left(\frac{d{Y}}{d{B}}\right)_{A}-\left(\frac{d{Y}}{d{A}}\right)_{B}\left(\frac{d{\mathcal{E}}}{d{B}}\right)_{A}\right]}d\mathcal{E}~{}.$
(10)

Inserting the above expressions into Eq. (3) immediately allows us to take the
derivative of the pressure with respect to the energy density with $Y$ kept
constant, that is

$\displaystyle
c_{Y}^{2}=\frac{\left(\frac{d{P}}{d{A}}\right)_{B}\left(\frac{d{Y}}{d{B}}\right)_{A}-\left(\frac{d{P}}{d{B}}\right)_{A}\left(\frac{d{Y}}{d{A}}\right)_{B}}{\left(\frac{d{\mathcal{E}}}{d{A}}\right)_{B}\left(\frac{d{Y}}{d{B}}\right)_{A}-\left(\frac{d{\mathcal{E}}}{d{B}}\right)_{A}\left(\frac{d{Y}}{d{A}}\right)_{B}}~{}.$
(11)

By identifying $Y=\\{(S/N),T\\}$, explicit expressions for Eqs. (1) and (2)
can be obtained, where it is useful to remember that we have

$\displaystyle dP=T~{}ds+n~{}d\mu~{},$ (12) $\displaystyle
d\mathcal{E}=T~{}ds+\mu~{}dn~{},$ (13) $\displaystyle
d(S/N)=d(s/n)=\frac{1}{n}~{}ds-\frac{s}{n^{2}}~{}dn~{}.$ (14)

##  Appendix 20.A The isentropic speed of sound $c_{\sigma}^{2}$ given
pressure as a function of $T$ and $\mu$

In this case we set $A=T$, $B=\mu$, and $Y=(S/N)$, and we can immediately
calculate

$\displaystyle\left(\frac{d{P}}{d{T}}\right)_{\mu}=s\hskip
15.0pt{\textrm{and}}\hskip 15.0pt\left(\frac{d{P}}{d{\mu}}\right)_{T}=n~{},$
(15)

as well as

$\displaystyle\left(\frac{d{\mathcal{E}}}{d{T}}\right)_{\mu}=T\left(\frac{d{s}}{d{T}}\right)_{\mu}+\mu\left(\frac{d{n}}{d{T}}\right)_{\mu}~{},$
(16)
$\displaystyle\left(\frac{d{\mathcal{E}}}{d{\mu}}\right)_{T}=T\left(\frac{d{s}}{d{\mu}}\right)_{T}+\mu\left(\frac{d{n}}{d{\mu}}\right)_{T}~{},$
(17)
$\displaystyle\left(\frac{d{Y}}{d{T}}\right)_{\mu}=\left(\frac{d{(S/N)}}{d{T}}\right)_{\mu}=\frac{1}{n}\left(\frac{d{s}}{d{T}}\right)_{\mu}-\frac{s}{n^{2}}\left(\frac{d{n}}{d{T}}\right)_{\mu}~{},$
(18)
$\displaystyle\left(\frac{d{Y}}{d{\mu}}\right)_{T}=\left(\frac{d{(S/N)}}{d{\mu}}\right)_{T}=\frac{1}{n}\left(\frac{d{s}}{d{\mu}}\right)_{T}-\frac{s}{n^{2}}\left(\frac{d{n}}{d{\mu}}\right)_{T}~{}.$
(19)

Inserting all od these into Eq. (11) yields

$\displaystyle
c_{\sigma}^{2}=\frac{sn\left(\frac{d{s}}{d{\mu}}\right)_{T}-s^{2}\left(\frac{d{n}}{d{\mu}}\right)_{T}-n^{2}\left(\frac{d{s}}{d{T}}\right)_{\mu}+sn\left(\frac{d{n}}{d{T}}\right)_{\mu}}{\left(sT+\mu
n\right)\left[\left(\frac{d{s}}{d{\mu}}\right)_{T}\left(\frac{d{n}}{d{T}}\right)_{\mu}-\left(\frac{d{s}}{d{T}}\right)_{\mu}\left(\frac{d{n}}{d{\mu}}\right)_{T}\right]}~{}.$
(20)

We note that from Gibbs relation we have

$\displaystyle sT+\mu n=\mathcal{E}+P~{},$ (21)

while from Maxwell’s relations

$\displaystyle\left(\frac{d{s}}{d{\mu}}\right)_{T}=\left(\frac{d{n}}{d{\mu}}\right)_{T}~{}.$
(22)

##  Appendix 20.B The isentropic speed of sound $c_{\sigma}^{2}$ given
pressure as a function of $T$ and $n$

In this case we set $A=T$, $B=n$, and $Y=(S/N)$. We assume that both

$\displaystyle\left(\frac{d{P}}{d{T}}\right)_{n}\hskip
15.0pt{\textrm{and}}\hskip 15.0pt\left(\frac{d{P}}{d{n}}\right)_{T}$ (23)

are known; for example they may be obtained by means of taking numerical
derivatives based on a table of the values of $P$ at different points $(T,n)$.
We also have

$\displaystyle\left(\frac{d{\mathcal{E}}}{d{T}}\right)_{n}=T\left(\frac{d{s}}{d{T}}\right)_{n}~{},$
(24)
$\displaystyle\left(\frac{d{\mathcal{E}}}{d{n}}\right)_{T}=T\left(\frac{d{s}}{d{n}}\right)_{T}+\mu~{},$
(25)
$\displaystyle\left(\frac{d{Y}}{d{T}}\right)_{n}=\left(\frac{d{(S/N)}}{d{T}}\right)_{n}=\frac{1}{n}\left(\frac{d{s}}{d{T}}\right)_{n}~{},$
(26)
$\displaystyle\left(\frac{d{Y}}{d{n}}\right)_{T}=\left(\frac{d{(S/N)}}{d{n}}\right)_{T}=\frac{1}{n}\left(\frac{d{s}}{d{n}}\right)_{T}-\frac{s}{n^{2}}~{}.$
(27)

Inserting all od these into Eq. (11) yields

$\displaystyle
c_{\sigma}^{2}=\frac{s\left(\frac{d{P}}{d{T}}\right)_{n}+n\left(\frac{d{P}}{d{n}}\right)_{T}\left(\frac{d{s}}{d{T}}\right)_{n}-n\left(\frac{d{P}}{d{T}}\right)_{n}\left(\frac{d{s}}{d{n}}\right)_{T}}{\big{(}sT+\mu
n\big{)}\left(\frac{d{s}}{d{T}}\right)_{n}}~{}.$ (28)

##  Appendix 20.C The isothermal speed of sound $c_{T}^{2}$ given pressure as
a function of $T$ and $\mu$

In this case we set $A=T$, $B=\mu$, and $Y=T$. We reuse Eqs. (15-17), while
for the other two derivatives we get trivial results

$\displaystyle\left(\frac{d{Y}}{d{T}}\right)_{\mu}=\left(\frac{d{T}}{d{T}}\right)_{\mu}=1~{},$
(29)
$\displaystyle\left(\frac{d{Y}}{d{\mu}}\right)_{T}=\left(\frac{d{T}}{d{\mu}}\right)_{T}=0~{}.$
(30)

Thus

$\displaystyle
c_{T}^{2}=\frac{n}{T\left(\frac{d{s}}{d{\mu}}\right)_{T}+\mu\left(\frac{d{n}}{d{\mu}}\right)_{T}}~{}.$
(31)

##  Appendix 20.D The isothermal speed of sound $c_{T}^{2}$ given pressure as
a function of $T$ and $n$

In this case we set $A=T$, $B=n$, and $Y=T$. We reuse Eqs. (24-25) , while for
the other two derivatives we again get trivial results

$\displaystyle\left(\frac{d{Y}}{d{T}}\right)_{n}=\left(\frac{d{T}}{d{T}}\right)_{n}=1~{},$
(32)
$\displaystyle\left(\frac{d{Y}}{d{n}}\right)_{T}=\left(\frac{d{T}}{d{n}}\right)_{T}=0~{}.$
(33)

Therefore

$\displaystyle
c_{T}^{2}=\frac{\left(\frac{d{P}}{d{n}}\right)_{T}}{T\left(\frac{d{s}}{d{n}}\right)_{T}+\mu}~{}.$
(34)

##  Appendix 20.E Showing the equivalency of formulas

The equivalence of the two formulas for the isentropic speed of sound squared,
Eqs. (20) and (28), as well as of the two formulas for the isothermal speed of
sound squared, Eqs. (31) and (34), can be easily shown. It is convenient to
start from Eq. (28). Using the same strategy as that leading to Eqs. (7) and
(8), but adapted a change of variables according to $(T,n)\to(T,\mu)$, we
arrive at

$\displaystyle\left(\frac{d{P}}{d{n}}\right)_{T}=\frac{n}{\left(\frac{d{n}}{d{\mu}}\right)_{T}}~{},$
(35)
$\displaystyle\left(\frac{d{P}}{d{T}}\right)_{n}=s-n\frac{\left(\frac{d{n}}{d{T}}\right)_{\mu}}{\left(\frac{\partial{n}}{\partial{\mu}}\right)_{T}}~{},$
(36)
$\displaystyle\left(\frac{d{s}}{d{n}}\right)_{T}=\frac{\left(\frac{d{s}}{d{\mu}}\right)_{T}}{\left(\frac{\partial{n}}{\partial{\mu}}\right)_{T}}~{},$
(37)
$\displaystyle\left(\frac{d{s}}{d{T}}\right)_{n}=\left(\frac{d{s}}{d{T}}\right)_{\mu}-\left(\frac{d{s}}{d{\mu}}\right)_{T}\frac{\left(\frac{d{n}}{d{T}}\right)_{\mu}}{\left(\frac{d{n}}{d{\mu}}\right)_{T}}~{}.$
(38)

The above formulas, inserted into Eq. (28), after some algebra yield Eq. (20).
The same allows to transform Eq. (34) into Eq. (31).

##  Appendix 20.F Behavior in the $\mu\to 0$ and $T\to 0$ limits

####  1 The limit of $\mu\to 0$

Let us consider the $\mu\to 0$ limit of the isentropic speed of sound, which
also implies $n\to 0$. Using Eq. (28) as well as Eqs. (36) and (38) together
with the fact that $\lim_{\mu\to 0}~{}\left(d{n}/d{T}\right)_{\mu}=0$, we
immediately get

$\displaystyle\lim_{\mu\to
0}c_{\sigma}^{2}=\frac{s}{T\left(\frac{d{s}}{d{T}}\right)_{\mu}}~{}.$ (39)

This can be rewritten by realizing that the specific heat at constant volume
is defined as

$\displaystyle
C_{V}=\left(\frac{dE}{dT}\right)_{V,N}=T\left(\frac{d{S}}{d{T}}\right)_{n}~{},$
(40)

where $E$ is the total energy of the system. Because at the same time

$\displaystyle S=V\left(\frac{d{P}}{d{T}}\right)_{\mu}~{},$ (41)

we can rewrite Eq. (40) as

$\displaystyle C_{V}$ $\displaystyle=$ $\displaystyle
TV\frac{d{}}{d{T}}\bigg{|}_{n}\left(\frac{d{P}}{d{T}}\right)_{\mu}=TV\frac{d{}}{d{T}}\bigg{|}_{\mu}\left(\frac{d{P}}{d{T}}\right)_{n}=TV\frac{d{}}{d{T}}\bigg{|}_{\mu}\left[s+n\left(\frac{d{\mu}}{d{T}}\right)_{n}\right]~{}.$
(42)

In the limit $n\to 0$ the second term in the square bracket drops out, so that
we get

$\displaystyle
c_{v}=\frac{C_{V}}{V}=T\left(\frac{d{s}}{d{T}}\right)_{\mu}~{}.$ (43)

We will obtain

$\displaystyle\lim_{\mu\to 0}c_{\sigma}^{2}=\frac{s}{c_{V}}$ (44)

as long as $\left(d{}/d{T}\right)_{\mu}=\left(d{}/d{T}\right)_{n}$. This
indeed is the case in the limit $\mu\to 0$, which can be seen as follows.

Let us consider quite generally the derivative

$\displaystyle\left(\frac{dA(T,\mu_{B})}{dC(T,\mu_{B})}\right)_{n_{B}}~{},$
(45)

where $A$ and $C$ are any functions of $T$ and $\mu_{B}$. We can always write
it out in the following way,

$\displaystyle\left(\frac{dA(T,\mu_{B})}{dC(T,\mu_{B})}\right)_{n_{B}}=\left(\frac{\left(\frac{dA}{dT}\right)_{\mu_{B}}dT+\left(\frac{dA}{d\mu_{B}}\right)_{T}d\mu_{B}}{\left(\frac{dC}{dT}\right)_{\mu_{B}}dT+\left(\frac{dC}{d\mu_{B}}\right)_{T}d\mu_{B}}\right)_{n_{B}}~{}.$
(46)

We can also write

$\displaystyle
dn_{B}=\left(\frac{dn_{B}}{dT}\right)_{\mu_{B}}dT+\left(\frac{dn_{B}}{d\mu_{B}}\right)_{T}d\mu_{B}~{}.$
(47)

If we demand that $dn_{B}=0$, then we can solve for $dT$,

$\displaystyle
dT=-\frac{\left(\frac{dn_{B}}{d\mu_{B}}\right)_{T}}{\left(\frac{dn_{B}}{dT}\right)_{\mu_{B}}}d\mu_{B}~{}.$
(48)

Inserting the above into Eq. (46) results in

$\displaystyle\left(\frac{dA(T,\mu_{B})}{dC(T,\mu_{B})}\right)_{n_{B}}$
$\displaystyle=$
$\displaystyle\frac{\left(\frac{dA}{d\mu_{B}}\right)_{T}\left(\frac{dn_{B}}{dT}\right)_{\mu_{B}}-\left(\frac{dA}{dT}\right)_{\mu_{B}}\left(\frac{dn_{B}}{d\mu_{B}}\right)_{T}}{\left(\frac{dC}{d\mu_{B}}\right)_{T}\left(\frac{dn_{B}}{dT}\right)_{\mu_{B}}-\left(\frac{dC}{dT}\right)_{\mu_{B}}\left(\frac{dn_{B}}{d\mu_{B}}\right)_{T}}~{}.$
(49)

We can now take the limit $\mu_{B}\to 0$. Remembering that for a gas with
antiparticles $\left({dn_{B}}/{dT}\right)_{\mu=0}=0$, we arrive at

$\displaystyle\lim_{\mu_{B}\to
0}\left(\frac{dA(T,\mu_{B})}{dC(T,\mu_{B})}\right)_{n_{B}}=\lim_{\mu_{B}\to
0}\frac{\left(\frac{dA}{dT}\right)_{\mu_{B}}}{\left(\frac{dC}{dT}\right)_{\mu_{B}}}=\lim_{\mu_{B}\to
0}\left(\frac{dA}{dC}\right)_{\mu_{B}}=\left(\frac{dA}{dC}\right)_{\mu_{B}=0}~{}.$
(50)

Thus the $\mu\to 0$ limit of the isentropic speed of sound $c_{\sigma}^{2}$ is
indeed given by Eq. (44).

####  2 The limit of $T\to 0$

In the limit $T\to 0$ we also have $s\to 0$, and using Eq. (20) we immediately
get

$\displaystyle\lim_{T\to
0}c_{\sigma}^{2}=\frac{n}{\mu\left(\frac{d{n}}{d{\mu}}\right)_{T}}~{}.$ (51)

Similarly, using Eq. (34) it is immediately clear that

$\displaystyle\lim_{T\to
0}c_{T}^{2}=\frac{\left(\frac{d{P}}{d{n}}\right)_{T}}{\mu}=\lim_{T\to
0}c_{\sigma}^{2}~{}.$ (52)

That is, the isothermal and isetropic speed of sound are identical at $T=0$.

Additionally, let us note that in the limit $T\to 0$, for a non-interacting
gas we have $\mu_{B}=\mu_{F}=\epsilon_{F}=\sqrt{p_{F}^{2}+m^{2}}$, as well as
$n={gp_{F}^{3}}/{6\pi^{2}}$, so that

$\displaystyle\frac{d{n}}{d{\mu_{B}}}=\frac{dp_{F}}{d\epsilon_{F}}\frac{d{n}}{d{p_{F}}}=\frac{g\epsilon_{F}p_{F}}{2\pi^{2}}$
(53)

and

$\displaystyle{c_{\sigma}^{2}}\Big{|}_{T=0}=\frac{1}{3}\frac{p_{F}^{2}}{\epsilon_{F}^{2}}~{}.$
(54)

In the nonrelativistic limit we have $\epsilon_{F}\approx m$ and thus

$\displaystyle{c_{\sigma}^{2}}\Big{|}_{\begin{subarray}{c}T=0\\\ \textrm{non-
rel}\end{subarray}}=\frac{1}{3}\frac{p_{F}^{2}}{m^{2}}~{}.$ (55)

Conversely, in the ultrarelativistic limit the speed of sound of a non-
interacting Fermi gas becomes

$\displaystyle{c_{\sigma}^{2}}\Big{|}_{\begin{subarray}{c}T=0\\\
\textrm{ultra-rel}\end{subarray}}=\frac{1}{3}~{},$ (56)

which is consistent with the conformal limit.

##  Chapter 21 Pressure derivatives in the VDF model

The pressure in the VDF model, Eq. (4), can be divided into a Fermi-gas–like
term and the interaction term,

$\displaystyle P_{(K)}=P_{\textrm{FG}}+P_{\textrm{int}}~{}.$ (1)

In the presence of antiparticles, the first term in the above equation is
given by

$\displaystyle P_{\textrm{FG}}$ $\displaystyle=$ $\displaystyle
P_{\textrm{FG}}^{\textrm{(particles)}}+P_{\textrm{FG}}^{\textrm{(antiparticles)}}$
(2) $\displaystyle=$ $\displaystyle
g\int\frac{d^{3}p}{(2\pi)^{3}}~{}T~{}\ln\Big{[}1+e^{-\beta\big{(}\epsilon_{\textrm{kin}}-\mu^{*}\big{)}}\Big{]}+g\int\frac{d^{3}p}{(2\pi)^{3}}~{}T~{}\ln\Big{[}1+e^{-\beta\big{(}\epsilon_{\textrm{kin}}+\mu^{*}\big{)}}\Big{]}~{},$

while the second term is simply

$\displaystyle
P_{\textrm{int}}=\sum_{k=1}^{K}C_{k}\left(\frac{b_{k}-1}{b_{k}}\right)n_{B}^{b_{k}}~{},$
(3)

where $n_{B}$ is the net baryon number density. With the distribution
functions for baryons and antibaryons given by

$\displaystyle
f_{\bm{p}}=\frac{1}{e^{\beta\big{(}\epsilon_{\textrm{kin}}-\mu^{*}\big{)}}+1}\hskip
15.0pt{\textrm{and}}\hskip
15.0pt\bar{f}_{\bm{p}}=\frac{1}{e^{\beta\big{(}\epsilon_{\textrm{kin}}+\mu^{*}\big{)}}+1}~{},$
(4)

respectively, we can define the following density-like integrals over powers
of the distribution functions,

$\displaystyle\mathcal{B}_{a}=g\int\frac{d^{3}p}{(2\pi)^{3}}~{}f_{\bm{p}}^{a}\hskip
15.0pt{\textrm{and}}\hskip 15.0pt\hbox to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{a}=g\int\frac{d^{3}p}{(2\pi)^{3}}~{}\bar{f}_{\bm{p}}^{a}~{}.$
(5)

In particular, the net baryon density, entering explicitly in the second term
in Eq. (1), is given by

$\displaystyle n_{B}\equiv\mathcal{B}_{1}-\hbox to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set<EMAIL_ADDRESS>(6)

In calculating $d{{}^{j}P_{(K)}}/d{n_{B}^{j}}$, the derivatives of the
interaction part of the pressure $P_{\textrm{int}}$ are straightforward,
leading to

$\displaystyle\frac{dP_{(K)}}{dn_{B}}=\frac{dP_{\textrm{FG}}}{dn_{B}}+\sum_{k=1}^{K}C_{k}\big{(}b_{k}-1\big{)}n_{B}^{b_{k}-1}~{},$
(7)
$\displaystyle\frac{d^{2}P_{(K)}}{dn_{B}^{2}}=\frac{d^{2}P_{\textrm{FG}}}{dn_{B}^{2}}+\sum_{k=1}^{K}C_{k}\big{(}b_{k}-1\big{)}^{2}n_{B}^{b_{k}-2}~{},$
(8)
$\displaystyle\frac{d^{3}P_{(K)}}{dn_{B}^{3}}=\frac{d^{3}P_{\textrm{FG}}}{dn_{B}^{3}}+\sum_{k=1}^{K}C_{k}\big{(}b_{k}-1\big{)}^{2}\big{(}b_{k}-2\big{)}n_{B}^{b_{k}-3}~{}.$
(9)

Below, we derive expressions for the derivatives of the Fermi-gas–like part of
the pressure $d{{}^{j}P_{\textrm{FG}}}/d{n_{B}^{j}}$.

We take $P_{\textrm{FG}}$ to be a function of temperature $T$ and effective
chemical potential $\mu^{*}$, $P_{\textrm{FG}}=P_{\textrm{FG}}(T,\mu^{*})$ (we
note that this choice is arbitrary and one can likewise use
$P_{\textrm{FG}}=P_{\textrm{FG}}(T,\mu_{B})$ or
$P_{\textrm{FG}}=P_{\textrm{FG}}(T,n_{B})$, although we do find that dealing
with $P_{\textrm{FG}}=P_{\textrm{FG}}(T,\mu^{*})$ is the most
straightforward). In this case we can write

$\displaystyle\frac{dP_{\textrm{FG}}}{dn_{B}}=\left(\frac{\partial{}}{\partial{n_{B}}}+\frac{d\mu^{*}}{dn_{B}}\frac{\partial{}}{\partial{\mu^{*}}}\right)P_{FG}~{}.$
(10)

There is no explicit dependence on $n_{B}$ in
$P_{FG}=P_{\textrm{FG}}(T,\mu^{*})$, so that the first term is zero. First, we
calculate

$\displaystyle\frac{\partial{P_{FG}}}{\partial{\mu^{*}}}$ $\displaystyle=$
$\displaystyle g\int\frac{d^{3}p}{(2\pi)^{3}}~{}T~{}\frac{\beta
e^{-\beta\big{(}\epsilon_{\textrm{kinetic}}-\mu^{*}\big{)}}}{1+e^{-\beta\big{(}\epsilon_{\textrm{kinetic}}-\mu^{*}\big{)}}}-g\int\frac{d^{3}p}{(2\pi)^{3}}~{}T~{}\frac{\beta
e^{-\beta\big{(}\epsilon_{\textrm{kinetic}}+\mu^{*}\big{)}}}{1+e^{-\beta\big{(}\epsilon_{\textrm{kinetic}}+\mu^{*}\big{)}}}$
(11) $\displaystyle=$ $\displaystyle
g\int\frac{d^{3}p}{(2\pi)^{3}}~{}f_{\bm{p}}-g\int\frac{d^{3}p}{(2\pi)^{3}}~{}\bar{f}_{\bm{p}}=\mathcal{B}_{1}-\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{1}=n_{B}~{},$

which is what we should expect. Then we want to calculate

$\displaystyle\frac{d\mu^{*}}{dn_{B}}=\left(\frac{dn_{B}}{d\mu^{*}}\right)^{-1}=\left(\frac{d}{d\mu^{*}}\big{(}\mathcal{B}_{1}-\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{1}\big{)}\right)^{-1}~{}.$
(12)

Let us see that in general

$\displaystyle\frac{d}{d\mu^{*}}\Big{(}\mathcal{B}_{k}\pm\hbox to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{k}\Big{)}=\frac{d}{d\mu^{*}}\left[g\int\frac{d^{3}p}{(2\pi)^{3}}~{}\frac{1}{\Big{[}e^{\beta\big{(}\epsilon_{\textrm{kin}}-\mu^{*}\big{)}}+1\Big{]}^{k}}\pm
g\int\frac{d^{3}p}{(2\pi)^{3}}~{}\frac{1}{\Big{[}e^{\beta\big{(}\epsilon_{\textrm{kin}}+\mu^{*}\big{)}}+1\Big{]}^{k}}\right]$
$\displaystyle\hskip 22.76219pt=\beta
k~{}\left[g\int\frac{d^{3}p}{(2\pi)^{3}}~{}\frac{e^{\beta\big{(}\epsilon_{\textrm{kin}}-\mu^{*}\big{)}}}{\Big{[}e^{\beta\big{(}\epsilon_{\textrm{kin}}-\mu^{*}\big{)}}+1\Big{]}^{k+1}}\mp
g\int\frac{d^{3}p}{(2\pi)^{3}}~{}\frac{e^{\beta\big{(}\epsilon_{\textrm{kin}}+\mu^{*}\big{)}}}{\Big{[}e^{\beta\big{(}\epsilon_{\textrm{kin}}+\mu^{*}\big{)}}+1\Big{]}^{k+1}}\right]$
$\displaystyle\hskip 22.76219pt=\beta
k~{}\left[g\int\frac{d^{3}p}{(2\pi)^{3}}~{}\Big{(}f_{\bm{p}}^{k}-f_{\bm{p}}^{k+1}\Big{)}\mp
g\int\frac{d^{3}p}{(2\pi)^{3}}~{}\Big{(}\bar{f}_{\bm{p}}^{k}-\bar{f}_{\bm{p}}^{k+1}\Big{)}\right]$
$\displaystyle\hskip 22.76219pt=\beta
k~{}\Big{[}\big{(}\mathcal{B}_{k}-\mathcal{B}_{k+1}\big{)}\mp\big{(}\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{k}-\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{k+1}\big{)}\Big{]}~{},$
(13)

where in going from the second to the third line we used the fact that the
numerators can be rewritten according to $x=x+1-1$. In particular, we can use
Eq. (13) to evaluate Eq. (12),

$\displaystyle\frac{d{\mu^{*}}}{d{n_{B}}}=\frac{T}{\Big{[}\big{(}\mathcal{B}_{1}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{1}\big{)}-\big{(}\mathcal{B}_{2}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{2}\big{)}\Big{]}}~{},$
(14)

and thus we obtain

$\displaystyle\frac{d{P_{FG}}}{d{n_{B}}}=\frac{Tn_{B}}{\Big{[}\big{(}\mathcal{B}_{1}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{1}\big{)}-\big{(}\mathcal{B}_{2}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{2}\big{)}\Big{]}}~{}.$
(15)

Using Eq. (15), we further calculate

$\displaystyle\frac{d^{2}P_{\textrm{FG}}}{dn_{B}^{2}}$ $\displaystyle=$
$\displaystyle\frac{T}{\big{(}\mathcal{B}_{1}+\hbox to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{1}\big{)}-\big{(}\mathcal{B}_{2}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{2}\big{)}}\left(1-n_{B}\frac{\frac{d}{dn_{B}}\Big{[}\big{(}\mathcal{B}_{1}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{1}\big{)}-\big{(}\mathcal{B}_{2}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{2}\big{)}\Big{]}}{\Big{[}\big{(}\mathcal{B}_{1}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{1}\big{)}-\big{(}\mathcal{B}_{2}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{2}\big{)}\Big{]}}\right)~{}.$
(16)

Eq. (13) allows us to compute that in general

$\displaystyle\frac{d\big{(}\mathcal{B}_{k}\pm\hbox to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{k}\big{)}}{dn_{B}}=\left(\frac{d\big{(}\mathcal{B}_{1}-\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{1}\big{)}}{d\mu^{*}}\right)^{-1}\frac{d\big{(}\mathcal{B}_{k}\pm\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{k}\big{)}}{d\mu^{*}}=k~{}\frac{\Big{[}\big{(}\mathcal{B}_{k}-\mathcal{B}_{k+1}\big{)}\mp\big{(}\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{k}-\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{k+1}\big{)}\Big{]}}{\Big{[}\big{(}\mathcal{B}_{1}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{1}\big{)}-\big{(}\mathcal{B}_{2}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{2}\big{)}\Big{]}}$
(17)

so that we have

$\displaystyle\frac{d}{dn_{B}}\Big{[}\big{(}\mathcal{B}_{1}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{1}\big{)}-\big{(}\mathcal{B}_{2}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{2}\big{)}\Big{]}=\frac{\big{(}\mathcal{B}_{1}-\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{1}\big{)}-3\big{(}\mathcal{B}_{2}-\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{2}\big{)}+2\big{(}\mathcal{B}_{3}-\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{3}\big{)}}{\Big{[}\big{(}\mathcal{B}_{1}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{1}\big{)}-\big{(}\mathcal{B}_{2}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{2}\big{)}\Big{]}}~{},$
(18)

with which Eq. (16) becomes

$\displaystyle\frac{d^{2}P_{\textrm{FG}}}{dn_{B}^{2}}=\frac{T}{\big{(}\mathcal{B}_{1}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{1}\big{)}-\big{(}\mathcal{B}_{2}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{2}\big{)}}\Bigg{(}1-n_{B}~{}\frac{\big{(}\mathcal{B}_{1}-\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{1}\big{)}-3\big{(}\mathcal{B}_{2}-\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{2}\big{)}+2\big{(}\mathcal{B}_{3}-\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{3}\big{)}}{\Big{[}\big{(}\mathcal{B}_{1}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{1}\big{)}-\big{(}\mathcal{B}_{2}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{2}\big{)}\Big{]}^{2}}\Bigg{)}~{}.$
(19)

Using the same approach, we can likewise calculate

$\displaystyle\frac{d^{3}P_{\textrm{FG}}}{dn_{B}^{3}}$ $\displaystyle=$
$\displaystyle-2T~{}\frac{\big{(}\mathcal{B}_{1}-\hbox to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{1}\big{)}-3\big{(}\mathcal{B}_{2}-\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{2}\big{)}+2\big{(}\mathcal{B}_{3}-\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{3}\big{)}}{\Big{[}\big{(}\mathcal{B}_{1}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{1}\big{)}-\big{(}\mathcal{B}_{2}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{2}\big{)}\Big{]}^{3}}$
(20) $\displaystyle\hskip
14.22636pt+~{}3Tn_{B}~{}\frac{\Big{[}\big{(}\mathcal{B}_{1}-\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{1}\big{)}-3\big{(}\mathcal{B}_{2}-\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{2}\big{)}+2\big{(}\mathcal{B}_{3}-\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{3}\big{)}\Big{]}^{2}}{\Big{[}\big{(}\mathcal{B}_{1}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{1}\big{)}-\big{(}\mathcal{B}_{2}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{2}\big{)}\Big{]}^{5}}$
$\displaystyle\hskip 28.45274pt-~{}Tn_{B}~{}\frac{\big{(}\mathcal{B}_{1}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{1}\big{)}-7\big{(}\mathcal{B}_{2}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{2}\big{)}+12\big{(}\mathcal{B}_{3}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{3}\big{)}-6\big{(}\mathcal{B}_{4}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{4}\big{)}}{\Big{[}\big{(}\mathcal{B}_{1}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{1}\big{)}-\big{(}\mathcal{B}_{2}+\hbox
to0.0pt{\hskip
1.04167pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{B}$}}_{2}\big{)}\Big{]}^{4}}~{}.$

##  Chapter 22 Cumulants in terms of derivatives of the chemical potential

In calculation of the cumulants, it may be more convenient to consider the
derivatives of the chemical potential instead of the derivatives of the
pressure. In particular, while the latter are easier to interpret based on the
expected behavior of the EOS, the former lead to simpler expressions. Here we
provide formulas for the first six cumulants,

$\displaystyle\kappa_{1}=Vn_{B}~{},$ (1)
$\displaystyle\kappa_{2}=VT\Bigg{[}\frac{1}{\Big{(}\frac{d\mu_{B}}{dn_{B}}\Big{)}_{T}}\Bigg{]}~{},$
(2)
$\displaystyle\kappa_{3}=VT^{2}\Bigg{[}-\frac{\left(\frac{d{{}^{2}\mu_{B}}}{d{n_{B}^{2}}}\right)_{T}}{\Big{(}\frac{d{\mu_{B}}}{d{n_{B}}}\Big{)}^{3}}\Bigg{]}~{},$
(3)
$\displaystyle\kappa_{4}=VT^{3}\Bigg{[}3\frac{\left(\frac{d{{}^{2}\mu_{B}}}{d{n_{B}^{2}}}\right)_{T}^{2}}{\Big{(}\frac{d{\mu_{B}}}{d{n_{B}}}\Big{)}_{T}^{5}}-\frac{1}{\Big{(}\frac{d{\mu_{B}}}{d{n_{B}}}\Big{)}_{T}^{4}}\left(\frac{d{{}^{3}\mu_{B}}}{d{n_{B}^{3}}}\right)_{T}\Bigg{]}~{},$
(4)
$\displaystyle\kappa_{5}=VT^{4}\Bigg{[}-15\frac{\left(\frac{d{{}^{2}\mu_{B}}}{d{n_{B}^{2}}}\right)_{T}^{3}}{\Big{(}\frac{d{\mu_{B}}}{d{n_{B}}}\Big{)}_{T}^{7}}+10\frac{\left(\frac{d{{}^{2}\mu_{B}}}{d{n_{B}^{2}}}\right)_{T}\left(\frac{d{{}^{3}\mu_{B}}}{d{n_{B}^{3}}}\right)_{T}}{\Big{(}\frac{d{\mu_{B}}}{d{n_{B}}}\Big{)}_{T}^{6}}-\frac{\left(\frac{d{{}^{4}\mu_{B}}}{d{n_{B}^{4}}}\right)_{T}}{\Big{(}\frac{d{\mu_{B}}}{d{n_{B}}}\Big{)}_{T}^{5}}\Bigg{]}~{},$
(5)
$\displaystyle\kappa_{6}=VT^{5}\Bigg{[}105\frac{\left(\frac{d{{}^{2}\mu_{B}}}{d{n_{B}^{2}}}\right)_{T}^{4}}{\Big{(}\frac{d{\mu_{B}}}{d{n_{B}}}\Big{)}_{T}^{9}}-105\frac{\left(\frac{d{{}^{2}\mu_{B}}}{d{n_{B}^{2}}}\right)_{T}^{2}\left(\frac{d{{}^{3}\mu_{B}}}{d{n_{B}^{3}}}\right)_{T}}{\Big{(}\frac{d{\mu_{B}}}{d{n_{B}}}\Big{)}_{T}^{8}}+10\frac{\left(\frac{d{{}^{3}\mu_{B}}}{d{n_{B}^{3}}}\right)_{T}^{2}}{\Big{(}\frac{d{\mu_{B}}}{d{n_{B}}}\Big{)}_{T}^{7}}$
$\displaystyle\hskip
85.35826pt+~{}15\frac{\left(\frac{d{{}^{2}\mu_{B}}}{d{n_{B}^{2}}}\right)_{T}\left(\frac{d{{}^{4}\mu_{B}}}{d{n_{B}^{4}}}\right)_{T}}{\Big{(}\frac{d{\mu_{B}}}{d{n_{B}}}\Big{)}_{T}^{7}}-\frac{\left(\frac{d{{}^{5}\mu_{B}}}{d{n_{B}^{5}}}\right)_{T}}{\Big{(}\frac{\partial{\mu_{B}}}{\partial{n_{B}}}\Big{)}^{6}}\Bigg{]}~{}.$
(6)

In the VDF model the derivatives of the chemical potential are given by

$\displaystyle\frac{d{{}^{j}\mu_{B}}}{d{n_{B}^{j}}}=\frac{d{{}^{j}\mu^{*}}}{d{n_{B}^{j}}}+\sum_{i}C_{i}\prod_{k=1}^{j}\big{(}b_{i}-j\big{)}n_{B}^{b_{i}-1-j}~{},$
(7)

and the consecutive derivatives of the effective chemical potential $\mu^{*}$
can be calculated using Eqs. (14) and (17) from Appendix 21.

##  Chapter 23 Basics of the kinetic theory

In this appendix, Sections 23.A and 23.B are based on Ref. [227], Section 23.C
is based on Refs. [228, 229], Section 23.D is based on Ref. [229], and Section
23.E is based on Refs. [229, 230].

##  Appendix 23.A Phase space

We consider a system of a large number $N$ of particles, described by
positions $\bm{q}_{i}$ and momenta $\bm{p}_{i}$, $i=1,...,N$, which are in a
given configuration
$(\bm{q}_{1},\dots,\bm{q}_{N},\bm{p}_{1},\dots,\bm{p}_{N})$; for simplicity of
notation we denote $\bm{q}=(\bm{q}_{1},\dots,\bm{q}_{N})$ and
$\bm{p}=(\bm{p}_{1},\dots,\bm{p}_{N})$, so that a particular configuration of
the system can be written as $(\bm{q},\bm{p})$. The considered system is
characterized by a given thermodynamic property $X$; in particular, $X$ may be
a collection of properties, e.g., for a system characterized by given values
of temperature and density we have $X=(T,n)$. We say that the system is in a
macrostate characterized by $X$, and the corresponding configuration
$(\bm{q},\bm{p})$ is called a microstate of the system. Even though the
macrostate of the system is a product of a specific microstate, there is a
large number of distinct microstates that also lead to the same macrostate
characterized by $X$. Moreover, while the macrostate of the system may remain
unchanged, the system in consideration is dynamic and over time it will
explore the available microstates (here “available” may for example mean
“microstates with the same energy”, etc.). This can be conveniently visualized
using the concept of the phase space $\Gamma$, which is a $6N$-dimensional
space of all possible positions and momenta of particles in the system. Within
$\Gamma$, there is a subset space $\Gamma_{X}$ corresponding to the macrostate
$X$. The thermodynamic postulate of equal a priori probabilities states that
any microstate $(\bm{q},\bm{p})$ consistent with a given macrostate $X$ has an
equal probability. If $X$ is an equilibrium state, the system will over time
move along a trajectory in phase space contained within $\Gamma_{X}$, and
given a sufficiently long time, it will explore all possible microstates
associated with $X$.

##  Appendix 23.B Statistical ensemble

For any thermodynamic quantity $X$, taking $S$ samples (measurements) of $X$
allows one to compute the average of $X$,

$\displaystyle\langle X\rangle=\frac{1}{S}\sum_{s=1}^{S}X_{s}~{}.$ (1)

If the measurements are consecutive in time, as is often the case for
macroscopic thermodynamic properties, the above equation may be also written
as

$\displaystyle\langle X\rangle=\frac{1}{S}\sum_{s=1}^{S}X(t_{s})~{},$ (2)

where $t_{s}$ are moments in time when the measurement is made; in particular,
in the limit of infinitesimal time intervals between the measurements we have

$\displaystyle\langle X\rangle=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}dt~{}X(t)~{}.$
(3)

At different instances of the measurement, the system is found in different
microstates in the phase space, leading to particular values of $X(t_{s})$ at
the moments of measurement $t_{1},t_{2},...,t_{S}$. Note that this measurement
of the states of the system over time is equivalent to considering $S$
snapshots of the system with particular values of
$\big{(}\bm{q}(t_{s}),\bm{p}(t_{s})\big{)}$, corresponding to a particular
value of $X(t_{s})=X\big{(}\bm{q}(t_{s}),\bm{p}(t_{s})\big{)}$. This in turn
is equivalent to considering $S$ copies of the system characterized by
particular configurations in phase space $\big{(}\bm{q}_{s},\bm{p}_{s}\big{)}$
and the corresponding values of $X\big{(}\bm{q}_{s},\bm{p}_{s}\big{)}$.

In fact, rather than calculating averages of observables over time, in general
it is more convenient to consider a hypothetical collection of copies of the
system corresponding to different phase space configurations, called a
statistical ensemble. The members of the ensemble (that is, hypothetical
copies of the system) reflect the possible phase space configurations that the
system may explore. One can then define the probability density
$f(\bm{q},\bm{p})$ for a member of the ensemble to occupy the volume element
$d\bm{q}~{}d\bm{p}$ in the phase space, normalized such that the number of
ensemble systems found within $d\bm{q}~{}d\bm{p}$ is given by

$\displaystyle f(\bm{q},\bm{p})~{}d\bm{q}~{}d\bm{p}~{}.$ (4)

(We note that the particular normalization used isn’t very important as it
ultimately drops out of any expressions of interest, and therefore one can
choose a normalization which is most convenient for a given discussion; an
alternative normalization to the one used here could prioritize that
$f(\bm{q},\bm{p})$ is a properly defined probability, i.e. that $\int
d\bm{q}~{}d\bm{p}~{}f(\bm{q},\bm{p})=1$ holds.) The ensemble average of an
observable $X(\bm{q},\bm{p})$ is then naturally given by

$\displaystyle\langle X\rangle=\frac{\int
d\bm{q}~{}d\bm{p}~{}X(\bm{q},\bm{p})~{}f(\bm{q},\bm{p})}{\int
d\bm{q}~{}d\bm{p}~{}f(\bm{q},\bm{p})}~{}.$ (5)

##  Appendix 23.C The Liouville theorem

Let us now consider a small region of phase space around the phase space point
$(\bm{q},\bm{p})$, whose boundaries extend from $\bm{q}$ to
$\bm{q}+\Delta\bm{q}$ and from $\bm{p}$ to $\bm{p}+\Delta\bm{p}$ and whose
volume is given by $\Delta V=\Delta\bm{q}\Delta\bm{p}$. This region contains a
number of ensemble members $N=N(\Delta V)$, which is given by Eq. (4).
Consequently, by definition, the probability density associated with this
subset of the ensemble members can be written as

$\displaystyle f(\Delta V)=\frac{N}{\Delta V}~{}.$ (6)

Each of the $N$ ensemble members considered here is characterized by a
particular set of coordinates and momenta $(\bm{q},\bm{p})$, and is in fact a
snapshot, taken at some time $t$, of a dynamical system whose positions and
momenta are evolving in time. The question that can arise here is: if we were
to consider the same $N$ ensemble members at some later time $t+dt$, what
would be the associated probability?

From the setup of the problem the number of considered ensemble members $N$ is
constant, therefore any change in the associated probability can only come
from the change in the phase space volume $\Delta V$ occupied by these
ensemble members, see Eq. (6). We then consider

$\displaystyle\frac{d\Delta
V}{dt}=\frac{d\big{(}\Delta\bm{q}\Delta\bm{p}\big{)}}{dt}$ $\displaystyle=$
$\displaystyle\frac{d\Delta\bm{q}}{dt}\Delta\bm{p}+\Delta\bm{q}\frac{d\Delta\bm{p}}{dt}$
(7) $\displaystyle=$
$\displaystyle\frac{d\Big{[}\big{(}\bm{q}+\Delta\bm{q}\big{)}-\bm{q}\Big{]}}{dt}\Delta\bm{p}+\Delta\bm{q}\frac{d\Big{[}\big{(}\bm{p}+\Delta\bm{p}\big{)}-\bm{p}\Big{]}}{dt}$
$\displaystyle=$
$\displaystyle\bigg{[}\frac{d\big{(}\bm{q}+\Delta\bm{q}\big{)}}{dt}-\frac{d\bm{q}}{dt}\bigg{]}\Delta\bm{p}+\Delta\bm{q}\bigg{[}\frac{d\big{(}\bm{p}+\Delta\bm{p}\big{)}}{dt}-\frac{d\bm{p}}{dt}\bigg{]}~{}.$

Because $\Delta q$ is small, we can approximate

$\displaystyle\frac{d\big{(}\bm{q}+\Delta\bm{q}\big{)}}{dt}\approx\frac{d\bm{q}}{dt}+\Delta\bm{q}\frac{d}{d\bm{q}}\frac{d\bm{q}}{dt}$
(8)

and similarly for the second term in Eq. (7) involving $\bm{p}$ (here we note
that we’re using a somewhat abstract notation, as the derivatives with respect
to the variables $\bm{q}=(\bm{q}_{1},\dots,\bm{q}_{N})$ and
$\bm{p}=(\bm{p}_{1},\dots,\bm{p}_{N})$ are really sums over derivatives with
respect to individual particle positions $\bm{q}_{i}$ and momenta
$\bm{p}_{i}$; introduction of such more explicit notation is trivial and we
omit it here for clarity). These approximations will become exact in the limit
$\Delta\bm{q}\to 0,\Delta\bm{p}\to 0$, leading to

$\displaystyle\frac{d\Delta
V}{dt}=\Delta\bm{q}\Delta\bm{p}\bigg{(}\frac{d}{d\bm{q}}\frac{d\bm{q}}{dt}+\frac{d}{d\bm{p}}\frac{d\bm{p}}{dt}\bigg{)}~{}.$
(9)

The equations of motion, ${d\bm{q}}/{dt}$ and ${d\bm{p}}/{dt}$, are given by
the Hamilton equations, yielding

$\displaystyle\frac{d\Delta
V}{dt}=\Delta\bm{q}\Delta\bm{p}\bigg{(}\frac{d}{d\bm{q}}\frac{dH}{d\bm{p}}-\frac{d}{d\bm{p}}\frac{dH}{d\bm{q}}\bigg{)}=0~{},$
(10)

where in the second equality we used the fact that the Hamiltonian is a well-
behaved function satisfying
$\left(d{{}^{2}H}/d{\bm{q}d\bm{p}}\right)=\left(d{{}^{2}H}/d{\bm{p}d\bm{q}}\right)$.
Thus we have shown that the volume of the phase space associated with the
chosen subset of the statistical ensemble does not change in time, and from
this it immediately follows that the associated probability density doesn’t
change as well,

$\displaystyle\frac{df(\Delta V)}{dt}=0~{}.$ (11)

This, in fact, is the Liouville theorem, usually written more explicitly as

$\displaystyle\frac{df(t,\bm{q},\bm{p})}{dt}=\frac{\partial{f}}{\partial{t}}+\frac{d{\bm{q}}}{d{t}}\frac{\partial{f}}{\partial{\bm{q}}}+\frac{d{\bm{p}}}{d{t}}\frac{\partial{f}}{\partial{\bm{p}}}=0~{},$
(12)

or, using the Poisson bracket, as

$\displaystyle\frac{df(t,\bm{q},\bm{p})}{dt}=\frac{\partial{f}}{\partial{t}}+\big{\\{}f,H\big{\\}}=0~{},$
(13)

and it simply states that the probability density associated with a given
volume element of the phase space does not change in time.

That the Liouville theorem is satisfied is extremely important to some of the
most basic thermodynamic assumptions. Let us take for example the assumption
of equal a priori probabilities, which states that all microstates
$(\bm{q},\bm{p})$ corresponding to a chosen macrostate $X$ have an equal
probability density. The Liouville theorem ensures that if this statement is
initially true, then it will also remain true with time evolution. More
generally, any state characterized by a given initial probability density will
over time evolve through states characterized by the same probability density.

Let us see what the Liouville theorem implies for the time evolution of
observable averages,

$\displaystyle\frac{d{\langle\mathcal{O}\rangle}}{d{t}}$ $\displaystyle=$
$\displaystyle\mathcal{N}~{}\frac{d{}}{d{t}}\int
d\bm{q}d\bm{p}~{}\mathcal{O}(\bm{q},\bm{p})~{}f(t,\bm{q},\bm{p})$ (14)
$\displaystyle=$ $\displaystyle\mathcal{N}~{}\int
d\bm{q}d\bm{p}~{}\mathcal{O}(\bm{q},\bm{p})~{}\frac{\partial{f(t,\bm{q},\bm{p})}}{\partial{t}}~{},$

where $\mathcal{N}$ is a normalization factor. Eq. (12) together with
Hamilton’s equations allows one to write this as

$\displaystyle\frac{d{\langle\mathcal{O}\rangle}}{d{t}}$ $\displaystyle=$
$\displaystyle-\mathcal{N}~{}\int
d\bm{q}d\bm{p}~{}\mathcal{O}(\bm{q},\bm{p})\left(\frac{\partial{H}}{\partial{\bm{p}}}\frac{\partial{f(t,\bm{q},\bm{p})}}{\partial{\bm{q}}}-\frac{\partial{H}}{\partial{\bm{q}}}\frac{\partial{f(t,\bm{q},\bm{p})}}{\partial{\bm{p}}}\right)~{},$
(15)

which can be further rewritten using integration by parts,

$\displaystyle\frac{d{\langle\mathcal{O}\rangle}}{d{t}}$ $\displaystyle=$
$\displaystyle\mathcal{N}~{}\int
d\bm{q}d\bm{p}~{}\left(\frac{\partial{\mathcal{O}(\bm{q},\bm{p})}}{\partial{\bm{q}}}\frac{\partial{H}}{\partial{\bm{p}}}-\frac{\partial{\mathcal{O}(\bm{q},\bm{p})}}{\partial{\bm{p}}}\frac{\partial{H}}{\partial{\bm{q}}}\right)~{}f(t,\bm{q},\bm{p})~{},$
(16)

where some of the terms disappear by the same arguments as used in Eq. (10).
Using the definition of the Poisson bracket we arrive at

$\displaystyle\frac{d{\langle\mathcal{O}\rangle}}{d{t}}=\mathcal{N}~{}\int
d\bm{q}d\bm{p}~{}\big{\\{}\mathcal{O}(\bm{q},\bm{p}),H\big{\\}}~{}f(t,\bm{q},\bm{p})=\langle\big{\\{}\mathcal{O}(\bm{q},\bm{p}),H\big{\\}}\rangle~{}.$
(17)

Interestingly, the Liouville theorem implies that it is impossible for any
closed system, initialized away from equilibrium, to come to an equilibrium.
This follows from the discussion above: If initially the system occupies a
region of phase space represented by a given microstate $(\bm{q},\bm{p})$
which is characterized by a particular probability density
$f(t,\bm{x},\bm{p})$ and corresponds to some non-equilibrium macrostate $X$,
then even after arbitrarily long evolution in the phase space the system will
be still described by the same probability density $f$ and therefore it will
correspond to the same macrostate $X$; that is, the system will only explore
the subspace $\Gamma_{X}$ of the full phase space.

Naturally, everyday experience tells us that in general systems do
equilibrate. First, this is because one can never prepare a truly isolated
system. Second, the Liouville theorem rests on the assumption that the
considered region of the phase space is infinitesimally small, see Eq. (8).
Any small but finite region of the phase space can be considered to consist of
many even smaller regions. While these regions will initially follow very
similar evolutions (and so the associated occupied volumes of the phase space
will be initially adjacent), over time their phase space trajectories will
spread out. This is true for any region of the phase space, so that after some
time any small but finite region of phase space will contain states that
originate from many different points in the phase space. Each of these
microstates maintains the same probability density as at the initial time, but
the probability density associated with a region spanned by any finite values
of $\Delta x$ and $\Delta p$ will be an average of the many probability
densities from different contributing microstates. Ultimately, the system
comes to an equilibrium when the probability densities become sufficiently
mixed, and the probability density associated with any finite phase space
volume element becomes a uniform average. Most importantly, the Liouville
theorem only considers the effects due to the drift of particles in the phase
space (note that the left-hand side of Eq. (12) is in fact a hydrodynamic
derivative, and the Liouville theorem states that the phase space density
behaves like an incompressible fluid), and in particular it does not take into
account the effects of particle collisions and particle transformations such
as decays or resonance formation. An isolated system in which particle
collisions and particle-number–changing processes take place will, over time,
come to an equilibrium.

##  Appendix 23.D The BBGKY hierarchy

The phase space distribution function $f(t,\bm{q},\bm{p})$ contains detailed
information about the system. Note that using an unabbreviated notation, the
phase space distribution function for $N$ particles can be explicitly written
as

$\displaystyle
f(t,\bm{q},\bm{p})=f_{N}\big{(}t,\bm{q}_{1},\bm{p}_{1},\bm{q}_{2},\bm{p}_{2},\dots,\bm{q}_{N},\bm{p}_{N}\big{)}~{}.$
(18)

This means that $f_{N}$ gives the probability that at time $t$ particle 1 has
position $\bm{q}_{1}$ and momentum $\bm{p}_{1}$, particle 2 has position
$\bm{q}_{2}$ and momentum $\bm{p}_{2}$, and so on. For most applications,
however, we are not interested in such detailed knowledge. For example,
calculating the pressure of the gas only requires knowing whether any particle
can be found at a given position $\bm{q}^{\prime}$ with a momentum of
$\bm{p}^{\prime}$. This information is contained in a one-particle
distribution function $f_{1}(t,\bm{q}^{\prime},\bm{p}^{\prime})$, which can be
obtained from the $N$-particle distribution function through

$\displaystyle f_{1}(t,\bm{q}^{\prime},\bm{p}^{\prime})$ $\displaystyle=$
$\displaystyle\int d\bm{q}_{1}d\bm{p}_{1}\int
d\bm{q}_{2}d\bm{p}_{2}~{}\dots\int
d\bm{q}_{N}d\bm{p}_{N}~{}f_{N}\big{(}t,\bm{q}_{1},\bm{p}_{1},\bm{q}_{2},\bm{p}_{2},\dots,\bm{q}_{N},\bm{p}_{N}\big{)}$
(19) $\displaystyle\hskip
14.22636pt\times~{}\left(\sum_{i=1}^{N}\delta^{3}\big{(}\bm{q}^{\prime}-\bm{q}_{i}\big{)}\delta^{3}\big{(}\bm{p}^{\prime}-\bm{p}_{i}\big{)}\right)~{}.$

By assuming that the distribution function $f_{N}$ is symmetric with respect
to permuting the particles, the above equation becomes

$\displaystyle f_{1}(t,\bm{q}^{\prime},\bm{p}^{\prime})$ $\displaystyle=$
$\displaystyle
N~{}\int\prod_{i=2}^{N}d\bm{q}_{i}d\bm{p}_{i}~{}f_{N}\big{(}t,\bm{q}_{1}=\bm{q}^{\prime},\bm{p}_{1}=\bm{p}^{\prime},\bm{q}_{2},\bm{p}_{2},\dots,\bm{q}_{N},\bm{p}_{N}\big{)}~{}.$
(20)

Similarly, a two-particle distribution is given by

$\displaystyle
f_{1}(t,\bm{q}^{\prime},\bm{p}^{\prime},\bm{q}^{\prime\prime},\bm{p}^{\prime\prime})$
$\displaystyle=$ $\displaystyle
N(N-1)~{}\int\prod_{i=3}^{N}d\bm{q}_{i}d\bm{p}_{i}$ (21) $\displaystyle\hskip
14.22636pt\times~{}f_{N}\big{(}t,\bm{q}_{1}=\bm{q}^{\prime},\bm{p}_{1}=\bm{p}^{\prime},\bm{q}_{2}=\bm{q}^{\prime\prime},\bm{p}_{2}=\bm{p}^{\prime\prime},\dots,\bm{q}_{N},\bm{p}_{N}\big{)}~{},$

and in general the $s$-particle distribution is defined by

$\displaystyle
f_{s}(t,\bm{q}_{1},\bm{p}_{1},\dots,\bm{q}_{s},\bm{p}_{s})=\frac{N!}{(N-s)!}\int\prod_{i=s+1}^{N}d\bm{q}_{i}d\bm{p}_{i}~{}f_{N}(t,\bm{q},\bm{p})~{}.$
(22)

The evolution equation for the $s$-particle distribution $f_{s}$ can be
derived within an approach named after the works of Nikolay Bogolyubov, Max
Born, Herbert Green, John Kirkwood, and Jacques Yvon, and often called the
Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy for reasons that will
become apparent. Consider a system of $N$ particles whose Hamiltonian contains
terms related to the kinetic energy $T$, the external potential $U$, and the
two-body interaction $V$,

$\displaystyle
H(\bm{q},\bm{p})=\sum_{i=1}^{N}T(\bm{p}_{i})+\sum_{i=1}^{N}U(\bm{q}_{i})+\frac{1}{2}\sum_{i=1}^{N}\sum_{j=1}^{N}V(\bm{q}_{i}-\bm{q}_{j})~{}.$
(23)

(Note that in principle one should further include terms corresponding to
three-body interactions, four-body interactions, and so on; in practice, these
terms can be often neglected, which is an excellent approximation, e.g., in
the case of dilute gases.) The strategy to compute the $s$-particle
distribution function is to divide the Hamiltonian $H(\bm{q},\bm{p})$, Eq.
(23), into a part related to the $s$ particles in question, a part related to
the remaining $N-s$ particles, and an interaction term between the two groups
of particles,

$\displaystyle H=H_{s}+H_{N-s}+H^{\prime}~{},$ (24)

with

$\displaystyle
H_{s}=\sum_{i=1}^{s}T(\bm{p}_{i})+\sum_{i=1}^{s}U(\bm{q}_{i})+\frac{1}{2}\sum_{i=1}^{s}\sum_{j=1}^{s}V(\bm{q}_{i}-\bm{q}_{j})~{},$
(25) $\displaystyle
H_{N-s}=\sum_{i=s+1}^{N}T(\bm{p}_{i})+\sum_{i=s+1}^{N}U(\bm{q}_{i})+\frac{1}{2}\sum_{i=s+1}^{N}\sum_{j=s+1}^{N}V(\bm{q}_{i}-\bm{q}_{j})~{},$
(26) $\displaystyle
H^{\prime}=\sum_{i=1}^{s}\sum_{j=s+1}^{N}V(\bm{q}_{i}-\bm{q}_{j})~{}.$ (27)

Using Eqs. (13) and (22) we can then write the time evolution of the $s$-body
distribution function as

$\displaystyle\frac{\partial{f_{s}}}{\partial{t}}$ $\displaystyle=$
$\displaystyle\frac{N!}{(N-s)!}\int\prod_{i=s+1}^{N}d\bm{q}_{i}d\bm{p}_{i}~{}\frac{\partial{f_{N}}}{\partial{t}}$
(28) $\displaystyle=$
$\displaystyle-\frac{N!}{(N-s)!}\int\prod_{i=s+1}^{N}d\bm{q}_{i}d\bm{p}_{i}~{}\bigg{[}\big{\\{}f_{N},H_{s}\big{\\}}+\big{\\{}f_{N},H_{N-s}\big{\\}}+\big{\\{}f_{N},H^{\prime}\big{\\}}\bigg{]}~{}.$

Note that the first of the Poisson brackets is explicitly given by

$\displaystyle\big{\\{}f_{N},H_{s}\big{\\}}=\sum_{a=1}^{N}\bigg{(}\frac{\partial{f_{N}}}{\partial{\bm{q}_{a}}}\cdot\frac{\partial{H_{s}}}{\partial{\bm{p}_{a}}}-\frac{\partial{f_{N}}}{\partial{\bm{p}_{a}}}\cdot\frac{\partial{H_{s}}}{\partial{\bm{q}_{a}}}\bigg{)}=\sum_{a=1}^{s}\bigg{(}\frac{\partial{f_{N}}}{\partial{\bm{q}_{a}}}\cdot\frac{\partial{H_{s}}}{\partial{\bm{p}_{a}}}-\frac{\partial{f_{N}}}{\partial{\bm{p}_{a}}}\cdot\frac{\partial{H_{s}}}{\partial{\bm{q}_{a}}}\bigg{)}~{},$
(29)

so that the integrations and differentiations present in the first term in Eq.
(28) are performed over different sets of variables, allowing us to exchange
their order and arrive at

$\displaystyle-\bigg{\\{}\frac{N!}{(N-s)!}\int\prod_{i=s+1}^{N}d\bm{q}_{i}d\bm{p}_{i}f_{N},H_{s}\bigg{\\}}=-\big{\\{}f_{s},H_{s}\big{\\}}~{}.$
(30)

The second term in Eq. (28) vanishes, which can be seen by writing the
integrand explicitly using Eq. (26),

$\displaystyle\int\prod_{i=s+1}^{N}d\bm{q}_{i}d\bm{p}_{i}~{}\sum_{a=s+1}^{N}\bigg{[}\frac{\partial{f_{N}}}{\partial{\bm{q}_{a}}}\cdot\frac{\partial{T(\bm{p}_{a})}}{\partial{\bm{p}_{a}}}-\frac{\partial{f_{N}}}{\partial{\bm{p}_{a}}}\cdot\bigg{(}\frac{\partial{U(\bm{q}_{a})}}{\partial{\bm{q}_{a}}}+\frac{1}{2}\sum_{j=s+1}^{N}\frac{\partial{V(\bm{q}_{s}-\bm{q}_{j})}}{\partial{\bm{q}_{a}}}\bigg{)}\bigg{]}~{},$
(31)

and performing integration by parts with respect to $\bm{q}_{a}$ in the first
term and with respect to $\bm{p}_{a}$ in the second term in the square
bracket. Finally, because $H^{\prime}=H^{\prime}(\bm{q})$, the third term in
Eq. (28) is explicitly

$\displaystyle-\frac{N!}{(N-s)!}\int\prod_{i=s+1}^{N}d\bm{q}_{i}d\bm{p}_{i}~{}\bigg{[}\bigg{(}\sum_{a=1}^{s}\frac{\partial{f_{N}}}{\partial{\bm{p}_{a}}}+\sum_{a=s+1}^{N}\frac{\partial{f_{N}}}{\partial{\bm{p}_{a}}}\bigg{)}\cdot\sum_{j=s+1}^{N}\frac{\partial{V(\bm{q}_{a}-\bm{q}_{j})}}{\partial{\bm{q}_{a}}}\bigg{]}~{},$
(32)

where the sum over all particles has been divided into two parts. The second
term disappears through integration by parts, and because the derivatives over
$\bm{q}_{a}$ are the same for each term in the sum over $j$, Eq. (32) is equal
to

$\displaystyle-(N-s)\frac{N!}{(N-s)!}\int\prod_{i=s+1}^{N}d\bm{q}_{i}d\bm{p}_{i}~{}\sum_{a=1}^{s}\frac{\partial{f_{N}}}{\partial{\bm{p}_{a}}}\cdot\frac{\partial{V(\bm{q}_{a}-\bm{q}_{s+1})}}{\partial{\bm{q}_{a}}}$
$\displaystyle=-\frac{N!}{\big{(}N-(s+1)\big{)}!}\sum_{a=1}^{s}\int
d\bm{q}_{s+1}d\bm{p}_{s+1}~{}\frac{\partial{V(\bm{q}_{a}-\bm{q}_{s+1})}}{\partial{\bm{q}_{a}}}\cdot\frac{\partial{}}{\partial{\bm{p}_{a}}}\int\prod_{i=s+2}^{N}d\bm{q}_{i}d\bm{p}_{i}~{}f_{N}$
$\displaystyle=-\sum_{a=1}^{s}\int
d\bm{q}_{s+1}d\bm{p}_{s+1}~{}\frac{\partial{V(\bm{q}_{a}-\bm{q}_{s+1})}}{\partial{\bm{q}_{a}}}\cdot\frac{\partial{f_{s+1}}}{\partial{\bm{p}_{a}}}~{},$
(33)

where in the last equality we have used Eq. (22). Altogether, Eq. (28) becomes

$\displaystyle\frac{\partial{f_{s}}}{\partial{t}}+\big{\\{}f_{s},H_{s}\big{\\}}=\sum_{a=1}^{s}\int
d\bm{q}_{s+1}d\bm{p}_{s+1}~{}\frac{\partial{V(\bm{q}_{a}-\bm{q}_{s+1})}}{\partial{\bm{q}_{a}}}\cdot\frac{\partial{f_{s+1}}}{\partial{\bm{p}_{a}}}~{}.$
(34)

It is easy to see that if there are no interactions between the particles
described by the $s$-particle distribution function, then we arrive at the
Liouville theorem, Eq. (13), for $f_{s}$. If, on the other hand, the
interactions with the other $N-s$ particles are non-zero, then the change of
$f_{s}$ in time is equal to the right-hand side of Eq. (34), often called the
collision term or collision integral because of the fact that it sums over
possible interactions (collisions) of particles from the group of $s$
particles with particles from the group of $N-s$ particles, weighted by the
probability $f_{s+1}$ of finding a particle from the $N-s$ group. This results
in a sequence, or hierarchy, of equations, where $df_{1}/dt$ depends on
$f_{2}$, $df_{2}/dt$ depends on $f_{3}$, and so on, which is the origin of the
name “BBGKY hierarchy of equations”.

As it is, the BBGKY hierarchy, Eq. (34), retains all the information
originally contained within the $N$-body distribution function $f_{N}$, and
solving for the consecutive $s$-body distribution functions $f_{s}$ using Eq.
(34) is as complicated as solving for the original $N$-body distribution
function $f_{N}$. However, in contrast to the Liouville equation specifying
$f_{N}$, the BBGKY hierarchy can be systematically truncated, leading to
solvable problems.

##  Appendix 23.E The Boltzmann equation

A detailed discussion of the derivation of the Boltzmann equation from the
BBGKY hierarchy in the dilute regime can be found in Ref. [229]; here we
briefly present the main steps. We focus on the first two equations in the
BBGKY hierarchy, obtained by setting $s=1$ and $s=2$ in Eq. (34). Using Eq.
(25) and assuming that the potential is symmetric,
$V(\bm{q}_{1}-\bm{q}_{2})=V(\bm{q}_{2}-\bm{q}_{1})$ (from which it follows
that
$\partial{V(\bm{q}_{1}-\bm{q}_{2})}/\partial{\bm{q}_{1}}=-\partial{V(\bm{q}_{2}-\bm{q}_{1})}/\partial{\bm{q}_{2}}$),
we arrive at

$\displaystyle\frac{\partial{f_{1}}}{\partial{t}}+\frac{\partial{T(\bm{p}_{1})}}{\partial{\bm{p}_{1}}}\cdot\frac{\partial{f_{1}}}{\partial{\bm{q}_{1}}}-\frac{\partial{U(\bm{q}_{1})}}{\partial{\bm{q}_{1}}}\cdot\frac{\partial{f_{1}}}{\partial{\bm{p}_{1}}}=\int
d^{3}\bm{q}_{2}d^{3}\bm{p}_{2}~{}\frac{\partial{V(\bm{q}_{1}-\bm{q}_{2})}}{\partial{\bm{q}_{1}}}\cdot\frac{\partial{f_{2}}}{\partial{\bm{p}_{1}}}$
(35)

and

$\displaystyle\frac{\partial{f_{2}}}{\partial{t}}+\left[\frac{\partial{T(\bm{p}_{1})}}{\partial{\bm{p}_{1}}}\cdot\frac{\partial{f_{2}}}{\partial{\bm{q}_{1}}}+\frac{\partial{T(\bm{p}_{2})}}{\partial{\bm{p}_{2}}}\cdot\frac{\partial{f_{2}}}{\partial{\bm{q}_{2}}}\right]-\left[\frac{\partial{U(\bm{q}_{1})}}{\partial{\bm{q}_{1}}}\cdot\frac{\partial{f_{2}}}{\partial{\bm{p}_{1}}}+\frac{\partial{U(\bm{q}_{2})}}{\partial{\bm{q}_{2}}}\cdot\frac{\partial{f_{2}}}{\partial{\bm{p}_{2}}}\right]$
$\displaystyle\hskip
42.67912pt-~{}\frac{\partial{V(\bm{q}_{1}-\bm{q}_{2})}}{\partial{\bm{q}_{1}}}\cdot\left[\frac{\partial{f_{2}}}{\partial{\bm{p}_{1}}}-\frac{\partial{f_{2}}}{\partial{\bm{p}_{2}}}\right]$
$\displaystyle\hskip 14.22636pt=\int
d^{3}\bm{q}_{3}d^{3}\bm{p}_{3}~{}\left[\frac{\partial{V(\bm{q}_{1}-\bm{q}_{3})}}{\partial{\bm{q}_{1}}}\cdot\frac{\partial{}}{\partial{\bm{p}_{1}}}+\frac{\partial{V(\bm{q}_{2}-\bm{q}_{3})}}{\partial{\bm{q}_{2}}}\cdot\frac{\partial{}}{\partial{\bm{p}_{2}}}\right]f_{3}~{}.$
(36)

All terms acting on $f_{s}$ in above equations have dimensions of inverse
time, and their magnitude can be estimated by dimensional analysis. We can
define the timescale $\tau_{U}$ associated with terms of the form

$\displaystyle\frac{\partial{U(\bm{q})}}{\partial{\bm{q}}}\cdot\frac{\partial{}}{\partial{\bm{p}}}\propto\frac{1}{\tau_{U}}\propto\frac{v}{L}~{},$
(37)

where $v$ is the typical particle velocity and $L$ is the length scale over
which the external potential $U$ varies. Likewise, we can define the timescale
$\tau_{V}$ associated with terms involving the particle-particle interactions
(collisions),

$\displaystyle\frac{\partial{V}}{\partial{\bm{q}}}\cdot\frac{\partial{}}{\partial{\bm{p}}}\propto\frac{1}{\tau_{V}}\propto\frac{v}{d},$
(38)

where $d$ is the effective range of the scattering potential. Finally, we note
that while a scattering term only occurs on the right-hand side of Eq. (35),
in case of Eq. (36) there are scattering terms both on the left- and right-
hand side of the equation. The relative magnitude of these terms will be
proportional to the ratio $f_{3}/f_{2}$. Over small volumes $\Delta V\approx
d^{3}$ corresponding to the effective range of the potential $d$, $f_{2}$ is
proportional to the number of particle pairs in $\Delta V$, while $f_{3}$ is
proportional to the number of particle triplets, therefore $f_{3}/f_{2}\propto
n$, where $n$ is the number density of particles. Consequently, the time scale
$\tau_{I}$ associated with the collision integral on the right-hand side of
Eq. (35) is given by

$\displaystyle\int
d^{3}\bm{q}_{3}d^{3}\bm{p}_{3}~{}\frac{\partial{V}}{\partial{\bm{q}}}\cdot\frac{\partial{}}{\partial{\bm{p}}}~{}\frac{f_{3}}{f_{2}}\propto\frac{1}{\tau_{I}}\propto\frac{nd^{3}}{\tau_{V}}\propto
nvd^{2}~{}.$ (39)

From the above equation one can see that $\tau_{I}$ is in fact the mean free
time, that is a typical period between subsequent collisions of the same
particle; in agreement with the basic intuition, the mean free time is
inversely proportional to particle density, the particle’s velocity, and the
effective “cross section area” of the particle given by the square of
interaction range $d$.

For dilute systems, the average number of particles in the volume associated
with the scattering range is very small, $nd^{3}\ll 1$. Therefore the
scattering term on the right-hand side of Eq. (35) will be negligible as
compared to the scattering term on the left-hand side of the same equation,
and it is reasonable to truncate the BBGKY hierarchy at $s=2$ by setting the
right-hand side of Eq. (35) to zero. Furthermore, typically the time scale
associated with scattering $\tau_{V}$ is much shorter than the time scale
associated with the variation of the external potential $\tau_{U}$, which
allows one to neglect terms proportional to $1/\tau_{U}$ in Eq. (35). Finally,
additionally assuming that $\partial{f}/\partial{t}\ll 1$ yields

$\displaystyle\left[\frac{\partial{T(\bm{p}_{1})}}{\partial{\bm{p}_{1}}}\cdot\frac{\partial{f_{2}}}{\partial{\bm{q}_{1}}}+\frac{\partial{T(\bm{p}_{2})}}{\partial{\bm{p}_{2}}}\cdot\frac{\partial{f_{2}}}{\partial{\bm{q}_{2}}}\right]-\frac{\partial{V(\bm{q}_{1}-\bm{q}_{2})}}{\partial{\bm{q}_{1}}}\cdot\left[\frac{\partial{f_{2}}}{\partial{\bm{p}_{1}}}-\frac{\partial{f_{2}}}{\partial{\bm{p}_{2}}}\right]=0~{}.$
(40)

The derivatives with respect to $\partial{}/\partial{\bm{q}_{1}}$ and
$\partial{}/\partial{\bm{q}_{2}}$ can be expressed in terms of the center-of-
mass and relative coordinate, $\bm{Q}=(\bm{q}_{1}+\bm{q}_{2})/2$ and
$\bm{q}=(\bm{q}_{2}-\bm{q}_{1})/2$, as we have $d\bm{q}_{1}=d\bm{Q}+d\bm{q}$
and $d\bm{q}_{2}=d\bm{Q}-d\bm{q}$. The two-body distribution function $f_{2}$
is expected to vary slowly along the $\bm{Q}$ direction and appreciably over
the $\bm{q}$ direction (this can be understood based on the fact that the
average position of two particles $\bm{Q}$ is governed by the long-range part
of the interaction, while their relative position $\bm{q}$ is governed by the
short range part of the interaction), so that
$\partial{}/\partial{\bm{q}_{1}}\approx-\partial{}/\partial{\bm{q}_{2}}\approx\partial{}/\partial{\bm{q}}$,
and we can rewrite Eq. (40) as

$\displaystyle\left[\frac{\partial{T(\bm{p}_{1})}}{\partial{\bm{p}_{1}}}-\frac{\partial{T(\bm{p}_{2})}}{\partial{\bm{p}_{2}}}\right]\cdot\frac{\partial{f_{2}}}{\partial{\bm{q}}}=\frac{\partial{V(\bm{q}_{1}-\bm{q}_{2})}}{\partial{\bm{q}_{1}}}\cdot\left[\frac{\partial{f_{2}}}{\partial{\bm{p}_{1}}}-\frac{\partial{f_{2}}}{\partial{\bm{p}_{2}}}\right]~{}.$
(41)

We can use the above relation on the right-hand side of Eq. (35), where we can
perform the substitution
$\partial{f_{2}}/\partial{\bm{p}_{1}}\to\partial{f_{2}}/\partial{\bm{p}_{1}}-\partial{f_{2}}/\partial{\bm{p}_{2}}$
because $\partial{f_{2}}/\partial{\bm{p}_{1}}$ is a complete derivative which
integrates to zero. Additionally, we use the fact that for a relativistic
system $\partial{T(\bm{p})}/\partial{\bm{p}}=\bm{p}/E=\bm{v}$, and in this way
we arrive at

$\displaystyle\frac{\partial{f_{1}}}{\partial{t}}+\frac{\bm{p}_{1}}{E_{1}}\cdot\frac{\partial{f_{1}}}{\partial{\bm{q}_{1}}}-\frac{\partial{U(\bm{q}_{1})}}{\partial{\bm{q}_{1}}}\cdot\frac{\partial{f_{1}}}{\partial{\bm{p}_{1}}}=\int
d^{3}\bm{q}d^{3}\bm{p}_{2}~{}\big{(}\bm{v}_{1}-\bm{v}_{2}\big{)}\cdot\frac{\partial{f_{2}(t,\bm{q}_{1},\bm{p}_{1},\bm{q},\bm{p}_{2})}}{\partial{\bm{q}}}~{}.$
(42)

On the right-hand side of the above equation we have a derivative of $f_{2}$
with respect to $\bm{q}$ along the direction of th relative motion
$\bm{p}=\bm{p}_{2}-\bm{p}_{1}$, and it is convenient to express $\bm{q}$ in a
new basis in which one axis, denoted with $a$, is parallel to
$\bm{p}=\bm{p}_{2}-\bm{p}_{1}$, while the remaining two coordinates form a
plane perpendicular to $\bm{p}$,

$\displaystyle\bm{q}\to\bm{a}+\bm{b}\hskip 15.0pt{\textrm{and}}\hskip
15.0ptd^{3}\bm{q}\to d\bm{a}~{}d^{2}\bm{b}~{};$ (43)

we note that $\bm{b}$ is an impact parameter vector, which in particular is
zero for a head-on collision where $\bm{q}_{1}-\bm{q}_{2}$ is parallel to
$\bm{p}_{1}-\bm{p}_{2}$. We can then attempt to integrate Eq. (42) over the
$a$ coordinate. As this integral is performed, we go from negative relative
distances, starting from some lower boundary value $a_{-}$, through a zero
relative distance which is where the collision takes place, and then through
positive relative distances up to some upper boundary value $a_{+}$ (on a side
note, $|a_{+}-a_{-}|$ can be taken to be small because $f_{2}$ changes
appreciably only over distances comparable with the collision range $d$). In
this way, we “observe” the system from some time before the collision to some
time after the collision. Importantly, the momenta of the individual particles
are changed throughout the collision even though the total momentum is
conserved,

$\displaystyle\bm{p}_{1}+\bm{p}_{2}=\bm{p}_{1}^{\prime}+\bm{p}_{2}^{\prime}~{}.$
(44)

Therefore the integral of Eq. (42) over the $a$ coordinate can be written as

$\displaystyle\frac{d{f_{1}}}{d{t}}=\int
d^{2}\bm{b}d^{3}\bm{p}_{2}~{}\big{|}\bm{v}_{1}-\bm{v}_{2}\big{|}\Big{[}f_{2}(t,\bm{q}_{1},\bm{p}_{1}^{\prime},\bm{b},a_{+},\bm{p}_{2}^{\prime})-f_{2}(t,\bm{q}_{1},\bm{p}_{1},\bm{b},a_{-},\bm{p}_{2})\Big{]}~{},$
(45)

where $|\bm{v}_{1}-\bm{v}_{2}|$ is the relative speed. As the next step, we
change the integration variable from the impact parameter vector to the solid
angle $\Omega$,

$\displaystyle\frac{d{f_{1}}}{d{t}}=\int d\Omega
d^{3}\bm{p}_{2}~{}\left|\frac{d\sigma}{d\Omega}\right|\big{|}\bm{v}_{1}-\bm{v}_{2}\big{|}\Big{[}f_{2}(t,\bm{q}_{1},\bm{p}_{1}^{\prime},\bm{b},a_{+},\bm{p}_{2}^{\prime})-f_{2}(t,\bm{q}_{1},\bm{p}_{1},\bm{b},a_{-},\bm{p}_{2})\Big{]}~{},$
(46)

where the Jacobian of the transformation $\left|{d\sigma}/{d\Omega}\right|$
has a dimension of area and is known as the differential cross section.
Finally, we use the assumption of molecular chaos, according to which the two-
body distribution function is given by a product of one-body distribution
functions,

$\displaystyle
f_{2}(t,\bm{q}_{1},\bm{p}_{1},\bm{b},a_{\pm},\bm{p}_{2})=f_{1}(t,\bm{q}_{1},\bm{p}_{1})f_{1}(t,\bm{q}_{1},\bm{p}_{2})~{},$
(47)

and which means that, except for the moment in which the collision take place,
the particles are assumed to be uncorrelated. This allows us to arrive at the
Boltzmann equation,

$\displaystyle\frac{\partial{f_{1}}}{\partial{t}}+\frac{\bm{p}_{1}}{E_{1}}\cdot\frac{\partial{f_{1}}}{\partial{\bm{q}_{1}}}-\frac{\partial{U(\bm{q}_{1})}}{\partial{\bm{q}_{1}}}\cdot\frac{\partial{f_{1}}}{\partial{\bm{p}_{1}}}$
$\displaystyle=\int d\Omega
d^{3}\bm{p}_{2}~{}\left|\frac{d\sigma}{d\Omega}\right|\big{|}\bm{v}_{1}-\bm{v}_{2}\big{|}\Big{[}f_{1}(t,\bm{q}_{1},\bm{p}_{1}^{\prime})f_{1}(t,\bm{q}_{1},\bm{p}_{2}^{\prime})-f_{1}(t,\bm{q}_{1},\bm{p}_{1})f_{1}(t,\bm{q}_{1},\bm{p}_{2})\Big{]}~{},$
(48)

which is an integrodifferential equation for the evolution of the one-body
distribution $f_{1}$.

Although various parts of the above derivation sketch have characteristics of
a deux ex machina, the form of Eq. (48) can be also argued on phenomenological
grounds. The terms on the left-hand side are connected to a motion of a single
particle in the external potential $U$. The collision term on the right-hand
side describes the change in the probability to find a particle of momentum
$\bm{p}_{1}$ at $\bm{q}_{1}$ that can happen due to a collision with another
particle of momentum $\bm{p}_{2}$, in result of which both particles acquire
new momenta $\bm{p}_{1}^{\prime}$ and $\bm{p}_{2}^{\prime}$. This probability
is proportional to the cross section for the scattering
$\left|{d\sigma}/{d\Omega}\right|\big{|}$, the flux of incident particles
$\big{|}\bm{v}_{1}-\bm{v}_{2}\big{|}$, the probability of finding the
particles at a position $\bm{q}$ with momenta $\bm{p}_{1}$ and $\bm{p}_{2}$,
and the probability of the final states with momenta $\bm{p}_{1}^{\prime}$ and
$\bm{p}_{2}^{\prime}$.

For describing relativistic heavy-ion collisions, it is useful to write Eq.
(48) in a manifestly covariant way,

$\displaystyle
p^{\mu}\frac{\partial{f}}{\partial{x^{\mu}}}+m\frac{\partial{\big{(}K^{\mu}f\big{)}}}{\partial{p^{\mu}}}=\int\frac{d^{3}\bm{p}_{2}}{E_{2}}\frac{d^{3}\bm{p}_{1}^{\prime}}{E_{1}^{\prime}}\frac{d^{3}\bm{p}_{2}^{\prime}}{E_{2}^{\prime}}~{}W\big{(}\bm{p}_{1}^{\prime},\bm{p}_{2}^{\prime}\leftarrow\bm{p},\bm{p}_{2}\big{)}\Big{(}f_{1}^{\prime}f_{2}^{\prime}-ff_{2}\Big{)}$
(49)

where $K^{\mu}$ is a four-force vector and
$W\big{(}\bm{p}_{1}^{\prime},\bm{p}_{2}^{\prime}\leftarrow\bm{p},\bm{p}_{2}\big{)}$
is a Lorentz invariant cross section for scattering of particles with momenta
$\bm{p}$, $\bm{p}_{2}$ into particles of momenta $\bm{p}_{1}^{\prime}$,
$\bm{p}_{2}^{\prime}$. Here, it is useful to adopt an interpretation of the
expression $\Big{(}f_{1}^{\prime}f_{2}^{\prime}-ff_{2}\Big{)}$ in which the
first term corresponds to particles scattering into the considered volume
element (gain term), while the second term corresponds to particles scattering
out of the considered volume element (loss term). Importantly, Boltzmann
equation can also include quantum effects, which was first studied by Edwin
Uehling and George Uhlenbeck [231], leading to

$\displaystyle
p^{\mu}\frac{\partial{f}}{\partial{x^{\mu}}}+m\frac{\partial{\big{(}K^{\mu}f\big{)}}}{\partial{p^{\mu}}}$
$\displaystyle=$
$\displaystyle\int\frac{d^{3}\bm{p}_{2}}{E_{2}}\frac{d^{3}\bm{p}_{1}^{\prime}}{E_{1}^{\prime}}\frac{d^{3}\bm{p}_{2}^{\prime}}{E_{2}^{\prime}}~{}$
(50)
$\displaystyle\times~{}W\big{(}\bm{p}_{1}^{\prime},\bm{p}_{2}^{\prime}\leftarrow\bm{p},\bm{p}_{2}\big{)}\Bigg{(}f_{1}^{\prime}f_{2}^{\prime}\big{(}1-af\big{)}\big{(}1-af_{2}\big{)}-ff_{2}\big{(}1-af_{1}^{\prime}\big{)}\big{(}1-af_{2}^{\prime}\big{)}\Bigg{)}~{},$

where $a=1$ for fermions and $a=-1$ for bosons. The additional terms in the
big round bracket can be easily understood: for example, in the case of
fermions, a particle in a state 2’ can only scatter into a state 2 if that
state is unoccupied, the probability of which is given by $1-f_{2}$. The above
equation is known as the Boltzmann-Uehling-Uhlenbeck (BUU) equation.

So far we have only dealt with the Boltzmann equation for one particle
species. To describe heavy-ion collisions one needs to include hundreds of
hadron species which can collide with each other, decay, and form resonances.
This means that the system is described by hundreds (as many as particle
species) of coupled Boltzmann equations. This extremely complex system can be
solved by means of Monte-Carlo simulations, where the particles are propagated
using the equations of motion entering the left-hand side of Eq. (50) and the
collision integral on the right-hand side is realized by explicitly simulating
collisions, decays, and resonance formation. For more details see Chapter 3.

##  Chapter 24 Pair distribution function and the second-order cumulant

The procedure to compute the radial distribution function $g_{i}(r)$, given by
Eq. (1), can be generalized to the case of a continuous system described by a
particle density distribution $n(\bm{r}^{\prime})$,

$\displaystyle g_{i}(r,\Delta r)$ $\displaystyle=$ $\displaystyle\int
d\bm{r}^{\prime}\Bigg{(}n(\bm{r}^{\prime})-1~{}\delta(\bm{r}_{i}-\bm{r}^{\prime})\Bigg{)}$
(1) $\displaystyle\hskip 42.67912pt\times~{}\theta\Big{(}r+\Delta
r-|\bm{r}_{i}-\bm{r}^{\prime}|\Big{)}\theta\Big{(}|\bm{r}_{i}-\bm{r}^{\prime}|-(r-\Delta
r)\Big{)}~{},$

where care must be taken to subtract the self-contribution from the reference
particle. Similarly, the pair distribution function $\widetilde{g}(r)$, Eq.
(2), can be rewritten as

$\displaystyle\widetilde{g}(r,\Delta r)$ $\displaystyle=$
$\displaystyle\frac{\mathcal{N}}{2}\int d\bm{r}^{\prime}\int
d\bm{r}^{\prime\prime}~{}n(\bm{r}^{\prime})\Big{(}n(\bm{r}^{\prime\prime})-\delta(\bm{r}^{\prime}-\bm{r}^{\prime\prime})\Big{)}$
(2) $\displaystyle\hskip 28.45274pt\times~{}\theta\Big{(}r+\Delta
r-|\bm{r}^{\prime}-\bm{r}^{\prime\prime}|\Big{)}\theta\Big{(}|\bm{r}^{\prime}-\bm{r}^{\prime\prime}|-(r-\Delta
r)\Big{)}$ $\displaystyle=$ $\displaystyle\frac{\mathcal{N}}{2}\int
d\bm{r}^{\prime}\int
d\bm{r}^{\prime\prime}~{}n(\bm{r}^{\prime})n(\bm{r}^{\prime\prime})~{}\theta\Big{(}r+\Delta
r-|\bm{r}^{\prime}-\bm{r}^{\prime\prime}|\Big{)}\theta\Big{(}|\bm{r}^{\prime}-\bm{r}^{\prime\prime}|-(r-\Delta
r)\Big{)}$ $\displaystyle\hskip 14.22636pt-~{}\frac{\mathcal{N}}{2}\int
d\bm{r}^{\prime}~{}n(\bm{r}^{\prime})~{}\theta\Big{(}\Delta r-r\Big{)}~{}.$

We note that the second term is only non-zero when $r<\Delta r$, which is
correct given that the self-contribution only needs to be subtracted if we
consider the pair distribution function within a distance $\Delta r$ around
the reference particles.

It is possible to establish a connection between the pair distribution
function and the second-order cumulant $\kappa_{2}$. For this, we consider the
pair distribution function $\widetilde{g}(r)$ at distances close to the
reference particle, that is we put $r=0$, by means of which Eq. (2) becomes

$\displaystyle\widetilde{g}(0,\Delta r)$ $\displaystyle=$
$\displaystyle\frac{\mathcal{N}}{2}\bigg{[}\int d\bm{r}^{\prime}\int
d\bm{r}^{\prime\prime}~{}n(\bm{r}^{\prime})n(\bm{r}^{\prime\prime})~{}\theta\Big{(}\Delta
r-|\bm{r}^{\prime}-\bm{r}^{\prime\prime}|\Big{)}-\int
d\bm{r}^{\prime}~{}n(\bm{r}^{\prime})\bigg{]}~{}.$ (3)

Let us assume that $\Delta r$ is small and that within the distance $\Delta r$
from $\bm{r}^{\prime}$ the density is smooth enough for
$n(\bm{r}^{\prime\prime})\approx n(\bm{r^{\prime}})$ to hold, in which case

$\displaystyle\widetilde{g}(0,\Delta r)$ $\displaystyle=$
$\displaystyle\frac{\mathcal{N}}{2}\bigg{[}\int
d\bm{r}^{\prime}~{}\big{[}n(\bm{r}^{\prime})\big{]}^{2}~{}\int
d\bm{r}^{\prime\prime}~{}\theta\Big{(}\Delta
r-|\bm{r}^{\prime}-\bm{r}^{\prime\prime}|\Big{)}-\int
d\bm{r}^{\prime}~{}n(\bm{r}^{\prime})\bigg{]}$ (4) $\displaystyle=$
$\displaystyle\frac{\mathcal{N}}{2}\left[V_{\Delta}\int
d\bm{r}^{\prime}~{}\big{[}n(\bm{r}^{\prime})\big{]}^{2}-\int
d\bm{r}^{\prime}~{}n(\bm{r}^{\prime})\right]~{},$

where $V_{\Delta}=(4/3)\pi r_{\Delta}^{3}$. Furthermore, let us divide the
volume of the system $V$ into cubes of volume $V_{\Delta}$,
$N_{\textrm{cubes}}=V/V_{\Delta}$, and assume that we can safely discretize
the remaining integrals in Eq. (4) according to $\int
d\bm{r}^{\prime}~{}f(\bm{r^{\prime}})\to\sum_{i=1}^{N_{\textrm{cubes}}}V_{\Delta}f(\bm{r}_{i})$,
where $\bm{r}_{i}$ points to the center of each cube. With this and taking the
number of particles in the $i$-th cube to be $N_{i}(\bm{r}_{i})\equiv
V_{\Delta}n(\bm{r}_{i})$, Eq. (4) becomes

$\displaystyle\widetilde{g}(0,\Delta r)$ $\displaystyle\approx$
$\displaystyle\mathcal{N}~{}\frac{1}{2}\left[\sum_{i=1}^{N_{\textrm{cubes}}}\big{[}N(\bm{r}_{i})\big{]}^{2}-\sum_{i=1}^{N_{\textrm{cubes}}}N(\bm{r}_{i})\right]~{}.$
(5)

Since the normalization can be freely chosen given that
$\widetilde{\rho}(0,\Delta r)$ should be compared to a reference distribution
for an ideal gas $\widetilde{\rho}_{0}(0,\Delta r)$, in particular we can take
$\mathcal{N}=2/N_{\textrm{cubes}}$, so that finally

$\displaystyle\widetilde{g}(0,\Delta r)$ $\displaystyle=$
$\displaystyle\frac{1}{N_{\textrm{cubes}}}\left[\sum_{i=1}^{N_{\textrm{cubes}}}\big{[}N(\bm{r}_{i})\big{]}^{2}-\sum_{i=1}^{N_{\textrm{cubes}}}N(\bm{r}_{i})\right]~{},$
(6)

where $N_{\textrm{cubes}}$ is determined by $\Delta r$.

It is clear from Eq. (6) that the radial distribution function of all distinct
particle pairs at distances close to the reference particles is

$\displaystyle\widetilde{g}(0,\Delta r)=M_{2}-M_{1}=F_{2}=\langle
N(N-1)\rangle~{},$ (7)

where $M_{i}$ and $F_{i}$ are moments and factorial moments of the
distribution, respectively. Moreover, assuming that the pair distribution
function for uncorrelated pairs $\widetilde{g}_{0}(0,\Delta r)$ is described
by the Poisson distribution, for which $\langle N\rangle=\lambda$ and $\langle
N^{2}\rangle=\lambda^{2}+\lambda$ (where $\lambda$ is the mean), we have

$\displaystyle\widetilde{g}_{0}(0,\Delta r)=\langle N\rangle^{2}~{}.$ (8)

Let us consider the deviation of the behavior of the pair distribution
function $\widetilde{g}(0,\Delta r)$ from the ideal case of
$\widetilde{g}_{0}(0,\Delta r)$, which can be conveniently done by considering
the measure

$\displaystyle R=\frac{\widetilde{g}~{}(0,\Delta
r)}{\widetilde{g}_{0}(0,\Delta r)}-1~{}.$ (9)

Using Eqs. (7) and (8) we can immediately rewrite this as

$\displaystyle R=\frac{\langle N^{2}\rangle-\langle N\rangle-\langle
N\rangle^{2}}{\langle
N\rangle^{2}}=\frac{\kappa_{2}-\kappa_{1}}{\kappa_{1}^{2}}~{}.$ (10)

In particular, provided that $\kappa_{1}>0$, we immediately obtain that $R$ is
bigger (smaller) than 0 if and only if the second-order cumulant ratio
$\kappa_{2}/\kappa_{1}$ is bigger (smaller) than 1, which can be alternatively
expressed as in Eqs. (11) and (12).

We would like to stress that the above relations hold for an arbitrary
distribution of particles, without any assumptions on the underlying physics,
provided that the corresponding uncorrelated system can be described by the
Poisson distribution. In any such system the sign of $[\widetilde{g}(r,\Delta
r)/\widetilde{g}_{0}(r,\Delta r)]-1$ at $r\to 0$ is the same as the sign of
$(\kappa_{2}/\kappa)-1$. In particular, it follows that
$\widetilde{g}(r,\Delta r)/\widetilde{g}_{0}(r,\Delta r)<1$ for systems where
a repulsive interaction dominates at short distances (leading to a
distribution more uniform than that of an ideal gas), while
$\widetilde{g}(r,\Delta r)/\widetilde{g}_{0}(r,\Delta r)>1$ for systems where
an attractive interaction dominates at short distances (which leads to a
distribution that is less uniform than that of an ideal gas).

##  Chapter 25 A posteriori application of the parallel ensembles method

Figure 1: A posteriori construction of the parallel ensembles mode for a
system undergoing nuclear spinodal fragmentation (for details see Section 1).
The upper row corresponds to the initialization time, while the lower row
corresponds to $t=100\ \textrm{fm}/c$. The left-most boxes in both rows show
the full ensemble event, in which test particles have been divided into
$N_{T}=4$ groups, marked with different colors. The remaining boxes in both
rows show the state of each of the parallel ensembles obtained by dividing the
original full ensemble event into $N_{T}=4$ separate events. It is evident
that the parallel ensemble systems mirror the evolution of the full ensemble.

The version of SMASH that we used did not have the option to run in a parallel
ensembles mode (this option has been recently added to SMASH [199] and is
currently being tested). However, for simulations with all collision and decay
channels turned off (such as we study in Chapter 4), the concept of parallel
ensembles can be safely employed a posteriori, that is at the analysis stage.
Specifically, in each event we divide the $N_{T}N_{B}$ test particles obtained
from a full ensemble SMASH simulation (where $N_{B}$ is the baryon number
evolved in the simulation and $N_{T}$ is the number of test particles per
particle) into $N_{T}$ separate groups; this process is shown in Fig. 1 for
$N_{T}=4$. We then treat these groups as separate events. Each of these a
posteriori constructed events is governed by $P_{N_{B}}(N_{i})$ (see Section
2). We stress that for a SMASH simulation run in the full ensemble mode with
$N_{\textrm{ev}}$ events and $N_{T}$ test particles per particle, the
corresponding calculation in the parallel ensembles mode will be characterized
by $N_{T}N_{\textrm{ev}}$ events with $N_{T}=1$ test particles per particle.

##  Chapter 26 The isothermal speed of sound at $\mu=0$

We want to compute the $\mu=0$ limit of Eq. (8), repeated here for
convenience,

$\displaystyle
c_{T}^{2}=\left(\frac{\mu}{T}\frac{\kappa_{2}}{\kappa_{1}}+\left(\frac{d\ln\kappa_{1}}{d\ln
T}\right)_{\mu}\right)^{-1}~{}.$ (1)

We will use the fact that at $\mu=0$ we have by symmetry, for all values of
the temperature $T$, $\kappa_{1}=0$ and $\kappa_{3}=0$, from which it follows
that $\left({d\kappa_{1}}/{dT}\right)_{\mu=0}=0$. We will also use the
l’Hospital rule, according to which the $x\to x_{0}$ limit of the ratio of two
functions $f(x)$ and $g(x)$ such that either $\lim_{x\to x_{0}}f(x)=\lim_{x\to
x_{0}}g(x)=0$ or $\lim_{x\to x_{0}}f(x)=\lim_{x\to x_{0}}g(x)=\infty$ is given
by

$\displaystyle\lim_{x\to x_{0}}\frac{f(x)}{g(x)}=\lim_{x\to
x_{0}}\frac{f^{\prime}(x)}{g^{\prime}(x)}~{},$ (2)

provided that $g^{\prime}(x)\neq 0~{}\forall~{}x\neq x_{0}$ and the limit
$\lim_{x\to x_{0}}\frac{f^{\prime}(x)}{g^{\prime}(x)}$ exists. We denote the
evoking of the l’Hospital rule with a subscript “LH”.

For the first term in Eq. (1), assuming that $\kappa_{2}\neq 0$ (which is
equivalent to assuming that there are finite fluctuations in the system), we
have

$\displaystyle\frac{1}{T}~{}\lim_{\mu\to
0}\frac{\mu\kappa_{2}}{\kappa_{1}}=\bigg{|}_{\textrm{LH}}\frac{1}{T}~{}\lim_{\mu\to
0}\frac{\left(\frac{d{(\mu\kappa_{2})}}{d{\mu}}\right)_{T}}{\left(\frac{d\kappa_{1}}{d\mu}\right)_{T}}=\frac{\kappa_{2}}{T}\lim_{\mu\to
0}~{}\frac{\kappa_{2}+\mu\frac{\kappa_{3}}{T}}{\frac{\kappa_{2}}{T}}=1~{}.$
(3)

Then we deal with the second term,

$\displaystyle\lim_{\mu\to
0}~{}\frac{T}{\kappa_{1}}\left(\frac{d\kappa_{1}}{dT}\right)_{\mu}$
$\displaystyle=$ $\displaystyle\bigg{|}_{\textrm{LH}}\lim_{\mu\to
0}~{}\frac{T}{\left(\frac{d\kappa_{1}}{d\mu}\right)_{T}}\frac{d}{d\mu}\bigg{|}_{T}\left(\frac{d\kappa_{1}}{dT}\right)_{\mu}=\lim_{\mu\to
0}~{}\frac{T}{\left(\frac{d\kappa_{1}}{d\mu}\right)_{T}}\frac{d}{dT}\bigg{|}_{\mu}\left(\frac{d\kappa_{1}}{d\mu}\right)_{T}$
(4) $\displaystyle=$ $\displaystyle\lim_{\mu\to
0}~{}\frac{T}{\frac{\kappa_{2}}{T}}\frac{d}{dT}\bigg{|}_{\mu}\left(\frac{\kappa_{2}}{T}\right)=\lim_{\mu\to
0}~{}\left(\frac{d\ln\left(\frac{\kappa_{2}}{T}\right)}{d\ln
T}\right)_{\mu}~{}.$

Here the limit can be now taken simply by taking the value at $\mu=0$,

$\displaystyle\left(\frac{d\ln\left(\frac{\kappa_{2}}{T}\right)}{d\ln
T}\right)_{\mu=0}=\frac{T^{2}}{\kappa_{2}\big{|}_{\mu=0}}\left[\frac{1}{T}\left(\frac{d\kappa_{2}}{dT}\right)_{\mu=0}-\frac{\kappa_{2}\big{|}_{\mu=0}}{T^{2}}\right]=\frac{T}{\kappa_{2}\big{|}_{\mu=0}}\left(\frac{d\kappa_{2}}{dT}\right)_{\mu=0}-1~{}.$
(5)

Altogether,

$\displaystyle\lim_{\mu\to
0}c_{T}^{2}=\left(\frac{T}{\kappa_{2}\big{|}_{\mu=0}}\left(\frac{d\kappa_{2}}{dT}\right)_{\mu=0}\right)^{-1}=\left(\frac{d\ln\kappa_{2}}{d\ln
T}\right)_{\mu=0}^{-1}~{}.$ (6)

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