Tsallis Statistics in High Energy Physics: Chemical and Thermal Freeze-Outs

J. Cleymans1, M. W. Paradza1,2

1 UCT-CERN Research Centre and Physics Department, University of Cape Town,
South Africa,

2 Centre for Postgraduate Studies, Cape Peninsula University of Technology,
Bellville 7535, South Africa

Abstract:

We present an overview of a proposal in relativistic proton-proton ($pp$)
collisions emphasizing the thermal or kinetic freeze-out stage in the
framework of the Tsallis distribution. In this paper we take into account the
chemical potential present in the Tsallis distribution by following a two step
procedure. In the first step we used the redundancy present in the variables
such as the system temperature, $T$, volume, $V$, Tsallis exponent, $q$,
chemical potential, $\mu$, and performed all fits by effectively setting to
zero the chemical potential. In the second step the value $q$ is kept fixed at
the value determined in the first step. This way the complete set of variables
$T,q,V$ and $\mu$ can be determined. The final results show a weak energy
dependence in $pp$ collisions at the centre-of-mass energy $\sqrt{s}=6$ GeV to
13 TeV. The chemical potential $\mu$ at kinetic freeze-out shows an increase
with beam energy. This simplifies the description of the thermal freeze-out
stage in $pp$ collisions as the values of $T$ and of the freeze-out radius $R$
vary only mildly over a wide range of beam energies.

##  1 Introduction

It has been estimated [1] that about 30,000 particles (pions, kaons, protons,
antiprotons)) are produced in a central heavy ion collision at the Large
Hadron Collider (LHC) at 5.02 TeV. Hence it is natural to use concepts from
statistical mechanics to analyze the produced particles. This procedure has a
long and proud history with contributions from three Nobel prize winners: E.
Fermi [2, 3], W. Heisenberg [4] and L.D. Landau [5]. To quote Landau:

> “ _Fermi originated the ingenious idea of considering the collision process
> at very high energies by the use of thermodynamic methods._ ”

This turned out to be useful also at much higher beam energies than those
initially envisaged. The main ingredient in the hadron resonance gas model
(referred to as thermal model here) is that all resonances listed in the
Review of Particle Physics [6] are in thermal and chemical equilibrium. This
reduces the number of available parameters and just a few thermodynamic
variables characterize the system.

The chemical freeze-out stage is well understood and is strongly supported by
experimental results (see e.g., [7] for a recent review) with a strong
connection to results obtained using Lattice Quantum Chromodynamics (LQCD) as
the chemical freeze-out temperature is consistent with the phase transition
temperature calculated in LQCD. Indeed, for the most central Pb-Pb collisions,
the best description of the ALICE data on yields of particles in one unit of
rapidity at mid-rapidity was obtained for a chemical freeze-out temperature
given by $T_{ch}=156.6\pm 1.7$ MeV [7, 8]. Remarkably, this value of $T_{ch}$
is close to the pseudo-critical temperature $T_{c}=156.5\pm 1.5$ MeV obtained
from first principles Lattice QCD (LQCD) calculations [9], albeit with the
possibility of a broad transition region [10].

For several decades, a well-established procedure using hydrodynamics [11] and
variations thereof has existed to describe this stage. In this paper we review
another possibility to describe the thermal freeze-out stage which has shown
considerable potential especially to describe the final state in proton–proton
(pp) collisions. Most of these approaches are based on variations of a
distribution proposed by Tsallis about 40 years ago [12] to describe entropy
by introducing an additional parameter called $q$. In the limit $q\rightarrow
1$ this reproduces the standard Boltzmann–Gibbs entropy. The advantage is that
thermodynamic variables like temperature, energy density, pressure and
particle density can still be used and thermodynamic consistency is
maintained.

This paper is an extension of [13]. For completeness and for the convenience
of the reader we have included the tables presented there and considerably
improved on them, the inclusion of the NA61/SHINE [14] is new and contributes
very much to the understanding of the energy dependence of the parameters,
also all figures are new.

##  2 Thermal Freeze-Out

We will focus here on one particular form of the Tsallis distribution,
satisfying thermodynamic consistency relations [15, 16] and given by:

$E\frac{d^{3}N}{d^{3}p}=gVE\frac{1}{(2\pi)^{3}}\left[1+(q-1)\frac{E-\mu}{T}\right]^{-\frac{q}{q-1}},$
(1)

where $V$ is the volume, $q$ is the Tsallis parameter, $T$ is the
corresponding temperature, $E$ is the energy of the particle, $p$ is the
momentum, $g$ is the degeneracy factor and $\mu$ is the chemical potential. In
terms of variables commonly used in high-energy physics, rapidity $y$,
transverse mass $m_{T}=\sqrt{p_{T}^{2}+m^{2}}$:

$\frac{d^{2}N}{dp_{T}dy}=gV\frac{p_{T}m_{T}\cosh
y}{(2\pi)^{2}}\left[1+(q-1)\frac{m_{T}\,\cosh\,y-\mu}{T}\right]^{-\frac{q}{q-1}}.$
(2)

In the limit where the parameter $q$ tends to unity one recovers the well-
known Boltzmann–Gibbs distribution (with $p_{T}$ being the particle transverse
momentum):

$\lim_{q\rightarrow 1}\frac{d^{2}N}{dp_{T}dy}=gV\frac{p_{T}m_{T}\cosh
y}{(2\pi)^{2}}\exp\left(-\frac{m_{T}\,\cosh\,y-\mu}{T}\right).$ (3)

The main advantage of Equation (2) over Equation (3) is that it has a
polynomial decrease with increasing $p_{T}$ which is what is observed
experimentally.

It was recognized early on [17] that there is a redundancy in the number of
parameters in this distribution, namely the four parameters $T,V,q$ and $\mu$
in Equation $(\ref{YieldNonZeroMu})$ can be replaced by just three parameters
$T_{0},V_{0},q$ with the help of the following transformation:

$\displaystyle T_{0}$ $\displaystyle=$ $\displaystyle
T\left[1-(q-1)\frac{\mu}{T}\right],\qquad\mu\leq\frac{T}{q-1},$ (4)
$\displaystyle V_{0}$ $\displaystyle=$ $\displaystyle
V\left[1-(q-1)\frac{\mu}{T}\right]^{\frac{q}{1-q}},$ (5)

leading to a transverse momentum distribution which can thus be written
equivalently as

$\frac{d^{2}N}{dp_{T}dy}=gV_{0}\frac{p_{T}m_{T}\cosh
y}{(2\pi)^{2}}\left[1+(q-1)\frac{m_{T}\,\cosh\,y}{T_{0}}\right]^{-\frac{q}{q-1}},$
(6)

where the chemical potential does not appear explicitly.

Corresponding to the volumes $V$ and $V_{0}$ defined in Equations (1) and (5)
we also introduce the corresponding radii $R$ and $R_{0}$

$\displaystyle V$ $\displaystyle=$ $\displaystyle\frac{4\pi}{3}R^{3},$ (7)
$\displaystyle V_{0}$ $\displaystyle=$ $\displaystyle\frac{4\pi}{3}R_{0}^{3}.$
(8)

It is to be noted that most previous analyses have confused the two Equations
(2) and (6) and reached conclusions that are incorrect, namely that at LHC
energies, different hadrons, $\pi,K,p,...$ cannot be described by the same
values of $T$ and $V$. As we will show this is based on using $T_{0}$ and
$V_{0}$ and not $T$ and $V$. Many authors have followed this conclusion
because at LHC energies equal numbers of particles and antiparticles are being
produced and, furthermore, at chemical equilibrium, one has indeed $\mu=0$ MeV
for all quantum numbers. However the equality of particle and antiparticle
yields, at thermal freeze-out, only implies that e.g., $\pi^{+}$ and $\pi^{-}$
have the same chemical potential but they are not necessarily zero. We
emphasize that Equations (2) and (6) carry a different meaning, notice the
difference in parameters: $T_{0}$ is not equal to $T$ and neither is $V$ equal
to $V_{0}$. Notice also that we do not have $\mu$ in Equation (6).

It is the purpose of the present paper to resolve this issue. The procedure we
choose is the following:

  1. 1.

Use Equation (6) to fit the transverse momentum distributions. This determines
the three parameters $T_{0}$, $q$ and $V_{0}$.

  2. 2.

Fix the parameter $q$ thus obtained.

  3. 3.

Perform a new fit to the transverse momentum distributions using Equation (2)
keeping $q$ as determined in the previous step. This determines the parameters
$T$ and $V$ and the chemical potential $\mu$.

  4. 4.

Check the consistency with Equations (4) and (5).

Each step in the fitting procedure thus involves only three parameters to
describe the transverse momentum distributions. This procedure was presented
in [13] and the present paper is an extension with more details in this paper,
some of the entries in Table 2 have been corrected.

We emphasize that the chemical potentials at kinetic freeze-out (described
here with a Tsallis distribution), are not related to those at chemical
freeze-out. At chemical freeze-out, where thermal and chemical equilibrium
have been well established the chemical potentials are zero. At kinetic
freeze-out however, there is no chemical equilibrium and the observed
particle-antiparticle symmetry only implies that the chemical potentials for
particles must be equal to those for antiparticles. However, due to the
absence of chemical equilibrium they do not have to be zero. The only
constraint is that they should be equal for particles and antiparticles.

We remind the reader here of the advantage of using the above distribution as
they follow a consistent set of thermodynamic relations (see e.g., [17]). From
this, it is thus clear that the parameter $T$ can indeed be considered as a
temperature in the thermodynamic sense since the relation below holds

$T=\left.\frac{\partial E}{\partial S}\right|_{V,N},$ (9)

where the entropy $S$ is the Tsallis entropy.

In the next section we include the chemical potential parameter in the Tsallis
fits to the transverse momentum spectra. Previously, it was first noted by
[17] that the variables $T,V,q$ and $\mu$ in the Tsallis distribution function
Equation $(\ref{Yield})$ have a redundancy for $\mu\neq 0$ MeV and recently
[18] considered the mass of a particle in place of chemical potential. This
necessitates work on determining the chemical potential from the transverse
momentum spectra.

##  3 Comparison of Fit Results

As mentioned in the introduction, we reproduce here for completeness the
values extracted from the results published by the ALICE Collaboration [19,
20, 21, 22]. The data at the centre-of-mass energy $\sqrt{s}=0.9$ TeV had the
smallest range in $p_{T}$ (for all the ALICE Collaboration results considered
here), of about an order of magnitude less than the experimental data at
$\sqrt{s}=2.76$ GeV and $\sqrt{s}=7$ TeV with ALICE.

In general the data were described very well; the figures showing the actual
fits results are not included in this paper since they form part of previous
publications. The least squares method was performed by the Minuit package
[23] as part of the fitting procedure in the code. There was no manual
selection in the choice of parameters, all parameters were initialized at the
beginning and the code returned the best fit parameter values. We did not
particularly fix the value of $T$ and tried to obtain the other parameters. In
particular, the value of $\mu$ did not affect $V$.

We did not observe any trend which suggested a deterioration of the fits with
the centre-of-mass energy. In Tables 1 and 2; we give the $\chi^{2}$ values.
Comparing the values of $\chi^{2}$ from 2.76 to 7.0 TeV, in Tables 1 and 2,
there was no clear trend with increasing energy.

Table 1: Fit results at $\sqrt{s}$ = 0.9 [19], 2.76 [20], 5.02 [21] and 7 TeV
[21, 22], using data from the ALICE Collaboration using Equations (6) and (7).

$\sqrt{s}$ (TeV) | Particle |  $R_{0}$ (fm) | $q$ |  $T_{0}$ (GeV) |  $\chi^{2}$/NDF
---|---|---|---|---|---
0.9 | $\pi^{+}$ | 4.83 $\pm$ 0.14 | 1.148 $\pm$ 0.005 | 0.070 $\pm$ 0.002 | 22.73/30
| $\pi^{-}$ | 4.74 $\pm$ 0.13 | 1.145 $\pm$ 0.005 | 0.072 $\pm$ 0.002 | 15.83/30
| $K^{+}$ | 4.52 $\pm$ 1.30 | 1.175 $\pm$ 0.017 | 0.057 $\pm$ 0.013 | 13.02/24
| $K^{-}$ | 3.96 $\pm$ 0.96 | 1.161 $\pm$ 0.016 | 0.064 $\pm$ 0.013 | 6.21/24
| $p$ | 42.7 $\pm$ 19.8 | 1.158 $\pm$ 0.006 | 0.020 $\pm$ 0.004 | 14.29/21
| $\bar{p}$ | 7.44 $\pm$ 3.95 | 1.132 $\pm$ 0.014 | 0.052 $\pm$ 0.016 | 13.82/21
2.76 | $\pi^{+}+\pi^{-}$ | 4.80 $\pm$ 0.10 | 1.149 $\pm$ 0.002 | 0.077 $\pm$ 0.001 | 20.64/60
| $K^{+}+K^{-}$ | 2.51 $\pm$ 0.13 | 1.144 $\pm$ 0.002 | 0.096 $\pm$ 0.004 | 2.46/55
| $p+\bar{p}$ | 4.01 $\pm$ 0.62 | 1.121 $\pm$ 0.005 | 0.086 $\pm$ 0.008 | 3.51/46
5.02 | $\pi^{+}+\pi^{-}$ | 5.02 $\pm$ 0.11 | 1.155 $\pm$ 0.002 | 0.076 $\pm$ 0.002 | 20.13/55
| $K^{+}+K^{-}$ | 2.44 $\pm$ 0.17 | 1.15 $\pm$ 0.005 | 0.099 $\pm$ 0.006 | 1.52/48
| $p+\bar{p}$ | 3.60 $\pm$ 0.55 | 1.126 $\pm$ 0.005 | 0.091 $\pm$ 0.009 | 2.56/46
7.0 | $\pi^{+}+\pi^{-}$ | 5.66 $\pm$ 0.17 | 1.179 $\pm$ 0.003 | 0.066 $\pm$ 0.002 | 14.14/38
| $K^{+}+K^{-}$ | 2.51 $\pm$ 0.15 | 1.158 $\pm$ 0.005 | 0.097 $\pm$ 0.005 | 3.11/45
| $p+\bar{p}$ | 3.07 $\pm$ 0.41 | 1.124 $\pm$ 0.005 | 0.101 $\pm$ 0.008 | 6.03/43

Table 2:  Fit results at $\sqrt{s}$ = 0.9 [19], 2.76 [20], 5.02 [21] and 7 TeV
[21, 22], using data from the ALICE Collaboration with $q$ from Table 1
following Equations (2) and (8).

$\sqrt{s}$ (TeV) | Particle |  $R$ (fm) |  $\mu$ (GeV) |  $T$ (GeV) |  $\chi^{2}$/NDF
---|---|---|---|---|---
0.9 | $\pi^{+}$ | 3.64 $\pm$ 0.21 | 0.055 $\pm$ 0.012 | 0.079 $\pm$ 0.002 | 3.66/30
| $\pi^{-}$ | 3.53 $\pm$ 0.21 | 0.059 $\pm$ 0.012 | 0.080 $\pm$ 0.002 | 2.18/30
| $K^{+}$ | 3.76 $\pm$ 0.33 | 0.029 $\pm$ 0.017 | 0.062 $\pm$ 0.003 | 5.31/24
| $K^{-}$ | 3.89 $\pm$ 0.35 | 0.003 $\pm$ 0.018 | 0.065 $\pm$ 0.003 | 3.38/24
| $p$ | 3.34 $\pm$ 0.27 | 0.233 $\pm$ 0.020 | 0.057 $\pm$ 0.007 | 7.44/21
| $\bar{p}$ | 3.93 $\pm$ 0.33 | 0.097 $\pm$ 0.024 | 0.065 $\pm$ 0.002 | 7.69/21
2.76 | $\pi^{+}+\pi^{-}$ | 4.32 $\pm$ 2.68 | 0.022 $\pm$ 0.130 | 0.080 $\pm$ 0.019 | 20.48/60
| $K^{+}+K^{-}$ | 4.75 $\pm$ 0.03 |  $-$0.140 $\pm$ 0.008 | 0.075 $\pm$ 0.004 | 2.48/55
| $p+\bar{p}$ | 4.47 $\pm$ 5.50 |  $-$0.071 $\pm$ 0.253 | 0.077 $\pm$ 0.030 | 3.52/46
5.02 | $\pi^{+}+\pi^{-}$ | 4.19 $\pm$ 2.64 | 0.038 $\pm$ 0.134 | 0.082 $\pm$ 0.021 | 20.14/55
| $K^{+}+K^{-}$ | 4.49 $\pm$ 0.03 |  $-$0.142 $\pm$ 0.009 | 0.078 $\pm$ 0.0005 | 1.52/48
| $p+\bar{p}$ | 4.00 $\pm$ 4.48 |  $-$0.075 $\pm$ 0.243 | 0.081 $\pm$ 0.031 | 2.56/46
7.0 | $\pi^{+}+\pi^{-}$ | 3.67 $\pm$ 0.02 | 0.081 $\pm$ 0.141 | 0.081 $\pm$ 0.003 | 14.15/38
| $K^{+}+K^{-}$ | 3.80 $\pm$ 0.22 |  $-$0.098$\pm$ 0.014 | 0.082 $\pm$ 0.002 | 3.13/55
| $p+\bar{p}$ | 4.07 $\pm$ 0.27 |  $-$0.127$\pm$ 0.018 | 0.085 $\pm$ 0.002 | 6.03/43

The fits to the transverse momentum distributions were then repeated using
Equation (2) but this time keeping the parameter $q$ fixed to the value
determined in the previous section and listed in Table 1. The results are
listed in Table 2, where we present the fit results for non-zero chemical
potential for $pp$ collisions at four different beam energies by the ALICE
Collaboration.

In the first case, we set the chemical potential as the mass of the respective
particle and compare our results to [18] for $pp$ collisions at $0.9$ TeV with
the CMS Collaboration, secondly we set the chemical potential as a free
parameter to fit the data and analysis of the fit results and lastly, we
calculated the chemical potential directly from Equation $(\ref{poland_mu})$.

In Table 3 we present the extracted values of $T,\,\,q\,,R$ and $\mu$ at four
different energies with the CMS Collaboration.

Table 3: The extracted values of $T,\,\,q\,,R,\,\mu\,\mathrm{and}\,\,\chi^{2}/NDF$ parameters, using the data published in [24, 26, 25] for $pp$ collisions with the CMS experiment. $\sqrt{s}$ (TeV) | Particle |  $T$ (MeV) | $q$ |  $R$ (fm) |  $\mu$ (MeV) | $\chi^{2}/NDF$
---|---|---|---|---|---|---
$0.9$ [24] | $\pi^{+}$ | $77\pm 1$ | $1.164\pm 0.004$ | $0.070\pm 0.102$ | $66\pm 4$ | $8.111/18$
| $K^{+}$ | $74\pm 1$ | $1.158\pm 0.008$ | $3.724\pm 0.126$ | $-25\pm 9$ | $2.123/13$
| $p^{+}$ | $71\pm 1$ | $1.139\pm 0.003$ | $3.536\pm 0.105$ | $94\pm 9$ | $9.596/23$
$2.76$ [25] | $\pi^{+}$ | $76\pm 1$ | $1.189\pm 0.005$ | $3.906\pm 0.100$ | $80\pm 5$ | $5.711/18$
| $K^{+}$ | $78\pm 1$ | $1.162\pm 0.008$ | $3.883\pm 0.019$ | $-5\pm 1$ | $2.447/13$
| $p^{+}$ | $67\pm 1$ | $1.166\pm 0.004$ | $3.508\pm 0.099$ | $107\pm 9$ | $27.43/23$
$7.0$ [25] | $\pi^{+}$ | $77\pm 1$ | $1.203\pm 0.005$ | $3.994\pm 0.105$ | $89\pm 1$ | $14.29/18$
| $K^{+}$ | $87\pm 1$ | $1.152\pm 0.009$ | $3.900\pm 0.135$ | $-96\pm 11$ | $2.074/13$
| $p^{+}$ | $67\pm 1$ | $1.184\pm 0.004$ | $3.509\pm 0.099$ | $84\pm 9$ | $12.22/23$
$13.0$ [26] | $\pi^{+}$ | $76\pm 2$ | $1.215\pm 0.008$ | $3.932\pm 0.157$ | $88\pm 3$ | $3.546/18$
| $K^{+}$ | $88\pm 3$ | $1.142\pm 0.0150$ | $4.044\pm 0.27$ | $-124\pm 22$ | $1.828/13$
| $p^{+}$ | $59\pm 1$ | $1.213\pm 0.008$ | $3.135\pm 0.130$ | $191\pm 14$ | $8.892/22$

In Figure 1 we compare the values for $T_{0}$ and $T$ at four different beam
energies. The results obtained for $T$ were more stable for different particle
types than the values obtained for $T_{0}$. We will come back to this with
more detail later in this paper.

An interesting proposal to determine the chemical potential was made in [27],
where the observation was made that the radius $R_{0}$ given in Table 1 is
larger than the one obtained from a femtoscopy analysis [28] by a factor
$\kappa$ estimated to be about 3.5, i.e.,

$R_{\rm femto}\approx\frac{1}{\kappa}R_{0}.$ (10)

Hence in [27] the suggestion is made to identify the corresponding volume
$V_{\rm femto}$ with the volume $V$ appearing in Equation (1).

Hence

$V_{0}\approx V\cdot\kappa^{3}.$ (11)

Combining this with Equations $(\ref{T0})$ and $(\ref{V0})$ this leads to a
chemical potential given by

$\mu=\frac{T_{0}}{q-1}\left(\kappa^{3(q-1)/q}-1\right),$ (12)

Hence, using this proposal, a knowledge of $T_{0}$ would lead to a
determination of $\mu$.

We compared the resulting values of the chemical potential $\mu$ using this
proposal [27] to the values using the procedure outlined above starting
Equation (2) and concluded that the results are very different; hence, our
results do not support this assumption and thus the volume $V$ appearing in
Equation (2) cannot be identified with the volume determined from femtoscopy.
The volume $V$ must be considered to be specific to the Tsallis distribution
as is the case with all the other variables used in this paper.

A clearer picture of the energy dependence emerges when including results from
the NA61/SHINE Collaboration [14] for $\pi^{-}$’s. The procedure outlined
above was repeated in this case using the data published in [14], first we
used Equation (6) and collect the results in Table 4. Next we fix the values
of $q$ obtained this way and repeat the fits using Equation (1); the results
are then collected in Table 5.

Figure 1: A comparison of the values of temperatures $T$ and $T_{0}$ of different hadron species for $pp$ collisions at $\sqrt{s}$ = 0.9 [19], 2.76 [20], 5.02 [21] and $7$ [22] TeV.  Table 4: The extracted values of $T_{0},\,\,q\,,R_{0}\,\,\mathrm{and}\,\,\chi^{2}/NDF$ parameters, using the data published in [14] for $pp$ collisions with the NA $61$ Collaboration. $\sqrt{s}$ (GeV) | Particle |  $T_{0}$ (MeV) | $q$ |  $R_{0}$ (fm) | $\chi^{2}/NDF$
---|---|---|---|---|---
$6.3$ | $\pi^{-}$ | $98\pm 6$ | $1.042\pm 0.015$ | $2.55\pm 0.14$ | $4.454/15$
$7.7$ | $\pi^{-}$ | $95\pm 3$ | $1.057\pm 0.008$ | $2.72\pm 0.09$ | $4.561/15$
$8.8$ | $\pi^{-}$ | $96\pm 2$ | $1.055\pm 0.006$ | $2.76\pm 0.06$ | $8.423/15$
$12.3$ | $\pi^{-}$ | $95\pm 2$ | $1.064\pm 0.006$ | $2.90\pm 0.06$ | $6.775/15$
$17.3$ | $\pi^{-}$ | $93\pm 3$ | $1.069\pm 0.006$ | $3.07\pm 0.08$ | $2.176/15$
Table 5: The extracted values of $T,\,\,\mu\,,R\,\,\mathrm{and}\,\,\chi^{2}/NDF$ parameters, using the data published in [14] for $pp$ collisions with the NA $61$ Collaboration. $\sqrt{s}$ (GeV) | Particle |  $R$ (fm) |  $\mu$ (GeV) |  $T$ (GeV) | $\chi^{2}/NDF$
---|---|---|---|---|---
$6.3$ | $\pi^{-}$ | $2.451\pm 0.399$ | $0.011\pm 0.046$ | $0.098\pm 0.003$ | $4.454/15$
$7.3$ | $\pi^{-}$ | $2.529\pm 0.223$ | $0.020\pm 0.024$ | $0.096\pm 0.002$ | $4.561/15$
$8.8$ | $\pi^{-}$ | $2.548\pm 0.016$ | $0.022\pm 0.002$ | $0.097\pm 0.001$ | $8.423/15$
$12.3$ | $\pi^{-}$ | $2.638\pm 0.171$ | $0.026\pm 0.018$ | $0.096\pm 0.001$ | $6.776/15$
$17.3$ | $\pi^{-}$ | $2.785\pm 0.216$ | $0.025\pm 0.021$ | $0.095\pm 0.002$ | $2.179/15$

The values for $T_{0}$ as a function of beam energy are shown in Figure 2. As
one can see a fairly strong energy dependence was present when comparing the
two sets of data.

However, this picture changes when plotting the temperature $T$ as a function
of beam energy as shown in Figure 3. The energy dependence becomes weaker and
the values of $T$ decrease with increasing beam energy from about 10 GeV all
the way up to 13,000 GeV. A similar decrease of the kinetic freeze-out energy
was also observed by the STAR collaboration [29] at the Brookhaven National
Laboratory.

Figure 2: The energy dependence of the temperature parameter $T_{0}$. The
triangular points are values of $T_{0}$ extracted from data in $pp$ collisions
obtained by the NA61/SHINE [14] Collaboration for pions (see Table 4). The
squares are the $T_{0}$ values in Table 1. All points were obtained by fits
using Equation (6). Figure 3:  The energy dependence of the temperature $T$
for pions in $pp$ collisions. The triangular points are values of $T$
extracted from data in $pp$ collisions obtained by the NA61/SHINE [14]
Collaboration for pions (see Table 5). The squares are the $T$ values in
Tables 2 and 3. All points were obtained by fits using Equation (2). The
straight line at $T$ = 0.1 GeV is there to guide the eye only.

Similarly when plotting the results obtained for the radius $R_{0}$ one sees a
strong dependence on the beam energy as seen in Figure 4.

Figure 4:  The energy dependence of the freeze-out radius $R_{0}$ of pions in
$pp$ collisions. The round (red) points are obtained from fits to the results
of the NA61/SHINE Collaboration [14] (see Table 4), the square points are for
the ALICE Collaboration data (see Table 1). The straight line at $R_{0}$ = 4
fm is there to guide the eye only.

However, similarly to the case with the temperatures $T$ and $T_{0}$, the
energy dependence s weakened when plotting the radius $R$ where only a very
mild energy dependence could be noticed, see Figure 5.

Figure 5:  The energy dependence of the freeze-out radius $R$ of pions in $pp$
collisions. The round (red) points are obtained from fits to the results of
the NA61/SHINE Collaboration [14] (see Table 5), the square (blue) points are
for the ALICE Collaboration data (see Table 2), while the round (black) points
are for the CMS Collaboration data (see Table 3). The straight line at $R$ = 4
fm is there to guide the eye only.

Finally the parameter which was most influenced by deviations from chemical
equilibrium, namely the chemical potential $\mu$ which is shown in Figure 6.
Here one sees a very clear increase with beam energy.

Figure 6:  The energy dependence of the freeze-out chemical potential $\mu$
for pions in $pp$ collisions. The round (red) points are obtained from fits to
the results of the NA61/SHINE Collaboration [14] (see Table 5), the square
(blue) points are for the ALICE Collaboration data (see Table 2), while the
round (black) points are for the CMS Collaboration data (see Table 3).

##  4 Conclusions

In this paper we have taken into account the chemical potential present in the
Tsallis distribution Equation (1) by following a two step procedure. In the
first step we used the redundancy present in the variables $T,V,q$ and $\mu$
expressed in Equations (4) and (5) and performed all fits using Equation (6),
i.e., effectively setting the chemical potential equal to zero. The only
variable which is common between Equations (1) and (6) is the Tsallis
parameter $q$; hence, in the second step of our procedure we fixed the value
of $q$ and performed all fits using Equation (1). This way we finally obtained
the set of variable $T,V$ and $\mu$. The results are shown in several figures.
It is to be noted that $T$ and $R$ (as deduced from the volume $V$) show a
weak energy dependence in proton-proton ($pp$) collisions at the centre-of-
mass energies from $\sqrt{s}$ = 20 GeV up to 7 and 13 TeV. This is not the
case for the variables $T_{0}$ and $V_{0}$. The chemical potential at kinetic
freeze-out shows an increase with beam energy as presented in Figure 6. This
simplifies the resulting description of the thermal freeze-out stage in $pp$
collisions as the values of $T$ and $R$ vary mildly over a wide range of beam
energies.

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# Conduction-radiation coupling between two
distant solids interacting in near-field regime

Marta Reina  Université Paris-Saclay, Institut d’Optique Graduate School,
CNRS, Laboratoire Charles Fabry, 91127, Palaiseau, France  ColibrITD, 91, rue
du Faubourg Saint-Honoré 75008 Paris  Chams Gharib Ali Barura  Université
Paris-Saclay, Institut d’Optique Graduate School, CNRS, Laboratoire Charles
Fabry, 91127, Palaiseau, France  Philippe Ben-Abdallah  Université Paris-
Saclay, Institut d’Optique Graduate School, CNRS, Laboratoire Charles Fabry,
91127, Palaiseau, France  Riccardo Messina  Université Paris-Saclay, Institut
d’Optique Graduate School, CNRS, Laboratoire Charles Fabry, 91127, Palaiseau,
France<EMAIL_ADDRESS>
(August 31, 2024)

###### Abstract

In the classical approach to deal with near-field radiative heat exchanges
between two closely spaced bodies no coupling between the different heat
carriers inside the materials and thermal photons is usually considered. Here
we make an overview of the current state of studies on this coupling between
solids of different sizes by paying a specific attention to the impact of the
conduction regime inside the solids on conduction-radiation coupling. We also
describe how the shape of solids affects this coupling. We show that this
coupling can be at the origin of a drastic change of temperature profiles
inside each body and of heat flux exchanged between them. These results could
have important implications in the fields of nanoscale thermal management,
near-field solid-state cooling and nanoscale energy conversion.

##  I Introduction

Radiative heat transfer is the phenomenon through which two bodies at
different temperatures can exchange energy even when separated by vacuum. A
milestone in the study of this effect, dating back to the 19th century, is
Stefan-Boltzmann’s law, setting an upper bound for the flux two bodies at
temperatures $T_{1}$ and $T_{2}$ can exchange: this upper limit, equal to
$\sigma(T_{1}^{4}-T_{2}^{4})$, $\sigma\simeq 5.67\cdot
10^{-8}\,\text{Wm}^{-2}\text{K}^{-1}$ being the Stefan-Boltzmann constant, can
be realized only in the ideal scenario ot two blackbodies (i.e. bodies
absorbing all the incoming radiation) exchanging heat. A second breakthrough
in the study of radiative heat transfer was set much later, in the 1970s, by
the development of fluctuational electrodynamics through the pioneering work
of Rytov, Polder and van Hove Rytov89 ; Polder71 . This theoretical framework
describes each body as a collection of fluctuating dipoles whose statistical
properties depend, by means of the fluctuation-dissipation theorem, on the
temperature and optical properties of the body they belong to. This theory
allowed to shine light on the very first experimental results Hargreaves69
demonstrating the possibility to beat the blackbody limit in near-field
regime, i.e. when their separation distance $d$ between the solids is small
compared to the thermal wavelength $\lambda_{\text{th}}=\hbar
c/k_{\text{B}}T$, of the order of 10$\,\mu$m at ambient temperature. More
specifically, this is prone to happen when the two bodies support resonant
modes of the electromagnetic field such as phonon-polaritons (for polar
materials) and plasmons (for metals) Joulain05  or even a continuum of
evanescent modes such as hyperbolic modes Biehs12 .

The unveiling of a near-field flux amplification paved the way to numerous
experiments (see Song15 ; Cuevas18 ; Biehs21  for some review papers) in a
variety of geometries (including e.g. plane–plane, sphere–plane and tip–plane)
and for several different materials. Parallel to the experimental
investigations, several ideas of applications have been put forward, ranging
from energy-conversion devices DiMatteo01 ; Narayanaswamy03 ; Basu07 ;
Fiorino18  to heat-assisted data recording Challener09 ; Stipe20 , infrared
spectroscopy DeWilde06 ; Jones12 , and thermotronics BenAbdallah13 ;
BenAbdallah15 , namely the conception of thermal equivalents of electrical
circuit elements.

Although the vast majority of experiments have confirmed the theoretical
predictions, some of them have observed deviations from it, both in the
extreme near-field scenario Kloppstech17  (nanometer and sub-nanometer range
of distances) and at tens of nanometers Kittel05 ; Shen09 . More specifically,
some experiments Kloppstech17  observed an amplification of the flux, whereas
others Kittel05 ; Shen09  highlighted a saturation effect. The explanation of
such inconsistencies between theory and experiment, to date unsolved, has
stimulated the theoretical investigations in several directions. First, is has
been suggested that non-local effects must be taken into account in order to
describe the energy exchange between metals at very short separation distances
Ford84 ; Chapuis08 . Moreover, in the extreme near field the participation of
other heat carriers (phonons and electrons) could affect significantly the
exchanged flux Zhang18 ; Alkurdi20 ; Volokitin19 ; Volokitin20 ; Volokitin21 ;
Tokunaga21 ; Tokunaga22 ; Guo22 ; GomezViloria23 , but these can only play a
role below a few nanometers. Finally, some works have also explored in this
sense the transition between conduction and radiation Joulain08 ; Chiloyan15 .

A further effect which could be at the origin of a deviation with respect to
the predictions of fluctuational electrodynamics is the coupling between
conduction acting inside each body and near-field radiative heat transfer
between them. In order to get an idea of the possible impact of this effect,
we can visualize the typical theoretical system, as shown in Fig. 1. Two
bodies are kept at temperatures $T_{L}$ and $T_{R}$ by two thermostats,
locally connected to them. In almost all theoretical works on near-field
radiative heat transfer, it is assumed that conduction inside each body is so
efficient (compared to the energy exchange mediated by radiation) that the
temperature can be assumed to be uniform in each body and equal to the one
imposed by the thermostat. This allows to define properly the radiative heat
transfer between two bodies at two given temperatures $T_{L}$ and $T_{R}$.
Nevertheless, the strong dependence of near-field radiative heat transfer on
the materials involved and, more importantly, on the separation distance
suggests that the two effects could compete in some ranges of parameters. This
would imply the existence of a temperature profile within each body (as
depicted in Fig. 1) and then, in turn, a modification of the flux exchanged
through radiation.

Figure 1: Configuration involging two arbitrarily-shaped bodies of finite size
and kept at different temperatures $T_{L}$ and $T_{R}$ by two thermostats. The
two bodies exchange heat radiatively, while conduction takes place inside each
of them. The size $\delta$ of each body compared to the phonon mean free path
$\Lambda$ dictates the conductive transport regime. For $\delta\ll\Lambda$
(left) the heat transport is ballistic (no collision events during phonon
trajectories), while for $\delta\gg\Lambda$ (right) the heat transport is
diffusive (many collision events). The coupling of conductive and radiative
heat transfer incudes, in general, two temperature profiles
$T_{1,2}(\mathbf{r})$ inside the two bodies. Reproduced from Reina2021a .

During the last years, we have performed a comprehensive study of the impact
of this coupling Messina2016a ; Messina2016b ; Jin17a ; Jin17b ; Reina2021a ;
Reina2021b ; GharibAliBarura22 , which is the topic of this review paper. More
specifically, we have first studied this effect in the simple geometry of two
parallel slabs and in the diffusive regime, as discussed in Sec. II. Following
this first analyis, we have investigated the role played by the size of the
two bodies. As a matter of fact, as shown pictorially in Fig. 1, this can have
an impact on the conduction tranport regime inside each body. These results
are discussed in Sec. III. Finally, in order to account for the variety of
geometries employed in experiments, we have also studied the same coupling
effect in different geometries, as discussed in Sec. IV. Finally, some
conclusions are given in Sec. V.

##  II Slab-slab configuration in the diffusive conduction regime

The simplest geometry to study the effect of conduction-radiation coupling is
indeed the one involving two parallel finite-thickness slabs separated by a
vacuum gap of thickness $d$, as represented in the inset of Fig. 2. To
describe the action of two thermostats connected to the two bodies, we assume
that the temperature in the first (second) body is fixed at $T_{L}$ ($T_{R}$)
except over a region of thickness $t_{a}$ ($t_{b}$).

Figure 2: Geometry of two parallel slabs separated by a $d$-thick vaxcuum gap.
In the left (right) slab the temperature can vary with respect to $T_{L}$
($T_{R}$) over a thickness $t_{a}$ ($t_{b}$). Temperature profile along the
left slab for two silica slabs with $t_{a}=t_{b}=100\,\mu$m and
$(T_{L},T_{R})=(600,300)\,$K. The lines correspond to $d=10\,$nm (black), 20
nm (red) and 50 nm (blue). The right inset shows the position-dependent
radiative flux $\phi(z)$ for $d=100\,$nm. Reproduced from Messina2016a .

In order to further simplify the problem, we also assume that the thickness of
the two slabs is large enough to safely treat the conduction problem in the
Fourier diffusive regime. In this case, the coupled equation to be solved
reads

$\frac{\partial}{\partial z}\left[\kappa(z)\frac{\partial}{\partial
z}T(z)\right]+\int\mathrm{d}z^{\prime}\,\varphi(z^{\prime},z)=0.$ (1)

In this equation $\kappa(z)$ is the bulk Fourier conductivity at point $z$,
respectively, whereas $\varphi(z^{\prime},z)$ representes the radiative power
per unit volume emitted at a point $z^{\prime}$ and absorbed at a point $z$.
At this stage, the expression of the radiative term $\varphi(z^{\prime},z)$ is
needed. This energy exchange can be, for example, calculated by means of a
framework introduced to calculate both Casimir forces and radiative heat
transfer and based on the knowledge of the scattering operators of the bodies
involved Messina11a ; Messina11b ; Messina14 ; Latella17 . In order to account
for the temperature profiles, we assume that each body is divided in slabs of
infinitesimal thickness, of which the scattering coefficients are known
analytically, and apply the scattering approach to deduce the radiative heat
transfer. Limiting ourselves to the contribution stemming from evanescent
waves in transverse magnetic polarization (dominating in the near field
between polar materials Joulain05 ) we write the flux as the frequency and
wavevector integral
$\varphi(z_{a},z_{b})=\int_{0}^{\infty}\mathrm{d}\omega\int_{\omega/c}^{\infty}\mathrm{d}\beta\leavevmode\nobreak\
\varphi_{a}(\omega,\beta;z_{a},z_{b})$, where the spectral flux can be
expressed as (see Messina2016a  for more details):

$\varphi(\omega,\beta;z_{a},z_{b})=\frac{4\beta}{\pi^{2}}(r^{\prime\prime}k_{zm}^{\prime\prime})^{2}\frac{e^{-2k_{z}^{\prime\prime}d}\,e^{-2k_{zm}^{\prime\prime}(z_{b}-d/2)}}{|1-r^{2}e^{-2k_{z}^{\prime\prime}d}|^{2}}\Bigl{(}N[\omega,T(z_{a})]-N[\omega,T(z_{b})]\Bigr{)},$
(2)

where $\beta$ is the parallel ($x$–$y$) component of the wavevector, while
$k_{z}=\sqrt{\omega^{2}/c^{2}-\beta^{2}}$ and
$k_{zm}=\sqrt{\varepsilon\omega^{2}/c^{2}-\beta^{2}}$ are the perpendicular
components in vacuum and inside the slabs, respectively. We also introduced
the Fresnel reflection coefficient of a slab, given by $r=(\varepsilon
k_{z}-k_{zm})/(\varepsilon k_{z}+k_{zm})$. Finally, in Eq. (2)
$a^{\prime\prime}$ represents the imaginary part of $a$ and and

$N(\omega,T)=\biggl{[}\exp\biggl{(}\frac{\hbar\omega}{k_{B}T}\biggr{)}-1\biggr{]}^{-1}.$
(3)

The coupled heat equation (1), combined with the flux expression (2), can be
solved numerically. Nevertheless, two further approximations can be performed
and allow us to obtain an analytical expression for both the temperature gap
$T_{a}-T_{b}$ between the two slabs (across the vacuum gap, see Fig. 2) and
the exchanged flux. We can first assume that the radiative energy exchange
takes places over a tiny thickness close to the vacuum interface of each body,
allowing us to treat it as a surface term, i.e. as a boundary condition.
Moreover, inspired by the results of fluctuational electrodynamics, we can
assume that the total flux exchanged radiatively can be expressed as
$\phi\simeq h_{0}(T_{a}-T_{b})/d^{2}$. In this simplified expression, the flux
depends only on the two temperatures at the interfaces and shows the known
$d^{-2}$ divergence. These approximations, whose validity has been verified
numerically, lead to the following analytical solutions

$\frac{T_{a}-T_{b}}{T_{L}-T_{R}}=\left(1+\frac{2th_{0}}{\kappa
d^{2}}\right)^{-1},\quad\frac{\varphi}{T_{L}-T_{R}}=\frac{h_{0}}{d^{2}}\left(\frac{T_{a}-T_{b}}{T_{L}-T_{R}}\right).$
(4)

The impact of conduction-radiation coupling is shown quantitatively in Figs. 2
and 3. Figure 2 concerns two silica slabs having $t_{a}=t_{b}=100\,\mu$m and
$(T_{L},T_{R})=(600,300)\,$K. The temperature profile is shown inside the left
slab for three different distances: it shows that the temperature can decrease
by more than 100 K in the left slab for the smallest distance considered.
Moreover, the inset of Fig. 2, showing the numerically-calculated distribution
of flux absorbed inside the slab, shows that it is highly peaked around the
vacuum interface, as assumed.

Figure 3: Flux $\varphi$ and temperature difference $T_{a}-T_{b}$ across the
vacuum gap (inset) as a function of $d$ between two silica slabs with
$T_{L}=600\,$K and $T_{R}=300\,$K. The solid lines correspond to different
thicknesses: 100 nm (black), $1\,\mu$m (red), $10\,\mu$m (brown), $100\,\mu$m
(blue) and $500\,\mu$m (green). The orange dashed line corresponds to the
absence of temperature gradients. Reproduced from Messina2016a .

Figure 3 shows the temperature difference across the gap $T_{a}-T_{b}$
(normalized with respect to $T_{L}-T_{R}$, inset) and the exchanged flux (main
part) for different slab thicknesses (see caption of Fig. 3). We clearly
observe that not only is a tempetaure profile inuced by the coupling, with an
effect growing with the slab thickness, but that also the flux is strongly
modified with respect to the scenario of absence of coupling (orange dashed
line in Fig. 3). The flux tends to saturate for $d$ going to zero, and both
the saturation value and the characteristic distance at which the distance-
dependent flux deviates from the no-coupling scenario depend strongly on the
thickness.

In order to make this more quantitative it is interesting to define a
characteristic coupling distance $\tilde{d}$, such that at this distance the
temperature gradient across the vacuum gap equals half of $T_{L}-T_{R}$. At
this distance we have $T_{a}-T_{b}=\frac{1}{2}(T_{L}-T_{R})$ and
$\varphi=\frac{1}{2}h_{0}(T_{L}-T_{R})/\tilde{d}^{2}$. This distance
$\tilde{d}=\sqrt{2th_{0}/\kappa}$ depends both on the thickness and on the
material-dependent $h_{0}/\kappa$ parameter, quantifying the competition
between radiative exchange (through the conductance $h_{0}$) and conductive
transport (through the conductivity $\kappa$). Figure 4 shows $\tilde{d}$ as a
function of this ratio for $t=100\,\mu$m. On top of this curve we highlight
some examples of materials, showing that this characteristic distance can vary
from some to tens or hundreds of nanometers, making the experimental
observation of conduction–radiation coupling in principle feasible for some
materials and thicknesses. We conclude this section mentioning that a previous
study was performed in this sense Wong14 , but applied only to a specific
configuration and not exploring the strong dependence on the choice of
materials and thicknesses.

Figure 4: Characteristic distance $\tilde{d}$ of conduction–radiation coupling
(see text) for $t_{a}=t_{b}=100\,\mu$m and different materials. AZO[1.2] and
AZO[0.05] denote aluminum zinc oxides of conductivities
$\kappa=1.2\,$W/m$\cdot$K and $\kappa=0.05\,$W/m$\cdot$K, respectively (see
Messina2016a  for more details). The inset shows $h_{0}$ as a function of
$\Delta T=T_{L}-300\,$K, for AZO (red), silica (blue) and SiC (black).
Reproduced from Messina2016a .

##  III The impact of slab thickness: from the diffusive to the ballistic
regime

The results presented in the previous section are based on the assumption that
the thickness of the two bodies exchanging heat is large compared to the mean
free path of phonons inside them, in the micron range for typical polar
materials. Nevertheless, because of the peculiar transport regimes which can
arise for conduction depending on the thickness, we have also extended our
study to the more general configuration of arbitrary thickness. Boltzmann’s
transport equation is the mathematical tool allowing us to fully grasp the
transition between the transport regimes, and more specifically between the
two extreme ones, namely ballistic and diffusion regimes.

At a given frequency $\omega$ (not explicitly shown) and in the relaxation
time approximation, this equation reads

$\frac{\partial f_{p}(t,\omega,\mathbf{r},\Omega)}{\partial
t}+\mathbf{v}_{g,p}(\omega)\cdot\nabla
f_{p}(t,\omega,\mathbf{r},\Omega)=-\frac{f_{p}(t,\omega,\mathbf{r},\Omega)-f_{0}(\omega)}{\tau_{p}(\omega,T(\mathbf{r}))}.$
(5)

The unknown of this equation is the distribution function $f$ associated to
the heat carriers within the solid for each polarization $p$, at time $t$,
frequency $\omega$, solid angle $\Omega$ and position $\mathbf{r}$. Moreover,
$\mathbf{v}_{g,p}(\omega)=\nabla_{\mathbf{k}}\omega_{p}$ is the group velocity
of carriers at polarization $p$ and frequency $\omega$, $f_{0}$ is the
equilibrium distribution (Fermi-Dirac for electrons and Bose-Einstein for
phonons) and $\tau_{p}$ the heat-carrier relaxation time.

In order to be solved, this equation has to be coupled to the one governing
the time evolution of the internal energy density $u$, which reads

$\frac{\partial u(\mathbf{r},t)}{\partial
t}=P_{\text{rad}}(\mathbf{r},t)+P_{\text{cond}}(\mathbf{r},t),$ (6)

$P_{\text{rad}}$ denoting the radiative power locally dissipated per unit
volume within a given body and coming from the other one, which can be
calculated within a fluctuational-electrodynamics approach (see Reina2021a
for more details). On the other hand, $P_{\text{cond}}$ denotes the conductive
power per unit volume at position $\mathbf{r}$, which is connected to the
distribution function $f$ through the relation

$\varphi_{\text{cond}}(t,\mathbf{r})=\sum_{p}\int_{4\pi}d\Omega\int
d\omega\,\hbar\omega\,\mathbf{v}_{g,p}(\omega)f_{p}(t,\omega,\mathbf{r},\Omega)\frac{D_{p}(\omega)}{4\pi},$
(7)

where $D_{p}(\omega)$ represents the density of states.

We have solved the two coupled equations in the simple geometry of two
parallel SiC slabs: the left one, denoted by index 1, is connected to a
thermostat at temperature $T_{L}=400\,$K, whereas slab 2 (on the right) is
connected to one at $T_{R}=300\,$K. Concerning the boundary conditions, two
different cases must be taken into account: for the edges in contact with
vacuum phonons hitting the surface are scattered specularly (specular
reflection), whereas phonons colliding against the thermostat are scattered in
all directions (diffuse reflection) ReinaPhD . Exploiting Boltzmann’s
equation, we are allowed to let the slab thickness vary in a wide range of
values, from the ballistic to the diffusive regime.

The results at two separation distances ($d=1$ and $5\,$nm) are shown in Fig.
5. The main part of the two plots shows the temperature profile inside slab 1,
normalized to the temperature difference across it $T_{1}(0)-T_{L}$. This
allows to highlight a signature of the conductive tranport regime in the shape
of the temperature profile. More specifically, while for the largest thickness
considered $T_{1}(z)$ becomes almost linear (as it should be in the strictly
diffusive regime, according to Fourier law), when decreasing the thickness we
observe a transition towards a significantly different behavior. In this
ballistic-like scenario the temperature profile tends to a uniform
distribution, excluding the region close to the thermostat ($z\simeq-\delta$),
where $T_{1}(z)$ is almost discontinuous (Casimir regime), and close to the
vacuum gap, where $T_{1}(z)$ shows a steep increase physically connected to
the fact that most to the radiative flux is absorbed close to the boundary.

Figure 5: (a) Steady-state temperature (inset) and normalized temperature
profile inside the left slab for different thicknesses and a separation
distance $d=1\,$nm. (b) Same as (a) for $d=5\,$nm. Reproduced from Reina2021a
.

While this discussion highlights the impact of transport regime on the shape
of the temperature profile, the information regarding the quantitative impact
of distance and thickness is contained in the insets of Fig. 5. It is clear
that an observable temperature profile (up to tens of degrees) can indeed
arise, but mainly for large thicknesses (tens of microns) and small distances
(below 5 nm).

In Fig. 6 we address the impact of the temperature profiles on the exchange
radiative flux. The exact result (solid black line) is compared to two
approximate configurations: the Polder and van Hove result (i.e. conventional
fluctuational electrodynamics ignoring the existence of a temperature profile,
red dashed line) and to the _modified Polder and van Hove_ configuration (blue
long-dashed line). The latter corresponds to a conventional fluctuational-
electrodynamics configuration assuming that the temperature inside each body
is uniform and equal to the temperature taken at the boundaries between it and
vacuum.

Figure 6: Radiative heat flux exchanged between two SiC slabs with respect to
their thickness for a separation distance of (a) $d=1\,$nm and (b) $d=5\,$nm.
We show the exact result (black line), the Polder and van Hove one (red dashed
line, uniform temperatures $T_{L}=300\,$K and $T_{R}=400\,$K) and the modified
Polder and van Hove flux (blue long-dashed line, uniform temperatures equal to
the temperatures at the boundaries with the vacuum gap). Insets: absolute
value of the error with respect to the PvH and modified PvH approaches.
Reproduced from Reina2021a .

These curves show that at the closest distance $d=1\,$nm the error in using
standard fluctuational electrodynamics (ignoring conduction–radiation
coupling) can be enormous, revealing once more the relevance of the coupling
effect for large thicknesses and small distances. On the contrary, the
modified approach reproduces relatively well the exact result. On one hand,
this confirms that radiative heat transfer is mainly a surface effect, and
thus almost entirely depends on the temperature at the boundaries between each
body and vacuum. Nevertheless, albeit its fundamental interest, this has no
direct practical use, since the knowledge of these boundary temperatures stems
from and thus needs the solution of the full problem in the presence of
coupling. Apart from being relevant for theory–experiment comparison, these
coupling effects could be relevant in some applications, involging e.g. the
thermalization of two bodies, as discussed in detail in Reina2021b .

##  IV The impact of geometry: the tip–plane configuration

After investigating the role played by the thickness of the two bodies, it is
interesting to address the impact of the geometrical configuration, not only
to unveil possible fundamental issues but also because experiments are only
rarely performed in the plane–plane configuration, which raises the serious
experimental challenge of ensuring parallelism.

In a first work Jin17b  we have studied conduction–radiation coupling between
two nanorods. In this configuration we have highlighted, the appearance of a
temperature profile, modifying in turn the exchanged radiative flux, along
with a deviation from a linear temperature profile, even in the diffusive
regime, due to the appearance of bulk polaritonic resonances, absent in the
case of two parallel planes.

We focus here more in detail on a more recent work GharibAliBarura22 , where
we have analyzed the effect of conduction–radiation coupling in the tip–plane
scenario, frequently employed in experiments. To this aim we have considered
the geometrical configuration sketched in Fig. 7 where two cylinders of radius
$R_{0}$ and $fR_{0}$ are placed in front of each other and separated by a
vacuum gap of thickness $d$. The former (latter) has thickness $\delta_{L}$
($\delta_{R}$) and is connected to a thermostat at temperature $T_{L}$
($T_{R}$). The modulation of the factor $f$ allows to switch from the
plane–plane configuration ($f=1$) to a tip–plane scenario, when $f$ tends to
zero.

Figure 7: Scheme of the two-cylinder employed to simulate the tip–plane
configuration. Reproduced from GharibAliBarura22 , with the permission of AIP
Publishing.

Inspired by the results discussed in Sec. III, we have considered large values
for the two thicknesses ($\delta_{L}=\delta_{R}=100\,\mu$m) such that we are
in the diffusive regime. We have then solved the heat equation in cylindrical
coordinates, by assuming that the two cylinders do not exchange any flux
across their lateral surfaces. Moreover, we have also assumed that radiative
flux can be treated as a surface term (i.e. as a boundary condution) and, by
following the Derjaguin approximation Derjaguin68  (also known as proximity-
force approximation), that twe two cylinders exchange heat radiatively only
across the surface of the smaller cylinder. In other words, for cylinder 1 we
have $\partial T(r,z_{2})/\partial z=0$ for $fR_{0}<r<R_{0}$ while $\partial
T(r,z_{i})/\partial z=-\varphi(r)/\kappa$ for $i=2,3$, $\kappa$ being the
thermal conductivity and $\varphi(r)$ the radiative flux locally exchanged at
coordinate $r$. The solution can be obtained analytically under the furthe
approximation that the exchanged flux can be written as

$\varphi\simeq\frac{\gamma[T(0,z_{2})-T(0,z_{3})]}{d^{2}},$ (8)

i.e. it is uniform, it depends only on the two temperatures at the center of
the two cylinder surfaces and on the separation distance as $d^{-2}$. This
allows to obtain analytical expressionf for the temperature profiles in the
cylinders

$T(r,z)=\begin{cases}T_{L}-\frac{\gamma(T_{L}-T_{R})}{\xi}\Biggl{[}f^{2}(z-z_{1})+2R_{0}f\sum_{k=1}^{\infty}\frac{J_{1}(f\alpha_{k})}{\alpha_{k}^{2}J_{0}^{2}(\alpha_{k})}\frac{\sinh\Bigl{[}\frac{\alpha_{k}(z-z_{1})}{R_{0}}\Bigr{]}}{\cosh\Bigl{[}\frac{\alpha_{k}(z_{2}-z_{1})}{R_{0}}\Bigr{]}}J_{0}\Bigl{(}\alpha_{k}\frac{r}{R}\Bigr{)}\Biggr{]},&z_{1}<z<z_{2},\\\
T_{R}+\frac{\gamma(T_{L}-T_{R})}{\xi}(z_{4}-z),&z_{3}<z<z_{4},\end{cases}$ (9)

and for the exchanged radiative flux

$\varphi(d,f,R_{0})=\frac{\frac{\gamma(T_{L}-T_{R})}{d^{2}}}{1+\frac{\gamma}{\kappa
d^{2}}\Bigl{[}f^{2}\delta_{L}+\delta_{R}+2R_{0}f\Gamma\Bigl{(}f,\frac{\delta_{L}}{R_{0}}\Bigr{)}\Bigr{]}},$
(10)

$\delta_{L}=z_{2}-z_{1}$ ($\delta_{R}=z_{4}-z_{3}$) being the height of the
larger (smaller) cylinder. In these expressions we have defined

$\begin{split}\xi&=\kappa d^{2}+\gamma(f^{2}\delta_{L}+\delta_{R})+2\gamma
R_{0}f\Gamma\Bigl{(}f,\frac{\delta_{L}}{R_{0}}\Bigr{)},\\\
\Gamma(f,\beta)&=\sum_{k=1}^{\infty}J_{1}(f\alpha_{k})\tanh(\alpha_{k}\beta)/[\alpha_{k}^{2}J_{0}^{2}(\alpha_{k})].\end{split}$
(11)

As expected, Eq. (10) allows to get back the results for two parallel slabs
Messina2016a  for $f=1$. As discussed more in detail in GharibAliBarura22 ,
going beyond the approximation described in Eq. (8) allows to get numerical
results in very good agreement, for the physical parameters taken into account
below, with the analytical expressions.

The results for a configuration with $\delta_{L}=\delta_{R}=100\,\mu$m,
$R_{0}=10\,\mu$m and $f=10^{-2}$ are shown in Fig. 8(a), where they are
compare to the configuration of absence of coupling and to the slab–slab
scenario (corresponding to $f=1$).

Figure 8: (a) Radiative heat flux as a function of $d$ in a tip - plane
configuration (solid lines), slab-slab scenario (dot-dashed lines), and in the
absence of coupling (dashed lines). The different colors correspond to
different materials: black for SiO2, red for SiC and blue for Au. (b) Ratio
between fluxes in the presence and absence of coupling as a function of
$R_{0}$ and $f$. The plot corresponds to two SiO2 cylinders having
$\delta_{L}=\delta_{R}=100\,\mu$m at distance $d=1\,$nm. Reproduced from
GharibAliBarura22 , with the permission of AIP Publishing.

More specifically, the figure shows the results for silica (SiO2), silicon
carbide (SiC) and gold bodies. In the latter scenario we observe that the
curves in the absence and in the presence of coupling (both for slabs and
cylinders) are indistinguishable. The reason behind this is that the radiative
exchange in the case of gold is weak compared to the case of polar materials,
mainly because the surface resonant modes (surface plasmons) supported by gold
bodies lie in the ultraviolet region of the spectrum, and are thus not excited
thermally around ambient temperature. As a result, no significant temperature
profile and no impact on the radiative flux are expected.

On the contrary, based on what we already discussed in Sec. II, for two polar
materials radiative heat flux in the nanometer range of distances is supposed
to compete with conduction. The results of Fig. 8(a) show that the results for
two cylinders qualitatively follow the ones for two slabs: the characteristic
distance at which the curve in the presence of coupling deviates from the one
in the absence of coupling is almost the same, while the value of the
saturated ($d\to 0$) flux is only slightly higher than the one for two slabs.
This allows to state that also in a tip–plane configuration the impact of
conduction–radiation coupling is supposed to be observable, at least for polar
materials, in the nanometer range of distances.

To conclude, we analyze in Fig. 8(b) the combined effect of radius $R_{0}$ and
radii fraction $f$. We observe that the ratio between flux in the presence
[$\varphi(d,f,R_{0})$] and in the absence [$\Phi(d)$] of coupling is of the
order of $10^{-3}$ in a wide range of both parameters. Finally, the black
dashed line in Fig. 8(b) indicates that the flux correction is significant for
all points defining a hypothetical tip radius of 100 nm, corresponding to an
experimentally reasonable value.

##  V Conclusions

We have reviewed our recent work on the coupling between conduction and
radiation for two solids out of thermal equiibrium interacting in near-field
regime. We have shown that, depending on the separation distance and the
materials nature under scrutiny, this coupling can be at the origin of a non-
negligible temperature profile inside each body (ignored in previous works),
which can in turn induce a saturation of the radiative heat flux exchanged in
the two bodies with respect to the predictions of conventional fluctuational
electrodynamics. Because of the current possibility to explore experimentally
distances in the nanometer range and below and of the ongoing miniaturization
of a variety of technological devices, these results show that this coupling
needs to be taken into account, both for the sake of theory–experiment
comparison and in view of the design of innovative devices operating at the
nanoscale.

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# Cost Efficient Repository Management for Cloud-Based On-Demand Video
Streaming

Mahmoud Darwich∗‡, Ege Beyazit∗, Mohsen Amini Salehi†‡, Magdy Bayoumi∗ ∗Center
for Advanced Computer Studies (CACS)
‡High Performance Cloud Computing (HPCC) Laboratory
†School of Computing and Informatics
University of Louisiana at Lafayette, Louisiana 70503
Email: {mkd1007, exb6143, amini<EMAIL_ADDRESS>
###### Abstract

Video transcoding is the process of converting a video to the format supported
by the viewer’s device. Video transcoding requires a huge storage and
computational resources, thus, many video stream providers choose to carry it
out on the cloud. Video streaming providers generally need to prepare several
formats of the same video (termed pre-transcoding) and stream the appropriate
format to the viewer. However, pre-transcoding requires enormous storage space
and imposes a significant cost to the stream provider. More importantly, pre-
transcoding proven to be inefficient due to long-tail access pattern to video
streams in a repository. To reduce the incurred cost, in this research, we
propose a method to partially pre-transcode video streams and re-transcode the
rest of it in an on-demand manner. We will develop a method to strike a trade-
off between pre-transcoding and on-demand transcoding of video streams to
reduce the overall cost. Experimental results show the efficiency of our
approach, particularly, when a high percentage of videos are accessed
frequently. In such repositories, the proposed approach reduces the incurred
cost by up to 70%.

###### Index Terms:

cloud, storage, video stream, partial pre-transcoding, re-transcoding

##  I Introduction

Video streaming through the Internet has become a common practice. Viewers
stream videos on variety of devices, from large screen TVs and desktops to
tablets and smart-phones. Based on the Global Internet Phenomena Report [1],
video streaming currently constitutes around 64% of the U.S. Internet traffic
and it is rocketing to 80% of the whole Internet traffic by 2019[2].

Video contents, either in the form of Video On Demand (VOD) (e.g.,
YouTube111https://www.youtube.com or Netflix222https://www.netflix.com) or
live-streaming (e.g., Livestream333https://livestreams.com), need to be
converted (i.e., transcoded) based on the device characteristics of viewers.
That is, the original video has to be converted to a supported characteristics
(e.g., frame rate, resolution, and network bandwidth) of the viewers’ devices
[3].

Video transcoding is a compute-intensive and time-consuming process. To
overcome the computational demand of transcoding, Video Stream Providers (VSP)
extensively utilize cloud computing services [4]. To make the VOD streams
readily available for viewers, VSPs commonly carry out the transcoding
operation of VODs in an off-line manner. That is, they transcode and store
multiple formats of the same video in the cloud to satisfy the requirements of
viewers with heterogeneous display devices. We call this approach _pre-
transcoding_. In practice, VSPs such as Netflix have to pre-transcode 40 to 50
different formats of a single video [5] and store them in their repositories.
This imposes a significant cost overhead to VSPs [6, 8].

Recent studies show that accessing videos of a VSP follows a long tail
distribution [7]. That is, there are few videos that are accessed very
frequently while there is a huge portion of videos that are rarely accessed.
Thus, research works have been undertaken (e.g.,  [4, 8]) to alleviate the
cost overhead of pre-transcoding by transcoding rarely-accessed videos in an
on-demand (i.e., lazy) manner. In this manner, one or few formats of a video
is stored and transcoding is performed on-the-spot upon request to access a
format of video that is not already pre-transcoded. This has become feasible
with the enormous computational capacity clouds offer. We term the lazy
transcoding of videos as _re-transcoding_ and storing of videos as _pre-
transcoding_.

Re-transcoding induces the computational cost of VSPs which is generally more
expensive than the storage cost [9], this is because computation power are
charged in on hourly basis in cloud. Therefore, the re-transcoding approach
would be beneficial to VSPs, only if it is applied on the rarely accessed
videos. Conversely, if it is applied on _frequently accessed videos_ (FAVs),
it increases the cost overhead significantly as we have to pay for every time
the video stream is transcoded.

The challenge is how to achieve partial pre-transcoding? That is, which parts
of the video should be re-transcoded and which part pre-transcoded?

To address these challenges, in this paper, we propose a method to perform
pre-transcoding on a portion of the video.

In summary, the contributions of this paper are: (A) Proposing method to
reduce the incurred cost of using cloud services through pre-transcoding, re-
transcoding, or partially-pretranscoding videos in the repository. (B)
Analyzing the impact of our proposed method when the rate of access to video
streams in a repository varies.

Experimental results demonstrate that our proposed partial-transcoding method
can reduce the cost overhead of VSPs significantly. The research outcome of
this paper can help VSPs to reduce their cost overhead without losing Quality
of Service (QoS) demanded by their viewers.

The rest of the paper is organized as follows: section II provides some
background on video streaming and transcoding. Section III presents the
partial pre-transcoding method. Experiment setup is detailed in section IV.
Experimental results are discussed in section V, related works are presented
in section VI and finally section VII concludes the paper.

##  II Background

###  II-A Video Stream Structure

A video stream is a set of sequences as shown in Fig. 1. Each sequence is
built of several _Group Of Pictures (GOPs)_. The first block of a sequence is
called a sequence header that contains some meta-data about that sequence.
Also, each GOP is constructed of a GOP header followed by several frame types,
starting with I (intra) frame, followed by a P (predicted) and B (bi-
directional) frames. Each frame is further comprised of slices that are formed
from macroblocks (MB) [17].

As each GOP can be processed independently, video transcoding operation is
commonly achieved at the GOP level [17]. That is, each GOP is considered as a
unit for processing or pre-transcoding.

Figure 1: Video Stream Structure

###  II-B Cloud Services for Video Stream Transcoding

Cloud providers offer different services in an on-demand manner and charge
their users in a pay-as-you-go manner. A cloud-based video streaming system
utilizes different cloud services as follows:

  * •

Computational Services: Computational services in clouds are generally
provided through Virtual Machines (VM) and users are charged in an hourly
basis.

  * •

Storage Services: Users are charged for storage services based on the volume
of their data stored on the cloud usually on a monthly basis.

  * •

Content Delivery Network (CDN) services: CDN is a technology that reduces the
delay to access different static content types, including video streams,
through the Internet. CDN technology replicates the content (e.g., video
content) in different geographical areas to minimize the network travel time
of content to users [10, 11].

Amazon Web Services444https://aws.amazon.com/ is a major cloud provider and
offers all the foregoing cloud services with a high reliability. Although this
study is independent of AWS technology and can work on any of cloud provider,
we consider AWS services, charging model, and costs for our evaluations.

Amazon Elastic Compute Cloud (Amazon EC2) provides computational services in
form of VMs. It offers various types of VMs to cover different computational
demands. General purpose t2-small VM type is the most common service used for
different type of processing and we utilize this VM type for our evaluations
as well. The hourly cost of t2-small VM is $0.026. Amazon Simple Storage
Service (Amazon S3) is the storage service of Amazon cloud. Amazon S3 costs
$0.03 for each Gigabyte of stored data in a month. Amazon also offers the CDN
service (called CloudFront555https://aws.amazon.com/cloudfront/). It delivers
the content to users (i.e., viewers) through a worldwide network of data
centers with minimum delay [12]. Amazon charges $0.085 for each Gigabyte of
data uses CloudFront for the first 10TB per month.

It is worth noting that CDN services are required, in addition to storage
services to perform pre-transcoded cloud-based video streaming. However, the
CDN cost is not applied when we provide on-demand video transcoding service.

##  III Proposed Partial Pre-Transcoding Method

###  III-A Video Streams Access Pattern

As re-transcoding cost is incurred every time a video stream is transcoded, it
is crucial to know the access pattern to videos. Previous studies show that
accessing video streams follow a long tail distribution. Other observations
(e.g.,  [15]) reveal that, within each video stream, the beginning segments
(GOPs) are watched more frequently than the rest of the video streams.

Miranda et al. [15] show that the distribution of the views within a video
stream follows a long tail distribution. More specifically, they show that the
distribution of accesses to GOPs of a video stream can be expressed by Power-
law [18] model. Therefore, if a video stream is accessed $V$ times, then we
can estimate the number of times each GOP is viewed. Let $GOP_{i}$ as the
$i$th GOP in a video stream, then the number of views for $GOP_{i}$, denoted
$P_{i}$, is estimated based on Equation 1. In this equation, $\alpha$ is a
constant with value 0.1.

$P_{i}=\dfrac{V}{GOP_{i}^{\alpha}}$ (1)

###  III-B Storage Cost of Video Streams

The storage cost of a video stream depends on the video size, and the cloud
storage unit price. The equation of cloud storage cost for video $v$ is
defined as:

$C_{S}=\dfrac{S_{v}\cdot P_{S}}{2^{10}}$ (2)

where $S_{v}$ is the size of video $v$ in MB, $P_{S}$ is the storage unit
price in dollar per GB, the term $2^{10}$ is used to convert the video size
unit from MB to GB. Equation 2, however, might be extended to compute the
storage of cost of each GOP in the video stream. The equation of storage cost
for $GOP_{i}$ is $C_{S_{i}}=\dfrac{P_{S}\cdot S_{GOP_{i}}}{2^{10}}$, where
$S_{GOP_{i}}$ is the size of $GOP_{i}$.

###  III-C Transcoding Cost of Video Streams

The transcoding cost is the cost of using virtual machines (VMs) that depends
on the time span of utilizing VMs and the type of VM utilized for transcoding.
Let $P_{T}$ the cost of using VM for an hour, and $\tau_{v}$ the estimated
transcoding time of video $v$ in seconds. Then, the cost of transcoding of
video $v$, denoted $C_{T}$, is obtained using Equation 3.

$C_{T}=\dfrac{P_{T}\cdot\tau_{v}}{3600}$ (3)

It is noteworthy that Equation 3 determines the cost for one time re-
transcoding of a video. However, if a video is re-transcoded $V$ times, the
total cost would be $V\cdotp C_{T}$.

Estimation of transcoding execution time ($\tau_{v}$ in Equation 3) can be
obtained based on historic transcoding execution times of the video in the
past666As we are dealing with VOD, we expect that videos have been watched,
thus, transcoded before. However, this is not the case when we deal with live
stream videos. In that case such historic execution information is not
available.. In particular, the estimated transcoding time of video $v$ is the
sum of transcoding time of all GOPs in that video, i.e.,
$\tau_{v}=\sum_{i=1}^{m}\tau_{i}$, where $\tau_{i}$ is the estimated
transcoding time of $GOP_{i}$ and $m$ is the total number of GOPs in video
$v$.

Accordingly, Equation 3 can be extended to compute the transcoding cost of
each GOP in the video stream. The equation of transcoding cost for $GOP_{i}$
is represented by $C_{T_{i}}=\dfrac{P_{T}\cdot\tau_{i}}{3600}$, where
$\tau_{i}$ is the estimated transcoding time of $GOP_{i}$.

###  III-D Algorithm 1

The goal of this algorithm is to minimize the incurred cost of using cloud
services for VOD transcoding. For that goal, the algorithm is executed for
each video in the VSP repository periodically (e.g., monthly). The pseudo code
for this algorithm is presented in Algorithm 1. It receives the video size,
estimated video transcoding time, storage unit price, transcoding unit price,
and number of views of the video in the last time period (e.g., last month) as
input values. The output of the algorithm is whether to pre-transcode the
whole video or re-transcode it upon request.

The algorithm calculates the cost of storage and transcoding according to what
was discussed in Equation 2 and 3 ( see steps 1 and 2 Algorithm 1). We
calculate the ratio of video storage cost to video transcoding cost in step 6
(called _cost ratio_). When the ratio is less than or equal 1, it means that
the cost of pre-transcoding is less than or equal the re-transcoding cost,
thus, we store that video format, otherwise, we call Algorithm 2 to decide
about partial pre-transcoding of the video. It is worth noting that for a new
video $V=0$, thus, we postpone its transcoding to when it is requested.

###  III-E Algorithm 2

Figure 2:  Long tail distribution of GOPs views in video stream and partial
pre-transcoding a video stream based on GOP threshold

When the cost ratio for a video is greater than 1, Algorithm 2 is called to
possibly pre-transcode a portion of the video (termed partially pre-transcoded
video). It is proven that the access pattern of GOPs within a video stream
follows a long tail distribution [15]. That is, the first GOPs of a video
stream are watched more often than the rest. Thus, the cost ratio of GOPs
constantly increases for the later GOPs in the video stream. Accordingly, it
is possible to find a boundary GOP so that all GOPs before it need to be pre-
transcoded and all GOPs after that re-transcoded. We call this dividing point
as GOP threshold ($GOP_{th}$). Formally, $GOP_{th}$ is defined as the first
GOP of a video stream that has its cost ratio greater than one.

Algorithm 2 receives GOPs of a video stream, GOP size, estimated transcoding
time of GOP, storage unit price, transcoding unit price and number of requests
for the video stream in the last time period as input values.

Algorithm 2 searches for the GOP threshold by calculating the cost ratio of
each GOP sequentially, starting from the first GOP in the video stream (step 1
to 5 in Algorithm 2). After finding the GOP threshold, Algorithm 2 pre-
transcodes the GOPs before GOPthreshold and re-transcodes the GOPs after it as
shown in Algorimth 2

Fig. 2 illustrates how Algorithm 2 functions based on the long tail access
pattern to GOPs within a video stream. Once we find the GOP threshold, all the
GOPs with cost ratio less than 1 (i.e., $R_{i}<1$) are pre-transcoded and the
rest of GOP are re-transcoded.

1

Input :

Size of video $v$: $S_{v}$

Transcoding time of video $v$: $\tau_{v}$

Transcoding unit price: $P_{T}$

Storage unit price: $P_{S}$

Number of views in the last time period: $V$

Output : fully pre-transcoding or fully re-transcode a video

2

3Calculate cost of transcoding of $v$:
$C_{T}\leftarrow\dfrac{P_{T}\cdot\tau_{v}}{3600}$

4 Calculate cost of storage of $v$: $C_{S}\leftarrow\dfrac{S_{v}\cdot
P_{S}}{2^{10}}$

5 if  _$V=0$_ then

6 Re-transcode the whole video $v$

7 end if

8 if  _$\dfrac{C_{S}}{V\cdot C_{T}}\leq 1$_ then

9 pre-transcode video $v$

10 else

11 Perform partial pre-transcoding using Algorithm 2

12 end if

Algorithm 1 Fully pre-transcode or fully re-transcode a video

1

Input :

video $v$ with $m$ $GOP$

Size of $GOP_{i}$: $S_{i}$

Transcoding time of video $GOP_{i}$: $\tau_{i}$

Transcoding unit price: $P_{T}$

Storage unit price: $P_{S}$

Views number in the last time period : $V$

Output : Partial pre-transcoding a video

2 For each $GOP_{i}$ in video stream $v$

3

4Estimated number of views for $GOP_{i}$: $P_{i}\leftarrow V\cdot
GOP_{i}^{-\alpha}$

5

6Calculate cost of transcoding of $GOP_{i}$:
$C_{T_{i}}\leftarrow\dfrac{P_{T}\cdot\tau_{i}}{3600}$

7 Calculate cost of storage of $GOP_{i}$:
$C_{S_{i}}\leftarrow\dfrac{S_{i}\cdot P_{S}}{2^{10}}$

8 Calculate cost ratio of $GOP_{i}$:
$R_{i}\leftarrow\dfrac{C_{S_{i}}}{P_{i}\cdot C_{T_{i}}}$

9

10if  _$R_{i} >1$_ then

11 $GOP_{th}\leftarrow GOP_{i}$

12 Break;

13 else

14 Continue;

15 end if

16 Pre-transcode $GOP_{1}$ to $GOP_{th-1}$

17 Re-transcode $GOP_{th}$ to $GOP_{m}$

18

Algorithm 2 Partial pre-transcoding a video

##  IV Experiment Setup

###  IV-A Videos Synthesis

VSPs have large video streams repository. However, we do not have access to
such repositories. Downloading a large quantity of videos and then transcoding
them is a long and costly process. Therefore, we simulate large repositories
by synthesizing videos.

To accurately synthesize videos, we need to know the distribution of
characteristics of videos. Specifically, GOP size, GOP transcoding time, and
number of GOPs for each video are required. We synthesize videos based on our
repository that includes 103 videos of a wide range of categories including
movies, animation, sports, documentary, news, and music. the size of the
videos are between 400MB and 2.5GB. In order to extract the characteristics of
those videos, we transcode them by implementing the method in [4] on a machine
with 2.8 GHz Intel Core i5, 8GB RAM, and 1TB Hard Disk. We analyze the
characteristic data extracted from videos and observe that GOP size and number
of GOPs follow a Gaussian curve, shown in Fig. 3(a) and 3(b). The parameters
of the Gaussian distributions are given in table I. Additionally, Li et
al.[19] proved through a regression analysis that there is a linear
relationship between GOP size and its transcoding time. By implementing the
same method we derive a linear equation between GOP sizes and their
corresponding transcoding times for all downloaded videos and the plot of the
analysis is shown in Fig. 4.

Having the statistics of GOP size, number of GOPs, and the linear equation for
GOP transcoding times, we generate 50,000 videos with 112.98 TB size for our
synthesized repository.

| GOP size (Kbytes) | number of GOPs
---|---|---
Mean | 655.08 | 1262.79
Standard Deviation | 201.44 | 271.46
Standard error mean | 0.57 | 26.74
Upper 95% Mean | 656.20 | 1315.85
Lower 95% Mean | 653.96 | 1209.74
Total no. of GOP | 124593 | 130068
Min. GOP size | 1.91 | 580
Max. GOP size | 2192.65 | 2018
TABLE I:  statistics of GOP size and number of GOPs in a video repository

(a) GOP size histogram

(b) GOPs of video histogram

Figure 3:  distribution of GOP size and numbers of GOPs in videos
$0$$1$$2$$3$$4$$5$$6$$7$$0$$5$$10$$15$$20$GOP size (Mb)Transcoding time
(s)data points$3.11\cdot x+1.14$ ($R^{2}=0.67$)  Figure 4: Linear regression
analysis of GOP size and transcoding time

###  IV-B View Pattern for Synthesized Videos

According to [7], we model the access pattern to repository by applying
Weibull distribution to generate long tail access pattern. The frequently
accessed videos (FAVs) that have high access rates, are located at the
beginning of the Weibull curve and those have less views are in the long tail.
We determine the percentage of FAVs number in the repository based on the
shape and scale coefficients $\alpha$ and $\beta$ respectively of the Weibull
distribution. As alpha decreases, the number of FAVs increases in the
repository and vice versa. The interval range of $\alpha$ used in the
experiment is $[0.4:2.4]$ while $\beta=1$

###  IV-C Storage and Computation Prices

The prices of storage and computation used in the experiments are modeled
based on Amazon Web Service (AWS) prices as shown in Table II

Transcoding | storage | CDN
---|---|---
t2-small | s3 | CloudFront
$0.026 /hour | $0.03 GB/month | $0.085 GB/month
TABLE II: AWS computation, storage, and Content Delivery Network (CDN)
services and their prices in US dollar $

###  IV-D Baseline Methods for Comparison

To evaluate our proposed algorithms, we compare them against two baseline
methods that are _Fully pre-transcoding_ (that stores the whole video streams)
and _Fully re-transcoding_ (that re-transcodes all video streams upon
request). These two methods ignore the metadata of processing and storage
costs of the video streams.

##  V  Experimental Results

###  V-A Impact of FAVs percentage in repository

We create several synthesized repositories of video streams to illustrate the
performance of the proposed algorithms. Each synthesized repository is made of
total 50,000 videos with a percentage of FAVs. The shape parameter in of
Weibull distribution ($\alpha$) controls the percentage of FAVs in the
repositories. The percentage of FAVs based on $\alpha$ and $\beta$ are shown
in Table III. The average view of videos in the repositories is 1.99. We
conduct each experiment 10 times and report the mean and 95% confidence
interval.

$\beta=1$ | $\alpha$ | FAVs
---|---|---
0.4 | 30%
0.6 | 25%
1 | 20%
1.4 | 15%
1.8 | 10%
| 2.4 | 5%
TABLE III: Percentage of FAVs in the repository varies by changing $\alpha$ in
the Weibull distribution

Fig. 5(a) shows the total cost of fully pre-transcoding, fully re-transcoding
and partial pre-transcoding methods when the percentage of FAVs vary. When
$\alpha$ is small, the percentage of FAVs increases and they, in turn,
increase the cost of transcoding process. As illustrated in the figure, the
fully storage cost does not change and it stays constant even when changing
the percentage of FAVs, because the fully storage cost is independent of the
number of views of video streams.

5%10%15%20%25%30%$1$$2$$3$$\times 10^{3}$percentage of FAVs in the
repositorytotal cost ($)Pre-transcoding MethodRe-transcoding MethodPartially
Pre-transcoding Method (a)

5%10%15%20%25%30%$2$$4$$6$$\times 10^{3}$percentage of FAVs in the
repositorytotal cost including CDN ($)Pre-transcoding MethodRe-transcoding
MethodPartially Pre-transcoding Method (b)

12345$1$$2$$\times 10^{3}$average number of views per videototal cost ($)Pre-
transcoding MethodRe-transcoding MethodPartially Pre-transcoding Method (c)

Figure 5: Incurred costs of fully pre-transcoding, fully re-transcoding, and
partially pre-transcoding methods: (5(a)) when the percentage of FAVs varies
in the repository; (5(b)) when the CDN cost is added to the cloud storage
cost; And (5(c)) when the average view for videos in the synthesized
repository is increased.

The experiment shows that our proposed method always reduces the incurred cost
comparing to fully re-transcoding method by up to 70% when 30% of the
repository is FAVs. Also, comparing to the fully pre-transcoding method, our
proposed approach reduces the cost up to 60%. As the percentage of FAVs
increases, our proposed method can reduce the total incurred cost
significantly.

Similarly, in Fig. 5(b), we includes the cost of Content Delivery Network
(CDN) to the cloud storage cost. The proposed partial re-transcoding method
reduces the cost up by 66%. In fact, when we consider CDN, the total cloud
storage cost is increased. We note that as the cost ratio is increased, the
number FAVs which are dependent on this ratio, decreases in the repository.

###  V-B Impact of average of number of views

In this experiment, we increase the average number of views of FAVs without
increasing percentage of FAVs, as shown in Fig. 5(c). In this experiment, the
repository includes 20% of FAVs. The average number of views of FAVs is
multiplied by constant 2, 3, 4, and 5 and the remaining videos in the
repository are divided by the same constants.

The cost of fully re-transcoding increases, because FAVs have high views which
has direct impact on the re-transcoding cost. The proposed method outperforms
the fully pre-transcoding and fully re-transcoding methods. As the views of
videos streams increases, the total cost of partial pre-transcoding method
reduces.

##  VI Related Work

Gao et al.[16] proposed a scheme that partially transcodes video contents in
the Cloud. Their approach aims to store the first segments of a video contents
which are more frequently viewed, while transcode the remaining video contents
online when requested, resulting storage and transcoding computation cost
efficiency. Their system model is based on a partial transcoding of a video
segments and a dynamic storing system that is adapted to the video access rate
change. They designed online algorithm to optimize the performance of their
scheme. They stated that the viewers play 20% of the video duration. they
adopted the video access pattern by implementing the truncated exponential
distribution to represent the video view rate. They demonstrated that their
method reduces 30% of operational cost by compared to storing the whole video.

Zhao et al.[14] proposed an approach to trade-off between computation and
storage costs and to minimize the cost for multi version videos. They utilized
the transcoding weight graph which is the transcoding relationships among
versions of a video, along with the popularity of those different versions of
the video. Based on the popularity and transcoding relationships among
different video versions, their method decides which versions of a video
should be stored in the repository or re-transcoded on demand. Their results
shows reduction in the cost when compared to storing all versions.

Jokhio et al.[13] developed a strategy to strike a trade off between the
computation and storage costs of a video. They estimated the computation cost,
the storage cost, and the video popularity information of individual
transcoded videos and then utilized this information to make decisions on how
long a video should be stored or how frequently it should be re-transcoded
from a given source video. They compared their proposal to semisynthetic and
realistic load patterns. Their results indicated that their strategy is more
cost-efficient than the two intuitive strategies.

##  VII Conclusion

In this paper, we propose a method to reduce the incurred cost of using cloud
services through pre-transcoding, re-transcoding, or partially pre-transcoding
of videos in the videos repository. We also analyze the impact of our proposed
method when the rate of access to video streams in a repository varies.
Experiment results show that as the percentage of FAVs in the repository
increases, the proposed method reduces the total incurred cost significantly.
In particular, by increasing the number of views of FAVs, our method reduces
the total incurred cost by up to 70%. Our future work will focus on the
prediction of the number of views of a video stream by applying machine
learning techniques. The method will operate based on views statistics over
the past and thus we can develop offline algorithms to estimate the cost of
video stream transcoding in the cloud. This method will help VSPs to optimize
the cost of using cloud resources.

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