# Features of hadronic and deconfined matter from AGS to LHC energies

M. Petrovici  National Institute for Physics and Nuclear Engineering - IFIN-
HH, P.O. Box MG-6,
Hadron Physics Department
Bucharest-Magurele, Romania  A. Pop  National Institute for Physics and
Nuclear Engineering - IFIN-HH, P.O. Box MG-6,
Hadron Physics Department
Bucharest-Magurele, Romania

###### Abstract

Previous extensive studies on the dependence of the average transverse
momentum, its slope as a function of the hadron mass and the average
transverse expansion on the particle multiplicity per unit rapidity and unit
transverse overlap area of the colliding partners are extended to the ratio of
the energy density to the entropy density.

The behaviour of the ratio between the average transverse momentum and the
square root of the particle multiplicity per unit rapidity and unit transverse
overlap area $\langle p_{T}\rangle/\sqrt{\langle dN/dy\rangle/S_{\perp}}$ as a
function of collision energy for a given centrality or as a function of
centrality for a given collision energy supports the predictions of CGC and
percolation based approaches. The dependence of the ratio of the energy
density $\langle dE_{T}/dy\rangle/S_{\perp}$ to the entropy density $\langle
dN/dy\rangle/S_{\perp}$ at different collision centralities for A-A collisions
from AGS, SPS, RHIC and LHC energies is presented. The trend of this ratio
towards a plateau at the highest RHIC energies followed by a steep rise at LHC
energies is in agreement with theoretical predictions made 40 years ago that
indicate this behaviour as a signature of a phase transition. This pattern
strongly depends on the collision geometry, converging towards the dependence
that characterizes the pp minimum bias (MB) collisions for the most peripheral
A-A collisions. Expected similarities between pp and Pb-Pb collisions at LHC
energies are confirmed.

##  I Introduction

The unprecedented amount of experimental information obtained at AGS, SPS,
RHIC up to the highest energies from LHC supports the theoretical predictions
made about 50 years ago on the possibility to produce very hot and dense
matter in heavy ion collisions [1]. Using QCD asymptotic freedom properties, a
transition from the hadronic phase to a high density ”quark soup” [2, 3] or
”quark-gluon plasma” [4] following the ”quark liberation” idea [3] is
expected. The first estimates of the transition from a gas of free nucleons to
hadronic matter and subsequently to deconfined matter as a function of density
were done within the percolation approach [5, 6]. Studying the production
mechanism and properties of tiny pieces of deconfined matter as a function of
temperature and density is a difficult task to the extent that the highly
inhomogeneous initial state, finite size effects, and violent dynamical
evolution must be taken into account to obtain a unique explanation of the
experimental observations. That is why a clean and unambiguous experimental
signature of deconfinement using collisions with heavy ions is rather
difficult to be established. Theoretical approaches, which combine sound
hypotheses for different snapshots of the formation and evolution of the
system produced in heavy ion collisions starting with the initial phase
described in the Color Glass Condensate (CGC) approach [7, 8, 9, 10], followed
by the process of equilibration described on the basis of the QCD kinetic
theory method and the hydrodynamic expansion were recently developed [11]. The
correlation between the multiplicity of particles and the average transverse
momentum ($\langle p_{T}\rangle$) has been proposed as a signature of the
transition from hadronic to deconfined matter [12]. Such phenomenological
models predict an increase followed by a plateau of $\langle p_{T}\rangle$ as
a function of entropy density, caused by the mixed phase corresponding to the
transition from pure hadronic matter to deconfined matter. At even larger
entropy density the $\langle p_{T}\rangle$ starts to increase again. Following
this idea and existing information on the average transverse mass ($\langle
m_{T}\rangle$) of charged hadrons identified in A-A collisions at AGS, SPS,
and RHIC, a saturation of $\langle m_{T}\rangle$ as a function of the charged
particle multiplicity per unit rapidity for the most central collisions was
evidenced [13]. In this study the $\langle m_{T}\rangle$ was estimated using
an exponential parameterization of the experimental $p_{T}$ spectra. Recently,
including more information obtained at RHIC in the Beam Energy Scan (BES)
program for Au-Au collisions and LHC data for the Pb-Pb collision at
$\sqrt{s_{NN}}$ = 2.76 TeV [14] the correlation ($\langle m_{T}\rangle-
m_{0})$-ln($\sqrt{s_{NN}}$) for the most central collisions was studied for
identified hadrons under the hypothesis that ($\langle m_{T}\rangle-m_{0})$
can be an approximation of the system temperature while ln
($\sqrt{s_{NN}}$)$\sim\langle dN/dy\rangle$ of its entropy. $m_{0}$ is the
rest mass of the particle. A dependence on mass was clearly observed. The
saturation range in ln($\sqrt{s_{NN}}$) decreases and ($\langle m_{T}\rangle-
m_{0}$) increases again more violently going from pions to protons. It is
worth mentioning that collective expansion and suppression phenomena can
contribute in such a correlation to the observed trend. A similar prediction
was obtained by including the relativistic hydrodynamic evolution of the
quark-gluon plasma [15, 16]. In these works it was suggested that the
correlation between the ratio of the energy density to entropy density is also
sensitive to the transition from hadronic to deconfined matter. The energy
density per unit transverse overlap area can be estimated using the Bjorken
invariance, while the entropy density per unit transverse overlap area,
assuming an isentropic expansion, is proportional to the particle multiplicity
per unit rapidity and unit transverse overlap area. A similar correlation,
using the charged particle multiplicity and the transverse energy per unit
pseudorapidity ($\langle dE_{T}/d\eta\rangle/\langle dN_{ch}/d\eta\rangle)$
for the most central collisions, with the assumption that
ln($\sqrt{s_{NN}}$)$\sim dN_{ch}/d\eta$ and a rather small dependence of the
transverse overlap area ($S_{\perp}$) on the collision energy, was plotted
using experimental information from SIS18 up to the highest energy from RHIC
[17], the results from the LHC for the Pb-Pb collision at $\sqrt{s_{NN}}$ =
2.76 TeV being included once the data became available [18]. As predicted by
the above mentioned models a plateau at the highest energies at RHIC and a
strong increase at the LHC energy were evidenced. The $\langle
dE_{T}/d\eta\rangle/\langle dN_{ch}/d\eta\rangle$-$\sqrt{s_{NN}}$ correlations
for the most central collisions were shown for data measured by the PHENIX
Collaboration at RHIC energies [19]. The $\langle dE_{T}/dy\rangle/\langle
dN_{ch}/dy\rangle$-$\sqrt{s_{NN}}$ correlations for different centralities
were represented for data measured by the STAR Collaboration at BES energies
[20]. Theoretical predictions show that the energy over entropy versus entropy
correlation is more sensitive to the equation of state than the average
transverse mass versus multiplicity density one [21].

In the present paper, the results for the $\langle dE_{T}/dy\rangle/\langle
dN/dy\rangle$-$\langle dN/dy\rangle/S_{\perp}$ correlation as a function of
centrality in A-A collisions at different energies and a comparison with pp
minimum bias and pp as a function of $\langle dN/dy\rangle/S_{\perp}$ at LHC
energies are presented. Results based on published data obtained in Au-Au
collisions at AGS and RHIC and Pb-Pb collisions at SPS and LHC, in terms of
$\langle dE_{T}/dy\rangle/\langle dN/dy\rangle$-$\langle
dN/dy\rangle/S_{\perp}$ for the most central collisions are presented in
Section II. Section III is dedicated to similar studies as a function of
centrality for RHIC and LHC energies. In Section IV the core contribution in
such correlations is presented. The dependence on $\langle
dN/dy\rangle$/$S_{\perp}$ of the Bjorken energy density times the interaction
time for central collisions and as a function of centrality is presented in
Section V. The comparison of A-A with MB pp collisions at the same energies
and for different charged particle multiplicities at LHC energies is dicsussed
in Section VI. Conclusions are presented in Section VII.

##  II $\langle dE_{T}/dy\rangle/\langle dN/dy\rangle-\langle
dN/dy\rangle/S_{\perp}$ correlation for central A-A collisions

In our previous studies [22, 23, 24] it was shown that the particle
multiplicity per unit rapidity and unit transverse overlap area, a good
estimate of the entropy density, turns out to be a scaling observable which
governs the behaviour of the average transverse expansion for identified
charged hadrons and hyperons, independent on the size of the colliding systems
and even for pp collisions at LHC energies. For collision energies higher than
$\sqrt{s_{NN}}$=39 GeV the scaling was also evident for the slope of $\langle
p_{T}\rangle$ as a function of mass. While such scaling also occurs below
$\sqrt{s_{NN}}$=39 GeV up to the mid-central collisions, a slight deviation of
$\sim$10% relative to the higher energies is observed towards central
collisions. As far as concerns $\langle p_{T}\rangle$, a very good scaling is
observed for the identified charged hadrons at RHIC energies. The linear
dependence of $\langle p_{T}\rangle$ on $\sqrt{\langle
dN/dy\rangle/S_{\perp}^{geom}}$ holds also for LHC energies, with a slightly
different offset relative to the RHIC energies and a tendency towards
saturation at the most central collisions. The geometrical transverse overlap
area ($S_{\perp}^{geom}$) of the two colliding nuclei for a given incident
energy and centrality was estimated on the basis of a Glauber Monte-Carlo (MC)
approach [25, 26, 27, 28] as explained in [22] where its values were compiled.
They are used in the estimates made in the present paper. These also the
experimental $\langle p_{T}\rangle$ values for positive pions, kaons and
protons and $\langle dN/dy\rangle$ compiled in [22], in Fig. 1a is represented
the ratio $\langle p_{T}\rangle/\sqrt{\langle dN/dy\rangle/S_{\perp}^{geom}}$
as a function of collision energy for different centralities while in Fig. 1b
as a function of centrality ($\langle N_{part}\rangle$) for different
collision energies.

Figure 1: $\langle p_{T}\rangle/\sqrt{(dN/dy)/S_{\perp}^{geom}}$ dependence
on: a) collision energy for different centralities and b) $\langle
N_{part}\rangle$ for different collision energies.

As can be seen in Fig. 1, the ratio between the average transverse momentum
and the square root of the hadron multiplicity per unit rapidity and unit
transverse overlap area of the colliding nuclei decreases towards central
collisions and higher energies, thus supporting the predictions based on
colour fields description of the small x degrees of freedom [7, 10, 29,
*Ianc2, 31, *KLN2, *KLN3, *KLN4, *KLN5, *KLN6], local parton-hadron duality
picture (LPHD) [37] and dimensionality argument [38, 39].

A similar behaviour is predicted by the string percolation scenario, inspired
by CGC [40, 41]. In a string percolation approach [41]:

$\frac{dN}{dy}=F(\eta)\bar{N}^{s}\mu_{0}$ (1) $\langle
p^{2}_{T}\rangle=\langle p^{2}_{T}\rangle_{1}/F(\eta)$ (2)

The colour reduction factor:

$F(\eta)\equiv\sqrt{\frac{1-e^{-\eta}}{\eta}}$ (3)

where $\eta$, the 2-dimensional transverse density of strings, is given by
$\eta\equiv(r_{0}/R)^{2}\bar{N}^{s}$, $r_{0}$ being the string radius and R
the radius of the transverse overlap region. $\bar{N}^{s}$ is the average
number of strings, $\mu_{0}$ and $\langle p^{2}_{T}\rangle_{1}$ are the
average particle density and respectively the average transverse momentum
squared of a single string. Therefore, $\sqrt{\langle
p^{2}_{T}\rangle}$/$\sqrt{\langle dN/dy\rangle/S_{\perp}}$$\sim
1/\sqrt{(1-e^{-\eta})}$ is decreasing with centrality for a given collision
energy and with collision energy for a given centrality. The effects of the
strong longitudinal colour fields on the identified charged hadrons and
hyperons $p_{T}$ distributions in pp collisions at $\sqrt{s}$=7 TeV have also
been investigated within the framework of the HIJING/B$\bar{B}$ v2.0 model
[42]. The experimental ratios of the $p_{T}$ distributions at different
charged particle multiplicities to the one corresponding to the MB pp
collision, each of them normalized to the corresponding charged particle
density, were well reproduced using an increased strength of the colour field,
characterized by the effective values of the string tension, from low to high
charged particle multiplicity. These results pointed out the necessity of
introducing a multiplicity (or energy density) dependence for the effective
value of the string tension. They also show that at the LHC energies the
global features of the interactions are mainly determined by the properties of
the initial chromoelectric flux tubes, the system size playing a minor role. A
rough estimate of the gluon number density and occupation number in the early
stage of relativistic heavy ion collisions, following [43], gives values by
2.3 (2.9) higher at $\sqrt{s_{NN}}$=2.76 TeV (5.02 TeV) Pb-Pb central
collisions relative to Au-Au central collisions at $\sqrt{s_{NN}}$=200 GeV.
For the largest charged particle multiplicity in pp collisions at 7 TeV the
values are a bit larger than those corresponding to Pb-Pb central collisions
at $\sqrt{s_{NN}}$=5.02 TeV.

The parton density evolution as a function of x and $Q^{2}$, addressed more
than 35 years ago [44] was experimentally confirmed at HERA [45]. The rise of
the structure function at low x is still visible at small values of $Q^{2}$
[46, 47] where the perturbative QCD does not work anymore. Therefore, rich and
short-lived partonic cascades become visible as the collision energy
increases, the string density and the percolation probability increasing too.
The approaches mentioned above, predict an increase of the average transverse
momentum at the same particle multiplicity per unit rapidity and unit
transverse overlap area with collision energy and system size independence of
the global features evidenced at LHC energies, the properties of the initial
color electromagnetic flux tubes playing the main role.

The $\langle dE_{T}/dy\rangle/\langle dN/dy\rangle$-$\langle
dN/dy\rangle/S_{\perp}$ correlation for the most central A-A collisions using
the experimental information collected over the years starting from AGS up to
the LHC energies, is presented in Fig. 2. In the case of RHIC and LHC,
experimental data from the STAR and ALICE collaborations were used mainly due
to the availability in terms of measured spectra for light-flavor hadrons. The
transverse energy per unit rapidity was considered in two ways. In case I
(upper plot of Fig. 2) published measured values were used when they were
available, while in case II (bottom plot of Fig. 2) an approximation similar
to that from [48] was applied for the $\langle dE_{T}/dy\rangle$ calculation.
For the AGS and SPS energies the $\langle dE_{T}/d\eta\rangle$ values and the
scaling factor for $\eta$ to y transition at midrapidity were taken from [49]
for all cases. For the BES energies the STAR data from [20] were used in case
I. It should be noted that these were obtained from the transverse momentum
spectra measured up to $\Lambda$ particles. For higher energies, STAR data
regarding $\langle dE_{T}/dy\rangle$ reported in [50] and [51] for Au-Au at
$\sqrt{s_{NN}}$ = 62.4 GeV and respectively [52] at $\sqrt{s_{NN}}$ = 200 GeV
as well as the results of the transverse energy measurement for Pb-Pb at
$\sqrt{s_{NN}}$ = 2.76 TeV from [53] were considered.

In case II, the full formula used to approximate $dE_{T}/dy$ was::

$\frac{dE_{T}}{dy}\simeq\frac{3}{2}(\langle
m_{T}\rangle\frac{dN}{dy})^{(\pi^{+}+\pi^{-})}+2(\langle
m_{T}\rangle\frac{dN}{dy})^{(p+\bar{p},\Xi^{-}+\bar{\Xi}^{+})}+(\langle
m_{T}\rangle\frac{dN}{dy})^{(K^{+}+K^{-},\Lambda+\bar{\Lambda},\Omega^{-}+\bar{\Omega}^{+})}+2(\langle
m_{T}\rangle\frac{dN}{dy})^{K^{0}_{S}}+2(\langle
m_{T}\rangle\frac{dN}{dy})^{(\Sigma^{+}+\Sigma^{-})}$ (4)

When the particle spectra were available, the average transverse mass was
calculated. For Pb-Pb at $\sqrt{s_{NN}}$ = 5.02 TeV in the case of strange and
multi-strange particles for which the transverse momentum distributions have
not yet been published, the approximation $\langle m_{T}\rangle=\sqrt{\langle
p_{T}^{2}\rangle+m_{0}^{2}}$ was used. In accordance with the convention made
in calorimetric measurements, $\langle m_{T}\rangle$ was replaced by $\langle
m_{T}\rangle-m_{0}$ for baryons and $\langle m_{T}\rangle+m_{0}$ for
antibaryons.

In both cases the particle multiplicity per unit rapidity, dN/dy was estimated
according to:

$\frac{dN}{dy}\simeq\frac{3}{2}\frac{dN}{dy}^{(\pi^{+}+\pi^{-})}+2\frac{dN}{dy}^{(p+\bar{p},\Xi^{-}+\bar{\Xi}^{+})}+\frac{dN}{dy}^{(K^{+}+K^{-},\Lambda+\bar{\Lambda},\Omega^{-}+\bar{\Omega}^{+})}+2\frac{dN}{dy}^{K^{0}_{S}}+2\frac{dN}{dy}^{(\Sigma^{+}+\Sigma^{-})}$
(5)

The experimental data in terms of transverse momentum spectra, yields and
average transverse momenta were taken from: [54] (AGS and SPS), [55, 56] (Au-
Au, BES), [48, 57] (Au-Au at $\sqrt{s_{NN}}$ = 62.4 GeV), [48, 58, 59] (Au-Au
at $\sqrt{s_{NN}}$ = 200 GeV), [60, 61, 62, *ALICE42] (Pb-Pb at
$\sqrt{s_{NN}}$ = 2.76 TeV) and [64, 65, 66, 67] (Pb-Pb at $\sqrt{s_{NN}}$ =
5.02 TeV).

Figure 2: The $\langle dE_{T}/dy\rangle/\langle dN/dy\rangle)$-$\langle
dN/dy\rangle/S_{\perp}$ correlation for the most central A-A collisions. Upper
plot: case I, bottom plot: case II. The continuous lines, used to guide the
eye, represent the results of a $4^{th}$ degree polynomial fits of the
experimental points.

Eqs. 4 and 5 were applied so that the same particles and anti-particles to be
taken into consideration in both formulas depending, from system to system and
case to case, on the most complete published experimental information. With
the increase of the collision energy, the contribution of strange and multi-
strange particles in the formulas becomes more and more important. Thus, for
Au-Au at $\sqrt{s_{NN}}$ = 62.4 and $\sqrt{s_{NN}}$ = 200 GeV and Pb-Pb at
both LHC energies, the contribution of $\Sigma$ particles was also considered
as explained in [68, 53].

In Fig. 2, $\langle dE_{T}/dy\rangle/\langle dN/dy\rangle$ as a function of
$\langle dN/dy\rangle/S_{\perp}$ for the most central collisions is shown. The
continuous lines in the figures, used to guide the eye, are the results of the
fits of experimental points using a $4^{th}$ degree polynomial. $\langle
dE_{T}/dy\rangle/\langle dN/dy\rangle$ increases with $\langle
dN/dy\rangle/S_{\perp}$ from AGS to SPS and RHIC energies, up to
$\sqrt{s_{NN}}$=39 GeV. Above $\sqrt{s_{NN}}$=39 GeV the slope changes and a
tendency towards saturation is evident. Even if the gap in the collision
energy between $\sqrt{s_{NN}}$=200 GeV, the highest one at RHIC, and the LHC
energies is rather large, it is obvious that the result of the extrapolation
of the trend of experimental data from low energies is below the real values
of the experimental data corresponding to $\sqrt{s_{NN}}$=2.76 TeV and 5.02
TeV LHC energies. Less influenced by processes such as collective expansion,
suppression or kinematic cuts, the behavior of such a representation supports
the theoretical predictions that identify it as a signature of the phase
transition [15, 16]. The obvious rise at LHC energies could be the result of
the combined contributions of a higher temperature of the deconfined medium
and a higher percolation probability in the regions of high density colour
electromagnetic fields.

##  III $\langle dE_{T}/dy\rangle/\langle dN/dy\rangle$ \- $\langle
dN/dy\rangle/S_{\perp}$ correlation for different centralities at RHIC and LHC
energies

The $\langle dE_{T}/dy\rangle/\langle dN/dy\rangle$-$\langle
dN/dy\rangle/S_{\perp}$ correlation depending on the centrality for several
collision energies is shown in Fig. 3 where the difference between the upper
and bottom plots is the same as in Fig. 2, explained in the previous chapter.
A close to linear dependence of $\langle dE_{T}/dy\rangle/\langle
dN/dy\rangle$ as a function of $\langle dN/dy\rangle/S_{\perp}$ can be seen.
The offsets are different, increasing with the collision energy.

Figure 3: The $\langle dE_{T}/dy\rangle/\langle dN/dy\rangle$ \- $\langle
dN/dy\rangle/S_{\perp}$ correlation for different collision energies and
different centralities. Upper plot: case I, bottom plot: case II.

The $\langle dE_{T}/dy\rangle/\langle dN/dy\rangle$-$\langle
dN/dy\rangle/S_{\perp}$ correlations for Au-Au at $\sqrt{s_{NN}}$=62.4 GeV
[50, 51] and 200 GeV [52] based on calorimetry measurements (case I) do not
follow the general trend what is not observed in case II (Fig. 3, bottom). In
Fig. 4 are shown the $\langle dE_{T}/dy\rangle/\langle dN/dy\rangle$-$\langle
dN/dy\rangle/S_{\perp}$ correlations for the ovelap transverse areas,
$S_{\perp}\approx$ 24, 64, 108 and 149 $fm^{2}$. The lines represent the
result of fitting the corresponding points with a 4th degree polynomial. A
clear dependence on the size of the transverse overlap area is observed. The
rise in $\langle dE_{T}/dy\rangle/\langle dN/dy\rangle$ at RHIC energies
becomes steeper and the range in $\langle dN/dy\rangle/S_{\perp}$,
corresponding to a close to a plateau trend decreases from central to
peripheral collisions [22] and is expected to converge towards the values
corresponding to minimum bias pp collisions in the same energy range, once the
contribution from nucleons suffering single collisions becomes predominant. At
each inflection point of the fit lines, the corresponding $\langle
dE_{T}/dy\rangle/\langle dN/dy\rangle$ value, related to the temperature of
the mixed phase up to the pressure contribution, was estimated. The
representation of these values as a function of $S_{\perp}$ shows a decrease
from the central collisions to the peripheral ones, as can be seen in Fig. 5.

Such a volume dependence was predicted for the Chiral phase transition, the
critical temperature decreasing towards smaller volume [69, 70]. It is worth
mentioning that a volume dependence has also been observed in the magnetic
phase transition [71]. The pressure term in the equation of state could also
contribute to the observed trend. Therefore, theoretical models taking into
account all possible contributions such to reproduce the experimental trend
are required.

Figure 4: The $\langle dE_{T}/dy\rangle/\langle dN/dy\rangle$ \- $\langle
dN/dy\rangle/S_{\perp}$ correlation for four values of $S_{\perp}^{geom}$ and
different collision energies. Upper plot: case I, bottom plot: case II. The
lines represent the results of a $4^{th}$ degree polynomial fits of the
experimental points.

Figure 5: $\langle dE_{T}/dy\rangle/\langle dN/dy\rangle$ calculated at the
inflection point in the $\langle dE_{T}/dy\rangle/\langle
dN/dy\rangle$-$\langle dN/dy\rangle/S_{\perp}$ correlation for four values of
$S_{\perp}^{geom}$ and different collision energies. The upper plot
corresponds to Fig. 4 \- up and the bottom plot to Fig. 4 \- bottom.

##  IV $(\langle dE_{T}/dy\rangle/\langle dN/dy\rangle)^{core}$-($\langle
dN/dy\rangle/S_{\perp})^{core}$ correlation

In a simple core-corona picture different observables in heavy ion collisions
as a function of centrality and collision energy can be seen as the result of
the combination between the contribution coming from the interactions of
nucleons undergoing only single collisions (corona) and that coming from the
multiple interactions of the rest of the nucleons (core).

Figure 6: The transverse overlap area of the colliding nuclei at different
energies estimated with the Glauber MC approach corresponding to all nucleons
($S_{\perp}^{geom}$) - open symbols and to nucleons suffering more than a
single collision ($(S_{\perp}^{geom})^{core}$) - full symbols as a function of
the number of participants, $\langle N_{part}\rangle$. Figure 7: $\langle
dN/dy\rangle$ estimated by considering only $\pi^{\pm}$, $K^{\pm}$, p and
$\bar{p}$ and the corresponding neutral hadrons for all A-A collisions at all
measured energies; open symbols - total, full symbols - core contribution.

Figure 8: Total $\langle dE_{T}/dy\rangle/\langle dN/dy\rangle)$-$\langle
dN/dy\rangle/S_{\perp}$ (upper plot) and core $(\langle
dE_{T}/dy\rangle/\langle dN/dy\rangle)^{core}$-$(\langle
dN/dy\rangle/S_{\perp})^{core}$ (bottom plot) correlations for four values of
the total and core transverse overlap area and different collision energies.
The lines are the results of the fit with a 4th order polynomial. Figure 9:
$\langle dE_{T}/dy\rangle/\langle dN/dy\rangle$ calculated at the inflection
point in the $\langle dE_{T}/dy\rangle/\langle dN/dy\rangle$-$\langle
dN/dy\rangle/S_{\perp}$ total and core correlations (total - blue symbols,
core - red symbols) for four values of the transverse overlap area and
different collision energies, as a function of the total transverse overlap
area.

The dependence on the collision energy and centrality of the core-corona
relative weight and its influence on different observables has been studied
based on different approaches [72, 73, 74, 75, *Aich2, *Aich3, 78, 79, 80,
81]. The transverse overlap area corresponding to interacting nucleons
suffering more than a single collision ($(S_{\perp}^{geom})^{core}$) were
estimated within the Glauber Monte-Carlo approach and compiled in [22]. Their
$\langle N_{part}\rangle$ dependence is presented in Fig. 6. As far as the
$p_{T}$ spectra are not yet published for all hadrons measured in MB pp
collisions, the studies, whose results are presented in this chapter, were
done by using yields and average transverse mass values only for $\pi^{\pm}$,
$K^{\pm}$, p and $\bar{p}$ and the corresponding neutral hadrons in Eqs. 4 and
5 for both cases, i.e. total and core. The core corresponding yield and
$\langle m_{T}\rangle$ for a given species have been estimated by subtracting
from the $p_{T}$ spectra for A-A at a given centrality the corresponding
$p_{T}$ spectra measured in the MB pp collision at the same energy, weighted
with the number of nucleons suffering only single collisions [81]. $\langle
dN/dy\rangle$ total and corresponding to the core contribution values are
shown in Fig. 7 as a function of total $\langle N_{part}\rangle$. As expected,
the corona contribution decreases from peripheral towards central collisions.
The total $\langle dE_{T}/dy\rangle/\langle dN/dy\rangle$ \- $\langle
dN/dy\rangle/S_{\perp}$ and core $(\langle dE_{T}/dy\rangle/\langle
dN/dy\rangle)^{core}$ \- ($\langle dN/dy\rangle/S_{\perp})^{core}$
correlations for four corresponding overlap areas are presented in Fig. 8
upper and bottom plots, respectively. Although they look similar, the
dependence on the fireball transverse area is more clearly observed especially
at lower values of the transverse overlap area, the tendency towards
saturation being enhanced for the core contribution.

The quantitative difference can be followed in Fig. 9 where are represented
the $\langle dE_{T}/dy\rangle/\langle dN/dy\rangle$ values corresponding to
the inflection points of the fit lines for the four values of $S_{\perp}$,
total and core. The decrease from central to peripheral collisions is similar
with the trend presented in the previous chapter where all hadrons were
considered. As expected, the main difference between total and core is at
peripheral collisions. Nevertheless, the decrease of $\langle
dE_{T}/dy\rangle/\langle dN/dy\rangle$ values corresponding to the inflection
points from central (larger $S_{\perp}$) to peripheral (lower $S_{\perp}$) is
also present when only the core contribution is considered.

##  V  Bjorken energy density as a function of $\langle
dN/dy\rangle/S_{\perp}$

The Bjorken energy density times the interaction time
($\epsilon_{Bj}\cdot\tau$), i.e. $\langle dE_{T}/dy\rangle/S_{\perp}$, as a
function of $\langle dN/dy\rangle/S_{\perp}$ for the most central A-A
collisions considered in the previous sections is presented in Fig. 10. The
upper and bottom plots were obtained in case I and II as discussed in Chapters
II and III.

Figure 10: The Bjoken energy density times the interaction time as a function
of $\langle dN/dy\rangle/S_{\perp}$ for the most central A-A collisions. Upper
plot: case I, bottom plot: case II. The green line is the result of a fit with
a 4th order polynomial, used to guide the eye.

Figure 11: The Bjoken energy density times the interaction time as a function
of $\langle dN/dy\rangle/S_{\perp}$ for different A-A collision energies and
centralities. The difference between the upper and bottom plots is the same as
it is explained for Fig. 10.

The $\epsilon_{Bj}\cdot\tau$ from AGS, SPS and RHIC energies has a linear
dependence on the particle multiplicity per unit rapidity and unit transverse
overlap area $\langle dN/dy\rangle/S_{\perp}$. The extrapolation to the
$\langle dN/dy\rangle/S_{\perp}$ values corresponding to the LHC energies
falls below the experimental results. In Fig. 11 are represented the
$\epsilon_{Bj}\cdot\tau$ values for different centralities and collision
energies. The upper and bottom plots were obtained for case I and II as
explained in Sections II and III.

For each collision energy $\epsilon_{Bj}\cdot\tau$ shows a linear dependence
on $\langle dN/dy\rangle/S_{\perp}$. The values of the slope of this
dependence, related to the temperature of the system, resulting from a linear
fit of this correlation for each system and energy, as a function of collision
energy are presented in Fig. 12.

Figure 12: The slope of the $\epsilon_{Bj}\cdot\tau$ linear dependence on
$\langle dN/dy\rangle/S_{\perp}$ as a function of the collision energy. The
line is the result of the fit with a 4th order polynomial in
ln($\sqrt{s_{NN}}$) and is used to guide the eye. Upper plot: case I, bottom
plot: case II.

The slope values increase from AGS to the highest RHIC energy. The values for
LHC energies are larger and not in-line with what should be expected from a
simple extrapolation of the trend observed at lower collision energies, thus
suggesting a saturation or an inflection point within the energy gap between
RHIC and LHC energies.

Following the considerations presented in Section II, the physics behind the
increasing trend at LHC energies could be related to the large increase of the
gluon density, therefore increasing the percolation probability, the ropes
decay being at the origin of increased average transverse momentum.

##  VI Similar studies for $pp$ MB for different collision energies and as a
function of $\langle dN/dy\rangle/S_{\perp}$ at LHC energies

The experimental data measured in pp MB collisions [48, 82, 83, 84, 85] at
$\sqrt{s}$ = 62.4 GeV, 200 GeV, 2.76 TeV and 5.02 TeV were used for obtaining
the $\langle dE_{T}/dy\rangle/\langle dN/dy\rangle)$-$\langle
dN/dy\rangle/S_{\perp}$ correlation considering only $\pi^{\pm}$, $K^{\pm}$, p
and $\bar{p}$ and the corresponding neutral hadrons in Eqs. 4 and 5. For the
BES energies the $p_{T}$ spectra of the same particles, obtained with an
interpolation procedure (INTP) from existing data in this range of energies as
explained in [81], were used. The transverse overlap area was estimated using
the Glauber MC approach at subnucleonic level [86]. In Fig. 13 this
correlation is presented on top of the results presented in Fig. 3, bottom
plot. It is seen that the correlation in A-A collisions at different collision
energies converges for very peripheral collisions, towards the one
corresponding to the pp MB collisions. This is expected as far as towards
peripheral A-A collisions the majority of interacting nucleons suffer only
single nucleon-nucleon interactions.

At LHC energies, studies of pp collisions up to very high charged particle
multiplicities, have shown similarities between pp and Pb-Pb in terms of the
behaviour of different observables, like near-side long range pseudorapidity
correlations [87], the ($\langle\beta_{T}\rangle$ \- $T_{kin}^{fo}$)
correlation as a function of charged particle multiplicity [88], azimuthal
angular correlations [89], geometrical scaling [22, 23, 24], etc. Experimental
data on transverse momentum spectra for light flavors as a function of the
average charged particle multiplicity at LHC energies were published in [90,
91, 92]. The transverse overlap area for pp collisions, $S_{\perp}^{pp}$=$\pi$
$r_{max}^{2}$ as a function of charged particle multiplicity was estimated
using the result of [93], computed in the IP-Glasma model, $r_{max}$ being the
maximal radius for which the energy density of the Yang-Mills fields is above
$\varepsilon=\alpha\Lambda_{QCD}^{4}$ ($\alpha\in{[1,10]}$). The model results
were fitted with:

$f_{pp}=\left\\{\begin{array}[]{rl}0.387+0.0335x+0.274x^{2}-0.0542x^{3}&\mbox{if
$x<3.4$}\\\ 1.538&\mbox{if $x\geq$ 3.4}\end{array}\right.$ (6)

Using the same recipe we fitted the $r_{max}$ values from Ref. [93] for
$\alpha$=10 with the following expression:

$f_{pp}=\left\\{\begin{array}[]{rl}-0.18+0.3976x+0.095x^{2}-0.028x^{3}&\mbox{if
$x<3.4$}\\\ 1.17&\mbox{if $x\geq$ 3.4}\end{array}\right.$ (7)

where x=$(dN/dy)^{1/3}$ and $r_{max}$=1fm$\cdot f_{pp}(x)$.

Figure 13: The $\langle dE_{T}/dy\rangle/\langle dN/dy\rangle)$-$\langle
dN/dy\rangle/S_{\perp}$ correlation for A-A collisions, case II, pp MB at the
same energies (the green line, the result of a 4th order polynomial, guides
the eye) and pp as a function of charged particle multiplicity at $\sqrt{s}$ =
7 and 13 TeV.  Figure 14: The same type of figure as Fig. 13, but for the
Bjorken energy density times the interaction time as a function of $\langle
dN/dy\rangle/S_{\perp}$.

The transverse energy per unit rapidity has been estimated using the same
procedure as the one used for A-A collisions at LHC explained in Chapter II.
The result in terms of $\langle dE_{T}/dy\rangle/\langle dN/dy\rangle$ \-
$\langle dN/dy\rangle/S_{\perp}$ correlations for pp collisions at
$\sqrt{s}$=7 TeV and 13 TeV is presented in Fig. 13 for $\alpha$=10 and
$\alpha$=1. The results corresponding to pp collision at $\sqrt{s}$=13 TeV
were obtained within the assumption that the above parameterization of the
transverse overlap area does not change significantly. It is seen that the
trend of the $\langle dE_{T}/dy\rangle/\langle dN/dy\rangle$ \- $\langle
dN/dy\rangle/S_{\perp}$ correlation for pp collisions at the LHC energies
follows the one of Pb-Pb at $\sqrt{s_{NN}}$=5.02 TeV, the quantitative
agreement being good for $\alpha$=10 within the same $\langle
dN/dy\rangle/S_{\perp}$ range. For the same values of $\langle
dN/dy\rangle/S_{\perp}$ slightly larger values of $\langle
dE_{T}/dy\rangle/\langle dN/dy\rangle$ in pp at $\sqrt{s}$=7 TeV relative to
Pb-Pb at $\sqrt{s_{NN}}$=5.02 TeV and in pp at $\sqrt{s}$=13 TeV relative to
$\sqrt{s}$=7 TeV are seen. As mentioned in the previous chapters, this is in
line with the expectations based on the string percolation approach. The
Bjorken energy density times the interaction time values for pp collisions at
the two LHC energies as a function of $\langle dN/dy\rangle/S_{\perp}$
presented in Fig. 14, using the same symbols as in Fig. 13, are in very good
agreement with the ones corresponding to Pb-Pb collision at
$\sqrt{s_{NN}}$=5.02 TeV. For pp collisions the slopes of
$\epsilon_{Bj}\cdot\tau$ as a function of $\langle dN/dy\rangle/S_{\perp}$ are
0.89$\pm$0.09 at $\sqrt{s}$ = 7 TeV and 0.92$\pm$0.07 at $\sqrt{s}$ = 13
TeV,respectively, for $\alpha$ =10, a bit larger values than that
corresponding to the Pb-Pb collision at $\sqrt{s_{NN}}$ = 5.02 TeV.

##  VII Conclusions

The behaviour of $\langle p_{T}\rangle/\sqrt{\langle dN/dy\rangle/S_{\perp}}$
as a function of the collision energy for a given centrality or as a function
of centrality for a given collision energy supports the predictions of CGC and
percolation based approaches. The dependence of the ratio of the energy
density to the entropy density as a function of entropy density at different
collision centralities for A-A collisions from AGS, SPS, RHIC and LHC energies
reveals a tendency towards saturation starting with the largest RHIC energies
and a steep rise at the LHC energies. A clear dependence of this behaviour on
the size of the transverse overlap area is highlighted. The increase in
$\langle dE_{T}/dy\rangle/\langle dN/dy\rangle$ at RHIC energies becomes
steeper and the $\langle dN/dy\rangle/S_{\perp}$ range, which corresponds to a
trend close to a plateau, decreases from central to peripheral collisions,
converging towards the behavior corresponding to MB pp collisions.

In a core-corona scenario, the corona contribution to the observed trends is
removed and similar correlations are presented for the core in the case of the
identified charged hadrons and their corresponding neutral particles.

The values of the energy density to the entropy density for the same entropy
density increase with the collision energy, supporting the expectation based
on the string percolation approach.

The Bjorken energy density times the interaction time has a linear dependence
on the particle multiplicity per unit rapidity and unit transverse overlap
area. The linear dependence also holds for the LHC energies, its slope
increasing significantly at these energies which is an expected trend in the
string percolation approach. The slopes as a function of the collision energy
show the same trend as the correlations discussed in the previous chapters.

The $\langle dE_{T}/dy\rangle/\langle dN/dy\rangle$-$\langle
dN/dy\rangle/S_{\perp}$ and $\epsilon_{Bj}\cdot\tau$-$\langle
dN/dy\rangle/S_{\perp}$ correlations for pp collisions at $\sqrt{s}$=7 TeV and
13 TeV follow qualitatively the ones corresponding to Pb-Pb collisions at
$\sqrt{s_{NN}}$=5.02 TeV. Within the error bars, there is also a quantitative
agreement if for $S_{\perp}$ for the pp collision as a function of particle
density only the region of overlap zone characterized by an energy density
larger than $\approx$ 2 GeV/$fm^{3}$ is considered. For pp collisions the
slopes corresponding to the $\epsilon_{Bj}\cdot\tau$-$\langle
dN/dy\rangle/S_{\perp}$ correlation have a bit larger values than that
corresponding to the Pb-Pb collision at $\sqrt{s_{NN}}$ = 5.02 TeV.

## ACKNOWLEDGMENTS

This work was carried out under the contracts sponsored by the Ministry of
Research, Innovation and Digitization: RONIPALICE-07/03.01.2022 (via IFA
Coordinating Agency) and PN-19 06 01 03.

## References

  * Chapline and Kerman [1978] G. F. Chapline and A. K. Kerman, MIT report CTP-695 (1978).
  * Collins and Perry [1975] J. C. Collins and M. J. Perry, Phys. Rev. Lett. 34, 1353 (1975).
  * Cabibbo and Parisi [1975] N. Cabibbo and G. Parisi, Phys. Lett. B 59, 67 (1975).
  * Shuryak [1978] E. V. Shuryak, Phys. Lett. B 78, 150 (1978).
  * Baym [1979] G. Baym, Physica A 96, 131 (1979).
  * Celik  _et al._ [1980] T. Celik _et al._ , Phys. Lett. B 97, 128 (1980).
  * McLerran and Venugopalan [1994a] L. McLerran and R. Venugopalan, Phys. Rev. D 49, 2233 (1994a).
  * McLerran and Venugopalan [1994b] L. McLerran and R. Venugopalan, Phys. Rev. D 49, 3352 (1994b).
  * McLerran and Venugopalan [1994c] L. McLerran and R. Venugopalan, Phys. Rev. D 50, 2225 (1994c).
  * Iancu  _et al._ [2001] E. Iancu, A. Leonidov, and L. McLerran, Nucl. Phys. A 692, 583 (2001).
  * Mazeliauskas [2019] A. Mazeliauskas, Nucl. Phys. A 982, 134 (2019), and references therein.
  * Van Hove [1982] L. Van Hove, Phys. Lett. B 118, 138 (1982).
  * Mohanty  _et al._ [2003] B. Mohanty _et al._ , Phys. Rev. C 68, 021901R (2003).
  * Adamczyk  _et al._ [2017a] L. Adamczyk _et al._ (STAR Collaboration), Phys. Rev. C 96, 044904 (2017a).
  * Blaizot and Ollitrault [1986] J. P. Blaizot and J. Y. Ollitrault, Nucl. Phys. A 458, 745 (1986).
  * Blaizot and Ollitrault [1987] J. P. Blaizot and J. Y. Ollitrault, Phys. Lett. B 191, 21 (1987).
  * Adler  _et al._ [2005a] S. S. Adler _et al._ (PHENIX Collaboration), Phys. Rev. C 71, 034908 (2005a).
  * Adam  _et al._ [2016a] J. Adam _et al._ (ALICE Collaboration), Phys. Rev. C 94, 034903 (2016a).
  * Adare  _et al._ [2016] A. Adare _et al._ (PHENIX Collaboration), Phys. Rev. C 93, 024901 (2016).
  * Sharma [2018] B. Sharma,  _Transverse energy analysis of Au+Au collisions at 7.7, 11.5, 19.6, 27, and 39 GeV through the use of identified particles spectra_ , Master’s thesis, University of Tennessee (2018), https://trace.tennessee.edu/utk_gradthes/5118.
  * Monnai and Ollitrault [2017] A. Monnai and J. Y. Ollitrault, Phys. Rev. C 96, 044902 (2017).
  * Petrovici  _et al._ [2018] M. Petrovici, A. Lindner, and A. Pop, Phys. Rev. C 98, 024904 (2018).
  * Petrovici  _et al._ [2019] M. Petrovici, A. Lindner, and A. Pop, AIP Conference Proceedings, 2076, 040001 (2019), EXOTIC NUCLEI AND NUCLEAR/PARTICLE ASTROPHYSICS (VII). PHYSICS WITH SMALL ACCELERATORS: Proceedings of the Carpathian Summer School of Physics 2018 (CSSP18).
  * Lindner  _et al._ [2021] A. Lindner, M. Petrovici, and A. Pop, Proceedings of Science (PoS), 380, 197 (2021), Particles and Nuclei International Conference (PANIC2021), https://pos.sissa.it/380/197/.
  * Glauber [1955] R. J. Glauber, Phys. Rev. 100, 242 (1955).
  * Franco and Glauber [1966] V. Franco and R. J. Glauber, Phys. Rev. 142, 119 (1966).
  * Miller  _et al._ [2007] M. L. Miller _et al._ , Ann. Rev. Nucl. Part. Sci. 57, 205 (2007).
  * Rybczynski  _et al._ [2014] M. Rybczynski, G. Stefanek, W. Broniowski, and P. Bozek, Comput. Phys. Commun. 185, 759 (2014).
  * Iancu  _et al._ [2002] E. Iancu, A. Leonidov, and L. McLarren, e-print arXiv:hep-ph/0202270 (2002).
  * Ferreiro  _et al._ [2002] E. Ferreiro, E. Iancu, A. Leonidov, and L. McLarren, Nucl. Phys. A 703, 489 (2002).
  * Kharzeev  _et al._ [2004a] D. Kharzeev, E. Levin, and M. Nardi, Nucl. Phys. A 730, 448 (2004a).
  * Kharzeev  _et al._ [2004b] D. Kharzeev, E. Levin, and M. Nardi, Nucl. Phys. A 743, 329 (2004b).
  * Kharzeev and Levin [2005] D. Kharzeev and E. Levin, Phys. Rev. C 71, 054903 (2005).
  * Kharzeev and Nardi [2001] D. Kharzeev and M. Nardi, Phys. Lett. B 507, 121 (2001).
  * Kharzeev  _et al._ [2005] D. Kharzeev, E. Levin, and L. McLarren, Nucl. Phys. A 748, 627 (2005).
  * Kharzeev  _et al._ [2003] D. Kharzeev, E. Levin, and L. McLarren, Phys. Lett. B 561, 93 (2003).
  * Dokshitzer  _et al._ [1991] Y. L. Dokshitzer, V. A. Khoze, and S. Troian, J. Phys. G 17, 1585 (1991).
  * Levin and Rezaeian [2011] E. Levin and A. H. Rezaeian, Phys. Rev. D 83, 114001 (2011).
  * Lappi [2011] T. Lappi, Eur. Phys. J. C 71, 1699 (2011).
  * Biro  _et al._ [1984] T. S. Biro, H. B. Nielsen, and J. Knoll, Nucl. Phys. B 245, 449 (1984).
  * J. Dias de Deus and C. Pajares [2011] J. Dias de Deus and C. Pajares, Phys. Lett. B 695, 211 (2011).
  * V. Topor Pop and Petrovici [2018] V. Topor Pop and M. Petrovici, Phys. Rev. C 98, 064903 (2018).
  * Mueller [2003] A. H. Mueller, Nucl. Phys. A 715, 20 (2003).
  * Gribov  _et al._ [1983] L. V. Gribov, E. M. Levin, and M. G. Ruskin, Phys. Rev. 100, 1 (1983), and references therein.
  * H1 Collaboration [1996] H1 Collaboration, Nucl. Phys. B 470, 3 (1996).
  * ZEUS Collaboration [1999] ZEUS Collaboration, Eur. Phys. J. C 7, 609 (1999), and references therein.
  * H1 Collaboration [2009] H1 Collaboration, Eur. Phys. J. C 64, 561 (2009).
  * Abelev  _et al._ [2009] B. I. Abelev _et al._ (STAR Collaboration), Phys. Rev. C 79, 034909 (2009).
  * Adler  _et al._ [2005b] S. S. Adler _et al._ (STAR Collaboration), Phys. Rev. C 71, 034908 (2005b).
  * R. Sahoo for the STAR Collaboration [2011] R. Sahoo for the STAR Collaboration, Indian J. Phys. 85, 897 (2011).
  * Sahoo  _et al._ [2006] R. Sahoo _et al._ , Rom. Rep. Phys. 58, 055 (2006).
  * Adams  _et al._ [2004] J. Adams _et al._ (STAR Collaboration), Phys. Rev. C 70, 054907 (2004).
  * Adam  _et al._ [2016b] J. Adam _et al._ (ALICE Collaboration), Phys. Rev. C 94, 034903 (2016b).
  * Chatterjee  _et al._ [2015] S. Chatterjee _et al._ , Advances in High Energy Physics 2015, 349013 (2015).
  * Adamczyk  _et al._ [2017b] L. Adamczyk _et al._ (STAR Collaboration), Phys. Rev. C 96, 044904 (2017b).
  * Adam  _et al._ [2020] J. Adam _et al._ (STAR Collaboration), Phys. Rev. C 102, 034909 (2020).
  * Aggarwal  _et al._ [2011] M. M. Aggarwal _et al._ (STAR Collaboration), Phys. Rev. C 83, 024901 (2011).
  * Adams  _et al._ [2007] J. Adams _et al._ (STAR Collaboration), Phys. Rev. Lett. 98, 062301 (2007).
  * Agakishiev  _et al._ [2012] G. Agakishiev _et al._ (STAR Collaboration), Phys. Rev. Lett. 108, 072301 (2012).
  * Abelev  _et al._ [2013a] B. Abelev _et al._ (ALICE Collaboration), Phys. Rev. C 98, 044910 (2013a).
  * Abelev  _et al._ [2013b] B. Abelev _et al._ (ALICE Collaboration), Phys. Rev. Lett. 111, 222301 (2013b).
  * Abelev  _et al._ [2014a] B. Abelev _et al._ (ALICE Collaboration), Phys. Lett. B 728, 216 (2014a).
  * Abelev  _et al._ [2014b] B. Abelev _et al._ (ALICE Collaboration), Phys. Lett. B 734, 409 (2014b).
  * Acharya  _et al._ [2020a] S. Acharya _et al._ (ALICE Collaboration), Phys. Rev. C 101, 044907 (2020a).
  * D. S. de Albuquerque [2019] D. S. de Albuquerque, Ph.D. thesis (2019), CERN-THESIS-2019-135.
  * P. Kalinak for the ALICE Collaboration [2017] P. Kalinak for the ALICE Collaboration, European Physical Society Conference on High Energy Physics, 5-12 July 2017, Venice, Italy, PoS(EPS-HEP2017)168 (2017), https://pos.sissa.it/314/168/pdf.
  * D. S. de Albuquerque for the ALICE Collaboration [2019] D. S. de Albuquerque for the ALICE Collaboration, Nucl. Phys. A 982, 823 (2019), XXVIIth International Conference on Ultrarelativistic Nucleus-Nucleus Collisions (Quark Matter 2018).
  * Subrata Pal and Scott Pratt [2004] Subrata Pal and Scott Pratt, Phys. Lett. B 518, 310 (2004).
  * Braun  _et al._ [2006] J. Braun _et al._ , Phys. Rev. D 73, 074010 (2006).
  * Bo-Liu Li  _et al._ [2019] Bo-Liu Li _et al._ , Nucl. Phys. B 938, 298 (2019).
  * Arnold  _et al._ [1994] Z. Arnold _et al._ , Solid State Communication 92, 807 (1994).
  * Bozek [2005] P. Bozek, Acta Phys. Pol. B 36, 3071 (2005).
  * Werner [2007] K. Werner, Phys. Rev. Lett. 98, 152301 (2007).
  * Becattini and Manninen [2008] F. Becattini and J. Manninen, J. Phys. G 35, 104013 (2008).
  * Aichelin and Werner [2009] J. Aichelin and K. Werner, Phys. Rev. C 79, 064907 (2009).
  * Aichelin and Werner [2010a] J. Aichelin and K. Werner, Phys. Rev. C 81, 029902(E) (2010a).
  * Aichelin and Werner [2010b] J. Aichelin and K. Werner, J. Phys. G 37, 094006 (2010b).
  * Bozek [2009] P. Bozek, Phys. Rev. C 79, 054901 (2009).
  * Gemard and Aichelin [2014] M. Gemard and J. Aichelin, Astronom. Nachr. 335, 660 (2014).
  * Petrovici  _et al._ [2017] M. Petrovici _et al._ , Phys. Rev. C 96, 014908 (2017).
  * [81] A. Pop and M. Petrovici, to be published.
  * Adare  _et al._ [2011] A. Adare _et al._ (PHENIX Collaboration), Phys. Rev. C 83, 064903 (2011).
  * Adams  _et al._ [2005] J. Adams _et al._ (STAR Collaboration), Phys. Lett. B 616, 8 (2005).
  * Adams  _et al._ [2006] J. Adams _et al._ (STAR Collaboration), Phys. Lett. B 637, 161 (2006).
  * Alper  _et al._ [1975] B. Alper _et al._ (British-Scandinavian Collaboration), Nucl. Phys. B 100, 237 (1975).
  * Bozek  _et al._ [2019] P. Bozek, W. Broniowski, M. Rybczynski, and G. Stefanek, Comput. Phys. Commun. 245, 106850 (2019).
  * CMS Collaboration [2010] CMS Collaboration, J. High Energ. Phys. 9, 91 (2010).
  * C. Andrei, ALICE Collaboration [2014] C. Andrei, ALICE Collaboration, Nucl. Phys. A 931, c888 (2014).
  * CMS Collaboration [2017] CMS Collaboration, Phys. Lett. B 765, 193 (2017).
  * Acharya  _et al._ [2019] S. Acharya _et al._ (ALICE Collaboration), Phys. Rev. C 99, 024906 (2019).
  * Acharya  _et al._ [2020b] S. Acharya _et al._ (ALICE Collaboration), Eur. Phys. J. C 80, 693 (2020b).
  * Acharya  _et al._ [2020c] S. Acharya _et al._ (ALICE Collaboration), Eur. Phys. J. C 80, 167 (2020c).
  * Bzdak  _et al._ [2013] A. Bzdak _et al._ , Phys. Rev. C 87, 064906 (2013).


# Dynamical elastic contact of a rope with the ground

Gregory Kozyreff<EMAIL_ADDRESS>Benoît Seron  Optique Nonlinéaire
Théorique, Université libre de Bruxelles (U.L.B.), CP 231, Campus de la
Plaine, 1050 Bruxelles, Belgium.

###### Abstract

A rope laid on the ground with one end subjected to time-dependent forcing is
proposed as a prototypical elastic dynamical contact problem, which we study
analytically, numerically, and experimentally. The dynamics is governed by an
infinite set of linear and nonlinear resonances. In the limit of weak bending
stiffness, the fundamental frequency is found to be independent of the rope
tension. A transition between a radiation-less and a wave radiating state
occurs via a series of grazing bifurcations, whereby new contacts between the
rope and the ground are formed. The grazing bifurcations form overlapping
Arnold tongues in the frequency-amplitude parameter space. Finally, for ropes
with large bending stiffness and when the geometric nonlinearity is important,
bistability is observed between several wave-making regimes.

Take a long piece of rope, lay it on the ground and give it a sudden vertical
shake at one extremity. If that was vigorous enough, you are likely to see a
bell-shape elevation travelling far away along the rope, until internal
friction makes it disappear. Such a wave is familiar to anyone and simple-
looking, but the underlying physics is more complicated than meets the eyes.
In this Letter, we shed some light on the dynamics underlying wave generation
by a combination of analytical, numerical, and experimental results. Aside
from its recreational aspect, this problem is an opportunity to gather basic
qualitative and quantitative knowledge into the theory of dynamical contacts
involving deformable bodies. Indeed, questions of this kind can usually be
approached only numerically Wriggers (2006); Doyen  _et al._ (2011). Even in
the static case, analytically tractable elastic contact problems are rare and
most often proceed from Hertz’s famous study of two spheres pressed against
one another Hertz (1896). On the other hand, the present problem has a
practical interest, as it may be connected to some aspects of cable laying on
ocean beds Zajac (1957); Jawed  _et al._ (2014), rails deformation under a
moving load Weitsman (1971), the intrusion of a rod into a cylinder in the
context of oil well drilling Miller  _et al._ (2015), or parasitic contacts in
rotating machines Mokhtar  _et al._ (2017). Importantly, and contrary to other
systems combining vibration and impact Astashev and Krupenin (2001); Murphy
and Morrison (2002), as in the atomic force microscope in the taping regime
Dick  _et al._ (2008), the contact point is not known in advance and becomes
non-unique on the occasion of wave emission.

In order to study wave generation in a systematic way, we set up an
experiment, schematically depicted in Fig. 1, where a long rope is excited
harmonically with a small amplitude. Intuitively, one would expect that below
a certain threshold frequency, the lifted part of the rope does no more than
gently following the motion imparted at the extremity and that, above, waves
are radiated along the rope. We determine such a frequency and find,
unexpectedly, that it is independent of the rope tension when bending
stiffness is negligible. Further analysis reveals that wave generation starts
at grazing bifurcations and that these form Arnold tongues in the parameter
space spanned by the forcing amplitude and frequency. Further, we show that
(i) the wave-less state can regain stability by increasing the driving
frequency above the fundamental frequency and (ii) waves can also be generated
below this frequency through nonlinear amplification. Finally, we
experimentally observe hysteresis between a regime of large wave emission and
another one where only small waves are radiated; here too, the boundary of
bistability indicates an Arnold tongue pattern.

Figure 1: Schematic of the rope with one end vibrated with relative amplitude
$\epsilon$ and pulsation $\omega$ around a fixed elevation $Z_{0}$. Prior to
vibrating the rope, a tension $T$ is applied and the resulting static
deformation is maintained by a weight placed at the other end of the rope. The
rope is sufficiently long to neglect reflection from the weight. For
$\epsilon$ sufficiently small, only one coordinate, $x_{c}(t)$, describes the
edge of the contact region between the rope and the ground (contact point).

In the experiment, one end of the rope, at $x=0$, is initially lifted to a
height $Z_{0}$ and then made to oscillate vertically and harmonically around
that position. The elevation $w(x,t)$ of the rope that is not in contact with
the ground obeys the beam equation Howell  _et al._ (2009)

$\rho\frac{\partial^{2}w}{\partial t^{2}}=T\frac{\partial^{2}w}{\partial
x^{2}}-B\frac{\partial^{4}w}{\partial x^{4}}-\rho g,$ (1)

where $\rho$ is the line density, $B$ is the bending stiffness and $g$ is the
acceleration due to gravity. We assume that the rope is under tension, $T$.
This tension could naturally arise from static friction with the ground, as
one lifts one extremity of the rope, or it could directly be applied, as in
our experimental set-up. Before shaking the rope, we manually apply a gentle
tension on the rope, on the order of $1$ N, and maintain the resulting static
deformation by placing a weight at the unlifted end of the rope (see Fig. 1).
This is done in order to maintain a small slope, which is required for the
validity of Eq. (1). In practice, due to space constraint, the rope made an
angle of approximately 30∘ with the horizontal at the actuated end. In what
follows, we assume for simplicity that $T$ is kept constant and uniform in
$x$, which neglects variations associate to stretch or slip along the ground
and requires the slope to remain small Coleman and Dill (1992). This is in
contrast with recent studies on the formation of rucks Kolinski  _et al._
(2009); Vella  _et al._ (2009) where that force is deduced through the
constrain of inextensibility. Note that rucks are essentially under
compression rather than in tension, leading to a completely different
dynamics. At $x=0$, one has

$\displaystyle w$ $\displaystyle=Z_{0}+\epsilon\Delta Z(t),$
$\displaystyle\Delta Z(t)/Z_{0}$ $\displaystyle=\cos\omega t,$
$\displaystyle\epsilon\ll 1,$ (2)

and, in the absence of applied moment, $\frac{\partial^{2}w}{\partial
x^{2}}=0$. At the contact point $x=x_{c}(t)$, the boundary conditions are
$w=\frac{\partial w}{\partial x}=\frac{\partial^{2}w}{\partial x^{2}}=0$. Our
aim is to describe the rope dynamics as a function of $\epsilon$ and $\omega$.

In spite of appearances, the above differential problem is strongly nonlinear:
The solution $w(x,t)$ of Eq. (1) nonlinearly depends on $x_{c}(t)$, which, in
turn is a functional of $w(x,t)$. A hint of nonlinear behaviour can already be
found from travelling-wave solutions of Eq. (1) far away down the rope in that
their speed depends on their amplitude Howell  _et al._ (2009). Before
embarking on the analysis, we note that a considerable simplification of Eq.
(1) can be made. In our experiments, we use stranded wires. We estimate their
bending rigidity by measuring the torque required to make them conform to a
quarter of circle of prescribed radius. For the thickest wire (diameter 6mm),
we find $B\approx 0.01$ Nm2. Hence, over a length scale of 1 m and with
$T\approx 1$ N, one clearly has $B\frac{\partial^{4}w}{\partial x^{4}}\ll
T\frac{\partial^{2}w}{\partial x^{2}}$ and the bending term can safely be
neglected everywhere except in boundary layers near $x=0$ and $x=x_{c}$.
Therefore, even though we numerically simulate Eq. (1) we focus, for the sake
of analytical investigation, on

$\displaystyle\frac{\partial^{2}w}{\partial t^{2}}$
$\displaystyle=c^{2}\frac{\partial^{2}w}{\partial x^{2}}-g,$ $\displaystyle
c^{2}=T/\rho,$ (3)

subjected to Eq. (2) and only $w=\frac{\partial w}{\partial x}=0$ at
$x=x_{c}$. Expanding $x_{c}$ as

$x_{c}\sim x_{0}+\epsilon x_{1}+\epsilon^{2}x_{2}+\ldots,$ (4)

the boundary conditions can be expanded as

$Y(x_{0},t)+\left(\epsilon x_{1}+\epsilon^{2}x_{2}\right)\left.\frac{\partial
Y}{\partial
x}\right|_{x_{0}}+\epsilon^{2}\frac{x_{1}^{2}}{2}\left.\frac{\partial^{2}Y}{\partial
x^{2}}\right|_{x_{0}}+\ldots=0,$ (5)

where $Y=w,\frac{\partial w}{\partial x}$. Furthermore, $w$ is also expanded
as

$w\sim w_{0}+\epsilon w_{1}+\epsilon^{2}w_{2}+\ldots.$ (6)

Substituting the above expansions into Eq. (3), the leading-order problem is a
static one and is easily solved:

$\displaystyle w_{0}$
$\displaystyle=\frac{g\left(x-x_{0}\right)^{2}}{2c^{2}},$ $\displaystyle
x_{0}^{2}$ $\displaystyle=\frac{2c^{2}Z_{0}}{g}=\frac{2TZ_{0}}{\rho g}.$ (7)

At the next order, we have $\left(\frac{\partial^{2}}{\partial
t^{2}}-c^{2}\frac{\partial^{2}}{\partial x^{2}}\right)w_{1}=0$, which has the
general solution $w_{1}=F(t-x/c)+G(t+x/c)$. Using Eq. (2), one has
$F(t)+G(t)=\Delta Z(t)$. Next, the two boundary conditions (5) yields
$F(t-x_{0}/c)+G(t+x_{0}/c)=0$ and
$-F^{\prime}(t-x_{0}/c)+G^{\prime}(t+x_{0}/c)=-gx_{1}(t)/c$. Eliminating the
functions $F$ and $G$ from these three equations, we find

$x_{1}(t)-x_{1}(t-2x_{0}/c)=(2c/g)\Delta Z^{\prime}(t-x_{0}/c).$ (8)

The above equation describes the linear response of the contact point to a
general small-amplitude excitation $\epsilon\Delta Z(t)$. In the particular
case of a harmonic excitation, $\Delta Z/Z_{0}=\cos\omega t$, one easily
finds, using Eq. (7), that

$x_{1}=\frac{x_{0}\cos\omega t}{2\operatorname{sinc}(\omega x_{0}/c)},$ (9)

where $\operatorname{sinc}(x)=\sin(x)/x$. An infinite linear response is thus
found at frequencies $\omega/(2\pi)=f_{p}$ where

$\displaystyle f_{p}$ $\displaystyle=p\times\sqrt{\frac{g}{8Z_{0}}},$
$\displaystyle p=1,2,\ldots.$ (10)

The set of these resonances make the rope equivalent to a resonator of length
$x_{0}$, in which waves can travel at speed $c$. What is surprising, however,
is that the tension $T$ is absent from the expression of the resonances. This
is because both $c$ _and_ $x_{0}$ increase in proportion to $\sqrt{T}$. The
fundamental resonance is akin to that of a classical pendulum of length
$2Z_{0}/\pi^{2}$. Alternatively, Eq. (1) can be recast in dimensionless form
to show that $T$ disappears from the mathematical formulation in the limit
$B/(Tx_{0}^{2})\to 0$. Indeed, with $\xi=x/x_{0}$, $W=w/Z_{0}$ and
$\tau=ct/x_{0}$, Eq. (1) becomes

$\displaystyle\frac{\partial^{2}W}{\partial\tau^{2}}$
$\displaystyle=\frac{\partial^{2}W}{\partial\xi^{2}}-2-\beta\frac{\partial^{4}W}{\partial\xi^{4}},$
$\displaystyle\beta=\frac{B}{Tx_{0}^{2}}=\frac{\rho gB}{2T^{2}Z_{0}}$ (11)

with $W-1-\epsilon\cos\Omega\tau=W_{\xi}=0$ at $\xi=0$ and
$W=W_{\xi}=W_{\xi\xi}=0$ at $\xi=\xi_{c}(\tau)$, and

$\Omega=\omega x_{0}/c=\omega\sqrt{2Z_{0}/g}.$ (12)

In the $\beta\to 0$ limit, the tension is thus scaled out of the problem. For
non zero $\beta$, the frequencies $f_{p}$ are not equispaced anymore and
$f_{1}$ increases slightly, up to 42% as $\beta\to\infty$ (see Supplemental
Material).

We have checked the independence of $f_{1}$ on $T$ experimentally. We mounted
a stepper motor (Nema 23) capable of delivering a torque of up to 3 Nm. The
motor was driven by a QGL-HQ MA860H pilot, whose signal came from an Arduino
Mega2560. The rotation of the motor was converted into vertical motion using a
home-made Scott Russel linkage. With this set-up, a precise motion $\Delta
Z(t)$ could be imparted on the rope. We used a thin stranded metallic wire
rope (diameter 2mm) with $\rho=15$ g/m, $B\approx 0.0005$ Nm2. The static
elevation $Z_{0}$ was 0.81 m, yielding a theoretical fundamental resonance
$\sqrt{g/8Z_{0}}\approx 1.23$ Hz. Note that the slope of the rope is not very
small at $x=0$. Nevertheless, the rope rapidly becomes horizontal as it nears
the ground. Additionally, we found that the static profile was almost
undistinguishable from a quadratic one, as in Eq. (7), away from the immediate
vicinity of the origin. Eqs. (1) and (3) thus appear to be reliable. In order
to avoid sideways motion of the rope, also known as “ponytail instability”
Belmonte  _et al._ (2001); Keller (2010), the rope was vertically oscillated
against a board. This produced some friction, but much less than the internal
one. According to the linear theory, a general driving $\Delta Z(t)/Z_{0}=\int
A(\omega)\exp(i\omega t)\mathrm{d}\omega$, yields a linear response

$x_{1}(t)=\frac{x_{0}}{2}\int\frac{A(\omega)e^{i\omega
t}}{\operatorname{sinc}(\omega x_{0}/c)}\,\mathrm{d}\omega.$ (13)

Hence, in order to excite the fundamental resonance only, $\Delta Z(t)$ was
gradually ramped from a vanishing amplitude to a constant sinusoidal
excitation in 20 periods of oscillation. In this way, $A(\omega)$ was strongly
peaked around a well-defined frequency and unwanted excitation of high order
poles in Eq. (13) was reduced. We increased the frequency until resonance was
detected, in the form of a transition to small waves being emitted down the
rope. Because of the very small amplitude of the waves, there was some
uncertainty on the frequency at the transition on the order of 0.02 Hz. We
repeated the experiment with different values of the tension $T$, measured by
the static part $x_{0}$ of the contact point. To measure $T$ directly was
difficult to implement but, from Eq. (7), $x_{0}$ increases monotonically with
$T$. Independence of $f_{1}$ on $x_{0}$ thus implies independence of $T$, all
other parameters being unchanged. The results, shown in Table. 1, confirm the
prediction.

Table 1: Experimental demonstration of independence of resonance frequency on the tension in the rope. $\rho=15$ g/m, $B\approx 0.0005$ Nm2, $\epsilon=0.05$, $Z_{0}=81$ cm, $\sqrt{g/8Z_{0}}\approx 1.23$ Hz. $x_{0}\pm 0.03$ (m): | 1.08 | 1.20 | 1.47 | 1.65 | 1.80 | 2.30 | 2.45
---|---|---|---|---|---|---|---
$f_{1}\pm 0.02$ (Hz): | 1.18 | 1.19 | 1.20 | 1.20 | 1.23 | 1.17 | 1.20

Eq. (9) indicates that, starting from a small frequency below $f_{1}$ and
increasing it, the response of the system diverges. One can anticipate that
the unbounded growth of the oscillation can only resolve itself into a train
of waves along the rope. On the other hand, in the range
$f_{1}<\omega/(2\pi)<f_{2}$, the linear response becomes small again,
suggesting that it is possible not to emit waves there. However, apart from
the divergence, the above linear theory fails to give details on how waves are
actually generated. Proceeding to higher orders of the analysis, we use the
coordinate $s=(x-x_{c}(t))/x_{0}$ and the reduced pulsation $\Omega=\omega
x_{0}/c$. For a purely harmonic forcing, we find (See Supplemental Material)
that

$w/Z_{0}\sim\sum_{j\geq
0}\left(\frac{\epsilon}{\sin\Omega}\right)^{j}F_{j}(s,\epsilon)\cos\left(j\omega
t\right),$ (14)

where $F_{0}\sim s^{2}+O(\epsilon^{2})$,

$F_{1}\sim\left(\Omega s-\sin\Omega
s\right)\left[1+\frac{\epsilon^{2}\Omega^{4}\cos^{2}\Omega}{32\sin^{4}\Omega}\left(1+2\frac{\tan\Omega}{\Omega}\right)\right]\\\
+\frac{\epsilon^{2}\Omega^{4}\sec\Omega}{16\sin^{3}\Omega}\left[\sin
2\Omega-\sin\left(2\Omega+\Omega s\right)\right]\sin\Omega
s+O\left(\epsilon^{4}\right)$ (15)
$F_{2}\sim\frac{\Omega^{2}}{8}\Big{[}\left(1-\cos\Omega
s\right)^{2}-\sin^{2}\Omega s-\\\ \frac{\cos 2\Omega}{\sin
2\Omega}\left(2\Omega s-\sin 2\Omega
s\right)\Big{]}+O\left(\epsilon^{2}\right)$ (16)

and

$F_{3}\sim\frac{\Omega^{4}}{32}\Big{[}\sin\Omega s+\sin 3\Omega s-2\sin
2\Omega s\\\ +\frac{\left(2-\cos 2\Omega+\cos 4\Omega\right)}{2\left(1+2\cos
2\Omega\right)\sin^{2}\Omega}\left(3\Omega s-\sin 3\Omega s\right)\\\
-4\frac{\cos 2\Omega}{\sin 2\Omega}\left(1-2\cos\Omega s\right)\sin^{2}\Omega
s\Big{]}+O\left(\epsilon^{2}\right)$ (17)

The leading order expressions of $F_{2}$ and $F_{3}$ indicate new resonances
when $\sin 2\Omega=0$ and when $1+2\cos 2\Omega=0$, that is when
$\omega/(2\pi)=f_{p}/2$ or $f_{p}/3$. Indeed, the nonlinearity in Eq. (5)
leads to higher harmonics of the $\cos\omega t$ forcing which, in turn, can
match the fundamental resonances. As one progresses to higher orders in the
analysis, more harmonics are found. In particular, the condition
$q\omega/(2\pi)=f_{p}$ yields the nonlinear resonances
$\omega_{p,q}/(2\pi)=f_{p}/q=(p/q)f_{1}$, similarly to Mokhtar  _et al._
(2017). Since $\\{\omega_{p,q}|p,q\in\mathbb{Z}\\}$ is a dense set, any
frequency is arbitrarily close to a nonlinear resonance and any harmonic
forcing should in principle lead to a divergent response in the absence of
dissipation.

Figure 2: Bifurcation diagram in the $(\omega,\epsilon)$ plane obtained by
simulation of Eq. (1) with $B/(Tx_{0}^{2})=0.01$. Orange dashed curves, from
left to right: $\epsilon=1.815|\Omega-\Omega_{0}/2|^{3/4}$,
$\epsilon=0.309|\Omega-\Omega_{0}|^{5/3}$, and
$\epsilon=0.077|\Omega-2\Omega_{0}|^{5/3}$, where $\Omega_{0}=3.2$. Inset:
analytical profile Eq. (18) at impact near the $\Omega_{0}$ resonance. Between
parentheses are indicated frequencies in Hz corresponding to an experiment
with $Z_{0}=81$ cm.

Let us study the limit $\omega/(2\pi)\to f_{p}$ of the above expansion.
Writing $\Omega=p\pi+\mu$, $\mu\ll 1$, Eqs. (14)-(17) yield, for sufficiently
small $\mu$:

$w/Z_{0}\sim
s^{2}+(-)^{p}\frac{\epsilon^{3}\left(p\pi\right)^{4}}{32\mu^{5}}\bigg{[}\cos\left(\omega
t\right)\left(p\pi s-\sin p\pi s\right)\\\ \left.+\frac{1}{3}\cos\left(3\omega
t\right)\left(3p\pi s-\sin 3p\pi s\right)\right].$ (18)

This function displays a local minimum at times $t$ given by $\omega
t=(2n+p)\pi$. One finds that, at such times, the rope makes contact with the
ground at $s=-1.13/(p\pi)$ with zero slope and zero velocity if

$\epsilon=\epsilon_{G}(\omega)\approx\frac{0.309}{p^{2}}\mu^{5/3}=\frac{0.309}{p^{2}}\left(\frac{\omega
x_{0}}{c}-p\pi\right)^{5/3}.$ (19)

Importantly, Arnold tongues given by the above formula become flatter as $p$
increases. Hence, the areas below these curves shrink as $1/p^{2}$. Similarly,
using the $O\left(\epsilon^{4}\right)$ expression for $F_{4}(s,\epsilon)$ (see
Supplemtal Material), one may investigate the region $\omega/(2\pi)\to
f_{1}/2$, by writing $\Omega=\pi/2+\mu$, $\mu\ll 1$. One then obtains that the
rope touches the ground before $x=x_{0}$ if, locally,

$\epsilon=\epsilon_{G}(\omega)\approx 1.815\mu^{3/4}=1.815\left(\frac{\omega
x_{0}}{c}-\frac{\pi}{2}\right)^{3/4}.$ (20)

Figure 3: (a) : Experimental bifurcation diagram in the $(\omega,\epsilon)$
parameter plane. $\epsilon_{LP}(\omega)$: limit-point curve above which large-
wave and small-wave regimes coexist. $\epsilon_{G}(\omega)$: grazing
bifurcation curve. (b) : snapshots of the large-wave oscillation. (c):
snapshot of the small-wave oscillation (extracted frames from video in
Supplemental Material.) Panels (b) and (c) both correspond to the same value
of $\omega$ and $\epsilon$ but different initial conditions [black square in
panel (a).] $\rho=48$ g/m, $B\approx 0.01$ Nm2, $Z_{0}=81$ cm

Eq. (18) is not rigorously valid because it breaks the asymptotic ordering of
terms in Eq. (14). Therefore, it shouldn’t be given any theoretical value
other than providing a qualitative trend. Nevertheless, the resulting
expression Eq. (19) is found to fit remarkably well with the appearance of
waves in numerical simulations. Numerical simulations of dynamical contact
problems are known to be challenging Doyen  _et al._ (2011). A practical issue
is the determination and proper processing of the contact set in the
discretized rope. We used an algorithm adapted from Liakou et al. Liakou  _et
al._ (2017). A description is given in the Supplemental Material. Fig. 2 shows
the boundary between the regimes of no radiation and radiation. Numerically,
the fundamental resonance is slightly shifted from $\Omega=\pi$ to
$\Omega=\Omega_{0}\approx 3.2$; this results from the small but non zero
bending stiffness [$B/(Tx_{0}^{2})=0.01$]. Strikingly, the numerical curve
contains a large number of sharp dips, in addition to those predicted by the
linear analysis. These are signatures of the nonlinear resonances discussed
above. In particular, a sharp resonance is seen at $\Omega_{0}/2$. As
anticipated by the theory, the wave-less state regains stability in the range
$\Omega_{0}<\omega x_{0}/c<2\Omega_{0}$ for sufficiently small $\epsilon$. The
analytical formulas derived above in the vicinity of $\Omega_{0}/2$,
$\Omega_{0}$ and $2\Omega_{0}$ convincingly fit the numerical curve.

The situation described by Eqs. (18) and (19) is reminiscent of a _grazing
bifurcation_. Such a bifurcation classically applies to a point mass attached
to a spring and subjected to a periodic force. Upon increasing the forcing
amplitude, the mass starts making contact with an obstacle. Beyond the grazing
bifurcation, the dynamics rapidly becomes chaotic Shaw and Holmes (1983). The
crucial difference here, of course, is that the rope is spatially distributed.
Nevertheless, the analogy is sufficiently strong to also call the threshold
identified by Eq. (19) a grazing bifurcation. Let us denote by $t^{*}$ and
$x^{*}$ the time and location of impact as determined above. If $\epsilon$
exceeds the grazing bifurcation threshold, the contact happens with a finite
speed, $-V$. In the limit of an infinitely rigid ground with restitution
coefficient $r$, the rope locally rebounds instantaneously with speed $rV$.
This amounts to a reaction force $\left(1+r\right)\rho
V\delta(t-t^{*})\delta(x-x^{*})$, which produces an elevation
$w^{*}(x,t)=0.5\left(1+r\right)Vx_{0}/c$ in the range
$-c(t-t^{*})<x-x^{*}<c(t-t^{*})$ Howell  _et al._ (2009). The perturbation
$w^{*}$ expands in both directions and adds itself to the elevation given by
Eq. (14). Ultimately, this provokes the detachment of the the bump between
$x^{*}$ and $x_{c}(t^{*})$ and its propagation at speed $c$ down the rope. In
this scenario, the portion of the rope ahead of $x^{*}$ which ultimately forms
the travelling wave, is initially given by Eq. (18) at $t^{*}$.

We now turn to the experimental demonstration of wave generation through
harmonic forcing at $x=0$. As previously mentioned, the computer-driven motor
was mounted on a table at a height $Z_{0}=81$cm. The angle of the rotor varied
in steps of 1.8∘. Given the length of the arm of the Russel linkage, this
translates into an uncertainty of $\Delta Z(t)$ of 4mm, hence an uncertainty
$|\Delta\epsilon|\approx 0.005$. The transitions were monitor by varying
$\epsilon$ for fixed $\omega$. Being computer-controlled, the uncertainty in
frequency is estimated to be well under 0.01 Hz and hence, negligible. From
what precedes, waves emitted at the grazing bifurcation point
$\epsilon_{G}(\omega)$ are of very small amplitude, the maximum being
approximately $0.085Z_{0}/(p\pi)^{2}$ (see inset of Fig. 2). Moreover, past
the bifurcation threshold, this amplitude does not grow rapidly as
$\left(\epsilon-\epsilon_{G}(\omega)\right)^{1/2}$, but, rather, only linearly
in $\epsilon-\epsilon_{G}(\omega)$, as in other impact systems di Bernardo
_et al._ (2008). Finally, the waves undergo rapid attenuation due to internal
friction between the strands of the rope. This makes the determination of the
transition experimentally challenging, and we focused on large frequencies,
near $2\Omega_{0}$, in order to benefit from a large amplification of the
oscillations.

Fig. 3(a) shows the experimental result. The no wave/small wave boundary is
the grazing bifurcation curve. It shows a minimum near 2.3 Hz, in good
agreement with the $f_{2}$ resonance of Fig. 2. However, quantitative
agreement between the numerical curve of Fig. 2 and the experimental one in
Fig. 3(a) is poor, even though the order of magnitude is the same. We
attribute this poor matching to the non-constancy of $T$. Indeed, $T$ is not
actively controlled in the experiment. The fact that the slope of the rope is
not everywhere small (see Supplemental Material) together with small parasitic
slippage on the ground both make $T$ vary in time and space in practice.

Next to the small-wave regime described thus far, we experimentally discover a
large-wave regime. The two regimes stably coexist over wide ranges of
parameters. Figs. 3(b) and (c) are snapshot of the two distinct dynamical
states, observed for the same values of $\omega$ and $\epsilon$ [black dot in
Fig. 3(a).] The limit of coexistence between the two regimes is classically
given by a curve of limit point $\epsilon_{LP}(\omega)$ in the
$(\omega,\epsilon)$ space.

Fortunately the curve $\epsilon_{LP}(\omega)$ is much more convenient to
determine than $\epsilon_{G}(\omega)$ as the transition between the two
regimes is visually clear-cut. We record $\epsilon_{LP}(\omega)$ in the
following way: (i) start with a value $\epsilon>\epsilon_{LP}(\omega)$ in the
state of large-wave emission. (ii) Decrease $\epsilon$ by small step and wait
for the system to relax to a stable operation. (iii) As soon as
$\epsilon<\epsilon_{LP}$, the large-wave state irreversably disappears after
only a few oscillations, giving way to the small-wave state. The most
important feature of the $\epsilon_{LP}(\omega)$ curve is its minimum. This
minimum attests of the resonant nature of that state and is an indirect
manifestation of the Arnold tongues described above.

Regarding the large-wave emitting state, we make two observations. Firstly, we
could not observe it with ropes of smaller cross sections, i.e. smaller
bending stiffness $B$. This suggests that bending stiffness plays an important
role in the existence of this dynamical regime. Secondly, we could not
reproduce the bistability between a large-wave and a small-wave making state
for any value of $B$ in Eq. (1). From this, we conclude that this model is
insufficient to describe this state and that a fully geometrically nonlinear
model is required Howell  _et al._ (2009); Belmonte  _et al._ (2001); Coleman
and Dill (1992). Geometrical nonlinearity couples the transverse motion
$w(x,t)$ to a longitudinal motion, making the displacement fully vectorial.
This could be the subject of further investigation.

Dynamical contacts between deformable bodies may display rich dynamical
behaviours. Using a simple rope as a prototype example, we have seen that the
dynamics combines the physics of free moving boundaries and that of non-smooth
dynamical systems. In particular, contacts are governed by the infinite set of
resonant deformation modes of the rope [see the multiple poles in Eq. (8)],
supplemented by their nonlinear harmonics. When analyzing more complicated
dynamical contact problems, where analytical results are out of reach, the
present study suggests to pay attention to linear resonances and their
nonlinear harmonics. In the vicinity of these resonances, nonlinear waves and
chaos are susceptible to arise via grazing bifurcations. Finally, the
stability regions are likely to be delimited by Arnold tongues in the
parameter space. Thanks to the slenderness of the rope, the elastic degrees of
freedom are reduced to their simplest expression, here. Next in complexity
would be the study of time-dependent contacts between two slender bodies and
the inclusion of out-of-plane degrees of freedom.

###### Acknowledgements.

B.S. is a Research Fellow and G.K. is a Research Associate of the Fonds de la
Recherche Scientifique - FNRS (Belgium.) We thank Pascal Damman and Fabian
Brau for stimulating discussions and Jamal Tahmaoui for fabricating the Scott
Russel linkage.

## References

  * Wriggers (2006) P. Wriggers,  _Computational Contact Mechanics_ (Springer, Berlin, Heidel-berg, New York, 2006).
  * Doyen  _et al._ (2011) D. Doyen, A. Ern, and S. Piperno, SIAM Journal on Scientific Computing 33, 223 (2011).
  * Hertz (1896) H. Hertz,  _Miscellanous Papers_ (London: Macmillan, New York, Macmillan and co, 1896).
  * Zajac (1957) E. E. Zajac, Bell Sys. Tech. J. 36, 1129 (1957).
  * Jawed  _et al._ (2014) M. K. Jawed, F. Da, J. Joo, E. Grinspun, and P. M. Reis, Proc. Natl. Acad. Sci. U.S.A. 111, 14663 (2014).
  * Weitsman (1971) Y. Weitsman, International Journal of Mechanical Sciences 13, 707 (1971).
  * Miller  _et al._ (2015) J. T. Miller, C. G. Mulcahy, J. Pabon, N. Wicks, and P. M. Reis, J. Appl. Mech. 82, 021003 (2015).
  * Mokhtar  _et al._ (2017) M. A. Mokhtar, A. K. Darpe, and K. Gupta, Journal of Sound and Vibration 401, 94 (2017).
  * Astashev and Krupenin (2001) V. Astashev and V. Krupenin, Doklady Physics 46, 522 (2001).
  * Murphy and Morrison (2002) K. D. Murphy and T. M. Morrison, The Journal of the Acoustical Society of America 111, 884 (2002).
  * Dick  _et al._ (2008) A. J. Dick, B. Balachandran, H. Yabuno, M. Numatsu, K. Hayashi, M. Kuroda, and K. Ashida, Nonlinear Dynamics 57, 335 (2008).
  * Howell  _et al._ (2009) P. Howell, G. Kozyreff, and J. Ockendon,  _Applied solid mechanics_ (Cambridge University Press, 2009).
  * Coleman and Dill (1992) B. D. Coleman and E. H. Dill, J. Acoust. Soc. Am. 91, 2663 (1992).
  * Kolinski  _et al._ (2009) J. M. Kolinski, P. Aussillous, and L. Mahadevan, Phys. Rev. Lett. 103 (2009).
  * Vella  _et al._ (2009) D. Vella, A. Boudaoud, and M. Adda-Bedia, Phys. Rev. Lett. 103, 174301 (2009).
  * Belmonte  _et al._ (2001) A. Belmonte, M. J. Shelley, S. T. Eldakar, and C. H. Wiggins, Phys. Rev. Lett. 87 (2001), 10.1103/physrevlett.87.114301.
  * Keller (2010) J. B. Keller, SIAM Journal on Applied Mathematics 70, 2667 (2010).
  * Liakou  _et al._ (2017) A. Liakou, V. Denoël, and E. Detournay, Journal of Engineering Mechanics 143, 04016116 (2017).
  * Shaw and Holmes (1983) S. Shaw and P. Holmes, Journal of Sound and Vibration 90, 129 (1983).
  * di Bernardo  _et al._ (2008) M. di Bernardo, C. J. Budd, A. R. Champneys, P. Kowalczyk, A. B. Nordmark, G. O. Tost, and P. T. Piiroinen, SIAM Review 50, 629 (2008).