Dinis Pestana, Antónia Turkman, João Branco, Luísa Canto e Castro e Ana Pires.

Actas do II Congresso da SPE, 1994, Luso; ed. M. Nazaré Mendes Lopes, M.
Esmeralda Gonçalves, M. Emília Nogueira, Ana Cristina Rosa e Helena Ferreira.

Bom Senso e Sensibilidade: Traves Mestras da Estatística [Common Sense and
Sensitivity: Main Supports of Statistics], Actas do III Congresso da SPE,
1995, Guimarães; ed. João Branco, Paulo Gomes e Jorge Prata.

A Estatística a Decifrar o Mundo [Statistics Deciphering the World], Actas do
IV Congresso da SPE, 1996, Funchal; ed. Rita Vasconcelos, Isabel Fraga Alves,
Luisa Canto e Castro e Dinis Pestana.

Estatística – a Diversidade na Unidade [Statistics – Diversity in Unity],
Actas do V Congresso Anual da Sociedade Portuguesa de Estatística, 1997,
Curia; ed. Manuela Souto de Miranda e Isabel Pereira.

Afirmar a Estatística: um Desafio para o Século XXI [Affirming Statistics: a
Challenge for the 21st Century], Atas do VI Congresso da SPE, 1999, Tomar; ed.
Carlos Daniel Paulino, António Pacheco, Ana Pires e Ferreira da Cunha.

Um Olhar sobre a Estatística [A Look at Statistics], Actas do VII Congresso da
SPE, 2001, Ofir; ed. Pedro Oliveira e Emília Athayde.

A Estatística em Movimento [Statistics in Motion], Actas do VIII Congresso da
SPE, 2001, Peniche; ed. Maria Manuela Neves, Jorge Cadima, Maria João Martins
e Fernando Rosado.

Novos Rumos em Estatística [New Directions in Statistics], Actas do IX
Congresso da SPE, 2002, Ponta Delgada; ed. Lucília Carvalho, Fátima Brilhante
e Fernando Rosado.

Literacia e Estatística [Literacy and Statistics], Actas do X Congresso da
SPE, 2003, Porto; ed. Paula Brito, Adelaide Figueiredo, Fernanda Sousa, Paulo
Teles e Fernando Rosado.

Estatística com Acaso e Necessidade [Statistics with Chance and Necessity],
Actas do XI Congresso da SPE, 2004, Faro; ed. Paulo M. M. Rodrigues, Efigénio
da Luz Rebelo e Fernando Rosado.

Estatística Jubilar [Jubilee Statistics], Actas do XII Congresso Anual da SPE,
2005, Évora; ed. Carlos Braumann, Paulo Infante, Manuela M. Oliveira, Russell
Alpízar-Jara e Fernando Rosado.

A Ciência Estatística [Statistical Science], Actas do XIII Congresso da SPE,
2006, Ericeira; ed. Luísa Canto e Castro, Eugénia Graça Martins, Cristina
Rocha, M. Fernanda Oliveira, Margarida Mendes Leal e Fernando Rosado.

Estatística, Ciência Interdisciplinar [Statistics, Interdisciplinary Science],
Actas do XIV Congresso da SPE, 2007, Covilhã; ed. Maria Eugénia Ferrão, Célia
Nunes e Carlos Braumann.

Estatística: da Teoria à Prática [Statistics: from Theory to Practice], Actas
do XV Congresso da SPE, 2008, Lisboa; ed. Manuela Magalhães Hill, Manuel
Alberto Ferreira, José G. Dias, Maria de Fátima Salgueiro, Helena Carvalho,
Paula Vicente e Carlos Braumann.

Estatística: Arte de Explicar o Acaso [Statistics: Art of Explaining Chance],
Actas do XVI Congresso Anual da SPE, 2009, Vila Real; ed. Irene Oliveira,
Elisabete Correia, Fátima Ferreira, Sandra Dias e Carlos Braumann.

Advances in Regression, Survival Analysis, Extreme Values, Markov Processes
and Other Statistical Applications, Selected Papers of the Statistical
Societies, Springer, XVII Congresso Anual da SPE, 2013, Sesimbra; Ed. João
Lita da Silva, Frederico Caeiro, Isabel Natário, Carlos A. Braumann, Manuel L.
Esquível and João Tiago Mexia.

Recent Developments in Modeling and Applications in Statistics, Selected
Papers of the Statistical Societies, Springer, XVIII Congresso Anual da SPE,
2013, São Pedro do Sul; ed, Paulo Eduardo Oliveira, Maria da Graça Temido,
Carla Henriques e Maurizio Vichi.

New Advances in Statistical Modeling and Applications, Selected Papers of the
Statistical Societies, Springer, XIX Congresso Anual da SPE, 2014, Nazaré; ed.
António Pacheco, Rui Santos, Maria do Rosário Oliveira e Carlos Daniel
Paulino.

Estatística: Novos Desenvolvimentos e Inspirações [Statistics: New
Developments and Inspirations], Actas do XX Congresso Anual da SPE, 2013,
Porto; ed. Manuela Maia, Pedro Campos e Pedro Duarte Silva.

Estatística: a ciência da incerteza [Statistics: the science of uncertainty],
Atas do XXI Congresso da SPE, 2014, Aveiro; ed. Isabel Pereira, Adelaide
Freitas, Manuel Scotto, Maria Eduarda Silva e Carlos Daniel Paulino.
https://www.spestatistica.pt/publicacoes/publicacao/estatIstica-ciencia-
incerteza

Estatística: progressos e aplicações [Statistics: Progresses and
Applications], Atas do XXII Congresso da SPE, 2015, Olhão; ed. Clara Cordeiro,
Conceição Ribeiro, Carlos Sousa, Maria Helena Gonçalves, Nelson Antunes e
Maria Eduarda Silva. https://www.spestatistica.pt/publicacoes/
publicacao/estatIstica-progressos-aplicacoes

Atas do XXIII Congresso da SPE [Proceedings of the XXIII SPE Congress], 2017,
Lisboa; ed. Maria de Fátima Salgueiro, Paula Vicente, Teresa Calapez, Catarina
Marques e Maria Eduarda Silva.
https://www.spestatistica.pt/publicacoes/publicacao/atas-do-xxiii-congresso-
da-spe

Estatística: Desafios Transversais às Ciências com Dados [Transversal
challenges to Sciences with Data], Atas do XXIV Congresso da Sociedade
Portuguesa de Estatística, 2021, Amarante ed. Paula Milheiro, António Pacheco,
Bruno de Sousa, Isabel Fraga Alves, Isabel Pereira, Maria João Polidoro e
Sandra Ramos. https://www.spestatistica.pt/publicacoes/publicacao/estatistica-
desafios-transversais-ciencias-com-dados

Recent Developments in Statistics and Data Science, Atas do XXV Congresso da
Sociedade Portuguesa de Estatística, 2021, Évora (ed. Regina Bispo, Lígia
Henriques-Rodrigues, Russell Alpizar-Jara e Miguel de Carvalho).
https://www.spestatistica.pt/publicacoes/publicacao/recent-developments-
statistics-and-data-science

New Frontiers in Statistics and Data Science, Springer Proceedings in
Mathematics & Statistics, Springer, XXVI Congresso da SPE, 2024, Guimarães
(ed. Lígia Henriques-Rodrigues, Raquel Menezes, Luís Meira Machado, Miguel de
Carvalho e Susana Faria), in print.

## Appendix 2

Index of the Manual de Estatística Médica (Manual of Medical Statistics) by
Carlos Santos Reis and Alexandre Sarmento.

Introduction | 3
---|---
1 – Fundamental Concepts | 6
2 – Statistical Methods | 14
1 – Definition | 14
2 – Objectives | 14
3 – Purpose | 14
4 – Feature | 14
5 – Observation methods | 14
6 – Applications | 14
7 – Limitations | 14
First part
Application of the statistical method | 25
3 – Planning | 26
4 – Organization | 28
5 – Preparation | 31
1 – The collection or collection of data | 32
1.1 – Harvest method | 33
1.2 – Observation methods | 34
1.3 – Scope of investigations | 36
1.4 – Investigation Modalities | 37
1.5 – Source of data | 42
1.6 – Preparation of questionnaires | 43
1.7 – Pilot survey | 46
1.8 – Sources of errors in surveys | 48
1.9 – Observation record | 52
6 – Data verification | 56
1 – Critique of results | 56
2 – Data ordering | 58
3 – Data classification | 61
4 – Organization of a frequency distribution | 64
5 – Enumeration of individuals | 68
6 – Coding of results | 75
7 – Presentation of data | 81
1 – Text presentation | 81
2 – Tabular presentation | 82
8 – Graphical presentation | 92
1 – General considerations | 92
2 – Main fundamentals and criticisms of graphic presentation | 93
3 – Rules for constructing graphs | 93
4 – Definition and major divisions in graphic presentation | 95

9 – Use | 113
---|---
1 – Absolute values | 114
2 – Derived values | 114
10 – Description of statistical series | 124
1 – Qualitative characters | 125
2 – Quantitative characters | 126
11 – Position measurements | 129
1 – Median | 130
2 – Arithmetic mean | 133
3 – Fashion | 137
4 – Geometric mean | 138
5 – Comparison of main position measurements | 142
12 – Dispersion and shape measurements | 145
1 – Dispersion measurements | 145
2 – Shape measurements | 152
3 – Presentation of results | 153
13 Simplified calculation of parameters | 154
1 – Fashion | 155
2 – Median | 156
3 – Arithmetic mean | 157
4 – Variance | 160
5 – Joint calculation of mean and variance | 164
14 – Interpretation of data. Statistical schemes | 167
15 – Some theoretical distributions | 174
1 – Notion of probability and law of large numbers | 175
2 – Binomial distribution | 178
3 – Normal distribution | 182
4 – Poisson Distribution | 189
5 – Variable transformation | 191
16 – Sampling | 193
1 – Sampling Deformation | 194
2 – Confidence Limits | 196
3 – Sample representativeness | 196
4 – Sampling Types | 197
5 – Number of observations | 202
6 – Sampling Problems | 204
17 – Estimation and confidence limits | 205
1 – Estimation and confidence limits of the mean | 207
2 – Estimation and confidence limits of variance | 210
3 – Estimation and confidence limits of a proportion | 210
18 – Compliance Tests | 215
1 – Conformity tests for parameters | 219
2 – Distribution Conformance Tests | 220
19 – Homogeneity problem | 232
1 – Comparison of two means | 234
2 – Comparison of two variances or two standard deviations | 245
3 – Comparison of two rates or proportions | 248

Appendix
---
Calculation of factorials | 257
20 – Comparison of multiple samples | 263
1 – Simultaneous comparison of multiple means | 265
2 – Simultaneous comparison of several variances | 304
3 – Simultaneous comparison of several proportions | 306
21 – Statistical dependence | 309
I – Relationships between two qualitative characters. Notions of independence and association | 309
1 – Presentation of results | 310
2 – Notion of $\ll$independence$\gg$ and $\ll$association$\gg$ | 311
3 – Association Issues | 312
4 – Existence of the association | 312
5 – Independence Tests | 314
6 – Different types of problems | 321
7 – Association Measures | 329
22 – Statistical dependence | 332
II – Relationships between two quantitative characters. Understanding correlation and regression | 332
1 – Understanding correlation and regression | 333
2 – Correlation Types | 335
3 – Presentation of results | 336
4 – Correlation form | 339
5 – Regression Line | 342
6 – Regression standard error | 350
7 – Correlation coefficient | 351
8 – Numerical calculation of correlation parameters | 353
9 – Interpretation of correlation coefficients | 364
10 – Significance of a correlation coefficient | 366
11 – Correlation applicability conditions | 368
12 – Advantages of correlation | 368
13 – Comparison of two correlation coefficients | 372
14 – Correlation ratio | 375
15 – Determination and alienation coefficients | 376
16 – Correlation of the number of cases | 376
17 – Position correlation | 377
18 – Conclusions | 378
23 – Statistical adjustment | 379
1 – Adjustment concept | 380
2 – Adjustment Methods | 380
3 – Linear adjustment | 388
4 – Curvy Fit | 393
5 – Adjustment to theoretical distributions | 400
24 – Checking the normality of a series | 406
1 – Graphic appreciation | 407
2 – Cumulative frequencies | 408
3 – Straight from Henri | 409
4 – Significance of kurtosis and asymmetry measurements | 411
5 – Fitting to a normal distribution | 415
6 – The $\ll$Probits$\gg$ | 420

25 – $\ll$Probits$\gg$ | 420
---|---
Method 1 – Principles of the | 420 method
2 – Checking the normality of a distribution | 424
3 – Calculation of lethal dose | 425
26 – Chronological series | 433
1 – Definition | 433
2 – Movements of a chronological series | 434
3 – Guidance to follow when studying these series | 434
4 – Data preparation | 435
5 – Graphical representation | 436
6 – Analysis of time series | 440
7 – Study of the general trend | 441
8 – Seasonal Trend | 444
9 – Cyclic movements | 453
10 – Comparison of time series | 453
27 – Index numbers | 456
1 – Concept | 457
2 – Types of indexes | 458
3 – Conditions for displaying an index number | 462
4 – Problem of creating an index | 462
5 – Random errors in index calculations | 464
Second part
Basic notions of biostatistics | 465
0 – Settings | 465
1 – Population | 466
1.1 – Population status | 466
1.2 – Population movement | 478
1.2.1 – Birth | 478
1.2.2 – Mortality | 486
1.2.3 – Demographic balance | 497
Third part
Annex I – Statistical terminology | 500
Annex II – Statistical symbols | 501
Bibliography | 503

## Appendix 3

Index of Introdução à Estatística (Introduction to Statistics), notes for the
course on General Mathematics for the degrees in Biological Sciences,
Geological Sciences, and Adjunct Professors, based on the lectures of
Professor Dias Agudo, compiled by Moreira Campos / Waldemar Nunes, Lisbon
1961/62

Chapter I – Fundamental Concepts of Probability Theory
---
1.1 – Introduction |
1.2 – A fundamental principle |
1.3 – “Arrangements and Combinations” |
1.4 – Event space and probability space |
1.5 – Probabilities of the sum and product of events. |
Conditional probabilities. |
Incompatible and independent events |
1.6 – Problem with repeated tests. |
Binomial formula |
Chapter II – Random Variables and Distribution Functions
2.1 – Casual variables |
2.2 – Distribution function |
2.3 – Classification of distributions |
2.4 – Functions of casual variables |
2.5 – Percentiles of a distribution |
Chapter III – Discrete Distributions
3.1 – Probability function; elementary probabilities |
3.2 – Mean value |
3.3 – Moments |
3.4 – Important examples of discrete distributions |
Poisson distribution |
Binomial Distribution |
Chapter IV – Continuous Distributions*
4.1 – Probability density |
4.2 – Mean value. Moments. Median |
4.3 – Important example of continuous distribution: Normal distribution |
4.3.1 – Reduced normal distribution |
4.3.2 – General normal distribution |
Chapter V – Sums of Random Variables
5.1 – Mean value and variance of the sum |
5.2 – Distribution of the sum of casual variables |
5.3 – Normal distribution as an approximation of binomials |
Chapter VI - Sampling
Data Presentation and Description |
6.1 – Introduction |
6.2 – Frequency distribution; histograms; data classification |
6.3 – Location and dispersion measurements |

Chapter VII – Parameter Estimation
---
7.1 – Introduction |
7.2 – Mean value estimation |
7.3 – Variance estimation |
Chapter VIII – Hypothesis Tests
8.1 – Introduction |
8.2 – Hypothesis tests on the mean value |
8.3 – Hypothesis tests on the type of distribution |
8.4 – Comparison of two populations |

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Acknowledgments. This work is partially financed by national funds through FCT
— Fundação para a Ciência e a Tecnologia under the project UIDB/00006/2020
https://doi.org/ 10.54499/UIDB/00006/2020.

The authors are thankful to Prof. Antónia Amaral Turkman, Maria de Fátima
Brilhante and Sandra Mendonça for observations that improved the presentation.


# Universal interrelation between dynamics and thermodynamics and a
dynamically-driven “c”-transition in fluids

C. Cockrell1,∗, V. V. Brazhkin2 and K. Trachenko1 1 School of Physics and
Astronomy, Queen Mary University of London, Mile End Road, London, E1 4NS, UK
$2$ Institute for High Pressure Physics, RAS, 108840, Troitsk, Moscow, Russia
$*$ Corresponding author<EMAIL_ADDRESS>
###### Abstract

Our first very wide survey of the supercritical phase diagram and its key
properties reveals a universal interrelation between dynamics and
thermodynamics and an unambiguous transition between liquidlike and gaslike
states. This is seen in the master plot showing a collapse of the data
representing the dependence of specific heat on key dynamical parameters in
the system for many different paths on the phase diagram. As a result, the
observed transition is path-independent. We call it a “c”-transition due to
the “c”-shaped curve parameterizing the dependence of the specific heat on key
dynamical parameters. The “c”-transition has a fixed inversion point and
provides a new structure to the phase diagram, operating deep in the
supercritical state (up to at least 2000 times the critical pressure and 50
times the critical temperature). The data collapse and path independence as
well as the existence of a special inversion point on the phase diagram are
indicative of either of a sharp crossover or a new phase transition in the
deeply supercritical state.

###### pacs:

65.20.De 65.20.JK 61.20Gy 61.20Ja

##  I Introduction

The basic phase diagram of matter charts the areas where solid, liquid and gas
exist as physically distinct phases separated by transition lines. The three
transition lines all emerge from the triple point, but two of three are finite
in length. The sublimation line terminates at zero temperature or pressure,
and the boiling line terminates at the critical point. The third line, the
melting line, extends to arbitrary temperatures and pressures, so long as the
system remains chemically unaltered. A large area of the phase diagram lies
above the critical point, representing the supercritical state of matter.

Until fairly recently, there was no reason to survey the state of matter well
above the critical point in any detail. This part of the phase diagram was
thought to be physically homogeneous with no discernible differences between
liquid-like and gas-like states landau ; deben . Not far above the critical
point, persisting critical anomalies can continue to conditionally separate
liquid-like and gas-like states. More recently, experiments and theory have
given indications that the entire supercritical state may in fact be
inhomonegenous and have states with qualitatively different properties.

Going deeply supercritical in the experiment is often unworkable and, when
possible, is limiting in terms of measuring key system properties nist . We
are much less limited in molecular dynamics (MD) simulations.

In this paper, we conduct the widest and most detailed survey of the
supercritical phase diagram undertaken so far: we extend the parameter range
from the melting point and up to 330$T_{c}$ and 8000$P_{c}$ ($T_{c}$ and
$P_{c}$ are critical temperature and pressure for Argon) and traverse it along
different isobars, isochores and isotherms. We calculate key thermodynamic and
dynamical properties along different paths on the phase diagram. We then show
that all data collapse on a single dynamic-thermodynamic “c”-shaped master
curve with a fixed inversion point. The “c”-transition provides an unambiguous
separation of liquidlike and gaslike fluids, extending deep into the
supercritical state and therefore challenging the existing view of this state
as homogeneous and lacking qualitative transitions. The data collapse and the
existence of a special inversion point on the phase diagram are consistent
with either a sharp crossover or a new phase transition (perhaps of higher-
order) operating in the supercritical state of matter.

##  II Simulation details

We simulate a commonly used and well-characterised Lennard-Jones system
describing Argon ($\sigma=3.4~{}\mathrm{\AA}$, $\epsilon=0.01032$ eV) along
three isobars, isotherms, and isochores in the deep supercritical state, plus
an additional isobar near the critical point. The supercritical paths are
plotted in Fig. 1. Our largest temperature and pressure, 330$T_{c}$ and
8000$P_{c}$ (about 50,000 K and 40 GPa) are below those where metallisation
and ionization start in Ar. Within the simulated range of state points, the
“c”-transition is seen up to 2000 times the critical pressure and 50 times the
critical temperature.

Equilibration was performed in the NPT ensemble with the Langevin thermostat
in order to generate the mean densities along the isobars and isotherms.
System sizes of 500, 4000, and 108,000 atoms were used, with no discrepancy in
calculated quantities or our results between these sized. Consistent with the
earlier ascertained insensitivity of viscosity to system size viscositysize ,
we find that the viscosity (and the heat capacity) for larger systems of up to
108,000 atoms coincides with that of the smaller systems we simulated. We
selected the timestep of 1 fs, which is under 100 times smaller than the
fastest oscillations in any system, and conserved total energy under the
Velocity-Verlet integrator in the NVE ensemble to one part in $10^{5}$.
Configurations at the target densities on all paths were then generated, which
were then equilibrated with the NVT ensemble for 50 ps. Following this
equilibration, we generated 20 independent initial conditions for each state
point using seeded velocities, and each of these initial conditions were run
for 1 ns in the NVE ensemble during which all properties were calculated. We
calculated $c_{V}$ in the NVE ensemble as Frenkel2001 :

Figure 1: Paths on the phase diagram and pressure and temperature points
explored in this work. Also labelled are the critical point and the state
points where the transition at the inversion point at $c_{V}=1.9$ takes place.

$\langle K^{2}\rangle-\langle
K\rangle^{2}=\frac{3}{2}NT^{2}\left(1-\frac{3}{2c_{V}}\right),$ (1)

with $K$ the total kinetic energy. From here on out, we set
$k_{\mathrm{B}}=1$. We plot the calculated heat capacities compared to NIST
data (where available) nist  in Fig. 2a-b, demonstrating good agreement. The
high-frequency shear modulus and shear viscosity were calculated using the
molecular stress autocorrelation function, from Green-Kubo theory Zwanzig1965
; Balucani1994 :

$G_{\infty}=\frac{V}{T}\langle\sigma^{xy}(0)^{2}\rangle,$ (2)
$\eta=\frac{V}{T}\int_{0}^{\infty}\rm{d}t\
\langle\sigma^{xy}(t)\sigma^{xy}(0)\rangle,$ (3)

with $\sigma^{xy}$ an off-diagonal component of the microscopic stress tensor.
The integration of the long-time tails of autocorrelation functions is a well
known issue with the practical implementation of Green-Kubo formulae Zhang2015
. The 20 independent initial conditions were used to address this, with the
autocorrelation function $\langle\sigma^{xy}(t)\sigma^{xy}(0)\rangle$ being
averaged over these initial conditions. The end result for viscosity was
insensitive to adding more initial conditions and also whether the
autocorrelation function was calculated in the NVT or NVE ensembles.

Figure 2: Comparison of (a) viscosities $\eta$ and (b) isochoric specifc heat
capacities $c_{V}$ ($k_{\rm B}=1$) calculated from simulated trajectories with
experimental data from NIST nist ; comparison of viscosities $\eta$ calculated
from simulated trajectories along the (c) 10 kbar isobar and (d) 500 K
isotherm, with experimental data from Abramson abramson .

##  III Results and Discussion

We zero in on the relationship between thermodynamic and dynamical properties
of the supercritical system. The specific heat $c_{V}$ is an informative
choice for a thermodynamic property. The choice of a dynamical property is
less obvious because there are several candidates. We first try viscosity,
$\eta$, because there is an obvious distinction between its behaviour in the
liquid and gas phases. Indeed, $\eta$ decreases and increases with temperature
in the liquidlike and gaslike regimes, respectively sciadv . In Fig. 2 we
observe a good agreement between calculated and experimental viscosity. This
includes good agreement with high-pressure data range abramson  where the NIST
data can be less reliable due to data interpolations used at high pressure.

We plot the dependence of $c_{V}$ on $\eta$ for each of our supercritical
phase diagram paths in Fig. 3. Along isobars and isochores, $c_{V}(\eta)$ has
clear turning points due to corresponding minima in $\eta$, whereas no such
minima exist along our isothermal paths. We observe strong path dependence in
the interrelation between $c_{V}$ and $\eta$, and will return to the issue of
path dependence below. Here we note the earlier work excess  relating
viscosity to excess entropy calculated from the virial expansion at low
density. This relation was discussed on empirical grounds and is unrelated to
our approach where we discuss the system properties on the basis of
excitations that the system possesses.

Figure 3: Specific heat $c_{v}$ ($k_{\rm B}=1$) as a function of viscosity
$\eta$, calculated along three (a) isobars; (b)isotherms; (c) isochores.

Our next important choice for the dynamical parameter is informed by the
Maxwell-Frenkel viscoelastic theory maxwell ; frenkel  where the combined
elastic and viscous response of a liquid is represented as:

$\frac{\mathrm{d}s}{\mathrm{d}t}=\frac{\sigma}{\eta}+\frac{1}{G_{\infty}}\frac{\mathrm{d}\sigma}{\mathrm{d}t}$
(4)

where $s$ is shear strain, $\eta$ is viscosity, $G$ is high-frequency shear
modulus and $P$ is shear stress.

Using (4), Frenkel modified the Navier-Stokes equation to include the elastic
response. A simplified form of the resulting equation reads ropp ; physrep :

$c^{2}\frac{\partial^{2}v}{\partial x^{2}}=\frac{\partial^{2}v}{\partial
t^{2}}+\frac{1}{\tau}\frac{\partial v}{\partial t}$ (5)

where $v$ is the velocity component perpendicular to $x$, $c$ is transverse
wave velocity $c=\sqrt{\frac{G}{\rho}}$, $\rho$ is density and
$\tau=\frac{\eta}{G_{\infty}}$ is the liquid relaxation time.

Seeking the solution of (5) as $v=v_{0}\exp\left(i(kx-\omega t)\right)$ gives
$\omega^{2}+\omega\frac{i}{\tau}-c^{2}k^{2}=0$ and
$\omega=-\frac{i}{2\tau}\pm\sqrt{c^{2}k^{2}-\frac{1}{4\tau^{2}}}$. If
$k<\frac{1}{2c\tau}$, $\omega$ has neither a real part nor transverse
propagating modes. For $k>k_{g}=\frac{1}{2c\tau}$,
$v\propto\exp\left(-\frac{t}{2\tau}\right)\exp(i\omega_{r}t)$, where
$\omega_{r}=\sqrt{c^{2}k^{2}-\frac{1}{4\tau^{2}}}$. Here,

$k_{g}=\frac{1}{2c\tau}$ (6)

defines a gapped momentum state seen in several distinct areas of physics
physrep . In liquids and supercritical fluids, $k_{g}$ is the important
parameter governing the existence of solid-like transverse waves physrep ; prl
.

It follows from the above discussion and Eq. (6) in particular that the
solution to the Maxwell-Frenkel viscoelasticity importantly depends on the
dynamical fluid elasticity length $\lambda_{d}=c\tau$ ropp . The physical
meaning of this length is that it sets the range of propagation of solid-like
transverse waves in the liquid because $\tau$ is the time during which the
shear stress is relaxed. Hence, $c_{V}$ should uniquely depend on
$\lambda_{d}$ because (a) $\lambda_{d}$ governs the propagation of transverse
modes and (b) each mode carries the energy $T$ (in harmonic classical case).
Therefore, we will use $\lambda_{d}$ in our subsequent analysis.

Before invoking $\lambda_{d}$, it is first instructive to look at $\tau(T)$
and $c_{v}(\tau)$ dependencies. We plot the calculated $\tau=\frac{\eta}{G}$
along our nine supercritical phase diagram paths in Fig. 4 and observe that
$\tau$ has minima along isobars and isotherms and has no minima along
isochores. To understand the minima, we recall that Frenkel’s theory relates
$\tau$ to the average time between molecular rearrangements in the liquid
frenkel . Backed by experiments and modelling jac ; iwa , this relation has
since become an accepted view dyre . In the low-temperature liquid regime,
$\tau$ decreases with temperature. In the high-temperature gaslike regime,
shear momentum transfer takes place via the collisions of kinetic theory, and
relaxation time $\tau$ is therefore interpreted as the mean time between
collisions in this regime frenkel ; chapman . Then, the minima represent a
crossover from a liquidlike relaxation time commanded by diffusion and
oscillation events zaccone  to a gaslike relaxation time commanded by
collisions ropp . This same crossover takes place along isochores, but
manifests differently in $\tau$ because $\tau$ can only decrease with
temperature on an isochore. Indeed, $\tau=\frac{L}{v_{\rm{th}}}$ in the
gaslike regime, where $L$ and $v_{\rm{th}}$ are the particle mean-free path
and thermal velocity in the gaslike state and $L=\frac{1}{nA}$, where $n$ is
particle density and $A$ is the particle cross-section area. Hence $\tau$
decreases with temperature at constant $n$ mostly because
$v_{\rm{th}}\propto\sqrt{T}$ increases ($A$ decreases with temperature
weakly). We note that the minima of both $\eta$ and $\tau$ depend on the path
taken on the phase diagram and path parameters.

Figure 4: Liquid relaxation time $\tau$ of molecular dynamics trajectories
along three (a) isobars; (b) isotherms; (c) isochores.

Figure 5: Specific heat $c_{v}$ ($k_{\rm B}=1$) as a function of relaxation
time $\tau$, calculated along three (a) isobars ; (b) isotherms; (c)
isochores. The solid lines in (c) represent the approach of $c_{V}$ and $\tau$
to their limiting ideal-gas value $c_{V}=3/2$ at large temperature and are
meant as guides for the eye.

We next investigate the relationship between dynamics and thermodynamics by
plotting $c_{V}$ as a function of $\tau$ along these nine paths in Fig. 5.
Along isobars and isotherms, $c_{V}(\tau)$ has clear turning points
corresponding to the minima in $\tau$. This turning point occurs close to
$c_{V}=2$, such that the crossover in $c_{V}$ from liquidlike to gaslike
corresponds to the dynamical transition in $\tau$. Along isochores, the
situation is again more subtle. However, since the ideal gas limit as
$T\rightarrow\infty$ corresponds to both $c_{V}\rightarrow 3/2$ and
$\tau\rightarrow 0$, the function $c_{V}(\tau)$ in its gaslike regime must
approach $3/2$ as $\tau\rightarrow 0$. Inspection of Fig. 5c reveals that this
is indeed the case, and that $c_{V}(\tau)$ settles into this limiting
behaviour again close to $c_{V}=2$.

Notably, Figures 3 and 5 show the significant path dependence of $c_{V}$ on
$\eta$ and $\tau$: $c_{V}$ depends differently on these parameters along
isochoric, isobaric and isothermic paths as well as different conditions for
each path. Moreover, switching the dependence of $c_{V}$ from $\eta$ to the
related $\tau$ completely changes the shape of the curves.

We now come to the pinnacle of these analyses. The minima of $\tau$ are path-
dependent as mentioned earlier. Instead of plotting $c_{V}(\tau)$, we plot
$c_{V}(c\tau)$ and as the function of the key parameter in the solution of Eq.
(5) which we have called the dynamical length $\lambda_{d}=c\tau$ ($c$ is
calculated as $c=\sqrt{\frac{G}{\rho}}$ as discussed below Eq. (5)). The
physical reason for this dependence is provided by Eq. (6): $c\tau$ governs
the propagation of collective modes (phonons) in liquids and supercritical
fluids. Since each phonon carries energy ($T$ in the classical harmonic case),
$c_{V}$ is governed by $c\tau$. Moreover, this dependence is predicted to be
unique and path-independent, in contrast to the dependence of $c_{v}$ on
$\eta$ or $\tau$ plotted earlier.

The result of this master plot in Fig. 6a is striking. Despite the clear
difference among paths of the same and different type seen in Figs. 3, 4 and
5, the functions $c_{V}(c\tau)$ along all deeply supercritical paths converge
into the same “c”-shaped curve in Fig. 6a.

---

Figure 6: (a) $c_{V}$ as a function of the dynamical length,
$\lambda_{d}=c\tau$ across 9 paths spanning the supercritical state up to 330
$T_{c}$ and 8000 $P_{c}$. $k_{\rm B}=1$. All these paths collapse onto a
single curve and undergo a unified dynamic-thermodynamic transition at the
path-independent point $c_{V}=1.88$ and $\lambda_{d}=1~{}\rm{\AA}$; (b)
Divergence in $\frac{\rm{d}c_{V}}{\rm{d}\lambda_{d}}$ as a function of $c_{V}$
at the point $c_{V}=1.88$. Curves are obtained by fitting $c_{V}(\lambda_{d})$
data from all nine supercritical paths.

The values of $c_{V}\approx 2$ and $\lambda_{d}\approx 1$ Å at the inversion
point in Fig. 6a are physically significant. Brillouin was first to recognize
the significance of $c_{V}=2$ bril . The solid energy can be written as the
sum of energies of one longitudinal and two transverse modes as $2\times
NT/2+2\times(NT/2+NT/2)$, where $NT/2$ is the kinetic and potential energy
component of each mode (the two components are equal according to the
equipartition theorem), $T$ is the energy of one mode in the harmonic
classical case and $N$ is the number of particles. Brillouin assumed that
liquids do not support solid-like transverse modes, implying that the
potential energy component of these modes, $NT/2$, becomes 0. This gives the
energy $2NT$ and $c_{V}=2$ bril . Since this contradicted the experimental
$c_{V}\approx 3$ of liquids at the melting point, a proposition was made that
liquids consist of crystallites with easy cleavage directions so that liquids
have $c_{V}=3$ and can flow at the same time. We now understand that (a)
liquids do, in fact, support solid-like transverse modes, albeit at high-$k$
only as in Eq. (6) (or at high-frequency for propagating modes
$\omega>\frac{1}{\tau}$ ropp ; physrep ) and (b) these modes do disappear
eventually, but only well above the melting point and at very high temperature
where $\tau$ becomes comparable to Debye vibration period. At this point,
$k_{g}$ in Eq. (6) becomes close to the largest wavevector set by the
interatomic separation in the system (ultaviolet, or UV, cutoff), and all
transverse waves disappear from the spectrum, resulting in $c_{V}=2$. This
disappearance of transverse modes and the condition $c_{V}=2$ correspond to
the Frenkel line, which separates two dynamical regimes of particle motion:
combined oscillatory and diffusive motion in the low-temperature “rigid”
liquid (rigid in a sense of its ability to support solid-like transverse
modes); and purely diffusive motion in the high-temperature non-rigid gaslike
fluid ropp . The loss of the oscillatory component at the Frenkel line
transition gives the second, dynamical, criterion of the line: the
disappearance of the minima of the velocity autocorrelation function (VAF).
The thermodynamic criterion $c_{V}=2$ and the dynamical VAF criterion give
coinciding lines on the phase diagram vaf .

A change in dynamics implies and is implied by a change in structure ropp ,
and, stimulated by the Frenkel line idea, the transitions at the Frenkel line
have been subsequently experimentally observed in supercritical Ne f1 , CH4 f2
, N2 f3 , C2H6 f4  and CO2 f5  using X-ray, neutron and Raman scattering
techniques. A crossover in structure has been seen across the FL in simulated
argon Wang2017 , using the same potential as in this work, however this
crossover is subtle as compared to the transition presented here.

As mentioned earlier, $c_{V}=2$ corresponds to the purely harmonic case where
the mode energy is $T$. Anharmonicity can change this result by a relatively
small amount ropp , and the disappearance of transverse modes corresponds to
$c_{V}=2$ approximately. We also note that similarly to solids, plane waves
decay in liquids. The decay mechanisms in solids include anharmonicity,
defects and structural disorder present in, for example, glasses. Despite this
decay, high-temperature specific heat in solids is governed by phonons. In
liquids, the additional decay mechanism is related to atomic jumps ropp .
Nevertheless, the propagation length of high-frequency excitations in liquids
and supercritical fluids is on the order of nanometers, as evidenced by
experiments and modelling hoso1 ; hoso2 ; hoso3 ; prl . This is similar to
room-temperature solids where the lifetime of high-frequency phonons is on the
order of picoseconds and the propagation range is on the order of nanometers
Jain2016  and where disorder and/or defects reduce these values further. This
is also similar to glasses which are structurally similar to liquids ruffle .
Therefore, phonon excitations govern the specific heat in liquids and
supercritical fluids to the same extent they do in solids.

The significance of the propagation range $\lambda_{d}$ reaching about 1 Å at
the inversion point in Fig. 6a is that it corresponds to the shortest distance
in the system (UV cutoff), the interatomic separation on the order of
Angstroms in condensed matter phases. This distance is fixed by fundamental
physical constants in the form of the Bohr radius which, together with the
Rydberg energy, sets the viscosity minimum sciadv . The UV cutoff puts the
upper limit for $k$-points for propagating waves in the system. Recall that
$k_{g}$ in (6) sets the range of transverse waves. When $k_{g}$ becomes
comparable to the largest, Debye, wavevector, all transverse modes disappear
from the system spectrum. We have calculated the Debye wavevector $k_{\rm
D}=(6\pi^{2}n)^{\frac{1}{3}}$ landau , where $n$ is particle density, to be
about 1.0 Å-1 at the inversion point. Equating it to $\frac{1}{2c\tau}$ in (6)
gives $\lambda_{d}=c\tau\approx$0.5 Å. This is consistent with about 1 Å at
the inversion point in Fig. 6a, given the approximate nature of Debye model
and that (6) applies to the linear part of the dispersion relation only but
not to the range where $\omega(k)$ flattens off close to the zone boundary at
$k=k_{\rm D}$.

In the gaslike state, the quantity $c\tau$ increases towards the ideal limit
of $c_{V}=3/2$ in Fig. 6a. In the gaslike state, $c\tau$ is related to the
particle mean free path since the speed of sound $c$ is proportional to the
thermal velocity $v_{\rm{th}}$ and $\tau$ is related to the average time
between particle collisions frenkel . The decrease of heat capacity from 2 to
3/2 in the gaslike regime of fluid dynamics can be explained in terms of
longitudinal modes at short distance in the fluid vanishing as $c\tau$ exceeds
their wavelength ropp . The collapse of all curves in the gaslike regime in
Fig. 6a implies that $c\tau$ is the only parameter necessary to characterise
the loss of these degrees of freedom in this regime.

We reiterate to make explicit the significance of this effect in Fig. 6a :
paths in the supercritical region, separated by orders of magnitude in
temperature and pressure, all sport the same $c_{V}$ vs $c\tau$ curves. The
heat capacity, usually defined as a function $c_{V}(T,P)$ of both temperature
and pressure, is here reduced to a near-universal function of the single
dynamical fluid elasticity length $\lambda_{d}=c\tau$ which sets the range of
propagation of solid-like transverse modes as discussed above. What’s more,
the apex, or the inversion point, of this unified curve takes place at
$c_{V}\approx 1.9$, close to the key value $c_{V}=2$ at the Frenkel line and
to the UV cutoff length of the system.

We plot temperature and pressure points at which $c_{V}=1.9$ at the inversion
point on all paths as stars in Fig. 1. We observe that the location of these
points varies by orders of magnitude on the phase diagram, yet they all
coincide on our master plot in Fig. 6a.

This curve $c_{V}(c\tau)$ therefore represents a dynamically driven transition
between liquidlike and gaslike states which is present across the
supercritical region and is path independent. We call this transition a
“c”-transition due to the shape of Fig. 6a. The inversion point of all
coinciding curves in Fig. 6a is well-defined and therefore serves as an
unambiguous transition point between liquidlike and gaslike states in the
supercritical region. The inversion point does not depend on a theory such as
that underlying the thermodynamic criterion of the Frenkel line $c_{V}=2$ and
the dynamical VAF criterion.

The transition is further illustrated in Fig. 6b: we fit the data in Fig. 6a,
calculate the derivative $\frac{\rm{d}c_{V}}{\rm{d}\lambda_{d}}$ and plot the
derivative vs $c_{V}$, observing the divergence at $c_{V}$ close to the key
value $c_{V}=2$ at the Frenkel line.

An alternative way to collapse the data from different paths is informed by
the theory of liquid thermodynamics. In this theory, the liquid energy at low
temperatures, where transverse waves exist below the Frenkel line, depends on
the ratio of $k_{g}$ in (6) and the Debye wavector $k_{\rm D}$:
$\frac{k_{g}}{k_{\rm{D}}}$ ropp ; prl ; jpcm  (or, alternatively, on
$\frac{\omega_{\rm F}}{\omega_{\rm{D}}}$ for propagating waves ropp , where
$\omega_{\rm F}=\frac{1}{\tau}$ and $\omega_{\rm{D}}$ is Debye frequency). The
physical picture here is that shorter $\tau$ at high temperature results in
progressive disappearance of transverse waves from the system spectrum ropp
(this theory has undergone a detailed and rigorous test fluids  and explains
the decrease of $c_{V}$ with temperature in Fig. 2 (b)). However, this
disappearance of transverse modes can only proceed up to the largest
$k$-point, $k_{\rm{D}}$, set by the UV cutoff in the system as discussed
earlier. As a result, $\frac{k_{g}}{k_{\rm{D}}}$ enters the equation for the
liquid energy. We calculate $k_{\rm{D}}$ as $(6\pi^{2}n)^{\frac{1}{3}}$ landau
and plot $c_{V}$ as a function of dimensionless product $c\tau k_{\rm{D}}$ in
Fig. 7. We observe that the curves from all paths collapse in the region of
large $c_{V}$ down to about $c_{V}=2$, showing that $c_{V}$ below the Frenkel
line depends on the product $c\tau k_{\rm{D}}$ only, as predicted
theoretically. This is followed by the divergence of $c_{V}$ below $c_{V}=2$
in the gaslike state along different paths, presumably because the solid-like
concepts underlying $k_{\rm{D}}$ become progressively less relevant in the
gaslike state at high temperature.

Figure 7: $c_{V}$ as a function of $c\tau k_{\rm{D}}$, where the curves
similarly converge in the liquidlike state at the path-independent point close
to $c_{V}=2$.

The key result of this work is visible in both scaling graphs in Figs. 6a and
7: the data collapse in either large or small-value range of $c_{V}$; the
curves diverge from each other in the other range; and this divergence takes
place near the key value $c_{V}=2$.

As mentioned earlier, the dynamical VAF criterion of the Frenkel line
corresponds to the disappearance of the oscillatory component of particle
motion. In Fig. 8, we plot the line calculated using the VAF criterion,
together with the “c”-transition line determined by ($P$,$T$) at the inversion
point where $c_{V}\approx 1.9$ in Fig. 6. We also plot the critical isochore
for comparison. The Frenkel line from the VAF criterion and the inversion
point are close and run parallel to each other. This serves as self-
consistency check for our theory and implies that the inversion point can
serve as a hallmark and a definition of the supercritical transition between
the liquidlike and gaslike states at the FL. As mentioned earlier, the
inversion point is unambiguously defined in Fig. 6 and does not depend on a
theory such as that underlying the thermodynamic criterion $c_{V}=2$ and the
dynamical VAF criterion.

Figure 8: Phase diagram showing the inversion points of the “c”-transition
corresponding to $c_{V}\approx 1.9$ in Fig. 6, the Frenkel line determined by
the VAF criterion vaf , the critical point, and the critical isochore.

We note that near the critical point, such as along the 100 bar isobar, the
dynamical and thermodynamical properties are strongly affected by near-
critical anomalies deben ; stanley , and the function $c_{V}(c\tau)$ is
affected as a result. The function $c_{V}(c\tau)$ calculated along this path
does not collapse onto the inversion point. On the other hand, our master
curve in Fig. 6 is deeply supercritical and is therefore free of the near-
critical anomalies.

The collapse of all curves up to the key value of about $c_{V}=2$ and
divergence of curves along different paths beyond this value has two further
implications. First, it suggests that the inversion point $c_{V}=2$ is a
special point on the phase diagram. Second, if a thermodynamic property has a
wide crossover, the behavior of different properties strongly depends on the
path taken on the phase diagram. On the other hand, the observed collapse of
all paths at the special inversion point close to $c_{V}=2$ and
$\lambda_{d}=1$ Å indicates either a sharp crossover or a dynamically driven
phase transition related to the “c”-transition between liquidlike and gaslike
states. By sharp we mean more abrupt that previous transitions observed over
the FL (see, e.g., Ref. f1 ), possibly involving discontinuities in higher-
order thermodynamic derivatives. Within the uncertainty set by fluctuations in
our simulations, we do not observe an anomaly of $c_{V}$ in Fig. 2b at
temperatures and pressures corresponding to the inversion point (our
simulations put an upper boundary of about 0.05$k_{\rm B}$ on the value of a
possible anomaly of $c_{V}$). This does not exclude a weak thermodynamic phase
transition, similar to a percolation transition, or a higher-order phase
transition seen in higher derivatives of thermodynamic functions.

In summary, we have discovered a universal, striking, and demonstrative inter-
relation between dynamics and thermodynamics using the specific heat $c_{V}$
and the dynamical parameter of the fluid elasticity length,
$\lambda_{d}=c\tau$. This connection provides a clear and path-independent
transition between liquidlike and gaslike supercritical states, which we call
a “c”-transition. Our “c”-shaped master curve provides an unambiguous and
path-independent criterion for the separation of liquidlike and gaslike
states, calculated from accessible quantities in molecular dynamics
simulations. The collapse onto this master curve occurs in the supercritical
state up to $T=330T_{c}$ and $P=8000P_{c}$, meaning the transition, and the
distinct states it separates, exist over a far larger range of temperatures
and pressures than the boiling line which separates subcritical liquids and
gases. The collapse is indicative of either a sharp crossover or a new phase
transition operating in the supercritical state.

##  IV Acknowledgements

We are grateful to EPSRC and Queen Mary University of London for support and
J. C. Dyre for discussions. This research utilized Queen Mary’s MidPlus
computational facilities, supported by QMUL Research-IT.

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