August 28, 2024

# QUASI-SOLID STATE MICROSCOPIC DYNAMICS
IN EQUILIBRIUM CLASSICAL LIQUIDS:
SELF-CONSISTENT RELAXATION THEORY

A. V. Mokshin  Kazan Federal University, 420008 Kazan, Russia  R. M.
Khusnutdinoff  Kazan Federal University, 420008 Kazan, Russia  Ya. Z. Vilf
Kazan Federal University, 420008 Kazan, Russia  B. N. Galimzyanov  Kazan
Federal University, 420008 Kazan, Russia

###### Abstract

In the framework of the concept of time correlation functions, we develop a
self-consistent relaxation theory of the transverse collective particle
dynamics in liquids. The theory agrees with well-known results in both the
short-wave (free particle dynamics) and the long-wave (hydrodynamic) limits.
We obtain a general expression for the spectral density $C_{T}(k,\omega)$ of
transverse particle current realized in the range of wave numbers $k$. In
domain of microscopic spatial scales comparable to action scale of effective
forces of interparticle interaction, the theory reproduces a transition from a
regime with typical equilibrium liquid dynamics to a regime with collective
particle dynamics where properties similar to solid-state properties appear:
effective shear stiffness and transverse (shear) acoustic waves. In the
framework of the corresponding approximations, we obtain expressions for the
spectral density of transverse particle current for all characteristic regimes
in equilibrium collective dynamics. We obtain expressions for dispersion law
for transverse (shear) acoustic waves and also relations for the kinematic
shear viscosity $\nu$, the transverse speed of sound $v^{(T)}$, and the
corresponding sound damping coefficient $\Gamma^{(T)}$. We compare the
theoretical results with the results of atomic dynamics simulations of liquid
lithium near the melting point.

##  I Introduction

The absence of shear stiffness is one of key physical-mechanical properties of
equilibrium classical liquids distinguishing liquids from solids Frenkel_book
; Bol_Phys_Enciclopedy ; Phys_Enc . This property is manifested in the
inability of liquids to maintain a definite volume form if there are no
boundary surfaces and the ability to take the shape of a filled vessel. The
absence of shear stiffness is also manifested in the fact that in contrast to
crystalline and amorphous solids, liquids are unable to support the
propagation of transverse (so-called shear) oscillation processes
Trachenko2016 ; Tareyeva2018 . In fact, everything occurs for equilibrium
classical liquids on that way only under conditions close to hydrodynamic,
when we consider macroscopic spatial scales and sufficiently large observation
times in corresponding experiments. Thus, for example, in the case of a high-
frequency shear-deformation perturbation under the action of a varying force,
a high-density liquid demonstrates an elastic response typical for solids.
This is clearly seen in experiments measuring the frequency-dependent shear
modulus hypersound_experiments . Such a reaction of a liquid to a perturbation
is fully rather understandable: in simple liquids, this reaction is most clear
expressed when external mechanical excitation has a time scale $\tau^{\prime}$
(where $\tau^{\prime}=2\pi/\omega^{\prime}$ and $\omega^{\prime}$ is the
excitation frequency) comparable to the structural relaxation time $\tau_{s}$.
Moreover, transverse oscillatory processes typical for solids can be
manifested in the collective dynamics of particles of a high-density liquid at
microscopic spatial scales. In this case, the manifestation of microscopic
shear stiffness is due to the action of effective attractive forces in the
interaction between particles of liquid, and the spatial scale $l_{g}$ at
which liquid quasistiffness is manifested must be determined by the size of
the action region of these forces.

The existence of propagating shear waves in compressed noble gases (so-called
Lennard-Jones liquids in the general case) near the triple point was first
confirmed using methods for modeling equilibrium molecular dynamics
Levesque/Verlet_1973 ; Sjogren1978 . The spectra of the computed spectral
density $C_{T}(k,\omega)$ of transverse molecule current for wave numbers $k$
corresponding to spatial scales of several effective linear sizes of the
molecule turned out to contain in expressed component in the terahertz region
with a maximum at the frequency $\omega_{c}^{(T)}\neq 0$ Levesque/Verlet_1973
. The dependence of $\omega_{c}^{(T)}$ on the wave number $k$ produces a
dispersion relation $\omega_{c}^{(T)}(k)$ characterizing the propagation of
transverse oscillatory processes in a liquid. As a rule, there is a gap with
zero values of $\omega_{c}^{(T)}$ in the dispersion law $\omega_{c}^{(T)}(k)$
in the domain of small wave numbers $k$. Obviously, the size $k_{g}$ of this
gap must be related to the spatial scale $l_{g}$ at which a quasistiff shear
response is manifested in a liquid.

The methods of classical and quantum mechanical molecular dynamics simulations
still remain the most appropriate tool for directly quantitatively estimating
almost all effects related to the shear oscillation dynamics of molecules in
both model Donko ; Khrapak2019 ; Ryltsev2013  and real liquids (see, e.g.,
Gonzalez2017 ; Jakse/Bryk2019 ; Jones2016 ; Fomin2019 ; Wang2019 ). Although
direct experimental measurements are difficult, some indirect information
about shear waves in high-density liquids can be obtained from inelastic
neutron scattering (INS) and inelastic x-ray scattering (IXS) experiments
Hosokawa_2015 . In particular, effects due to the transverse oscillation
dynamics of particles can be manifested in the contour of the dynamic
structure factor $S(k,\omega)$ measured in INS and IXS experiments
Egelstaff_book ; Burkel_review .

We note that key components (a central Rayleigh and two symmetric Brillouin
components) of the scattering law in a liquid at microscopic spatial scales
accessible by INS and IXS experimental techniques are not separated, and the
contribution to the general scattering law $S(k,\omega)$ due to transverse
oscillation dynamics is weakly expressed and is not recognized in the form of
a separate component. Features typical for transverse oscillatory processes
must obviously appear in the spectra $S(k,\omega)$ in a frequency range
between the Rayleigh and Brillouin peaks. In fact, only continuations of the
Brillouin and Rayleigh peaks are observed in the experimental spectra
$S(k,\omega)$ in this frequency range. Consequently, directly interpreting the
experimental data is complicated, and the resulting conclusions from this
interpretation can be ambiguous Hosokawa_Ga ; Hosokawa2013 ; Monaco_PNAS ;
Monaco_PRB .

We mention some currently well-known theoretical models and approaches
connected with describing transverse collective dynamics: the viscoelasticity
model MacPhail/Kivelson ; Bryk_PRE2000 , the generalized collective mode
method Mryglod_CMP1994 , the Maxwell-Frenkel model for describing the
dispersion relation of transverse collective modes Trachenko2017 ;
Baggioli2020 , and the approach for recovering the dispersion dependences of
longitudinal and transverse collective modes based on separate and joint mode
analysis (the “two oscillator model”) Yurchenko2019 ; Yurchenko_SC_2019 ;
Yurchenko_1 . It is also remarkable that the presence of transverse
excitations can lead to the appearance of so-called positive dispersion of
longitudinal sound: the observed/measured speed of sound in a certain range of
finite wave numbers is higher than the hydrodynamic values Fomin_JPCM_2016 ;
Brazhkin_2018 .

Here, we show that the transverse collective dynamics in simple liquids can be
described in the framework of a self-consistent relaxation theory. This
theoretical description is a direct development of the idea of the time-scale
invariance of relaxation processes in liquids Mokshin_PRE2001 ; Mokshin2002 ,
based on which a theory of local density fluctuations in liquids was developed
and the use of this theory for analyzing INS and IXS experimental data was
demonstrated in liquid lithium Mokshin_JPCM2018 , sodium Mokshin_JPCM2003 ;
Mokshin_JCP2004 , cesium Mokshin_PRE2001 ; Mokshin2002 , aluminum
Mokshin_JETP2006 ; Mokshin_JPCM2007 , and gallium Khusnutdinoff2020 . We
present the theoretical formalism in Sec. II and compare the theoretical
results with data from equilibrium atomic dynamics simulations of liquid
lithium in Sec. III.

##  II Theoretical formalism

In the framework of the general statistical mechanics approach, we can regard
an equilibrium liquid as a multiparticle system, an ensemble of atoms and
molecules Ryzhov_UFN2008 . In this case, it is convenient to use the
mathematical apparatus of correlation functions, distribution functions,
moments, and cumulants of these functions to describe both structural and
dynamical properties of this liquid Hansen/McDonald ; Zwanzig2001 ;
Mokshin/Yulmetyev ; Klumov .

We consider an isotropic system comprising $N$ identical classical particles
of mass $m$ in a volume $V$. The coordinates $\mathbf{r}_{j}$ and velocities
$\mathbf{v}_{l}$, $j,\,l=1,\,\ldots,\,N$, of these particles determine a
$6N$-dimensional phase space, where we can define a certain dynamical variable
$A(\mathbf{r}_{1},\,\ldots,\,\mathbf{r}_{N},\,\mathbf{v}_{1},\,\ldots,\,\mathbf{v}_{N})$.
Let the system Hamiltonian $H$ be given. We can then determine the mean
$\left\langle A\right\rangle$ in terms of the phase space density distribution
$\rho\propto\exp\\{-(H-\mu N)/(k_{B}T)\\}$, where $T$ and $\mu$ are the
temperature and chemical potential. Moreover, the Hamiltonian $H$ determines
the evolution of the dynamical variable $A$:

$\frac{\partial}{\partial
t}A(t)=\left\\{H,A(t)\right\\}=i\hat{\mathcal{L}}A(t),$ (1)

where $\\{...\\}$ is Poisson bracket, $\hat{\mathcal{L}}$ is the Liouville
operator, which is Hermitian Zwanzig2001 ; Balucani2003 . In the case where
the interaction potential $u(r)$ between particles is pairwise and spherical,
we can write the Liouville operator in the form

$\hat{\mathcal{L}}=-i\left\\{\frac{1}{m}\sum_{j=1}^{N}\left(\mathbf{p}_{j}\cdot\nabla_{j}\right)-\sum_{l>j=1}^{N}\nabla_{j}u(r_{jl})\left(\nabla_{\mathbf{p}_{j}}-\nabla_{\mathbf{p}_{l}}\right)\right\\}.$
(2)

where $\mathbf{p}_{j}$ is the momentum of the $j$th particle and $\nabla_{j}$
and $\nabla_{\mathbf{p}_{j}}$ are the coordinate and momentum gradients.

###  II.1 Dynamical variables

Let the particle current occur in some region of the inverse space of the wave
vectors k and be determined by the velocities of all particles belonging to
that region. We can write the expression for the full particle current

$\displaystyle\textbf{j}_{k}(t)=\sum_{l=1}^{N}\textbf{v}_{l}(t)\;\mathrm{e}^{-i\textbf{k}\cdot\textbf{r}_{l}(t)},$
(3)

where $k=\left|\textbf{k}\right|$ is the wave number and $\textbf{r}_{l}(t)$
is the radius vector characterizing the trajectory of the $l$th particle.
Assuming that the wave vector k is directed along the coordinate axis $OZ$, we
can introduce the corresponding expressions for the longitudinal current

$\displaystyle
j_{k}^{(z)}(t)=\sum_{l=1}^{N}v_{l}^{(z)}(t)\;\mathrm{e}^{-i\textbf{k}\cdot\textbf{r}_{l}(t)}$
(4)

and the transverse current

$\displaystyle
j_{k}^{(x)}(t)=\sum_{l=1}^{N}v_{l}^{(x)}(t)\;\mathrm{e}^{-i\textbf{k}\cdot\textbf{r}_{l}(t)},$
(5)

and

$\displaystyle
j_{k}^{(y)}(t)=\sum_{l=1}^{N}v_{l}^{(y)}(t)\;\mathrm{e}^{-i\textbf{k}\cdot\textbf{r}_{l}(t)},$
(6)

where $v^{(x)}$, $v^{(y)}$ and $v^{(z)}$ are the velocity components for the
$l$th particle,
$\mathbf{v}_{l}=v_{l}^{(x)}\mathbf{i_{1}}+v_{l}^{(y)}\mathbf{i_{2}}+v_{l}^{(z)}\mathbf{i_{3}}$
and
$\textbf{j}_{k}=j_{k}^{(x)}\mathbf{i_{1}}+j_{k}^{(y)}\mathbf{i_{2}}+j_{k}^{(z)}\mathbf{i_{3}}$.
The quantities $j_{k}^{(z)}$, $j_{k}^{(x)}$ and $j_{k}^{(y)}$ and similarly
for $\textbf{j}_{k}$ have the meaning of dynamical variables characterizing
the collective dynamics of the multiparticle system. The transverse current
components $j_{k}^{(x)}$ and $j_{k}^{(y)}$ in a structurally isotropic system
(dense gases, liquids, amorphous solids) are obviously equal.

It is convenient in considering transverse collective dynamics to choose the
transverse current $A_{1}^{T}$ as initial dynamical variable:

$A_{1}^{T}(k)=j_{k}^{(x)}=j_{k}^{(y)}.$ (7)

Using the orthogonalization procedure Reed_Book

$\displaystyle A_{l+1}^{T}(k)$ $\displaystyle=$ $\displaystyle
i\hat{\mathcal{L}}A_{l}^{T}(k)+\frac{\left\langle\left|A_{l+1}^{T}(k)\right|^{2}\right\rangle}{\left\langle\left|A_{l}^{T}(k)\right|^{2}\right\rangle}A_{l-1}^{T}(k),$
(8) $\displaystyle l$ $\displaystyle=$ $\displaystyle 1,\;2,\;3,...;\ \
A_{0}^{T}\equiv 0,$

we obtain the set

$\textbf{A}^{T}(k)=\left\\{A_{1}^{T}(k),\;A_{2}^{T}(k),\;A_{3}^{T}(k),\;\ldots,\;A_{l}^{T}(k),\;\ldots\right\\},$
(9)

of dynamical variables satisfying the orthonormalization condition

$\displaystyle\left\langle A_{i}^{T}(-k)A_{j}^{T}(k)\right\rangle$
$\displaystyle=$
$\displaystyle\delta_{i,j}\left\langle\left|A_{j}^{T}(k)\right|^{2}\right\rangle,$
(10) $\displaystyle i,\;j$ $\displaystyle=$ $\displaystyle
1,\;2,\;3,\;\ldots.$

where the angle brackets denote averaging over the statistical ensemble and
$\delta_{i,j}$ is the Kronecker symbol.

It follows from expressions (7) and (8) that the dynamical variable
$A_{2}^{T}(k)$ is related to the acceleration and kinetic energy in the
transverse collective particle dynamics:

$A_{2}^{T}(k)=\sum_{l=1}^{N}\dot{v}_{l}^{(x)}\;\mathrm{e}^{-i\mathbf{k}\cdot\mathbf{r}_{l}}-i\sum_{l=1}^{N}k_{x}\left(v_{l}^{(x)}\right)^{2}\;\mathrm{e}^{-i\mathbf{k}\cdot\mathbf{r}_{l}},$
(11)

For the third dynamical variable, we obtain

$\displaystyle A_{3}^{T}(k)$ $\displaystyle=$
$\displaystyle\sum_{l=1}^{N}\ddot{v}_{l}^{(x)}\;\mathrm{e}^{-i\mathbf{k}\cdot\mathbf{r}_{l}}-3i\sum_{l=1}^{N}k_{x}\dot{v}_{l}^{(x)}v_{l}^{(x)}\;\mathrm{e}^{-i\mathbf{k}\mathbf{r}_{l}}$
(12) $\displaystyle-$
$\displaystyle\sum_{l=1}^{N}k_{x}^{2}\left(v_{l}^{(x)}\right)^{3}\;\mathrm{e}^{-i\mathbf{k}\cdot\mathbf{r}_{l}}+\frac{\left\langle\left|A_{2}^{T}(k)\right|^{2}\right\rangle}{\left\langle\left|A_{1}^{T}(k)\right|^{2}\right\rangle}\;\sum_{l=1}^{N}{v}_{l}^{(x)}\;\mathrm{e}^{-i\mathbf{k}\cdot\mathbf{r}_{l}}.$

The dynamic variable $A_{3}^{T}(k)$ characterizes the change in the particle
acceleration.

###  II.2 Correlation functions, frequency moments, and relaxation parameters

We define time correlation functions (TCFs) for the set $\textbf{A}^{T}(k)$ of
dynamical variables:

$M_{l}^{T}(k,t)=\frac{\left\langle
A_{l}^{T}(-k,0)A_{l}^{T}(k,t)\right\rangle}{\left\langle\left|A_{l}^{T}(k,0)\right|^{2}\right\rangle},\
\ \ l=1,\;2,\;\ldots,$ (13)

where

$M_{1}^{T}(k,t)\equiv C_{T}(k,t).$

The TCFs $M_{l}^{T}(k,t)$ have the properties Plakida2005 ; Lee2000

$M_{l}^{T}(k,t)\bigg{|}_{t=0}=1,$ (14a) $\left\\{\begin{matrix}\text{
}\cfrac{d^{p}}{dt^{p}}M_{l}^{T}(k,t)\bigg{|}_{t=0}=0,\text{$p$ is odd,}\\\
\cfrac{d^{p}}{dt^{p}}M_{l}^{T}(k,t)\bigg{|}_{t=0}\neq 0,\text{$p$ is
even,}\end{matrix}\right.$ (14b) $\left|M_{l}^{T}(k,t)\right|\leq 1,$ (14c)
$\lim_{t\rightarrow\infty}M_{l}^{T}(k,t)=0.$ (14d)

Properties (14a), (14b) and (14c) reveal that the TCF $M_{l}^{T}(k,t)$ takes
its maximum value $1$ at the initial instant. Property (14d) is related to
ergodicity and indicates that the correlation weakens with time.

The TCF of the transverse current is then

$\displaystyle M_{1}^{T}(k,t)=C_{T}(k,t)=\frac{\left\langle
j_{-k}^{(x)}(0)j_{k}^{(x)}(t)\right\rangle}{\left\langle\left|j_{k}^{(x)}(0)\right|^{2}\right\rangle}.$
(15)

This quantity is related to the spectral density of transverse current TCF by

$\displaystyle
C_{T}(k,\omega)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\;C_{T}(k,t)\;\mathrm{e}^{i\omega
t}\;dt.$ (16)

The quantity $\left\langle\left|j_{k}^{(x)}(0)\right|^{2}\right\rangle$ in
(15) is equal to

$\left\langle\left|j_{k}^{(x)}(0)\right|^{2}\right\rangle=\omega_{T}^{(0)}(k)=\frac{1}{N}\sum_{l=1}^{N}\left(v_{l}^{(x)}\right)^{2}=\frac{k_{B}T}{m}.$
(17)

We define the frequency moments of the spectral density $C_{T}(k,\omega)$ as

$\displaystyle\left\langle\omega_{T}^{(l)}(k)\right\rangle=\left(-i\right)^{l}\left.\frac{d^{l}}{dt^{l}}C_{T}(k,t)\right|_{t=0}=\frac{\int_{-\infty}^{\infty}\omega^{l}C_{T}(k,\omega)\;d\omega}{\int_{-\infty}^{\infty}C_{T}(k,\omega)\;d\omega},\
\ \ l=1,\;2,\;\ldots.$ (18)

As follows from property (14b), only even frequency moments
$\omega_{T}^{(2)}(k)$, $\omega_{T}^{(4)}(k)$, $\omega_{T}^{(6)}(k)$, $\ldots$
take nonzero values.

We introduce the frequency relaxation parameters

$\displaystyle\Delta_{T,\;l+1}^{2}(k)=\frac{\left\langle\left|A_{l+1}^{T}(k)\right|^{2}\right\rangle}{\left\langle\left|A_{l}^{T}(k)\right|^{2}\right\rangle},\
\ \ l=1,\;2,\;3,\;\ldots.$ (19)

The super- and subscripts $T$ mean that parameters are computed based on the
dynamical variables related to the transverse dynamics, and the index $l$ is
the order number of the parameter according to set $\mathbf{A}^{T}(k)$ of
dynamical variables.

The parameters $\Delta_{T,\;l}^{2}(k)$ have the following properties
Mokshin_JPCM2018 .111The parameters $\Delta_{T,\;l}^{2}(k)$ are called static
correlation functions in the Mori formalism Balucani2003  and denote recursion
coefficients in the M.H. Lee’s recurrent relation method Lee2000 . First, the
$l$th-order frequency relaxation parameter characterizes the relaxation time
$\tau_{T,\;l}(k)$ of a process related to the dynamical variable
$A_{l}^{T}(k)$:

$\tau_{T,\;l}(k)=1/\sqrt{\Delta_{T,\;l}^{2}(k)}.$ (20)

Hence, for example, as follows from (7), (11) and (19), the frequency
relaxation parameter $\Delta_{T,\;2}^{2}(k)$ in the short-wave limit related
to free particle dynamics characterizes mean travel time of a particle at the
spatial scale $\ell=2\pi/k$:

$\tau_{T,\;2}(k)=1/\sqrt{k_{x}^{2}\;(k_{B}T)/m}=1/\sqrt{(k_{x}\bar{v}_{x})^{2}},$
(21)

where $k_{x}\equiv k$ and $\bar{v}_{x}$ is the mean velocity of the particle.
This property was discussed in detail in paper Mokshin2005 .

Secondly, the frequency relaxation parameters are related to the frequency
moments Mokshin2015 :

$\Delta_{T,\;2}^{2}(k)=\langle\omega_{T}^{(2)}(k)\rangle,$ (22a)
$\Delta_{T,\;3}^{2}(k)=\frac{\langle\omega_{T}^{(4)}(k)\rangle}{\langle\omega_{T}^{(2)}(k)\rangle}-\langle\omega_{T}^{(2)}(k)\rangle,$
(22b)
$\Delta_{T,\;4}^{2}(k)=\frac{\langle\omega_{T}^{(6)}(k)\rangle\langle\omega_{T}^{(2)}(k)\rangle-(\langle\omega_{T}^{(4)}(k)\rangle)^{2}}{\langle\omega_{T}^{(4)}(k)\rangle\langle\omega_{T}^{(2)}(k)\rangle-(\langle\omega_{T}^{(2)}(k)\rangle)^{3}}.$
(22c)

Expressions (22) are also called the sum rules of the spectrum
$C_{T}(k,\omega)$. The mathematical meaning of this property is that a unique
set of frequency relaxation parameter values can be related to a particular
form of the spectrum $C_{T}(k,\omega)$. Consequently, the sum rules can be
used to verify the correctness of a proposed theoretical model of the spectrum
$C_{T}(k,\omega)$. In this case, the frequency relaxation parameter values
obtained based on some theoretical model are related to the parameter values
obtained from the definition of the frequency moments by numerically computing
the integral expressions for the frequency moments based on an experimental
spectrum $C_{T}(k,\omega)$ [see the second equality in (18)].

Finally, as follows from (7), (11) and (12), the frequency relaxation
parameters can be expressed in terms of microscopic characteristics of the
system, such as the mean particle velocity, the interaction potential between
particles, and particle distribution functions Balucani2003 . Hence, for
example, in the case of the set of dynamical variables defined by (7), (11)
and (12), the frequency relaxation parameter $\Delta_{T,\;2}^{2}(k)$ is

$\Delta_{T,\;2}^{2}(k)=\left(\frac{k_{B}T}{m}\right)^{2}k^{2}+\frac{\rho}{m}\int\left[1-\cos(kz)\right]\frac{\partial^{2}u(r)}{\partial
x^{2}}g(r)\;d^{3}\mathbf{r},$ (23)

where $\rho$ is the quantitative particle density and $g(r)$ is the radial
distribution function of particles in the system. We can also obtain
microscopic expressions for the other frequency relaxation parameters
$\Delta_{T,\;3}^{2}(k)$, $\Delta_{T,\;4}^{2}(k)$, $\ldots$ Balucani2003 .
These microscopic expressions are usually not used for direct numerical
calculations, because they contain multiparticle distribution functions and
the well-known problem of decoupling an equation chain for the particle
distribution functions must be solved for corresponding computations
Bogolyubov_book . We note that these frequency relaxation parameters can also
be computed based on basic definition (19) with configuration data
(trajectories and particle velocities) obtained, for example, from molecular
dynamic modeling (see the appendix in Mokshin_JPCM2018 ). The following
analysis of the frequency relaxation parameters values allows defining such
physical characteristic of the system as the sound propagation velocity, the
sound damping coefficient, the mean particle velocity, etc. Mokshin_JPCM2018 .

###  II.3 The function $C_{T}(k,t)$ in the hydrodynamic limit

In the case of transverse collective dynamics in a liquid, solutions are
generally known in only two limit cases: in the hydrodynamic limit ($k\to 0$)
and in the free-moving particle limit (large wave numbers $k$).

In the hydrodynamic limit, the time evolution of the transverse current
$j_{k}^{(x)}$ is given by

$\frac{\partial}{\partial t}j_{k}^{(x)}(t)=-\nu k^{2}j_{k}^{(x)}(t),$ (24)

where $\nu$ is kinematic shear viscosity. The solution of this equation has
the form

$\displaystyle j_{k}^{(x)}(t)=j_{k}^{(x)}(0)\;\exp{(-\nu k^{2}t)}.$ (25)

Multiplying (24) by the complex conjugate quantity

$j_{-k}^{(x)}(0)\frac{1}{\left\langle\left|j_{k}^{(x)}(0)\right|^{2}\right\rangle}$

and averaging over the ensemble, we obtain an equation for the TCF
$C_{T}(k,t)$

$\frac{\partial}{\partial t}C_{T}(k,t)=-\nu k^{2}C_{T}(k,t).$ (26)

The solution of (26) has the form

$\displaystyle C_{T}(k,t)=\exp{(-\nu k^{2}t)}.$ (27)

Acting on (27) with the Fourier transform operator

$FT\\{f\\}(\omega)=f(\omega)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\exp(i\omega
t)f(t)\,dt,$

we obtain an expression for spectral density of the TCF of the transverse
current:

${C}_{T}(k,\omega)=\frac{1}{\pi}\frac{\nu k^{2}}{\omega^{2}+\left(\nu
k^{2}\right)^{2}}.$ (28)

Therefore, in the hydrodynamic limit, the TCF $C_{T}(k,t)$ is characterized by
an exponentially decaying time-dependence with the relaxation time
$\tau=1/(\nu k^{2})$ and a Gaussian dependence on $k$. The absence of high-
frequency components in the spectrum ${C}_{T}(k,\omega)$ defined by (28)
indicates the absence of transverse acoustic waves in the hydrodynamic mode.

###  II.4 The function $C_{T}(k,t)$ in the short-wave limit

We consider the short-wave limit, where the particle dynamics occurs on
spatial scales comparable to the mean free path. In this case, the time and
spatial scales are so small that particles in fact demonstrate free motion. In
the case of free motion of an arbitrary $i$th particle, its velocity is
constant:

$v_{i}^{(x)}(t)=v_{i}^{(x)}.$

For the TCF of the transverse current, we have Hansen/McDonald

$\displaystyle C_{T}(k,t)$ $\displaystyle=$
$\displaystyle\frac{m}{k_{B}T}\frac{1}{N}\sum_{i,j=1}^{N}\;\left(v_{i}^{(x)}\right)^{2}\;\exp\left(-i\textbf{k}\cdot\left[\textbf{r}_{j}(t)-\textbf{r}_{i}(0)\right]\right)$
(29) $\displaystyle=$
$\displaystyle\exp{\left(-\frac{k_{B}T}{2m}\;k^{2}t^{2}\right)}.$

For the spectral density ${C}_{T}(k,\omega)$, we find

$\displaystyle{C}_{T}(k,\omega)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{m}{2\pi\;k_{B}T\;k^{2}}}\;\exp{\left(-\frac{m\;\omega^{2}}{2k_{B}Tk^{2}}\right)}$
(30) $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2\pi}}\frac{1}{\bar{v}_{x}k}\;\exp{\left(-\frac{\omega^{2}}{2\bar{v}_{x}^{2}k^{2}}\right)}.$

Similarly to (28), relation (30) defines a spectrum with no high-frequency
components.

###  II.5 Transverse current in a range of wave numbers

From the Liouville equations of motion for the dynamical variables
$A_{1}^{T}(k)$, $A_{2}^{T}(k)$, $A_{3}^{T}(k)$ and $A_{4}^{T}(k)$ [see
expressions (7), (11) and (12)] we can obtain exactly two interrelated kinetic
integro-differential equations containing the TCFs $C_{T}(k,t)$,
$M_{2}^{T}(k,t)$, $M_{3}^{T}(k,t)$ and $M_{4}^{T}(k,t)$ of these variables
Mokshin2015 ; Lee2000 ; Zwanzig2001 :

$\frac{\partial^{2}}{\partial
t^{2}}C_{T}(k,t)+\Delta_{T,\;2}^{2}(k)C_{T}(k,t)+\Delta_{T,\;3}^{2}(k)\int_{0}^{t}M_{3}^{T}(k,t-\tau)\frac{\partial}{\partial\tau^{\prime}}C_{T}(k,{\tau}^{\prime})d\tau=0,$
(31a) $\frac{\partial^{2}}{\partial
t^{2}}M_{2}^{T}(k,t)+\Delta_{T,\;3}^{3}(k)M_{2}^{T}(k,t)+\Delta_{T,\;4}^{2}(k)\int_{0}^{t}M_{4}^{T}(k,t-\tau)\frac{\partial}{\partial\tau^{\prime}}M_{2}^{T}(k,{\tau}^{\prime})d\tau=0.$
(31b)

Applying the Laplace transformation

$LT\\{f\\}(s)=\tilde{f}(s)=\int_{0}^{\infty}\exp(-st)f(t)dt,\ \ \ s=i\omega,$
(32)

to Eqs. (31) and expressing the Laplace image of the transverse current TCF
$\widetilde{C}_{T}(k,s)$, we find

$\displaystyle\widetilde{C}_{T}(k,s)=\cfrac{1}{s+\cfrac{\Delta_{T,\;2}^{2}(k)}{s+\cfrac{\Delta_{T,\;3}^{2}(k)}{s+\Delta_{T,\;4}^{2}(k)\widetilde{M}_{4}^{T}(k,s)}}}=\cfrac{1}{s+\cfrac{\Delta_{T,\;2}^{2}(k)}{s+\cfrac{\Delta_{T,\;3}^{2}(k)}{s+\cfrac{\Delta_{T,\;4}^{2}(k)}{s+\ddots}}}}.$
(33)

The quantity $\widetilde{C}_{T}(k,s)$ is related to the spectral density
$C_{T}(k,\omega)$ by

$C_{T}(k,\omega)=\frac{1}{\pi}\;\mathrm{Re}\left[\widetilde{C}_{T}(k,s=i\omega)\right].$
(34)

The key idea of a self-consistent relaxation theory of equilibrium liquids is
that the description of the equilibrium dynamics in the whole range of wave
numbers $k$ is realized using a dynamical variable set $\textbf{A}(k)$ related
to the variables in the hydrodynamic theory Mokshin_JPCM2018 . Namely, the
dynamical variable set $\textbf{A}(k)$ either corresponds exactly to the
hydrodynamic quantity or contains the hydrodynamic quantity in its expression
(see, e.g., the discussion in Sec. II.1)). In the case of describing
transverse collective dynamics, the variables in the corresponding
hydrodynamic equations are related to the dynamical variables $A_{1}^{T}(k)$
and $A_{2}^{T}(k)$.

In the hydrodynamic limit $k\to 0$, variables such as current and energy vary
slowly with large relaxation times Gotze2009 . This allows excluding other
variables from consideration using a simple Markov approximation in the
kinetic equations Resibua/Lener . In accordance with the self-consistent
relaxation theory, typical relaxation times of processes related to the
hydrodynamic variables and other variables also differ, and this difference
can be significant. But the description is reduced not by neglecting
“nonhydrodynamic” variables but by assuming the their characteristic time
scales are comparable:

$\displaystyle
1/\sqrt{\Delta_{T,\;l}^{2}(k)}=1/\sqrt{\Delta_{T,\;l+1}^{2}(k)}=1/\sqrt{\Delta_{T,\;l+2}^{2}(k)},\
\ \ l\in\mathbb{N}^{*}.$ (35)

In case of transverse collective particle dynamics with the extended variable
set containing the energy current $A_{3}^{T}(k)$ in addition to $A_{1}^{T}(k)$
and $A_{2}^{T}(k)$, we have $l=4$. We note that a completely equivalent
variable set (the density $A_{0}(k)$, longitudinal current $A_{1}(k)$, energy
$A_{2}(k)$, and energy current $A_{3}(k)$ is realized in describing
longitudinal collective particle dynamics in equilibrium liquids
Mokshin_JPCM2018 .

From (33) with (32) taken into account, we obtain expressions for the Laplace
images of the TCFs:

$\displaystyle\widetilde{M}_{3}^{T}(k,s)=\frac{-s\pm\sqrt{s^{2}+4\Delta_{T,\;4}^{2}(k)}}{2\Delta_{T,\;4}^{2}(k)},$
(36a)
$\widetilde{M}_{2}^{T}(k,s)=\frac{2\Delta_{T,\;4}^{2}(k)}{s[2\Delta_{T,\;4}^{2}(k)-\Delta_{T,\;3}^{2}(k)]\pm\Delta_{T,\;3}^{2}(k)\sqrt{s^{2}+4\Delta_{T,\;4}^{2}(k)}},$
(36b) and $\displaystyle\widetilde{C}_{T}(k,s)$ $\displaystyle=$
$\displaystyle\widetilde{M}_{1}^{T}(k,s)=$ $\displaystyle=$
$\displaystyle\frac{s[2\Delta_{T,\;4}^{2}(k)-\Delta_{T,\;3}^{2}(k)]\pm\Delta_{T,\;3}^{2}(k)\sqrt{s^{2}+4\Delta_{T,\;4}^{2}(k)}}{2\Delta_{T,\;2}^{2}(k)\Delta_{T,\;4}^{2}(k)+s^{2}[2\Delta_{T,\;4}^{2}(k)-\Delta_{T,\;3}^{2}(k)]\pm
s\Delta_{T,\;3}^{2}(k)\sqrt{s^{2}+4\Delta_{T,\;4}^{2}(k)}}.$

From (36) and (34), we then directly obtain an expression for the spectral
density of the transverse particle current:

$\displaystyle
C_{T}(k,\omega)=\frac{1}{2\pi}\;\frac{\Delta_{T,\;2}^{2}(k)\Delta_{T,\;3}^{2}(k)}{\Delta_{T,\;4}^{2}(k)-\Delta_{T,\;3}^{2}(k)}\;\frac{\pm\sqrt{4\Delta_{T,\;4}^{2}(k)-\omega^{2}}}{\omega^{4}+\mathcal{A}_{1}^{T}(k)\omega^{2}+\mathcal{A}_{2}^{T}(k)},$
(37a) where
$\displaystyle\mathcal{A}_{1}^{T}(k)=\frac{\Delta_{T,\;3}^{4}(k)+\Delta_{T,\;2}^{2}(k)\Delta_{T,\;3}^{2}(k)-2\Delta_{T,\;2}^{2}(k)\Delta_{T,\;4}^{2}(k)}{\Delta_{T,\;4}^{2}(k)-\Delta_{T,\;3}^{2}(k)},$
(37b)
$\displaystyle\mathcal{A}_{2}^{T}(k)=\frac{\Delta_{T,\;2}^{4}(k)\Delta_{T,\;4}^{2}(k)}{\Delta_{T,\;4}^{2}(k)-\Delta_{T,\;3}^{2}(k)}.$
(37c)

Figure 1: Top: Schematic illustrating the dispersion law of similar transverse
acoustic oscillations $\omega_{c}^{(T)}(k)$ and the static structure factor
$S(k)$ of some equilibrium liquid. Bottom: Schematic showing the form of the
spectrum of the transverse current $C_{T}(k,\omega)$ for the domain of
intermediate values of the wave number $k$, for transition regime, and for the
domain of generalized hydrodynamics (the small-$k$ limit).

As follows from (37), the contour $C_{T}(k,\omega)$ of the frequency spectrum
is determined by the frequency parameters $\Delta_{T,\;2}^{2}(k)$,
$\Delta_{T,\;3}^{2}(k)$ and $\Delta_{T,\;4}^{2}(k)$ and is consequently
determined by the structural features of the system and the interparticle
interaction and also depends on the thermodynamic state of the system.

We can distinguish characteristic regimes in the transverse collective
particle dynamics of an equilibrium liquid, which are conveniently considered
in correspondence with the dispersion law of similar transverse acoustic
oscillations $\omega_{c}^{(T)}(k)$ and with a static structure factor $S(k)$
(see Fig. 1 (top)). We have the following c regimes as the wave number
increases on the $k$ scale: the hydrodynamic regime ($k\to 0$), generalized
hydrodynamics, the transition regime, the domain of intermediate values of
$k$, and the short-wave limit (not shown in Fig. 1).

Short-wave limit. The particle dynamics in the short-wave limit is
characterized by a single parameter, the mean particle velocity $\bar{v}_{x}$.
The frequency relaxation parameters are related by the recurrence relation

$\Delta_{T,\;l+2}^{2}(k)=(l+1)\;\Delta_{T,\;2}^{2}(k),\ \ \ l=1,\;2,\;\ldots,$
(38)

where the equality

$\Delta_{T,\;2}^{2}(k)=(k\bar{v}_{x})^{2}$

follows from (23) if we neglect the interparticle interaction. In this case,
expression (37) for the spectral density $C_{T}(k,\omega)$ with (38) taken
into account becomes

$C_{T}(k,\omega)=\sqrt{\frac{1}{2\pi\;\Delta_{T,\;2}^{2}(k)}}\;\exp{\left(-\frac{\omega^{2}}{2\Delta_{T,\;2}^{2}(k)}\right)},$
(39)

which corresponds exactly to the correct model result (30).

Domain of intermediate values of k. The domain of intermediate values of $k$
is realized for wave numbers corresponding to spatial scales comparable to the
effective action scale of the interaction potential between particles. This
wave-number domain is shown in dark gray in Fig. 1 (top). At these spatial
scales, shear stiffness appears in a liquid, and quasi-solid-state collective
dynamics supporting transverse oscillatory processes are realized. In this
regime, the spectral density $C_{T}(k,\omega)$ is a high-frequency doublet:
two expressed symmetric maximums located at the frequencies
$\pm\;\omega_{c}^{(T)}$ and a minimum at the frequency zero ($\omega=0$) (Fig.
1 (bottom)). Moreover, the dispersion $\omega_{c}^{(T)}=\omega_{c}^{(T)}(k)$
of transverse collective excitations in an equilibrium high-density liquid
near the melting point must be extrapolated to the domain of small wave
numbers in the form of the usual linear relation

$\omega_{c}^{(T)}(k)=v^{(T)}k,$ (40)

where $v^{(T)}$ is propagation velocity of transverse sound waves.

The frequency relaxation parameter $\Delta_{T,\;4}^{2}(k)$ characterizes the
minimum time scale $\tau_{T,\;4}(k)=1/\sqrt{\Delta_{T,\;4}^{2}(k)}$ at which
the transverse collective particle dynamics is realized. Consequently, for the
frequency domain in which we are interested, we can write

$\omega^{2}\ll\Delta_{T,\;4}^{2}(k).$ (41)

Expression (37) then becomes

$C_{T}(k,\omega)=\frac{1}{\pi}\;\frac{\Delta_{T,\;2}^{2}(k)\Delta_{T,\;3}^{2}(k)\sqrt{\Delta_{T,\;4}^{2}(k)}}{\Delta_{T,\;4}^{2}(k)-\Delta_{T,\;3}^{2}(k)}\;\frac{1}{\omega^{4}+\mathcal{A}_{1}^{T}(k)\omega^{2}+\mathcal{A}_{2}^{T}(k)},$
(42)

where the coefficients $\mathcal{A}_{1}^{T}(k)$ and $\mathcal{A}_{2}^{T}(k)$
are defined by (37b) and (37c).

We note several important properties of expression (42). First, because the
spectral density can take only nonnegative values, $C_{T}(k,\omega)\geq 0$,
the relation between the frequency parameters $\Delta_{T,\;3}^{2}(k)$ and
$\Delta_{T,\;4}^{2}(k)$

$\Delta_{T,\;4}^{2}(k)\geq\Delta_{T,\;3}^{2}(k),$ (43)

must be satisfied, which completely agrees with condition (41) for the minimum
time scale.

Second, for a fixed $k$, the form of the spectrum $C_{T}(k,\omega)$ is defined
by the biquadratic polynomial in the denominator in (42). Almost all features
of the spectrum $C_{T}(k,\omega)$ can be rather easily determined in the
framework of the usual mathematical analysis of (42).

Further, solving the problem of the existence of extremums, we obtain the
dispersion law from (42):

$\omega_{c}^{(T)}(k)=\sqrt{-\frac{\mathcal{A}_{1}^{T}(k)}{2}}$ (44a) or
$\omega_{c}^{(T)}(k)=\sqrt{\Delta_{T,\;2}^{2}(k)-\frac{\Delta_{T,\;3}^{2}(k)[\Delta_{T,\;3}^{2}(k)-\Delta_{T,\;2}^{2}(k)]}{2[\Delta_{T,\;4}^{2}(k)-\Delta_{T,\;3}^{2}(k)]}}.$
(44b)

As can be seen from (44a), the high-frequency doublet in the spectrum
$C_{T}(k,\omega)$ appears under the condition $\mathcal{A}_{1}^{T}(k)<0$ [see
Fig. 1], which is similar to the existence condition for high-frequency
excitations in accordance with the viscoelastisity model (see p. 268 in
Hansen/McDonald ):

$\omega_{c}^{(T)}(k)=\sqrt{\Delta_{T,\;2}^{2}(k)-\frac{1}{2\tau_{t}^{2}(k)}}$
(45)

with the relaxation time $\tau_{t}(k)$. Comparing (44b) and (45), we find the
definition of the relaxation time in the framework of the self-consistent
relaxation theory with expression (42):

$\tau_{t}^{2}(k)=\frac{\Delta_{T,\;3}^{2}(k)[\Delta_{T,\;3}^{2}(k)-\Delta_{T,\;2}^{2}(k)]}{\Delta_{T,\;4}^{2}(k)-\Delta_{T,\;3}^{2}(k)}.$
(46)

It is important that the quantity $\tau_{t}^{2}(k)$ remains undefined in the
viscoelasticity model. An interesting consequence of the obtained expression
(44a) for the dispersion law that the dispersion of transverse collective
excitations is determined by the interaction potential $u(r)$, between
particles and also by two-, three-, and four-particle correlations. This
conclusion follows directly from the microscopic expressions for the frequency
relaxation parameters in (44a).

Finally, expression (42) allows estimating the values of physical quantities
characterizing the propagation of shear oscillations in a liquid: the
propagation velocity $v^{(T)}$ and the damping coefficient $\Gamma^{(T)}$ of
transverse sound oscillations. Hence, for the velocity $v^{(T)}$, from
dispersion relation (44b), we obtain

$v^{(T)}k=\lim_{k\to
0}\sqrt{\Delta_{T,\;2}^{2}(k)-\frac{\Delta_{T,\;3}^{2}(k)[\Delta_{T,\;3}^{2}(k)-\Delta_{T,\;2}^{2}(k)]}{2[\Delta_{T,\;4}^{2}(k)-\Delta_{T,\;3}^{2}(k)]}}$
(47a) or $v^{(T)}k=\lim_{k\to
0}\sqrt{\frac{2\Delta_{T,\;2}^{2}(k)\Delta_{T,\;4}^{2}(k)}{2\Delta_{T,\;4}^{2}(k)-\Delta_{T,\;3}^{2}(k)}}.$
(47b)

Expression (47b) is obtained as a result of solving the dispersion equation
solution according to the approximation scheme proposed by Mountain in
Mountain_1966  (also see Mokshin_JPCM2018 ). Expressions (47a) and (47b) must
give close values for $v^{(T)}$. For the damping coefficient of transverse
sound, we obtain

$\displaystyle\Gamma^{(T)}k^{2}=\lim_{k\to
0}\;\frac{\Delta_{T,\;3}^{2}(k)\sqrt{\Delta_{T,\;4}^{2}(k)}}{2\Delta_{T,\;4}^{2}(k)-\Delta_{T,\;3}^{2}(k)}.$
(48)

Therefore, we can respectively compute the values of $v^{(T)}$ and
$\Gamma^{(T)}$ from (47a) or (47b) and (48). For these computations, we must
use values of the frequency parameters corresponding to wave numbers in the
domain of intermediate values of $k$, where we have quasi-solid-state
properties of the liquid. We note that (48) is also interesting because direct
experimental measurements of $\Gamma^{(T)}$ are rather complicated.

Transition regime. Expression (42) for the spectral density $C_{T}(k,\omega)$
with a dispersion law $\omega_{c}^{(T)}(k)$ of form (44a) indicates the
existence of regime in the transverse collective particle dynamics, where we
have a transition from quasi-solid-state dynamics, discussed above, to the
dynamics of an ordinary equilibrium liquid. In Fig 1 (top), the wave number
domain corresponding to this regime is shown in gray. Here, the high-frequency
doublet in the spectrum $C_{T}(k,\omega)$ is transformed into a single
broadened spectral component with the form of smoothed trapezoid centered on
the frequency $\omega=0$ (see Fig. 1 (bottom)). The tangents to the dispersion
curve $\omega_{c}^{(T)}(k)$ for the corresponding wave numbers are
extrapolated to finite values $k$ as $\omega_{c}^{(T)}\to 0$ (see Fig. 1
(top)).

The existence of this regime is quite understandable because the transition to
quasi-solid-state dynamics occurs in some domain of spatial scales and
consequently in some wave number domain. An analysis of expression (42) for
$C_{T}(k,\omega)$ shows that such a transition is realized for the
correspondence between frequency relaxation parameters

$\displaystyle\Delta_{T,\;3}^{2}(k)$ $\displaystyle=$ $\displaystyle
3\Delta_{T,\;2}^{2}(k),$ (49) $\displaystyle\Delta_{T,\;4}^{2}(k)$
$\displaystyle=$ $\displaystyle
2\Delta_{T,\;3}^{2}(k)=6\Delta_{T,\;2}^{2}(k).$

In this case, for the coefficients $\mathcal{A}_{1}^{T}(k)$ and
$\mathcal{A}_{2}^{T}(k)$ in (42), we obtain

$\mathcal{A}_{1}^{T}(k)=0,$ (50a)
$\mathcal{A}_{2}^{T}(k)=2\Delta_{T,\;2}^{4}(k),$ (50b)

and expression (42) becomes

$\displaystyle
C_{T}(k,\omega)=\frac{1}{\pi}\;\frac{\Delta_{T,\;2}^{2}(k)\sqrt{6\Delta_{T,\;2}^{2}(k)}}{\omega^{4}+2\Delta_{T,\;2}^{4}(k)}.$
(51)

In fact, correspondence law (49) for the frequency relaxation parameters is
equivalent to relation (50a) and represents existence condition for this
transition mode.

In turn, as can be seen from (51), transverse collective dynamics is
determined by only the frequency relaxation parameter $\Delta_{T,\;2}^{2}(k)$
and consequently by the interaction potential $u(r)$ between particles, the
particle radial distribution function $g(r)$, and the mean particle velocity
$\bar{v}_{x}$. This regime is related to the transition from the collective
dynamics description in terms of microscopic characteristics to the
description where hydrodynamics parameters such as the generalized viscosity
$\nu(k)$ and the thermal conductivity $D_{T}(k)$ can be used.

Generalized hydrodynamics regime. The generalized hydrodynamics regime
corresponds to small wave numbers and passes into the usual hydrodynamic
regime in the limit $k\to 0$. Here, we have typical liquid collective
dynamics, and transverse acoustic waves are absent, i.e.,
$\omega_{c}^{(T)}=0$. In Fig. 1 (top), the wave number domain corresponding to
this regime is shown in light gray. Here, as in the case of the hydrodynamic
regime, the spectrum $C_{T}(k,\omega)$ has only a single component at the
frequency zero and is reproduced by a Lorentz function (see expression (28)
and Fig. 1 (bottom)). The width of this spectral component is determined by
the generalized shear viscosity $\nu(k)$, whose values depend on the wave
number $k$. In passing to the hydrodynamic limit $k\to 0$, $\nu(k)$ is
extrapolated to the hydrodynamic kinematic shear viscosity,

$\nu=\lim_{k\to 0}\;\nu(k).$

In this regime, the transverse collective particle dynamics is described by
the hydrodynamic variables $A_{1}^{T}(k)$ and $A_{2}^{T}(k)$. Consequently,
the equalization of the time scales given by relation (35) occurs for $l=3$:

$\displaystyle
1/\sqrt{\Delta_{T,\;3}^{2}(k)}=1/\sqrt{\Delta_{T,\;4}^{2}(k)}=1/\sqrt{\Delta_{T,\;5}^{2}(k)}=\ldots,$
(52)

and this time scale equality naturally realizes the transition from the
extended variable set $A_{1}^{T}(k)$, $A_{2}^{T}(k)$ and $A_{3}^{T}(k)$ to the
usual hydrodynamic variables $A_{1}^{T}(k)$ and $A_{2}^{T}(k)$. Taking (52)
into account, from (37a), we obtain

$\displaystyle
C_{T}(k,\omega)=\frac{1}{\pi}\;\frac{\Delta_{T,\;2}^{2}(k)\sqrt{\Delta_{T,\;3}^{2}(k)}}{\Delta_{T,\;3}^{2}(k)-\Delta_{T,\;2}^{2}(k)}\;\frac{1}{\omega^{2}+\mathcal{A}_{2}^{T}(k)},$
(53a) where
$\mathcal{A}_{2}^{T}(k)=\frac{\Delta_{T,\;2}^{4}(k)}{\Delta_{T,\;3}^{2}(k)-\Delta_{T,\;2}^{2}(k)}.$
(53b)

Expression (53a) is characterized by the following properties. First, this
expression determines the spectral density $C_{T}(k,\omega)$ in the frequency
domain

$\omega^{2}\ll\Delta_{T,\;3}^{2}(k),$ (54)

where $\tau_{T,\;3}(k)=1/\sqrt{\Delta_{T,\;3}^{2}(k)}$ is the minimum time
scale in the collective particle dynamics for wave numbers corresponding to
the generalized hydrodynamics.

Second, expression (53a) yields the correspondence between the frequency
parameters

$\Delta_{T,\;3}^{2}(k)\geq\Delta_{T,\;2}^{2}(k),$ (55)

which is analogous to relation (43) and which just like condition (54)
reflects the fact that the existence of oscillatory processes related to
transverse collective particle dynamics in the generalized hydrodynamic regime
are characterized by frequencies not exceeding $\sqrt{\Delta_{T,\;3}^{2}(k)}$.

Third, expression (53a) corresponds to the known hydrodynamic expression (28)
for the spectral density $C_{T}(k,\omega)$ of the of transverse current, and
coefficient $\mathcal{A}_{2}^{T}(k)$ is related to the generalized shear
viscosity $\nu(k)$:

$\nu(k)=\frac{1}{k^{2}}\sqrt{\mathcal{A}_{2}^{T}(k)}=\frac{1}{k^{2}}\frac{\Delta_{T,\;2}^{2}(k)}{\sqrt{\Delta_{T,\;3}^{2}(k)-\Delta_{T,\;2}^{2}(k)}}.$
(56)

Fourth, expression (53a) becomes identical to hydrodynamic expression (28)
under the relation between the frequency parameters

$\Delta_{T,\;3}^{2}(k)\gg\Delta_{T,\;2}^{2}(k),$ (57)

which is equivalent to the assumption that the time scales are separated and
to the Markov approximation, which are used in hydrodynamic theory
Hansen/McDonald  and lead to Eq. (26) and expression (28) for
$C_{T}(k,\omega)$.

##  III Transverse collective dynamics of liquid lithium near the melting
point

A self-consistent relaxation theory of longitudinal collective particle
dynamics in simple liquids was presented in Mokshin_JPCM2018 . As an example,
the theoretically calculated spectra of the dynamic structure factor
$S(k,\omega)$ and the dispersion law $\omega_{c}^{(L)}(k)$ of longitudinal
sound waves were compared with experimental data on inelastic x-ray scattering
in liquid lithium at the temperature $T=475$ K (the melting temperature is
$T_{m}=453.65$ K) Scopigno2000 . Moreover, in the framework of the theory, the
sound propagation velocity $v^{(L)}=6000\pm 1200$ m/s, sound damping
coefficient $\Gamma^{(L)}=18.4\pm 3.5$ nm2/ps and thermal conductivity
$D_{T}=21.5\pm 3.5$ nm2/ps were computed. Here, we also quantitatively compute
various characteristics of the transverse collective dynamics for liquid
lithium at $T=475$ K and the number density of atoms $\rho=0.0\,445$
$\textrm{\AA}^{-3}$. We compare the theoretical results with the results of
the equilibrium atomic dynamics simulations in an $NVT$ ($N$ is the number of
particles, $V$ is the volume, $T$ is the temperature)-ensemble Murtazayev1999
.

The modeling details are analogous to those presented in Mokshin_JPCM2018 ;
Khusnutdinoff2018 . We performed the simulation calculations for a system of
$N=16\,000$ atoms enclosed in cubic box with periodic boundary conditions. The
interaction between lithium atoms was given by a spherical pseudopotential
Gonzalez1993 . As we previously showed Khusnutdinoff2018 , this model of the
potential agrees best with experimental data on elastic and inelastic x-ray
scattering compared with known multiparticle potentials of the EAM (embedded
atom model) type. The equations of motion were integrated according to the
velocity Verlet algorithm with the time step $\tau^{\prime}=0.01$ ps
Mokshin2012 . To bring the system to the thermodynamic equilibrium and to
calculate the spectral characteristics, we respectively performed $200\;000$
and $100\;000$ time steps.

Figure 2: Frequency relaxation parameter $\Delta_{T,\;2}^{2}(k)$,
$\Delta_{T,\;3}^{2}(k)$, $\Delta_{T,\;4}^{2}(k)$ and $\Delta_{T,\;5}^{2}(k)$
of the transverse collective atom dynamics in equilibrium melted lithium at
the temperature $T=475$ K: the results were obtained based on configurational
data for the atomic dynamics simulated. Vertical closed intervals denote the
confidence intervals obtained as a result of numerical calculations with the
configurational data at different instants. The dashed line is the
experimental static structure factor $S(k)$ lithium at the given temperature
Waseda . Figure 3: Top: Dispersion law of transverse oscillatory dynamics of
atoms of equilibrium melted lithium at the temperature $T=475$ K: circles
($\circ\leavevmode\nobreak\ \circ\leavevmode\nobreak\ \circ$) denote data from
atomic dynamics simulations data; squares joined by a solid line are the
results of theoretical calculations with Eq. (44b); the straight line (T) (T)
in the small wave number range is the extrapolated hydrodynamic result
$\omega_{c}^{(T)}(k)=v^{(T)}k$, where $v^{(T)}=2\,390$ m/s is the transverse
speed of sound. Bottom: An enlarged fragment of Fig. 2 showing the frequency
relaxation parameters $\Delta_{T,\;2}^{2}(k)$, $\Delta_{T,\;3}^{2}(k)$,
$\Delta_{T,\;4}^{2}(k)$ and $\Delta_{T,\;5}^{2}(k)$ in the range of small
finite wave numbers: vertical dashed lines separate different regimes of the
oscillatory dynamics.

Based on the configurational data representing an array data structure of atom
coordinates and velocities at an arbitrary fixed instant, we calculated
frequency moments and also frequency relaxation parameters
$\Delta_{T,\;n}^{2}(k)$, $n=2$, $3$, $4$ and $5$, in a wide range of wave
numbers $k\in[0.072,\;6.481]$ Å-1 [see expressions (19) and (22)]. We note
that basic definition (19) and microscopic expression (23) for the frequency
parameter $\Delta_{T,\;2}^{2}(k)$ give identical results. As can be seen from
Fig. 2, the calculated frequency parameters have a similar monotonic
dependence on $k$ in the wave number range $k>0.25$ Å-1. Just as expected, the
typical time scales $\tau_{T,\;n}(k)=1/\sqrt{\Delta_{T,\;n}^{2}(k)}$ shorten
as $k$ increases. The minimum values are taken by
$\tau_{T,\;4}(k)=1/\sqrt{\Delta_{T,\;4}^{2}(k)}$ and
$\tau_{T,\;5}(k)=1/\sqrt{\Delta_{T,\;5}^{2}(k)}$, determining time scales less
than $0.01$ ps. Such time scales are even shorter than the step
$\tau^{\prime}$ for integrating the atomic equation of motion in the
simulation calculations. The time $\tau_{T,\;4}(k)$ can therefore indeed be
regarded as a threshold quantity in the hierarchy of time scales for the
relaxation process Bogolyubov_book . In the wave number range $k\leq 3.5$ Å-1,
the frequency parameters $\Delta_{T,\;4}^{2}(k)$ and $\Delta_{T,\;5}^{2}(k)$
take comparable values and show a difference only as $k$ increases further,
approaching correspondence (38) characteristic of the free-moving particle
regime.

Figure 4: Spectra ${C}_{T}(k,\omega)$ of transverse current of melted lithium
at the temperature $T=475$ K for different wave numbers: circles
($\circ\leavevmode\nobreak\ \circ\leavevmode\nobreak\ \circ$) are denote the
atomic dynamics simulations data; the solid line is the theoretical results
with general expression (37); the dotted line shows theoretical results for
different regimes, Eq. (53a) for $k=0.18$ Å-1; Eq. (51) for $k=0.27$ Å-1, and
Eq. (42) for $k=0.54$ Å-1, $0.81$ Å-1, $1.08$ Å-1 and $1.80$ Å-1.

The obtained frequency relaxation parameter values allow calculating the
dispersion relation $\omega_{c}^{(T)}(k)$ of the transverse oscillatory
dynamics based on Eq. (44b). As can be seen from Fig. 3, the theoretical
expression for the dispersion relation completely reproduces the modeling
results. Moreover, the typical regimes of collective oscillatory dynamics in
melted lithium at the given temperature can be clearly seen in the plot of
$\omega_{c}^{(T)}(k)$. The generalized hydrodynamics regime corresponds to
small $0.25$ Å${}^{-1}<k<0.4$ Å-1, the transition regime is realized on the
interval $k\geq 0.4$ Å-1, and the domain of intermediate values of $k$, where
shear oscillatoty processes occur, is $k\geq 0.4$ Å-1.

In Fig. 4, we compare the spectra $C_{T}(k,\omega)$ obtained based on the
atomic dynamics simulations data and calculated according to general
expression (37). Here, we also give the theoretical spectra $C_{T}(k,\omega)$
calculated by formulas (53a), (51) and (42) or the corresponding typical
collective atomic dynamics regimes of lithium. As can be seen from the figure,
the theoretical and modeling results agree completely. The theory reproduces
all features of the spectra $C_{T}(k,\omega)$ for different wave numbers,
correctly reproducing the transition from dynamics the expressed high-
frequency oscillation modes [see the spectra $C_{T}(k,\omega)$ in Fig. 4] for
wave numbers from $k=0.54$ Å-1 to $k=1.80$ Å-1] to the usual transverse
collective dynamics for a liquid [see the spectrum $C_{T}(k,\omega)$ at
$k=0.18$ Å-1 in Fig. 4].

Based on Eqs. (47b) and (48), we determined transverse speed of sound
$v^{(T)}\simeq 2\,390$ m/s and the damping coefficient $\Gamma^{(T)}\simeq 17$
Å2/ps of the transverse sound mode. Numerical calculations with Eq. (56) show
that the generalized kinematic shear viscosity depends on $k$ as

$\nu(k)=\frac{\nu}{1+\alpha^{2}k^{2}},$ (58)

where the coefficient $\alpha$ and kinematic shear viscosity $\nu$ take the
values $\alpha=1.1$ and $\nu=105\pm 11$ Å2/ps. It is remarkable that the
obtained viscosity value agrees well with the experimental value:
$\nu_{exp}=111$ Å2/ps link_1 .

##  IV Concluding commentary

In conclusion, we note the following points.

The presented self-consistent relaxation theory of transverse collective
particle dynamics in equilibrium liquids is a realization of Bogoliubov’s idea
about a reduced description of relaxation processes in liquids. This theory is
completely consistent with the theoretical description of longitudinal
collective dynamics in liquids, which was previously given in detail in
Mokshin2015 ; Mokshin_JPCM2018 ; Mokshin_JPCM2018/Corrigendum . A similar
consistent, albeit more simplified, description until now could only be
realized in the framework of the viscoelasticity model Copley/Lovesey .

The theory yields the proper asymptotic behavior in both the short-wave limit,
corresponding to free particle motion, and the long-wave hydrodynamic limit
and allows calculating hydrodynamic characteristics. The key condition
determing the correspondence between time scales of relaxation processes in
liquids allows acounting for the presence of high-speed processes, which are
usually ignored in hydrodynamic theory and some of its generalizations, in the
framework of the presented theoretical description. The theory satisfies a
complete set of so-called sum rules of the spectral density $C_{T}(k,\omega)$.
It generalizes the known viscoelasticity model Copley/Lovesey , whose key
parameters can be calculated in the framework of the presented theory (also
see Sec. 3.1. in Mokshin_JPCM2018 ).

In this theory, we did not use any assumptions that some particular
“relaxation” and “oscillation” regimes (or modes) form any features of the
transverse collective particle dynamics, in turn determining some regions of
the spectral density $C_{T}(k,\omega)$ of the transverse particle current.
Therefore, it is totally unnecessary to separate or isolate any components of
this spectral density for interpreting an experimentally measured or a
numerically modeled spectral density.

## References

  * (1) J. Frenkel, Kinetic Theory of Liquids, Clarendon Press, Oxford (1946)
  * (2) A. M. Prokhorov et al., eds., Encyclopedic Dictionary of Physics [in Russian], Sovet. Entsiklopediya, Moscow (1983).
  * (3) A. M. Prokhorov et al., eds., Encyclopedia of Physics [in Russian], Vol. 2, Bol’shaya Rossiiskaya Entsiklopediya, Moscow (1998).
  * (4) K. Trachenko and V.V. Brazhkin, Rep. Prog. Phys., 79 (2016), 016502, 36 pp.
  * (5) E. E. Tareyeva, Yu. D. Fomin, E. N. Tsyok, and V. N. Ryzhov, Theor. Math. Phys., 194:1 (2018), 148-156.
  * (6) A.V. Granato, J. Phys. IV France, 06 (1996), C8, 9 pp.
  * (7) D. Levesque, L. Verlet, J. Kürkijarvi, Phys. Rev. A, 7:5 (1973), 1690-1700.
  * (8) L. Sjögren, Ann. Phys., 110:1 (1978), 173-179.
  * (9) Z. Donkó, G.J. Kalman, P. Hartmann, J. Phys.: Condens. Matter, 20:41 (2008), 413101, 35 pp.
  * (10) S.A. Khrapak, A.G. Khrapak, N.P. Kryuchkov, S.O. Yurchenko, J. Chem. Phys., 150:10 (2019), 104503, 8 pp.
  * (11) R.E. Ryltsev, N.M. Chtchelkatchev, V.N. Ryzhov, Phys. Rev. Lett., 110:2 (2013), 025701, 5 pp.
  * (12) B.G. del Rio, L.E. González, Phys. Rev. B, 95:22 (2017), 224201, 15 pp.
  * (13) Noël Jakse, T. Bryk, J. Chem. Phys., 151:3 (2019), 034506, 8 pp.
  * (14) M. Ropo, J. Akola, R.O. Jones, J. Chem. Phys., 145:18 (2016), 184502, 8 pp.
  * (15) Y.D. Fomin, E.N. Tsiok, V.N. Ryzhov, V.V. Brazhkin, J. Molec. Liq., 287 (2019), 110992, 4 pp.
  * (16) L. Wang, C. Yang, M.T. Dove, A.V. Mokshin, V.V. Brazhkin, K. Trachenko, Sci. Rep., 9 (2019), 755, 9 pp.
  * (17) S. Hosokawa, M. Inui, Y. Kajihara, S. Tsutsui, A.Q.R. Baron, J. Phys.: Condens. Matter, 27:19 (2015), 194104, 7 pp.
  * (18) P.A. Egelstaff, An Introduction to the Liquid State, Academic Press, New York, 1967.
  * (19) E. Burkel, H. Sinn, J. Phys.: Condens. Matter, 6 (1994), A225-A225.
  * (20) S. Hosokawa, M. Inui, Y. Kajihara, K. Matsuda, T. Ichitsubo, W.C. Pilgrim, H. Sinn, L.E. Gonz$\acute{a}$lez, D.J. Gonz$\acute{a}$lez, S. Tsutsui, A.Q.R. Baron, Phys. Rev. Lett., 102:10 (2009), 105502, 4 pp.
  * (21) S. Hosokawa, S. Munejiri, M. Inui, Y. Kajihara, W.-C. Pilgrim, Y. Ohmasa, S. Tsutsui, A.Q.R. Baron, F. Shimojo and K. Hoshino, J. Phys. Condens. Matter, 25:11 (2013) 112101, 5 pp.
  * (22) V.M. Giordano, G. Monaco, Proc. Natl. Acad. Sci. USA, 107:51 (2010), 21985-21989.
  * (23) V.M. Giordano, G. Monaco, Phys. Rev. B, 84:5 (2011), 052201, 4 pp.
  * (24) R.A. MacPhail, D. Kivelson, J. Chem. Phys., 80:5 (1984), 2102-2114.
  * (25) T. Bryk, I. Mryglod, Phys. Rev. E, 62:2 (2000), 2188-2199.
  * (26) I.P. Omelyan, I.M. Mryglod, Condens. Matter Phys., N4 (1994), 128-160.
  * (27) K. Trachenko, Phys. Rev. E, 96:6 (2017), 062134, 5 pp.
  * (28) M. Baggioli, M. Vasin, V. Brazhkin and K. Trachenko, Phys. Rep., 865 (2020), 1-44.
  * (29) N.P. Kryuchkov, L.A. Mistryukova, V.V. Brazhkin, S.O. Yurchenko, Sci. Rep., 9 (2019), 10483, 12 pp.
  * (30) N.P. Kryuchkov, V.V. Brazhkin, S.O. Yurchenko, J. Phys. Chem. Lett., 10 (2019), 4470-4475.
  * (31) E.V. Yakovlev, N.P. Kryuchkov, P.V. Ovcharov, A.V. Sapelkin, V.V. Brazhkin, S.O. Yurchenko, J. Phys. Chem. Lett., 11 (2020), 1370-1376.
  * (32) Yu.D. Fomin, V.N. Ryzhov, E.N. Tsiok, V.V. Brazhkin, K. Trachenko, J. Phys.: Condens. Matter, 29 (2017), 059501, 1 pp.
  * (33) V.V. Brazhkin, Yu.D. Fomin, V.N. Ryzhov, E.N. Tsiok, K. Trachenko, Physica A: Statistical Mechanics and its Applications, 509 (2018), 690-702.
  * (34) R.M. Yulmetyev, A.V. Mokshin, P. Hänggi, V.Yu. Shurygin, Phys. Rev. E, 64:5 (2001), 057101, 4 pp.
  * (35) R.M. Yulmetyev, A.V. Mokshin, P. Hänggi, V.Yu. Shurygin, JETP Lett., 76:3 (2002), 147-150.
  * (36) A.V. Mokshin, B.N. Galimzyanov, J. Phys.: Condens. Matter, 30 (2018), 085102, 17 pp.
  * (37) R.M. Yulmetyev, A.V. Mokshin, T. Scopigno, P. Hänggi, J. Phys.: Codens. Matter, 15 (2003), 2235-2257.
  * (38) A.V. Mokshin, R.M. Yulmetyev, P. Hänggi, J. Chem. Phys., 121:15 (2004), 7341-7346.
  * (39) A. V. Mokshin, R. M. Yulmetyev, R. M. Khusnutdinov, and P. Hänggi, JETP, 103, 841-849, (2006).
  * (40) A.V. Mokshin, R.M. Yulmetyev, R.M. Khusnutdinoff, P. Hänggi, J. Phys.: Condens. Matter, 19 (2007), 046209, 16 pp.
  * (41) R.M. Khusnutdinoff, C. Cockrell, O.A. Dicks, A.C.S. Jensen, M.D. Le, L. Wang, M.T. Dove, A.V. Mokshin, V.V. Brazhkin, and K. Trachenko, Phys. Rev. B, 101:21 (2020), 214312, 9 pp.
  * (42) V. N. Ryzhov, A. F. Barabanov, M. V. Magnitskaya, and E. E. Tareeva, Phys. Usp., 51, 1077-1083 (2008).
  * (43) J.P. Hansen, I.R. McDonald, Theory of Simple Liquids Academic Press, London, 2006.
  * (44) R. Zwanzig, Nonequilibrium statistical mechanics, Clarendon Press, Oxford, 2001.
  * (45) A. V. Mokshin and R. M. Yulmetyev, Microscopic Dynamics of Simple Liquids [in Russian], Tsentr Innovatsionnykh Tekhnologii, Kazan (2006).
  * (46) B. A. Klumov, Phys. Usp., 53, 1053-1065 (2010).
  * (47) U. Balucani, M.H. Lee, V. Tognetti, Phys. Rep., 373:6 (2003), 409-492.
  * (48) M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis, Acad. Press, New York (1972).
  * (49) A. A. Vladimirov, D. Ihle, and N. M. Plakida, Theor. Math. Phys., 145, 1576-1589 (2005).
  * (50) M.H. Lee, Phys. Rev. E, 62:2 (2000), 1769-1772.
  * (51) A.V. Mokshin, R.M. Yulmetyev, P. Hänggi, Phys. Rev. Lett., 95:20 (2005), 200601, 4 pp.
  * (52) A. V. Mokshin, Theor. Math. Phys., 183, 449-477 (2015).
  * (53) N. N. Bogolyubov, Problems of Dynamical Theory in Statistical Physics [in Russian], Gostekhizdat, Moscow (1946); English transl. (Stud. Statist. Mech., Vol. 1), North-Holland, Amsterdam (1962).
  * (54) W. Götze, Complex Dynamics of Glass-Forming liquids: A Mode-Coupling Theory, Oxford Univ. Press, Oxford, 2009.
  * (55) P. Resibua and M. De Lener, Classical Kinetic Theory of Liquids and Gases [in Russian], Moscow, Mir, (1980).
  * (56) R. Mountain, Rev. Mod. Phys., 38:1 (1966), 205-214.
  * (57) T. Scopigno, U. Balucani, G. Ruocco and F. Sette, J. Phys.: Condens. Matter, 12 (2000), 8009-8034.
  * (58) I. K. Kamilov, A. K. Murtazaev, and Kh. K. Aliev, Phys. Usp., 42, 689-709 (1999)
  * (59) R. M. Khusnutdinoff, B. N. Galimzyanov, and A. V. Mokshin, JETP, 126, 83-89 (2018).
  * (60) L.E. Gonz$\acute{a}$lez, D.J. Gonz$\acute{a}$lez, M. Silbert and J.A. Alonso, J. Phys.: Condens. Matter, 5 (1993), 4283-4298.
  * (61) A. V. Mokshin, A. V. Chvanova, and R. M. Khusnutdinov, Theor. Math. Phys, 171, 541-552 (2012).
  * (62) Y. Waseda, The Structure of Non-Crystalline Materials: Liquids and Amorphous Solids, McGraw-Hill, New York, 1980.
  * (63) R.W. Ohse (Ed.), Handbook of Thermodynamic and Transport Properties of Alkali Metals, Intern. Union of Pure and Applied Chemistry Chemical Data Series No. 30. Oxford: Blackwell Scientific Publ., 1985, pp. 987.
  * (64) A.V. Mokshin, B.N. Galimzyanov, J. Phys.: Condens. Matter, 31:20 (2019), 209501, 1 pp.
  * (65) J.R.D. Copley, S.W. Lovesey, Rep. Prog. Phys., 38 (1975), 461-563.


# A note on extremal intersecting linear Ryser systems

Adrián Vázquez-Ávila
Subdirección de Ingeniería y Posgrado
Universidad Aeronáutica en Querétaro
<EMAIL_ADDRESS>
###### Abstract

A famous conjecture of Ryser states that any $r$-partite set system has
transversal number at most $r-1$ times their matching number. This conjecture
is only known to be true for $r\leq 3$ in general, for $r\leq 5$ if the set
system is intersecting, and for $r\leq 9$ if the intersecting set system is
linear. In this note, we deal with Ryser’s Conjecture for intersecting
$r$-partite linear systems; that is, if $\tau$ is the transversal number for
an intersecting $r$-partite linear system, then Ryser’s Conjecture states that
$\tau\leq r-1$. If this conjecture is true, this is known to be sharp for $r$
for which there exists a projective plane of order $r-1$. There has also been
considerable effort to find intersecting $r$-partite set systems whose
transversal number is $r-1$.

In this note, the following is proved: if $r\geq 4$ is an even integer, then
$f_{l}(r)\geq 3(r-2)+1$, where $f_{l}(r)$ is the minimum number of lines of an
intersecting $r$-partite linear system whose transversal number is $r-1$. This
lower bound gives an exact value for $f_{l}(r)$, for some small values of $r$.
Also, we prove that any $r$-partite linear system satisfies $\tau\leq r-1$ if
$\nu_{2}\leq r$ for all $r\geq 3$ odd integer and $\nu_{2}\leq r-1$ for all
$r\geq 4$ even integer, where $\nu_{2}$ is the maximum cardinality of a subset
of lines $R\subseteq\mathcal{L}$ such that every triplet of different elements
of $R$ does not have a common point.

Keywords. Ryser’s Conjecture, linear systems, transversal number, 2-packing
number, projective planes

##  1 Introduction

A _set system_ is a pair $(X,\mathcal{F})$ where $\mathcal{F}$ is a finite
family of subsets on a ground set $X$. A set system can be also thought of as
a hypergraph, where the elements of $X$ and $\mathcal{F}$ are called
_vertices_ and _hyperedges_ respectively. The set system $(X,\mathcal{F})$ is
called _$r$ -uniform_, when all subsets of $\mathcal{F}$ has size $r$. The set
system $(X,\mathcal{F})$ is _$r$ -partite_ if the elements of $X$ can be
partitioned into $r$ sets $X_{1},\ldots,X_{r}$, called the _sides_ , such that
each element of $\mathcal{F}$ contain exactly one element of $X_{i}$, for all
$i=1,\ldots,r$, that is $|F\cap X_{i}|=1$, where $F\in\mathcal{F}$. Thus, an
$r$-partite set system is an $r$-uniform set system.

Let $(X,\mathcal{F})$ be a set system. A subset $T\subseteq X$ is a
_transversal_ of $(X,\mathcal{F})$ if for any element $F\in\mathcal{F}$
contain at least an element of $T$, that is, $T\cap F\neq\emptyset$, for every
$F\in\mathcal{F}$. The _transversal number_ of $(X,\mathcal{F})$,
$\tau=\tau(X,\mathcal{F})$, is the smallest possible cardinality of a
transversal of $(X,\mathcal{F})$. The transversal number has been studied in
the literature in many different contexts and names. For example, with the
name of _piercing number_ and _covering number_ , see for instance [8, 9, 7,
11, 14, 16, 17, 18, 19].

Let $(X,\mathcal{F})$ be a set system. A subset
$\mathcal{E}\subseteq\mathcal{F}$ of a set system $(X,\mathcal{F})$ is called
a _matching_ if $F\cap\hat{F}=\emptyset$ for every $F,\hat{F}\in\mathcal{E}$.
The matching number of $(X,\mathcal{F})$, $\nu=\nu(X,\mathcal{F})$, is the
cardinality of the largest matching of $(X,\mathcal{F})$. A set system is
called _intersecting_ if $\nu=1$; that is, $F\cap\hat{F}\neq\emptyset$, for
every $F,\hat{F}\in\mathcal{F}$.

It is not hard to see that any $r$-uniform set system $(X,\mathcal{F})$
satisfies the inequality $\tau\leq r\nu$. It is well-known this bound is
sharp, as shown by the family of all subsets of size $r$ in a ground set of
size $kr-1$, which has $\nu=k-1$ and $\tau=(k-1)r$. On the other hand, if
$\nu=1$, any projective plane of order $r-1$ (denoted by $\Pi_{r-1}$ in this
paper), where $r-1$ is a prime power, satisfies $\tau=r$. However, for
$r$-partite set systems, Ryser conjectured in the 1960’s that the upper bound
could be improved.

Ryser’s Conjecture: Any $r$-partite set system satisfies $\tau\leq(r-1)\nu.$

For the special case $r=2$, Ryser’s Conjecture is equivalent to Kőnig’s
Theorem. The only other known general case of the conjecture is when $r=3$,
see [2]. However, Ryser’s conjecture is also known to be true in some special
cases. Tuza verified [21] Ryser’s Conjecture for $r\leq 5$ if the set system
is assumed intersecting. Furthermore, Francetić et al. [13] verified Ryser’s
Conjecture for $r\leq 9$ if the set system is a linear system, that is, a set
system $(X,\mathcal{F})$ is _linear_ if it satisfies $|E\cap F|\leq 1$, for
every pair of distinct subsets $E,F\in\mathcal{F}$. In this note every linear
system is denoted by $(P,\mathcal{L})$, where the elements of $P$ and
$\mathcal{L}$ are called _points_ and _lines_ , respectively. In the rest of
this paper, only linear systems are considered. Most of the definitions can be
generalized for set systems. Thus, we deal with Ryser’s Conjecture for
intersecting $r$-partite linear systems.

Intersecting linear Ryser’s Conjecture: Every intersecting $r$-partite linear
system satisfies $\tau\leq r-1$.

In case the conjecture would be true, it is tight in the sense that for
infinitely many $r$’s there are constructions of intersecting $r$-partite
linear systems with $\tau=r-1$. For example, if $r-1$ is a prime power,
consider the finite projective plane of order $r-1$ as a linear system,
$\Pi_{r-1}$. This linear system is $r$-uniform and intersecting. To make it
$r$-partite, one just needs to delete one point from the projective plane.
This truncated projective plane gives an intersecting $r$-partite linear
system with $\tau\geq r-1$, and $r(r-1)$ points and $(r-1)^{2}$ lines, and it
will be denoted by $\Pi^{\prime}_{r-1}$. However, the construction obtained
from the projective plane is not the “optimal” extremal. Although the
projective plane construction only contains $r(r-1)$ points (which is an
optimal number of points), it has a lot of lines. Let $f(r)$ be the minimum
integer so that there exists an intersecting $r$-partite set system
$(X,\mathcal{F})$ with $\tau=r-1$ and $|\mathcal{F}|=f(r)$ lines. Analogously,
let $f_{l}(r)$ be the minimum integer so that there exists an intersecting
$r$-partite linear system $(P,\mathcal{L})$ with $\tau=r-1$ and
$|\mathcal{L}|=f_{l}(r)$ lines. It is probably that $f_{l}(r)$ does not exist
for some values of $r$ (if $r-1$ is a prime power, then $\Pi^{\prime}_{r-1}$
is known to exist, providing proof that $f_{l}(r)$ is well-defined). Hence, if
$f_{l}(r)$ does exist, for some $r$, then $f(r)\leq f_{l}(r)$. It is not
difficult to prove that $f_{l}(2)=1$ and $f_{l}(3)=3$, see [15]. Furthermore,
Mansour et al. [15] proved that $f_{l}(4)=6$ and $f_{l}(5)=9$. On the other
hand, Aharoni et al. [3] proved that $f_{l}(6)=13$ and $f(7)=17$ (even when
the truncated projective plane does not exist, since it has been proved that
finite projective planes of order six do not exist, see [20]); however
Francetić proved [13] that $f_{l}(7)$ does not exist, that is, there is no
intersecting $7$-partite linear system such that $\tau=6$. Abu-Khazneha et al.
[1] constructed new infinite family of intersecting $r$-partite set systems
extremal to Ryser’s Conjecture, which exist whenever a projective plane of
order $r-2$ exists. That construction produces a large number of non-
isomorphic extremal set systems. Finally, Aharoni et al. [3] gave a lower
bound on $f(r)$ when $r\to\infty$, showing that $f(r)\geq$3.052$r+O(1)$, this
lower bound is an improvement since Mansour proved [15] that
$f(r)\geq(3-\frac{1}{\sqrt{18}})r(1-o(1))\approx$2.764$(1-o(1))$, when
$r\to\infty$. In this paper, a lower bound for $f_{l}(r)$ is given for $r\geq
4$ an even integer.

######  Theorem 1.1.

If $r\geq 4$ is an even integer, then $3(r-2)+1\leq f_{l}(r)$.

Notice that, if $r\in\\{2,4,6\\}$, then $f_{l}(r)=3(r-2)+1$. Aharoni et al.
[3] proved $18\leq f(8)$ and $24\leq f(10)$. Hence, by Theorem 1.1 we have:
$19\leq f_{l}(8)$ and $25\leq f_{l}(10)$.

##  2 Main Results

Let $(P,\mathcal{L})$ be a linear system and $p\in P$ be a point. It is
denoted by $\mathcal{L}_{p}$ to be the set of lines incident to $p$. The
_degree_ of $p$ is defined as $deg(p)=|\mathcal{L}_{p}|$ and the maximum
degree over all points of the linear system is denoted by
$\Delta=\Delta(P,\mathcal{L})$.

A subset $R$ of lines of a linear system $(P,\mathcal{L})$ is a _$2$ -packing_
of $(P,\mathcal{L})$ if the elements of $R$ are triplewise disjoint, that is,
if any three elements are chosen in $R$, then they have not a common point.
The 2-packing number of $(P,\mathcal{L})$, $\nu=\nu_{2}(P,\mathcal{L})$, is
the maximum cardinality of a 2-packing of $(P,\mathcal{L})$. There are some
works that study this new parameter, see [4, 5, 10, 6, 22, 24, 23, 25, 26].

######  Theorem 2.1.

[5] Let $(P,\mathcal{L})$ be a linear system with $p,q\in P$ be two points
such that $\Delta=deg(p)$ and $\Delta^{\prime}=\max\\{deg(x):x\in
P\setminus\\{p\\}\\}$. If $|\mathcal{L}|\leq\Delta+\Delta^{\prime}+\nu_{2}-3$,
then $\tau\leq\nu_{2}-1$.

######  Lemma 2.2.

[22] Let $(P,\mathcal{L})$ be an intersecting $r$-uniform linear system, with
$r\geq 3$ be an odd integer. If $\tau=r$, then $\nu_{2}=r+1$.

######  Corollary 2.3.

Let $(P,\mathcal{L})$ be an intersecting $r$-partite linear system, with
$r\geq 3$ be an odd integer. If $\nu_{2}\leq r$ then $\tau\leq r-1$.

Now, let deal with the case when $r$ is an even integer.

######  Lemma 2.4.

[24] Let $(P,\mathcal{L})$ be an $r$-uniform intersecting linear system with
$r\geq 2$ be an even integer. If $\nu_{2}=r+1$ then $\tau=\frac{r+2}{2}$.

Hence, if $\tau>\frac{r+2}{2}$ then $\nu_{2}\leq r$ and $r\geq 4$.

######  Lemma 2.5.

[24] Let $(P,\mathcal{L})$ be an intersecting $r$-uniform linear system, with
$r\geq 4$ be an even integer. If $\tau=r$, then $\nu_{2}=r$.

######  Lemma 2.6.

Let $(P,\mathcal{L})$ be an intersecting $r$-uniform linear system with $r\geq
4$ be an even integer. If $\tau=r-1$, then $\nu_{2}=r$.

Proof. Let $(P,\mathcal{L})$ be an intersecting $r$-partite linear system. Let
$p,q\in P$ be two points such that $\Delta=deg(p)$ and
$\Delta^{\prime}=\max\\{deg(x):x\in P\setminus\\{p\\}\\}$. By Theorem 2.1 if
$|\mathcal{L}|\leq\Delta+\Delta^{\prime}+\nu_{2}-3\leq 3(r-2)+1$, then
$\tau\leq\nu_{2}-1$, which implies that $\nu_{2}=r$.

######  Corollary 2.7.

Let $(P,\mathcal{L})$ be an intersecting $r$-partite linear system with $r\geq
4$ be an even integer. If $\nu_{2}\leq r-1$ then $\tau\leq r-2$.

By Lemmas 2.5 and 2.6, we have:

######  Corollary 2.8.

Let $(P,\mathcal{L})$ be an intersecting $r$-uniform linear system with $r\geq
4$ be an even number. If $\tau\in\\{r-1,r\\}$, then $\nu_{2}=r$.

By Corollaries 2.3 and 2.7, and Lemma 2.5, we have:

######  Theorem 2.9.

Let $r\geq 3$ be an integer. Then every intersecting $r$-partite linear system
satisfies

  1. 1.

$\tau\leq r-1$ If $\nu_{2}\leq r$ and $r\geq 3$ be an odd integer; and

  2. 2.

$\tau\leq r-2$ if $\nu_{2}\leq r-1$ and $r\geq 4$ be an even integer.

The intersecting linear Ryser’s Conjecture is almost proved; it only remains
to analyzes two cases concerning the 2-packing number:

######  Conjecture 2.10.

Let $r\geq 3$ be an integer. Then every intersecting $r$-partite linear system
satisfies:

  1. 1.

$\tau\leq r-1$ if $\nu_{2}=r+1$, with $r\geq 3$ an odd number.

  2. 2.

$\tau\leq r-1$ if $\nu_{2}=r$, with $r\geq 4$ an even number.

######  Theorem 2.11.

Let $r\geq 4$ be an even integer, then $3(r-2)+1\leq f_{l}(r)$.

Proof. Suppose that $\nu_{2}\leq r-1$ (by Corollary 2.8). Let $p,q\in P$ be
two points such that $deg(p)=\Delta$ and
$\Delta^{\prime}=\deg(q)=\max\\{deg(x):x\in P\setminus\\{p\\}\\}$. By Theorem
2.1 if $|\mathcal{L}|\leq\Delta+\Delta^{\prime}+\nu_{2}-3\leq 3(r-2)$, then
$\tau\leq\nu_{2}-1\leq r-2$. Therefore, $3(r-2)+1\leq f_{l}(r)$.

Acknowledgment

Research was partially supported by SNI and CONACyT.

## References

  * [1] A. Abu-Khazneha, J. Barát, A. Pokrovskiy and T. Szabó, _A family of extremal hypergraphs for Ryser’s conjecture_ , J. Combin. Theory Ser. A 161, 164–177 (2019)
  * [2] R. Aharoni, _Ryser’s conjecture for 3-graphs_ , Combinatorica 21 (1), 1–4 (2001)
  * [3] R. Aharoni, J. Barát and I.M. Wanless, _Multipartite hypergraphs achieving equality in Ryser’s conjecture_ , Graphs Combin. 32, 1–15 (2016)
  * [4] C. Alfaro and A. Vázquez-Ávila, _A note on a problem of Henning and Yeo about the transversal number of uniform linear systems whose 2-packing number is fixed_ , Discrete Math. Lett., 3, 61–66 (2020)
  * [5] C. Alfaro, G. Araujo-Pardo, C. Rubio-Montiel and A. Vázquez-Ávila, _On transversal and $2$-packing numbers in uniform linear systems_, AKCE J. Graphs Comb., 17 (1), 335-3341 (2020)
  * [6] C. Alfaro, C. Rubio-Montiel and A. Vázquez-Ávila, _Covering and 2-degree-packing number in graphs_ , J. Comb. Math. Comb. Comput. Accepted.
  * [7] N. Alon, G. Kalai, J. Matoušek, R. Meshulam, _Transversal numbers for hypergraphs arising in geometry_ , Advances in Applied Math. 29, 79-101 (2002)
  * [8] N. Alon, D.J. Kleitman, _Piercing convex sets and the Hadwiger Debrunner (p,q)-problem_ , Advances in Math. 96, 103-112 (1992)
  * [9] N. Alon, D.J. Kleitman, _Piercing convex sets_ , Bull. Of the Amér. Math. Soc. 29, 252-256 (1992)
  * [10] G. Araujo-Pardo, A. Montejano, L. Montejano and A. Vázquez-Ávila, _On transversal and $2$-packing numbers in straight line systems on $\mathbb{R}^{2}$_, Util. Math. 105, 317–336 (2017)
  * [11] J. Eckhoff, _A Survey of Hadwiger-Debrunner (p,q)-problem_ , Discrete and Computational Geometry, The Goodman-Pollack Festschrift Algorithms and Combinatorics, 25, 347-377 (2003)
  * [12] A. Eustis, and J. Verstraëte, _On the independence number of Steiner systems_ , Combin. Probab. Comput. 22 (2), 241–252 (2013)
  * [13] N. Francetić, S. Herke, B. D. McKay and I. M. Wanless, _On Ryser’s conjecture for linear intersecting multipartite hypergraphs_ , Eur. J. Comb. 61, 91–105 (2017)
  * [14] M. Huicochea, J. Jerónimo-Castro, L. Montejano and D. Oliveros, _About the piercing number of a family of intervals_ , Discrete Math., 338 (12), 2545–2548 (2015)
  * [15] T. Mansour, C. Song and R. Yuster, _A comment on Ryser’s conjecture for intersecting hypergraphs_ , Graphs Combin. 25, 101–109 (2009)
  * [16] L. Montejano, P. Soberón, _Piercing numbers for balanced and unbalanced families_ , Discrete Computational Geometry 45 (2), 358–364 (2011)
  * [17] A. Noga, K. Gil, J. Matousek and R. Meshulam, _Transversal numbers for hypergraphs arising in geometry_ , Adv. in Appl. Math. 29 (1), 79–101 (2002)
  * [18] A. Noga and D. J. Kleitman, _Piercing convex sets_ , Bull. Amer. Math. Soc. (N.S.) 27 (2), 252–256 (1992)
  * [19] D. Oliveros, C. O’Neill and S. Zerbib, _The geometry and combinatorics of discrete line segment hypergraphs_ , Discrete Math. 343 (6), 111825 (2020)
  * [20] G. Tarry, _Le probléme de 36 officiers_ , Compte Rendu de l’Assoc. Francais Avanc. Sci. Naturel 2, 170–203 (1901)
  * [21] Z. Tuza, _Ryser’s conjecture on transversals of $r$-partite hypergraphs_, Ars. Combin. 16, 201–209 (1983)
  * [22] A. Vázquez-Ávila, _A note on domination and 2-packing numbers in intersecting linear systems_ , Appl. Math. E-Notes 19, 310–314 (2019)
  * [23] A. Vázquez-Ávila, _Domination and 2-degree-packing number in graphs_ , J. Comb. Math. Comb. Comput. Accepted.
  * [24] A. Vázquez-Ávila, _On domination and 2-packing numbers in intersecting linear systems_ , Ars. Comb. Accepted.
  * [25] A. Vázquez-Ávila, _On intersecting straight line systems_ , J. Discret. Math. Sci. Cryptogr. Accepted.
  * [26] A. Vázquez-Ávila, _On transversal numbers of intersecting straight line systems and intersecting segment systems_. Bol. Soc. Mat. Mex. Accepted.