Editing Frenkel line

The '''Frenkel line''' is a line of change of microscopic dynamics of
fluids. Below the Frenkel line the fluids are "rigid" and
"solid-like" while above it fluids are "soft" and "gas-like".

Two types of approaches to the behavior of liquids are present in the
literature. The most common one is due to [[Johannes Diderik van der Waals|van der Waals]]. It treats
the liquids as dense structureless gases. Although this approach
allows one to explain many principle features of fluids, in
particular, the [[Gas-liquid phase transitions|liquid-gas phase transition]], it fails to
explain  other important issues such as, for example,
the existence in liquids of transverse collective excitations such as
phonons.

Another approach to fluid properties was proposed by Jacov Frenkel
<ref>Jacov Frenkel "Kinetic Theory of Liquids", Oxford University Press (1947)</ref>. 
It is based on the assumption that at moderate
[[temperature]]s the particles of liquid behave in a similar manner as a
crystal, ''i.e.'' the particles demonstrate oscillatory motions.
However, while in crystal they oscillate around theirs nodes, in
liquids after several periods the particles change the nodes. This
approach is based on postulation of some similarity between crystals
and liquids,  providing insight into many important properties of the
latter: transverse collective excitations, large [[heat capacity]] and
so on.

From the discussion above one can see that the microscopic
behavior of particles of moderate and high temperature fluids is
qualitatively different. If one [[heat]]s up a fluid from a
temperature close to the [[Melting curve|melting point]]  up to some high temperature, a
crossover from the solid-like to gas-like regime appears. The line
of this crossover was named Frenkel line after J. Frenkel.

Several methods to locate the Frenkel line were proposed in the
literature. The most detailed reviews of the methods are given in
Refs. 
<ref name="ufn"> [http://dx.doi.org/10.3367/UFNe.0182.201211a.1137 Vadim V. Brazhkin, Aleksandr G Lyapin, Valentin N. Ryzhov, Kostya Trachenko, Yurii D. Fomin and Elena N. Tsiok "Where is the supercritical fluid on the phase diagram?",  Physics-Uspekhi '''55''' pp. 1061-1079 (2012)]</ref>,
<ref name="frpre"> [http://dx.doi.org/10.1103/PhysRevE.85.031203 V. V. Brazhkin, Yu. D. Fomin, A. G. Lyapin, V. N. Ryzhov, and K. Trachenko "Two liquid states of matter: A dynamic line on a phase diagram", Physical Review E '''85''' 031203 (2012)]</ref>. 
The exact criterion of Frenkel line is the one based on comparison of characteristic times in fluids. One can
define a 'jump time' via 

:<math> \tau_0=\frac{a^2}{6D} </math>, 

where <math> a </math> is the particles size and <math> D </math> is the [[Diffusion|diffusion coefficient]]. This is the time necessary for a particle to move a distance comparable to it's own size. The second characteristic time corresponds to the shortest period of transverse oscillations of particles within the fluid, <math> \tau^* </math>. When these two time
scales are roughly equal one cannot distinguish between the oscillations of the particles and theirs jumps to another position. Thus
the criterion for Frenkel line is given by <math> \tau_0 \approx \tau^* </math>.

There are several approximate criteria to locate the Frenkel line
on the [[Phase diagrams: Pressure-temperature plane|pressure-temperature plane]] 
(see Refs. 
<ref name="ufn"> </ref>, 
<ref name="frpre"> </ref>, 
<ref name="frprl"> [http://dx.doi.org/10.1103/PhysRevLett.111.145901 V. V. Brazhkin, Yu. D. Fomin, A. G. Lyapin, V. N. Ryzhov, E. N. Tsiok, and Kostya Trachenko "“Liquid-Gas” Transition in the Supercritical Region: Fundamental Changes in the Particle Dynamics", Physical Review Letters '''111''' 145901 (2013)]</ref>). 
One of these criteria is based
on the velocity [[autocorrelation]] function (vacf): below the Frenkel
line the vacf demonstrates oscillatory behaviour, while above it the vacf
monotonically decays to zero. The second criteria is based on the
fact that at moderate temperatures liquids can sustain transverse
excitations, which disappear upon heating. One further
criteria is based on [[Heat capacity#At constant volume|isochoric heat capacity]] measurements. 
The isochoric heat capacity per particle of a monatomic liquid near
to the melting line is close to <math> 3 k_B </math> (where <math> k_B  </math> is the
[[Boltzmann constant]]). The contribution to the heat capacity due to potential part
of transverse excitations is <math> 1 k_B </math>. Therefore at the Frenkel
line, where transverse excitations vanish, the isochoric heat
capacity per particle should be <math> c_V=2 k_B </math>.

Crossing the Frenkel line leads also to some structural changes in fluids 
<ref> [http://dx.doi.org/10.1063/1.4844135  Dima Bolmatov, V. V. Brazhkin, Yu. D. Fomin, V. N. Ryzhov, and K. Trachenko "Evidence for structural crossover in the supercritical state", Journal of Chemical Physics '''139''' 234501 (2013)]</ref>.
Currently Frenkel lines of several [[Idealised models|idealised liquids]], such as [[Lennard-Jones model|Lennard-Jones]] and [[Soft sphere potential|soft spheres]] 
<ref name="ufn"> </ref>, <ref name="frpre"> </ref>, <ref name="frprl"> </ref>  
as well as [[realistic models]] such as
[[iron|liquid iron]] <ref>[http://dx.doi.org/10.1038/srep07194  Yu. D. Fomin, V. N. Ryzhov, E. N. Tsiok, V. V. Brazhkin, and K. Trachenko "Dynamic transition in supercritical iron", Scientific Reports '''4''' 7194 (2014)]</ref>,
[[hydrogen]] <ref> [http://dx.doi.org/10.1103/PhysRevE.89.032126 K. Trachenko, V. V. Brazhkin, and D.Bolmatov, "Dynamic transition of supercritical hydrogen: Defining the boundary between interior and atmosphere in gas giants", Physical Review E '''89''' 032126 (2014)]</ref>, 
[[water]] <ref name="kostya3"> [http://dx.doi.org/10.1103/PhysRevE.91.012112  C. Yang, V. V. Brazhkin, M. T. Dove, and K. Trachenko "Frenkel line and solubility maximum in supercritical fluids", Physical Review E '''91''' 012112 (2015)]</ref>, 
[[carbon dioxide]] <ref name="kostya3"> </ref>, 
and [[methane]] <ref name="kostya3"> </ref> 
have been reported in the literature.
==Related Lines==
*[[Widom line]]
*[[Fisher-Widom line]]

==References==
<references />
[[Category:statistical mechanics]]