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Frenkel line

2015-02-25T16:15:58Z

Fomin314:

'''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 present in the
literature. The most common one is due to van der Waals. It treats
the liquids as dense structureless gases. Although this approach
allows to explain many principle features of fluids, in
particular, the liquid-gas phase transition, it fails in
explanation of other important issues, such as, for example,
existence in liquids of transverse collective excitations such as
phonons.

Another approach to fluid properties was proposed by J. Frenkel
<ref>[J. Frenkel, Kinetic Theory of Liquids (Oxford University Press, London, 1947]</ref>. It is based on an assumption that at moderate
temperatures the particles of liquid behave similar to the case of
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 based on postulation of some similarity between crystals
and liquids allows to explain many important properties of the
later: transverse collective excitations, large hear 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 heats up a fluid from a
temperature close to the melting one 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://iopscience.iop.org/1063-7869/55/11/R01/ V.V. Brazhkin, A.G. Lyapin, V.N. Ryzhov, K. Trachenko, Yu.D. Fomin, E.N. Tsiok, Phys. Usp. 55, 1061 (2012) ]</ref>, <ref name="frpre"> [http://journals.aps.org/pre/abstract/10.1103/PhysRevE.85.031203 V. V. Brazhkin, Yu. D. Fomin, A. G. Lyapin, V. N. Ryzhov, and K. Trachenko, Phys. Rev. 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> - diffusion coefficient. This is the time necessary for a particle to move to it's own size. The second characteristic time is the shortest period of transverse oscillations of particles of fluid: <math> \tau^* </math>. When these two time
scales become comparable one cannot distinguish the oscillations of the particles and theirs jumps to another position. Therefore
the criterion for Frenkel line is given by <math> \tau_0 \approx \tau^* </math>.

There are several approximate criteria to locate the Frenkel line
in <math> (P,T) </math> Refs. <ref name="ufn"> </ref>, <ref name="frpre"> </ref>, <ref name="frprl"> [http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.111.145901 V. V. Brazhkin, Yu. D. Fomin, A. G. Lyapin, V. N. Ryzhov, E. N. Tsiok, and Kostya Trachenko, Phys. Rev. Lett. 111, 145901 (2013)]</ref>. One of these criteria is based
on velocity autocorrelation function (vacf): below the Frenkel
line vacf demonstrate oscillation behavior while above it vacfs
monotonically decay to zero. The second criterion is based on the
fact that at moderate temperature liquids can sustain transverse
excitations which disappear on heating the liquid up. One more
criterion is based on isochoric heat capacities measurements. The
isochoric heat capacity per particle of a monatomic liquid close
to the melting line is close to <math> 3 k_B </math> ( <math> k_B </math> is Boltzmann
constant). The contribution to the heat capacity of 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://scitation.aip.org/content/aip/journal/jcp/139/23/10.1063/1.4844135 D. Bolmatov, V. V. Brazhkin, Yu. D. Fomin, V. N. Ryzhov and K. Trachenko, J. Chem. Phys. 139, 234501 (2013)] </ref>.

Currently Frenkel lines of several model liquids (Lennard-Jones
and soft spheres <ref name="ufn"> </ref>, <ref name="frpre"> </ref>, <ref name="frprl"> </ref> and real ones (liquid
iron <ref> [http://www.nature.com/srep/2014/141126/srep07194/fig_tab/srep07194_F1.html Yu. D. Fomin, V. N. Ryzhov, E. N. Tsiok, V. V. Brazhkin and K. Trachenko, Scientific Reports, 4, 7194 (2014)] </ref>, hydrogen <ref> [http://journals.aps.org/pre/abstract/10.1103/PhysRevE.89.032126 K. Trachenko, V. V. Brazhkin, and D.Bolmatov, Phys. Rev. E 89, 032126 (2014)] </ref>, water
<ref name="kostya3"> [http://journals.aps.org/pre/abstract/10.1103/PhysRevE.91.012112 C. Yang, V. V. Brazhkin, M. T. Dove, and K. Trachenko, Phys. Rev. E, 91, 012112 (2015)] </ref>, <math> CO_2 </math> <ref name="kostya3"> </ref>, <math> CH_4 </math> <ref name="kostya3"> </ref> were reported in the literature.

==Related Lines==
*[[Widom line]]
*[[Fisher-Widom line]]

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

Statistical mechanics

2015-02-25T16:05:12Z

Fomin314:

{{columns-list|3|
==B==
*[[BBGKY hierarchy]]
*[[Boltzmann average]]
*[[Boltzmann constant]]
*[[Boltzmann distribution]]
*[[Boltzmann factor]]
*[[Born-Green equation]]
*[[Brownian motion]]

==C==
*[[Canonical ensemble]]
*[[Chemical potential]]
*[[Compressibility equation]]
*[[Critical exponents]]
*[[Critical points]]
*[[Curie's law]]

==D==
*[[Darwin-Fowler method]]
*[[de Broglie thermal wavelength]]

==E==
*[[Energy equation]]
*[[Entropy]]
*[[Ensembles in thermostatistics]]
*[[Ergodic hypothesis]]

==F==
*[[Fermi-Pasta-Ulam experiment]]
*[[Fisher–Ruelle stability criteria]]
*[[Fisher-Widom line]]
*[[Four-body function]]
*[[Frenkel line]]

==G==
*[[Gibbs distribution]]
*[[Gibbs ensemble]]
*[[Gibbs measures]]
*[[Gibbs paradox]]
*[[Goldstone modes]]
*[[Grand canonical ensemble]]

==I==
*[[Information theory]]
*[[Internal energy]]
*[[Intermolecular pair potential]]
*[[Isoenthalpic–isobaric ensemble]]
*[[Isothermal-isobaric ensemble]]
==J==
*[[Joule-Thomson effect]]
==K==
*[[Kirkwood superposition approximation]]
*[[Kolmogorov–Arnold–Moser theorem]]

==L==
*[[Law of corresponding states]]
*[[Liouville's theorem]]
*[[Loschmidt's paradox]] (Umkehreinwand)
*[[Lyapunov exponents]]

==M==
*[[Maxwell velocity distribution]]
*[[Maxwell's demon]]
*[[Mean field models]]
*[[Melting curve]]
*[[Microcanonical ensemble]]
*[[Microstate]]
*[[Mixing systems]]

==P==
*[[Pair distribution function]]
*[[Parrondo's paradox]]
*[[Partition function]]
*[[Phase space]]
*[[Phase transitions]]
*[[Poincaré theorem]]
*[[Pressure]]
*[[Pressure equation]] (aka. virial equation)

==Q==
*[[Quantum statistics]]
==R==
*[[Radial distribution function]]
*[[Ross melting rule]]
==S==
*[[Semi-grand ensembles]]
*[[Stochastic transition]]
*[[Structure factor]]
*[[Surface tension]]

==T==
*[[Temperature]]
*[[Thermodynamic limit]]
*[[Transfer matrices]]
==W==
*[[Widom line]]
*[[Wigner's distribution function]]
==Y==
*[[Yang-Yang anomaly]]
==Z==
*[[Zermelo’s paradox]] (Wiederkehreinwand)
}}
==General reading==
'''Books'''
* Donald A. McQuarrie "Statistical Mechanics", University Science Books (1984) (Re-published 2000) ISBN 978-1-891389-15-3
* L. D. Landau and E. M. Lifshitz "Statistical Physics", Course of Theoretical Physics volume 5 Part 1 3rd Edition (1984) ISBN 0750633727
* Terrell L. Hill "Statistical Mechanics: Principles and Selected Applications" (1956) ISBN 0486653900
* Terrell L. Hill "An Introduction to Statistical Thermodynamics" (1986) ISBN 0486652424
'''Papers'''
*[http://dx.doi.org/10.1103/RevModPhys.27.289 D. Ter Haar "Foundations of Statistical Mechanics", Reviews of Modern Physics '''27''' pp. 289 - 338 (1955)]
*[http://dx.doi.org/10.1088/0034-4885/42/12/002 O. Penrose "Foundations of statistical mechanics", Reports on Progress in Physics '''42''' pp. 1937-2006 (1979)]
*[http://dx.doi.org/10.1126/science.177.4047.393 P. W. Anderson "More Is Different", Science '''177''' pp. 393-396 (1972)]
*[http://dx.doi.org/10.1080/00268970600965835 J. S. Rowlinson "The evolution of some statistical mechanical ideas", Molecular Physics '''104''' pp. 3399 - 3410 (2006)]
*[http://dx.doi.org/10.1016/S0378-4371(98)00517-2 Elliott H. Lieb "Some problems in statistical mechanics that I would like to see solved", Physica A '''263''' pp. 491-499 (1999)]
*[http://www.pro-physik.de/Phy/pdfstart.do?mid=3&articleid=25753&recordid=25793 Joel L. Lebowitz "Emergent Phenomena", Physik Journal '''6''' #9 pp. 41-46 (2007)]
[[category: statistical mechanics]]
__NOTOC__

Frenkel line

2015-02-25T15:56:33Z

Fomin314: Created page with "'''Frenkel line''' is a line of change of microscopic dynamics of fluids. Below the Frenkel line the fluid are "rigid" and "solid-like" while above it fluids are "soft" and "g..."

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

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

Another approach to fluid properties was proposed by J. Frenkel
<ref>[J. Frenkel, Kinetic Theory of Liquids (Oxford University Press, London, 1947]</ref>. It is based on an assumption that at moderate
temperatures the particles of liquid behave similar to the case of
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 based on postulation of some similarity between crystals
and liquids allows to explain many important properties of the
later: transverse collective excitations, large hear 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 heats up a fluid from a
temperature close to the melting one 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://iopscience.iop.org/1063-7869/55/11/R01/ V.V. Brazhkin, A.G. Lyapin, V.N. Ryzhov, K. Trachenko, Yu.D. Fomin, E.N. Tsiok, Phys. Usp. 55, 1061 (2012) ]</ref>, <ref name="frpre"> [http://journals.aps.org/pre/abstract/10.1103/PhysRevE.85.031203 V. V. Brazhkin, Yu. D. Fomin, A. G. Lyapin, V. N. Ryzhov, and K. Trachenko, Phys. Rev. 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> - diffusion coefficient. This is the time necessary for a particle to move to it's own size. The second characteristic time is the shortest period of transverse oscillations of particles of fluid: <math> \tau^* </math>. When these two time
scales become comparable one cannot distinguish the oscillations of the particles and theirs jumps to another position. Therefore
the criterion for Frenkel line is given by <math> \tau_0 \approx \tau^* </math>.

There are several approximate criteria to locate the Frenkel line
in <math> (P,T) </math> Refs. <ref name="ufn"> </ref>, <ref name="frpre"> </ref>, <ref name="frprl"> [http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.111.145901 V. V. Brazhkin, Yu. D. Fomin, A. G. Lyapin, V. N. Ryzhov, E. N. Tsiok, and Kostya Trachenko, Phys. Rev. Lett. 111, 145901 (2013)]</ref>. One of these criteria is based
on velocity autocorrelation function (vacf): below the Frenkel
line vacf demonstrate oscillation behavior while above it vacfs
monotonically decay to zero. The second criterion is based on the
fact that at moderate temperature liquids can sustain transverse
excitations which disappear on heating the liquid up. One more
criterion is based on isochoric heat capacities measurements. The
isochoric heat capacity per particle of a monatomic liquid close
to the melting line is close to <math> 3 k_B </math> ( <math> k_B </math> is Boltzmann
constant). The contribution to the heat capacity of 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://scitation.aip.org/content/aip/journal/jcp/139/23/10.1063/1.4844135 D. Bolmatov, V. V. Brazhkin, Yu. D. Fomin, V. N. Ryzhov and K. Trachenko, J. Chem. Phys. 139, 234501 (2013)] </ref>.

Currently Frenkel lines of several model liquids (Lennard-Jones
and soft spheres <ref name="ufn"> </ref>, <ref name="frpre"> </ref>, <ref name="frprl"> </ref> and real ones (liquid
iron <ref> [http://www.nature.com/srep/2014/141126/srep07194/fig_tab/srep07194_F1.html Yu. D. Fomin, V. N. Ryzhov, E. N. Tsiok, V. V. Brazhkin and K. Trachenko, Scientific Reports, 4, 7194 (2014)] </ref>, hydrogen <ref> [http://journals.aps.org/pre/abstract/10.1103/PhysRevE.89.032126 K. Trachenko, V. V. Brazhkin, and D.Bolmatov, Phys. Rev. E 89, 032126 (2014)] </ref>, water
<ref name="kostya3"> [http://journals.aps.org/pre/abstract/10.1103/PhysRevE.91.012112 C. Yang, V. V. Brazhkin, M. T. Dove, and K. Trachenko, Phys. Rev. E, 91, 012112 (2015)] </ref>, <math> CO_2 </math> <ref name="kostya3"> </ref>, <math> CH_4 </math> <ref name="kostya3"> </ref> were reported in the literature.

==Related Lines==
*[[Widom line]]
*[[Fisher-Widom line]]

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

Fermi-Jagla model

2014-01-24T11:29:05Z

Fomin314:

The '''Fermi-Jagla model''' is a smooth variant of the [[Ramp model | Jagla model]]. It is given by (Eq. 1 in <ref>[http://dx.doi.org/10.1021/jp205098a Joel Y. Abraham, Sergey V. Buldyrev, and Nicolas Giovambattista "Liquid and Glass Polymorphism in a Monatomic System with Isotropic, Smooth Pair Interactions", Journal of Physical Chemistry B '''115''' pp. 14229-14239 (2011)]</ref>):

:<math>\Phi_{12}(r) = \epsilon_0 \left[ \left( \frac{a}{r} \right)^n + \frac{A_0}{1+\exp \left[ \frac{A_1}{A_0} \frac{r}{a-A_2} \right]} - \frac{B_0}{1+\exp \left[ \frac{B_1}{B_0} \frac{r}{a-B_2} \right]} \right]</math>

There is a relation between Fermi function and hyperbolic tangent:

:<math>\frac{1}{e^x+1}=\frac{1}{2}-\frac{1}{2}\tanh \frac{x}{2}</math>

Using this relation one can show that Fermi-Jagla model is equivalent to [[Fomin potential]] introduced earlier.
==References==
<references/>
;Related reading
*[http://dx.doi.org/10.1063/1.4790404 Shaina Reisman and Nicolas Giovambattista "Glass and liquid phase diagram of a polyamorphic monatomic system", Journal of Chemical Physics '''138''' 064509 (2013)]

[[category: models]]

Fomin potential

2014-01-24T11:27:08Z

Fomin314:

The '''Fomin potential''' was firstly introduced as <ref>[http://dx.doi.org/10.1063/1.2965880 Yu. D. Fomin, N. V. Gribova, V. N. Ryzhov, S. M. Stishov, and Daan Frenkel "Quasibinary amorphous phase in a three-dimensional system of particles with repulsive-shoulder interactions", Journal of Chemical Physics '''129''' 064512 (2008)]</ref> (Eq. 2):

:<math>
\Phi_{12}\left( r \right) = \left( \frac{d}{r} \right)^n + \frac{\epsilon}{2} (1 - \tanh ( k_0 (r-\sigma_s )))
</math>

where <math> \Phi_{12}\left( r \right) </math> is the [[intermolecular pair potential]], <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math> is the distance between site 1 and site 2, <math>n=14</math> and <math>k_0=10</math>. <math>d</math> is the diameter of the hard core, <math>\sigma_s</math> is the width of the repulsive shoulder, <math>\epsilon</math> is the height of the shoulder. As such, this model can be viewed as a softened [[square shoulder model]].

Later it was generalized to the form <ref>[http://dx.doi.org/10.1063/1.3530790 Yu. D. Fomin, E. N. Tsiok and V. N. Ryzhov, "Complex phase behavior of the system of particles with smooth potential with repulsive shoulder and attractive well", Journal of Chemical Physics '''134''' 044523 (2011)]</ref> :

:<math>
\Phi_{12}\left( r \right) = \left( \frac{d}{r} \right)^n + \lambda_0 + \sum_{i=1}^{i=i_{max}} \lambda_i \tanh ( k_i (r-\sigma_i )))
</math>

By varying coefficients <math> \lambda_i </math> one can add repulsive shoulders or attractive wells to the potential.
==References==
<references/>
[[category: models]]

Fomin potential

2014-01-24T11:25:41Z

Fomin314:

The '''Fomin potential''' was firstly introduced as <ref>[http://dx.doi.org/10.1063/1.2965880 Yu. D. Fomin, N. V. Gribova, V. N. Ryzhov, S. M. Stishov, and Daan Frenkel "Quasibinary amorphous phase in a three-dimensional system of particles with repulsive-shoulder interactions", Journal of Chemical Physics '''129''' 064512 (2008)]</ref> (Eq. 2):

:<math>
\Phi_{12}\left( r \right) = \left( \frac{d}{r} \right)^n + \frac{\epsilon}{2} (1 - \tanh ( k_0 (r-\sigma_s )))
</math>

where <math> \Phi_{12}\left( r \right) </math> is the [[intermolecular pair potential]], <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math> is the distance between site 1 and site 2, <math>n=14</math> and <math>k_0=10</math>. <math>d</math> is the diameter of the hard core, <math>\sigma_s</math> is the width of the repulsive shoulder, <math>\epsilon</math> is the height of the shoulder. As such, this model can be viewed as a softened [[square shoulder model]].

Later it was generalized to the form <ref>[http://dx.doi.org/10.1063/1.3530790 Yu. D. Fomin, E. N. Tsiok and V. N. Ryzhov, "Complex phase behavior of the system of particles with smooth potential with repulsive shoulder and attractive well", Journal of Chemical Physics '''134''' 044523 (2011)]</ref> :

:<math>
\Phi_{12}\left( r \right) = \left( \frac{d}{r} \right)^n + \lambda_0 \sum_{i=1}^{i=imax} \lambda_i \tanh ( k_i (r-\sigma_i )))
</math>

By varying coefficients <math> \lambda_i </math> one can add repulsive shoulders or attractive wells to the potential.
==References==
<references/>
[[category: models]]

Fermi-Jagla model

2014-01-22T17:42:36Z

Fomin314:

The '''Fermi-Jagla model''' is a smooth variant of the [[Ramp model | Jagla model]]. It is given by (Eq. 1 in <ref>[http://dx.doi.org/10.1021/jp205098a Joel Y. Abraham, Sergey V. Buldyrev, and Nicolas Giovambattista "Liquid and Glass Polymorphism in a Monatomic System with Isotropic, Smooth Pair Interactions", Journal of Physical Chemistry B '''115''' pp. 14229-14239 (2011)]</ref>):

:<math>\Phi_{12}(r) = \epsilon_0 \left[ \left( \frac{a}{r} \right)^n + \frac{A_0}{1+\exp \left[ \frac{A_1}{A_0} \frac{r}{a-A_2} \right]} - \frac{B_0}{1+\exp \left[ \frac{B_1}{B_0} \frac{r}{a-B_2} \right]} \right]</math>

There is a relation between Fermi function and hyperbolic tangent:

:<math>\frac{1}{1+e^x}=\frac{1}{2}-\frac{1}{2}tanh(x/2)</math>

Using this relation one can deduce Fermi-Jagla model to Fomin potential introduced earlier and described in another section of this site.

==References==
<references/>
;Related reading
*[http://dx.doi.org/10.1063/1.4790404 Shaina Reisman and Nicolas Giovambattista "Glass and liquid phase diagram of a polyamorphic monatomic system", Journal of Chemical Physics '''138''' 064509 (2013)]

[[category: models]]