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1-dimensional hard rods

2012-08-15T15:45:44Z

Franzl aus tirol: chemical potential

'''1-dimensional hard rods''' (sometimes known as a ''Tonks gas'' <ref>[http://dx.doi.org/10.1103/PhysRev.50.955 Lewi Tonks "The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic Spheres", Physical Review '''50''' pp. 955- (1936)]</ref>) consist of non-overlapping line segments of length <math>\sigma</math> who all occupy the same line which has length <math>L</math>. One could also think of this model as being a string of [[hard sphere model | hard spheres]] confined to 1 dimension (not to be confused with [[3-dimensional hard rods]]). The model is given by the [[intermolecular pair potential]]:

: <math> \Phi_{12}(x_{i},x_{j})=\left\{ \begin{array}{lll}
0 & ; & |x_{i}-x_{j}|>\sigma\\ \infty & ; & |x_{i}-x_{j}|<\sigma \end{array}\right. </math>

where <math> \left. x_k \right. </math> is the position of the center of the k-th rod, along with an external potential. Thus, the [[Boltzmann factor]] is

: <math>e_{ij}:=e^{-\beta\Phi_{12}(x_{i},x_{j})}=\Theta(|x_{i}-x_{j}|-\sigma)=\left\{ \begin{array}{lll} 1 & ; & |x_{i}-x_{j}|>\sigma\\ 0 & ; & |x_{i}-x_{j}|<\sigma \end{array}\right. </math>

The whole length of the rod must be inside the range:

: <math> V_{0}(x_i) = \left\{ \begin{array}{lll} 0 & ; & \sigma/2 < x_i < L - \sigma/2 \\
\infty &; & {\mathrm {elsewhere}}. \end{array} \right. </math>

== Canonical Ensemble: Configuration Integral ==
The [[statistical mechanics]] of this system can be solved exactly.
Consider a system of length <math> \left. L \right. </math> defined in the range <math> \left[ 0, L \right] </math>. The aim is to compute the [[partition function]] of a system of <math> \left. N \right. </math> hard rods of length <math> \left. \sigma \right. </math>.
Consider that the particles are ordered according to their label: <math> x_0 < x_1 < x_2 < \cdots < x_{N-1} </math>;
taking into account the pair potential we can write the canonical partition function
([http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 configuration integral])
of a system of <math> N </math> particles as:

:<math>\begin{align}
\frac{Z\left(N,L\right)}{N!} & =\int_{\sigma/2}^{L-\sigma/2}dx_{0}\int_{\sigma/2}^{L-\sigma/2}dx_{1}\cdots\int_{\sigma/2}^{L-\sigma/2}dx_{N-1}\prod_{i=1}^{N-1}e_{i-1,i}\\
& =\int_{\sigma/2}^{L+\sigma/2-N\sigma}dx_{0}\int_{x_{0}+\sigma}^{L+\sigma/2-N\sigma+\sigma}dx_{1}\cdots\int_{x_{i-1}+\sigma}^{L+\sigma/2-N\sigma+i\sigma}dx_{i}\cdots\int_{x_{N-2}+\sigma}^{L+\sigma/2-N\sigma+(N-1)\sigma}dx_{N-1}.
\end{align}</math>

Variable change: <math> \left. \omega_k = x_k - \left(k+\frac{1}{2}\right) \sigma \right. </math> ; we get:

:<math>\begin{align}
\frac{Z\left(N,L\right)}{N!} & =\int_{0}^{L-N\sigma}d\omega_{0}\int_{\omega_{0}}^{L-N\sigma}d\omega_{1}\cdots\int_{\omega_{i-1}}^{L-N\sigma}d\omega_{i}\cdots\int_{\omega_{N-2}}^{L-N\sigma}d\omega_{N-1}\\
& =\int_{0}^{L-N\sigma}d\omega_{0}\cdots\int_{\omega_{i-1}}^{L-N\sigma}d\omega_{i}\frac{(L-N\sigma-\omega_{i})^{N-1-i}}{(N-1-i)!}=\int_{0}^{L-N\sigma}d\omega_{0}\frac{(L-N\sigma-\omega_{0})^{N-1}}{(N-1)!}
\end{align}</math>

Therefore:
:<math>
\frac{ Z \left( N,L \right)}{N!} = \frac{ (L-N\sigma )^{N} }{N!}.
</math>

: <math>
Q(N,L) = \frac{ (L-N \sigma )^N}{\Lambda^N N!}.
</math>

== Thermodynamics ==
[[Helmholtz energy function]]
: <math> \left. A(N,L,T) = - k_B T \log Q \right. </math>

In the [[thermodynamic limit]] (i.e. <math> N \rightarrow \infty; L \rightarrow \infty</math> with <math> \rho = \frac{N}{L} </math>, remaining finite):

:<math> A \left( N,L,T \right) = N k_B T \left[ \log \left( \frac{ N \Lambda} { L - N \sigma }\right) - 1 \right]. </math>

== Equation of state ==
Using the [[thermodynamic relations]], the [[pressure]] (''linear tension'' in this case) <math> \left. p \right. </math> can
be written as:

:<math>
p = - \left( \frac{ \partial A}{\partial L} \right)_{N,T} = \frac{ N k_B T}{L - N \sigma};
</math>

The compressibility factor is

:<math>
Z = \frac{p L}{N k_B T} = \frac{1}{ 1 - \eta} = \underbrace{1}_{Z_{\mathrm{id}}}+\underbrace{\frac{\eta}{1-\eta}}_{Z_{\mathrm{ex}}},
</math>

where <math> \eta \equiv \frac{ N \sigma}{L} </math>; is the fraction of volume (i.e. length) occupied by the rods. 'id' labels the ideal and 'ex' the excess part.

It was shown by van Hove <ref>[http://dx.doi.org/10.1016/0031-8914(50)90072-3 L. van Hove, "Sur L'intégrale de Configuration Pour Les Systèmes De Particules À Une Dimension", Physica, '''16''' pp. 137-143 (1950)]</ref> that there is no [[Solid-liquid phase transitions |fluid-solid phase transition]] for this system (hence the designation ''Tonks gas'').

== Chemical potential ==
The chemical potential is given by

:<math>
\mu=\left(\frac{\partial A}{\partial N}\right)_{L,T}=k_{B}T\left(\ln\frac{\rho\Lambda}{1-\rho\sigma}+\frac{\rho\sigma}{1-\rho\sigma}\right)=k_{B}T\left(\ln\frac{\rho\Lambda}{1-\eta}+\frac{\eta}{1-\eta}\right)
</math>

with ideal and excess part separated:

:<math>
\beta\mu=\underbrace{\ln(\rho\Lambda)}_{\beta\mu_{\mathrm{id}}}+\underbrace{\ln\frac{1}{1-\eta}+\frac{\eta}{1-\eta}}_{\beta\mu_{\mathrm{ex}}}
</math>

== Isobaric ensemble: an alternative derivation ==
Adapted from Reference <ref>J. M. Ziman ''Models of Disorder: The Theoretical Physics of Homogeneously Disordered Systems'', Cambridge University Press (1979) ISBN 0521292808</ref>. If the rods are ordered according to their label: <math> x_0 < x_1 < x_2 < \cdots < x_{N-1} </math> the canonical [[partition function]] can also be written as:
: <math>
Z=
\int_0^{x_1} d x_0
\int_0^{x_2} d x_1
\cdots
\int_0^{L} d x_{N-1}
f(x_1-x_0)
f(x_2-x_1)
\cdots
f(L-x_{N-1}),
</math>
where <math>N!</math> does not appear one would have <math>N!</math> analogous expressions
by permuting the label of the (distinguishable) rods. <math>f(x)</math> is the [[Boltzmann factor]]
of the hard rods, which is <math>0</math> if <math>x<\sigma</math> and <math>1</math> otherwise.

A variable change to the distances between rods: <math> y_k = x_k - x_{k-1} </math> results in
: <math>
Z =
\int_0^{\infty} d y_0
\int_0^{\infty} d y_1
\cdots
\int_0^{\infty} d y_{N-1}
f(y_1)
f(y_2)
\cdots
f(y_{N-1}) \delta \left( \sum_{i=0}^{N-1} y_i-L \right):
</math>
the distances can take any value as long as they are not below <math>\sigma</math> (as enforced
by <math>f(y)</math>) and as long as they add up to <math>L</math> (as enforced by the [[Dirac_delta_distribution | Dirac delta]]). Writing the later as the inverse [[Laplace transform]] of an exponential:
: <math>
Z =
\int_0^{\infty} d y_0
\int_0^{\infty} d y_1
\cdots
\int_0^{\infty} d y_{N-1}
f(y_1)
f(y_2)
\cdots
f(y_{N-1})
\frac{1}{2\pi i } \int_{-\infty}^{\infty} ds \exp \left[ - s \left(\sum_{i=0}^{N-1} y_i-L \right)\right].
</math>
Exchanging integrals and expanding the exponential the <math>N</math> integrals decouple:
:<math>
Z =
\frac{1}{2\pi i } \int_{-\infty}^{\infty} ds
e^{ L s }
\left\{
\int_0^{\infty} d y f(y) e^{ - s y }
\right\}^N.
</math>
We may proceed to invert the Laplace transform (e.g. by means of the residues theorem), but this is not needed: we see our configuration integral is the inverse Laplace transform of another one,
:<math>
Z'(s)= \left\{ \int_0^{\infty} d y f(y) e^{ - s y } \right\}^N, </math>
so that
:<math>
Z'(s) = \int_0^{\infty} ds e^{ L s } Z(L).
</math>
This is precisely the transformation from the configuration integral in the canonical (<math>N,T,L</math>) ensemble to the isobaric (<math>N,T,p</math>) one, if one identifies
<math>s=p/k T</math>. Therefore, the [[Gibbs energy function]] is simply <math>G=-kT\log Z'(p/kT) </math>, which easily evaluated to be <math>G=kT N \log(p/kT)+p\sigma N</math>. The [[chemical potential]] is <math>\mu=G/N</math>, and by means of thermodynamic identities such as <math>\rho=\partial p/\partial \mu</math> one arrives at the same equation of state as the one given above.
==Confined hard rods==
<ref>[http://dx.doi.org/10.1080/00268978600101521 A. Robledo and J. S. Rowlinson "The distribution of hard rods on a line of finite length", Molecular Physics '''58''' pp. 711-721 (1986)]</ref>
==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1016/0031-8914(49)90059-2 L. van Hove "Quelques Propriétés Générales De L'intégrale De Configuration D'un Système De Particules Avec Interaction", Physica, '''15''' pp. 951-961 (1949)]
*[http://dx.doi.org/10.1063/1.1699116 Zevi W. Salsburg, Robert W. Zwanzig, and John G. Kirkwood "Molecular Distribution Functions in a One-Dimensional Fluid", Journal of Chemical Physics '''21''' pp. 1098-1107 (1953)]
*[http://dx.doi.org/10.1063/1.1699263 Robert L. Sells, C. W. Harris, and Eugene Guth "The Pair Distribution Function for a One-Dimensional Gas", Journal of Chemical Physics '''21''' pp. 1422-1423 (1953)]
*[http://dx.doi.org/10.1063/1.1706788 Donald Koppel "Partition Function for a Generalized Tonks' Gas", Physics of Fluids '''6''' 609 (1963)]
*[http://dx.doi.org/10.1103/PhysRev.171.224 J. L. Lebowitz, J. K. Percus and J. Sykes "Time Evolution of the Total Distribution Function of a One-Dimensional System of Hard Rods", Physical Review '''171''' pp. 224-235 (1968)]
*[http://dx.doi.org/10.3390/e10030248 Paolo V. Giaquinta "Entropy and Ordering of Hard Rods in One Dimension", Entropy '''10''' pp. 248-260 (2008)]

[[Category:Models]]
[[Category:Statistical mechanics]]

Hard sphere model

2012-08-15T09:51:12Z

Franzl aus tirol: Phase diagram

[[Image:sphere_green.png|thumb|right]]
[[Image:Hard-sphere phase diagram pressure vs packing fraction.png|thumb|right|Phase diagram (pressure vs packing fraction) of hard sphere system (Solid line - stable branch, dashed line - metastable branch)]]
The '''hard sphere model''' (sometimes known as the ''rigid sphere model'') is defined as

: <math>
\Phi_{12}\left( r \right) = \left\{ \begin{array}{lll}
\infty & ; & r < \sigma \\
0 & ; & r \ge \sigma \end{array} \right.
</math>

where <math> \Phi_{12}\left(r \right) </math> is the [[intermolecular pair potential]] between two spheres at a distance <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>, and <math> \sigma </math> is the diameter of the sphere.
The hard sphere model can be considered to be a special case of the [[hard ellipsoid model]], where each of the semi-axes has the same length, <math>a=b=c</math>.
==First simulations of hard spheres (1954-1957)==
The hard sphere model, along with its two-dimensional manifestation [[hard disks]], was one of the first ever systems studied using [[computer simulation techniques]] with a view
to understanding the thermodynamics of the liquid and solid phases and their corresponding [[Phase transitions | phase transition]]
<ref>[http://dx.doi.org/10.1063/1.1740207 Marshall N. Rosenbluth and Arianna W. Rosenbluth "Further Results on Monte Carlo Equations of State", Journal of Chemical Physics '''22''' pp. 881-884 (1954)]</ref>
<ref>[http://dx.doi.org/10.1063/1.1743956 W. W. Wood and J. D. Jacobson "Preliminary Results from a Recalculation of the Monte Carlo Equation of State of Hard Spheres", Journal of Chemical Physics '''27''' pp. 1207-1208 (1957)]</ref>
<ref>[http://dx.doi.org/10.1063/1.1743957 B. J. Alder and T. E. Wainwright "Phase Transition for a Hard Sphere System", Journal of Chemical Physics '''27''' pp. 1208-1209 (1957)]</ref>, much of this work undertaken at the Los Alamos Scientific Laboratory on the world's first electronic digital computer ENIAC <ref>[http://ftp.arl.army.mil/~mike/comphist/eniac-story.html The ENIAC Story]</ref>.
==Liquid phase radial distribution function==
The following are a series of plots of the hard sphere [[radial distribution function]] <ref>The [[total correlation function]] data was produced using the [http://www.vscht.cz/fch/software/hsmd/hspline-8-2004.zip computer code] written by [http://www.vscht.cz/fch/en/people/Jiri.Kolafa.html Jiří Kolafa]</ref> shown for different values of the number density <math>\rho</math>. The horizontal axis is in units of <math>\sigma</math> where <math>\sigma</math> is set to be 1. Click on image of interest to see a larger view.
:{| border="1"
|-
|<math>\rho=0.2</math> [[Image:HS_0.2_rdf.png|center|220px]] ||<math>\rho=0.3</math> [[Image:HS_0.3_rdf.png|center|220px]] || <math>\rho=0.4</math> [[Image:HS_0.4_rdf.png|center|220px]]
|-
|<math>\rho=0.5</math> [[Image:HS_0.5_rdf.png|center|220px]] ||<math>\rho=0.6</math> [[Image:HS_0.6_rdf.png|center|220px]] || <math>\rho=0.7</math> [[Image:HS_0.7_rdf.png|center|220px]]
|-
|<math>\rho=0.8</math> [[Image:HS_0.8_rdf.png|center|220px]] ||<math>\rho=0.85</math> [[Image:HS_0.85_rdf.png|center|220px]] || <math>\rho=0.9</math> [[Image:HS_0.9_rdf.png|center|220px]]
|}
The value of the radial distribution at contact, <math>{\mathrm g}(\sigma^+)</math>, can be used to calculate the [[pressure]] via the [[equations of state |equation of state]] (Eq. 1 in <ref name="Tao1"> [http://dx.doi.org/10.1103/PhysRevA.46.8007 Fu-Ming Tao, Yuhua Song, and E. A. Mason "Derivative of the hard-sphere radial distribution function at contact", Physical Review A '''46''' pp. 8007-8008 (1992)]</ref>)
:<math>\frac{p}{\rho k_BT}= 1 + B_2 \rho {\mathrm g}(\sigma^+)</math>
where the [[second virial coefficient]], <math>B_2</math>, is given by
:<math>B_2 = \frac{2\pi}{3}\sigma^3</math>.
Carnahan and Starling <ref>[http://dx.doi.org/10.1063/1.1672048 N. F.Carnahan and K. E.Starling,"Equation of State for Nonattracting Rigid Spheres" Journal of Chemical Physics '''51''' pp. 635-636 (1969)]</ref> provided the following expression for <math>{\mathrm g}(\sigma^+)</math> (Eq. 3 in <ref name="Tao1" > </ref>)
:<math>{\mathrm g}(\sigma^+)= \frac{1-\eta/2}{(1-\eta)^3}</math>
where <math>\eta</math> is the [[packing fraction]].

Over the years many groups have studied the radial distribution function of the hard sphere model:
<ref>[http://dx.doi.org/10.1063/1.1747854 John G. Kirkwood, Eugene K. Maun, and Berni J. Alder "Radial Distribution Functions and the Equation of State of a Fluid Composed of Rigid Spherical Molecules", Journal of Chemical Physics '''18''' pp. 1040- (1950)]</ref>
<ref>[http://dx.doi.org/10.1103/PhysRev.85.777 B. R. A. Nijboer and L. Van Hove "Radial Distribution Function of a Gas of Hard Spheres and the Superposition Approximation", Physical Review '''85''' pp. 777 - 783 (1952)]</ref>
<ref>[http://dx.doi.org/10.1063/1.1742004 B. J. Alder, S. P. Frankel, and V. A. Lewinson "Radial Distribution Function Calculated by the Monte-Carlo Method for a Hard Sphere Fluid", Journal of Chemical Physics '''23''' pp. 417- (1955)]</ref>
<ref>[http://dx.doi.org/10.1063/1.1727245 Francis H. Ree, R. Norris Keeler, and Shaun L. McCarthy "Radial Distribution Function of Hard Spheres", Journal of Chemical Physics '''44''' pp. 3407- (1966)]</ref>
<ref>[http://dx.doi.org/10.1080/00268977000101421 W. R. Smith and D. Henderson "Analytical representation of the Percus-Yevick hard-sphere radial distribution function", Molecular Physics '''19''' pp. 411-415 (1970)]</ref>
<ref>[http://dx.doi.org/10.1080/00268977100101331 J. A. Barker and D. Henderson "Monte Carlo values for the radial distribution function of a system of fluid hard spheres", Molecular Physics '''21''' pp. 187-191 (1971)]</ref>
<ref>[http://dx.doi.org/10.1080/00268977700102241 J. M. Kincaid and J. J. Weis "Radial distribution function of a hard-sphere solid", Molecular Physics '''34''' pp. 931-938 (1977)]</ref>
<ref>[http://dx.doi.org/10.1103/PhysRevA.43.5418 S. Bravo Yuste and A. Santos "Radial distribution function for hard spheres", Physical Review A '''43''' pp. 5418-5423 (1991)]</ref>
<ref>[http://dx.doi.org/10.1080/00268979400100491 Jaeeon Chang and Stanley I. Sandler "A real function representation for the structure of the hard-sphere fluid", Molecular Physics '''81''' pp. 735-744 (1994)]</ref>
<ref>[http://dx.doi.org/10.1063/1.1979488 Andrij Trokhymchuk, Ivo Nezbeda and Jan Jirsák "Hard-sphere radial distribution function again", Journal of Chemical Physics '''123''' 024501 (2005)]</ref>
<ref>[http://dx.doi.org/10.1063/1.2201699 M. López de Haro, A. Santos and S. B. Yuste "On the radial distribution function of a hard-sphere fluid", Journal of Chemical Physics '''124''' 236102 (2006)]</ref>
==Liquid-solid transition==
The hard sphere system undergoes a [[Solid-liquid phase transitions |liquid-solid]] [[First-order transitions |first order transition]] <ref name="HooverRee">[http://dx.doi.org/10.1063/1.1670641 William G. Hoover and Francis H. Ree "Melting Transition and Communal Entropy for Hard Spheres", Journal of Chemical Physics '''49''' pp. 3609-3617 (1968)]</ref>, sometimes referred to as the Kirkwood-Alder transition <ref name="GastRussel">[http://dx.doi.org/10.1063/1.882495 Alice P. Gast and William B. Russel "Simple Ordering in Complex Fluids", Physics Today '''51''' (12) pp. 24-30 (1998)]</ref>.
The liquid-solid coexistence densities (<math>\rho^* = \rho \sigma^3</math>) are given by
:{| border="1"
|-
| <math>\rho^*_{\mathrm {solid}}</math> || <math>\rho^*_{\mathrm {liquid}}</math> || Reference
|-
| 1.041|| 0.945 || <ref name="HooverRee"> </ref>
|-
| 1.0376|| 0.9391 || <ref name="FrenkelSmitBook">Daan Frenkel and Berend Smit "Understanding Molecular Simulation: From Algorithms to Applications", Second Edition (2002) (ISBN 0-12-267351-4) p. 261.</ref>
|-
| 1.0367(10) || 0.9387(10) || <ref name="Fortini">[http://dx.doi.org/10.1088/0953-8984/18/28/L02 Andrea Fortini and Marjolein Dijkstra "Phase behaviour of hard spheres confined between parallel hard plates: manipulation of colloidal crystal structures by confinement", Journal of Physics: Condensed Matter '''18''' pp. L371-L378 (2006)]</ref>
|-
| 1.0372 || 0.9387 || <ref name="VegaNoya"> [http://dx.doi.org/10.1063/1.2790426 Carlos Vega and Eva G. Noya "Revisiting the Frenkel-Ladd method to compute the free energy of solids: The Einstein molecule approach", Journal of Chemical Physics '''127''' 154113 (2007)]</ref>
|-
| 1.0369(33) || 0.9375(14) || <ref name="Noya"> [http://dx.doi.org/10.1063/1.2901172 Eva G. Noya, Carlos Vega, and Enrique de Miguel "Determination of the melting point of hard spheres from direct coexistence simulation methods", Journal of Chemical Physics '''128''' 154507 (2008)]</ref>
|-
| 1.037 || 0.938 || <ref>[http://dx.doi.org/10.1063/1.476396 Ruslan L. Davidchack and Brian B. Laird "Simulation of the hard-sphere crystal–melt interface", Journal of Chemical Physics '''108''' pp. 9452-9462 (1998)]</ref>
|-
| 1.035(3) || 0.936(2) || <ref name="Miguel"> [http://dx.doi.org/10.1063/1.3023062 Enrique de Miguel "Estimating errors in free energy calculations from thermodynamic integration using fitted data", Journal of Chemical Physics '''129''' 214112 (2008)]</ref>
|}
The coexistence [[pressure]] is given by
:{| border="1"
|-
| <math>p (k_BT/\sigma^3) </math> || Reference
|-
| 11.567|| <ref name="FrenkelSmitBook"> </ref>
|-
| 11.57(10) || <ref name="Fortini"> </ref>
|-
| 11.54(4) || <ref name="Noya"> </ref>
|-
| 11.50(9) || <ref>[http://dx.doi.org/10.1103/PhysRevLett.85.5138 N. B. Wilding and A. D. Bruce "Freezing by Monte Carlo Phase Switch", Physical Review Letters '''85''' pp. 5138-5141 (2000)]</ref>
|-
| 11.55(11) || <ref>[http://dx.doi.org/10.1088/0953-8984/9/41/006 Robin J. Speedy "Pressure of the metastable hard-sphere fluid", Journal of Physics: Condensed Matter '''9''' pp. 8591-8599 (1997)]</ref>
|-
| 11.48(11) || <ref name="Miguel"> </ref>
|-
| 11.43(17) || <ref>[http://dx.doi.org/10.1063/1.3244562 G. Odriozola "Replica exchange Monte Carlo applied to hard spheres", Journal of Chemical Physics '''131''' 144107 (2009)]</ref>
|}
The coexistence [[chemical potential]] is given by
:{| border="1"
|-
| <math>\mu (k_BT) </math> || Reference
|-
| 15.980(11) || <ref name="Miguel"> </ref>
|}
The [[Helmholtz energy function]] (in units of <math>Nk_BT</math>) is given by
:{| border="1"
|-
| <math>A_{\mathrm {solid}}</math> || <math>A_{\mathrm {liquid}}</math> || Reference
|-
| 4.887(3) || 3.719(8) || <ref name="Miguel"> </ref>
|}

==Helmholtz energy function==
Values for the [[Helmholtz energy function]] (<math>A</math>) are given in the following Table:
:{| border="1"
|-
| <math>\rho^*</math> || <math>A/(Nk_BT)</math>|| Reference
|-
| 0.25 || 0.620 <math>\pm</math> 0.002 || Table I <ref name="Schilling"> [http://dx.doi.org/10.1063/1.3274951 T. Schilling and F. Schmid "Computing absolute free energies of disordered structures by molecular simulation", Journal of Chemical Physics '''131''' 231102 (2009)]</ref>
|-
| 0.50 || 1.541 <math>\pm</math> 0.002 || Table I <ref name="Schilling"> </ref>
|-
| 0.75 || 3.009 <math>\pm</math> 0.002 || Table I <ref name="Schilling"> </ref>
|-
| 1.04086 || 4.959 || Table VI <ref name="VegaNoya"> </ref>
|-
| 1.099975 || 5.631 || Table VI <ref name="VegaNoya"> </ref>
|-
| 1.150000 || 6.274 || Table VI <ref name="VegaNoya"> </ref>
|}
==Interfacial Helmholtz energy function==
The [[Helmholtz energy function]] of the solid–liquid [[interface]] has been calculated using the [[cleaving method]] giving (Ref. <ref>[http://dx.doi.org/10.1063/1.3514144 Ruslan L. Davidchack "Hard spheres revisited: Accurate calculation of the solid–liquid interfacial free energy", Journal of Chemical Physics '''133''' 234701 (2010)]</ref> Table I):
:{| border="1"
|-
| || [[work]] per unit area/<math>(k_BT/\sigma^2)</math>
|-
| <math>\gamma_{100}</math> || 0.5820(19)
|-
| <math>\gamma_{110}</math> || 0.5590(20)
|-
| <math>\gamma_{111}</math> || 0.5416(31)
|-
| <math>\gamma_{120}</math> || 0.5669(20)
|}
==Solid structure==
The [http://mathworld.wolfram.com/KeplerConjecture.html Kepler conjecture] states that the optimal packing for three dimensional spheres is either cubic or hexagonal close [[Lattice Structures | packing]], both of which have maximum densities of <math>\pi/(3 \sqrt{2}) \approx 74.048%</math> <ref>[http://dx.doi.org/10.1038/26609 Neil J. A. Sloane "Kepler's conjecture confirmed", Nature '''395''' pp. 435-436 (1998)]</ref>
<ref>[http://dx.doi.org/10.1103/PhysRevE.52.3632 C. F. Tejero, M. S. Ripoll, and A. Pérez "Pressure of the hard-sphere solid", Physical Review E '''52''' pp. 3632-3636 (1995)]</ref>. However, for hard spheres at close packing the [[Building up a face centered cubic lattice |face centred cubic]] phase is the more stable
<ref>[http://dx.doi.org/10.1039/a701761h Leslie V. Woodcock "Computation of the free energy for alternative crystal structures of hard spheres", Faraday Discussions '''106''' pp. 325 - 338 (1997)]</ref>
*See also: [[Equations of state for crystals of hard spheres]]
==Direct correlation function==
For the [[direct correlation function]] see:
<ref>[http://dx.doi.org/10.1080/00268970701725021 C. F. Tejero and M. López De Haro "Direct correlation function of the hard-sphere fluid", Molecular Physics '''105''' pp. 2999-3004 (2007)]</ref>
<ref>[http://dx.doi.org/10.1080/00268970902784934 Matthew Dennison, Andrew J. Masters, David L. Cheung, and Michael P. Allen "Calculation of direct correlation function for hard particles using a virial expansion", Molecular Physics pp. 375-382 (2009)]</ref>
==Bridge function==
Details of the [[bridge function]] for hard sphere can be found in the following publication
<ref>[http://dx.doi.org/10.1080/00268970210136357 Jiri Kolafa, Stanislav Labik and Anatol Malijevsky "The bridge function of hard spheres by direct inversion of computer simulation data", Molecular Physics '''100''' pp. 2629-2640 (2002)]</ref>
== Equations of state ==
:''Main article: [[Equations of state for hard spheres]]''
==Virial coefficients==
:''Main article: [[Hard sphere: virial coefficients]]''
== Experimental results ==
Pusey and van Megen used a suspension of PMMA particles of radius 305 <math>\pm</math>10 nm, suspended in poly-12-hydroxystearic acid <ref>[http://dx.doi.org/10.1038/320340a0 P. N. Pusey and W. van Megen "Phase behaviour of concentrated suspensions of nearly hard colloidal spheres", Nature '''320''' pp. 340-342 (1986)]</ref>
For results obtained from the [http://exploration.grc.nasa.gov/expr2/cdot.html Colloidal Disorder - Order Transition] (CDOT) experiments performed on-board the Space Shuttles ''Columbia'' and ''Discovery'' see Ref. <ref>[http://dx.doi.org/10.1016/S0261-3069(01)00015-2 Z. Chenga, P. M. Chaikina, W. B. Russelb, W. V. Meyerc, J. Zhub, R. B. Rogersc and R. H. Ottewilld, "Phase diagram of hard spheres", Materials & Design '''22''' pp. 529-534 (2001)]</ref>
==Mixtures==
*[[Binary hard-sphere mixtures]]
*[[Multicomponent hard-sphere mixtures]]
== Related systems ==
*[[Quantum hard spheres]]
*[[Dipolar hard spheres]]
*[[Lattice hard spheres]]
Hard spheres in other dimensions:
* 1-dimensional case: [[1-dimensional hard rods | hard rods]].
* 2-dimensional case: [[Hard disks | hard disks]].
* [[Hard hyperspheres]]
==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1007/978-3-540-78767-9 "Theory and Simulation of Hard-Sphere Fluids and Related Systems", Lecture Notes in Physics '''753/2008''' Springer (2008)]
*[http://dx.doi.org/10.1063/1.3506838 Laura Filion, Michiel Hermes, Ran Ni and Marjolein Dijkstra "Crystal nucleation of hard spheres using molecular dynamics, umbrella sampling, and forward flux sampling: A comparison of simulation techniques", Journal of Chemical Physics '''133''' 244115 (2010)]

==External links==
*[http://www.smac.lps.ens.fr/index.php/Programs_Chapter_2:_Hard_disks_and_spheres Hard disks and spheres] computer code on SMAC-wiki.
[[Category:Models]]
[[category: hard sphere]]

File:Hard-sphere phase diagram pressure vs packing fraction.png

2012-08-15T09:45:55Z

Franzl aus tirol: Phase diagram of hard sphere system (Solid line - stable branch, dashed line - metastable branch): Reduced pressure <math>\beta P\sigma^3</math> as a function of the volume fraction (or packing fraction) <math>\eta</math>. With inverse thermal energy <ma

Phase diagram of hard sphere system (Solid line - stable branch, dashed line - metastable branch): Reduced pressure <math>\beta P\sigma^3</math> as a function of the volume fraction (or packing fraction) <math>\eta</math>.

With inverse thermal energy <math>\beta=(kT)^{-1}</math>, sphere diameter <math>\sigma=2R</math>, packing fraction <math>\eta=V_0\rho</math>, volume of a single sphere <math>V_0=4\pi R^3/3=\pi\sigma^3/6</math>, number density <math>\rho=N/V</math>, system volume <math>V</math>, total particle number <math>N</math>.

Vertical grey lines indicate:
* freezing: <math>\eta_\mathrm{f}\approx 0.494</math>
* melting: <math>\eta_\mathrm{m}\approx 0.545</math>
* random close packing: <math>\eta_\mathrm{rcp}\approx 0.644</math>
* close packing: <math>\eta_\mathrm{cp}=\pi \sqrt{2}/6 \approx 0.74048</math>

Reference:
* A. Mulero: ''Theory and Simulation of Hard-Sphere Fluids and Related Systems''. Springer 2008, ISBN 978-3-540-78766-2

1-dimensional hard rods

2012-08-15T09:39:12Z

Franzl aus tirol: Boltzmann factor, Configuration Integral

'''1-dimensional hard rods''' (sometimes known as a ''Tonks gas'' <ref>[http://dx.doi.org/10.1103/PhysRev.50.955 Lewi Tonks "The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic Spheres", Physical Review '''50''' pp. 955- (1936)]</ref>) consist of non-overlapping line segments of length <math>\sigma</math> who all occupy the same line which has length <math>L</math>. One could also think of this model as being a string of [[hard sphere model | hard spheres]] confined to 1 dimension (not to be confused with [[3-dimensional hard rods]]). The model is given by the [[intermolecular pair potential]]:

: <math> \Phi_{12}(x_{i},x_{j})=\left\{ \begin{array}{lll}
0 & ; & |x_{i}-x_{j}|>\sigma\\ \infty & ; & |x_{i}-x_{j}|<\sigma \end{array}\right. </math>

where <math> \left. x_k \right. </math> is the position of the center of the k-th rod, along with an external potential. Thus, the Boltzmann factor is

: <math>e_{ij}:=e^{-\beta\Phi_{12}(x_{i},x_{j})}=\Theta(|x_{i}-x_{j}|-\sigma)=\left\{ \begin{array}{lll} 1 & ; & |x_{i}-x_{j}|>\sigma\\ 0 & ; & |x_{i}-x_{j}|<\sigma \end{array}\right. </math>

The whole length of the rod must be inside the range:

: <math> V_{0}(x_i) = \left\{ \begin{array}{lll} 0 & ; & \sigma/2 < x_i < L - \sigma/2 \\
\infty &; & {\mathrm {elsewhere}}. \end{array} \right. </math>

== Canonical Ensemble: Configuration Integral ==
The [[statistical mechanics]] of this system can be solved exactly.
Consider a system of length <math> \left. L \right. </math> defined in the range <math> \left[ 0, L \right] </math>. The aim is to compute the [[partition function]] of a system of <math> \left. N \right. </math> hard rods of length <math> \left. \sigma \right. </math>.
Consider that the particles are ordered according to their label: <math> x_0 < x_1 < x_2 < \cdots < x_{N-1} </math>;
taking into account the pair potential we can write the canonical partition function
([http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 configuration integral])
of a system of <math> N </math> particles as:

:<math>\begin{align}
\frac{Z\left(N,L\right)}{N!} & =\int_{\sigma/2}^{L-\sigma/2}dx_{0}\int_{\sigma/2}^{L-\sigma/2}dx_{1}\cdots\int_{\sigma/2}^{L-\sigma/2}dx_{N-1}\prod_{i=1}^{N-1}e_{i-1,i}\\
& =\int_{\sigma/2}^{L+\sigma/2-N\sigma}dx_{0}\int_{x_{0}+\sigma}^{L+\sigma/2-N\sigma+\sigma}dx_{1}\cdots\int_{x_{i-1}+\sigma}^{L+\sigma/2-N\sigma+i\sigma}dx_{i}\cdots\int_{x_{N-2}+\sigma}^{L+\sigma/2-N\sigma+(N-1)\sigma}dx_{N-1}.
\end{align}</math>

Variable change: <math> \left. \omega_k = x_k - \left(k+\frac{1}{2}\right) \sigma \right. </math> ; we get:

:<math>\begin{align}
\frac{Z\left(N,L\right)}{N!} & =\int_{0}^{L-N\sigma}d\omega_{0}\int_{\omega_{0}}^{L-N\sigma}d\omega_{1}\cdots\int_{\omega_{i-1}}^{L-N\sigma}d\omega_{i}\cdots\int_{\omega_{N-2}}^{L-N\sigma}d\omega_{N-1}\\
& =\int_{0}^{L-N\sigma}d\omega_{0}\cdots\int_{\omega_{i-1}}^{L-N\sigma}d\omega_{i}\frac{(L-N\sigma-\omega_{i})^{N-1-i}}{(N-1-i)!}=\int_{0}^{L-N\sigma}d\omega_{0}\frac{(L-N\sigma-\omega_{0})^{N-1}}{(N-1)!}
\end{align}</math>

Therefore:
:<math>
\frac{ Z \left( N,L \right)}{N!} = \frac{ (L-N\sigma )^{N} }{N!}.
</math>

: <math>
Q(N,L) = \frac{ (L-N \sigma )^N}{\Lambda^N N!}.
</math>

== Thermodynamics ==
[[Helmholtz energy function]]
: <math> \left. A(N,L,T) = - k_B T \log Q \right. </math>

In the [[thermodynamic limit]] (i.e. <math> N \rightarrow \infty; L \rightarrow \infty</math> with <math> \rho = \frac{N}{L} </math>, remaining finite):

:<math> A \left( N,L,T \right) = N k_B T \left[ \log \left( \frac{ N \Lambda} { L - N \sigma }\right) - 1 \right]. </math>

== Equation of state ==
Using the [[thermodynamic relations]], the [[pressure]] (''linear tension'' in this case) <math> \left. p \right. </math> can
be written as:

:<math>
p = - \left( \frac{ \partial A}{\partial L} \right)_{N,T} = \frac{ N k_B T}{L - N \sigma};
</math>

:<math>
Z = \frac{p L}{N k_B T} = \frac{1}{ 1 - \eta},
</math>

where <math> \eta \equiv \frac{ N \sigma}{L} </math>; is the fraction of volume (i.e. length) occupied by the rods.

It was shown by van Hove <ref>[http://dx.doi.org/10.1016/0031-8914(50)90072-3 L. van Hove, "Sur L'intégrale de Configuration Pour Les Systèmes De Particules À Une Dimension", Physica, '''16''' pp. 137-143 (1950)]</ref> that there is no [[Solid-liquid phase transitions |fluid-solid phase transition]] for this system (hence the designation ''Tonks gas'').
== Isobaric ensemble: an alternative derivation ==
Adapted from Reference <ref>J. M. Ziman ''Models of Disorder: The Theoretical Physics of Homogeneously Disordered Systems'', Cambridge University Press (1979) ISBN 0521292808</ref>. If the rods are ordered according to their label: <math> x_0 < x_1 < x_2 < \cdots < x_{N-1} </math> the canonical [[partition function]] can also be written as:
: <math>
Z=
\int_0^{x_1} d x_0
\int_0^{x_2} d x_1
\cdots
\int_0^{L} d x_{N-1}
f(x_1-x_0)
f(x_2-x_1)
\cdots
f(L-x_{N-1}),
</math>
where <math>N!</math> does not appear one would have <math>N!</math> analogous expressions
by permuting the label of the (distinguishable) rods. <math>f(x)</math> is the [[Boltzmann factor]]
of the hard rods, which is <math>0</math> if <math>x<\sigma</math> and <math>1</math> otherwise.

A variable change to the distances between rods: <math> y_k = x_k - x_{k-1} </math> results in
: <math>
Z =
\int_0^{\infty} d y_0
\int_0^{\infty} d y_1
\cdots
\int_0^{\infty} d y_{N-1}
f(y_1)
f(y_2)
\cdots
f(y_{N-1}) \delta \left( \sum_{i=0}^{N-1} y_i-L \right):
</math>
the distances can take any value as long as they are not below <math>\sigma</math> (as enforced
by <math>f(y)</math>) and as long as they add up to <math>L</math> (as enforced by the [[Dirac_delta_distribution | Dirac delta]]). Writing the later as the inverse [[Laplace transform]] of an exponential:
: <math>
Z =
\int_0^{\infty} d y_0
\int_0^{\infty} d y_1
\cdots
\int_0^{\infty} d y_{N-1}
f(y_1)
f(y_2)
\cdots
f(y_{N-1})
\frac{1}{2\pi i } \int_{-\infty}^{\infty} ds \exp \left[ - s \left(\sum_{i=0}^{N-1} y_i-L \right)\right].
</math>
Exchanging integrals and expanding the exponential the <math>N</math> integrals decouple:
:<math>
Z =
\frac{1}{2\pi i } \int_{-\infty}^{\infty} ds
e^{ L s }
\left\{
\int_0^{\infty} d y f(y) e^{ - s y }
\right\}^N.
</math>
We may proceed to invert the Laplace transform (e.g. by means of the residues theorem), but this is not needed: we see our configuration integral is the inverse Laplace transform of another one,
:<math>
Z'(s)= \left\{ \int_0^{\infty} d y f(y) e^{ - s y } \right\}^N, </math>
so that
:<math>
Z'(s) = \int_0^{\infty} ds e^{ L s } Z(L).
</math>
This is precisely the transformation from the configuration integral in the canonical (<math>N,T,L</math>) ensemble to the isobaric (<math>N,T,p</math>) one, if one identifies
<math>s=p/k T</math>. Therefore, the [[Gibbs energy function]] is simply <math>G=-kT\log Z'(p/kT) </math>, which easily evaluated to be <math>G=kT N \log(p/kT)+p\sigma N</math>. The [[chemical potential]] is <math>\mu=G/N</math>, and by means of thermodynamic identities such as <math>\rho=\partial p/\partial \mu</math> one arrives at the same equation of state as the one given above.
==Confined hard rods==
<ref>[http://dx.doi.org/10.1080/00268978600101521 A. Robledo and J. S. Rowlinson "The distribution of hard rods on a line of finite length", Molecular Physics '''58''' pp. 711-721 (1986)]</ref>
==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1016/0031-8914(49)90059-2 L. van Hove "Quelques Propriétés Générales De L'intégrale De Configuration D'un Système De Particules Avec Interaction", Physica, '''15''' pp. 951-961 (1949)]
*[http://dx.doi.org/10.1063/1.1699116 Zevi W. Salsburg, Robert W. Zwanzig, and John G. Kirkwood "Molecular Distribution Functions in a One-Dimensional Fluid", Journal of Chemical Physics '''21''' pp. 1098-1107 (1953)]
*[http://dx.doi.org/10.1063/1.1699263 Robert L. Sells, C. W. Harris, and Eugene Guth "The Pair Distribution Function for a One-Dimensional Gas", Journal of Chemical Physics '''21''' pp. 1422-1423 (1953)]
*[http://dx.doi.org/10.1063/1.1706788 Donald Koppel "Partition Function for a Generalized Tonks' Gas", Physics of Fluids '''6''' 609 (1963)]
*[http://dx.doi.org/10.1103/PhysRev.171.224 J. L. Lebowitz, J. K. Percus and J. Sykes "Time Evolution of the Total Distribution Function of a One-Dimensional System of Hard Rods", Physical Review '''171''' pp. 224-235 (1968)]
*[http://dx.doi.org/10.3390/e10030248 Paolo V. Giaquinta "Entropy and Ordering of Hard Rods in One Dimension", Entropy '''10''' pp. 248-260 (2008)]

[[Category:Models]]
[[Category:Statistical mechanics]]