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In [[statistical mechanics]], the '''hard hexagon model''' is a 2-dimensional [[Lattice model (physics)|lattice model]] of a gas, where particles are allowed to be on the vertices of a [[hexagonal lattice|triangular lattice]] but no two particles may be adjacent.


The model was solved by {{harvs|authorlink=Rodney J. Baxter|last=Baxter|year=1980|txt}}, who found that it was related to the [[Rogers-Ramanujan identities]].


==The partition function of the hard hexagon model==
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For a triangular lattice with ''N'' sites, the [[grand partition function]] is
 
:<math>\displaystyle \mathcal Z(z) = \sum_n z^n g(n,N) = 1+Nz+ \tfrac{1}{2}N(N-7)z^2+\cdots</math>
 
where ''g''(''n'', ''N'') is the number of ways of placing ''n'' particles on distinct lattice sites such that no 2 are adjacent. The variable ''z'' is called the '''activity''' and larger values correspond roughly to denser configurations. The function &kappa; is defined by
 
:<math>\kappa(z) = \lim_{N\rightarrow\infty} \mathcal Z(z)^{1/N} = 1+z-3z^2+\cdots</math>
 
so that log(&kappa;) is the free energy per unit site. Solving the hard hexagon model means (roughly) finding an exact expression for &kappa; as a function of ''z''.
 
The '''mean density''' &rho; is given for small ''z'' by
 
:<math>\rho= z\frac{d\log(\kappa)}{dz}  =z-7z^2+58z^3-519z^4+4856z^5+\cdots.</math>
 
The vertices of the lattice fall into 3 classes numbered 1, 2, and 3, given by the 3 different ways to fill space with hard hexagons. There are 3 local densities &rho;<sub>1</sub>, &rho;<sub>2</sub>, &rho;<sub>3</sub>, corresponding to the 3 classes of sites. When the activity is large the system approximates one of these 3 packings, so the local densities differ, but when the activity is below a critical point the three local densities are the same. The critical point  separating the low-activity homogeneous phase from the high-activity ordered phase is ''z''<sub>''c''</sub> = (11&nbsp;+&nbsp;5<sup>3/2</sup>)/2 = 11.0917....  Above the critical point the local densities  differ and in the phase where most hexagons are on sites of type 1 can be expanded as
 
:<math> \rho_1 = 1-z^{-1}-5z^{-2}-34z^{-3}-267z^{-4}-2037z^{-5}-\cdots</math>  
:<math> \rho_2=\rho_3 = z^{-2} + 9z^{-3} + 80z^{-4} +  965z^{-5}-\cdots.</math>
 
==Solution==
The solution is given for small values of ''z''&nbsp;<&nbsp;''z''<sub>''c''</sub> by
 
:<math>\displaystyle z=\frac{-xH(x)^5}{G(x)^5}</math>
 
:<math>
\kappa = \frac{H(x)^3 Q(x^5)^2} {G(x)^2}
\prod_{n\ge 1} \frac{(1-x^{6n-4})(1-x^{6n-3})^2(1-x^{6n-2})}
{(1-x^{6n-5})(1-x^{6n-1})(1-x^{6n})^2}
</math>
 
:<math>\rho =\rho_1=\rho_2=\rho_3= \frac{-xG(x)H(x^6)P(x^3)}{P(x)}</math>
 
where
:<math>G(x) = \prod_{n\ge 1}\frac{1}{(1-x^{5n-4})(1-x^{5n-1})}</math>
 
:<math>H(x) = \prod_{n\ge 1}\frac{1}{(1-x^{5n-3})(1-x^{5n-2})}</math>
 
:<math>P(x) = \prod_{n\ge 1}(1-x^{2n-1}) = Q(x)/Q(x^2)</math>
 
:<math>Q(x) = \prod_{n\ge 1}(1-x^n).</math>
 
For large ''z''&nbsp;>&nbsp;''z''<sub>''c''</sub> the solution (in the phase where most occupied sites have type 1) is given by
:<math>\displaystyle z=\frac{G(x)^5}{xH(x)^5}</math>
 
:<math>
\kappa = \frac{G(x)^3 Q(x^5)^2} {H(x)^2}
\prod_{n\ge 1} \frac{(1-x^{3n-2})(1-x^{3n-1})}
{(1-x^{3n})^2}
</math>
 
:<math>\rho_1 = \frac{H(x)Q(x)(G(x)Q(x)+x^2H(x^9)Q(x^9))}{Q(x^3)^2}</math>
 
:<math>\rho_2=\rho_3 = \frac{x^2H(x)Q(x)H(x^9)Q(x^9)}{Q(x^3)^2}</math>
 
:<math>R=\rho_1-\rho_2= \frac{Q(x)Q(x^5)}{Q(x^3)^2}.</math>
 
The functions ''G'' and ''H'' turn up in the [[Rogers-Ramanujan identities]], and the function ''Q'' is closely related to the [[Dedekind eta function]]. If ''x'' = e<sup>2&pi;i&tau;</sup>, then ''q''<sup>&minus;1/60</sup>''G''(''x''), ''x''<sup>11/60</sup>''H''(''x''), ''x''<sup>&minus;1/24</sup>''P''(''x''), ''z'', &kappa;, &rho;, &rho;<sub>1</sub>,  &rho;<sub>2</sub>, and &rho;<sub>3</sub> are [[modular function]]s of &tau;, while ''x''<sup>1/24</sup>''Q''(''x'') is a modular form of weight 1/2. Since any two modular functions are related by an algebraic relation, this implies that the functions ''&kappa;'', ''z'', ''R'', ''&rho;'' are all algebraic functions of each other (of quite high degree) {{harv|Joyce|1988}}.
 
==References==
*{{Citation | doi=10.1073/pnas.78.9.5290 | authorlink=George E. Andrews | last1=Andrews | first1=George E. | title=The hard-hexagon model and Rogers-Ramanujan type identities | url=http://www.pnas.org/content/78/9/5290.abstract?ck=nck | id={{MathSciNet | id = 629656}} | year=1981 | journal=[[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]] | issn=0027-8424 | volume=78 | issue=9 | pages=5290&ndash;5292 | pmid=16593082 | pmc=348728}}
*{{Citation | last1=Baxter | first1=Rodney J. | title=Hard hexagons: exact solution | doi=10.1088/0305-4470/13/3/007  | id={{MathSciNet | id = 560533}} | year=1980 | journal=Journal of Physics. A. Mathematical and General | issn=0305-4470 | volume=13 | issue=3 | pages=L61&ndash;L70}}
*{{Citation | last1=Baxter | first1=Rodney J. | title=Exactly solved models in statistical mechanics | url=http://physics.anu.edu.au/theophys/_files/Exactly.pdf| publisher=Academic Press Inc. [Harcourt Brace Jovanovich Publishers] | location=London | isbn=978-0-12-083180-7 | id={{MathSciNet | id = 690578}} | year=1982}}
*{{Citation | last1=Joyce | first1=G. S. | title=Exact results for the activity and isothermal compressibility of the hard-hexagon model | doi=10.1088/0305-4470/21/20/005  | id={{MathSciNet | id = 966792}} | year=1988 | journal=Journal of Physics. A. Mathematical and General | issn=0305-4470 | volume=21 | issue=20 | pages=L983&ndash;L988}}
 
==External links==
*{{mathworld|urlname=HardHexagonEntropyConstant|title=Hard Hexagon  Entropy Constant}}
 
[[Category:Exactly solvable models]]
[[Category:Statistical mechanics]]
[[Category:Lattice models]]
[[Category:Modular forms]]

Latest revision as of 11:49, 7 February 2014


By investing in a premium Word - Press theme, you're investing in the future of your website. Offshore expert Word - Press developers high level of interactivity, accessibility, functionality and usability of our website can add custom online to using. Your parishioners and certainly interested audience can come in to you for further information from the group and sometimes even approaching happenings and systems with the church. They found out all the possible information about bringing up your baby and save money at the same time. The number of options offered here is overwhelming, but once I took the time to begin to review the video training, I was amazed at how easy it was to create a squeeze page and a membership site.

Creating a website from scratch can be such a pain. Infertility can cause a major setback to the couples due to the inability to conceive. A Wordpress plugin is a software that you can install into your Wordpress site. Now, I want to anxiety that not every single query will be answered. Now a days it has since evolved into a fully capable CMS platform which make it, the best platform in the world for performing online business.

Here are a few reasons as to why people prefer Word - Press over other software's. The following piece of content is meant to make your choice easier and reassure you that the decision to go ahead with this conversion is requited with rich benefits:. After age 35, 18% of pregnancies will end in miscarriage. You can allow visitors to post comments, or you can even allow your visitors to register and create their own personal blogs. Article Source: Stevens works in Internet and Network Marketing.

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