In
statistical mechanics, the hard hexagon model is a 2-dimensional
lattice model of a gas, where particles are allowed to be on the vertices of a
triangular lattice
The hexagonal lattice or triangular lattice is one of the five two-dimensional Bravais lattice types. The symmetry category of the lattice is wallpaper group p6m. The primitive translation vectors of the hexagonal lattice form an angle of 120° ...
but no two particles may be adjacent.
The model was solved by , who found that it was related to the
Rogers–Ramanujan identities In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions. The identities were first discovered and proved by , and were subsequently rediscovered (without a proof) by Sriniv ...
.
The partition function of the hard hexagon model
The hard hexagon model occurs within the framework of the
grand canonical ensemble
In statistical mechanics, the grand canonical ensemble (also known as the macrocanonical ensemble) is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that are in thermodynamic equilibriu ...
, where the total number of particles (the "hexagons") is allowed to vary naturally, and is fixed by a
chemical potential
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
. In the hard hexagon model, all valid states have zero energy, and so the only important thermodynamic control variable is the ratio of chemical potential to temperature ''μ''/(''kT''). The exponential of this ratio, ''z'' = exp(''μ''/(''kT'')) is called the activity and larger values correspond roughly to denser configurations.
For a triangular lattice with ''N'' sites, the
grand partition function is
:
where ''g''(''n'', ''N'') is the number of ways of placing ''n'' particles on distinct lattice sites such that no 2 are adjacent. The function κ is defined by
:
so that log(κ) is the free energy per unit site. Solving the hard hexagon model means (roughly) finding an exact expression for κ as a function of ''z''.
The mean density ρ is given for small ''z'' by
:
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 ρ
1, ρ
2, ρ
3, 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
with
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
''φ''. 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
:
:
Solution
The solution is given for small values of ''z'' < ''z''
''c'' by
:
:
:
where
:
:
:
:
For large ''z'' > ''z''
''c'' the solution (in the phase where most occupied sites have type 1) is given by
:
:
:
:
:
The functions ''G'' and ''H'' turn up in the
Rogers–Ramanujan identities In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions. The identities were first discovered and proved by , and were subsequently rediscovered (without a proof) by Sriniv ...
, and the function ''Q'' is the
Euler function
In mathematics, the Euler function is given by
:\phi(q)=\prod_^\infty (1-q^k),\quad , q, A000203
On account of the identity \sum_ d = \sum_ \frac, this may also be written as
:\ln(\phi(q)) = -\sum_^\infty \frac \sum_ d.
Also if a,b\in\mathbb^ ...
, which is closely related to the
Dedekind eta function
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string ...
. If ''x'' = e
2πiτ, then ''x''
−1/60''G''(''x''), ''x''
11/60''H''(''x''), ''x''
−1/24''P''(''x''), ''z'', κ, ρ, ρ
1, ρ
2, and ρ
3 are
modular function
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of ...
s of τ, while ''x''
1/24''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 ''κ'', ''z'', ''R'', ''ρ'' are all algebraic functions of each other (of quite high degree) . In particular, the value of ''κ''(1), which
Eric Weisstein
Eric Wolfgang Weisstein (born March 18, 1969) is an American mathematician and encyclopedist who created and maintains the encyclopedias ''MathWorld'' and ''ScienceWorld''. In addition, he is the author of the '' CRC Concise Encyclopedia of M ...
dubbed the hard hexagon entropy constant , is an
algebraic number of degree 24 equal to 1.395485972... ().
Related models
The hard hexagon model can be defined similarly on the square and honeycomb lattices. No exact solution is known for either of these models, but the critical point ''z''
c is near for the square lattice and for the honeycomb lattice; ''κ''(1) is approximately 1.503048082... () for the square lattice and 1.546440708... for the honeycomb lattice .
References
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Exton, H. (1983), ''q-Hypergeometric Functions and Applications'', New York: Halstead Press, Chichester: Ellis Horwood
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External links
*{{mathworld, urlname=HardHexagonEntropyConstant, title=Hard Hexagon Entropy Constant
Exactly solvable models
Statistical mechanics
Lattice models
Modular forms
Algebraic numbers