Consider a regular tetrahedron with the correlation functions for empty, point, pair, triplet and tetrahedron being u0, u1, u2, u3 and u4. Let the corresponding cluster interactions being e0, e1, e2, e3 and e4. Take the multiplicities of the clusters be m0, m1, m2, m3 and m4. Using cluster expansion, the energy of the system can be expressed as
H1= e0 m0 u0+ e1 m1 u1 + e2 m2 u2+ e3 m3 u3 + e4 m4 u4.
For the same system, the internal energy can be expressed in terms of cluster energies as:
H2= EAAAA yAAAA + 4 EAAAB yAAAB + 6 EAABB yAABB + 4 EABBB yABBB + EBBBB yBBBB.
In the above expression Eijkl and yijkl represent the cluster energies and the cluster variables respectively.
Based on the relation between the correlations ui and the clyster variables yi, both the expressions can be made equal and the relation between the ei and Eijkl can be obtained.
Based on whether the multiplicity of the tetrahedron cluster is considered or not in H2, the relation between the ei and Eijkl gets altered and thereby the related thermodynamic properties and its derived properties.
Should the multiplicity be considered in H2 or not?
With a little investigation and calculations, it is found that the multiplicity of the cluster need not be included in H2.
Details:
Start with H1.
The correlation functions (u) can be expressed in terms of the cluster variables (y) using appropriate basis. Then collect the terms of y in the transformed expressed. In this transformed expression, it is found that the multiplicity of the maximal cluster does not get factored. So,
while expressing the internal energy using cluster energies, the multiplicity should not be included.
- Badges
- Science topic
- Similar topics
- Geoscience
- Atmospheric Sciences
- Climate Science
- Climate Change