Hi,
I apologize for this apparently silly question, but please, could you point me out if there is an underlying relationship between the defect driven phase transition and the directed percolation?
Secondly, is it possible to have a system which undergoes a KT transition at T1 generating free vortices, and subsequent by a spatial spreading of disorder via directed percolation at T2?
Please, if there is any relevant examples and materials, do let me know.
Many thanks.
Wang Zhe
No, there's no relation between the KT transition and percolation. Percolation processes are not thermal but geometric transitions.
Dear Zhe,
Yes, you can have a relation between percolation and the correlation length, where the correlation length is defined to be the distance at which the probability of a site being connected to the origin falls to a level 1/e . In other words, the correlation function G(r) gives the average correlation between two lattice sites (one at the origin, the other at position r). That is, loosely speaking, it describes how much more likely it is for the site at position r to belong to the same cluster as the origin than it would be for a site chosen randomly from across the whole lattice.
As you go above the critical point pc, the probability of being in a large, finite cluster gets smaller and smaller, and it is only likely to be in an infinite cluster, or in a very small finite cluster. Obviously, the average finite cluster size decreases until p=1 when it becomes zero (everything is in the infinite cluster)
Dear Zhe,
The translation of thermodynamic and geometrical factors was made a long time ago for magnetic models
1. L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys.Rev. 65 (1944) 117.
2. S. Istrail, Statistical Mechanics, Three-Dimensionality and NP-Completeness: I. Universality of Intractability of the Partition Functions of the Ising Model Across Non-Planar Lattices, In 32nd ACM Symposium on the Theory of Computing (STOC00), ACM Press, Portland, Oregon (2000) 87–96
3.P.W. Kasteleyn and C.M. Fortuin J. Phys. Soc. Jpn., 26 (1969)(Suppl.) 11.
4.. C.M. Fortuin and P.W. Kasteleyn, On the random-cluster model: I. Introduction and relation to other models, Physica (Amsterdam) 57 (1972) 536–564
I don't know if that was made exactly in Kosterlitz-Thouless phases, but in principle it seems to be possible.
- Badges
- Science topic
- Similar topics
- Physics
- Condensed Matter Physics