Conductivity of Coniglio-Klein clusters.

Area of Science:

• Statistical Physics
• Condensed Matter Physics
• Computational Physics

Background:

• The q-state Potts model is a fundamental model in statistical mechanics.
• Understanding conductivity exponents is crucial for characterizing critical phenomena in disordered systems.
• Coniglio-Klein clusters exhibit complex structures relevant to transport properties.

Purpose of the Study:

• To compute the reduced conductivity exponent (t/ν) for critical Coniglio-Klein clusters in two dimensions.
• To investigate the dependence of t/ν on the Potts model parameter q.
• To propose a conjecture for the behavior of t/ν for noninteger q values.

Main Methods:

• Numerical simulations of the q-state Potts model.
• Calculation of conductivity at criticality and its size dependence.
• Analysis of random walk diffusion on cluster backbones.

Main Results:

• Precise estimations of the reduced conductivity exponent t/ν for q=1, 2, 3, and 4.
• Values obtained for t/ν are 0.986 ± 0.012, 0.877 ± 0.014, 0.785 ± 0.015, and 0.658 ± 0.030, respectively.
• A conjecture for the dependence of t/ν on q for 0 ≤ q ≤ 4 is proposed: 40t/ν = 72 + 20g - 3g(2).

Conclusions:

• The study provides accurate numerical results for the reduced conductivity exponent in the q-state Potts model.
• The proposed conjecture offers a new theoretical insight into the behavior of transport properties for noninteger q.
• These findings contribute to a deeper understanding of critical phenomena and conductivity in disordered systems.