Topics in Statistic Mechanics

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Homework 8 Markovian-chain Monte Carlo 1.Introduction 1.1 Simple Sampling 1.2 Importance Sampling 2.Markov Process 3.MCMC application on 2D Ising model Class Homework 1 Class Homework 2
Adv. Sta. Phy. Homework 1 Percolation Li,Zimeng PB06203182 Percolation Edited by Li,Zimeng Contents: 1.Definition 2.Application 3.Exact Solution on Cayley Tree 4.Exact Solution on Square Lattice The phase transition of Percolation is a typical Non-Hamiltonian System phase transition. It shows us property of continuous phase transition, and to some extent, in deep contact with the q-state Potts Model. This article would showcase a brief survey of a Percolation Model and its applications. 1.Definition Suppose there is a hypercubic lattice G, with sites and bonds combining the nearest [1] Bond Percolation The probability is p for a bond to be occupied, thus 1-p for those unoccupied. [2] Lattice Percolation The probability is p for a site to be occupied, thus 1-p for those unoccupied. In both models above, clusters are formed when nearest occupied bonds or sites are joined together. We define P(p) as the probability that an infinitely large cluster is formed. It is easy to see and also by computer simulation, that P(p) = 0 when p < , and p significantly increases when it passes p , later reaches 1 when p = 1. We call pc percolation threshold. It transpires that P(p) can be seen as the analogue of the order parameter of magnetic systems, which I will not proved here, q-states Potts Model. 2.Application [1]Forest Fires Percolation can be used to predict how long it takes a fire to penetrate the forest or to be extinguished. [2]Oil Fields Percolation can be used as an idealized simple model for the distribution of oil or gas
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Topics in Statistic Mechanics

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