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Understanding the Ising Model: Probability Calculations and Spin Configurations
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[QUOTE="oasis2007, post: 1901432, member: 143449"] A simple and famous model of certain magnetic materials is the Ising model in which each of N atoms is modeled as a simple spin which can point upwards or downwards. The magnetization of the material is proportional to the number of up spins minus the number of down spins (each spin has a certain magnetization associated with it). Let's consider N spins in a row (as shown below). Let the "magnetization" be M, where M is the number of up spins minus the number of down spins. Each spin is constantly changing its direction from up to down and vice versa according to some complex dynamics. Assume the dynamics of one spin are independent of other spins. ↑↓↓↓↑↑↓↑↓↓↑↓↓↑↓↑↑↓↓↓↑↓ (i) Clearly state a postulate on the spin configurations which will help us calculate probabilities of different spin configurations. (ii) Using (i), calculate the probability of a certain M value P(M) exactly. (iii) Show P(M) is a Gaussian in the large N limit and sketch it. (iv) Estimate the probability that M is larger than 0.1% of N when N takes the value of Avogadro’s constant, 6x1023 [/QUOTE]
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Understanding the Ising Model: Probability Calculations and Spin Configurations
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