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p847mc1 - Monte Carlo Methods in Statistical Physics A...

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Monte Carlo Methods in Statistical Physics: A Five-Minute Primer The basic idea of the Monte Carlo method is to generate configurations at random in such a way that a given configuration will occur with the right probability for the ensemble and Hamiltonian of interest. To make the approach clear, I will describe the method (briefly) for the Ising Hamiltonian. As you recall, that Hamiltonian has the form H = - J summationdisplay <ij> S i S j , (1) where J > 0 is a parameter of the Hamiltonian, the sum runs over pairs of nearest-neighbor sites i and j , and the variables S i can take on the values ± 1. These are usually called spin variables. Clearly, for an N-particle system, there are 2 N states in the system. Also, we can see, for J > 0, that the system has a lower energy if neighboring spins are parallel (both up or both down). Also, notice that for this Hamiltonian, the ground state has energy E = - JNz/2, where z is the number of nearest neighbor sites (thus, z = 4 for a square lattice, for example).
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