The sum is largest in case i when all of the spins

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Unformatted text preview: stings algorithm has its roots in statistical physics. A typical problem is the Ising model of ferromagnetism. Space is represented by a two dimensional grid ⇤ = { L, . . . L}2 . If we made the lattice three dimensional, we could think of the atoms in an iron bar. In reality each atom has a spin which can point in some direction, but we simplify by supposing that each spin can be up +1 or down 1. The state of the systems is a function ⇠ : ⇤ ! { 1, 1} i.e., a point in the product space { 1, 1}⇤ . We say that points x and y in ⇤ are neighbors if y is one of the four points x + (1, 0), x + ( 1, 0), x + (0, 1), x + (0, 1). See the picture: + +++ +++ + ++ + ++ y+ + +yxy +y+ + ++ + Given an interaction parameter , which is inversely proportional to the temperature, the equilibrium state is ! X 1 ⇡ (x) = exp ⇠x ⇠y Z( ) x,y ⇠x where the sum is over all x, y 2 ⇤ with y a neighbor of x, and Z ( ) is a constant that makes the probabilities sum to one. At the boundaries of the square...
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