Advanced Statistical Physics: Lecture Notes (Wintersemester by Johannes Berg, Gerold Busch

By Johannes Berg, Gerold Busch

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2) i It is called O(n)-model because the Hamiltonian is invariant under rotation. , q}. 1: 2d lattice 25 where neighbouring spins give a non-zero contribution only if they are in the same configuration. , +s. A general nearest-neighbour Hamiltonian is J1 si sj + J2 (si sj )2 + ... 4) i For s = 1/2 we get back the Ising model. Models which are mathematically equivalent to the Ising model or others also appear in contexts different from ferromagnetism. An example is a lattice gas where si = −1 means that particles are absent and si = 1 that particles are present).

We then yield: D x, t˜) bz ∂t˜p˜(˜ x, t˜) = b2 ∂x˜2 p˜(˜ 2 D ⇒ ∂t˜p˜(˜ x, t˜) = b2−z ∂x˜2 p˜(˜ x, t˜) 2 ˜ = b2−z D is the new effective diffusion constant. For the choice with p˜(˜ x, t˜) = p(x(˜ x), t(t˜)). D ˜ z = 2, D = D. 2: One example for the ballistic deposition model. Particle A is released randomly and sticks at the first site where it has an occupied nearest neighbor. Particle B can fall to the ground. 1 A simple model: Ballistic deposition (BD) The following growth models work a bit as tetris: We release a particle from a randomly chosen point above the surface.

E. (−1)nc . 3 Walks on the lattice and transfer matrices We show how walks on the lattice can be enumerated using transfer matrices. For now we study unweighted walks, x, y|W |x , y . The appropriate phase factor depending on the number of crossings will be introduced in the next section. Define the adjacency matrix of a lattice 1 if nodes i,j are neighbours 0 otherwise Tij = Walks on the lattice of length l can be generated by raising the adjacency matrix to the l-th power. Example: 1d   0 1 0 ···  1 0 1 ···     T=  0 1 0 ...

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