Glauber dynamics

In the Ising model, we have say N particles that can spin up (+1) or down (-1). Say the particles are on a 2D grid. We label each with an x and y coordinate. Glauber's algorithm becomes:[1]

1. Choose a particle ${\displaystyle \sigma _{x,y}}$ at random.
2. Sum its four neighboring spins. ${\displaystyle S=\sigma _{x+1,y}+\sigma _{x-1,y}+\sigma _{x,y+1}+\sigma _{x,y-1}}$.
3. Compute the change in energy if the spin x, y were to flip. This is ${\displaystyle \Delta E=2\sigma _{x,y}S}$ (see the Hamiltonian for the Ising model).
4. If ${\displaystyle \Delta E<0}$ flip the spin. That is if flipping reduces the energy, then do it.
5. Else flip the spin with probability ${\displaystyle e^{-\Delta E/T}}$ where T is the temperature.
6. Display the new grid. Repeat the above N times.

This tries to approximate how the spins change over time. The fancy term is that it is part of nonequilibrium statistical mechanics, which roughly studies the time-dependent behavior of statistical mechanics.[1]

The algorithm is named after Roy J. Glauber.[1]

• Metropolis algorithm
• Ising model
• Monte Carlo algorithm
• Simulated annealing