Glauber dynamics

In statistical physics, Glauber dynamics is a way to simulate the Ising model (a model of magnetism) on a computer. It is a type of Markov Chain Monte Carlo algorithm.

The algorithm

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

History

The algorithm is named after Roy J. Glauber, Nobel Prize winner and a Harvard physicist who worked at Los Alamos.

Related pages

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