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.[1]

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]

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]

History

The algorithm is named after Roy J. Glauber, Nobel Prize winner and a Harvard physicist who worked at Los Alamos.[1]

References

1. "Glauber's dynamics | bit-player". Retrieved 2019-07-21.