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

Glauber Dynamics Media

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  The probability distribution according to Glauber Dynamics for the change in energy that would result from flipping the some spin s for different temperatures, T.

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  The probability distribution according to Metropolis-Hastings Dynamics for the change in energy that would result from flipping some spin s for different temperatures, T. P(\Delta E \leqslant 0) = 1.

Related pages

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