So far, in these lectures, we have concentrated on equilibrium statistical mechanics and related computational-physics approaches, notably the equilibrium Monte Carlo method. These and other
approaches allowed us to determine partition functions, energies, superfluid densities, etc. Physical time played a minor role, as the observables were generally time-independent. Likewise, Monte Carlo time was
treated as of secondary interest, if not a nuisance: we strove only to make things happen as quickly as possible, that is, to have algorithms converge rapidly.

In this lecture, we reach beyond equilibrium statistical mechanics, and explore time-dependent phenomena such as the crystallization of hard spheres after a sudden increase in pressure or the magnetic response
of Ising spins to an external field switched on at some initial time. The local Monte Carlo algorithm often provides an excellent framework for studying dynamical phenomena.

We must be sure to understand the difference in paradigm between the role of the Monte Carlo time in
equilibrium methods and in dynamic algorithms: In the former, it is often unphysical, in the latter, the
Monte Carlo time is taken as the model for the physical time, often the time-scale for diffusion.

Example: a single spin in a field

A single spin in a field
We consider a single spin σ in an external field h, and use the Metropolis algorithm as a dynamic model:

This transition probability satisfies detailed balance and ensures that at large times, the two spin configurations
appear with their Boltzmann weights. Note that, if at time t the spin is opposite to the field, it will be aligned with it at time t+1. The sampling problem is non-trivial only if the spin σ = + 1.

Dynamical simulation

For fun, let us now simulate the above model at a special temperature βh = 1/2 log 6, where the above
A boy playing with a flip-die.
simulate this problem with the naive-throw algorithm (SMAC algorithm 7.5 [1] ).

We can also sample the time at which a flip + -> - will take place. This time is given by

which leads to the SMAC algorithm 7.6 (fast-throw). This algorithm can simulate up to a given MC time t in less than t steps, which is why it is called a "faster-than-the-clock" algorithm.

Ising model, the n-fold way (BKL) algorithm

For the single-spin model, the "faster-than-the-clock" algorithm, besides having a nice name, is actually quite efficient. It is therefore tempting to apply it to a non-trivial case, such as the Ising model.

The probability to do nothing

In the one-spin model, although the probability to flip was 1/6, the relevant parameter was
λ = 5/6, the probability "to do nothing". Let us therefore consider a spin configuration, σ, and the same configuration, with spin k flipped, σk. The Metropolis probability to flip spin k is

and the probability to do nothing is

This equation expresses the fact that to determine the probability to do nothing we must know all the N flipping probabilities.

The n-fold way

Here we explain an algorithm due to Bortz, Kalos, and Lebowitz [2]

Classes of the n-fold way algorithm for the Ising model .

Here we show the ten classes for the two-dimensional Ising model with periodic boundary conditions. Permuting neighbors does not change the class. The probability λ (probability to do nothing) can be computed from the number of members of each class...
Change of classes for the n-fold way algorithm as we flip a spin in the Ising model .
The probability to do nothing can be computed from the number of members of each class, but if a spin flips, a number of spins change class, and the number of members are recomputed... This involves some book-keeping.


In a dynamic simulation with the algorithm dynamic-ising, and in many other dynamic models which might be attacked with a faster-than-the-clock approach, we may soon be disappointed by the futility of the system's dynamics, even though it has no rejections. Intricate behind-the-scenes bookkeeping makes improbable moves or flips happen. The system climbs up in energy, but then takes the first opportunity to slide back down in energy to where it came from. We have no choice but to diligently undo all
bookkeeping, before the same unproductive back-and-forth motion starts again elsewhere in the system.

Lifting & Faster-than-the-clock: the explosive mix!

The ideas of lifting, treated in our second lecture, combine in a very interesting way with the faster than-the-clock idea.


  1. ^ Krauth W. (2006) Statistical Mechanics: Algorithms and Computations, Chap. 7
  2. ^ Bortz A. B., Kalos M. H., Lebowitz J. L. (1975) A new algorithm for Monte Carlo simulations of Ising spin systems Journal of Chemical Physics 17, 10