Dynamic+Monte+Carlo

=Introduction=

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

math p(\sigma \to -\sigma) = \begin{cases}1 & \text{if } \sigma = -1 \\ \exp(- 2 \beta h) & \text{if } \sigma = + 1 \end{cases} math

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 // β h// = 1/2 log 6, where the above simulate this problem with the naive-throw algorithm (SMAC algorithm 7.5 ).

We can also sample the time at which a flip + -> - will take place. This time is given by math [\frac{5}{6}]^t < ran(0,1) < [\frac{5}{6}]^{t-1} math 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, ** σ ** k. The Metropolis probability to flip spin //k// is math p(\boldsymbol{\sigma} \to \boldsymbol{\sigma}^k) = \frac{1}{N} \min[1, \exp(-\beta \Delta_E) math

and the probability to do nothing is math \lambda = 1 - \sum_{k=1}^N p( \boldsymbol{\sigma} \to \boldsymbol{\sigma}^ )k math

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



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... 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.

=Futility=

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. math \text{rejection rate: } = 1 - \min [ 1, \exp (-\beta \Delta E ) ] = 1 - \exp (-\max(0,\beta \Delta E) ) = - \exp ( - \beta \Delta E^+ = \beta \Delta E^+ math

math \text{acceptance rate: } = \min [ 1, \exp (-\beta \Delta E ) ] = \exp (-\max(0,\beta \Delta E) ) = \exp ( - \beta \Delta E^+ math

=References=