Introduction

So far, in these lectures, we have concentrated on equilibrium statistical mechanics and related computational-physics approaches, notably the equilibrium Monte method. These and other approaches allowed us to 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 response of Ising 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 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.

The metropolis dynamics for a spin embedded in its molecular field

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

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

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

Ising model and the (BKL) algorithm

For the single- 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.

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

$p(\boldsymbol{\sigma} \to \boldsymbol{\sigma}^k) = \frac{1}{N} \min[1, \exp(-\beta \Delta_E)$

The first term, 1/N, gives the probability of selecting spin k, followed by the Metropolis probability of accepting a flip of that spin. In the Ising model and its variants at low , most spin-flips are rejected. It can then be interesting to implement a faster-than-the-clock algorithm which first samples the time of the next spin-flip, and then the spin to be flipped, just as in the earlier deposition problem. Rejections are avoided altogether, although the method is not unproblematic.

The probability to do nothing is

$\lambda = 1 - \sum_{k=1}^N p( \boldsymbol{\sigma} \to \boldsymbol{\sigma}^K )$

This expresses that to determine the probability of doing nothing, we must know all the probabilities for N flipping spins. Naively, we
can recalculate λ after each step, and sample the waiting time as. After finding out when to flip the next spin, we must decide on which of the N spins to flip. This problem is solved through a second application of tower sampling. However, each spin-flip requires of the order of N operations.

We can actually deal with this cumbersome constructions using an algorithm due to Bortz, Kalos, and Lebowitz [1] . The algorithm is based on classes and is not in the course .

References

1. ^ 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