A common example to introduce Monte-Carlo methods in statistical mechanics is to work out the properties of the 2D square lattice Ising model and compare the obtained results with Onsager's exact solution. One generally works with single "spin flip" dynamics (SFD) to update the spin configurations in the 2D Ising model. However, other types of updates can be used as long as they satisfy ergodicity and detailed balance. Now my question is the following:

Since different types of updates lead to the same equilibrium state, can we ascribe any physical significance to a specific type of update ?

I've seen comments like: this model is glassy under SFD.

But does that tell us anything about the equilibrium properties ?

Finally, some non-local algorithm (so-called worms algorithms) can be used to equilibrate constrained models (such as ice type models). Do these "worm" carry any physical significance for the real physical system ?

Yet another example (again from spin systems) are the so-called facilitated spin models, where the low energy excitations are some non-trivial plaquette moves. I believe that these models are glassy under SFP dynamics.

  • $\begingroup$ From my very limited experience I think this is an archetypical question and the answer (in general, though specific cases can have real dynamical significance) is no. The goal of detailed balance and ergodicity is to let you get uncorrelated and unbiased samples of your configuration space. That's it. $\endgroup$
    – user12029
    Commented Sep 28, 2014 at 20:23
  • $\begingroup$ Single-flip moves mimic the real-life thermal sampling of the Boltzmann distribution. Worm moves don't really have a physical interpretation, though. $\endgroup$ Commented Feb 3, 2017 at 3:03

1 Answer 1


One could interpret the update steps as possible discrete steps in a fictitious time and in that case the transitions represent dynamics on the state space of a Markov chain. As an example, there is the relaxational non-conservative Glauber dynamics and the magnetization conserving Kawasaki dynamics which are used to simulate Ising and related systems. The issue is that there is no time scale in the problem (as we are studying only equilibrium systems), and unless we know apriori the mechanisms involved in the dynamics, the Monte-Carlo updates are just what they are, a way to sample phase space in an unbiased fashion. It is important to understand that the "dynamics" does not dictate the equilibrium. The equilibrium is set by the ensemble and the macroscopic observables that are kept fixed, nothing else. So in principle, using different update schemes will give you the same results, in the limit of infinite steps (infinite MC "time"). This is one of the major issues in computation, deciding when equilibrium has been achieved. Sometimes, the dynamics is slow and the system might get caught in local minima, making the dynamics glassy and equilibration can be a difficult task (but in principle is guaranteed to be achieved).

There are other faster ways of sampling the configuration space, which include cluster algorithms and biased sampling techniques (which reproduce the Gibbs measure though). In these cases one typically works with larger groups of spins (clusters) or works with evolving domain walls. These methods are typically faster as they involve global changes, but the sampling and energetic weightage given to each state is nontrivial.


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