How is Ising Model a Markov Chain?

[marschmellow]

The title says it all. It looks like the configuration probability only depends on where you want to go, not what state you are in now. Yet when I watch simulations, there is clearly a dependence on the previous state. Is there something pretty basic I'm misunderstanding about configuration probability? Is it only equilibrium configuration probability or something?

 

[Dickfore]

I have never seen any dynamics associated with the Ising model. As far as I know, one calculates equilibrium properties of the system. If you ever observe any dynamical change, it might be due to an artifice of using some Monte Carlo technique to calculate equilibrium properties.

You might be confusing the Markov property of the Monte Carlo algorithm with that of an Ising model.

 

[atyy]

http://mae.ucdavis.edu/dsouza/Talks/msri-June06.pdf
Take a look at slide 12, on Glauber dynamics, where the transition matrix is given.

http://pages.uoregon.edu/dlevin/AMS_shortcourse/ams_ising.pdf
"The (single-site) Glauber dynamics ... is a Markov chain ..."

 

[Dickfore]

Take a look at the title of slide 10.

 

[atyy]

Take a look at the title of slide 10.

I think the relationship is that those stochastic dynamics produce long run probabilities which are the correct ensemble equilibrium probabilities. Something like what's described in http://arxiv.org/abs/0911.3984

"We shall consider a molecular system in terms of a Markov model ... One class of master equations is particularly important: Its stationary distribution satisfies detailed balance ... The following results are well known: (i) The system has ... entropy S, a total internal energy U, and a free energy F ..."

 

[Dickfore]

I think the relationship is that those stochastic dynamics produce long run probabilities which are the correct ensemble equilibrium probabilities.

And that's precisely what I said. The Monte Carlo is a markov chain, not the Ising model.

 

[marschmellow]

Okay great that helped clear things up. I still have more questions, however. What does the configuration probability distribution mean? A configuration is a microstate, correct? What does it mean for a state "equilibrium" to have a certain microstate? Don't microstates rapidly change in equilibrium but with the macrostate fluctuating only mildly around some stable state? What exactly is in equilibrium? Is it temperature or is it the magnetization? How does time factor into the Ising model, if at all?

Thanks.

 

[atyy]

Okay great that helped clear things up. I still have more questions, however. What does the configuration probability distribution mean? A configuration is a microstate, correct? What does it mean for a state "equilibrium" to have a certain microstate? Don't microstates rapidly change in equilibrium but with the macrostate fluctuating only mildly around some stable state? What exactly is in equilibrium? Is it temperature or is it the magnetization? How does time factor into the Ising model, if at all?

Thanks.

The Ising model is a statistical mechanics model, which is basically a kludge. The true derivation must come from deterministic dynamics, and a distribution of initial conditions. But this is too hard, so people use models in which the dynamics are stochastic. Glauber dynamics are a form of stochastic dynamics whose long run probabilities are the same as the probability distribution in the statistical mechanics model. In statistical mechanics, one averages over ensembles to get the macrostate, while in the stochastic dynamics approach one averages over time to get the macrostate. The stochastic dynamics approach is also a kludge, since the true dynamics are deterministic, but they can be a good model for the approach to equilibrium, which is impossible to examine in the statistical mechanics approach - eg. "critical slowing down" can be studied in the stochastic approach, but not the statistical mechanics approach.

The section on "Dynamical Ising classes" on p8 of Odor's Universality classes in nonequilibrium lattice systems might be useful.