# Can Chaos Theory be used to explain the Ising model in paramagnetic phase?

Is it possible? How can I explain the randomness of spins in the paramagnetic phase with chaos theory? In this case, is the randomness apparent?

If yes, I think the temperature would be a reasonable parameter change that is mentioned in chaotic systems, which changing the temperature leads to phase transition and consequently making it disordered. I had a brief discussion with a professor of mine regarding this, and he mentioned that temperature can't be the parameter and the reason was something like "temperature is also random or too many possible states for a specific temperature." (He was speaking in another language that I'm not so good at.)

And can the Ising model be considered a nonlinear system? How do I find the time-evolution equation of the Ising model?

Are there any experts or physicists in this field who give some comments on these? I'm quite interested in phase transitions as well as chaos theory. I'm hoping to bridge and link these two fields in my mind.

• I don't think it has anything to do directly with chaos theory. For a fixed temperature the flips of the spin are given by random fluctuations as at high temperature is enough to make the ernergy gap between up and down spins negligible. – Mauricio Nov 29 '17 at 21:26

$$\bf 1$$) Can Chaos theory be used to explain the Ising model [...] And can Ising model be considered as a nonlinear system?

The Ising model is certainly a complex system, but (deterministic) chaos per se doesn't have much to do with it in principle, especially because the effect of temperature is usually modeled as a random noise (by definition non-deterministic, but see more about it in the answer to (2)).

Edit: The Appendix A39 - Statistical mechanics recycled of the ChaosBook.org actually does describe a much more deep connection between spin systems and chaos theory:

The geometrization of spin systems strengthens the connection between statistical mechanics and dynamical systems.
[...] A spin system with long-range interactions can be converted into a chaotic dynamical system that is differentiable and low-dimensional. The thermodynamic limit quantities of the spin system are then equivalent to longtime averages of the dynamical system. In this way the spin system averages can be recast as the cycle expansions. If the resulting dynamical system is analytic, the convergence to the thermodynamic limit is faster than with the standard transfer matrix techniques.

There is also a concept of chaos in Ising spin glasses, where the dominant spin configuration is extremely sensitive to small variations in the bond strength or in the temperature. Notice, though, that these are global perturbations, so I'd say that's more a structural instability rather than the instability of orbits of traditional chaos. Some references are Chaos in a Two-Dimensional Ising Spin Glass, Chaos and universality in two-dimensional Ising spin glasses, and Temperature chaos in 3D Ising Spin Glasses is driven by rare events.

$$\bf 2$$) How can I explain the randomness of spins in the paramagnetic phase with chaos theory? In this case, is the randomness apparent?

The unpredictability is real, it's a reflection of the preponderance of thermal fluctuations in the dynamics, so the answer is: the disorder is chaotic or random to the same degree to which the thermal fluctuations are chaotic or random. In classical physics, they are chaotic, in quantum physics, it depends on the interpretation you pick.

If enough care is taken with respect to the relevant statistical properties, chaotic processes may be modeled as random, and vice-versa. In particular, chaotic variables have been used in place of random numbers in Monte Carlo simulations of systems such as the Ising and Potts models (see, e.g., Monte Carlo simulation of classical spin models with chaotic billiards).

$$\bf 3$$) How do I find the time-evolution equation of the Ising model?

Although one is often interested in the equilibrium states when studying systems such as the Ising model (for which time doesn't play a role), the dynamics of how these equilibria are reached is also important. A reference on the description of the evolution from a paramagnetic state to an ordered one once the system is abruptly brought to a subcritical temperature is Bray's Theory of Phase Ordering Kinetics (arxiv - without figures, e-print) and a summary can be found in this answer.