Dynamical correlated spin model

Previously we already wrote about a work of our colleague, Julius Ruseckas, in which he proposed an elementary model, which reproduces q-Gaussian distribution. Recently we introduced temporal dynamics into that static model [1]. In this text we briefly discuss the dynamical version of the model.

Let us remind you that the correlated spin model describes possible configurations of the spin chain. In this spin chain neighboring spins are usually coaligned, meaning that nearby spins point in the same direction, though there are fixed number of cases \( d-1 \) (in this text \( d \) has slightly different meaning) where spins are antialigned. Consequently we have \( d \) distinct domains inside of which spins are aligned. We are interested in the total spin, \( M = \sum_i s_i \), of such system.

If \( d \) is an even number, then ends of the chain may by connected (periodic boundary conditions may be applied), leading to a differing interpretation of \( d \) as it now represents both the number of domains and number of boundaries between them. Though if \( d \) is odd, then we have to use the chain interpretation (which is equivalent to fixing one boundary in place). Note that in case of the spin chain, if the global spin-flips of the whole chain are forbidden, the stationary distribution will be asymmetric Beta distribution and not a q-Gaussian distribution (which is obtained the topology is ring).

In this model we allow the boundaries to move! During each time tick all of the boundaries may move to the left or to the right with certain probability \( \gamma \Delta t \). But we assume that the number of domains needs to be conserved. Thus if due to the movement of boundaries any single domain would disappear, then the movement is canceled.

Below you should see an interactive app, which may be used to test this model. Namely, observe that for even \( d \) the dynamical model generates a q-Gaussian PDF of total spin. For odd \( d \) small deviation (to the right side) from the q-Gaussian PDF will be observed. In this app, for the convenience sake, we have fixed the following parameters \( \gamma = 0.005 \) and \( \Delta t=1 \). These parameters influence only the speed with which model converges to stationary distribution.