Kawasaki dynamics

One of the traditional models that come up time and time again is the Ising model. Originally developed to study ferromagnets, the model is considered one of the simplest models that exhibits “complex” behavior and has been applied to study a variety of phenomena such as opinion dynamics, neural dynamics, segregation, and even criminal systems.

The Ising model consists of a collection of nodes $S = \{ s_1, \dots, s_n \}$ which contains a alphabet $X = \{-1, 1 \}$. That is, each “vertex” or “node” contains a “spin” or binary state representing up or down, opinion A or B, firing or non-firing.

The system dynamics of the Ising model occurs traditionally by Glauber update. Each time step $t$ a spins is chosen according to some function $g(S)$. Traditionally, $g(S)$ is chosen to be uniform over the nodes in the system. The energy of the sampled node is then computed compared to its opposite state. That is, a proposal state is drawn uniformly from $X$ and “accepted” using Metropolis-Hasting sampling

\begin{equation} \begin{aligned} A(s_i \to s_i’) = \frac{ p(s_i’) }{ p(s_i) } = \Bigg \{ \begin{aligned} \exp( -\beta \Delta E) & \textrm{ if } \Delta E < 0 \\ 1 & \textrm{ otherwise }.\\ \end{aligned} \end{aligned} \end{equation}

The difference in energy is given as

$$ \Delta E= \mathbb{H}(S’) - \mathbb{H}(S)$$

with $\mathbb{H}(S) = - \sum_{ij} J_{ij} s_i s_j - \sum_{i} h_i s_i$, where $h_i$ represents some external magnetic field.

For a single spin difference results in $$ \Delta E= \mathbb{H}(s_i’) - \mathbb{H}(s_i)$$ as the difference in energy for all other spins $s_j \in S$ cancels.

There are, however, different update schemes. One of which is the Kawasaki dynamic. In this way, the magnetization of the system remains constant. That is, each node gets assigned a state and does not change as a function of time. Each simulation step, a radomly chosen spin may swap its state with its neighbor. That is, a spin $s_i$ may choose its next state $s_i’$ by swapping its state with some neighbor $s_j$ such that $s_i’ = s_j$ and $s_j’ = s_i$ with transition $A(s_i, s_j \to s_i’, s_j’)$

\begin{equation} \begin{aligned} A(s_i, s_j \to s_i’, s_j’)_{\textrm{Kawasaki}} = \frac{ p(S’) }{ p(S) } = \Bigg \{ \begin{aligned} \exp( -\beta \Delta E) & \textrm{ if } \Delta E < 0 \\ 1 & \textrm{ otherwise }.\\ \end{aligned} \end{aligned} \end{equation}

Note that here the $\Delta E$ is computed over a proposed state $S’$ where the states of $s_i$ and $s_j$ are swapped.

Kawasaki dynamics ensures that the average magnetization remains constant. The fraction of positive spins will remain constant over time. In contrast, for Glauber dynamics the ratio between positive and negative spins may change, depending on the temperature $\beta = \frac{1}{T}$ in the system. For $T < T_{C}$ tends to magnetize the system in Ising spin systems with Glauber dynamics and Metropolis-Hasting upates. This means that if the system starts with an equal propotion of positive and negative spins, the system will tend to a state in which all spins are aligned. For Kawasaki dynamics, the spins “move” through the space. Clustering will occur, similar to the Ising dynamics with Glauber updates, but no majority will win. Kawasaki dynamic can therefore be used to study things like segregation, gang-turf demarcation, echo-chambers and so on. A difference between the two dynamics is shown can be seen in fig. 1 with $T = 1$ and for a 4 state potts model in fig. 2.