papers/lux-field-protocol/sections/02-electromagnetic-analogy.tex

\section{Electromagnetic Analogy}

\label{sec:analogy}

\subsection{Ising and Potts Models}

The binary metastable protocols (Slush, Flare, Wave) are naturally modeled by the
\emph{Ising model}: $n$ spins on a graph, each taking values $\pm 1$, evolving through
local majority dynamics (Glauber dynamics). Below a critical temperature, the system
spontaneously magnetizes.

For $d > 2$, the natural generalization is the \emph{Potts model}: each spin takes
one of $d$ values, and the energy function penalizes disagreement between neighbors.
The Field protocol implements mean-field Potts dynamics through random sampling.

\subsection{From Spins to Vectors}

Rather than discrete Potts spins, we use continuous preference vectors
$\mathbf{p}_i \in \Delta^{d-1}$ on the probability simplex, where component $p_i^{(c)}$
represents validator $i$'s degree of preference for value $c$. The ``hard'' preference
is $c^*(i) = \arg\max_c p_i^{(c)}$.

This continuous representation enables smooth dynamics: instead of discrete flips,
preferences rotate gradually toward the sampled majority, providing better convergence
properties and natural handling of uncertainty.

\begin{definition}[Field Potential]
\label{def:potential}
The field potential over the ensemble $\mathbf{P} = \{\mathbf{p}_1, \ldots, \mathbf{p}_n\}$ is:
\begin{equation}
\Phi(\mathbf{P}) = -\frac{1}{n^2} \sum_{i=1}^{n} \sum_{j=1}^{n} \mathbf{p}_i \cdot \mathbf{p}_j
= -\frac{1}{n^2} \left\| \sum_{i=1}^{n} \mathbf{p}_i \right\|^2
\label{eq:potential}
\end{equation}
where $\mathbf{p}_i \cdot \mathbf{p}_j = \sum_{c=1}^{d} p_i^{(c)} p_j^{(c)}$ is the inner product.
\end{definition}

The potential $\Phi$ is minimized when all preference vectors are identical (unanimous
agreement: $\Phi_{\min} = -1$) and maximized when preferences are uniformly distributed
($\Phi_{\max} = -1/d$ for $d$ equally supported values).

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