We study a generalized Ehrenfest urn model that interpolates between the Ehrenfest and voter dynamics through a mixing parameter $\alpha$. This model can be interpreted in two different ways: adding noise to the voter model, where $\alpha$ represents the intensity of the noise; or adding interaction to the Ehrenfest model, where $(1-\alpha )$ represents the level of the interaction. We focus on a thermodynamic limit where the system size $N\to \infty$ and $\alpha \to 0$ with $N\alpha$ held constant, and show that the stationary distribution converges to a $q$-Gaussian law. In this regime, the entropic index $q$ is determined explicitly by the constant $N\alpha$. The definition of $q$-Gaussians with compact support is extended to include boundary-singular but integrable densities, thereby allowing two equivalent representations: a compact-support and a real-line $q$-Gaussian, establishing a duality between them. Moreover, after a suitable change of variable, we prove that the extended version of the $q$-Gaussian is the symmetric beta distribution. The analysis also reveals an order-disorder phase transition structure, being the Cauchy's distribution, (supported on the entire real line); and the arcsine distribution, (their equivalent with compact support), the limit distributions when the susceptibility becomes maximal. These results provide a direct microscopic link between interacting urn models and generalized entropies.

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• Noise-induced transitions
• Phase transitions by order
• Random walks
• Stochastic processes
• Thermodynamics