Ehrenfest urn model with interaction

We studied the Ehrenfest urn model in which particles in the same urn interact with each other. Depending on the nature of interaction, the system undergoes a first- or second-order phase transition. The relaxation time to the equilibrium state, the Poincaré cycles of the equilibrium state and the most far-from-equilibrium state, and the duration time of the states during first-order phase transition are calculated. It was shown that the scaling behavior the Poincaré cycles could serve as an indication to the nature of phase transition, and the behavior of the ratio of duration time of the states could be strong evidence of the metastability during first-order phase transition.

Figure 1

Schematic diagram to illustrate the transition in our model.

Figure 2

The relation between the saddle point $y_{\mathrm{sp}}$ in Eq. (17) and the coupling constant $g$. $\frac{1}{N}\left(n_{\mathrm{sp}}-\frac{N}{2}\right)$ as a function of coupling constant $g$ for different $p$. As $g\le g_{\mathrm{sp}}$, two saddle points arise where $n_{\mathrm{sp},-}<n_{\mathrm{sp},+}$. Inset: $g_{\mathrm{sp}}$ as a function of $p-\frac{1}{2}$.

Figure 3

The relation between the saddle point $y_{\mathrm{sp}}$ in Eq. (17) and $p$. $\frac{1}{N}\left(n_{\mathrm{sp}}-\frac{N}{2}\right)$ as a function of $p-\frac{1}{2}$ for different $g$.

Figure 4

The proportion of particle number in the left urn, ${\langle n\rangle }_s/N$, as a function of time step $s$ for $g>g_{\mathrm{sp}}$ at different $p$ and $g$. The initial value $n_0$ is chosen to be the most far-from-equilibrium state. Solid lines represent the corresponding result by Eq. (25).

Figure 5

The proportion of particle number in the left urn, $(\langle |n-n_0{|\rangle }_s+n_0)/N$, as a function of time step $s$ for $g<g_{\mathrm{sp}}$ at different $p$ and $g$. The initial value $n_0$ is chosen to be the most far-from-equilibrium state. Solid lines represent the corresponding result by Eq. (29).