Nonequilibrium ensemble inequivalence and large deviations of the density in the $ABC$ model

We consider the one-dimensional driven $ABC$ model under particle-conserving and particle-nonconserving processes. Two limiting cases are studied: (a) The rates of the nonconserving processes are vanishingly slow compared with the conserving processes in the thermodynamic limit and (b) the two rates are comparable. For case (a) we provide a detailed analysis of the phase diagram and the large deviations function of the overall density, $\mathcal{G}\left(r\right)$. The phase diagram of the nonconserving model, derived from $\mathcal{G}\left(r\right)$, is found to be different from the conserving one. This difference, which stems from the nonconvexity of $\mathcal{G}\left(r\right)$, is analogous to ensemble inequivalence found in equilibrium systems with long-range interactions. An outline of the analysis of case (a) was given in an earlier letter. For case (b) we show that, unlike the conserving model, the nonconserving model exhibits a moving density profile in the steady state with a velocity that remains finite in the thermodynamic limit. Moreover, in contrast with case (a), the critical lines of the conserving and nonconserving models do not coincide. These are new features which are present only when the rates of the conserving and nonconserving processes are comparable. In addition, we analyze $\mathcal{G}\left(r\right)$ in case (b) using macroscopic fluctuations theory. Much of the derivation presented in this paper is applicable to any driven-diffusive system coupled to an external particle bath via a slow dynamics.

Figure 1

Effective potential exerted on the $A$ particles, $V_{A,i}={\sum }_{k=1}^{i-1}{B}_k-C_k$, and on the $B$ particles, $V_{B,i}={\sum }_{k=1}^{i-1}{C}_k-A_k$, drawn for a specific configuration of a system of size $L=15$. The configuration itself is written above the two figures.

Figure 2

Phase diagram of the conserving (a) and nonconserving (b) $ABC$ models with equal densities, $N_A=N_B=N_C$, plotted in the ($\mu ,1/\beta$) plane. The two diagrams show the same second order transition line (thin solid line). In the nonconserving model the transition line turns into a first order (thick solid line) at a tricritical point ($☆$). The boundaries of the coexistence region, where both the homogenous and the ordered profiles are locally stable, are denoted in the nonconserving model by dashed lines.

Figure 3

Phase diagram of the conserving (a) and nonconserving (b) $ABC$ models for in two-equal-densities case, $r_A=r_B=r/3-0.01$ and $r_C=r/3+0.02$, or, equivalently, $a_0=0.01\sqrt{3}{e}^{i\pi /3}$, plotted in the ($\mu ,1/\beta$) plane. The two diagrams show the same second order transition line (thin solid line). In the conserving model the transition line turns at the tricritical point ($•$) into a first order transition line, which cannot be computed without knowing the full large deviations function of the $ABC$ model. In the nonconserving model the transition line turns into first order (thick solid line) at a critical end point ($\blacksquare$), which appears at a higher value of $1/\beta$ than the tricritical point ($☆$). Both diagrams are supplemented by an illustration of the region where the first and second order transition lines meet. The boundaries of the coexistence regions, where both the homogenous and the ordered profiles are locally stable, are denoted by dashed lines.

Figure 4

The $m=1$ ordered density profile obtained from the solution of Eq. (29) for $r_A=r_B=7/30$ and $r_C=16/30$ and $\beta =25$.

Figure 5

The large deviations function of the overall density, $r$, for $\beta =40$, $r_A=r_B=r/3-0.025$, and $\mu =-0.0515$. For this choice of parameters $\mathcal{G}\left(r\right)$ has two local minima, corresponding to a disordered and an ordered phase. The dashed line denotes the critical value of $r$, given in Eq. (21).

Figure 6

Chemical potential calculated in the conserving model for the case of $r_A=r_B=r/3-0.01$. Panels (a),(b),(c),(d) correspond to $\beta =25,33.15,40,250$, respectively. The critical point is denoted by $•$. First order transition points, computed via Maxwell's construction, are denoted in (b) and (c) by dashed lines. The vertical scale in (b) is chosen to be extremely small in order to display to the variations in $\mu \left(r\right)$ around the first order transition point between the two ordered phases.

Figure 7

Velocity of the unstable mode on the critical point for $r_A=0.18+x$, $r_B=0.18-x$, $r_C=0.24$, and $\varphi =1$. The horizontal axis represents the value of $x$.

Figure 8

Tricritical lines in the conserving and the nonconserving models. The dashed and solid thick lines represent points where $G_6<0$ and $G_6>0$, respectively. In the latter case the tricritical points are preempted by a CEP. The thin dashed gray lines denote the smallest possible values of $r$. The case of two equal densities, $r_A=r_B=r/3-\Delta$ and $r_C=r/3+2\Delta$ or, equivalently, $a_0=\Delta \sqrt{3}{e}^{i\pi /3}$, is shown in (a) and (b). Panels (c) and (d) show the fully nonequal-densities case, where $r_A=r/3-\Delta +0.007$, $r_B=r/3-\Delta +0.007$, and $r_C=r/3+2\Delta$.