Nonequilibrium phase transitions in feedback-controlled three-dimensional particle dynamics

We consider point particles moving inside spherical urns connected by cylindrical channels whose axes both lie along the horizontal direction. The microscopic dynamics differ from that of standard 3D billiards because of a kind of Maxwell's demon that mimics clogging in one of the two channels, when the number of particles flowing through it exceeds a fixed threshold. Nonequilibrium phase transitions, measured by an order parameter, arise. The coexistence of different phases and their stability, as well as the linear relationship between driving forces and currents, typical of the linear regime of irreversible thermodynamics, are obtained analytically within the proposed kinetic theory framework, and are confirmed with remarkable accuracy by numerical simulations. This purely deterministic dynamical system describes a kind of experimentally realizable Maxwell's demon, that may unveil strategies to obtain mass separation and stationary currents in a conservative particle model.

Figure 1

A container $\mathrm{\Omega }$ made of two 3D spherical urns and two coaxial cylindrical channels house $N$ noninteracting particles. These move on straight lines from collision to collision with the walls of $\mathrm{\Omega }$, where they are elastically reflected. Periodic boundary conditions are imposed to form a circuit with the passive channel. Inside the active channel, a Maxwell's demon pushes particles back, when their number in a gate exceeds a given threshold.

Figure 2

Trajectory of a single particle in the absence of the bounce-back mechanism, for $r=v=1, {\ell }_a=1, w_a=0.15$, and $w_p=0$.

Figure 3

Plots of $\eta$, Eq. (5), (solid line), and $\partial \eta /\partial {\lambda }_2$ (dashed line) at fixed $N=2000$ and for fixed geometrical parameters, as functions of the normalized variable ${\lambda }_2/\lambda$ and for different values of $\mathrm{\Theta }$ in [(a)–(f)], with $r=v=1, {\ell }_a=1.5, w_a=0.15, w_p=0.05$, hence $\lambda \approx 14$ and $C=1/9$. Zeroes of the black curve identify stationary states. Stable states, in the various panels, are identified by red circles.

Figure 4

Critical line in the $(C,\lambda )$ plane, corresponding to the graph of the function $\mathcal{C}(\lambda -1,\lambda )$ (black solid line), along with solutions of the interface equation (7) evaluated for different values of $\mathrm{\Theta }$ (colored solid lines). The bullets correspond to analytical solutions of the same equation for $C=0$.

Figure 5

Steady-state density profiles in the $x$-radial plane, using the parameters explored in Fig. 3, with ${\ell }_p=0.1$, for (a) $\mathrm{\Theta }=4$, (b) $\mathrm{\Theta }=6$, (c) $\mathrm{\Theta }=14$, and (d) $\mathrm{\Theta }=20$.

Figure 6

(a) Dynamical behavior of $\chi \left(t\right)$ as a function of time for various $\mathrm{\Theta }$ at $N=2000$ and same geometry as in Fig. 3, supplemented by ${\ell }_a=1.5, {\ell }_p=0.1$ and $\chi \left(0\right)=1$. (b) $N=2000, \mathrm{\Theta }=6$, same geometry as in Fig. 3. The dashed black horizontal line marks the theoretical value.

Figure 7

Phase portrait of the 3D urn system in the $(\mathrm{\Theta },\lambda )$ plane, where $\mathrm{\Theta }$ is integer valued. (a) Stationary values of the mass spread $\overline{\chi }$ for a system with a single channel ($C=0$); (b) $\overline{\chi }$ for a system with two channels, and (c) the net current per particle $\overline{J}$ as functions of the parameters $\lambda$ and $\mathrm{\Theta }$ for $N=2000$ and $C=1/9$ in (b) and (c). The initial condition corresponds to $\chi \left(0\right)=1$. The long dashed black-white line is the solution of the interface equation (7) (assuming continuous $\mathrm{\Theta }$), while the dotted line marks the onset of nonequilibrium states. The dotted line and the dashed line meeting on the left delimit the coexistence region. Panel (d) shows $\overline{\chi }(\chi \left(0\right)=1)-\overline{\chi }(\chi \left(0\right)=0)$.

Figure 8

Exact numerical results (symbols) in the absence of the bounce-back mechanism for $r=v=1, {\ell }_a=1, {\ell }_p=0.1$, various $w_a$ and $w_p$. Symbols with identical color have identical $w_p$, but different $w_a$. (a) $p_a/v\delta$ vs $C_a/4V_u$ shows that (B2) does not hold. (b) $p_a/v\delta$ vs $\tilde{w}/\pi {\tilde{A}}_u$ confirms the coefficient $\kappa =1/\pi$, as well as the expressions for ${\tilde{w}}_c$ and $p_a/v\delta$, (B13), (B14), and (B15). (c) $C=p_p/p_a$ vs $w_p/w_a$ is seen to be in agreement with $C=C_p/C_a={(w_p/w_a)}^2$, (B16), and thus also in agreement with $C={\tilde{w}}_p/{\tilde{w}}_a$.