Deterministic model of battery, uphill currents, and nonequilibrium phase transitions

We consider point particles in a table made of two circular cavities connected by two rectangular channels, forming a closed loop under periodic boundary conditions. In the first channel, a bounce-back mechanism acts when the number of particles flowing in one direction exceeds a given threshold $T$. In that case, the particles invert their horizontal velocity, as if colliding with vertical walls. The second channel is divided in two halves parallel to the first but located in the opposite sides of the cavities. In the second channel, motion is free. We show that, suitably tuning the sizes of cavities of the channels and of $T$, nonequilibrium phase transitions take place in the $N\to \infty$ limit. This induces a stationary current in the circuit, thus modeling a kind of battery, although our model is deterministic, conservative, and time reversal invariant.

Figure 1

The billiard table: the $N$ point particles are represented as small disks, and velocities are represented by arrows. The two gray-shaded regions, in the first channel, are the gates in which the bounce-back mechanism separately acts. The horizontal component of the velocity of the particles contained in one gate and moving toward the other is reversed whenever their number is larger than the prescribed threshold value $T$.

Figure 2

Stationary values of the mass spread $\chi$ (left panel) and net current (right panel) for $N={10}^3, w=0.3, r=\ell ={\ell }^{\prime }=1$. The initial condition yields $\chi \left(0\right)=0$. In the left panel, the red pixels denote the homogeneous phase, whereas the other pixels refer to the inhomogeneous phase. The white lines mark the theoretical boundary between homogeneous and inhomogeneous steady states [see Eq. (6)].

Figure 3

Currents as functions of time, for $r=\ell ={\ell }^{\prime }=1, 0.01=w^{\prime }\ll w=0.3, N={10}^3$, and initial datum such that $\chi \left(0\right)=0$. For $T=2$ (a) and $T=18$ (c), a homogeneous steady state with zero net current is reached. For $T=7$ (b), a stationary net current with $N_1\ne N_2$ arises Disks (squares) represent numerically computed flows out of urn 1 (urn 2) in channel 1; triangles (diamonds) represent flows out of urn 1 (urn 2) in channel 2. Black dashed lines are the theoretical values (obtained from the probabilistic model) of the stationary currents; solid red (blue) lines denote the net currents in the first (second) channel. The parameter $\eta$ is plotted in the insets as a function of $N_2/N$, while the red disks indicate the stable state reached by the deterministic dynamics.

Figure 4

Mass spread with the same values of the parameters and of the threshold considered in Fig. 3: Homogeneous and inhomogeneous states are both stable (blue curve, $T=2$), only the inhomogeneous state is stable (red curve, $T=7$), and only the homogeneous state is stable (black curve, $T=18$). The dotted and dashed lines indicate the theoretical values of the mass spread for the inhomogeneous states at $T=2$ and $T=7$, respectively. The initial datum is such that $\chi \left(0\right)=1$.

Figure 5

Net currents as a function of $T/N$ (left panel) and of $\chi$ (right panel) for $N={10}^3, w=0.3, r=\ell ={\ell }^{\prime }=1, w^{\prime }=0.0102$ (circles), 0.0204 (squares), 0.0306 (triangles), and 0.0612 (diamonds). The dashed lines correspond to the stable solutions of $\eta =0$; see Eq. (5).

Figure 6

The billiard table considered in the numerical simulations.

Figure 7

Stationary currents as functions of $T/N$, with $N=500$ and $N=1000$ (left and right panel, respectively), for $r=\ell =1$, and $w=0.1$ (squares), 0.2 (circles), and 0.3 (triangles), numerically obtained from the simulation of the deterministic dynamics. Dotted and dashed lines correspond to the theoretical values of the current with $N=500$ and $N=1000$, respectively, obtained from Eq. (A4).

Figure 8

Difference between the theoretical and the numerical values of the currents, divided by the corresponding theoretical values, as a function of $T/N$ with $N=500$ and $N=1000$ (left and right panel, respectively), with the same values of the parameters of Fig. 7.

Figure 9

Stationary currents, rescaled by $N$, as functions of $T/N$, with $N=500$ and $N=1000$ (open and and filled symbols, respectively), with the same values of the parameters of Fig. 7. Dotted and dashed lines correspond to the theoretical values of the current rescaled by $N$, with $N=500$ and $N=1000$, respectively.