Unbinding Transition of Probes in Single-File Systems

Single-file transport, arising in quasi-one-dimensional geometries where particles cannot pass each other, is characterized by the anomalous dynamics of a probe, notably its response to an external force. In these systems, the motion of several probes submitted to different external forces, although relevant to mixtures of charged and neutral or active and passive objects, remains unexplored. Here, we determine how several probes respond to external forces. We rely on a hydrodynamic description of the symmetric exclusion process to obtain exact analytical results at long times. We show that the probes can either move as a whole, or separate into two groups moving away from each other. In between the two regimes, they separate with a different dynamical exponent, as $t^{1/4}$. This unbinding transition also occurs in several continuous single-file systems and is expected to be observable.

Figure 1

Two probes submitted to opposite forces in a single file system. (a) General scheme: two probes (red) in a single file system (bath particles in grey) initially at a distance $L$ are submitted to opposite forces $\mp f$. (b) Possible moves and transition rates in the SEP with biased probes (red). (c) Example of trajectories for $f=0.5$ (blue) and $f=1.5$ (red), for ${\rho }_{\infty }=0.5$. The dashed black lines are the theoretical predictions. Distances are given in units of $a$ and forces in units of $k_{B}T/a$. (d) Rescaled trajectory of the probe 2 in log-log scale for $f=0.5$, $P({\rho }_{\infty })\simeq 0.69$, 1.5, and $L=10$, 20, 50, 100, 200, 500 (red to blue). (e) Separation of the probes in the bound [$f<P({\rho }_{\infty })$, filled circle] and unbound [$f>P({\rho }_{\infty })$, filled square] regimes. Points are the results of numerical simulations and the lines are the theoretical results (6) and (7).

Figure 2

Two probes submitted to arbitrary forces. (a) Two probes located at $X_1<X_2$ are submitted to arbitrary forces $f_1$ and $f_2$. (b) Two examples of trajectories for $f_1=0$, $f_2=1$ (blue) and $f_1=-1$, $f_2=2$ (red). (c) Phase diagram: bound (filled circle) and unbound (circle) configurations, the line is the theoretical prediction (12), (13). (d) Separation in the bound (filled circle) and unbound (filled square) regimes for $F=f_1+f_2=1$ as a function of the force difference $\mathrm{\Delta }f=f_2-f_1$; $\mathrm{\Delta }{f}^{*}$ is the critical force difference [Eqs. (12), (13)]. (e) Motion of each probe, and of the center of mass (c.m.) for the same parameters.

Figure 3

Five probes submitted to arbitrary forces (a). (b),(c) Simulated trajectories and theoretical predictions (black dashed lines) for forces $(1,1,-1,-1,1)$ (b) and $(1,-1,-1,1,1)$ (c). (d) Two probes phase diagram for $\rho =0.5$; it shows that all the possible divisions remain bound in case $b$ (filled circle) and that the groups (1,2,3) and (4,5) unbind in case $c$ (circle).

Figure 4

Two probes submitted to opposite forces in the Tonks’ gas (a) and the dipolar gas (b). Trajectory of the probe 2 in the Tonks’ gas of hard rods (a) and in the dipolar gas with $1/r^3$ interactions (b) at ${\rho }_{\infty }=0.2$, for different values of the force $f$ (solid lines). In the Tonks’ gas, $P({\rho }_{\infty })=0.25$; in the dipolar gas, $P({\rho }_{\infty })\simeq 0.265$. The dashed black lines are the theoretical predictions [20].