Feedback between interacting transport channels

A model of particle transport through a large number of channels is introduced. Interactions among the particles can lead to a strong suppression of fluctuations in the particle number statistics. Within a mean-field-type limit, this becomes equivalent to a time-dependent (nonautonomous) collective feedback control mechanism. The dynamics can be interpreted as a diffusive spreading of a feedback signal across the channels that displays scaling, can be quantified via the flow of information, and becomes visible, e.g., in the spectral function of the particle noise.

Figure 1

(a) Particle transport between source and drain reservoirs at rates ${\gamma }_{\pm }$ through connectors in channels at positions ${\mathbf{R}}_l$. (b) Connectors could have internal states. (c) Autonomous feedback via interactions $\left(g>0\right)$ among drain reservoirs reduces fluctuations of particle numbers $n_l$, represented as Brownian particles on a line. This is compared to the nonfeedback case $g=0$ during time evolution from $t_0$ to $t_1$. (d) Active, nonautonomous measurement-based feedback for a single channel [34].

Figure 2

(a) Diagonal second cumulant $C_{ll}$ Eq. (20) multiplied with interaction parameter $g$ and (b) information current ${\iota }_{1^{\prime }\to 1}$, Eq. (38), as a function of time $t$ multiplied with $g\gamma$. (c) Stationary diagonal noise spectrum [$l=l^{\prime }$, Eq. (23)] as a function of scaled frequency $\mathrm{\Omega }\equiv \omega /\left(g\gamma \right)$. Curves for infinite range interaction model Eq. (19) with channel numbers $N=50$ (magenta dotted), $N=500$ (red dashed), $N=5000$ (blue thick line), diffusive approximation Eq. (22) in $d=2$ dimensions with finite $N$ in (a), and nearest-neighbor model Eq. (21) in $d=1$ dimension with $N\to \infty$ (thin black line).