Transport Statistics of Bistable Systems

We consider the transport statistics of classical bistable systems driven by noise. The stochastic path integral formalism is used to investigate the dynamics and distribution of transmitted charge. Switching rates between the two stable states are found from an instanton calculation, leading to an effective two-state system on a long time scale. In the bistable current range, the telegraph noise dominates the distribution, whose logarithm is found to be universally described by a tilted ellipse.

Figure 1

A schematic of the average current flowing through the left and right connectors as a function of the charge in the central node. While linear connectors (the dashed line) have only one crossing point, nonlinear devices may have three crossing points, two stable, $Q_{1,2}$, and one unstable, $Q_u$. Inset: the stochastic network. The direction of current flow through the network is indicated.

Figure 2

The stability profile. The Hamiltonian is given as a phase space density plot. The system has three fixed points, two stable at $Q=Q_1,Q_2$ and one unstable at $Q=Q_u$. For long time, the system is forced to the zero energy lines in black, ${\lambda }_0,{\lambda }_{\mathrm{i}\mathrm{n}}$. A decay from the unstable state may happen with zero action, but to hop from either stable state to the unstable state requires finite action, the area underneath the curves shown.

Figure 3

The log distribution of transmitted charge, divided by $t\sqrt{{\Gamma }_1{\Gamma }_2}$. Within the bistable range, the distribution is well fit by a tilted ellipse, connecting to the distribution dominated by state 1 or 2. The average current is denoted by $⟨I⟩$. Inset: the same distribution, $\ln P[It]/t$, showing the comparatively flat bistable distribution with the left and right asymptotics. ${\Gamma }_2/{\Gamma }_1=4,F_{1,2}=(1,1.5)\times {10}^{-2}《I^2》$.