Fluctuation loops in noise-driven linear dynamical systems

Understanding the spatiotemporal structure of most probable fluctuation pathways to rarely occurring states is a central problem in the study of noise-driven, nonequilibrium dynamical systems. When the underlying system does not possess detailed balance, the optimal fluctuation pathway to a particular state and relaxation pathway from that state may combine to form a looplike structure in the system phase space called a fluctuation loop. Here, fluctuation loops are studied in a linear circuit model consisting of coupled $RC$ elements, where each element is driven by its own independent noise source. Using a stochastic Hamiltonian approach, we determine the optimal fluctuation pathways, and analytically construct corresponding fluctuation loops. To quantitatively characterize fluctuation loops, we study the time-dependent area tensor that is swept out by individual stochastic trajectories in the system phase space. At long times, the area tensor scales linearly with time, with a coefficient that precisely vanishes when the system satisfies detailed balance.

Figure 1

Two-element, capacitively coupled network of $RC$ circuit elements, each with its own independent noise source.

Figure 2

Numerically generated segment of stochastic trajectory passing through the target region (marked by a cross) and showing a clockwise looping structure. Noise intensities are $d_1=0.01$ and $d_2=0.1$ so that the system violates detailed balance.

Figure 3

(a) Numerically generated segment of stochastic trajectory passing through the target region under detailed balance conditions with noise intensities are $d_1=d_2=0.1$. (b) Segment of stochastic trajectory computed for parameters $d_1=0.1$ and $d_2=0.09$. The system is nondetailed balance but the expected looping structure is not discernible.

Figure 4

(a) Calculated most-probable fluctuational and relaxational paths for clockwise vorticity: $d_1=0.01$, $d_2=0.1$, and $\mu =0.1$. (b) Calculated most-probable fluctuational and relaxational paths for counterclockwise vorticity: $d_1=0.1$, $d_2=0.01$, and $\mu =0.1$.

Figure 5

Most probable escape path (MPEP) and relaxation path (RP) determined by averaging 744 individual trajectories using a target ball centered at $(0.1,0.1)$. System parameters are $\mu =0.1$, $\epsilon =0.01,d_1=0.01$, and $d_2=0.1$.

Figure 6

Area swept by a diffusing particle. The striped region represents positive area, while the dotted region is negative area. The total area swept is the area of the striped region minus the area of the dotted region.

Figure 7

Numerically determined stochastic area function for three cases: clockwise vorticity ($d_1=0.01,d_2=0.1$), detailed balance ($d_1=d_2=0.1$), and counterclockwise vorticity ($d_1=0.1,d_2=0.01$). The inset shows expected quadratic dependence for short times. These plots are generated using Euler-Maruyama iteration of (17) to determine the area function using (29). The area was then averaged over 1000 trials.

Figure 8

Here we show the swept area divided by time for one long stochastic trial, i.e., there is no ensemble averaging in this case. Note that this scenario is relatively easy to implement for a real experiment.