Stochastic Model and Optimal Control of an Active Tracking Particle with Information Processing

Abstract

Living systems often function with regulatory interactions, but the question of how activity, stochasticity and regulations work together for achieving different goals still remains puzzling. We propose a stochastic model of an active tracking particle with information processing, where the entropy production and information flow are discussed, with the generalised fluctuation theorem serving as a benchmark for verifying the probability setup. Based on the model, the system performance, in terms of the first passage steps and the total energy consumption, are analysed in the variable space of (measurement error, control field), leading to discussions on optimal controls of the system. Not only elucidating the basic concepts involved in a stochastic active system with information processing, this prototypical model could also inspire more elaborated modelings of natural smart organisms and industrial designs of controllable active systems with desired physical performances in the future.

Figures (3)

• Figure 1: (a) Evolution of the system consisting of a physical subsystem $\mathcal{X}$ in charge of the particle locomotion and a control (measurement - feedback) subsystem $\mathcal{C}$ in charge of the information-based regulation, and (b) zoom-in of the system dynamics during the time step, $t_{k}\rightarrow t_{k+1}$.
• Figure 2: Mean first passage steps $\langle \mathcal{T}\rangle$ as a function of the magnetic field $\hat{B}$, for $\epsilon=0$ (red), 1/2 (blue) and 1 (brown). The black solid line is obtained by $\langle \mathcal{T}\rangle /\mathcal{D} \simeq 2/[\hat{B} (1 -u_{p})]$ for $\hat{B}\rightarrow0$, and the round circles for different $\epsilon$ denote the values obtained by $\langle \mathcal{T}\rangle/\mathcal{D} \simeq2/(1+u_{p}-2 \epsilon u_{p})$ for $\hat{B}\rightarrow \infty$. The parameter, $u_{p}=\exp[-k_{0}\Delta t]=0.9$ is taken for illustration.
• Figure 3: Mean energy consumption per time step, $\ln[\langle \mathcal{E}\rangle /(\mathcal{D}E_{A}^{0})]$, as a function of the measurement error and the control field for the parameters of (a) $\mathcal{B}=1/2, \mathcal{M}=1$, (b) $\mathcal{B}=1/2$, $\mathcal{M}=30$, and (c) $\mathcal{B}=10$, $\mathcal{M}=0.1$, where the black circles denote the local minima and the red triangle denote the global minimum. The control parameter, $\mathcal{B}=E_{B}^{0}/E_{A}^{0}$, compares the magnetic-control energy scale to the active (running) energy $E_{A}^{0}$, and the measurement parameter, $\mathcal{M}=E_{M}^{0}/E_{A}^{0}$, compares the measurement energy scale to the active energy $E_{A}^{0}$. (d) Optimal measurement error and (e) optimal field as a function of the measurement parameter, $\mathcal{M}$, for the control parameter of $\mathcal{B}=1/2$ (black line), $10$ (red line) and $50$ (blue line). Squares and circles in (e) denote the values of the optimal field at the limit of $\mathcal{M}\rightarrow0$ and $\mathcal{M}\rightarrow\infty$, respectively. (f) Optimal field at the limit of $\mathcal{M}\rightarrow0 ~ (\epsilon=0)$, $\hat{B}^{\mathrm{opt}}_{\epsilon=0}$, and (g) at the limit of $\mathcal{M}\rightarrow\infty ~(\epsilon=1)$, $\hat{B}^{\mathrm{opt}}_{\epsilon=1}$, as a function of the control parameter, $\mathcal{B}$. Inset of (f) denote the values of the optimal field for the full range of $\mathcal{B}\in(10^{-6},5\times 10^{1})$.