Nonequilibrium uncertainty principle from information geometry

With a statistical measure of distance, we derive a classical uncertainty relation for processes traversing nonequilibrium states both transiently and irreversibly. The geometric uncertainty associated with dynamical histories that we define is an upper bound for the entropy production and flow rates, but it does not necessarily correlate with the shortest distance to equilibrium. For slowly driven systems, we show that our uncertainty lower bounds the rate of energy fluctuations. For a model one-bit memory device, we find that expediting the erasure protocol increases the maximum dissipated heat and geometric uncertainty. A driven version of Onsager's three-state model shows that a set of dissipative, high-uncertainty initial conditions, some of which are near equilibrium, scar the state space.

Figure 1

(a) Kinetic scheme for an asymmetric three-state model. The time-dependent transition rates are $k^{+}\left(t\right)=4\arctan \left({\omega }_{1}t\right)$ and $k{\left(t\right)}^{-}=4\arctan \left({\omega }_{2}t\right)$ with ${\omega }_1=4$ and ${\omega }_2=6$. Using $\arctan \left({\omega }_{i}t\right)$ ensures that all initial conditions evolve to the stationary distribution $p_x^{\infty }=(1/3,1/3,1/3)$. Each initial condition in (b) and (c) is marked by a circle in (d), which shows the positive octant of a sphere colored by $\dot{S}(t=t_f)$. Here $t_f$ is the initial time for each path to reach the stationary distribution (white square). At $t=0, \dot{S}\left(t_0\right)=0$. The scar on the state space in (d) emerges over time from initial conditions that ultimately dissipate and have higher path uncertainties. One initial condition is highlighted with a circle and corresponds to the dashed lines in (b) and (c). (c) Upper bound for $\dot{S}\left(t\right), \mathcal{C}-{\sigma }^2\tau /2$, for the five select initial conditions as a function of the time interval $\tau =t-t_0$.

Figure 2

Accelerating the erasure of information (destroying state $x=0$), by increasing $\omega$, generates more (a) uncertainty (${\sigma }^2\tau /2$, dashed lines) about the erasure path and maximum heat dissipated $[\langle \langle q(t)\rangle \rangle$, solid lines]. (b) Heat dissipated $\langle \langle q\left(t\right)\rangle \rangle$ follows the geodesic (${\sigma }^2\approx 0$) up until the maximum amount of heat is being released. Beyond that point, the system undergoes entropic deceleration. The time series are from $p\left(t_0\right)=[0.5,0.5]$ to $p\left(t_f\right)=[1,0]$ for $\omega =[0.05,0.2,0.3]$ shown in blue, black, and red lines, respectively.