Mathematical, Thermodynamical, and Experimental Necessity for Coarse Graining Empirical Densities and Currents in Continuous Space

We present general results on fluctuations and spatial correlations of the coarse-grained empirical density and current of Markovian diffusion in equilibrium or nonequilibrium steady states on all timescales. We unravel a deep connection between current fluctuations and generalized time-reversal symmetry, providing new insight into time-averaged observables. We highlight the essential role of coarse graining in space from mathematical, thermodynamical, and experimental points of view. Spatial coarse graining is required to uncover salient features of currents that break detailed balance, and a thermodynamically “optimal” coarse graining ensures the most precise inference of dissipation. Defined without coarse graining, the fluctuations of empirical density and current are proven to diverge on all timescales in dimensions higher than one, which has far-reaching consequences for the central-limit regime in continuous space. We apply the results to examples of irreversible diffusion. Our findings provide new intuition about time-averaged observables and allow for a more efficient analysis of single-molecule experiments.

Figure 1

(a) Diffusive trajectory traversing an observation window $U_{\mathbf{0}}^h(x,y)=1$ if $|x|,|y|\le 1/2$ and $U_{\mathbf{0}}^h(x,y)=0$ otherwise, with time running from dark to bright. Arrows denote contributions $\delta {\mathbf{x}}_i^s=(\delta x_i^s,\delta y_i^s)$ of the two sojourns in $U_{\mathbf{0}}^h$ between times ${\tau }_i^{-}$ and ${\tau }_i^{+}$ (see Eq. (a1) in Appendix pp1). (b) Corresponding $t\overline{{\rho }_{\mathbf{0}}^U}(t)$ and components of $t\overline{{\mathbf{J}}_{\mathbf{0}}^U}(t)$ from Eq. (1) as functions of $t$. (c) Two trajectories $({\mathbf{x}}_{\tau })$ (gray lines) of length $t=5$ in confined rotational flow with $\mathrm{\Omega }=5$ (arrows depict the steady-state current ${\mathbf{j}}_{\text{s}}$). The red cross is the reference point ${\mathbf{x}}_R=(1,0)$ considered in (d)–(f). Coarse-grained density (d) and $x$ current (e) for a Gaussian window $U_{{\mathbf{x}}_R}^h$ with $h=0.3$. Fluctuations of $\overline{{\rho }_{\mathbf{x}}^U}$ and $\overline{J_{x,y}^U}\equiv (\overline{{\mathbf{J}}_{{\mathbf{x}}_R}^U}{)}_{x,y}$ encode violations of detailed balance even where $\overline{J_x^U}$ vanishes. (f) Squared relative error of $\overline{J_y^U}$ for $U_{{\mathbf{x}}_R}^h$ as a function of $h$ (gray) bounded by the thermodynamic uncertainty relation (TUR; blue). A variance diverging as $h^{-2}$ (dashed) as $h\to 0$ and vanishing mean for $h\gg 1$ allow for intermediate $h$ optimizing the TUR-bound and thus the inferred dissipation.

Figure 2

(a) Two sample trajectories in a shear flow ${\mathbf{F}}_{\mathrm{sh}}(\mathbf{x})$ (gray arrows) with Stratonovich displacements $\circ d{\mathbf{x}}_t$ in the initial ${\mathbf{x}}_{t_1}=\mathbf{z}$ and final point ${\mathbf{x}}_{t_2}={\mathbf{z}}^{\prime }$ for fixed $t_1<t_2$ depicted by purple and yellow arrows, respectively. Time is running from dark to bright. (b) Trajectories as in (a) but running from ${\mathbf{x}}_{t_1}={\mathbf{z}}^{\prime }$ to ${\mathbf{x}}_{t_2}=\mathbf{z}$. (c) As in (b) but with the inverted shear flow $-{\mathbf{F}}_{\mathrm{sh}}({\mathbf{x}}^{\prime })$ (blue background arrows) and initial and final increments depicted by gray and blue arrows. (d) Ensemble of paths from ${\mathbf{x}}_{t_1}=\mathbf{z}$ to ${\mathbf{x}}_{t_2}={\mathbf{z}}^{\prime }$ contributing to $P_{\mathbf{z}}({\mathbf{z}}^{\prime },t_2-t_1)$. The average initial displacement ${⟨\circ d{\mathbf{x}}_t⟩}_{{\mathbf{x}}_{t_1}=\mathbf{z}}^{{\mathbf{x}}_{t_2}={\mathbf{z}}^{\prime }}$ is depicted by the black-purple arrow, and the mean path $\mathbf{z}\to {\mathbf{z}}^{\prime }$ in time $t_2-t_1$ by the gray gradient line. (e) As in (d) but corresponding to (b) instead of (a). (f) As in (e) but with the reversed shear flow as in (c). (g), (h) Since the shear flow breaks time-reversal symmetry, initial-point increments in (a) cannot be obtained by inverting final-point increments in (b). By dual-reversal symmetry initial-point increments follow from inverting the final-point increments in the inverted shear flow in (c), which explains initial point increments $\circ d{\mathbf{x}}_{t_1}$ in current-density correlations and current (co)variances via the easier and more intuitive final point increments $\circ d{\mathbf{x}}_{t_2}^{-{\mathbf{j}}_{\text{s}}}$.

Figure 3

$t{\mathrm{var}}_{\mathbf{J}}^{\mathbf{x}}$ as a function of the radius $|\mathbf{x}|$ in the harmonically confined rotational flow in Fig. 1 for increasing $\mathrm{\Omega }$ with Gaussian $U_{\mathbf{x}}^h$ with width $h$ at (a) $t=0.2$ and (b) $t=1$; lines depict Eq. (5) and symbols simulations [80]. (c) $t{\mathrm{var}}_{\mathbf{J}}^{\mathbf{x}}$ at $t=1$ for $\mathrm{\Omega }=10$ (full lines) and equilibrium $\mathrm{\Omega }=0$ (dashed lines), for various $h$ decreasing along the arrow. Inset: divergence of ${\mathrm{var}}_{\mathbf{J}}^{\mathbf{x}}$ as $h\to 0$ at $|\mathbf{x}|=1$; the dashed line depicts Eq. (6). Note the logarithmic scales. (d) ${\mathrm{var}}_{\mathbf{J}}^{\mathbf{x}}$ as a function of $t$ for very strong driving $\mathrm{\Omega }=50$; inset: (d) on logarithmic scales alongside the central-limit scaling $\propto t^{-1}$.