Thermodynamic Uncertainty Relation Bounds the Extent of Anomalous Diffusion

In a finite system driven out of equilibrium by a constant external force the thermodynamic uncertainty relation (TUR) bounds the variance of the conjugate current variable by the thermodynamic cost of maintaining the nonequilibrium stationary state. Here we highlight a new facet of the TUR by showing that it also bounds the timescale on which a finite system can exhibit anomalous kinetics. In particular, we demonstrate that the TUR bounds subdiffusion in a single file confined to a ring as well as a dragged Gaussian polymer chain even when detailed balance is satisfied. Conversely, the TUR bounds the onset of superdiffusion in the active comb model. Remarkably, the fluctuations in a comb model evolving from a steady state behave anomalously as soon as detailed balance is broken. Our work establishes a link between stochastic thermodynamics and the field of anomalous dynamics that will fertilize further investigations of thermodynamic consistency of anomalous diffusion models.

Figure 1

Anomalous diffusion in finite systems. (a) Single file on a ring driven by a force $F$. (b) Tagged-particle diffusion in a harmonic chain. (c) Biased diffusion in a finite (periodic) comb. The experimental observable is the unbounded displacement $x_t$ in the direction of the force $F$. (d) The TUR, ${\sigma }_x^2(t)\ge Ct$, delivers a threshold time $t^{*}$ that imposes an upper bound on the duration of subdiffusion (dashed blue line) or the earliest possible onset of superdiffusion (dotted green line). The star denotes $K_{\alpha }(t^{*}{)}^{\alpha }=C{t}^{*}$ in Eq. (3).

Figure 2

Variance of particle-displacement in a single file on a ring [see Fig. 1]. (a) All $N$ particles are pulled by a force $F$ (here $F=0$); (b) only the tagged particle is pulled by a force $F\mathrm{\Omega }\equiv f\times k_{B}T$ (the inset depicts the effect of $F$) with $N=10$. Symbols represent the centralized TAMSD [49] extracted from a long trajectory $\tau ={10}^3\times D/{\mathrm{\Omega }}^2$ for each $N$. The lines are deduced from a modified Jepsen mapping (see Supplemental Material [72]). Parameters: $D=k_{B}T=\mathrm{\Omega }=1$ and $d=0$, i.e., time is measured in units of $D/{\mathrm{\Omega }}^2$ and displacements in units of $\mathrm{\Omega }=l-Nd$.

Figure 3

(a) ${\sigma }_x^2(t)$ from Eq. (4) for a dragged Gaussian chain with $N=100$ beads, where we tag the $k$th particle ($k=1,2,10,50$). The TUR bound is shown as the dashed black line. Taking, e.g., $k=10$ we find transient subdiffusion ${\sigma }_x^2(t)\simeq K_{\alpha }{t}^{\alpha }$ (solid black line) in the vicinity of $t\sim t_{\mathrm{ref}}\equiv {10}^1$; using Eq. (4) yields the exponent $\alpha \equiv t{\partial }_t\ln {\sigma }^2(t){|}_{t=t_{\mathrm{ref}}}\approx 0.508$ with $K_{\alpha }\equiv t_{\mathrm{ref}}^{-\alpha }{\sigma }^2(t_{\mathrm{ref}})$. The rectangle denotes the upper bound on the extent of subdiffusion $t^{*}$ while the vertical arrow highlights the actual time at which the subdiffusive regime for $k=10$ terminates. (b) ${\sigma }_x^2(t)$ of the first bead ($k=1$) for increasing $N$. Symbols denote the TUR bound.

Figure 4

${\sigma }_x^2$ in the driven comb model [see Fig. 1]. We consider various driving forces $F$ and side branches with length $L=10$ separated by a distance $l=3$ yielding a steady state probability in the ring ${\varphi }_m=l/(l+2L)=3/23\approx 0.13$ with $D=\beta =1$. The force-free case $F=0$ coincides with the bound $Ct$ in Eq. (2). The thick lines correspond to $K_{\alpha }{t}^{\alpha }$ with the maximal exponent $\alpha \equiv {\max }_{t}t{\partial }_t\ln {\sigma }_x^2(t)$ depicted in the inset. Symbols denote the time $t^{*}$ in Eq. (3), where $K_{\alpha }{t}^{\alpha }$ and $Ct$ intersect. Long times $t\to \infty$ and strong driving $\beta Fl\gg 1$ yield ${\sigma }_x^2\simeq 2D{\varphi }_{m}t+(\beta Fl{)}^2(1-{\varphi }_m{)}^{3}t/6$.