Observation-Time-Induced Crossover in Driven Anomalous Transport

Abstract

We investigate how a weak constant force becomes detectable through fluctuations in anomalous transport in strongly heterogeneous media. Rather than focusing on the mean drift, we show that the key signature of the force appears in the variance of the particle displacement. As representative models, we study a biased continuous-time random walk (CTRW) with nearest-neighbor jumps and a biased quenched trap model (QTM) with a power-law waiting-time tail. By analysing the force dependence of the displacement variance, we quantify how fluctuations respond to weak driving. We find that for $α<2$, the response exhibits an observation-time-induced crossover: at fixed bias, the variance initially follows its unbiased scaling and only at later times crosses over to a force-dominated nonequilibrium regime. Equivalently, at fixed observation time $t$, there exists a threshold bias $\varepsilon_c(t)$ separating an apparently equilibrium-like regime from a detectable nonequilibrium response. This threshold decreases with increasing $t$, implying that weaker forces become observable over longer measurement windows. Quenched disorder further lowers the detection threshold compared with CTRW, and the crossover reflects a competition between the finite observation time and the intrinsic relaxation time of the driven heterogeneous system.

Figures (7)

• Figure 1: Normalized transport response of the CTRW for different observation times. (a) $\alpha=0.6$, (b) $\alpha=1.5$, and (c) $\alpha=2.5$. The solid lines show the asymptotic behaviors of the variance under an external force, obtained Eqs. \ref{['eq:varofx(t)_CTRW']}, \ref{['eq:Var_CTRW_alp12']}, and \ref{['eq:Var_CTRW_alpgt2']}. The dashed horizontal line indicates $R=1$ as a reference.
• Figure 2: Normalized transport response of the QTM for different observation times. (a) $\alpha=0.6$, (b) $\alpha=1.5$, and (c) $\alpha=2.5$. For $\alpha<2$, the solid lines represent the leading-order contribution to the displacement variance in the large-bias limit, corresponding to the second term of Eq. \ref{['eq:VarXt_Nt']}, $\left(\mathrm{Var}(N_t)_{\varepsilon>0}-\Braket{N_t}_{\varepsilon>0}\right)\varepsilon^{2}$. For $\alpha>2$, where an observation-time induced crossover does not occur, the solid lines depict the asymptotic expression valid over the entire $\varepsilon$ range, given by Eq. (\ref{['eq:Var_QTM_gt2']}). The dashed horizontal line indicates $R=1$ as a reference.
• Figure 3: Scaling exponent $\nu$ of the threshold bias $\varepsilon_c$ for the driven CTRW and the driven QTM. (a) $0<\alpha<1$, (b) $1<\alpha<2$. The solid and dashed lines correspond to the scaling exponents for the driven CTRW and the driven QTM, respectively.
• Figure 4: Heatmap of the normalized transport response for (a) the driven CTRW and (b) the driven QTM. The solid lines indicate the theoretical crossover threshold given by Eq. \ref{['eq:DCepsCTRW']} for the CTRW and Eq. \ref{['eq:e_t_QTM']} for the QTM. The coefficient $C_\alpha$ is obtained numerically.
• Figure 5: Convergence of the variance under bias, $\mathrm{Var}[x(t)]_{\varepsilon>0}$, to the nonequilibrium scaling in the CTRW. The black solid line represents the asymptotic behavior of the unbiased variance, $\mathrm{Var}[x(t)]_{\varepsilon=0}$, whereas the other solid lines represent the asymptotic behaviors of the biased variance, $\mathrm{Var}[x(t)]_{\varepsilon>0}$, for each value of $\varepsilon$.