Transport and fluctuation-dissipation relations in asymptotic and preasymptotic diffusion across channels with variable section

We study the asymptotic and preasymptotic diffusive properties of Brownian particles in channels whose section varies periodically in space. The effective diffusion coefficient $D_{\mathrm{eff}}$ is numerically determined by the asymptotic behavior of the root mean square displacement in different geometries, considering even cases of steep variations of the channel boundaries. Moreover, we compared the numerical results to the predictions from the various corrections proposed in the literature to the well known Fick-Jacobs approximation. Building an effective one-dimensional equation for the longitudinal diffusion, we obtain an approximation for the effective diffusion coefficient. Such a result goes beyond a perturbation approach, and it is in good agreement with the actual values obtained by the numerical simulations. We discuss also the preasymptotic diffusion which is observed up to a crossover time whose value, in the presence of strong spatial variation of the channel cross section, can be very large. In addition, we show how the Einstein's relation between the mean drift induced by a small external field and the mean square displacement of the unperturbed system is valid in both asymptotic and preasymptotic regimes.

Figure 1

Sketch of the periodic structures considered in this work. (a) Smooth channel (Sm); (b) sharp channel (Sh). We identify $H$ regions (humps) of the channel such that $R/2<\left|y\right|\le \omega \left(x\right)$ and the $S$ region (shaft) $\left|y\right|\le R/2$. Diffusion regimes inside these channels depend on the ratio $Q=\mathcal{A}/R$ controlling the importance of the $H$ regions over the $S$ region. When not specified, we consider channels with fixed shaft parameters $L=10$, $R=4$, whereas, the size of the $H$ regions is changed to obtain stronger or weaker lateral particle trapping.

Figure 2

Plot of Eq. (18) vs $Q=\mathcal{A}/R$ (black line) together with the simulated $D_{\mathrm{eff}}$ data obtained by the slope of $\langle \Delta x^2\left(t\right)\rangle$ in the linear regime. Symbols correspond to values $Q=2^{-2},2^1,2^3,...2^{13}$; circles refer to the Sm channel with $L=10,R=4,\gamma =10$, while squares refer to the Sh channel with $L=10,R=4,\Delta =1.23$. Since the channels are equivalent (same area), the data from both structures are supposed to collapse onto each other. The percentage error is within 4%; the inset instead shows the deviation $|D_{\mathrm{eff}}\left(\mathrm{th}\right)-D_{\mathrm{eff}}\left(\mathrm{sim}\right)|$ of the black line from the open circles in the main panel. For a further comparison, we report the $D_{\mathrm{eff}}$ of the Sm channel computed by Eq. (10) using the local diffusion coefficients mentioned in the text: $D_{RR}\left(x\right)$ (red), $D_{KP}\left(x\right)$ (green), $D_{Zw}\left(x\right)$ (blue), and $D_{FJ}\left(x\right)$ (orange).

Figure 3

Comparison between simulated data of $D_{\mathrm{eff}}$ (circles) and the theoretical prediction (18) (black line) for the unbiased diffusion in a Sm channel, with $\gamma =1$ and the other parameters as in Fig. 2. The percentage error on simulation data is within 4%. The accuracy of Eq. (18) worsens at increasing $Q=\mathcal{A}/R$. Dashed lines represents the predictions of $D_{\mathrm{eff}}$ yielded by the generalized FJ approach involving the same $D\left(x\right)$ expressions of Fig. 2.

Figure 4

Sketch of a Sm channel with even (solid) and odd (shaded) $\gamma$. Notice the concavity change of the profile when $\gamma$ turns from even to odd. For $\gamma$ odd, the $H$ region of the channel becomes predominant, whereas the $S$ domain remains ill identified as it is restricted to pointlike regions marked by the letter $P$. In this case, the structure becomes equivalent to a septate channel [37] of section $R+\mathcal{A}$ (dashed lines) for which Eq. (15) fails.

Figure 5

Examples of sharp channels ${\text{Sh-T}}_1$ and ${\text{Sh-T}}_2$ with a boundary profile $\omega \left(x\right)$ which is a multivalued function of the longitudinal position; in that case, the generalized FJ approximation cannot be used. In both structures, the parameters, $h,\Delta ,{\ell }_1,{\ell }_2$ are tuned such that $\mu \left(H\right)$ coincides with the hump area of the boundaries used to generate Fig. 2.

Figure 6

Plot of $D_{\mathrm{eff}}/D_0$, theory (dashed line) and simulation data (symbols) vs $Q=\mu \left(H\right)/\left(R\Delta \right)$ referred to the structures ${\text{Sh-T}}_1$ and ${\text{Sh-T}}_1$ drawn in Fig. 5. We considered channels with two $\Delta$ values and tuned the other parameters ($h,{\ell }_1,{\ell }_2$) to maintain the same hump area in both structures. For this reason different data sets collapse onto each other. The agreement between theory and simulation results is convincing despite the geometric complexity of the channels. The percentage error on data is about 4% leaving the bars well below the symbol size.

Figure 7

Comparison between the relaxation of longitudinal $\langle \Delta x{\left(t\right)}^2\rangle$ and transversal $\langle \Delta y^2\left(t\right)\rangle$ diffusion for the same channel of Fig. 8 with two different values of $Q=2^5$ and $Q=2^{13}$. The system is prepared uniform over $S\cup H$, but the large $Q$ values prevents the fast lateral homogenization. In the case $Q=2^5$, the homogenization takes place at times ${\tau }_{\parallel }<{\tau }_{\perp }$, while for $Q=2^{13}$ the homogenization becomes particularly slow (${\tau }_{\perp }\simeq 3^9$) and it delays the onset of the standard diffusion along the $x$ axis, ${\tau }_{\perp }\simeq {10}^3$. Moreover, as long as $t<{\tau }_{\perp }$, the lateral diffusion satisfies $\langle \Delta y^2\left(t\right)\rangle =2D_{0}t$, supporting the derivation of Eqs. (24) and (25). The horizontal dashed line marks the value $2D_0{\tau }_{\perp }(S\cup H)=R^2{(1+Q)}^2/6$ obtained from the last line of Eq. (21).

Figure 8

Log-log plot of $\langle \Delta x^2\left(t\right)\rangle$ in a Sh channel with $Q=2^5$ as a function of the time for the system starting from three initial conditions: ${\mathbf{r}}_0\in H$ (squares), ${\mathbf{r}}_0\in H\cup S$ (triangles), ${\mathbf{r}}_0\in S$ (circles). Symbols refer to the simulation data, whereas the dashed lines represent their fit by Eq. (25) upon tuning the free parameter ${\tau }_0$. Parameters $R,L,\Delta$ are as in Fig. 2. The inset shows the same results for $Q=2^2$.

Figure 9

Log-log plot of the MSD versus the elapsed time for the channel sketched in Fig. 1 with increasing values of $\mathcal{A}$. For $\mathcal{A}/R=2^8$, the diffusion is anomalous ${\langle \Delta x^2\left(t\right)\rangle }_0\sim t^{1/2}$ with the same exponent of the random walk on a comb, $2\nu =1/2$, supporting the view that the structure in Fig. 1 virtually behaves as a comb at enough large $\mathcal{A}$.

Figure 10

Check of the FDR for the diffusion in Sh channels (13). Log-log plot of ${\langle \Delta x\left(t\right)\rangle }_f$ vs ${\langle \Delta x^2\left(t\right)\rangle }_0$; symbols are the simulation results and dashed lines represent the theoretical expectation (26), which is verified regardless of the $\mathcal{A}$ value and turns out to be independent of the initial particle distribution within the channel. The FDR is exactly satisfied also in smooth channels (not shown).

Figure 11

Sketch of the regular comb lattice used for its analogy to channels with narrow and long hump regions. The backbone plays the role of the channel $S$ region while the teeth act as $H$ regions.