Fluctuations of absorption of interacting diffusing particles by multiple absorbers

We study fluctuations of particle absorption by a three-dimensional domain with multiple absorbing patches. The domain is in contact with a gas of interacting diffusing particles. This problem is motivated by living cell sensing via multiple receptors distributed over the cell surface. Employing the macroscopic fluctuation theory, we calculate the covariance matrix of the particle absorption by different patches, extending previous works which addressed fluctuations of a single current. We find a condition when the sign of correlations between different patches is fully determined by the transport coefficients of the gas and is independent of the problem's geometry. We show that the fluctuating particle flux field typically develops vorticity. We establish a simple connection between the statistics of particle absorption by all the patches combined and the statistics of current in a nonequilibrium steady state in one dimension. We also discuss connections between the absorption statistics and (i) statistics of electric currents in multiterminal diffusive conductors and (ii) statistics of wave transmission through disordered media with multiple absorbers.

Figure 1

A sketch of our system at $t=0$. A gas of particles (black dots) surrounds a three-dimensional domain. The reflecting part ${\mathrm{\Omega }}_r$ of the domain boundary is shown in dark gray, the absorbing patches ${\mathrm{\Omega }}_1$, ${\mathrm{\Omega }}_2$, and ${\mathrm{\Omega }}_3$ are shown in light gray.

Figure 2

(a) The vertical cross section $y=0$ of the optimal density field (60) of a gas of RWs in contact with a sphere, conditioned on an enhanced particle absorption by the northern hemisphere, $n_1/{\overline{n}}_1=2$, and a reduced absorption by the southern hemisphere, $n_2/{\overline{n}}_2=0.5$, where ${\overline{n}}_1={\overline{n}}_2=2\pi D_0{\rho }_{0}R$. The rest of the parameters are $R=1,n_0=1$, and $D_0=1$. The black segment is the equator. (b) The density difference ${\rho }_{\text{RWs}}\left(\mathbf{x}\right)-{\overline{\rho }}_{\text{RWs}}\left(\mathbf{x}\right)$. Notice the enhanced density next to the northern hemisphere and reduced density next to the southern hemisphere.

Figure 3

The flux field ${\mathbf{J}}_{\text{RWs}}\left(\mathbf{x}\right)$ from Eq. (62) for the same setting as in Fig. 2. The arrows represent the field lines of the flux.

Figure 4

The vorticity norm $|\mathbf{\nabla }\times {\mathbf{J}}_{\text{RWs}}(r,\theta )|$, see Eq. (67), for the same setting as in Figs. 2 and 3. Shown is the vorticity norm vs $r$ in a vertical cross section for different $\theta$. (a) $\theta =\pi /2$. The vorticity diverges at $r\to R$, as predicted by Eq. (69). The dashed line in the inset is the $r^{-5}$ asymptotic (68). (b) $\theta =\pi /3$, $\pi /4$, and $\pi /6$ (marked by $a$, $b$, and $c$, respectively). Along these directions the vorticity vanishes on the sphere, as again predicted by Eq. (69). The dashed line in the inset is the $r^{-5}$ asymptotic (68) for $\theta =\pi /6$.