Entropic stochastic resonance of finite-size particles in confined Brownian transport

We demonstrate the existence of entropic stochastic resonance (ESR) of passive Brownian particles with finite size in a double- or triple-circular confined cavity, and compare the similarities and differences of ESR in the double-circular cavity and triple-circular cavity. When the diffusion of Brownian particles is constrained to the double- or triple-circular cavity, the presence of irregular boundaries leads to entropic barriers. The interplay between the entropic barriers, a periodic input signal, the gravity of particles, and intrinsic thermal noise may give rise to a peak in the spectral amplification factor and therefore to the appearance of the ESR phenomenon. It is shown that ESR can occur in both a double-circular cavity and a triple-circular cavity, and by adjusting some parameters of the system, the response of the system can be optimized. The differences are that the spectral amplification factor in a triple-circular cavity is significantly larger than that in a double-circular cavity, and compared with the ESR in a double-circular cavity, the ESR effect in a triple-circular cavity occurs within a wider range of external force parameters. In addition, the strength of ESR also depends on the particle radius, and smaller particles can induce more obvious ESR, indicating that the size effect cannot be safely neglected. The ESR phenomenon usually occurs in small-scale systems where confinement and noise play an important role. Therefore, the mechanism that is found could be used to manipulate and control nanodevices and biomolecules.

Figure 1

Sketch of a double-circular cavity, where the forces $F\left(t\right)$ and $G$ are applied on the overdamped particles (the orange and green balls in the cavity). The orange dotted line represents the effective boundary that can be reached by the particle center with limited size. The radius of the circular cavity is $R_1$, and the width of the hole between the two circular cavities is $2a_1$.

Figure 2

Sketch of a triple-circular cavity, where the forces $F\left(t\right)$ and $G$ are applied on the overdamped particles (the orange and green balls in the cavity). The orange dotted line represents the effective boundary that can be reached by the particle center with limited size. The radius of the circular cavity is $R_2$, and the width of the hole between two adjacent circular cavities is $2a_2$.

Figure 3

Schematic diagram of effective potential function $A_1\left(x\right)$ and effective potential functions $A_{11}\left(x\right)$ and $A_{12}\left(x\right)$ under two limit cases when three different noise intensities $D$ are taken in the double-circular cavity.

Figure 4

Schematic diagram of effective potential function $A_2\left(x\right)$ and effective potential functions $A_{21}\left(x\right)$ and $A_{22}\left(x\right)$ under two limit cases when three different noise intensities $D$ are taken in the triple-circular cavity.

Figure 5

Schematic diagram of the change of spectral amplification $\eta$ with noise intensity $D$ when taking different periodic driving signal frequencies $\sigma \left(\sigma =0.01,0.001,0.0001\right)$, for the input signal amplitude $A={10}^{-4}$, the transversal force $G=1$, and for the radius of double-circular cavity $R_1=1$. For small driving frequencies and amplitudes, the spectral amplification $\eta$ is calculated by the two-state approximation and Brownian dynamics simulation. The corresponding results of the solid lines with different colors are obtained by the two-state approximation, i.e., Eqs. (19, 20, 21), while the corresponding results of the symbols with different shapes are calculated by Brownian dynamics simulation.

Figure 6

(a) The dependence of the spectral amplification $\eta$ on noise intensity $D$ when different amplitudes $A\left(A=1.0,2.0,3.0\right)$ are taken, for the transversal force $G=5.5$ and the periodic driving signal frequency $\sigma =0.1$. (b) The dependence of spectral amplification $\eta$ on noise level $D$ at three different input signal frequencies $\sigma \left(\sigma =0.1,0.3,0.5\right)$, at a constant input amplitude $A=1.0$ and transversal force $G=5.5$. Different from the driving frequencies $\sigma$ and amplitudes $A$ in Fig. 5, for larger $\sigma$ and $A$, the spectral amplification $\eta$ is calculated by the one-dimensional modeling and Brownian dynamics simulation. The solid line corresponds to the result of numerical integration through the one-dimensional probability density equation (23), and different types of symbols are marked as the results obtained by the Brownian dynamics simulation method.

Figure 7

The dependence of spectral amplification $\eta$ on noise level $D$ in the triple-circular cavity (TC) and the double-circular cavity (DC) at different external force parameters (including transversal force $G$, input signal amplitude $A$, and frequency $\sigma$). For the double- and triple-circular cavities, the radius of the cavity, $R_i(i=1,2)$, and the half width of the hole connecting adjacent circular cavities $a_i$ are the same, which are $R_i=1$ and $a_i=0.3R_i$, respectively. The radius of the Brownian particles diffused in the two cavities is $r_p=0.3a_i$.

Figure 8

The dependence of spectral amplification $\eta$ of Brownian particles of different sizes on noise level $D$ in the triple-circular cavity (TC) with the same external force parameters (including transversal force $G$, input signal amplitude $A$, and frequency $\sigma$). In the four subgraphs, the radii of the Brownian particles diffused in the cavity are $0.1a_3,0.3a_3,0.7a_3,0.9a_3$, respectively.

Figure 9

The dependence of the maximum value of spectral amplification ${\eta }_{max}$ on particle radius $r_p$ in the triple-circular cavity (TC). The external force parameters (including transversal force $G$, input signal amplitude $A$, and frequency $\sigma$) are consistent with those in Fig. 8.

Figure 10

Realization of reflection boundary conditions. (a) Check whether the particles move outside the channel. (b) Calculate the collision point, and the tangent vector and normal vector at the collision point when the particles diffuse outside the channel. (c) Calculate the new position coordinates of the particles after collision with the wall according to the reflection boundary condition.