Figure 1

Brownian ratchet and entropic barriers. (a) Local biased diffusion due to confinement, described in terms of an effective entropic potential $S\left(x\right)$. (b) Two-state model for a Brownian ratchet: In state 1 particles slide along the potential $V_1$ and jump with rate ${\omega }_{1,2}$ to state 2, where they diffuse until they jump back with rate ${\omega }_{2,1}$. The jump rates operate in different regions along the potential period, breaking detailed balance. (c) A Brownian motor moving in a confined environment will be sensitive to the free energy $A_1$ (solid) generated by ratchet potential $V_1$ (dotted-dashed) and entropic potential generated by the channel shown in (a) (dashed).Reuse & Permissions

Figure 2

Rectification of a Brownian motor in a symmetric ratchet and symmetric corrugated channel. (a), (b) Velocity, in units of $D_0/L$, where $D_0=k_{B}T/6\pi \eta R$ and $L$ stands for the potential period, of a flashing ratchet (squares) or a processive (circles), nonprocessive (triangles) motor moving according to the two-state model as a function of the phase shift $\Delta \phi$ for different values of the parameter $\beta /\gamma =0.7,0.8,0.9$ (the larger the symbol size, the larger $\beta$) and fixed $\gamma$ being the energetic potential $\Delta V_1=4.0$ and $\Delta Q=10$ for the flashing ratchet. (c) Free-energy profile for different $\Delta \phi$; the dotted box is the region where ${\omega }_{12}\ne 0$, and the vertical dashed-dotted lines mark the position of the maxima of $A_1\left(x\right)$ in the absence of rectification. (d), (e) Rectified velocity as a function of the entropic barrier height upon variation of $R$ (solid points) and $\beta$ (open points) for flashing ratchet (squares) or a processive (circles), nonprocessive (triangles) motor moving according to the two-state model. Larger points stand for larger $\Delta \phi$, being $\Delta \phi =0.1,0.2$ for the nonprocessive two-state model and flashing ratchet, and $\Delta \phi =0.1,0.3$ for the processive two-state model.Reuse & Permissions

Figure 3

Rectification of a Brownian motor in an asymmetric ratchet and symmetric corrugated channel. (a), (b) Velocity is normalized by $v_0$, which is the intrinsic velocity of a particle with the same radius $R$ under the action of the ratchet in the case of a flat channel with the same volume; symbols and parameters values as in Fig. 2. (c), (d) Velocity as a function of $\beta$ (open points) or $R$ (solid points); points as in Fig. 2. $\Delta \phi =0.2,0.3,0.5,0.6$ for the flashing ratchet and $\Delta \phi =0.1,0.8,0.9$ for the two-state model; the larger the symbols, the larger $\Delta \phi$.Reuse & Permissions