Figure 1

The asymptotic probability distribution function of the particle based on numerical simulation for chaotic fluctuations from the $G^{\left(2\right)}$ map (solid curve with square markers), the $G^{\left(3\right)}$ map (dashed curve with triangle markers), and the $G^{\left(4\right)}$ map (dashed-dotted curve with circle markers). The parameters in dimensionless units are $\tau =10.0$, $\gamma =10.0$, $k_{B}T=0.25$. The ensemble size used in the numerical simulation is $1\times {10}^5$, with an iteration length of 50.Reuse & Permissions

Figure 2

The spatially periodic and asymmetric potential given by Eq. (30) with $\mu =1$ and $A=-2$.Reuse & Permissions

Figure 3

The directed current $J_d$ against the asymmetric factor $A$ of the spatially periodic and asymmetric potential. The solid line represents analytical results from Eq. (33), while the $*$'s represents numerical solutions. The parameters in dimensionless units are $\tau =1$, $\gamma =2\times {10}^3$, $k_{B}T=0.2$, $\mu =1$. The ensemble size used in the simulation is $1\times {10}^3$, with an iteration length of $1\times {10}^7$.Reuse & Permissions

Figure 4

The directed current $J$ versus the turn on time $n_{\text{on}}$ of the constant flashing ratchet. We have assumed $n_{\text{on}}=n_{\text{off}}$. The curves from top to bottom are as follows: (solid line) analytical result, chaotic noise for $A=2$; (dashed line) analytical result, Gaussian noise for $A=2$; (dotted line) analytical result, chaotic noise for $A=1$; (dash-dotted line) analytical result, Gaussian noise for $A=1$; ($+$) numerical result, chaotic noise for $A=2$; ($\circ$) numerical result, Gaussian noise for $A=2$; ($*$) numerical result, chaotic noise for $A=1$; and ($▵$) numerical result, Gaussian noise for $A=1$. Note that the analytical results are given by Eq. (39), with $J_d=0$ in the case of Gaussian noise. The parameters in dimensionless units are $\tau =1$, $\gamma =2\times {10}^3$, $k_{B}T=0.2$, $\mu =1$. The ensemble size used in the simulation is $1\times {10}^3$, with an iteration length of $1\times {10}^6$.Reuse & Permissions

Figure 5

A comparison of the relation between the directed current $J$ and the asymmetric factor $A$ for the constant flashing ratchet when $n_{\text{on}}=1500$ and $n_{\text{off}}=500$. The curves are given as follows: (solid line) analytical result for chaotic noise; (dash-dotted line) analytical result for Gaussian noise; ($*$) numerical result for chaotic noise; and ($\circ$) numerical result for Gaussian noise. The analytical results are based on Eq. (39), with $J_d$ set to zero for the case of Gaussian noise. The parameters in dimensionless units are $\tau =1$, $\gamma =2\times {10}^3$, $k_{B}T=0.2$, $\mu =1$. The ensemble size used in the simulation is $1\times {10}^3$, with an iteration length of $1\times {10}^6$.Reuse & Permissions

Figure 6

A comparison of the relation between the directed current $J$ and the asymmetric factor $A$ for the dichotomous flashing ratchet. The curves are given as follows: ($*$) numerical result for chaotic noise; and ($\circ$) numerical result for Gaussian noise. The parameters in dimensionless units are $\nu =50$, $\varepsilon =0.01$, $\tau =1$, $\gamma =2\times {10}^3$, $k_{B}T=0.2$, $\mu =1$. The ensemble size used in the simulation is $2\times {10}^3$, with an iteration length of $2\times {10}^6$.Reuse & Permissions

Figure 7

The directed current $J_d$ versus the phase $\phi$ of the state dependent diffusion. The potential is spatially periodic and symmetric; that is, $A=0$. The curves are as follows: (solid line) analytical results for chaotic noise; ($*$) numerical results for chaotic noise; (dash-dotted line) analytical results for Gaussian noise; and ($\square$) numerical results for Gaussian noise. The parameters in dimensionless units are $\tau =1$, $\gamma =2\times {10}^3$, $k_{B}T=0.2$, $\mu =1$, $\epsilon =0.2$. The ensemble size used in the simulation is $1\times {10}^3$, with an iteration length of $1\times {10}^7$.Reuse & Permissions

Figure 8

The directed current $J_d$ versus the phase $\phi$ of the state dependent diffusion for the case of chaotic noise. The solid line in Fig. 7 has been replotted here as a thick solid line for the purpose of comparison. The rest of the curves are as follows: (dash-dotted line) analytical results for $A=-2$; ($\square$) numerical results for $A=-2$; (thin solid line) analytical results for $A=2$; and ($*$) numerical results for $A=2$. The details of the simulation parameters employed are the same as those used in Fig. 7.Reuse & Permissions

Figure 9

The directed current $J_d$ versus the phase $\phi$ of the state dependent diffusion for the case of Gaussian noise. The dash-dotted line in Fig. 7 has been replotted here as a thick solid line for the purpose of comparison. The rest of the curves are as follows: (dash-dotted line) analytical results for $A=-2$; ($\square$) numerical results for $A=-2$; (thin solid line) analytical results for $A=2$; and ($*$) numerical results for $A=2$. The details of the simulation parameters employed are the same as those used in Fig. 7.Reuse & Permissions

Figure 10

The directed current $J_d$ versus the load $F_L$. The potential is spatially periodic and symmetric; that is, $A=0$. The curves are as follows: (solid line) analytical results for chaotic noise; ($*$) numerical results for chaotic noise; (dash-dotted line) analytical results for Gaussian noise; and ($\square$) numerical results for Gaussian noise. The parameters in dimensionless units are $\tau =1$, $\gamma =2\times {10}^3$, $k_{B}T=0.2$, $\mu =1$. The ensemble size used in the simulation is $1\times {10}^3$, with an iteration length of $1\times {10}^7$.Reuse & Permissions

Figure 11

The directed current $J_d$ versus the load $F_L$. The potential is spatially periodic and asymmetric. The curves are as follows: (dotted line) analytical results for chaotic noise with $A=-2$; ($+$) numerical results for chaotic noise with $A=-2$; (dashed line) analytical results for Gaussian noise with $A=-2$; ($▵$) numerical results for Gaussian noise with $A=-2$; (solid line) analytical results for chaotic noise with $A=2$; ($*$) numerical results for chaotic noise with $A=2$; (dash-dotted line) analytical results for Gaussian noise with $A=2$; and ($\square$) numerical results for Gaussian noise with $A=2$. The details of the simulation parameters employed are the same as those used in Fig. 10.Reuse & Permissions