1. Introduction
Through fascinating machines such as molecular motors, nature can achieve directed transport in cells or can also promote bacteria in the diffusive environment. This kind of device is often referred to as Brownian motors in physics, and they can convert random Brownian fluctuations into directed net motion. For example, some intracellular cargos can be transported along microtubules by kinesin-1, and the operation of this molecular motor is realized by energy input and the breaking of symmetry [1, 2]. Brownian motors operating in a nonequilibrium state serve as models for both biological and artificial machines at the submicron scale.
A fascinating application deals with transport phenomena: the noise is used to induce directed motion, i.e. unbiased fluctuations caused by a thermal environment can be rectified to a net current in the system [3]. The key mechanism that enables the transformation of isotropic thermal noise into average unidirectional motion is the ratchet effect. The ratchet mechanism not only plays an important role in related theoretical research, but also in relevant biological research. For example, the ratchet is considered a valid theoretical model for describing the step motion of the two-headed kinesin [4, 5]; protein biosynthesis on ribosomes is suggested to be accomplished by the ratchet mechanism [6, 7]. Recently, the Brownian ratchet has also been used to simulate the action of RNA and DNA polymerases in the process of RNA and DNA replication [8, 9].
In essence, the operation of both natural and artificial motors requires an asymmetric ratchet-shaped potential as well as an unbiased external driving force to push the system out of equilibrium [10]. Experimental studies [11] have revealed that the mechanism of symmetry breaking in molecular motors may be attributable to their intrinsic structure. Thus, symmetry breaking can be achieved by using the internal degrees of freedom, such as regulating the relative distance between the dimer components, periodically adjusting the interaction parameters of Brownian particles [12] and applying different forces on the dimer's two components [13, 14]. Moreover, theoretical studies of coupled Brownian particles in periodic structures are of great significance in various research domains, including molecular motors [15], surface diffusion [16], polymer physics [17] and colloids [18]. In particular, dimers are suited for modeling these systems due to their inherent degree of freedom and simplicity [14]. The ratchet systems based on different friction environments belong to another type of system whose internal symmetry is broken. In a coupled system, the effect of asymmetric friction on coupled particles can induce the internal symmetry breaking of a dimer. For example, in under-damped systems, self-propulsion can be realized by applying friction to the internal degrees of freedom [19], even without the external potential [20]. A macroscopic mechanical device which has the characteristics of friction asymmetry has been realized experimentally, and the device can convert fluctuating motion into unidirectional rotation [21].
In the asymmetrical potential, the directed transport of the ratchet has been discussed widely. However, with regard to inertial Brownian ratchets, there are still few studies on the transport behavior induced by internal symmetry breaking. In the present study, we examine a simple under-damped model composed of two inertial elastically coupled Brownian particles subjected to independent friction in a symmetric potential. This type of friction asymmetry is different from two-state systems that use protein friction to model molecular motors [22], where the irregular fluctuating motion can be rectified by switching between the high and low frictional states. The two states mentioned above correspond to the periodic attachment to the track. In our inertia ratchet model, the friction coefficient is independent of both time and space, and it is a fixed quantity that can produce a non-zero average velocity. The results show that the dimer velocity can obtain the maximum of both the optimal coupling strength and friction damping forces.
Unlike the molecular mechanism of the motility, the random and nonequilibrium behavior of kinesin brings difficulties in understanding the efficiency of a motor. Therefore, the aim of this paper is also to explore the efficiency of molecular motors further. The energetics of single-molecule motors has been analyzed in studies that have measured their stall forces [23]. Experimentally, the stall force of the kinesin was about 7 pN, which means that the maximum work per 8 nm step is about 56 pN nm, and this maximum work is less than the approximation of the physiological free energy change per ATP molecule hydrolyzed (about 85 pN nm). Therefore, the efficiency of kinesins has attracted widespread attention [24–27]. In particular, different kinds of efficiencies have been defined for motor proteins. For example, if the work against an external force is concerned, thermodynamic efficiency (energy conversion efficiency) is proposed. However, if the work against viscous friction is considered, Stokes efficiency or generalized efficiency is more appropriate [28–31]. Interestingly, Spiechowicz et al demonstrated that a symmetric periodic potential system driven by temporally symmetric periodic force exhibits higher efficiency in the presence of asymmetric Poissonian white noise of mean $\left\langle \eta (t)\right\rangle =F$ [32, 33].
In this study, we investigate the impact of friction on the efficiency of inertial frictional ratchets. To assess the energy conversion efficiency of these ratchets, we apply periodic forcing and an external constant force (load) that opposes the ratchet current. Furthermore, we analyze our inertial frictional ratchets using the theory of stochastic thermodynamics [34, 35]. In this study, we discuss the averages of diverse thermodynamic quantities, such as the average velocity of the center of mass and the efficiency of energy conversion over a large number of stochastic trajectories. Interestingly, we observe some counterintuitive results, for example, the efficiency increases with the increase in the externally applied load. In particular, under certain circumstances, the dimer with different friction will move in opposite directions and can be separated. Our results can be applied practically for controlling and separating inertial Brownian particles as an impactful method.
2. The model of inertial frictional ratchets
We consider the transport behavior of two under-damped coupled Brownian particles in symmetric feedback ratchets. When the dimer driven by time-periodic forcing $F(t)$ and subjected to a Gaussian thermal fluctuating force $\xi (t)$ of zero mean moves in a cosinusoid potential ${V}_{P}(x),$ it can be described by the dimensionless Langevin equation [35, 36]
$\begin{eqnarray}\begin{array}{c}{\ddot{x}}_{{i}}+{\gamma }_{{i}}{\dot{x}}_{{i}}=-\beta (t){{\rm{d}}}_{{x}_{{i}}}{V}_{{\rm{P}}}({x}_{{i}})\\ \,-\,{\partial }_{{x}_{{i}}}{V}_{{I}}({x}_{1},{x}_{2};l)+F(t)-f+\sqrt{2{\gamma }_{{i}}D}{\xi }_{{i}}\left(t\right).\end{array}\end{eqnarray}$
In equation (1 ), ${x}_{{i}}$ ($i=1,2$) represents the position of the coupled particles at time $t,$ and ${\gamma }_{i}$ is the friction coefficient of the respective dimer components. The ratio of the friction coefficients of the two components can be merged into a dimensionless asymmetry parameter $\alpha ,$ allowing us to identify $\alpha =\tfrac{{\gamma }_{2}}{{\gamma }_{1}}.$ The biased force is the static perturbation denoted as $f,$ and $D$ is the noise intensity. Thermal fluctuations due to the coupling of the particles with a thermal bath are modelled by $\delta $-correlated Gaussian white noise ${\xi }_{i}(t)$ of zero mean and unit intensity, i.e. $\left\langle {\xi }_{i}(t)\right\rangle =0,$ $\left\langle {\xi }_{i}\left(t){\xi }_{j}(s\right)\right\rangle ={\delta }_{{ij}}\delta (t-s)$ for $i,j\in (1,2)$.
The rhs of equation (1 ) describes the influence of various forces on the Langevin dynamics. Note that the external forces are zero on average, e.g. $\left\langle {V^{\prime} }_{P}(x)\right\rangle =0$ over a spatial period $L,$ $\left\langle F(t)\right\rangle =0$ over a time period $\tau ,$ and symmetric thermal noise $\left\langle {\xi }_{i}(t)\right\rangle =0$ over its realizations. If the spatial reflection symmetry of ${V}_{P}(x)$ and/or the temporal reflection symmetry of $F(t)$ are broken, directed transport can always arise for an arbitrary value of $f$ [37]. For the general case, if ${V}_{P}(x)$ and $F(t)$ are symmetric, the particles cannot produce directed transport when $f=0.$ However, due to the friction asymmetry (different friction coefficients) between the two coupled particles, the directed transport discussed above still occurs in our inertial frictional ratchets.
For the potential ${V}_{P}(x),$ here we select the cosine function with a spatial period $L,$ 3 ), ${\rm{\Theta }}$ represents the Heaviside function. If the average force $g(t)$ is positive, the ratchet potential is on and the potential is off in all other instances.
$\begin{eqnarray}{V}_{P}(x)=\displaystyle \frac{1}{2}\left[1-\,\cos \left(\displaystyle \frac{2\pi }{L}x\right)\right].\end{eqnarray}$
Here, we set the period of the potential $L=1,$ i.e. ${V}_{P}(x+1)={V}_{P}(x).$ A significant increase in the transportation of the inertial frictional ratchets can be obtained when feedback on the state of the system is applied by the protocol that switches the ratchet potential on or off [38–40]. Here, both contributions, ${V}_{P}({x}_{1})$ and ${V}_{P}({x}_{2}),$ are synchronously switched on or off by a control protocol $\beta (t).$ The parameter $\beta (t),$ which switches the ratchet potential on or off, is given by [38–40] $\begin{eqnarray}\beta (t)={\rm{\Theta }}(g(t))=\left\{\begin{array}{cc}0, & g(t)\leqslant 0\\ 1, & g(t)\gt 0\end{array}\right.,\end{eqnarray}$
where $g(t)$ indicates the ensemble average of the force that the coupled particles are affected by the ratchet potential ${V}_{p}(x).$ As a ‘control target' and due to the potential switch on, we consider the average force $g(t)$ as $\begin{eqnarray}g(t)=-\displaystyle \frac{1}{2}\displaystyle \sum _{i=1}^{2}\displaystyle \frac{{\rm{d}}}{{\rm{d}}{x}_{i}}{V}_{p}({x}_{i}).\end{eqnarray}$
In equation (The coupling interaction ${V}_{I}({x}_{1},{x}_{2};l)$ is assumed to be elastic with the spring constant $k$ and natural length $l,$ which can be written as
$\begin{eqnarray}{V}_{I}({x}_{1},{x}_{2};l)=\displaystyle \frac{1}{2}k{({x}_{1}-{x}_{2}-l)}^{2}.\end{eqnarray}$
This means that there is no internal force between the components if ${x}_{2}-{x}_{1}=l.$ Furthermore, the potential is rocked by a time-dependent force $F(t).$ The external time-dependent force $F(t)$ can be of any form, both deterministic or stochastic [41, 42]. This force causes the ratchet to be in a nonequilibrium state and serves as an energy source for the movement of protein motors. In biological motors, this force originates from chemical reactions. In this context, the dimer is driven by unbiased time-periodic forces, namely $\begin{eqnarray}F(t)=A\,\cos \,\omega t,\end{eqnarray}$
where $A$ indicates the amplitude of the harmonic force, $\omega =2\pi /\tau $ is the frequency and $\tau $ denotes the rocking period.To analyze the directed transport of inertial frictional ratchets, the center-of-mass mean velocity of the dimer is discussed in detail. It is determined by [37, 43]
$\begin{eqnarray}\left\langle {V}_{\rm{cm}}\right\rangle =\mathop{\mathrm{lim}}\limits_{T\to \infty }\displaystyle \frac{1}{2T}\displaystyle \sum _{i=1}^{2}\displaystyle {\int }_{0}^{T}{\dot{x}}_{i}(t){\rm{d}}t.\end{eqnarray}$
Furthermore, the efficiency is also a crucial performance characteristic in the transport dynamics of inertial frictional ratchets. For frictional ratchets, the energy conversion efficiency [44] $\eta $ is defined as the ratio of the output to the input power
$\begin{eqnarray}\eta =\displaystyle \frac{{P}_{{\rm{out}}}}{{P}_{{\rm{in}}}},\end{eqnarray}$
where the total power output, defined as the work performed on the load per unit of time, is determined by $\begin{eqnarray}{P}_{\rm{out}}=\displaystyle \sum _{i=1}^{2}f\cdot \left\langle {V}_{i}\right\rangle ,\end{eqnarray}$
and $\langle {V}_{i}\rangle $ is the mean velocity of the ith particle.To do useful work, a load $f$ is applied against the direction of the current. As long as the load is less than the stopping force ${f}_{s},$ the current flows against the load and the inertial ratchet does work. Thus, in the operating range of the load, i.e. $0\lt f\lt {f}_{s},$ the dimer moves in the direction opposite to the load and the frictional ratchets do useful work. Therefore, the energy conversion efficiency of the dimer is always positive and $\eta \lt 1.$
For each trajectory, corresponding to one initial position $x(0),$ the thermodynamic work done by the external force $F(t)$ on the system (or the input energy) is quantified by utilizing the stochastic energetics theory proposed by Sekimoto [34]
$\begin{eqnarray}W(0,N\tau )=\displaystyle {\int }_{0}^{N\tau }\displaystyle \frac{\partial {U}_{i}({x}_{i},t)}{\partial t}{\rm{d}}t,\end{eqnarray}$
where $N$ is a large integer representing the number of periods required to reach the final point of the trajectory. The effective potential ${U}_{i}({x}_{i},t)$ acting on the ith particle is expressed as $\begin{eqnarray}{U}_{i}({x}_{i},t)=\beta (t)\left[{V}_{P}\left({x}_{i}\right)\right]+{V}_{I}({x}_{1},{x}_{2};l)-{x}_{i}F(t)+{x}_{i}f.\end{eqnarray}$
For a given trajectory, the mean input energy per period is determined as11 ). For the ith particle,
$\begin{eqnarray}\bar{W}=\displaystyle \frac{1}{N}W(0,N\tau ).\end{eqnarray}$
Since ${V}_{P}({x}_{i}),$ ${V}_{I}({x}_{1},{x}_{2};l),$ and $f$ does not have an explicit time dependency, all of the contribution to $W(0,N\tau )$ comes from the third term in equation ( $\begin{eqnarray}{W}_{i}(0,N\tau )=-\displaystyle {\int }_{0}^{N\tau }{x}_{i}\displaystyle \frac{\partial F(t)}{\partial t}{\rm{d}}t.\end{eqnarray}$
Therefore, for the ith particle, the work done by $F(t)$ on the system (or equivalently, the energy dissipated by the system to the environment), in a period $\tau ,$ is calculated as $\begin{eqnarray}\begin{array}{rcl}{\bar{W}}_{i} & = & \displaystyle \frac{1}{N}{W}_{i}(0,N\tau )\\ & = & \displaystyle \frac{1}{N}\displaystyle {\int }_{0}^{N\tau }\displaystyle \frac{\partial {U}_{i}({x}_{i},t)}{\partial t}{\rm{d}}t=\displaystyle \frac{A\omega }{N}\displaystyle {\int }_{0}^{N\tau }{x}_{i}\,\sin \,\omega t{\rm{d}}t.\end{array}\end{eqnarray}$
The total input power ${P}_{\rm{in}}$ of the dimer obtained from harmonic force is given by $\begin{eqnarray}{P}_{{\rm{in}}}=\displaystyle \frac{1}{\tau }\displaystyle \sum _{i=1}^{2}{\bar{W}}_{i}.\end{eqnarray}$
3. Numerical results and discussion
The investigation on the transport characteristics of inertial frictional ratchets in typical molecular motors, such as kinesins, has previously been undertaken via experimental research and inertial Langevin dynamics simulations. It is apparent that there are no analytical techniques for analyzing equation (1 ) in the presence of inertia. As a result, we performed extensive numerical analyses using the stochastic Runge–Kutta method with a time step $h={10}^{-3}.$ The initial conditions for the position $x(0)$ were chosen based on the random distribution over the periodic ratchet potential given in equation (2 ). Initial velocities $v(0)$ were consistently set to zero in this work. We calculated the outcomes over 500 distinct trajectories, each of which evolved over $1\,\times \,{10}^{4}$ driving-periods $\tau .$ The evolution time was sufficiently large to guarantee the value of the computed mean velocity close to its asymptotic value. Hereinafter, dimensionless variables will be employed in our simulations.
It is interesting to investigate the dependence of the motion on other important parameters, such as the coupling free length, amplitude, coupling strength and friction coefficient ratio, which indicate the influence of the internal parameters of the inertial frictional ratchets on directed transport performance. In this section, we systematically examine these effects on the average velocity and the energy conversion efficiency using numerical simulation and provide physical interpretations of the findings by using the stochastic energetics theory.
3.1. Effects of the coupling free length
From equations (2 ) and (5 ), it can be found that the potential energy related to the spatial position ${V}_{s}({x}_{1},{x}_{2};l)\,={V}_{p}(x)+{V}_{I}({x}_{1},{x}_{2};l)$ naturally depends on the position of the coupled particles and the dimer length $l.$ Clearly, any ${V}_{I}({x}_{1},{x}_{2},l)$ and $l$ are of dependent quantities. Therefore, the average velocity of the dimer could be affected by those fundamental parameters and we will discuss these effects first. Taking into account the spatial character of ${V}_{s}({x}_{1},{x}_{2};l),$ there are three different symmetry transformations: (a) translational invariance, ${V}_{s}({x}_{1},{x}_{2};l)={V}_{s}({x}_{1}+nL,{x}_{2}+mL,l+nL-mL)$, where $n,m\in {\mathbb{Z}};$ (b) inversion symmetry, ${V}_{s}({x}_{1},{x}_{2};l)={V}_{s}(-{x}_{1},-{x}_{2};-l);$ and (c) exchange symmetry, ${V}_{s}({x}_{1},{x}_{2};l)\,={V}_{s}({x}_{2},{x}_{1};-l).$ In particular, for the coupling free length, a change from $l$ to $-l$ can be interpreted as the reversal of the dimer's orientation. It is noted that the translational invariance makes the average velocity a periodic function, $\left\langle {V}_{{\rm{c}}{\rm{m}}}(l)\right\rangle =\left\langle {V}_{\rm{cm}}(l+kL)\right\rangle ,$ where $k\in {\mathbb{Z}}.$ Therefore, we focus on the averaged velocity $\left\langle {V}_{{\rm{c}}{\rm{m}}}\right\rangle $ of the dimer for only one full period. This fundamental characteristic obtained in our inertial frictional ratchets is the same as the ratchet effect of two coupled particles investigated by Reimann et al [45]. Furthermore, the changes in $\left\langle {V}_{\rm{cm}}\right\rangle $ with the increase in dimer length $l$ are investigated, and the symmetric results are shown in figure 1(a).
Figure 1. The curves of (a) the center-of-mass average velocity $\left\langle {V}_{{\rm{c}}{\rm{m}}}\right\rangle $, and (b) the energy conversion efficiency $\eta $ varying with the natural length $l$ for different values of amplitude $A,$ where ${\gamma }_{1}=5,$ $\alpha =0.3,$ $k=10,$ $\omega =3,$ $f=0.02$ and $D=0.1$. |
Besides the above-mentioned symmetry properties of $\left\langle {V}_{\rm{cm}}(l)\right\rangle ,$ figure 1(a) includes the variations in another parameter: the driving amplitude $A.$ For a fixed spring length, e.g. $l=L/2=0.5,$ it can be found that the behavior of the velocity becomes more complex, showing distinct non-monotonic characteristics for different amplitudes $A.$ For a fixed coupling length, e.g. $l=0.5,$ with the increase in amplitude $A,$ the minimum valley of the center-of-mass average velocity $\left\langle {V}_{\rm{cm}}(A=4,5)\right\rangle $ may became the maximum peak of velocity $\left\langle {V}_{{\rm{c}}{\rm{m}}}(A=7,8)\right\rangle .$ As observed, the average velocity demonstrated the strong dependence on the driving amplitude. This effect will be discussed in the next section. Hence, the optimal $l$ and $A$ can enhance the directed transport in inertial frictional ratchets.
To comprehend the transport efficiency from the viewpoint of energy, we explore how the energy conversion efficiency $\eta $ varies with the coupling free length $l$ for different driving amplitudes $A.$ According to equations (8 ) and (9 ), the energy conversion efficiency $\eta $ is roughly proportional to $\left\langle {V}_{{\rm{c}}{\rm{m}}}\right\rangle ,$ and the velocity behavior displays a symmetric distribution, as depicted in figure 1(a). Therefore, we indeed find that $\eta $ changes periodically as $l$ increases and the nearly symmetric relation $\left\langle \eta (l)\right\rangle =\left\langle \eta (l+kL)\right\rangle ,$ where $k\in {\mathbb{Z}}$ holds for different amplitudes $A.$ In addition, it can be found that the energy conversion efficiency $\eta $ can still obtain one local maximum in each period, as shown in figure 1(b). This observation implies that the ability to drag the load for the frictional ratchets can reach the maximum for different optimal coupling lengths $l.$ However, the energy conversion efficiency $\eta $ decreases with the increase in amplitude $A.$ It can be understood that the total input power ${P}_{\rm{in}}$ of inertial ratchets is obtained from the external harmonic force according to the stochastic energetics theory (see equations (14 ) and (15 )), and ${P}_{\rm{in}}$ of frictional ratchets increases with the increase in the amplitude. Therefore, the energy conversion efficiency of inertial ratchets could be reduced with the increase in amplitude $A.$
3.2. Effects of the amplitude
Figure 2(a) illustrates the dependency of the average velocity $\left\langle {V}_{\rm{cm}}\right\rangle $ on the amplitude $A.$ This figure also presents a very rich structure. It is interesting to note that the behavior of $\left\langle {V}_{\rm{cm}}\right\rangle $ presents a multi-peaked structure, and the peak values decrease with the increase in amplitude $A.$ The multi-peaked formation in the transport of inertial frictional ratchets is related to the breaking of symmetry between the two coupled particles. This is due to the fact that the inertial system has the friction symmetry breaking ($\alpha ={\gamma }_{2}/{\gamma }_{1}\ne 1.0$) and the coupling interaction, and these two factors cooperate and compete with each other to strengthen the transport of the frictional ratchets. Consequently, there are more local maxima of the average velocity.
Figure 2. The curves of (a) the center-of-mass average velocity $\left\langle {V}_{{\rm{c}}{\rm{m}}}\right\rangle $ and (b) the energy conversion efficiency $\eta $ varying with the amplitude $A,$ where ${\gamma }_{1}=5,$ $\alpha =0.3,$ $l=0.1,$ $k=1.0,$ $\omega =3,$ $f=0.02$ and $D=0.1$. |
In addition, in the complicated case, the two damping forces of the dimer are different and the change in the energy conversion efficiency $\eta $ with the increase in amplitude $A$ is shown in figure 2(b). It has been demonstrated that $\eta $ as a function of $A$ displays monotonic dependence. According to the similar analysis used in figure 1(b), it can be concluded that the efficiency could decrease with the increase in $A.$ Therefore, as for the two coupled particles of different damping forces, varying the driving amplitude $A$ is another way of promoting the transport performance.
3.3. Effects of the coupling strength
Figure 3(a) presents the dependency of the center-of-mass velocity on the coupling constant $k$ for varying friction coefficient ratios $\alpha .$ One can distinguish an optimal value of the coupling strength that leads to a local maximum of the current for a suitable coupling free length, e.g. $l=0.4.$ The result leads to the conclusion that the suitable coupling strength ${k}^{{\rm{o}}{\rm{p}}{\rm{t}}}$ can enhance the current. For this optimal value of the coupling constant, it can be understood that the friction symmetry breaking ($\alpha ={\gamma }_{2}/{\gamma }_{1}\ne 1.0$) and the coupling interaction may cooperate with each other to promote the transport of inertial frictional ratchets. However, for a strong coupling constant $k\gt 30$ (figure not shown), there is almost no correlation between the center-of-mass velocity and the coupling constant, and the velocity tends to a constant value. This suggests that, for the strong coupling, the influence of the spring is weak compared to the friction coefficient ratio. In addition, it can be clearly observed that the average velocity of the dimer decreases with the increase in the friction coefficient ratio $\alpha ={\gamma }_{2}/{\gamma }_{1},$ (i.e. ${\gamma }_{2}$ is increased). The qualitative behavior of the velocity can be explained by the fact that with the increase in $\alpha ,$ the friction damping of the dimer component ${x}_{2}$ increases; therefore, the centre-of-mass velocity of the dimer decreases. Recent investigations have revealed that the coupling behavior plays an important role in facilitating directed transport [46].
Figure 3. The curves of (a) the center-of-mass average velocity $\left\langle {V}_{\rm{cm}}\right\rangle $ and (b) the energy conversion efficiency $\eta $ varying with the coupling constant $k$ for different friction coefficient ratios $\alpha ,$ where ${\gamma }_{1}=5,$ $l=0.4,$ $A=4,$ $\omega =3,$ $f=0.02$ and $D=0.1$. |
Figure 3(b) illustrates the variation in the energy conversion efficiency $\eta $ with the increase in the coupling constant $k$ for different friction coefficient ratios $\alpha .$ It is easy to find that the dependence between the energy conversion efficiency $\eta $ and coupling strength $k$ is not monotonic. Nevertheless, the efficiency $\eta $ can show an obvious peak; therefore, there is an optimal coupling constant ${k}^{{\rm{o}}{\rm{p}}{\rm{t}}}$ that can make the energy conversion efficiency obtain the local maximum ${\eta }^{\max }.$ Meanwhile, the ${\eta }^{\max }$ also decreases as the friction coefficient ratio increases. Note that the energy conversion efficiency depends on $\alpha ,$ and the effect will be discussed thoroughly in the next section. The conclusion once again indicates that there is an optimal coupling strength in the inertial frictional ratchets, which can enable the dimer to achieve more effective energy conversion and transportation.
3.4. Effects of the friction coefficient ratio
For the case of different external force $f,$ $\left\langle {V}_{{\rm{c}}{\rm{m}}}\right\rangle $ as a function of the friction coefficient ratio $\alpha $ is discussed, and the result is shown in figure 4(a). It has been demonstrated that the center-of-mass average velocity of coupled particles can obtain the extremum with the increase in the friction coefficient ratio. This indicates that an optimal friction coefficient ratio may enhance the directed transportation of inertial frictional ratchets. These phenomena can be interpreted as follows. On the one hand, when the friction coefficient ratio $\alpha ={\gamma }_{2}/{\gamma }_{1}\to 0,$ i.e. ${\gamma }_{2}\to 0$ (${\gamma }_{1}$ is constant) and the damping force ${\gamma }_{2}{v}_{2}\to 0,$ it means vanishing friction of the dimer component ${x}_{2},$ and the ${x}_{2}$ can move more easily. For the convenience of analysis, the term ‘$-\beta (t){\rm{d}}{V}_{P}({x}_{i})+A\,\cos (\omega t)-f$' on the right-hand side of equation (1 ) can be regarded as the equivalent external force on the particles. When the equivalent external force is positive, it means that the shape of the corresponding effective external potential is left-high and right-low, and the equivalent external force is more likely to cause the dimer to move in the $x$-positive direction. When the particle escapes from the potential well to the next barrier (crossing the positive slope of the external potential, $\tfrac{{\rm{d}}{V}_{p}(x)}{{\rm{d}}x}\gt 0$) , on average, $g(t)$ in equation (4 ) is less than 0. Therefore, the feedback switch $\beta (t)=0,$ i.e. the term $-\beta (t){\rm{d}}{V}_{P}({x}_{i})$ vanishes, implying that the coupled particles are not subjected to the external periodic potential and have a higher probability of moving in the $x$-positive direction. By contrast, when the equivalent external force is negative, the shape of the potential is left-low and right-high, and a similar analysis shows that the equivalent external force causes the dimer to move in the $x$-negative direction and the feedback switch of the external potential is opened. In this case, the coupled particles need to consume energy to cross the potential barrier; therefore, the dimer produces a smaller negative current. Combining the above two factors, the total particle current of the dimer can exhibit a positive finite value when the friction coefficient ratio $\alpha $ tends to zero.
Figure 4. The curves of (a) the center-of-mass average velocity $\left\langle {V}_{{\rm{c}}{\rm{m}}}\right\rangle $ and (b) the energy conversion efficiency $\eta $ varying with the friction coefficient ratio $\alpha $ for different external force $f,$ where ${\gamma }_{1}=5,$ $k=10,$ $l=0.1,$ $A=4,$ $\omega =3$ and $D=0.1$. |
On the other hand, when the friction coefficient ratio $\alpha \to \infty ,$ i.e. ${\gamma }_{2}\to \infty ,$ the movement of ${x}_{2}$ in the dimer is strongly hindered. Due to the coupling between ${x}_{1}$ and ${x}_{2},$ the motion of ${x}_{1}$ will also be inhibited in this case; thus the coupled particles are not able to easily produce the directed motion. Therefore, the center-of-mass mean velocity of the dimer $\left\langle {V}_{\rm{cm}}\right\rangle \to 0$ when the friction coefficient ${\gamma }_{2}\to \infty .$ With the change in $\alpha ,$ the positive and negative particle currents compete with each other. Therefore, the average velocity of the dimer can be maximized for a suitable friction coefficient ratio.
However, it can be found that the frictional ratchets present anomalous transport behavior; that is, a very weak negative driving (the negative bias force in equation (1 )) can cause the inertial ratchets to produce a positive average velocity ($\left\langle {V}_{{\rm{c}}{\rm{m}}}\right\rangle \gt 0$) of the dimer. As seen in figure 4(a), the dimer can move forward at a positive velocity over a large range of bias $f.$ Even if the external force $f$ is increased to 0.2, the anomalous transport phenomenon can still be clearly observed. Nevertheless, when the bias force is further increased, the inertial frictional ratchets could show normal transport ($\left\langle {V}_{\rm{cm}}\right\rangle \lt 0$). Thus, unlike an earlier model [14, 45], the result of anomalous transport can still be obtained from our inertial frictional dimer.
In figure 4(b) the energy conversion efficiency $\eta $ is plotted as a function of the friction coefficient ratio $\alpha $ for different values of the load $f$ with the same parameters shown in figure 4(a). The results of figure 4(b) demonstrate that there is an optimal dependence relationship between the energy conversion efficiency and the friction coefficient ratio, which means that each maximum energy conversion efficiency corresponds to an optimal friction coefficient ratio ${\alpha }^{{\rm{o}}{\rm{p}}{\rm{t}}}.$ However, it is also interesting to observe that the energy conversion efficiency increases with the increase in the load. This counterintuitive behavior appears over the whole $\alpha $ region despite the fact that $\left\langle {V}_{{\rm{c}}{\rm{m}}}\right\rangle $ consistently decreases with the increasing load $f$ for various friction coefficient ratios $\alpha ,$ as shown in figure 4(a). These interesting results obtained in our inertial frictional ratchets are similar to the transport performances investigated by Mahato et al [47]. According to equations (8 ), (9 ), (14 ) and (15 ), the input power ${P}_{{\rm{i}}{\rm{n}}}$ is independent of the load $f;$ therefore, the energy conversion efficiency $\eta $ is essentially proportional to the output power ${P}_{{\rm{o}}{\rm{u}}{\rm{t}}}.$ Consequently, $\eta $ of complex inertial frictional ratchets closely mimic the behavior of the average velocity for different values of the load $f.$ Furthermore, it can be found from equation (9 ) that the output power ${P}_{{\rm{o}}{\rm{u}}{\rm{t}}}$ increases with the increase in the load $f,$ and the corresponding decrease in $\left\langle {V}_{{\rm{c}}{\rm{m}}}\right\rangle $ is smaller in the region of $\alpha $ on average (see figure 4(a)). Therefore, under certain circumstances, the load $f$ could enhance the energy conversion efficiency. These results lead to the conclusion that the greater the load of a frictional ratchet, the greater its efficiency with some appropriate parameters.
When the friction coefficient ${\gamma }_{1}$ of the dimer component ${x}_{1}$ is changed, the dependence of the center-of-mass mean velocity $\left\langle {V}_{{\rm{c}}{\rm{m}}}\right\rangle $ on the friction coefficient ratio $\alpha $ is also investigated. From figure 5(a), it can be seen that the average velocity of the inertial frictional ratchets can reach the maximum for the optimal value of the friction coefficient ratio $\alpha .$ Furthermore, it has been demonstrated that the peaked transport behavior is similar to that in figure 4(a), and the behavior can be analyzed by a similar method. Therefore, the directed transport of inertial frictional ratchets could be optimized and manipulated by adjusting the friction coefficient ratio. However, when ${a}_{2}$ is fixed, one can observe that the transport decreases with the increase in the friction coefficient ${\gamma }_{1}.$ It can be understood that the increasing of ${\gamma }_{1}$ in the dimer component ${x}_{1}$ will be accompanied with the enlargement of the damping force ${\gamma }_{1}{v}_{1}$ and ${\gamma }_{2}{v}_{2}$ (i.e. ${\gamma }_{2}=\alpha {\gamma }_{1}$) for the dimer and, correspondingly, all the curves are observed to be diminished for different friction coefficients. Therefore, the rectification transportation of frictional ratchets could be enhanced by selecting the appropriate friction coefficient of each component of the dimer.
Figure 5. The curves of (a) the center-of-mass average velocity $\left\langle {V}_{{\rm{c}}{\rm{m}}}\right\rangle $ and (b) the energy conversion efficiency $\eta $ varying with the friction coefficient ratio $\alpha $ for different friction coefficients ${\gamma }_{1},$ where $k=10,$ $l=0.1,$ $A=4,$ $\omega =3,$ $f=0.02$ and $D=0.1$. |
To thoroughly understand the transport properties when the friction of each dimer component is different, we also calculate the energy conversion efficiency $\eta $ as a function of $\alpha $ for different values of the friction coefficient ${\gamma }_{1},$ as shown in figure 5(b). Similar to the efficiency maximization with respect to the external force $f$ shown in figure 4(b), it is also demonstrated that the energy conversion efficiency is a peaked function of the friction coefficient ratio $\alpha $ for different values of ${\gamma }_{1}.$ This means that an optimal value ${\alpha }^{\rm{opt}}$ of the friction coefficient ratio exists at which the energy conversion efficiency is maximal. The reason for this result can also be explained by the similar method used in figure 4(b). Nevertheless, it is noted that the energy conversion efficiency with small friction (e.g. ${\gamma }_{1}=5$) is greater than that with large friction (e.g. ${\gamma }_{1}=8$). These results suggest that a smaller friction coefficient should be used in the frictional ratchets if we want the inertial coupled dimer to be able to use the input energy more efficiently to drag the load. In addition, these conclusions imply that the breaking of friction symmetry is an important factor inducing the ratchet effects of the dimer.
3.5. Current reversal phenomenon
At present, the average velocity and energy conversion efficiency of a dimer have been discussed, especially in the large friction environment, i.e. ${\gamma }_{1}\geqslant 5.$ Now we focus on another characteristic of inertial frictional ratchets, the current reversal phenomenon [37], which plays an important role in the design of the separation devices for microscopic particles [10]. In our present case of feedback frictional ratchets, the current reversal could be obtained under the conditions of a weak damping force.
For the more complicated case, the change in $\left\langle {V}_{{\rm{c}}{\rm{m}}}\right\rangle $ with the increase in the friction coefficient ${\gamma }_{1}$ for different values of the external force $f$ is investigated. As seen in figure 6, there is an optimal friction coefficient ${\gamma }_{1}^{\rm{opt}}$ at which the average velocity $\left\langle {V}_{{\rm{c}}{\rm{m}}}\right\rangle $ reaches the local maximum. It shows that the directional transport of inertial ratchets can be enhanced under the optimal friction coefficient conditions. However, a classic example of an anomalous transport phenomenon is that particles always move in the opposite direction to the bias force. It has been demonstrated that the velocity $\left\langle {V}_{{\rm{c}}{\rm{m}}}\right\rangle $ has the positive-valued maxima at different optimal coefficient ${\gamma }_{1}^{\rm{opt}}$ for an external force $f\leqslant 0.8.$ The results show that the positive transport of the inertial frictional ratchets could be maximized by selecting the appropriate bias force and friction coefficients. Meanwhile, it is found that the maximum of $\left\langle {V}_{{\rm{c}}{\rm{m}}}\right\rangle $ decreases with the increase in the load $f.$ In particular, it is also interesting to observe that there are twice the current reversal phenomena when the load $f$ is 0.8. It can be understood that both the damping force and bias force may cooperate and compete with each other to induce the reversed motion of the inertial frictional ratchets. Furthermore, for $f\leqslant 0.8,$ the peak can be explained by the negative mobility effect [48, 49]. Basically, the dimer with different friction coefficients moves to different directions and could be separated under certain conditions. However, the inertial ratchets display normal transport behavior ($\left\langle {V}_{{\rm{c}}{\rm{m}}}\right\rangle \lt 0$) when the load force $f$ continues to increase and reach 0.85. Therefore, the inertial frictional ratchets can achieve the current reversal by selecting the appropriate external bias force and friction.
Figure 6. The curves of the center-of-mass average velocity $\left\langle {V}_{{\rm{c}}{\rm{m}}}\right\rangle $ varying with the friction coefficient ${\gamma }_{1}$ for different external force $f,$ where $\alpha =0.3,$ $k=10,$ $l=0.1,$ $A=4,$ $\omega =3$ and $D=0.1$. |
3.6. Resonant steps in inertial frictional ratchets
In addition to the driving amplitude $A,$ the frequency of the harmonic force is another important factor to induce the formation of resonant steps. Figure 7 shows the scaled average velocity $2\pi \left\langle {V}_{{\rm{c}}{\rm{m}}}\right\rangle /\omega $ of the dimer as a function of frequency $\omega .$ It can be found that the scaled current could present a series of defined resonance steps. The values of these steps can be given by the ratio $n/m,$ where $n$ and $m$ are integers. The resonance steps in figure 7 can actually be understood as the synchronization regions under the periodic driving effects of harmonic force. The detailed structure of resonance steps in over-damped open-loop ratchets [50] and under-damped deterministic feedback ratchets [51] has been reported. It should be noted that the resonance step is, theoretically, a series of defined steps, and its theoretical prediction has been given in [51]. However, it can be observed only for a part in this inertial frictional ratchet. The property can be well interpreted as that the friction coefficients between the dimer components are different, thus breaking the dynamics of the inertial frictional systems under weak noise intensity (e.g. $D=0.001$) circumstances. Therefore, we can only clearly see part of the steps displayed by the particle current. In addition, as seen in figure 7, the presence of friction has an additional dependency on the resonant steps. This means that the lower the damping force, the more resonance steps. In addition, the decrease in the friction coefficient means that the inertia of the ratchet motion is increased, and the oscillatory fluctuation effect of the velocity caused by the inertial term is strengthened. The oscillation and the mode-locking effect caused by periodic external forces can lead to more resonant steps of directed current. As expected, with the increase in friction, e.g. ${\gamma }_{1}=10,$ the scaled average velocity exhibits a smooth curve and the structure of the resonant steps is broken. This means that some high-order resonance steps will be smoothed out. Nevertheless, the current values scaled in this way are very large under the low-frequency driving conditions. Due to the fact that the average velocity of the dimer is a finite value, the scale current $2\pi \left\langle {V}_{{\rm{c}}{\rm{m}}}\right\rangle /\omega $ increases as the driving frequency decreases. When the frequency of external driving increases, e.g. $\omega \to \infty ,$ it means that the change in external driving is very fast and, at this time, the two coupled particles are generally unable to feel the continuous driving in a short period of time. Therefore, the directional transport of the two coupled particles occurs only under the effects of external loading $f,$ and the corresponding average velocity of the dimer decreases with the increase in the external driving frequency $\omega .$
Figure 7. The curves of the scaled average velocity $2\pi \left\langle {V}_{{\rm{c}}{\rm{m}}}\right\rangle /\omega $ varying with the harmonic force frequency $\omega $ for different friction coefficients ${\gamma }_{1},$ where $\alpha =0.3,$ $k=10,$ $l=0.1,$ $A=4,$ $f=0.02$ and $D=0.001$. |
4. Conclusions and final remarks
We have studied the directional motion of inertial frictional ratchets in which the friction coefficients of the elastic coupling component are different and the two coupled particles are subjected to a feedback flashing potential. Due to the breaking of friction symmetry, coupling interaction and the effects of different external forces, the inertial frictional ratchets present richer transport behavior. The ratchet motion is observed and interpreted in terms of changes in the free length, coupling strength, frictional asymmetry, bias force and driving amplitude, and frequency.
Due to the friction symmetry breaking and the coupling interaction of the individual particles, the center-of-mass average velocity can be maximized with respect to the driving amplitude, coupling strength, damping force, etc. In this work, we have also investigated the performance of inertial frictional ratchets by using the energy conversion efficiency. Thus, in the present discussion of inertial frictional ratchets, the energy conversion efficiency measures how well the energy input from the harmonic forces can be used to complete useful work. Spiechowicz et al studied a similar ratchet [52], but our inertial feedback ratchet has an advantage in that we can investigate the effects of friction. In some cases, the inertial frictional ratchets could also achieve maximum energy conversion efficiency under the optimal values of some parameters, such as natural length and coupling strength. In particular, the energy conversion efficiency of inertial ratchets against an applied load shows interesting behavior. It can be found that the efficiency increases as the applied load increases. According to the theory of stochastic energetics, the input power ${P}_{\rm{in}}$ is independent of the load. Therefore, the interesting behavior is possible under certain circumstances.
Another important result is the relationship between bias force and the average velocity of the dimer. The conclusions indicate that the current reversal phenomenon can also occur in our inertial frictional ratchets. Nevertheless, for a weak bias, the directed transport of frictional ratchets can be reversed twice by modulating the suitable friction of the dimer. Furthermore, a series of resonant steps could be obtained for different particle currents. These resonance steps are induced by the synchronous phenomena driven by harmonic forces. These results indicate that the breaking of friction symmetry can also induce the so-called ratchet effect of the dimer. Meanwhile, the above conclusion also reveals that, for a ratchet, the friction asymmetry is an important factor for generating complex transport in a non-uniform damping environment.
In general, the conclusions of this article could be used to improve current technologies from data storage to disease prevention [53], as well as to enhance the energy conversion efficiency of biomolecular motors [33]. Experimentally, more abundant particle currents could be obtained by applying the flow reversal technique, and the transport direction of ratchets could be used as a function of different system parameters for real-time control [54].