When a system at rest is perturbed by an external force, it usually responds by moving in the direction of that force. However, in many situations, the quite surprising contrary behavior in the form of a permanent motion against a (not too large) static force of whatever direction is known as absolute negative mobility. The absolute negative mobility is impossible in a heat-balanced system because it violates the second law of the thermodynamics. Therefore, the study of the absolute negative mobility in the ratchet has aroused much attention in theoretical and experimental physics.^[1–7]

The phenomena of the absolute negative mobility are showed in many different cases. In a spatially symmetric and periodic system, which is far from thermal equilibrium, it is discovered that the average particle velocity of the single Brownian particle is positive for negative force and negative for positive force.^[1] When several inertial Brownian particles move in a symmetric periodic potential under the influence of a time periodic and a constant, biasing driving force, thermal equilibrium fluctuations can cause the absolute negative mobility.^[2] The overdamped motion of Brownian particles is analysed in the one-dimensional system with a symmetric serration potential subjected to both unbiased thermal noise and spatially inhomogeneous three-level colored noise. It is found that the appearance of the absolute negative mobility depends on the parameters of nonequilibrium and thermal noises such as amplitude, temperature, switching rate.^[8] In addition, the absolute negative mobility can be induced by the spatial asymmetry in the shape of the transported particles,^[4] white Poissonian noise,^[6] the colored thermal fluctuations.^[5]

The absolute negative mobility has been experimentally detected and theoretically studied in p-modulation-doped multiple quantum wells,^[9] semiconductor superlattices,^[10] and tunnel junction in superconductor devices.^[11] In addition, negative conductances or even absolute negative conductances have been theoretically interpreted in the Josephson junctions^[12–15] according to electric transportation.

However, in contrast to the regular potential, the shape of the substrate potential, which deviates from the standard one is more common in the real physical systems. The deformation of the potential may have a great influence on the transport characteristics of the particle. In the physical situations, application of standard Frenkel-Kontorova (FK) model could be very restricted, and real physical systems could not be “exactly” described by standard models.^[16] For wider generality, Remoissenet and Peyrard^[17–18] obtained in a control manner by an adequate choice of parameters rich variety of deformable potentials related to the physical systems such as Josephson junctions, charge-density wave condensates, and crystals with dislocations, these deformable potentials allow the modeling of many specific physical situations without employing perturbation methods. Therefore, it is necessary to study the influence of the deformation of the potential on average particle velocity. In this paper, we extend the previous work from standard sinusoidal potentials to the asymmetric deformable potentials. We will focus on how the deformation of the potential affects the directed transport. Since the inertial of the particle affects the anomalous transport feature, we will discuss the underdamped motion of a Brownian particle.

We consider underdamped dynamics of a Brownian particle in the presence of the one-dimensional asymmetric deformable potential.^[17,19] The particle is subjected to both an unbiased time-periodic external force A cos(Ωt) and a constant force F. Its dynamic can be described by the following inertial Langevin equation:^[2,20]

The asymmetric deformable potential can be given by^[17,19]

Here we will use Brownian dynamic simulations to study the transport of the particle. The average velocity of the particle can be obtained from the following formula:

During the numerical simulations, we set the step time Δt to be 0.001 and the total integration time to be more than 10⁷. Unless otherwise noted, our simulations are under the parameter sets: γ = 0.9, ω = 4.9, a = 4.2, ΔU = 1.0, and D₀ = 0.001.

Figure 1 shows the average velocity as a function of the static force f for different values of r. When the force f and r are both small, the absolute negative mobility is presented. However, when f increases, the current shows a reversal, which means the absolute negative mobility disappears. When the force f is larger than 0.175, the average velocity is always positive and increases with the increasing of f. It is noted that there are two current reversals presented for r = 0.015. Namely, the average velocity is firstly positive, then negative, and finally positive. From Fig. 1, we can also obtain that the behaviors of are very different for the different values of r. When r = 0, it is equivalent to the Brownian particle moving in the simply sinusoidal potential, the average velocity is negative for f < 0.17. It means that the absolute negative mobility can be observed in the range of f < f_stall (f_stall ≈ 0.17).^[2] When r increases from zero but less than 0.017, the absolute negative mobility is gradually suppressed. When r is equal to 0.017, the absolute negative mobility disappears and the curve of shows fluctuation behavior (Fig. 1(b)). When r ≥ 0.03, the average velocity is firstly equal to zero for f ≤ 0.15, then it exponentially increases with the increasing of f (Fig. 1(c)). Therefore, we can obtain the current reversal by changing the deformable parameter. Now we will give the physical interpretation.

Fig. 1

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	Fig. 1 (Color online) Average velocity as a function of the static force f for different values of r. (a) 0 ≤ r ≤ 0.013, (b) 0.015 ≤ r ≤ 0.017, (c) r ≥ 0.03.

When r is not equal to zero, the deformable potential has two inequivalent wells with the flat and sharp bottoms. When the particle stays in the flat well, it can easily pass through the barriers although the force acting on it is very small. When the particle stays in the sharp well, although the acting force is very large, it is also hard for the particle to pass through the barriers. Therefore, the flat well is beneficial to the particle’s transport while the sharp well impede the particle’s transport. When r increases but it is smaller, the effect of the flat well on particle’s transport is larger than that of the sharp one. At this time, the flat well dominates the particle’s transport and the absolute negative mobility is suppressed. On the further increasing r, the particle in the sharp well is trapped more strongly. When the force is not large enough, the particle can not pass through the potential, so the average velocity tends to zero. However, when the force is large enough, the transport of the particle is dominated by the static force, so the average velocity increases with the increasing of the force.

In Fig. 2, we study the average velocity versus the noise intensity D₀ for different values of r and f. From Figs. 2(a) and 2(b), we can see the absolute negative mobility is presented when D₀ and r are both small. When D₀ increases, the current shows a reversal from negative to positive and the average velocity increases with the increasing of D₀. In addition, there is a small fluctuation in the range of 0.1 ≤ D₀ ≤ 1. We also find that the potential deformation has stronger influences on the absolute negative mobility. When r increases, the absolute negative mobility is suppressed. When r ≥ 0.03, the absolute negative mobility disappears and the average velocity is firstly equal to zero then increases with the D₀ increasing (Figs. 2(a) and 2(b)). These results denote that the potential deformation affects the absolute negative mobility. In these cases, when D₀ → 0, the particle cannot pass across the barrier of the potential, so there is no net current. When D₀ increases from zero but is still small. At the same time, for r ≤ 0.01, the probability of the particle in the flat well is larger than in the sharp well and the particle can pass across the potential barrier. So the average velocity is not zero and the absolute negative mobility presents. With the increasing of D₀, the particle can jump freely between the flat well and sharp well, so the average velocity goes to zero, reverses its direction, and increases monotonically to a certain value. It should be noted that the above phenomena occur when the force is small. From Fig. 2(c), we can find the average velocity is always positive when the force is equal to 0.2. It indicates that the large force can drive the particle to move in its direction.

Fig. 2

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	Fig. 2 (Color online)Average velocity as a function of the noise intensity D0 for different values of r and f. (a) f = 0.05, (b) f = 0.1, (c) f = 0.2.

It is well known that the system is in underdamped state when the friction coefficient γ is small. The system will move with the periodic oscillation and the amplitude of the oscillation logarithmically decreases with the time increasing. However, when γ is large enough, the system is in overdamped state and it will slowly move to the equilibrium position without any oscillation. In Fig. 3, we display the average velocity as a function of the friction coefficient γ for different values of r and f. We can find that the behavior of the average velocity (r ≠ 0.1) shows the oscillation motion and presents several peaks as what the underdamped system transport curve shows. However, when r = 0.1, the behavior of shows the same as the overdamped system transport (Fig. 3(a)). Very interestingly, contrary to Figs. 1 and 2, the average velocity (r = 0 and r = 0.01) shows three reversals with the increasing of γ, which means that there are two intervals of the absolute negative mobility. For r ≥ 0.03, the absolute negative mobility disappears and the average velocity is equal to zero or larger than zero. In addition, the average velocity behaviors are also different from Fig. 2 for r = 0 and r = 0.01. The absolute negative mobility always presents when the force is equal to 0.2. The force f only affects the amplitude of and the absolute negative mobility is determined by r and γ.

Fig. 3

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	Fig. 3 (Color online)Average velocity as a function of the friction coefficient γ for different values of r and f. (a) f = 0.05, (b) f = 0.1, (c) f = 0.2.

In summary, we numerically study the transport of a particle in a one-dimensional asymmetric deformable potential. It is found that the competition between the force and the deformation of the potential can result in the current reversals and the absolute negative mobility suppressed. Remarkably, when the deformable parameter r = 0.015, there are two current reversals presenting. When r ≤ 0.016, the absolute negative mobility is suppressed with the increasing of r. When r > 0.016, the absolute negative mobility disappears. When the noise intensity, the force, and the potential deformation are small, the absolute negative mobility and current reversals present, whereas the absolute negative mobility disappears when both the force and the potential deformation are large. Very interestingly, the average velocity behaviors as a function of the friction coefficient for different values of r and f are very different. These show the oscillation motion and three reversals with the increasing of γ, which means that there exists two intervals of the absolute negative mobility when r is small. Therefore, we can obtain the current reversals and control the particle’s transport direction by changing the potential deformation. The results we have presented may be another way to manipulate the transport of particles in complex environments. Furthermore, we expect our results can help people develop a new route to control particle systems based on the underdamped dynamics.