Brownian Motor Research Articles

A composite micromotor driven by self-thermophoresis and Brownian rectification* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11874397 and 11674365).

Brownian motors and self-phoretic microswimmers are two typical micromotors, for which thermal fluctuations play different roles. Brownian motors utilize thermal noise to acquire unidirectional motion, while thermal fluctuations randomize the self-propulsion of self-phoretic microswimmers. Here we perform mesoscale simulations to study a composite micromotor composed of a self-thermophoretic Janus particle under a time-modulated external ratchet potential. The composite motor exhibits a unidirectional transport, whose direction can be reversed by tuning the modulation frequency of the external potential. The maximum transport capability is close to the superposition of the drift speed of the pure Brownian motor and the self-propelling speed of the pure self-thermophoretic particle. Moreover, the hydrodynamic effect influences the orientation of the Janus particle in the ratched potential, hence also the performance of the composite motor. Our work thus provides an enlightening attempt to actively exploit inevitable thermal fluctuations in the implementation of the self-phoretic microswimmers.

Arcsine law and multistable Brownian dynamics in a tilted periodic potential.

Multistability is one of the most important phenomena in dynamical systems, e.g., bistability enables the implementation of logic gates and therefore computation. Recently multistability has attracted a greatly renewed interest related to memristors and graphene structures, to name only a few. We investigate tristability in velocity dynamics of a Brownian particle subjected to a tilted periodic potential. It is demonstrated that the origin of this effect is attributed to the arcsine law for the velocity dynamics at the zero temperature limit. We analyze the impact of thermal fluctuations and construct the phase diagram for the stability of the velocity dynamics. It suggests an efficient strategy to control the multistability by changing solely the force acting on the particle or temperature of the system. Our findings for the paradigmatic model of nonequilibrium statistical physics apply to, inter alia, Brownian motors, Josephson junctions, cold atoms dwelling in optical lattices, and colloidal systems.

Cooperative dynamics of coupled motor–cargoes system with stochastic interactions in the crowded environment

In this study, we investigate the cooperative transport behaviors of coupled motor–cargoes system, in which multiple passive cargoes stochastically interact with one active Brownian motor. The environment with stochastic interactions is characterized by the concentration (reflecting the cargo’s number in unit volume) and switching rate (reflecting the interacting stability between motor and cargoes), based on which the stationary multiple-state process can be employed to describe the fluctuating-cargo state in the coupled system. By analyzing the average probability current of decoupled system in the thermodynamic limit, we effectively study the possibility of cooperative transport through stochastic cargoes to behave rich dynamical behaviors, including the directed current, current reversal, stochastic resonance (SR) and stochastic inhibition (SI), inverse SR and SI, even without the effect of external driving force. Based on numerical results, we systematically discuss the transport dependence on various parameters, including the cargo concentration in the crowded environment, cargo capacity of the motor, driving amplitude of external periodic force, and medium temperature. Obviously, the sensitivity of transport process to parameter changes can be used by the environment to regulate its cargo traffic, which also provides latent support for manipulating the transport performance and optimizing the coupled structure in artificial nano-machines.

Brownian motors arranged on nontrivial networks to achieve fast transport

Brownian motors arranged on nontrivial networks to achieve fast transport

Adiabatic Brownian motor with a stepwise potential perturbed by a dichotomous harmonic sygnal

We obtained an analytical expression for the average motion velocity of an adiabatic Brownian motor (ratchet), which operates due to small dichotomous spatially harmonic fluctuations of a stepwise potential. The symmetry properties of the average velocity as a functional of the stationary and fluctuating components of the nanoparticle potential energy are revealed, and the ranges of values of the system parameters that ensure the rightward and leftward motion of the motor are determined. We showed that the average motor velocity is a non-monotonic function of the stepwise potential height. For a singular (infinitely high and narrow) potential barrier, the average velocity depends non-monotonically on the «power» of this barrier (the barrier width multiplied by the exponent of the ratio of the barrier height to the thermal energy). The article continues the further development of theoretical methods of symmetry analysis by applying the general approaches proposed by the authors to specific motor systems.

Rotation-translation coupling of a double-headed Brownian motor in a traveling-wave potential

Considering a double-headed Brownian motor moving with both translational and rotational degrees of freedom, we investigate the directed transport properties of the system in a traveling-wave potential. It is found that the traveling wave provides the essential condition of the directed transport for the system, and at an appropriate angular frequency, the positive current can be optimized. A general current reversal appears by modulating the angular frequency of the traveling wave, noise intensity, external driving force and the rod length. By transforming the dynamical equation in traveling-wave potential into that in a tilted potential, the mechanism of current reversal is analyzed. For both cases of Gaussian and Levy noises, the currents show similar dependence on the parameters. Moreover, the current in the tilted potential shows a typical stochastic resonance effect. The external driving force has also a resonance-like effect on the current in the tilted potential. But the current in the traveling-wave potential exhibits the reverse behaviors of that in the tilted potential. Besides, the currents obviously depend on the stability index of the Levy noise under certain conditions.

Nanometer-Scale-Resolution Multichannel Separation of Spherical Particles in a Rocking Ratchet with Increasing Barrier Heights

We present a nanoparticle size-separation device based on a nanofluidic rocking Brownian motor. It features a ratchet-shaped electrostatic particle potential with increasing barrier heights along the particle transport direction. The sharp drop of the particle current with barrier height is exploited to separate a particle suspension into multiple subpopulations. By solving the Fokker-Planck equation, we show that the physics of the separation mechanism is governed by the energy landscape under forward tilt of the ratchet. For a given device geometry and sorting duration, the applied force is thus the only tunable parameter to increase the separation resolution. For the experimental conditions of 3.5 V applied voltage and 20 s sorting, we predict a separation resolution of approximately $2$ nm, supported by experimental data for separating spherical gold particles of nominal diameters of 80 and 100 nm.

Symmetry properties of a Brownian motor with a sawtooth potential perturbed by harmonic fluctuations

We present a study of general symmetry properties of a Brownian ratchet model. The study is based both on constructing chains of symmetry transformations reflecting explicit and hidden symmetries of the average ratchet velocity as a functional of the spatially periodic potential energy of a nanoparticle and on taking into account the symmetry types of periodic functions that are components of the potential energy of an additive-multiplicative form. A ratchet with a sawtooth stationary potential profile, dichotomously perturbed by a spatially harmonic signal, is investigated. Conclusions are made on both the possibility of occurrence of the ratchet effect and its direction for given values of the asymmetry parameter of the sawtooth profile, phase shifts of the control component, and frequencies of temporal fluctuations. These conclusions have been obtained only on the basis of symmetry transformations; that demonstrates the predictive value of the approach presented. The results of the symmetry analysis are confirmed by numerical simulation of the functioning of a ratchet with dichotomous stochastic spatially periodic fluctuations of the nanoparticle potential energy.

Infima statistics of entropy production in an underdamped Brownian motor.

The second law of thermodynamics states that the entropy never decreases for isolated macroscopical systems, which defines the arrow of time. For small systems, although the entropy increases on average, due to strong fluctuation, it may encounter a temporary decrease. The probability of negative entropy production follows the fluctuation theorem. Recently, it has been demonstrated theoretically the infima law that there exists a lower bound for the average values of the minima of the negative entropy production, which is -k_{B}. In this paper, we have constructed a horizontal Brownian motor immersed in a granular gas, whose dynamics is governed by the underdamped stochastic process. By recording the angular motion of the motor and measuring the key parameters of the system, we experimentally demonstrate that, despite the nonideal elements in the experiments and that the complex underlying dynamics, the average value of the minima of the negative entropy production is still bounded by -k_{B}, which may invoke further theoretical investigations of the applicability of the infima law in nonideal realistic small systems.

Ratchet motion and current reversal of Brownian motors coupled by birth-death interactions in the crowded environment

In this study, we investigate the ratchet transport behaviors of coupled Brownian motors with stochastic interactions in the crowded environment, which is characterized by two parameters, including particle concentration and interacting stability. Thus stationary birth-death process is employed to describe the process of fluctuating particles in the coupled system, and it is a more realistic and accurate description than the conventional variable-mass systems on stochastic phenomena of absorbing and desorbing particles in some complex mediums. It is found that the stochastic interactions could induce the normal or reversal current by influencing the asymmetry and barrier height of effective ratchet potential, and in strong-coupling regime, the dynamical mechanism could be well interpreted. By the numerical simulations, we systematically study the dependence of directed current, stochastic resonance (SR) and coherent resonance (CR) behaviors of transport velocity on various parameters, including particle concentration, interacting stability, coupling strength, spring’s free length, driving amplitude, ratchet asymmetry, and medium temperature. The results also provide the latent support for manipulating transport behaviors and optimizing the coupled structure in artificial nanomachines.

In Russian

in Russian

In Russian

A relevant and important question in studying the transport of nanoparticles is the possibility and methods of controlling the generated currents. One of the possibilities is to use the ratchet effect, i.e. the occurrence of directed motion as a result of the influence of nonequilibrium fluctuations of various nature when one or more symmetries in the system are broken. To implement the ratchet effect, a deterministic dichotomous process is often used, which can be modeled by two alternating states that have constant characteristics. Usually, the main factor determining the direction of motion of a Brownian motor is the spatial asymmetry of the potential profile. In certain cases, for example, for a double-well potential profile, one can relatively easily investigate the conditions for reversal of the direction of motor motion. In this paper, using the idea of Parrondo’s paradox games consisting of alternating strategies of a game that provides an average winning, we simulated the ratchet effect for a diffusion hopping model of an adiabatic Brownian motor with an asymmetric double-well on-off potential. The conditions affecting the direction of nanoparticle motion are investigated, the possibility of temperature control of the motion direction is shown, and an estimate of the generated average velocity of the Brownian motor in the adiabatic approximation is obtained. The motor functioning is simulated in terms of the game theory and the average trajectories of capital accumulation are obtained, which correspond to the trajectories of the average displacement of a Brownian particle due to the ratchet effect. For the chosen model, it is shown that, at low temperatures, a particle moves to the right in accordance with the simplest on-off ratchet model, while, at high temperatures, the motion reverses. A comparison of the simulation results with the values of the ratchet velocity obtained within the adiabatic approximation shows that this approximation becomes valid at sufficiently large values of the lifetimes of the states of the dichotomous process, and it turns out to be much more accurate in the high-temperature region than in the low-temperature one.

Adiabatic Ratchet Effect in Systems with Discrete Variables

An adiabatic theory of the ratchet effect in various systems described by random transitions between discrete states has been developed. The theory is based on the development of a discrete analog of the Parrondo lemma, which is one of the fundamental relations of the theory of ratchet systems allowing the calculation of integral fluxes in time intervals from the initial to final (equilibrium) distributions. Correspondence between the transition probabilities and the parameters of the potential profile, in which the hopping motion is described by the developed theory and is a low-temperature limit of continuous motion, has been proposed. The rate of the ratchet effect calculated by the proposed theory is well confirmed by a numerical simulation. The developed approach makes it possible to study the characteristics of the operation of Brownian motors of various natures by simple methods.

Directed transport of coupled Brownian motors in a two-dimensional traveling-wave potential**Project supported by the National Natural Science Foundation of China (Grant No. 11875135), Quanzhou Scientific and Technological Foundation, China (Grant No. 2018C085R), the Key Project of Beijing Institute of Graphic Communication, China (Grant No. Ea201702), and the International Ability Improvement Project of Teaching Staff of Beijing Institute of Graphic Communication,

Considering an elastically coupled Brownian motors system in a two-dimensional traveling-wave potential, we investigate the effects of the angular frequency of the traveling wave, wavelength, coupling strength, free length of the spring, and the noise intensity on the current of the system. It is found that the traveling wave is the essential condition of the directed transport. The current is dominated by the traveling wave and varies nonmonotonically with both the angular frequency and the wavelength. At an optimal angular frequency or wavelength, the current can be optimized. The coupling strength and the free length of the spring can locally modulate the current, especially at small angular frequencies. Moreover, the current decreases rapidly with the increase of the noise intensity, indicating the interference effect of noise on the directed transport.

Phonon-Induced Ratchet Motion of a Water Nanodroplet on a Supported Black Phosphorene.

Phonons are not supposed to carry any physical momentum as lattice vibrational modes; thus, it is believed no mass transport could be induced by phonons. In this Letter, we show that a ratchet motion of a water nanodroplet could be induced on a two-dimensional puckered lattice like black phosphorene (BP) by exciting its flexural phonons through a moving substrate. The water nanodroplet exhibits a forward motion along the armchair or a backward motion along the zigzag directions on a BP lattice that is supported on a substrate possessing a relative armchair or zigzag forward motion with BP. Through the analysis of the structure and vibrational density states of BP, it is found that in-plane lattice displacement asymmetry and the in-plane vibration asymmetry are induced by the excited flexural phonons, which determine the water nanodroplet motion as an anisotropic Brownian motor. Simulations of the nanodroplet motion as functions of the substrate relative motion speed and direction and also the substrate coupling strength with BP are performed. Results of the nanodroplet ratchet motion exhibit good agreement with the theoretical predications from calculating the Brownian motor asymmetry. Our findings reveal a promising mass transport strategy and a further understanding of phonon-related interactions in crystalline solids.

In Russian

In spite of the widespread opinion about the negative effect of chaos on the nature and human life, there exist examples of its constructive role in various processes. Among them, a prominent position is occupied by the processes in which thermal noise causes not only Brownian motion but also drift of nanoparticles as a result of unbiased non-equilibrium perturbations of various nature in systems with broken spatial and/or temporal symmetry. Such processes are relevant to the operation of Brownian motors, or ratchets, actively studied in the past decades. In the present paper, the working principles of systems of this kind are explained based on the author’s approach which unifies the treatment of different ratchet effect manifestations in the models of fluctuating potential, catalytic wheel, and electroconformational coupling. Another manifestation of this effect is provided by the paradoxical games proposed by Parrondo in which chaos appears in random tosses of a die and the ratchet effect arises from a certain alternation of game strategies ensuring an average win. The paper presents the simplest version of Parrondo’s game; it consists in the alternation of two antisymmetric games, with the rules depending on the parity of the capital possessed by the player before the next toss. The dependence of the average win on the number of tosses is calculated by computer simulation and compared with the result of the catalytic wheel model valid in the adiabatic approximation. The theoretical win calculated by this model agrees well with the numerical simulation result for the game concerned. The attractiveness of the treatment of the ratchet effect in terms of game theory stems from the possibility to visualize the theoretical framework and to study the laws of Brownian motor operation by simple modelling methods.

Resonant phenomena and mechanism in vibrated granular systems.

We were motivated to perform this research by the investigation of Brownian motors in excited granular materials converting the chaotic motion of granules into the oriented motion of motors. We conducted experimental studies to explore the horizontal motion of granules in vertically vibrated annular granular systems, including mixed and pure granular systems with an asymmetrical periodic structure on the bottom. The variations of the horizontal granular flow caused by the height, vibrating parameters, and mixing ratio were described in detail. Our results revealed considerable changes in the horizontal flow of different granular systems. Most importantly, resonance was induced in the horizontal granular flow by the vertical vibration; that is, the horizontal flow reached its maximum at specific vibrating parameters. A collisional model of rigid objects was constructed to probe the flowing resonances in these granular systems and provided a qualitative agreement with the experimental results obtained. We conclude that when a flowing resonance occurs, the granular system oscillates horizontally with a natural frequency under periodic external excitation. The frequency matching between the external excitation and the horizontal oscillation is responsible for the flowing resonance. Our results could improve the current understanding of the dynamic properties of granular systems under external excitation.

Ionized gas Thermoelectric Generator

The behavior of charged particles mixed with neutral particles laid in Circular Ratchet Potential (CRP) inside circular tube container was studied, showing a net Counter-clockwise flow of charged particles and other net flow of neutral particles equal in magnitude and opposite in direction (Clockwise), the neutral flow of particles will not cancel the positive particles flow; by Causality Principle. These two opposite flow of particles will reach terminal intensity at some stage inside the circular path; the charged particles are confined to move in a semi-circular (Nut Shape) trajectory according to the CRP, while the neutral particles can escape out of the container in tangential trajectory, hence producing perpetual disequilibrium in the thermodynamic properties of the system.Allowing the neutral particles to escape through outer tangential slots, and allow new neutral particles to enter the container through inner slots to compensate neutral particles. This gives the basic idea for a new concept of Brownian Motor (Plyukhin, 2013) [1] or Thermoelectric Generator (TEG) (Bqoor, 2013) [2], then the excess in charged particles flow will reduce the neutral particle flow and produce micro-turbulences. This mechanism will allow extracting power from the ambient thermal energy.Several considerations including efficiency, manufacturing and improvement aspects were evaluated and discussed in this paper. However, this Ionized Gas TEG can be assumed as a Perpetual Motion Machine (Rao, 2004) [3] of the second type, hence questioning the validity of the “empirical” Second Law of Thermodynamics, were several phenomena showing the necessity for new formula amending the Second Law of Thermodynamics (Crosignani and Di Porto, 2001; Wang et al., 2002) [4,5].

Investigation on the directional transportation of coupled Brownian motors with asymmetric friction

<sec>Biological molecular motors exist in cells widely. They can make use of intracellular free energy to complete all kinds of internal biological transports by transforming chemical energy into mechanical energy. The kind of directional movement of biological molecular motors plays a very important role in intracellular material transportation. In order to study the transport mechanism of molecular motors further, a large number of ratchet models are proposed, such as rocking ratchets and flashing ratchets. By investigating various kinds of ratchets we can not only understand the directional movement mechanism of Brownian particles, but can find suitable conditions in which the performance of Brownian motors’ directional transportation could be enhanced. Meanwhile, the investigation of ratchets could also be applied in manufacturing nanometer devices.</sec><sec>At present, the directional transportation of Brownian ratchet has attracted extensive interests of researchers. In general, most friction factors of Brownian ratchet models are considered unit. In fact, the concentration of solutions and cell fluid impurity affect the actual frictional damping conditions, so the real frictional coefficient of Brownian motors is often changed. In addition, lots of experimental studies have shown that the movement of Brownian motors is collectively directed motion and the kind of directional movement is induced by intermolecular coupling interaction. As a result, it is more valuable to investigate the transporting performance of coupled Brownian particles that existed in different frictional damping conditions. In order to enhance the transporting performance of Brownian ratchet in different frictional damping conditions, we discuss how the frictional damping factor influences the directional movement of coupled Brownian particles deeply when Brownian particles drag loads.</sec><sec> In this paper, we established the overdamped frictional ratchets, and then we investigated how frictional damping coefficient ratio, coupling strength and external force amplitude affect the transportation of coupled Brownian ratchets. On the basis of the investigation, some interesting results are found. The directional transport of frictional ratchets can be promoted by adjusting the frictional damping factor. Besides, the transportation can obtain the maximum under the appropriate friction factor case. In addition, under certain frictional damping condition, the directional transportation of the friction ratchets present multi-peak structure as the external force amplitude increases. Meanwhile, the appropriate free length and coupling strength can also enhance the transportation characteristic of frictional ratchets. All conclusions obtained in this paper can not be applied in selecting suitable frictional damping conditions experimentally to improve the directional transportation of coupled Brownian ratchets, but they can also be used in developing and manufacturing nanometer devices.</sec>

Crystalline chitin hydrolase is a burnt-bridge Brownian motor

Motor proteins are essential units of life and are well-designed nanomachines working under thermal fluctuations. These proteins control moving direction by consuming chemical energy or by dissipating electrochemical potentials. Chitinase A from bacterium Serratia marcescens (SmChiA) processively moves along crystalline chitin by hydrolysis of a single polymer chain to soluble chitobiose. Recently, we directly observed the stepping motions of SmChiA labeled with a gold nanoparticle by dark-field scattering imaging to investigate the moving mechanism. Time constants analysis revealed that SmChiA moves back and forth along the chain freely, because forward and backward states have a similar free energy level. The similar probabilities of forward-step events (83.5%=69.3%+14.2%) from distributions of step sizes and chain-hydrolysis (86.3%=(1/2.9)/(1/2.9+1/18.3)×100) calculated from the ratios of time constants of hydrolysis and the backward step indicated that SmChiA moves forward as a result of shortening of the chain by a chitobiose unit, which stabilizes the backward state. Furthermore, X-ray crystal structures of sliding intermediate and molecular dynamics simulations showed that SmChiA slides forward and backward under thermal fluctuation without large conformational changes of the protein. Our results demonstrate that SmChiA is a burnt-bridge Brownian ratchet motor.