Previous works have shown that time asymmetric forcing on the one hand, as well as non-Gaussian noises on the other, can separately enhance the efficiency and current of a Brownian motor. Here, we study the result of subjecting a Brownian motor to both effects simultaneously. Our results have been compared with those obtained for the Gaussian white noise regime in the adiabatic limit. We find that, although the inclusion of the time asymmetry parameter increases the efficiency value up to a certain extent, for the present case this increase is much less appreciable than in the white noise case. We also present a comparative study of the transport coherence in the context of colored noise. Though the efficiency in some cases becomes higher for the non-Gaussian case, the Péclet number is always higher in the Gaussian colored noise case than in the white noise as well as non-Gaussian colored noise cases.
We study a genuine Brownian motor by hard disk molecular dynamics and calculate analytically its properties, including its drift speed and thermal conductivity, from microscopic theory.
The efficiency of different types of Brownian motors is calculated analytically and numerically. We find that motors based on flashing ratchets present a low efficiency and an unavoidable entropy production. On the other hand, a certain class of motors based on adiabatically changing potentials, named reversible ratchets, exhibit a higher efficiency and the entropy production can be arbitrarily reduced.
Recently, a thermal Brownian motor was introduced [Van den Broeck, Kawai and Meurs, Phys. Rev. Lett. (2004)], for which an exact microscopic analysis is possible. The purpose of this paper is to review some further properties of this construction, and to discuss in particular specific issues including the relation with macroscopic response and the efficiency at maximum power.
In this paper we present a model of a symmetric Brownian motor (SBM) which changes the sign of its velocity when the temperature gradient is inverted. The velocity, external work and efficiency are studied as a function of the temperatures of the baths and other relevant parameters. The motor shows a current reversal when another parameter (a phase shift) is varied. Analytical predictions and results from numerical simulations are performed and agree very well. Generic properties of this type of motors are discussed.
An asymmetric object, undergoing dissipative collisions with surrounding particles, acquires a nonzero average velocity. The latter is calculated analytically by an expansion of the Boltzmann equation and the result is compared with Monte Carlo simulations.
Recent studies on Brownian motors driven by colored non Gaussian noises have shown that the departure of the noise distribution from Gaussian behavior induces an enhancement of its current and efficiency. Here we discuss some new aspects of this phenomenon focusing in some analytical results based in an adiabatic approximation, and in the analysis of the long probability distribution tails' role.
We study analytically a thermal Brownian motor model and calculate exactly the Onsager coefficients. We show how the reciprocity relation holds and that the determinant of the Onsager matrix vanishes. Such condition implies that the device is built with tight coupling. This explains why Carnot's efficiency can be achieved in the limit of infinitely slow velocities. We also prove that the efficiency at maximum power has the maximum possible value, which corresponds to the Curzon-Alhborn bound. Finally, we discuss the model acting as a Brownian refrigerator.
We investigate the motion of a wedge-shaped object (a granular Brownian motor), which is restricted to move along the x-axis and cannot rotate, as gas particles collide with it. We show that its steady-state drift, resulting from inelastic gas-motor collisions, is dramatically affected by anisotropy in the velocity distribution of the gas. We identify the dimensionless parameter providing the dependence of this drift on shape, masses, inelasticity, and anisotropy: the anisotropy leads to a dramatic breaking of equipartition, which should easily be visible in experimental realizations.
We discuss an autonomous motor based on a Brownian particle driven from thermal equilibrium by periodic in time variation of the internal potential through which the particle interacts with molecules of the surrounding thermal bath. We demonstrate for such a motor the absence of a linear response regime: The average driving force and drift velocity are shown to be quadratic in both the frequency and amplitude of the variation. The adiabatic approximation (of an infinitely slow variation) and the leading correction to it (linear in the variation's frequency) both lead to zero drift and are insu…
We study transport properties of an inertial Brownian particle moving in viscous symmetric periodic structures and driven by an oscillating signal of two harmonic components. We analyze the influence of symmetric, antisymmetric and asymmetric signals on directed transport and reveal the shift symmetry of the stationary averaged velocity of the Brownian particle with respect to the relative phase of two components of the signal. The shift symmetry holds true in all regimes.
The problem of Brownian motion in a periodic potential, under the influence of external forcing, which is either random or periodic in time, is studied in this paper. Multiscale techniques are used to derive general formulae for the steady state particle current and the effective diffusion tensor. These formulae are then applied to calculate the effective diffusion coefficient for a Brownian particle in a periodic potential driven simultaneously by additive Gaussian white and colored noise. Our theoretical findings are supported by numerical simulations.
In systems possessing spatial or dynamical symmetry breaking, Brownian motion combined with symmetric external input signals, deterministic or random, alike, can assist directed motion of particles at the submicron scales. In such cases, one speaks of "Brownian motors". In this review the constructive role of Brownian motion is exemplified for various one-dimensional setups, mostly inspired by the cell molecular machinery: working principles and characteristics of stylized devices are discussed to show how fluctuations, either thermal or extrinsic, can be used to control diffusive particle tra…
We consider a primary model of ac-driven Brownian motors, i.e., a classical particle placed in a spatial-time periodic potential and coupled to a heat bath. The effects of fluctuations and dissipations are studied by a time-dependent Fokker-Planck equation. The approach allows us to map the original stochastic problem onto a system of ordinary linear algebraic equations. The solution of the system provides complete information about ratchet transport, avoiding such disadvantages of direct stochastic calculations as long transients and large statistical fluctuations. The Fokker-Planck approach …
Brownian motors are devices which "rectify" Brownian motion, i.e. they can generate a current of particles out of unbiased fluctuations. Brownian motors are important for the understanding of molecular motors, and are also promising for the realization of new nanolelectronic devices. Among the different systems that can be used to study Brownian motors, cold atoms in optical lattices are quite an unusual one: there is no thermal bath and both the potential and the fluctuations are determined by laser fields. In this article recent experimental implementations of Brownian motors using cold atom…
Transport phenomena in spatially periodic systems far from thermal equilibrium are considered. The main emphasize is put on directed transport in so-called Brownian motors (ratchets), i.e. a dissipative dynamics in the presence of thermal noise and some prototypical perturbation that drives the system out of equilibrium without introducing a priori an obvious bias into one or the other direction of motion. Symmetry conditions for the appearance (or not) of directed current, its inversion upon variation of certain parameters, and quantitative theoretical predictions for specific models are revi…
Emergence, optimization and stability of a motor-like motion in a fluctuating environment are analyzed. The emergence of motion is shown to be a general phenomenon. A motor converges to the state with the minimum of effective temperature and with the corresponding minimum in the rate of conformation changes similarly as some stochastic processes converge to the states with minimum diffusion activity. This mechanism is important to bacterial foraging (chemotaxis). This work, therefore, raises an analogy between chemotaxis and the emergence of living-like systems. The implications include the de…
We study fluctuating tilt Brownian ratchets based on fractional subdiffusion in sticky viscoelastic media characterized by a power law memory kernel. Unlike the normal diffusion case the rectification effect vanishes in the adiabatically slow modulation limit and optimizes in a driving frequency range. It is shown also that anomalous rectification effect is maximal (stochastic resonance effect) at optimal temperature and can exhibit a surprisingly good quality. Moreover, subdiffusive current can flow in the counter-intuitive direction upon a change of temperature or driving frequency. The depe…
In order to optimize the directed motion of an inertial Brownian motor, we identify the operating conditions that both maximize the motor current and minimize its dispersion. Extensive numerical simulation of an inertial rocked ratchet displays that two quantifiers, namely the energetic efficiency and the Péclet number (or equivalently the Fano factor), suffice to determine the regimes of optimal transport. The effective diffusion of this rocked inertial Brownian motor can be expressed as a generalized fluctuation theorem of the Green -- Kubo type. Addendum and Erratum The expression for the e…
A solvable model is proposed and analyzed to reveal the mechanism underlying the diffusion enhancement recently reported for a model of molecular motors and predicted to be observed in the biological rotary motor $\rm F_1$-ATPase. It turns out that the diffusion enhancement for the present model can approximately described by a random walk in which the waiting time for a step to occur is exponentially distributed and it takes nonzero time to proceed forward by the step. It is shown that the diffusion coefficient of such a random walk can significantly be increased when the average waiting time…
We consider the translationally invariant but parity (left-right symmetry) breaking Ising chain Hamiltonian \begin{equation} {\cal H} = -U_2\sum_{k} s_{k}s_{k+1} - U_3\sum_{k} s_{k}s_{k+1}s_{k+3} \nonumber \end{equation} and let this system evolve by Kawasaki spin exchange dynamics. Monte Carlo simulations show that perturbations forcing this system off equilibrium make it act as a Brownian molecular motor which, in the lattice gas interpretation, transports particles along the chain. We determine the particle current under various different circumstances, in particular as a function of the ra…
Human-created engines and evolutionarily optimized molecular motors exhibit sophisticated design in order to harvest chemical or thermal energy for generating unidirectional motion. The complexity of these motors makes their random emergence unlikely. Molecules capable of locomotion, however, seem to be essential to the creation of the first self-replicator and initiation of Darwinian evolution, posing a question of the physical mechanism that can facilitate emergence of directed motion in an isotropic environment. Here we show a universal thermodynamic mechanism for spontaneous emergence of m…
A model of an autonomous isothermal Brownian motor with an internal propulsion mechanism is considered. The motor is a Brownian particle which is semi-transparent for molecules of surrounding ideal gas. Molecular passage through the particle is controlled by a potential similar to that in the transition rate theory, i.e. characterized by two stationary states with a finite energy difference separated by a potential barrier. The internal potential drop maintains the diode-like asymmetry of molecular fluxes through the particle, which results in the particle's stationary drift.
We study a noisy drive mechanism for efficiency enhancement of Brownian motors operating on the micro-scale domain. It was proven [J. Spiechowicz et al., J. Stat. Mech. P02044, (2013)] that biased noise $η(t)$ can induce normal and anomalous transport processes similar to those generated by a static force $F$ acting on inertial Brownian particles in a reflection-symmetric periodic structure in presence of symmetric unbiased time-periodic driving. Here, we show that within selected parameter regimes, noise $η(t)$ of the mean value $\langle η(t) \rangle = F$ can be significantly more effective t…
We review a family of models recently introduced to describe Brownian motors under the influence of Coulomb friction, or more general non-linear friction laws. It is known that, if the heat bath is modeled as the usual Langevin equation (linear viscosity plus white noise), additional non-linear friction forces are not sufficient to break detailed balance, i.e. cannot produce a motor effect. We discuss two possibile mechanisms to elude this problem. A first possibility, exploited in several models inspired to recent experiments, is to replace the heat bath's white noise by a ``collisional noise…
We investigated three models of Brownian motors which convert rotational diffusion into directed translational motion by switching on and off a potential. In the first model a spatially asymmetric potential generates directed translational motion by rectifying rotational diffusion. It behaves much like a conventional flashing ratchet. The second model utilizes both rotational diffusion and drift to generate translational motion without spatial asymmetry in the potential. This second model can be driven by a combination of a Brownian motor mechanism (diffusion driven) or by powerstroke (drift d…
A Brownian Motor is a nanoscale or molecular device that combines the effects of thermal noise, spatial or temporal asymmetry, and directionless input energy to drive directed motion. Because of the input energy, Brownian motors function away from thermodynamic equilibrium and concepts such as linear response theory, fluctuation dissipation relations, and detailed balance do not apply. The {\em generalized} fluctuation-dissipation relation, however, states that even under strongly thermodynamically non-equilibrium conditions the ratio of the probability of a transition to the probability of th…
We study a model consisting of $N$ nonlinear oscillators with {\em global periodic} coupling and {\em local multiplicative} and additive noises. The model was shown to undergo a nonequilibrium phase transition towards a broken-symmetry phase exhibiting noise-induced "ratchet" behavior. A previous study \cite{[7]} focused on the relationship between the character of thehysteresis loop, the number of ``homogeneous'' mean-field solutions and the shape of the stationary mean-field probability distribution function. Here we show --as suggested by the absence of stable solutions when the load force …
We address a recently introduced model describing a system of periodically coupled nonlinear phase oscillators submitted to multiplicative white noises, wherein a ratchet-like transport mechanism arises through a symmetry-breaking noise-induced nonequilibrium phase transition. Numerical simulations of this system reveal amazing novel features such as negative zero-bias conductance and anomalous hysteresis, explained resorting to a strong-coupling analysis in the thermodynamic limit. Using an explicit mean-field approximation we explore the whole ordered phase finding a transition from anomalou…
The noise-assisted, directed transport in a one-dimensional dissipative, inertial Brownian motor of the rocking type that is exposed to an external bias is investigated. We demonstrate that the velocity-load characteristics is distinctly non-monotonic, possessing regimes with a {\em negative differential mobility}. In addition, we evaluate several possible efficiency quantifiers which are compared among each other. These quantifiers characterize the mutual interplay between the viscous drag and the external load differently, weighing the inherent rectification features from different physical …
The energetics of a Brownian heat engine and heat pump driven by position dependent temperature, known as the Büttiker-Landauer heat engine and heat pump, is investigated by numerical simulations of the inertial Langevin equation. We identify parameter values for optimal performance of the heat engine and heat pump. Our results qualitatively differ from approaches based on the overdamped model. The behavior of the heat engine and heat pump, in the linear response regime is examined under finite time conditions and we find that the efficiency is lower than that of an endoreversible engine worki…
Brownian motors and Feymann ratchets have been intensively studied in the past decades due to their significance to the foundation of statistical physics. In this work we propose a new type of Brownian motor, i.e., a self-driven motor that only utilizes the temperature difference inside and outside the motor. The motor is a container with asymmetric geometry; when filling of gas, directional motion occurs if its temperature is different from the environment. The essentially new mechanism is that the asymmetric geometry of the container may induce anisotropic thermalization, which results in a …
We study a Brownian motor driven by a colored non Gaussian noise source with a $q$-dependent probability distribution, where $q$ is a parameter indicating the departure from Gaussianity. For $q=1$ the noise is Gaussian (Ornstein--Uhlenbeck), while, for $q>1$, the probability distribution falls like a power law. In the latter case, we find a marked enhancement of both the current and the efficiency of the Brownian motor in the overdamped regime. We also analyze the case with inertia and show that, again for $q > 1$, a remarkable increase of the ratchet's mass separation capability is obtained.
Two losing games, when alternated in a periodic or random fashion, can produce a winning game. This paradox occurs in a family of stochastic processes: if one combines two or more dynamics where a given quantity decreases, the result can be a dynamic system where this quantity increases. The paradox could be applied to a number of stochastic systems and has drawn the attention of researchers from different areas. In this paper we show how the phenomenon can be used to design Brownian or molecular motors, i.e., thermal engines that operate by rectifying fluctuations. We briefly review the liter…
With this work we investigate an often neglected aspect of Brownian motor transport: The rôle of fluctuations of the noise-induced current and its consequences for the efficiency of rectifying noise. In doing so, we consider a Brownian inertial motor that is driven by an unbiased monochromatic, time-periodic force and thermal noise. Typically, we find that the asymptotic, time- and noise-averaged transport velocities are small, possessing rather broad velocity fluctuations. This implies a corresponding poor performance for the rectification power. However, for tailored profiles of the ratchet …
The directed transport of Brownian particles requires a system with an asymmetry and with non-equilibrium noise. We here investigate numerically alternative ways of fulfilling these requirements for a two-state Brownian motor, realised with Brownian particles alternating between two phase-shifted, symmetric potentials. We show that, besides the previously known spatio-temporal asymmetry based on unequal transfer rates between the potentials, inequalities in the potential depths, the frictions, or the equilibrium temperatures of the two potentials also generate the required asymmetry. We also s…
A system of periodically coupled nonlinear phase oscillators submitted to both additive and multiplicative white noises has been recently shown to exhibit ratchetlike transport, negative zero-bias conductance, and anomalous hysteresis. These features stem from the "asymmetry" of the stationary probability distribution function, arising through a noise-induced nonequilibrium phase transition which is "reentrant" as a function of the multiplicative noise intensity. Using an explicit mean-field approximation we analyze the effect of the multiplicative noises being coloured, finding a contraction …
The rectification of noise into directed movement or useful energy is utilized by many different systems. The peculiar nature of the energy source and conceptual differences between such Brownian motor systems makes a characterization of the performance far from straightforward. In this work, where the Brownian motor consists of atoms interacting with dissipative optical lattices, we adopt existing theory and present experimental measurements for both the efficiency and the transport coherence. We achieve up to 0.3% for the efficiency and 0.01 for the Péclet number.
We extend the Langevin dynamics so that particles can be exchanged with a particle reservoir. We show that grand canonical ensembles are realized at equilibrium and derive the relations of thermodynamics for processes between equilibrium states. As an application of the proposed evolution rule, we devise a simple model of Brownian motors driven by particle exchange. KEYWORDS: Langevin Dynamics, Thermodynamics, Open Systems
In systems possessing a spatial or dynamical symmetry breaking thermal Brownian motion combined with unbiased, non-equilibrium noise gives rise to a channelling of chance that can be used to exercise control over systems at the micro- and even on the nano-scale. This theme is known as ``Brownian motor'' concept. The constructive role of (the generally overdamped) Brownian motion is exemplified for a noise-induced transport of particles within various set-ups. We first present the working principles and characteristics with a proof-of-principle device, a diffusive temperature Brownian motor. Ne…
Based on recent work [L. Machura, M. Kostur, P. Talkner, J. Luczka, and P. Hanggi, Phys. Rev. Lett. 98, 040601 (2007)], we extend the study of inertial Brownian motors to the case of an asymmetric potential. It is found that some transport phenomena appear in the presence of an asymmetric potential. Within tailored parameter regimes, there exists two optimal values of the load at which the mean velocity takes its maximum, which means that a load can facilitate the transport in the two parameter regimes. In addition, the phenomenon of multiple current reversals can be observed when the load is …
We study Brownian motors driven by colored non Gaussian noises, both in the overdamped regime and in the case with inertia, and analyze how the departure of the noise distribution from Gaussian behavior can affect its behavior. We analyze the problem from two alternative points of view: one oriented mainly to possible technological applications and the other more inspired in natural systems. In both cases we find an enhancement of current and efficiency due to the non-Gaussian character of the noise. We also discuss the possibility of observing an enhancement of the mass separation capability …
We introduce a general framework to study the processivity of molecular motors moving along a polar filament and discuss the average time spent attached to the filament as a function of a tangentially applied load. Our study of specific models suggests that the attachment time of a motor decreases with increasing ATP concentration and that double-headed motors such as kinesins lose their processivity under forcing conditions while processive single-headed motors are less sensitive to tangential forcing.
In this work we experimentally study a granular rotor which is similar to the famous Smoluchowski-Feynman device and which consists of a rotor with four vanes immersed in a granular gas. Each side of the vanes can be composed of two different materials, creating a rotational asymmetry and turning the rotor into a ratchet. When the granular temperature is high, the rotor is in movement all the time, and its angular velocity distribution is well described by the Brownian Limit discussed in previous works. When the granular temperature is lowered considerably we enter the so-called Single Kick Li…
A reversal of the particle current in rocking Brownian motors was predicted more than 20 years ago; however, an experimental verification and a deeper insight into the underlying mechanisms remained elusive. Here, we investigate the high frequency behaviour of a rocking Brownian motor for charged nanoparticles based on electrostatic interactions in a 3D shaped nanofluidic slit and electro-osmotic forcing of the particles. A sub ms temporal and $\approx\,10\,$nm spatial resolution of the 60 nm gold spheres allows us to measure the time-resolved and frequency dependent evolution of the particle …
We have realized real-time steering of the directed transport in a Brownian motor based on cold atoms in optical lattices, and demonstrate drifts along pre-designed paths. The transport is induced by spatiotemporal asymmetries in the system, where we can control the spatial part, and we show that the response to changes in asymmetry is very fast. In addition to the directional steering, a real-time control of the magnitude of the average drift velocity and an on/off switching of the motor are also demonstrated. We use a non-invasive real-time detection of the transport, enabling a feedback con…
Onsager symmetry implies that a Brownian motor, driven by a temperature gradient, will also perform a refrigerator function upon loading. We analytically calculate the corresponding heat flux for an exactly solvable microscopic model and compare it with molecular dynamics simulations.
We present the exact analysis of a chiral Brownian motor and heat pump. Optimization of the construction predicts, for a nanoscale device, frequencies of the order of kHz and cooling rates of the order of femtojoule per second.
We explore the transport features of a Brownian particle that walks in a periodic ratchet potential that is coupled with a spatially varying temperature background. Since the viscous friction of the medium decreases as the temperature of the medium increases, any reasonable exploration regarding the thermodynamic features of the Brownian engine should take into account the role of temperature on the viscosity of the fluid. In this work, we study this effect of temperature by considering a viscous friction that decreases exponentially as the background temperature increases. Our result depicts …
We investigate directed thermal heat flux across 1D homogenous nonlinear lattices when no net thermal bias is present on average. A nonlinear lattice of Fermi-Pasta-Ulam-type or Lennard-Jones-type system is connected at both ends to thermal baths which are held at the same temperature on temporal average. We study two different modulations of the heat bath temperatures, namely: (i) a symmetric, harmonic ac-driving of temperature of one heat bath only and (ii) a harmonic mixing drive of temperature acting on both heat baths. While for case (i) an adiabatic result for the net heat transport can …
We study the steady state behaviour of a confined quantum Brownian particle subjected to a space-dependent, rapidly oscillating time-periodic force. To leading order in the period of driving, the result of the oscillating force is an effective static potential which has a quantum dissipative contribution, $V_{QD}$, which adds on to the classical result. This is shown using a coherent state representation of bath oscillators. $V_{QD}$ is evaluated exactly in the case of an Ohmic dissipation bath. It is strongest for intermediate values of the damping, where it can have pronounced effects.
In this work we experimentally study the behavior of a freely-rotating asymmetric probe immersed in a vibrated granular medium. For a wide variety of vibration conditions the probe exhibits a steady rotation whose direction is constant with respect to the asymmetry. By changing the vibration amplitude and by filtering the noise in different frequency bands we show that the velocity of rotation does not depend only on the RMS acceleration $Γ$, but also on the amount of energy provided to two separate frequency bands which are revealed to be important for the dynamics of the granular medium: the…
A quantum Smoluchowski equation is put forward that consistently describes thermal quantum states. In particular, it notably does not induce a violation of the second law of thermodynamics. This so modified kinetic equation is applied to study {\it analytically} directed quantum transport at strong friction in arbitrarily shaped ratchet potentials that are driven by nonthermal two-state noise. Depending on the mutual interplay of quantum tunneling and quantum reflection these quantum corrections can induce both, either a sizable enhancement or a suppression of transport. Moreover, the threshol…
The Carnot cycle imposes a fundamental upper limit to the efficiency of a macroscopic motor operating between two thermal baths. However, this bound needs to be reinterpreted at microscopic scales, where molecular bio-motors and some artificial micro-engines operate. As described by stochastic thermodynamics, energy transfers in microscopic systems are random and thermal fluctuations induce transient decreases of entropy, allowing for possible violations of the Carnot limit. Despite its potential relevance for the development of a thermodynamics of small systems, an experimental study of micro…
We extend the analysis of a thermal Brownian motor reported in Phys. Rev. Lett. 93, 090601 (2004) by C. Van den Broeck, R. Kawai, and P. Meurs to a three-dimensional configuration. We calculate the friction coefficient, diffusion coefficient, and drift velocity as a function of shape and present estimates based on physically realistic parameter values.
We develop a numerical algorithm for computing the effective drift and diffusivity of the steady-state behavior of an overdamped particle driven by a periodic potential whose amplitude is modulated in time by multiplicative noise and forced by additive Gaussian noise (the mathematical structure of a flashing Brownian motor). The numerical algorithm is based on a spectral decomposition of the solution to the Fokker-Planck equation with periodic boundary conditions and the cell problem which result from homogenization theory. We also show that the numerical method of Wang, Peskin, Elston (WPE, 2…
Two headed motor proteins, such as kinesin and dynein, hidrolyze environmental ATP in order to propel unidirectionally along cytoskeletal filaments such as microtubules. In the case of kinesin, protein heads bind primarily on the alpha tubulin site of asymmetric alpha-beta 8nm-long tubulin dimers that constitute the microtubular protofilaments. Kinesin dimers overcome local binding forces up to 5pN and are known to move on protofilaments with ATP concentration-dependent speeds while hydrolizing on average one ATP molecule per 8nm step. The salient features of protein trajectories are the disti…
The motility of certain gram-negative bacteria is mediated by retraction of type IV pili surface filaments, which are essential for infectivity. The retraction is powered by a strong molecular motor protein, PilT, producing very high forces that can exceed 150 pN. The molecular details of the motor mechanism are still largely unknown, while other features have been identified, such as the ring-shaped protein structure of the PilT motor. The surprisingly high forces generated by the PilT system motivate a model investigation of the generation of large forces in molecular motors. We propose a si…
