A channel Brownian pump powered by an unbiased external force

Optimizing Work Extraction in the Presence of an Entropic Potential: An Entropic Stochastic Resonance

We study the nontrivial thermodynamic responses of an overdamped Brownian system driven by an unbiased driving force when the particle is confined inside a bilobal irregular structure. The spatial irregularity of the confinement results in an effective entropic bistable potential along the direction of transport. We calculate the thermodynamic response functions in terms of the averaged work done and the absorbed heat over a cycle of driving. We find that the thermodynamic responses are influenced by the nonlinearity of the effective entropic potential, the frequency of the external periodic driving force, and the random thermal fluctuations in a nontrivial way. In the presence of an optimal amount of thermal noise and a favoring driving frequency, the process exhibits a resonance-like precedent in terms of both output work and absorbed heat. We explore the conditions to get best synchronized work extraction (or absorbed heat), which can be utilized as a potential quantifier of an entropic stochastic resonance phenomenon. Finally, we identify a hallmark of entropy dominance over an analogous energy-driven scenario in terms of output work.

Geometric Brownian information engine: Essentials for the best performance

We investigate a geometric Brownian information engine (GBIE) in the presence of an error-free feedback controller that transforms the information gathered on the state of Brownian particles entrapped in monolobal geometric confinement into extractable work. Outcomes of the information engine depend on the reference measurement distance x_{m}, the feedback site x_{f}, and the transverse force G. We determine the benchmarks for utilizing the available information in an output work and the optimum operating requisites for best achievable work. Transverse bias force (G) tunes the entropic contribution in the effective potential and hence the standard deviation (σ) of the equilibrium marginal probability distribution. We recognize that the amount of extractable work reaches a global maximum when x_{f}=2x_{m} with x_{m}∼0.6σ, irrespective of the extent of the entropic limitation. Because of the higher loss of information during the relaxation process, the best achievable work of a GBIE is lower in an entropic system. The feedback regulation also bears the unidirectional passage of particles. The average displacement increases with growing entropic control and is maximum when x_{m}∼0.81σ. Finally, we explore the efficacy of the information engine, a quantity that regulates the efficiency in utilizing the information acquired. With x_{f}=2x_{m}, the maximum efficacy reduces with increasing entropic control and shows a crossover from 2 to 11/9. We discover that the condition for the best efficacy depends only on the confinement lengthscale along the feedback direction. The broader marginal probability distribution accredits the increased average displacement in a cycle and the lower efficacy in an entropy-dominated system.

First-passage times in conical varying-width channels biased by a transverse gravitational force: Comparison of analytical and numerical results

We study the crossing time statistic of diffusing point particles between the two ends of expanding and narrowing two-dimensional conical channels under a transverse external gravitational field. The theoretical expression for the mean first-passage time for such a system is derived under the assumption that the axial diffusion in a two-dimensional channel of smoothly varying geometry can be approximately described as a one-dimensional diffusion in an entropic potential with position-dependent effective diffusivity in terms of the modified Fick-Jacobs equation. We analyze the channel crossing dynamics in terms of the mean first-passage time, combining our analytical results with extensive two-dimensional Brownian dynamics simulations, allowing us to find the range of applicability of the one-dimensional approximation. We find that the effective particle diffusivity decreases with increasing amplitude of the external potential. Remarkably, the mean first-passage time for crossing the channel is shown to assume a minimum at finite values of the potential amplitude.

Geometric Brownian information engine: Upper bound of the achievable work under feedback control

We design a geometric Brownian information engine by considering overdamped Brownian particles inside a two-dimensional monolobal confinement with irregular width along the transport direction. Under such detention, particles experience an effective entropic potential which has a logarithmic form. We employ a feedback control protocol as an outcome of error-free position measurement. The protocol comprises three stages: measurement, feedback, and relaxation. We reposition the center of the confinement to the measurement distance (x_p) instantaneously when the position of the trapped particle crosses x_p for the first time. Then, the particle is allowed for thermal relaxation. We calculate the extractable work, total information, and unavailable information associated with the feedback control using this equilibrium probability distribution function. We find the exact analytical value of the upper bound of extractable work as (53-2ln2)k_BT. We introduce a constant force G downward to the transverse coordinate (y). A change in G alters the effective potential of the system and tunes the relative dominance of entropic and energetic contributions in it. The upper bound of the achievable work shows a crossover from (53-2ln2)k_BT to 12k_BT when the system changes from an entropy-dominated regime to an energy-dominated one. Compared to an energetic analog, the loss of information during the relaxation process is higher in the entropy-dominated region, which accredits the less value in achievable work. Theoretical predictions are in good agreement with the Langevin dynamics simulation studies.

Two-dimensional diffusion biased by a transverse gravitational force in an asymmetric channel: Reduction to an effective one-dimensional description

We focus on the derivation of a general position-dependent effective diffusion coefficient to describe two-dimensional (2D) diffusion in a narrow and smoothly asymmetric channel of varying width under a transverse gravitational external field, a generalization of the symmetric channel case using the projection method introduced earlier by Kalinay and Percus [P. Kalinay and J. K. Percus, J. Chem. Phys. 122, 204701 (2005)10.1063/1.1899150]. To this end, we project the 2D Smoluchowski equation into an effective one-dimensional generalized Fick-Jacobs equation in the presence of constant force in the transverse direction. The expression for the diffusion coefficient given in Eq. (34) is our main result. This expression is a more general effective diffusion coefficient for narrow 2D channels in the presence of constant transverse force, which contains the well-known previous results for a symmetric channel obtained by Kalinay, as well as the limiting cases when the transverse gravitational external field goes to zero and infinity. Finally, we show that diffusivity can be described by the interpolation formula proposed by Kalinay, D_{0}/[1+(1/4)w^{'2}(x)]^{-η}, where spatial confinement, asymmetry, and the presence of a constant transverse force can be encoded in η, which is a function of channel width (w), channel centerline, and transverse force. The interpolation formula also reduces to well-known previous results, namely, those obtained by Reguera and Rubi [D. Reguera and J. M. Rubi, Phys. Rev. E 64, 061106 (2001)10.1103/PhysRevE.64.061106] and by Kalinay [P. Kalinay, Phys. Rev. E 84, 011118 (2011)10.1103/PhysRevE.84.011118].

Dimensional reduction of a general advection-diffusion equation in 2D channels

Diffusion of point-like particles in a two-dimensional channel of varying width is studied. The particles are driven by an arbitrary space dependent force. We construct a general recurrence procedure mapping the corresponding two-dimensional advection-diffusion equation onto the longitudinal coordinate x. Unlike the previous specific cases, the presented procedure enables us to find the one-dimensional description of the confined diffusion even for non-conservative (vortex) forces, e.g. caused by flowing solvent dragging the particles. We show that the result is again the generalized Fick-Jacobs equation. Despite of non existing scalar potential in the case of vortex forces, the effective one-dimensional scalar potential, as well as the corresponding quasi-equilibrium and the effective diffusion coefficient [Formula: see text] can be always found.

One-dimensional description of driven diffusion in periodic channels

Diffusion of point-like particles driven by a constant longitudinal force in two-dimensional channels of periodically varying width is studied. The dynamics of such systems can be effectively described by the one-dimensional Smoluchowski(-Fick-Jacobs) equation in the longitudinal coordinate x, extended by a space dependent effective diffusion coefficient D(x). Our paper is focused on calculation of this function for an arbitrary channel shaping function h(x). Unlike the previous algorithms based on scaling of the transverse lengths, the method presented here uses periodicity of the channel. Instead of complicated expansion containing higher order derivatives of h(x), the proposed algorithm results in an integral formula for D(x), enabling us to study the system for wide range of the driving force and various (periodic) shaping functions h(x).

Landauer's blow-torch effect in systems with entropic potential

We consider local heating of a part of a two-dimensional bilobal enclosure of a varying cross section confining a system of overdamped Brownian particles. Since varying cross section in higher dimension results in an entropic potential in lower dimension, local heating alters the relative stability of the entropic states. We show that this blow-torch effect modifies the entropic potential in a significant way so that the resultant effective entropic potential carries both the features of variation of width of the confinement and variation of temperature along the direction of transport. The reduced probability distribution along the direction of transport calculated by full numerical simulations in two dimensions agrees well with our analytical findings. The extent of population transfer in the steady state quantified in terms of the integrated probability of residence of the particles in either of the two lobes exhibits interesting variation with the mean position of the heated region. Our study reveals that heating around two particular zones of a given lobe maximizes population transfer to the other.

Capturing the Landauer bound through the application of a detailed Jarzynski equality for entropic memory erasure

The states of an overdamped Brownian particle confined in a two-dimensional bilobal enclosure are considered to correspond to two binary values: 0 (left lobe) and 1 (right lobe). An ensemble of such particles represents bits of entropic information. An external bias is applied on the particles, equally distributed in two lobes, to drive them to a particular lobe erasing one kind of bit of information. It has been shown that the average work done for the entropic memory erasure process approaches the Landauer bound for a very slow erasure cycle. Furthermore, the detailed Jarzynski equality holds to a very good extent for the erasure protocol, so that the Landauer bound may be calculated irrespective of the time period of the erasure cycle in terms of the effective free-energy change for the process. The detailed Jarzynski equality applied to two subprocesses, namely the transition from entropic memory state 0 to state 1 and the transition from entropic memory state 1 to state 1, connects the work done on the system to the probability to occupy the two states under a time-reversed process. In the entire treatment, the work appears as a boundary effect of the physical confinement of the system not having a conventional potential energy barrier. Finally, an analytical derivation of the detailed and classical Jarzynski equality for Brownian movement in confined space with varying width has been proposed. Our analytical scheme supports the numerical simulations presented in this paper.

Asymptotic solution of the diffusion equation in slender impermeable tubes of revolution. I. The leading-term approximation

The anisotropic 3D equation describing the pointlike particles diffusion in slender impermeable tubes of revolution with cross section smoothly depending on the longitudinal coordinate is the object of our study. We use singular perturbations approach to find the rigorous asymptotic expression for the local particles concentration as an expansion in the ratio of the characteristic transversal and longitudinal diffusion relaxation times. The corresponding leading-term approximation is a generalization of well-known Fick-Jacobs approximation. This result allowed us to delineate the conditions on temporal and spatial scales under which the Fick-Jacobs approximation is valid. A striking analogy between solution of our problem and the method of inner-outer expansions for low Knudsen numbers gas kinetic theory is established. With the aid of this analogy we clarify the physical and mathematical meaning of the obtained results.

Rectification of confined diffusion driven by a sinusoidal force

A particle diffusing in an asymmetric periodic channel, driven by a sinusoidal force F(t)=F0cosωt (the rocking ratchet) is considered. The asymptotic solution of the generalized Fick-Jacobs equation describing the system is studied in the nonadiabatic regime. The leading term of the rectified current, appearing in the order ∼F02, is derived. The method presented enables us to solve the problem analytically for a sawtooth channel and also to look for approximative formulas applicable in a wide range of frequencies ω. Even the simplest approximation qualitatively reproduces the current reversal at higher frequencies as the result of growing phase lag of the rocking density behind the driving force.

Control of logic gates by dichotomous noise in energetic and entropic systems

We consider the stochastic response of a nonlinear dynamical system towards a combination of input signals. The response can assume binary values if the state of the system is considered to be the output and the system can make transitions between two states separated by an energetic or entropic barrier. We show how the input-output correspondence can be controlled by an external exponentially correlated dichotomous noise optimizing the logical response which exhibits a maximum at an intermediate value of correlation time. This resonance manifests itself as a "logical" resonance correlation effect and sets the condition for performance of the stochastic system as a logic gate. The role of asymmetry of the dichotomous noise is examined and the results on numerical simulations are correlated with a two-state model using a master equation approach.

When is the next extending of Fick-Jacobs equation necessary?

Applicability of the effective one-dimensional equations, such as Fick-Jacobs equation and its extensions, describing diffusion of particles in 2D or 3D channels with varying cross section A(x) along the longitudinal coordinate x, is studied. The leading nonstationary correction to Zwanzig-Reguera-Rubí equation [R. Zwanzig, J. Phys. Chem. 96, 3926 (1992); D. Reguera and J. M. Rubí, Phys. Rev. E 64, 061106 (2001)] is derived and tested on the exactly solvable model, diffusion in a 2D linear cone. The effects of such correction are demonstrated and discussed on elementary nonstationary processes, a time dependent perturbation of the stationary flow and calculation of the mean first passage time.

Current control in a two-dimensional channel with nonstraight midline and varying width

Transport of overdamped Brownian particles in a two-dimensional channel with nonstraight midline and narrow varying width is investigated in the presence of an asymmetric unbiased external force. In the adiabatic limit, we obtain the analytical expression of the directed current. It is found that the current is manipulated by changing the phase shift between the top and bottom walls of the channel. As the phase shift is increased from 0 to π, the variation of the channel width decreases and the current also decreases. Remarkably, the current is always zero when the phase shift is equal to π, where the entropic barrier disappears. In addition, the temporal asymmetric parameter of the unbiased force not only determines the direction of the current but also affects its amplitude.

Logic gates for entropic transport

We consider a Brownian particle that is confined in a two-dimensional enclosure and driven by a combination of input signals. It has been shown that the logic gates can be formed by considering the state of the particle experiencing an entropic barrier as the output signal. For a consistent logical output, it is necessary to optimize the strength of the noise driving the particle for a given system size. The variation of the logical output behavior exhibits a turnover at an optimal value of system size parameter, implying a size resonance condition in entropic transport. The role of a transverse bias field used to tune the transport between the entropy dominated regime and the energy dominated regime is elucidated.

Effective one-dimensional description of confined diffusion biased by a transverse gravitational force

Diffusion of pointlike noninteracting particles in a two-dimensional channel of varying cross section is considered. The particles are biased by a constant force in the transverse direction. A recurrence mapping procedure is applied, which enables the derivation of an effective one-dimensional (1D) evolution equation that governs the 1D density of the particles in the channel. In the limit of stationary flow, an extended Fick-Jacobs equation is reached, which is corrected by an effective diffusion coefficient D(x) that depends on the longitudinal coordinate x. The result is an approximate formula for D(x) that also involves the influence of the transverse force. The calculations are verified by the stationary diffusion in a linear cone, which is exactly solvable.

Analytical treatment of biased diffusion in tubes with periodic dead ends

Effective mobility and diffusion coefficient of a particle in a tube with identical periodic dead ends characterize the motion on large time scale, when the particle displacement significantly exceeds the tube period. We derive formulas that show how these transport coefficients depend on the driving force and the geometric parameters of the system. Numerical tests show that values of the transport coefficients obtained from Brownian dynamics simulations are in excellent agreement with our theoretical predictions.

Communication: Turnover behavior of effective mobility in a tube with periodic entropy potential

Using Brownian dynamics simulations, we study the effective mobility and diffusion coefficient of a point particle in a tube formed from identical compartments of varying diameter, as functions of the driving force applied along the tube axis. Our primary focus is on how the driving force dependences of these transport coefficients are modified by the changes in the compartment shape. In addition to monotonically increasing or decreasing behavior of the effective mobility in periodic entropy potentials reported earlier, we now show that the effective mobility can even be nonmonotonic in the driving force.

Mapping of diffusion in a channel with soft walls

We study diffusion of pointlike particles biased toward the x axis by a quadratic potential U(x,y)=κ(x)y². This system mimics a channel with soft walls of some varying (effective) cross section A(x), depending on the stiffness κ(x). We show that diffusion in this geometry can also be mapped rigorously onto the longitudinal coordinate x by a procedure known for channels with hard walls [P. Kalinay and J. K. Percus, Phys. Rev. E 74, 041203 (2006)]; i.e., we arrive at a one-dimensional evolution equation of the Fick-Jacobs type. On the other hand, the calculation presented serves as a prototype for mapping of the Smoluchowski equation with a wide class of potentials U(x,y) varying in both the longitudinal as well as the transverse directions, which is necessary for understanding, e.g., stochastic resonance.

Biased diffusion in tubes formed by spherical compartments

We study the effect of the driving force on brownian motion of a point particle in a tube formed by identical spherical compartments, which create periodic entropy potential for the motion along the tube axis. The focus is on (i) the effective mobility and diffusion coefficient of the particle as functions of the driving force, (ii) localization of the particle in the central part of the tube induced by the driving force, and (iii) transit time of the particle between the openings connecting neighboring compartments. Some of the results at very small and large driving force are obtained analytically, while the majority of the results are obtained from brownian dynamics simulations.

Diffusion over an entropic barrier: non-Arrhenius behavior

We have examined the temperature dependence of noise-induced escape rate of brownian particles between two distinct regions in a two-dimensional enclosure through a narrow bottleneck. Varying cross section of the confinement results in an effective entropic barrier in reduced dimension. The strong non-Arrhenius dependence of the escape rate is observed, which can be explained in terms of a crossover between entropy-dominated regime and energy-dominated regime in presence of an interplay of gravitational bias and thermal motion.

Communications: Drift and diffusion in a tube of periodically varying diameter. Driving force induced intermittency

We show that the effect of driving force F on the effective mobility and diffusion coefficient of a particle in a tube formed by identical compartments may be qualitatively different depending on the compartment shape. In tubes formed by cylindrical (spherical) compartments the mobility monotonically decreases (increases) with F and the diffusion coefficient diverges (remains finite) as F tends to infinity. In tubes formed by cylindrical compartments, at large F there is intermittency in the particle transitions between openings connecting neighboring compartments.

Heat conduction in driven Frenkel-Kontorova lattices: thermal pumping and resonance

Heat conduction through the Frenkel-Kontorova chain under the influence of an ac driving force applied locally at one boundary is studied by nonequilibrium molecular dynamics simulations. We observe the occurrence of thermal resonance, namely, there exists a value of the driving frequency at which the heat flux takes its maximum value. The resonance frequency is determined by the dynamical parameters of the model, which has been numerically explored. Remarkably, the heat can be pumped from the low-temperature heat bath to the high temperature one by suitably adjusting the frequency of the ac driving force. By examining effects of the driving amplitude on heat conduction, we show that the amplitude threshold for nonlinear supratransmission is absent when the system is in contact with heat baths, namely, the heat flux smoothly increases with the increasing of amplitude.

Diffusion in a tilted periodic potential with entropic barriers

Diffusion of Brownian particles in a periodic channel is investigated in the presence of a tilted spatially periodic potential. Reduction of spatial dimensionality from two or three dimensions to an effective one-dimensional system entails the appearance of not only an entropic barrier but also an effective diffusion coefficient. It is found that diffusion exhibits striking features which are different from those observed in the previous cases. The interplay between the potential barriers and entropic barriers makes the phenomena richer. Remarkably, two temperature values exist at which the Peclet number takes its maximum.

Mapping of forced diffusion in quasi-one-dimensional systems

Diffusion in an external potential in a two-dimensional channel of varying cross section is considered. We show that a rigorous mapping procedure applied on the corresponding Smoluchowski equation yields a one-dimensional evolution equation of the Fick-Jacobs type corrected by an effective coefficient D(x). The procedure enables us to derive this function within a recurrence scheme. We test this result on a model of stationary diffusion in a linear cone in a homogeneous potential, which is exactly solvable. Extension of the approximate formulas for D(x) derived for the diffusion alone is discussed.

Entropic effects in channel-facilitated transport: interparticle interactions break the flux symmetry

We analyze transport through conical channels that is driven by the difference in particle concentrations on the two sides of the membrane. Because of the detailed balance, fluxes of noninteracting particles through the same channel, inserted into the membrane in two opposite orientations, are equal. We show that this flux symmetry is broken by particle-particle interactions so that one of the orientations can be much more efficient for transport under the same external conditions. The results are obtained analytically using a one-dimensional diffusion model and confirmed by three-dimensional Brownian dynamics simulations.

Pore structure and function of synthetic nanopores with fixed charges: tip shape and rectification properties

We present a complete theoretical study of the relationship between the structure (tip shape and dimensions) and function (selectivity and rectification) of asymmetric nanopores on the basis of previous experimental studies. The theoretical model uses a continuum approach based on the Nernst-Planck equations. According to our results, the nanopore transport properties, such as current-voltage (I-V) characteristics, conductance, rectification ratio, and selectivity, are dictated mainly by the shape of the pore tip (we have distinguished bullet-like, conical, trumpet-like, and hybrid shapes) and the concentration of pore surface charges. As a consequence, the nanopore performance in practical applications will depend not only on the base and tip openings but also on the pore shape. In particular, we show that the pore opening dimensions estimated from the pore conductance can be very different, depending on the pore shape assumed. The results obtained can also be of practical relevance for the design of nanopores, nanopipettes, and nanoelectrodes, where the electrical interactions between the charges attached to the nanostructure and the mobile charges confined in the reduced volume of the inside solution dictate the device performance in practical applications. Because single tracks are the elementary building blocks for nanoporous membranes, the understanding and control of their individual properties should also be crucial in protein separation, water desalination, and bio-molecule detection using arrays of identical nanopores.