Diffusion over an entropic barrier: non-Arrhenius behavior

Particle-wall alignment interaction and active Brownian diffusion through narrow channels

We numerically examine the impacts of particle-wall alignment interactions on active species diffusion through a structureless narrow two-dimensional channel. We consider particle-wall interaction to depend on the self-propulsion velocity direction whereby some specific particle's alignments with respect to the boundary walls are stabilized more. Further, the alignment interaction is meaningful as long as particles are close to the confining boundaries. Unbiased diffusion of active particles for various possible stable velocity alignments against the walls has been examined. We show that for the most stable configuration leading to the self-propulsion velocity direction perpendicular to the wall, diffusivity becomes inversely proportional to the square of the alignment interaction torque. On the other hand, when the self-propulsion velocity direction making an acute angle to the channel walls is the most stable configuration, diffusion exponentially grows with strengthening alignment interaction. Hence, particle-wall interaction plays a pivotal role in the transport control of active particles through narrow channels. Moreover, the impacts of the alignment interactions on diffusion largely depend on the particle's self-propulsion properties and its chirality. Our simulation results can potentially be used to understand unbiased diffusion of artificial or living micro/nano-objects (such as virus, bacteria, Janus particles, etc.) though narrow confined structures.

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 of a diffusing particle under stochastic resetting in bounded domains with spherical symmetry

We investigate the first passage properties of a Brownian particle diffusing freely inside a d-dimensional sphere with absorbing spherical surface subject to stochastic resetting. We derive the mean time to absorption (MTA) as functions of resetting rate γ and initial distance r of the particle to the center of the sphere. We find that when r>r_{c} there exists a nonzero optimal resetting rate γ_{opt} at which the MTA is a minimum, where r_{c}=sqrt[d/(d+4)]R and R is the radius of the sphere. As r increases, γ_{opt} exhibits a continuous transition from zero to nonzero at r=r_{c}. Furthermore, we consider that the particle lies between two two-dimensional or three-dimensional concentric spheres with absorbing boundaries, and obtain the domain in which resetting expedites the MTA, which is (R_{1},r_{c_{1}})∪(r_{c_{2}},R_{2}), with R_{1} and R_{2} being the radii of inner and outer spheres, respectively. Interestingly, when R_{1}/R_{2} is less than a critical value, γ_{opt} exhibits a discontinuous transition at r=r_{c_{1}}; otherwise, such a transition is continuous. However, at r=r_{c_{2}} the transition is always continuous.

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.

Escape kinetics of self-propelled particles from a circular cavity

We numerically investigate the mean exit time of an inertial active Brownian particle from a circular cavity with single or multiple exit windows. Our simulation results witness distinct escape mechanisms depending on the relative amplitudes of the thermal length and self-propulsion length compared to the cavity and pore sizes. For exceedingly large self-propulsion lengths, overdamped active particles diffuse on the cavity surface, and rotational dynamics solely governs the exit process. On the other hand, the escape kinetics of a very weakly damped active particle is largely dictated by bouncing effects on the cavity walls irrespective of the amplitude of self-propulsion persistence lengths. We show that the exit rate can be maximized for an optimal self-propulsion persistence length, which depends on the damping strength, self-propulsion velocity, and cavity size. However, the optimal persistence length is insensitive to the opening windows' size, number, and arrangement. Numerical results have been interpreted analytically based on qualitative arguments. The present analysis aims at understanding the transport controlling mechanism of active matter in confined structures.

Entropic stochastic resonance induced by a transverse driving force

The phenomenon of entropic stochastic resonance (ESR) is investigated with the presence of a time-periodic force in the transverse direction. Simulation results manifest that the ESR can survive even if there is no static bias force in any direction, just if a transverse driving field is applied. In the weak noise region, the transverse driving force leads to a giant-suppression of the escape rate from one well to another, i.e. the entropic trapping. The increase in noise intensity will eliminate this suppression and induce the ESR phenomenon. An alternative quantity, called the mean free flying time, is also proposed to characterize the ESR as well as the conventional spectral power amplification. The ESR can be modulated conveniently by the transverse periodic force, which implies an alternative method for controlling the dynamics of small-scale systems. This article is part of the theme issue 'Vibrational and stochastic resonance in driven nonlinear systems (part 2)'.

Entropic vibrational resonance

We demonstrate the existence of vibrational resonance associated with the presence of an uneven boundary. When the motion of a Brownian particle is confined in a region with an uneven boundary, constrained to a double cavity, a high-frequency signal may produce a peak in the spectral power amplification of the other low-frequency signal and therefore to the appearance of the vibrational resonance phenomenon. The mechanism of vibrational resonance in constrained boundaries is different from that in energetic potentials and is termed entropic vibrational resonance (EVR). The EVR can be observed even if the bias force is absent in any direction. Through careful analysis, we clarify two types of mechanisms of the EVR. The one mechanism is ascribed to the transition from a bistable system to a monostable system, and the other corresponds to the match between the escape rate and the natural frequency of the low-frequency signal. Our work merges the vibrational resonance with an uneven boundary, thus extending the scope of the vibrational resonance and shedding new light on the concept of resonance.

Field-driven tracer diffusion through curved bottlenecks: fine structure of first passage events

Using scaling arguments and extensive numerical simulations, we study the dynamics of a tracer particle in a corrugated channel represented by a periodic sequence of broad chambers and narrow funnel-like bottlenecks enclosed by a hard-wall boundary.

Brownian dynamics of self-regulated particles with additional degrees of freedom: Symmetry breaking and homochirality

We consider the Brownian motion of a collection of particles each with an additional degree of freedom. The degree of freedom of a particle (or, in general, a molecule) can assume distinct values corresponding to certain states or conformations. The time evolution of the additional degree of freedom of a particle is guided by those of its neighbors as well as the temperature of the system. We show that the local averaging over these degrees of freedom results in emergence of a collective order in the dynamics in the form of selection or dominance of one of the isomers leading to a symmetry-broken state. Our statistical model captures the basic features of homochirality, e.g., autocatalysis and chiral inhibition.

Composition- and temperature-dependent liquid structures in Al-Cu alloys: an ab initio molecular dynamics and x-ray diffraction study

The composition- and temperature-dependent liquid structures in eight Al_rich-Cu binary alloys (from hypoeutectic Al₉₃Cu₇ to hypereutectic Al₇₀Cu₃₀) have been experimentally and computationally studied by x-ray diffraction (XRD) experiments and ab initio molecular dynamics (AIMD) simulations. The remarkable agreements of structure factors for all liquid Al_rich-Cu alloys obtained from high-temperature high-energy XRD measurements and AIMD simulations have been achieved, which consolidates the analyses of structural evolutions in Al_rich-Cu liquids during the cooling processing by AIMD simulations. The heat capacity of liquid Al_rich-Cu alloys continuously increases and presents no abnormal peak when reducing the temperature, which differs from the reported prediction for 55-atom Al_rich-Cu nanoliquids. The diffusivities of Al and Cu undergo an increasing deviation from Arrhenius behavior by tuning Cu concentration from 7 to 30 atomic percentages, correlated to the local ordering in these liquids by means of coordination number, bond-angle distribution, Honeycutt-Andersen index, bond-orientational order and Voronoi tessellation analyses. Upon cooling, the microstructure of the liquid Al_rich-Cu alloys inclines to form Al₂Cu crystal-like local atomic ordering, especially in the hypereutectic liquids. The favorable short-range ordering between Cu and Al atoms could cause the non-Arrhenius diffusion behavior.

Stochastic resonance during a polymer translocation process

We have studied the occurrence of stochastic resonance when a flexible polymer chain undergoes a single-file translocation through a nano-pore separating two spherical cavities, under a time-periodic external driving force. The translocation of the chain is controlled by a free energy barrier determined by chain length, pore length, pore-polymer interaction, and confinement inside the donor and receiver cavities. The external driving force is characterized by a frequency and amplitude. By combining the Fokker-Planck formalism for polymer translocation and a two-state model for stochastic resonance, we have derived analytical formulas for criteria for emergence of stochastic resonance during polymer translocation. We show that no stochastic resonance is possible if the free energy barrier for polymer translocation is purely entropic in nature. The polymer chain exhibits stochastic resonance only in the presence of an energy threshold in terms of polymer-pore interactions. Once stochastic resonance is feasible, the chain entropy controls the optimal synchronization conditions significantly.

Ratchet rectification effect on the translocation of a flexible polyelectrolyte chain

We report a three dimensional Langevin dynamics simulation of a uniformly charged flexible polyelectrolyte chain, translocating through an asymmetric narrow channel with periodically varying cross sections under the influence of a periodic external electric field. When reflection symmetry of the channel is broken, a rectification effect is observed with a favored direction for the chain translocation. For a given volume of the channel unit and polymer length, the rectification occurs below a threshold frequency of the external periodic driving force. We have also observed that the extent of the rectification varies non-monotonically with increasing molecular weight and the strength of geometric asymmetry of the channel. Observed non-monotonicity of the rectification performance has been interpreted in terms of a competition between two effects arising from the channel asymmetry and change in conformational entropy. An analytical model is presented with predictions consistent with the simulation results.

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.

Entropic memory erasure

We have considered a Brownian particle confined in a two-dimensional bilobal enclosure where the state of the particle represents a bit of information having binary value 0 (left lobe) or 1 (right lobe). A time linear force is applied on the particle, driving it selectively to a particular lobe, and thus erasing one bit of information. We explore the statistics of heat and work associated with memory erasure to realize the Landauer limit in the entropic domain. Our results suggest that the mean value of work done associated with the complete erasure procedure satisfies the Landauer bound even when the memory is purely entropic in nature.

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.

Analytical estimates of free brownian diffusion times in corrugated narrow channels

The diffusion of a suspended brownian particle along a sinusoidally corrugated narrow channel is investigated to assess the validity of two competing analytical schemes, both based on effective one-dimensional kinetic equations, one continuous (entropic channel scheme) and the other discrete (random walker scheme). For narrow pores, the characteristic diffusion time scale is represented by the mean first exit time out of a channel compartment. Such a diffusion time has been analytically calculated in both approximate schemes; the two analytical results coincide in leading order and are in excellent agreement with the simulation data.

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.

Brownian transport in corrugated channels with inertia

Transport of suspended Brownian particles dc driven along corrugated narrow channels is numerically investigated in the regime of finite damping. We show that inertial corrections cannot be neglected as long as the width of the channel bottlenecks is smaller than an appropriate particle diffusion length, which depends on the the channel corrugation and the drive intensity. With such a diffusion length being inversely proportional to the damping constant, transport through sufficiently narrow obstructions turns out to be always sensitive to the viscosity of the suspension fluid. The inertia corrections to the transport quantifiers, mobility, and diffusivity markedly differ for smoothly and sharply corrugated channels.

Entropic dynamical hysteresis in a driven system

We show that the application of a time periodic field driving a Brownian particle between the two lobes of a two-dimensional bilobal enclosure results in a hysteresis loop in the variation of integrated probability of residence of the particle as a function of the field. The confinement of the particle is characterized by symmetry breaking of the hysteresis loop, and the area of the loop exhibits a turnover with variation of frequency of the field. This dynamical hysteresis is geometry controlled, entropic in nature, and amenable to theoretical analysis with a two-state model.

Driven Brownian transport through arrays of symmetric obstacles

We numerically investigate the transport of a suspended overdamped Brownian particle which is driven through a two-dimensional rectangular array of circular obstacles with finite radius. Two limiting cases are considered in detail, namely, when the constant drive is parallel to the principal or the diagonal array axes. This corresponds to studying the Brownian transport in periodic channels with reflecting walls of different topologies. The mobility and diffusivity of the transported particles in such channels are determined as functions of the drive and the array geometric parameters. Prominent transport features, like negative differential mobilities, excess diffusion peaks, and unconventional asymptotic behaviors, are explained in terms of two distinct lengths, the size of single obstacles (trapping length), and the lattice constant of the array (local correlation length). Local correlation effects are further analyzed by continuously rotating the drive between the two limiting orientations.

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.

Geometric stochastic resonance in a double cavity

Geometric stochastic resonance of particles diffusing across a porous membrane subject to oscillating forces is characterized as a synchronization process. Noninteracting particle currents through a symmetric membrane pore are driven either perpendicular or parallel to the membrane, whereas, harmonic-mixing spectral current components are generated by the combined action of perpendicular and parallel drives. In view of potential applications to the transport of colloids and biological molecules through narrow pores, we also consider the role of particle repulsion as a controlling factor.

Enhancement of entropic transport by intermediates

Brownian particles confined in a two-dimensional enclosure that give rise to a bistable entropic potential are considered. With the introduction of an intermediate lobe, the mean first passage time from one lobe to another through the intermediate shows a turnover behavior with the variation of the stability of the entropic intermediate. The mean escape time shows a minimum for an optimal value of the barrier height of the intermediate state. A three-state model is proposed to explain the nonmonotonic behavior of the entropic transport.

Entropic particle transport: higher-order corrections to the Fick-Jacobs diffusion equation

Transport of point-size Brownian particles under the influence of a constant and uniform force field through a planar three-dimensional channel with smoothly varying, axis-symmetric periodic side walls is investigated. Here we employ an asymptotic analysis in the ratio between the difference of the widest and the most narrow constriction divided through the period length of the channel geometry. We demonstrate that the leading-order term is equivalent to the Fick-Jacobs approximation. By use of the higher-order corrections to the probability density we show that in the diffusion-dominated regime the average transport velocity is obtained as the product of the zeroth-order Fick-Jacobs result and the expectation value of the spatially dependent diffusion coefficient D(x), which substitutes the constant diffusion coefficient in the common Fick-Jacobs equation. The analytic findings are corroborated with the precise numerical results of a finite element calculation of the Smoluchowski diffusive particle dynamics occurring in a reflection symmetric sinusoidal-shaped channel.

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.