Ratchet-induced variations in bulk states of an active ideal gas

Ratchet-induced variations in bulk states of an active ideal gas

Jeroen Rodenburg a.j.rodenburg@uu.nl Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands    Siddharth Paliwal Soft Condensed Matter Group, Debye Institute for Nanomaterials Science, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands    Marjolein de Jager Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands    Peter G. Bolhuis Van ’t Hoff Institute for Molecular Sciences, University of Amsterdam, P.O. Box 94157, 1090 GD Amsterdam, The Netherlands    Marjolein Dijkstra Soft Condensed Matter Group, Debye Institute for Nanomaterials Science, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands    René van Roij r.vanroij@uu.nl Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands
Abstract

We study the distribution of active, noninteracting particles over two bulk states separated by a ratchet potential. By solving the steady-state Smoluchowski equations in a flux-free setting, we show that the ratchet potential affects the distribution of particles over the bulks, and thus exerts an influence of infinitely long range. As we show, crucial for having such a long-range influence is an external potential that is nonlinear. We characterize how the difference in bulk densities depends on activity and on the ratchet potential, and we identify power law dependencies on system parameters in several limiting cases. While weakly active systems are often understood in terms of an effective temperature, we present an analytical solution that explicitly shows that this is not possible in the current setting. Instead, we rationalize our results by a simple transition state model, that presumes particles to cross the potential barrier by Arrhenius rates modified for activity. While this model does not quantitatively describe the difference in bulk densities for feasible parameter values, it does reproduce - in its regime of applicability - the complete power law behavior correctly.

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I Introduction

Over the last few years, active matter has emerged as a testing ground for nonequilibrium statistical physics Pietzonka, Kleinbeck, and Seifert (2016); Pietzonka and Seifert (2017); Solon et al. (2018a); Takatori and Brady (2016); Menzel et al. (2016); Liluashvili, Ónody, and Voigtmann (2017); Clewett et al. (2016); Krinninger, Schmidt, and Brader (2016). Its relevance comes from the fact that experimental realizations exist Bechinger et al. (2016); Schwarz-Linek et al. (2016); Drescher et al. (2011) of relatively simple active matter models, such as active Brownian particles (ABPs) and run-and-tumble (RnT) particles Solon, Cates, and Tailleur (2015). While describing these systems can be very challenging when they are far from thermodynamic equilibrium Solon et al. (2018b); Dijkstra et al. (2018), for small activity they are well understood by effective equilibrium approaches Farage, Krinninger, and Brader (2015); Rein and Speck (2016); Marconi and Maggi (2015); Marconi, Paoluzzi, and Maggi (2016); Trefz et al. (2016). In particular, it is well established that noninteracting particles at small activity can be described as an equilibrium system at an effective temperature Loi, Mossa, and Cugliandolo (2008); Wang and Wolynes (2011); Marconi and Maggi (2015); Fily and Marchetti (2012); Szamel (2014). For example, inserting the effective temperature in the Einstein relation yields the enhanced diffusion coefficient of an active particle, and using the effective temperature in the Boltzmann distribution gives the distribution of weakly active particles in a gravitational field Palacci et al. (2010); Enculescu and Stark (2011); Ginot et al. (2015); Wolff, Hahn, and Stark (2013); Szamel (2014); Solon, Cates, and Tailleur (2015); Ginot et al. (2015); Stark (2016).

\includegraphics

[width=]SawtoothPotential.pdf

Figure 1: (Dimensionless) ratchet potential , as a function of the Cartesian -coordinate in units of the diffusive length scale . The ratchet can be characterized by its height , the width of its left side , and its asymmetry .

However, even weakly active systems can display behavior very different from equilibrium systems Lindner et al. (1999); Galajda et al. (2007); Angelani, Constanzo, and Di Leonardo (2011); Reichhardt and Reichhardt (2016); McDermott, Reichhardt, and Reichhardt (2016); Nikola et al. (2016); Ai and Li (2017). For instance, a single array of funnel-shaped barriers, that is more easily crossed from one lateral direction than from the other, can induce a steady state with ratchet currents that span the entire system Reichhardt and Reichhardt (2016). Alternatively, when the boundary conditions deny such a system-wide flux, the result is a steady state with a higher density on one side of the array than on the other Galajda et al. (2007); Reichhardt and Reichhardt (2016). As the system can be arbitrarily long in the lateral direction, the presence of the funnels influences the density profile at arbitrarily large distance.
Needless to say, characterizing such a long-range effect is a challenge, and the natural place to start is in a setting as simple as possible. As we shall show, having an external potential with a long-range influence on the density profile in steady state is only possible with the key ingredients of (1) activity, and (2) an external potential that is nonlinear. Therefore, a good candidate for a minimal model is to study the distribution of active particles over two bulks separated by a potential barrier that is only piecewise linear. Here, we focus on a sawtooth-shaped barrier, known as a ratchet potential (see Fig. 1). As we will see, the asymmetry of the ratchet induces a flux-free steady state with different densities in both bulks. Since the bulk sizes can be arbitrarily large, the influence of the ratchet potential is indeed of infinite range. This system has actually already been studied, both experimentally Koumakis et al. (2013) and theoretically Koumakis, Maggi, and Di Leonardo (2014). However, the former study was performed at high degree of activity, and the latter study neglected Brownian fluctuations, such that the degree of activity could not be quantified. Thereby, the regime of weak activity, where the statistical physics generally seems best understood Farage, Krinninger, and Brader (2015); Rein and Speck (2016); Marconi and Maggi (2015); Marconi, Paoluzzi, and Maggi (2016); Trefz et al. (2016); Dijkstra et al. (2018), remains largely unexplored.
In this work, we study the long-range influence of the external potential with as few complications as possible. To this end, we investigate how a ratchet potential affects active particles that also undergo translational Brownian motion, such that the degree of activity can be quantified. We ask the questions: can we characterize how the external potential influences the density distribution as a function of activity? And can we understand this distribution in the limit of weak activity?

The article is organized as follows. In section II, we introduce two active particle models, as well as the ratchet potential. In section III, we numerically solve the density and polarization profiles of these active particles in the ratchet potential, and we study how the difference in bulk densities depends on activity, and on the ratchet potential. In section IV, we specialize to the limit of weak activity, and provide an analytical solution that explicitly shows that the nonzero difference in bulk densities cannot be understood by the use of an effective temperature. Instead, in section V, we propose to understand the density difference in terms of a simple transition state model. We end with a discussion, in section VI, on what ingredients are necessary to have the external potential affect the densities in such a (highly) nonlocal way, and with concluding remarks in section VII.

Ii Models

ii.1 2D ABPs

In order to investigate the behavior of active particles in a ratchet potential, we consider the widely employed model of active Brownian particlesRomanczuk et al. (2012) (ABPs) in two dimensions. For simplicity, we consider spherical, noninteracting particles. Every particle is represented by its position , where and are Cartesian unit vectors and is time, as well as by its orientation . Its time evolution is governed by the overdamped Langevin equations

(1a)
(1b)

Eq. (1a) expresses that a particle’s position changes in response to (i) a propulsion force, that acts in the direction of , and that gives rise to a propulsion speed , (ii) an external force, generated by the external potential , and (iii) the unit-variance Wiener process , that gives rise to translational diffusion with diffusion coefficient . Here is the friction coefficient. Note that is an inverse energy scale, and that in thermodynamic equilibrium the Einstein relation implies , where is the Boltzmann constant and the temperature. Eq. (1b) expresses that the orientation of a particle changes due to the unit-variance Wiener process , which leads to rotational diffusion with diffusion coefficient .
The stochastic Langevin equations (1) induce a probability density , whose time evolution follows the Smoluchowski equation

(2)

Here is the two-dimensional spatial gradient operator. Two useful functions to characterize the probability density are the density and the polarization . Their time-evolutions follow from the Smoluchowski equation (2) as

(3)

where is the identity matrix, and where is the nematic alignment tensor. Due to the appearance of , Eqs. (3) are not closed. Therefore, solving Eqs. (3), rather than the full Smoluchowski Eq. (2), requires a closure, an example of which we discuss in section II.2.
We consider a planar geometry that is invariant in the -direction, i.e. , such that , , etc. The geometry consists of two bulks, located at and . These bulk systems are separated by the ratchet potential

(4)

where and are both positive. This sawtooth-shaped potential is illustrated in Fig. 1. Note that the potential is generally asymmetric, the degree of which is characterized by the asymmetry factor . Without loss of generality, we only consider ratchets for which , such that .

The complete problem is specified by four dimensionless parameters. We use the rotational time , and the diffusive length scale , to obtain the Peclet number

(5)

We caution the reader that the factor is often not in the definition of the Peclet number; it is included here to connect to the model described below.

ii.2 1D RnT

The fact that there is only one nontrivial dimension in the problem suggests a simpler, one-dimensional model with the same physical ingredients. In this model, which we refer to as the 1D Run and Tumble (RnT) model, particles are characterized by a position , as well as by an orientation that points in either the positive or the negative -direction, i.e. . The orientation can flip with probability per unit time. Every particle performs overdamped motion driven by (i) a propulsion force, that acts in the direction of its orientation, (ii) an external force, generated by the ratchet potential (4), and (iii) Brownian motion, with associated diffusion constant . The problem can be specified in terms of probability density functions to find particles with orientation . For our purposes, it is more convenient to consider the density , and polarization . These fields evolve as

(6)

Note the similarity of Eqs. (6) with Eqs. (3) of the 2D ABP model. In fact, if we define the Peclet number for the 1D RnT model as , then supplying the 2D ABP model with the closure maps Eqs. (3) to the 1D RnT model. The mapping is such that if one uses the same values for the dimensionless parameters , , and , then both models yield equal density profiles and polarization profiles . As the closure is exact in the limit of weak activity, i.e. , this mapping is expected to give good agreement between the two models for small values of the Peclet number Pe.

Iii Numerical solutions

iii.1 Density and mean orientation profiles

\includegraphics

[width=]TypicalProfiles.pdf

Figure 2: (a) Density profiles and (b) mean orientation profiles of 2D ABPs, and 1D RnT particles, as indicated, in a ratchet potential of height , width , and asymmetry . The dashed, vertical lines indicate the positions of the barrier peak () and the ratchet sides ( and . The bulk density to the left of the ratchet is . Passive particles () are distributed isotropically (), with a density profile given by the Boltzmann weight . Consequently, the densities and in the bulks on either side of the ratchet are equal. Active particles () display much richer behaviour, with an accumulation of particles at either side of the ratchet, with a mean orientation towards the barrier peak, with a depletion of particles near the top of the ratchet, and with the right bulk density exceeding the left bulk density .

We study steady state solutions of both 2D ABPs and 1D RnT particles in the ratchet potential (4). To find the solutions, for the 2D ABP model we numerically solve Eq. (2) with , whereas for the 1D model we numerically solve Eqs. (6) with . We impose the following three boundary conditions.

  1. To the left of the ratchet, we imagine an infinitely large reservoir that fixes the density to be at , i.e. we impose for the 2D case, and , for the 1D case.

  2. To the right of the ratchet, we assume an isotropic bulk that is thermodynamically large, yet finite, such that its density follows from the solution of the equations. In technical terms, at we impose for the 2D case, and , for the 1D case.

  3. Additionally, for the 2D case we assume periodic boundary conditions, i.e. and for all .

Typical solutions are shown in Fig. 2. The considered ratchet potential, with height , width , and asymmetry , is shown as the dashed line in Fig. 2(a). We consider both a passive system () and an active system (). The resulting density profiles and mean orientation profiles are shown in Fig. 2(a) and Fig. 2(b), respectively. For the passive system, the solution is isotropic (i.e. and everywhere), and given by the Boltzmann weight . One checks that these solutions indeed solve Eqs. (2) and (6) when the propulsion speed equals . Thus, in accordance with this Boltzmann distribution, the density in the passive system is lower in the ratchet region than in the left bulk, and its value in the right bulk satisfies , with the density in the left bulk. This is a necessity in thermodynamic equilibrium, even for interacting systems: the equality of the external potential implies equal densities of the bulks.
For the active case (), the behavior is much richer. Firstly, the solution is anisotropic in the ratchet region, even though the external potential is isotropic. Indeed, Fig. 2(b) shows a mean orientation of particles directed towards the barrier on either side of the ratchet. This is consistent with the finding that active particles tend to align against a constant external force Enculescu and Stark (2011); Ginot et al. (), but is also reminiscent of active particles near a repulsive wall. Indeed, at walls particles tend to accumulate with a mean orientation towards the wall Elgeti and Gompper (2013); Ni, Cohen Stuart, and Bolhuis (2015), and a similar accumulation is displayed by the density profiles of Fig. 2(a) at the ratchet sides and . The overall result is an accumulation of particles at the ratchet sides, a depletion of particles near the center of the ratchet, and, remarkably, a density in the right bulk that is higher than the density in the left bulk.
The fact that the difference in bulk densities is positive is caused by the asymmetry of the ratchet: due to their propulsion force, particles can cross the potential barrier more easily from the shallower, left side than from the steeper, right side. This argument is easily understood in the absence of translational Brownian motion (), i.e. when the only force that makes particles move (apart from the external force) is the propulsion force. Indeed, in this case, one can even think of ratchet potentials whose asymmetry is such that particles can climb it from the shallow side, but not from the steep side Koumakis, Maggi, and Di Leonardo (2014). For such a ratchet potential, all particles eventually end up on the right side of the ratchet, such that clearly the right bulk density exceeds the left bulk density . The effect of having nonzero translational Brownian motion () is that particles always have some probability to climb also the steep side of the ratchet. This leads to a density difference that is smaller than in the case. Yet, as long as the ratchet is asymmetric, the density difference always turns out positive for any positive activity Pe.
We stress that the fact that is actually quite remarkable. The reason is that, whereas the ratchet potential is localized around , the right bulk can be arbitrarily large. Since our results clearly show that the right bulk density is influenced by the ratchet, this means that the range of influence of the external potential is in some sense infinitely large.

\includegraphics

[width=]DensityDifferences.pdf

Figure 3: Normalized density difference as a function of (a) activity Pe, (b) barrier height , (c) barrier width , and (d) barrier asymmetry . Results are shown for both the 2D ABP and 1D RnT models, as indicated. In the limiting cases of small and large values of its arguments, the density difference shows power law behavior. The corresponding exponents are listed in Table 1. Additionally, (a) shows the density difference obtained analytically in the limit of weak activity (see section IV), for the same ratchet parameters as used for the numerical solutions. The analytical and numerical solutions show good agreement up to .
exponent
base limit
numerical
solution
Pe
solution
transition state
model
Pe Pe 2 2 2
Pe -4
3 3
0 0 0
2 2 2
* -3
1 1 1
0 0 0

* depends on Pe. For , this exponent equals .

Table 1: Power laws , for limiting values of the base. Here the base denotes either the activity Pe, the barrier height , the barrier width , or the barrier asymmetry . Exponents were obtained numerically for the 1D RnT and 2D ABP models (yielding consistent exponents), analytically for the case of small activity , and for a simple transition state model. Exponents are shown only in limits where the corresponding solution is applicable.

iii.2 Scaling of the bulk density difference

Next, we examine, one by one, how the density difference depends on activity Pe, the barrier height , the barrier width , and on the barrier asymmetry . The results are shown in Figs. 3(a)-(d), for both the 2D ABP and the 1D RnT models. In all cases, both models give density differences that are quantitatively somewhat different, but qualitatively similar, as they are both consistent with identical power lawsFoo ().
Fig. 3(a) shows the density difference as a function of activity Pe, for two different ratchet potentials. For small Pe, the figure shows that the density difference increases as . For large Pe, the density difference decreases again, to decay to in the limit . The reason for this decrease is that particles with high activity can easily climb either side of the ratchet potential, such that they hardly notice the presence of the barrier at all. As shown by Fig. 3(a), this decay follows the power law . Whereas the prefactors of these power laws are different for the two different ratchet potentials considered, the exponents were found to be independent of the ratchet parameters, which was tested for many more values of , , and .
Fig. 3(b) shows the density difference as a function of the barrier height . The barrier width, , and asymmetry, , are kept fixed, and two levels of activity, and , are considered. For all cases, we find the power law , up to values of the barrier height . Exploring the behavior for large values of the barrier height was numerically not feasible, but the fact that the curves for activity level off for barrier heights seems consistent with the asymptotic behavior for that we shall obtain, in section IV, in the limit of weak activity.
Fig. 3(c) shows the density difference as a function of the width of the left side of the ratchet. Here the barrier height and asymmetry are fixed, at and , respectively, whereas the degree of activity is varied as and . For small barrier widths, i.e. for , the curves show the power law , independent of the activity Pe. For very wide barriers, i.e. for , the curves show power law behavior with an exponent that does depend on the activity Pe. For the smallest degree of activity, , this exponent is found to equal . This scaling, for large widths , will also be obtained analytically in section IV for the case of weak activity.
Finally, Fig. 3(d) shows the density difference as a function of the barrier asymmetry . The barrier height and width are fixed, at and , respectively, and the degree of activity is varied as and . For nearly symmetric ratchets, i.e. for , all curves show , whereas for large asymmetries the curves suggest asymptotic behavior, i.e. . This asymptotic behavior can be understood on physical grounds, as the limit corresponds to a ratchet whose right slope is vertical, a situation that we expect to lead to a finite density difference indeed.
All discussed scalings are summarized in Table 1. Of these, the scaling for small activity can be regarded as trivial. The reason is that, in an expansion of the density difference around , the quadratic term is the first term to be expected on general grounds: (i) Eqs. (2) and (6) are invariant under a simultaneous inversion of the self-propulsion speed () and the orientation (, and hence ), such that the expansion of the density difference contains only even powers of Pe, and (ii) for the passive case (), the density difference equals , such that the zeroth order term is absent. Along a similar reasoning, the scaling for small asymmetry is as expected. However, all other scalings listed in Table 1 cannot be predicted by such general arguments, and are therefore nontrivial findings.
We emphasize that these results have been obtained and verified by multiple approaches independently. While the presented results have been obtained by numerically solving the differential equations (2) and (6) as explained above, both the 2D ABP model and the 1D RnT model were also solved by separate approaches. For the 2D ABP model, results were additionally obtained by numerically integrating the Langevin equations (1) in particle-based computer simulations. For the 1D RnT model, results were also obtained by solving a lattice model, where particles can hop to neighbouring lattice sites, and change their orientation, with probabilities that reflect the same physical processes of self-propulsion, external forcing, translational Brownian motion, and tumblingde Jager (2017). For both the 2D ABP and the 1D RnT model, the two alternative approaches showed full agreement with the presented results.

Iv Weak activity limit

Having characterized how the ratchet potential influences the densities of the adjoining bulks, we now turn to the question whether we can better understand this long-range influence. We first try to answer this question for the simplest case possible, and therefore focus on the limit of weak activity, i.e. . Recall that in this limit the 2D ABP model and the 1D RnT model are equivalent. In this section, we present an analytical solution for the limit. In the next section, we propose to rationalize its results by a simple transition state model, that is valid for, but not limited to, weak activity.

\includegraphics

[width=]SmallPeSolutions.pdf

Figure 4: (a) Normalized polarization profiles and (b) deviations of the density from the passive solution , for a ratchet potential of height , width , and asymmetry . The dashed, vertical lines indicate the positions of the barrier peak () and the ratchet sides ( and . Results are shown for the analytical Pe solution, and for the numerical solutions to the 1D RnT model, for activity levels Pe and . The polarizations and density deviations are divided by Pe and Pe, respectively, such that the curves for the analytical solution are independent of Pe.

In case of a small propulsion force, i.e. of , the density can be expanded as , and the polarization as . Here , and are assumed to be independent of Pe. We used the arguments that the density is an even function of Pe, and the polarization an odd function of Pe, as explained in section III.2. With these expansions, Eqs. (6) can be solved perturbatively in Pe, separately for each region where the ratchet potential (4) is defined. The solutions within one region are

(7)

Here we defined the non-dimensionalized external force , such that outside the barrier, on the left slope of the barrier, and on its right slope. Furthermore, we defined . The integration constants and are found separately for each region, by applying the boundary conditions , , and the appropriate continuity conditions at the region boundaries , and . Applying these conditions to the solutions in Eq. (IV) shows that the leading order solution is given by the Boltzmann weight, i.e. for all . Clearly, this is the correct passive solution. The higher order solutions that follow, i.e. the polarization profile and the density correction , are plotted in Fig. 4. Qualitatively, these plots show the same features as displayed by the numerical solutions in Fig. 2: an accumulation of particles facing the barrier at the ratchet sides and , and a right bulk density that exceeds the left bulk density . To allow for a quantitative comparison, Fig. 4 also shows polarization profiles and density corrections that were obtained for the 1D RnT model numerically. While the ratchet potential is fixed, with barrier height , width , and asymmetry , the comparison is made for several degrees of activity, namely . The analytical and numerical results show good agreement for , reasonable agreement for , and deviate significantly for . All of these observations are as expected, since the analytical solutions (IV) are obtained under the assumption .
The most interesting part of solution (IV) is the density correction , as this correction contains the leading order contribution to the difference in bulk densities . To gain some understanding for the meaning of the various terms contributing to , we point out that for small activity, i.e. for , active particles are often understood as passive particles at an effective temperature Loi, Mossa, and Cugliandolo (2008); Wang and Wolynes (2011); Marconi and Maggi (2015); Fily and Marchetti (2012); Palacci et al. (2010); Szamel (2014). In our convention, this effective temperature reads . Therefore, one might think that for our weakly active system the density profile is given by Boltzmann weight at this effective temperature, i.e. by within one region. Here the prefactor can depend on the activity Pe. Expanding this effective Boltzmann weight for small Pe yields the passive solution , and the terms on the first line of in Eq. (IV). However, it does not reproduce the final two terms that contribute to in Eq. (IV). Precisely these last two terms are crucial to obtain a nonzero difference in bulk densities. Indeed, a density profile given solely by the effective Boltzmann weight necessarily yields equal bulk densities , as the external potential is equal on either side of the ratchet.

\includegraphics

[width=]AnalyticalDensityDifferences.pdf

Figure 5: Normalized leading order coefficient in the expansion of the density difference for small activity Pe, as found from the analytical solution and as predicted by the transition state model, (a) as a function of the barrier height , at fixed barrier width and asymmetry , (b) as a function of the barrier width , at fixed barrier height and asymmetry , and (c) as a function of the asymmetry , at fixed barrier height and barrier width . The power laws shown by the transition state model in its regime of applicability, i.e. for and , have exponents that agree with the power laws of the analytical solution. These exponents can be found in Table 1. The analytical and transition state solution do not agree quantitavely for these parameter values.

The analytical expression for the difference in bulk densities , implied by the solutions (IV), is rather lengthy and intransparent, and is therefore not shown here. Instead, we show the dependence of on the activity Pe graphically, in Fig. 3(a), for the same two ratchet potentials as used for the numerical solutions. As the density difference follows from the correction , it scales as , just like the numerical solutions for Pe . As shown by Fig. 3(a), the analytical and numerical solutions agree quantitatively up to , as also found in Fig. 4. Before we illustrate how the density difference depends on the ratchet potential, we extract its dependence on activity Pe by considering , i.e. the leading order coefficient in an expansion of around . The coefficient is independent of Pe, but still depends on the barrier height , the barrier width , and the asymmetry . Its dependence on these ratchet parameters is plotted in Figs. 5(a)-(c), respectively. These figures display all the power law behavior that was obtained numerically in section III. The power laws are summarized in Table 1.

V Transition State Model

As argued in the previous section, the nonzero difference in bulk densities cannot be accounted for by the effective temperature that is often employed in the weak activity limit. Instead, to understand the behavior of the bulk density difference better, we propose the following simple transition state model. The model consists of four states, designed to mimic the 1D RnT model in a minimal way. Particles in the bulk to the left of the ratchet, with an orientation in the positive (negative) -direction, are said to be in state , whereas particles in the bulk to the right of the ratchet, with positive (negative) -orientation, are in state . This setting is illustrated in Fig. 6. Particles can change their orientation, i.e. transition from to , and from to , with a rate . Furthermore, particles can cross the potential barrier and transition between the - and -states. The associated rate constants are assumed to be given by modified Arrhenius rates Friddle (2008); A. Tobolsky (1943); Bell (1978), where the effect of self-propulsion is to effectively increase or decrease the potential barrier. For example, the rate to transition from to is

(8)

As the propulsion force helps the particle to cross the barrier, it effectively lowers the potential barrier by the work that the propulsion force performs when the particle climbs the left slope of the ratchet. This modified Arrhenius rate is expected to be valid under the assumptions (a) of a large barrier height , which is a condition for the Arrhenius rates to be valid even for passive systems Kramers (1940), (b) of a ratchet potential that is typically crossed faster than a particle reorients, which can be achieved by making the barrier width sufficiently small, and (c) that the work performed by the propulsion force is much smaller than the barrier height . We point out that assumption (c) can be rewritten as . This means that if assumptions (a) and (b) are satisfied, which imply that , then assumption (c) is not much further restrictive on the activity Pe. The remaining rate constants follow along a similar reasoning as

(9)

For large bulks on either side of the ratchet, the attempt frequencies in the rate expressions (8) and (9) are inversely proportional to the size of the bulk that is being transitioned from. This size is denoted by for the left bulk, and by for the right bulk. Therefore, the factors and are independent of the bulk sizes and , and can only depend on the shape of the rachet potential, i.e. on its height , on its width , and on its asymmetry .

\includegraphics

[width=]1Dgassetup5-crop.pdf

Figure 6: Illustration of the states in the transition state model. Particles in the left bulk with positive (negative) -orientation are in state (). Similarly, particles in the right bulk are in state or . Within one bulk, particles can change their orientation with rate constant . Between the bulks, particles can transition by crossing the potential barrier with the effective Arrhenius rates of Eqs. (8) and (9), where the effect of self-propulsion is to shift the potential barrier by the work () performed by the propulsion force when a particles climbs the left (right) slope of the ratchet.

We denote the number of particles in the and states by and , respectively. The time evolution of these particle numbers follows from the rates outlined above. For example, the number of particles in state evolves according to the rate equation

(10)

Similar equations hold for the particle numbers , and . These rate equations can be solved in steady state, i.e. when , for the particle numbers and . We consider infinitely large bulks, i.e. . In this case, the solutions show that and , such that the and states correspond to isotropic bulks. Furthermore, the solution shows that the bulk densities and differ by an amount given by

(11)

where we recall that . We point out that the ratio can generally depend on the ratchet parameters , , and . However, in the following we simply assume , which is justified for nearly symmetric ratchets.
To enable a comparison with the analytical solution of the previous section, we now focus on the limit of weak activity, i.e. of . This ensures assumption (c) to be satisfied, but we emphasize that the transition state model is not limited to weak activity. We expand the density difference (11) as , and compare the coefficient with the same coefficient obtained in section III for the analytical solution in the weak activity limit. The coefficient is plotted in Figs. 5(a)-(c), as a function of the of the barrier height , the barrier width , and the barrier asymmetry , respectively. Fig. 5(a) merely illustrates that the density difference (11) is independent of the barrier height . This independency agrees with the asymptotic behavior displayed by the analytical solution for large barrier heights . Note that the regime is indeed assumed for the modified Arrhenius rates (assumption (b)). Fig. 5(b) illustrates that the density difference predicted by the transition state model scales quadratically with the barrier width, i.e. that . This scaling agrees with the scaling of the analytical solution for the regime of small barrier widths . Again, this regime is assumed for the modified Arrhenius rates, as having a small barrier width is required for having particles cross the ratchet faster than they typically reorient (assumption (c)). Finally, Fig. 5(c) illustrates that the density difference predicted by the transition state model scales linearly with the barrier asymmetry for nearly symmetric ratchets, i.e. for , and asymptotically for very asymmetric ratchets, i.e. for . Both scalings are also displayed by the analytical solution. All these power laws can again be found in Table 1.
Of course, the transition state model reproduces only the power laws that lie inside its regime of applicability. However, the fact this simple model does reproduce all these power laws is quite remarkable, since, as discussed in section III, most of these scalings are nontrivial. Furthermore, we note that the transition state model can also be solved for finite bulk sizes, which in fact predicts a turnover of the density difference as a function of activity Pe, as observed in Fig. 3(a).
Quantitatively, Fig. 5 clearly shows that the predictions of the transition state model typically differ from the analytical solution by an order of magnitude. A possible reason for this disagreement is that these plots are made for parameters values that do not satisfy assumptions (a) and (b) that underly the modified Arrhenius rates. In fact, it turned out to be impossible to satisfy these assumptions simultaneously with feasible parameter values. The root of the difficulty is that the time it takes a particle to cross the potential barrier increases with the barrier height . As a consequence, having a barrier that is simultaneously very high (assumption (a)), and typically crossed faster than a particle reorients (assumption (b)), turns out to require unrealistically small barrier widths . The quantitative mismatch of the transition state model with the full solution for small activity might also be attributed to the assumption that the prefactors and in the rate expressions (8) and (9) are not exactly identical, but in fact might depend on the precise shape of the barrier. However, this possibility goes beyond the current scope of this paper, and we leave it for future study.
We conclude that, whereas it was not possible to test the predictions of the transition state model in its regime of applicability quantitatively, the model does reproduce the complete power law behavior of this regime correctly.

Vi Discussion

The most interesting aspect of the studied system is that the external potential has a long-range influence on the density profile. This is in sharp contrast to an ideal gas in equilibrium, whose density profile is only a function of the local external potential. So what ingredients are necessary to obtain this effect? To answer this question, we consider the 1D RnT model subject to a general external potential . Furthermore, we introduce the particle current and the orientation current that appear in the evolution equations (6), i.e.

(12)

We focus on a state that is steady, such that , and flux-free, such that . Then Eqs. (6) and (12) can be recast into the first order differential equation

(13)

for the three (non-dimensionalized) unknowns . The coefficient matrix in Eq. (13) is given by

(14)

where is the dimensionless external force, that is now a function of position . For a passive system (Pe ), Eqs. (13) and (14) show that the density equation decouples. In this case, the density profile is solved by the Boltzmann weight, i.e. , as required in thermodynamic equilibrium. For the general case, we observe that, if the coefficient matrix commutes with its integral , then Eq. (13) is solved by

(15)

where the integration constants , and are to be determined from boundary conditions. Here is an arbitrary reference position. By virtue of , the solution (15) is a local function of the external potential. An explicit calculation of the commutator shows that if and only if , i.e. if the external potential is a linear function of . Therefore, for linear potentials, the density profile is a local function of the external potential. This explains why in a gravitational field the density profile can be found as a local function of the external potential, and why sedimentation profiles stand a chance to be described in terms of an effective temperature in the first placeTailleur and Cates (2008, 2009); Nash et al. (2010); Palacci et al. (2010); Enculescu and Stark (2011); Wolff, Hahn, and Stark (2013); Szamel (2014); Solon, Cates, and Tailleur (2015); Ginot et al. (2015); Stark (2016); Hermann and Schmidt (2018); Ginot et al. (). However, for nonlinear external potentials, e.g. for the ratchet studied here that is only piecewise linear, the solution (15) is not valid, and a nonlocal dependence on the external potential is to be expected. Therefore, for the ratchet potential (4), the kinks at , and are crucial to have a density that depends nonlocally on the external potential. Indeed, in the analytical solution for weak activity, presented in section IV, the nonlocal dependence of the right bulk density on the external potential enters through the fact that the integration constants in Eq. (IV) are found from continuity conditions that are applied precisely at the positions of these kinks.
Summarizing, in order to have the external potential influence the steady-state density of ideal particles in a nonlocal way, one needs to have (1) particles that are active (such that the system is out of thermodynamic equilibrium), and (2) an external potential that is nonlinear. Thereby, the 1D RnT particles in the ratchet potential (4) illustrate the nonlocal, and even long-range, influence of the external potential in a most minimal way.

Vii Conclusions

We have studied the distribution of noninteracting, active particles over two bulks separated by a ratchet potential. The active particles were modelled both as two-dimensional ABPs, and as one-dimensional RnT particles. Our numerical solutions to the steady state Smoluchowski equations show that the ratchet potential influences the distribution of particles over the bulks, even though the potential is short-ranged itself. Thus, the external potential exerts a long-range influence on the density profile. We have shown that such a (highly) nonlocal influence can occur for noninteracting particles only when they are (1) active, and (2) subject to an external potential that is nonlinear. Thereby, the piecewise linear setup considered in this work captures this long-range influence in a most minimal way.
To characterize the long-range influence of the external potential, we have described how the difference in bulk densities depends on activity, as well as on the ratchet potential itself. Both models of active particles showed consistent power law behavior that is summarized in Table 1.
To understand the long-range influence of the potential in the simplest case possible, we focussed on the limit of weak activity. While weakly active systems are often described by an effective temperature, our analytical solution explicitly shows that the long-range influence of the ratchet potential cannot be rationalized in this way. Instead, we propose a simple transition state model, in which particles can cross the potential barrier by Arrhenius rates with an effective barrier height that depends on the degree of activity. While the model could not be tested quantitatively, as its underlying assumptions could not be simultaneously satisfied for feasible parameter values, it does reproduce - in its regime of applicability - the complete power law behavior of the distribution of particles over the bulks.
Future questions are whether the power law behavior can be understood also outside the regime where the transition state model applies, and whether the power laws also hold for potential barriers of more generic shape than the sawtooth of Fig. 1. Our work illustrates that even weakly active, noninteracting particles pose challenges that are fundamental to nonequilibrium systems, and, moreover, that an external potential can exert a long-range influence in such systems. We expect that incorporating such long-range and nonlocal effects will be part of a more generic statistical mechanical description of nonequilibrium systems.

Viii Acknowledgments

This work is part of the D-ITP consortium, a program of the Netherlands Organisation for Scientific Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW). We acknowledge funding of a NWO-VICI grant. S.P. and M.D. acknowledge the funding from the Industrial Partnership Programme ‘Computational Sciences for Energy Research’ (Grant No. 14CSER020) of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organization for Scientific Research (NWO). This research programme is co-financed by Shell Global Solutions International B.V.

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