Single-file transport in periodic potentials: The Brownian asymmetric exclusion process

TL;DR

This paper investigates the collective dynamics of Brownian particles in periodic potentials, revealing how particle size and density influence steady-state currents and phase behavior, with implications for understanding single-file transport.

Contribution

It introduces a comprehensive analysis of the Brownian asymmetric simple exclusion process, including general properties, phase predictions, and the impossibility of current reversals in certain driven systems.

Findings

Average currents vary with particle size and density.

Extremal current principles predict nonequilibrium phases.

Current reversals are impossible in systems driven by constant drag or traveling waves.

Abstract

Single-file Brownian motion in periodic structures is an important process in nature and technology, which becomes increasingly amenable for experimental investigation under controlled conditions. To explore and understand generic features of this motion, the Brownian asymmetric simple exclusion process (BASEP) was recently introduced. The BASEP refers to diffusion models, where hard spheres are driven by a constant drag force through a periodic potential. Here, we derive general properties of the rich collective dynamics in the BASEP. Average currents in the steady state change dramatically with the particle size and density. For an open system coupled to particle reservoirs, extremal current principles predict various nonequilibrium phases, which we verify by Brownian dynamics simulations. For general pair interactions we discuss connections to single-file transport by traveling-wave potentials and prove the impossibility of current reversals in systems driven by a constant drag and by traveling waves.

Equations

$$\frac{\text{differential}{x}_i}{\text{differential}t}=\mu \left(f+f_i^{\mathrm{int}}-\frac{\partial U(x_i)}{\partial x_i}\right)+\sqrt{2D}{\eta }_i(t),$$

$$\frac{\partial \varrho (x,t)}{\partial t}=-\frac{\partial j(x,t)}{\partial x},$$

$$j(x,t)=\mu [f^{\mathrm{ext}}(x)+f^{\mathrm{int}}(x,t)]\varrho (x,t)-D\frac{\partial \varrho (x,t)}{\partial x},$$

$$f^{\mathrm{int}}(x,t)$$

$$={\int }_0^L\text{differential}y{f}^{(2)}(x,y)\varrho (y|x;t),$$

$$f^{\mathrm{int}}(x,t)=k_{\mathrm{B}}T\frac{{\varrho }^{(2)}(x,x-\sigma ,t)-{\varrho }^{(2)}(x,x+\sigma ,t)}{\varrho (x,t)}.$$

$$j_{\mathrm{st}}(\rho ,\sigma )=\frac{\mu \left(f+\frac{1}{\lambda }{\int }_0^{\lambda }\text{differential}x{f}_{\mathrm{st}}^{\mathrm{int}}(x)\right)}{\frac{1}{\lambda }{\int }_0^{\lambda }\text{differential}x\frac{1}{{\varrho }_{\mathrm{st}}(x)}}=\frac{\mu \left(f+\overline{f_{\mathrm{st}}^{\mathrm{int}}}\right)}{\overline{{\varrho }_{\mathrm{st}}^{-1}}},$$

$$\frac{\text{differential}{x}_i}{\text{differential}t}=-\mu \frac{\partial U(x_i-v_{\mathrm{w}}t)}{\partial x_i}+\mu f_i^{\mathrm{int}}+\sqrt{2D}{\eta }_i(t).$$

$${\varrho }^{\mathrm{TW}}(x,t)$$

$$j^{\mathrm{TW}}(x,t)$$

$${\varrho }^{\mathrm{TW}}(x,t)$$

$$=⟨\sum _{i=1}^N\delta (x-v_{\mathrm{w}}t-x_i^{\prime }(t))⟩=\varrho (x-v_{\mathrm{w}}t,t).$$

$${\varrho }^{(2),\mathrm{TW}}(x,y,t)$$

$$f^{\mathrm{int},\mathrm{TW}}(x,t)=f^{\mathrm{int}}(x-v_{\mathrm{w}}t,t).$$

$$j^{\mathrm{TW}}(x,t)$$

$$-D\frac{\partial {\varrho }^{\mathrm{TW}}(x,t)}{\partial x}$$

$$=\mu \left(f^{\mathrm{int}}(x-v_{\mathrm{w}}t,t)-\frac{\partial U(x-v_{\mathrm{w}}t)}{\partial x}\right)\varrho (x-v_{\mathrm{w}}t,t)$$

$$-D\frac{\partial \varrho (x-v_{\mathrm{w}}t,t)}{\partial x}$$

$$=j(x-v_{\mathrm{w}}t,t)-\mu f\varrho (x-v_{\mathrm{w}}t,t)$$

$$=j(x-v_{\mathrm{w}}t,t)+v_{\mathrm{w}}\varrho (x-v_{\mathrm{w}}t,t),$$

$${\varrho }_{\mathrm{st}}^{\mathrm{TW}}(x,t)$$

$$j_{\mathrm{st}}^{\mathrm{TW}}(x,t)$$

$$\frac{\dot{S}(t)}{k_{\mathrm{B}}}$$

$$J_i=J_i(x_1,\dots ,x_N,t)=\mu f^{\mathrm{ext}}(x_i)p_N-D\frac{\partial p_N}{\partial x_i},$$

$$\frac{\dot{S}(t)}{k_{\mathrm{B}}}$$

$$=\sum _{i=1}^N{\int }_{\Omega }{\text{differential}}^{N}x\frac{J_i^2}{D{p}_N}-\frac{1}{k_{\mathrm{B}}T}\sum _{i=1}^N{\int }_{\Omega }{\text{differential}}^{N}x{J}_i{f}^{\mathrm{ext}}(x_i)$$

$$=\frac{{\dot{S}}_{\mathrm{tot}}(t)}{k_{\mathrm{B}}}-\frac{{\dot{S}}_{\mathrm{m}}(t)}{k_{\mathrm{B}}}.$$

$${\dot{S}}_{\mathrm{tot}}$$

$$=\frac{j_{\mathrm{st}}M}{T}{\int }_0^{\lambda }\text{differential}x\left(f-\frac{\partial U(x)}{\partial x}\right)=\frac{j_{\mathrm{st}}fL}{T}>0.$$

$${\dot{S}}_{\mathrm{m}}^{\mathrm{TW}}(t)$$

$$=-\frac{1}{T}\sum _{i=1}^N{\int }_0^L\text{differential}{x}_i\frac{\partial U(x_i-v_{\mathrm{w}}t)}{\partial x_i}{j}_i^{\mathrm{TW}}(x_i,t).$$

$$\overline{{\dot{S}}_{\mathrm{tot}}^{\mathrm{TW}}}$$

$$=\frac{M{v}_{\mathrm{w}}}{T}{\int }_0^{\lambda }\text{differential}x\left(-\frac{\text{differential}U(x)}{\text{differential}x}{\varrho }_{\mathrm{st}}(x)\right)>0.$$

$$-\frac{\text{differential}U(x)}{\text{differential}x}{\varrho }_{\mathrm{st}}(x)$$

$$+k_{\mathrm{B}}T\frac{\text{differential}{\varrho }_{\mathrm{st}}(x)}{\text{differential}x}.$$

$$\overline{{\dot{S}}_{\mathrm{tot}}^{\mathrm{TW}}}=\frac{v_{\mathrm{w}}L}{\mu T}\left[\overline{j_{\mathrm{st}}^{\mathrm{TW}}}-\mu \overline{f_{\mathrm{st}}^{\mathrm{int}}(x){\varrho }_{\mathrm{st}}(x)}\right]>0.$$

$$\overline{f_{\mathrm{st}}^{\mathrm{int}}(x){\varrho }_{\mathrm{st}}(x)}$$

$$=\frac{1}{L}{\int }_0^L\text{differential}x{\varrho }_{\mathrm{st}}(x)f_{\mathrm{st}}^{\mathrm{int}}(x)$$

Full text

I Introduction

Single-file transport refers to collective motion in strongly confined geometries, where the particles cannot overtake each other. In most chemical and biophysical settings, single-file dynamics is severely affected by thermal fluctuations and takes place far from thermodynamic equilibrium. In cell biology, examples of single-file transport are the directed motion of motor proteins along microtubules or actin filaments Lipowsky et al. (2001); Frey and Kroy (2005); Hille (2001), ion migration through membrane channels Hille (2001), and protein synthesis by ribosomes MacDonald et al. (1968). With the steadily increasing quality of experimental techniques capable to control and detect particle motion on molecular scales, single-file transport in periodic (free) energy landscapes becomes of increasing importance also for applications, as, e.g., transport in carbon nanotubes Zeng et al. (2018), zeolites Van de Voorde and Sels (2017), mesoporous materials Hartmann (2005); Yiu et al. (2001), and nanofluidic devices Ma et al. (2015).

A much investigated and well-understood feature in single-file diffusion is the anomalous subdiffusion of a tracer particle in the long-time limit Taloni et al. (2017). This was first proven in the mathematical literature Harris (1965) and later detected experimentally in zeolites Hahn et al. (1996); Chmelik et al. (2018) and in nanotubes Cheng and Bowers (2007); Dvoyashkin et al. (2014) by using nuclear magnetic resonance techniques. Further direct observation was possible in colloidal experiments by optical imaging Wei et al. (2000). The aspect of subdiffusion was further elaborated in connection with general theories of anomalous transport Taloni et al. (2010), including descriptions in terms of fractional Brownian motion Sanders and Ambjörnsson (2012), effects of initial and boundary conditions Leibovich and Barkai (2013); Lizana and Ambjörnsson (2008), external force fields Barkai and Silbey (2009), time-varying potentials Ryabov and Chvosta (2011), first-passage time distributions Ryabov and Chvosta (2012); Ryabov (2013); Locatelli et al. (2016), statistics of residence times Krapivsky et al. (2015), partial overtaking of particles Mon and Percus (2002); Ooshida et al. (2018); Ahmadi et al. (2019), and large deviation functions for the position of a tracer Krapivsky et al. (2014).

As for collective transport properties, single-file motion has been mainly investigated in lattice models, which reflect a periodic structure in a coarse-grained manner. These lattice models can be considered as variants of the so-called asymmetric simple exclusion process (ASEP) and have found applications in particular in the modeling of biological traffic Schadschneider et al. (2010); Chou et al. (2011); Kolomeisky (2013); Appert-Rolland et al. (2015).

The minimal ASEP model, where particles perform nearest-neighbor hops between lattice sites with a bias in one direction and under the constraint that only one particle can occupy a lattice site has become a reference model for studying fundamental questions of statistical physics out of equilibrium Derrida (1998); Schütz (2001). For this model, exact results for microstate distributions in nonequilibrium steady states could be obtained Blythe and Evans (2007). When coupled to particle reservoirs, three different phases of nonequilibrium steady states appear in dependence of the reservoir densities Krug (1991); Parmeggiani et al. (2003). Studies with nearest-neighbor interactions between particles showed richer phase diagrams Antal and Schütz (2000); Dierl et al. (2011). They led to a clarification of the role of system-reservoir couplings in open systems for the topology of nonequilibrium phase diagrams and of the meaning of particle-hole symmetry Dierl et al. (2012, 2013). This clarification turned out to be useful also for understanding collective particle dynamics in lattice models with time-varying site energies Dierl et al. (2014).

Many further interesting results were reported for the ASEP, as, e.g., singularities in large deviation functions for time-averaged currents Bertini et al. (2005); Lazarescu (2015); Baek et al. (2017), and condensation transitions for nonuniform hopping rates Evans (1996); Concannon and Blythe (2014). More recently, new universality classes in the hydrodynamic limit of nonlinear hydrodynamics were discovered for multi-lane variants of the model Popkov et al. (2015). Predictions for long-time tails could be proven for a specific microscopic model Chen et al. (2018).

As the hopping transitions in the ASEP can be considered to reflect rare events of thermally activated barrier crossings, one may conjecture that continuous single-file motion in a periodic potential exhibits features similar to the ASEP. However, in a recent study Lips et al. (2018) we showed that a much richer behavior occurs in continuous Brownian motion because of additional length scales associated with particle-particle interactions. The simplest class of models is that of hardcore interacting particles, which we refer to as the Brownian asymmetric simple exclusion process (BASEP). In this BASEP, hard rods with diameter $\sigma$ are driven through a periodic potential with wavelength $\lambda$ by a constant drag force $f$.

The BASEP is particularly interesting in connection with recent experiments utilizing advanced techniques of microfluidics and optical and/or magnetic micromanipulation Arzola et al. (2017); Skaug et al. (2018); Schwemmer et al. (2018); Stoop et al. (2019); Misiunas and Keyser (2019), which includes setups where the particles are not driven by a constant drag force but a traveling-wave potential Straube and Tierno (2013). This is because the Brownian motion of a particle in a traveling-wave potential $U(x-v_{\rm w}t)$ is mapped onto that in a periodic potential $U(x)$ with a constant drag force $f=-v_{\rm w}/\mu$ after a coordinate transformation $x\to x-v_{\rm w}t$; $\mu$ is the bare mobility of the particles.

Furthermore, the BASEP should allow one to understand under which conditions a coarse-grained description in terms of lattice models will be appropriate. Generally, a more detailed understanding of the connection between the BASEP and corresponding lattice models is necessary to explain why certain effects are seen in one description but not in the other. For example, steady-state currents opposite to the external bias, so-called current-reversals, were reported for lattice models Jain et al. (2007); Slanina (2009a, b); Chaudhuri and Dhar (2011); Dierl et al. (2014) and recently seen experimentally in a rocking Brownian motor Schwemmer et al. (2018), but they were not found in an analogous setting with continuous-space dynamics Chaudhuri et al. (2015). Current reversals in space-continuous models were reported earlier for a constant and “flashing” asymmetric sawtooth-shaped external potential Derényi and Vicsek (1995); Derenyi and Ajdari (1996), and in a recent work with time-discontinuous driving of a single potential barrier along a ring Rana et al. (2018). In the BASEP, a barrier reduction and an exchange symmetry effect were identified as decisive mechanisms for the characteristics of single-file transport, but it is yet unclear to which extent these effects have a counterpart in lattice descriptions. In this study, we will provide further insight into these issues.

A further goal of this study is to gain an extended description of the average steady-state current $j_{\rm st}$ in the BASEP and an explicit verification of the phases of nonequilibrium steady states in an open system coupled to particle reservoirs. We believe that this provides a useful basis for future investigations of model variants where further details can be included to capture specific experimental conditions.

As for the extended description of the average steady-state current $j_{\rm st}$ (which we often refer to as “the current” in the following), we note that our results reported in Ref. Lips et al. (2018) focused on a limited regime of particle densities $\rho$, which we defined as the (dimensionless) filling factor of the potential wells. The maximal possible $\rho$ is equal to $\lambda/\sigma$ and the yet unexplored regime is that of high densities in the range $1<\rho\leq\lambda/\sigma$. In fact, by making use of a mapping of currents for different particle sizes, we give a description of $j_{\rm st}=j_{\rm st}(\rho,\sigma)$ for all particle sizes and densities.

Moreover, we investigate the influence of the temperature on $j_{\rm st}$, and prove that current reversals cannot occur in closed systems driven by a constant drag force or a traveling wave. This leads to rigorous upper bounds of $|j_{\rm st}|$ and of $|\overline{j_{\rm st}^{\rm\scriptscriptstyle TW}}|$, where $\overline{j_{\rm st}^{\rm\scriptscriptstyle TW}}$ denotes the period-averaged steady-state current in a traveling-wave driven system.

Our simulations are supported by an analytical treatment extending the derivations in the supplemental material of Ref. Lips et al. (2018). This yields an approximate expression for the current in the linear-response regime, which shows qualitative agreement with the simulation results. We show that quantitative deviations are mainly due to the neglect of an interaction-mediated effective drift term, which stems from the interplay between the external periodic potential and interparticle interactions in the steady state. This mean interaction force has a clear physical interpretation, yet its quantitative analytical description is challenging.

In the open BASEP coupled to particle reservoirs, we introduce a simple scheme of particle injection and ejection, which allows us to demonstrate four of the five possible phases predicted by the extremal current principles in Ref. Lips et al. (2018). The missing phase corresponds to a small region in the phase diagram appearing at very high reservoir densities. All simulated phases are demonstrated by corresponding density profiles.

As our results for analytical expressions of the current, current reversals and current bounds are valid for quite general pair interactions, we first discuss collective Brownian single-file transport for this wider class of systems. From Sec. III.4 onward, we focus on the BASEP with hardcore-interacting particles.

II Single-file Brownian motion

We consider single-file Brownian motion of $N$ particles in a periodic potential $U(x)=U(x+\lambda)$. The particles are driven by a constant drag force $f$ and interact via pair forces $f^{(2)}(x,y)$, i.e., $f_{i}^{\rm\scriptscriptstyle int}=\sum_{j\neq i}f^{(2)}(x_{i},x_{j})$ is the interaction force on the $i$th particle. For overdamped Brownian motion, the particle dynamics are described by the Langevin equations

[TABLE]

where $\mu$ and $D=\mu k_{\rm\scriptscriptstyle B}T$ are the bare mobility and diffusion coefficient, and $k_{\rm\scriptscriptstyle B}T$ is the thermal energy. The $\eta_{i}(t)$ are independent and $\delta$-correlated Gaussian white noise processes with zero mean and unit variance, $\langle\eta_{i}\rangle=0$ and $\langle\eta_{i}(t)\eta_{j}(t^{\prime})\rangle=\delta_{ij}\delta(t-t^{\prime})$.

For the BASEP, the hardcore interactions imply the boundary conditions $|x_{i}-x_{j}|\geq\sigma$, i.e., overlaps between neighboring particles are forbidden. Taking into account these boundary conditions, the interaction force $f_{i}^{\rm\scriptscriptstyle int}$ can be set to zero in Eq. (1). We define the density as a (dimensionless) filling factor of the potential wells, i.e., by $\rho=N/M$, where $M$ denotes the total number of periods of $U(x)$. The system length is $L=M\lambda$ and the number density is $\rho/\lambda$ with the upper bound $1/\sigma$. The model is sketched in Fig. 1 for the cosine periodic potential $U(x)$ discussed thoroughly in Secs. IV and V.

In the following we will distinguish between the closed BASEP, where periodic boundary conditions are applied with particle coordinates in sequential order $x_{1}\leq x_{2}\leq\ldots\leq x_{N}$ 111To implement the periodic boundary conditions, we assume an ordered initial configuration $0\leq x_{1}\leq x_{2}\ldots\leq x_{N}<L$, and introduce two fictive particles with enslaved coordinates $x_{0}=x_{N}-L$ and $x_{N+1}=x_{1}+L$, which implies $x_{N}-x_{1}<L-\sigma$. This implementation, where the $x_{i}$ can assume any real value, is convenient for discussing transformation properties of the Langevin equations. Alternatively, one could consider the particle positions to be confined to a ring of size $L$, corresponding to a mapping $x_{i}\to x_{i}\,\mathrm{mod}\,L$. However, then the sequential order of the particles can no longer be expressed by $x_{1}\leq x_{2}\ldots\leq x_{N}$ in the course of time., and the open BASEP, where the left and right boundaries are coupled to particle reservoirs with in general different densities. In the closed BASEP, the drag force $f$ leads to a steady state with a constant particle current $j_{\rm st}$ and a $\lambda$-periodic local density profile $\varrho_{\rm st}(x)$. The dependence of the steady-state current $j_{\rm st}$ on $\rho$ and $\sigma$ is denoted explicitly, i.e., $j_{\rm st}=j_{\rm st}(\rho,\sigma)$, while other dependencies on $T$, $f$ etc. are omitted in the notation. In the open BASEP, specification of the way of how particles are exchanged with the reservoirs is a subtle issue, which will be discussed in Secs. V.1 and V.3.

In simulations of the BASEP, we used numerical algorithms specifically developed for Brownian dynamics of hard-sphere systems Tao et al. (2006); Scala et al. (2007); Scala (2012); Behringer and Eichhorn (2012). These algorithms use the Euler method and differ in the implementation of the hardcore (excluded volume) constraints. Specifically, we applied the two algorithms developed in Refs. Scala (2012) and Behringer and Eichhorn (2012). Our results are not affected by the choice of any of these algorithms, and they showed agreement with exact analytical findings for specific cases.

III Current in closed systems: General results

III.1 Continuity equation and steady-state current

The conservation of the particle number in a closed system is described by the continuity equation

[TABLE]

where $\varrho(x,t)=\langle\sum_{i=1}^{N}\delta(x-x_{i})\rangle$ is the local density and $\langle\ldots\rangle$ refers to the ensemble average at time $t$ (for an arbitrary given initial condition). The local particle current is Lips et al. (2018)

[TABLE]

where $f^{\rm\scriptscriptstyle ext}(x)=f-\partial U(x)/\partial x$, and $f^{\rm\scriptscriptstyle int}(x,t)$ is the mean force at position $x$ at time $t$ due to the interactions. With the two-particle density $\varrho^{(2)}(x,y,t)=\langle\sum_{i=1}^{N}\sum_{k=1,k\neq i}^{N}\delta(x-x_{i})\delta(y-x_{k})\rangle$, this force can be written as

[TABLE]

where $\varrho(y|x;t)=\varrho^{(2)}(x,y,t)/\varrho(x,t)$ is the local density at position $y$ at time $t$ under the condition that a particle is at position $x$ at time $t$.

For hardcore interactions, an exact treatment based on the many-particle Smoluchowski equation given in the supplemental material of Ref. Lips et al. (2018) leads to

[TABLE]

That expression follows also when formally setting $f^{(2)}(x,y)=k_{\rm\scriptscriptstyle B}T[\delta(y-(x-\sigma))-\delta(y-(x+\sigma))]$ in Eq. (4). Intuitively, this can be understood as follows: For a particle at position $x$ there is a positive and a negative force on contacts with other particles at positions $x-\sigma$ and $x+\sigma$ that correspond to the two $\delta$-functions. The amplitude in front of the $\delta$-functions must be an energy on dimensional reasons, for which $k_{\rm\scriptscriptstyle B}T$ is the only relevant scale.

In a steady state, the density profile $\varrho(x,t)$ is time-independent, $\varrho(x,t)=\varrho_{\rm st}(x)$, and the current homogeneous, $j(x,t)=j_{\rm st}$. For periodic boundary conditions, the $\lambda$ periodicity of the external force $f^{\rm\scriptscriptstyle ext}$ entails that the steady-state solution is as well $\lambda$ periodic, i.e., $\varrho_{\rm st}(x)=\varrho_{\rm st}(x+\lambda)$. Dividing Eq. (3) by $\varrho_{\rm st}(x)$ and integrating over one period, we obtain for the steady-state current

[TABLE]

where $\overline{\ldots\phantom{|}}=\lambda^{-1}\int_{0}^{\lambda}\differential x\ldots$ denotes a period-averaging.

III.2 Relation to traveling-wave driving

Overdamped Brownian motion in a traveling-wave potential with wave velocity $v_{\rm w}$ is described by the Langevin equations

[TABLE]

If the pair interaction force $f^{(2)}(x,y)$ is a function of the pair distance $(x-y)$ only, then the Galilean transformation $x_{i}\to x^{\prime}_{i}=x_{i}-v_{\rm w}t$ reduces these equations to Eqs. (1) with the constant drag force $f=-v_{\rm w}/\mu$ (the primed coordinates corresponding to the system with constant drag force). Note that also hardcore constraints remain invariant under the Galilean transformation, $|x_{i+1}-x_{i}|=|x^{\prime}_{i+1}-x^{\prime}_{i}|\geq\sigma$.

The correspondence implies the relations

[TABLE]

between local particle densities and currents, where the superscript “TW” marks the quantities in the traveling-wave system. Equation (8a) is quite obvious and it follows formally by transforming to the coordinates $x_{i}^{\prime}(t)=x_{i}(t)-v_{\rm w}t$ of the comoving frame,

[TABLE]

In the same manner, one finds (and corresponding relations hold for many-particle densities of higher order)

[TABLE]

Given Eqs. (9) and (10), the local mean interaction forces in the two systems satisfy an analogous relation [cf. Eq. (4)],

[TABLE]

For the transformation between the local currents we thus obtain [cf. Eq. (3)]

[TABLE]

which agrees with Eq. (8b). For steady states, Eqs. (8a) and (8b) become

[TABLE]

The relations given in this section can be of particular interest in experimental setups with optical or magnetic tweezers, where traveling-wave potentials represent a feasible way to mimic the constant drift force Straube and Tierno (2013). They are also of interest in the following discussion of current reversal.

III.3 Absence of current reversals and implications for current bounds

In systems with a constant drag force $f$, a current reversal is not possible in the steady state, i.e., $j_{\rm st}$ must have the same sign as $f$. This can be expected from the second law of thermodynamics, which forbids the system to continuously extract heat from its environment and to perform work against the external force. In systems driven by a traveling wave, the impossibility of current reversals is less expected, in particular because such reversals were seen in corresponding lattice models Chaudhuri and Dhar (2011); Dierl et al. (2014). An absence of current reversals in traveling-wave systems with continuous space dynamics was conjectured in Ref. Chaudhuri et al. (2015) based on Brownian dynamics simulations for different pair potentials and a perturbative expansion of the single-particle density around its period-averaged value. Here we give rigorous proofs for the BASEP for both types of driving.

To this end we consider the entropy production

[TABLE]

in the system, where $p_{N}=p_{N}(x_{1},\ldots,x_{N},t)$ is the $N$-particle joint probability density and $\Omega$ specifies the allowed configuration space (for hardcore interactions, this is restricted due to the conditions $|x_{i+1}-x_{i}|\geq\sigma$). Using the Smoluchowski equation $\partial p_{N}/\partial t=-\sum_{i=1}^{N}\partial J_{i}/\partial x_{i}$ with the probability currents

[TABLE]

the time derivative of $p_{N}$ in Eq. (14) can be written as

[TABLE]

Here we took into account the periodic boundary conditions in the partial integration and expressed $\partial\ln p_{\scriptscriptstyle N}/\partial x_{i}$ using Eq. (15). In the last step, we identified the resulting two terms as the production of total entropy and entropy in the surrounding medium in the framework of stochastic thermodynamics Seifert (2012). Note that the total entropy production is positive out of equilibrium ($\sum_{i=1}^{N}J_{i}^{2}>0$) in agreement with the second law of thermodynamics.

In the nonequilibrium steady state, the entropy production in the system is zero, $\dot{S}=0$, and the (time-independent) total entropy production $\dot{S}_{\rm tot}>0$ equals the entropy production $\dot{S}_{\rm m}$ in the medium. Because the partial current $j_{i}(x,t)$ of the $i$th particle is $J_{i}$ integrated over all but the $i$th coordinate, i.e., $j_{i}(x,t)=\int_{\Omega}\differential^{N}\!x\,J_{i}(x_{1},\ldots,x_{N},t)\delta(x-x_{i})$, and $j_{i}(x,t)=j_{\rm st}/N$ in the steady state, we obtain from Eq. (16)

[TABLE]

Here we have used the periodicity of the potential and $L=M\lambda$. The total entropy production has the Onsager form of current $j_{\rm st}$ times the thermodynamic force $fL/T$. Its positive definiteness implies that $j_{\rm st}$ and $f$ must have the same sign.

Likewise, it can be shown that the total entropy production in the steady state of a traveling-wave driven system obeys an analogous Onsager type relation, if period-averaged quantities are considered. Using the general expression $\dot{S}_{\rm m}=T^{-1}\sum_{i=1}^{N}\int_{\Omega}\differential^{N}\!x\,J_{i}f^{\rm\scriptscriptstyle ext}(x_{i})$ for the entropy production in the medium, see Eq. (16), we obtain for a traveling-wave system

[TABLE]

Time averaging of the total entropy production $\dot{S}_{\rm tot}^{\rm\scriptscriptstyle TW}=\dot{S}_{\rm m}^{\rm\scriptscriptstyle TW}+\dot{S}^{\rm\scriptscriptstyle TW}$ over one period $\tau=\lambda/v_{\rm w}$ in the steady state yields, under consideration of $S(t+\tau)=S(t)$ and $j_{{\rm st},i}^{\rm\scriptscriptstyle TW}(x,t)=j^{\rm\scriptscriptstyle TW}_{\rm st}{(x,t)}/N=j_{\rm st}+v_{\rm w}\varrho_{\rm st}(x-v_{\rm w}t)$ [cf. Eq. (13b)],

[TABLE]

Here, $\overline{\ldots\phantom{|}}=\tau^{-1}\int_{0}^{\tau}\differential t\ldots$ denotes the time averaging. Using Eq. (3) in the steady state with $\mu f=-v_{\rm w}$, the integrand in the last expression of Eq. (19) can be written as

[TABLE]

Inserting this into Eq. (19) gives, when utilizing the $\lambda$ periodicity of $\varrho_{\rm st}(x)$, and taking into account that $\overline{j_{\rm st}^{\rm\scriptscriptstyle TW}}=j_{\rm st}+v_{\rm w}\overline{\varrho_{\rm st}}$ [cf. Eq. (13b)]

[TABLE]

Note that in the system with constant drag force, the bar always refers to a spatial period averaging, while in the corresponding traveling-wave system, time and space averaging over one period yield the same results in the steady state.

The average $\overline{f^{\rm\scriptscriptstyle int}_{\rm st}(x)\varrho_{\rm st}(x)}$ vanishes for pair interaction forces obeying the principle of actio and reactio, $f^{(2)}(x,y)=-f^{(2)}(y,x)$ (for hardcore interactions one can use $f^{(2)}(x,y)=k_{\rm\scriptscriptstyle B}T[\delta(y-x+\sigma)-\delta(y-x-\sigma)]$, see the discussion after Eq. (5)):

[TABLE]

Here we have inserted $f^{\rm\scriptscriptstyle int}_{\rm st}(x)$ from Eq. (4) and used that the double integral is zero because the two-particle density is a symmetric function, $\varrho_{\rm st}^{(2)}(x,y)=\varrho_{\rm st}^{(2)}(y,x)$. Hence,

[TABLE]

This has an Onsager form fully analogous to Eq. (17) but now for the period-averaged quantities. It follows that $\overline{j_{\rm st}^{\rm\scriptscriptstyle TW}}$ and $v_{\rm w}$ must have the same sign for traveling-wave potentials. The above derivations can be generalized to systems with short-range conservative interaction forces.

The absence of current reversals imply upper bounds for the magnitudes of the currents: From Eq. (23) we obtain $\overline{j_{\rm st}^{\rm\scriptscriptstyle TW}}v_{\rm w}=-(j_{\rm st}-\mu f\overline{\varrho_{\rm st}})\mu f>0$ and with $j_{\rm st}f>0$ this implies $|j_{\rm st}|<\mu|f|\overline{\varrho_{\rm st}}=\mu|f|\rho/\lambda$. From Eq. (17) it follows $j_{\rm st}f=-(\overline{j_{\rm st}^{\rm\scriptscriptstyle TW}}-v_{\rm w}\overline{\varrho_{\rm st}})v_{\rm w}/\mu>0$ and with $\overline{j_{\rm st}^{\rm\scriptscriptstyle TW}}v_{\rm w}>0$ this implies $|\overline{j_{\rm st}^{\rm\scriptscriptstyle TW}}|<\overline{\varrho_{\rm st}}|v_{\rm w}|=\rho|v_{\rm w}|/\lambda$. These bounds can be expected: In a system with constant drag force, it means that the magnitude of the current can never exceed that of independent particles in a flat potential ($U=0$). In a traveling-wave system, it means that the magnitude of the current can never exceed that of particles coherently co-moving with the wave.

III.4 Recurrent dynamics in periodicity intervals of the particle diameter in the BASEP

In the BASEP, the complete range of densities (filling factors) $\rho$ and particle diameters $\sigma$ covers the range $0\leq\sigma<\infty$ and $0\leq\rho\leq\lambda/\sigma$. We now discuss that for a closed system with periodic boundary conditions, it is sufficient to know the behavior in the reduced range $0\leq\sigma<\lambda$ because of a recurrent dynamics in successive intervals separated by integer multiples of the diameter, i.e., for $m\lambda<\sigma\leq(m+1)\lambda$, $m=1,2,\ldots$ Lips et al. (2018).

The reason for this is as follows. Let us consider a system with particle diameter $\sigma^{\prime}>\lambda$ and denote by $m=\mathrm{int}(\sigma^{\prime}/\lambda)$ the integer number of periods fitting into $\sigma^{\prime}$. By applying the coordinate transformation $x_{i}^{\prime}\to x_{i}=x_{i}^{\prime}-im\lambda$ to the Langevin equations (1), the external forces remain the same, $f^{\rm\scriptscriptstyle ext}(x_{i}^{\prime})=f^{\rm\scriptscriptstyle ext}(x_{i}+im\lambda)=f^{\rm\scriptscriptstyle ext}(x_{i})$, and the hardcore constraints become $|x_{i+1}-x_{i}|\geq\sigma^{\prime}-m\lambda=\sigma$. Moreover, the confinement $(x_{N}^{\prime}-x_{1}^{\prime})<(L^{\prime}-\sigma^{\prime})$ transforms into $(x_{N}-x_{1})<L^{\prime}-(N-1)m\lambda-\sigma^{\prime}=(L^{\prime}-Nm\lambda)-\sigma$, corresponding to a change $L^{\prime}\to L=L^{\prime}-Nm\lambda$ of the system length.

Hence there is a one-to-one mapping of probabilities of paths $\{x_{i}^{\prime}(t)\}$ and $\{x_{i}(t)=x_{i}^{\prime}(t)-im\lambda\}$ between a system with $\sigma^{\prime}$ and $L^{\prime}$ and a system with $\sigma=\sigma^{\prime}-m\lambda$ and $L=L^{\prime}-Nm\lambda$ (for fixed particle number $N$),

[TABLE]

The arguments after the semicolon indicate the respective system length and particle diameter. Knowing the behavior of an observable for diameters $0\leq\sigma<\lambda$ and for all densities $0\leq\rho\leq\lambda/\sigma$, one can infer the behavior of related observables for all particle diameters in the range $m\lambda\leq\sigma<(m+1)\lambda$ for $m\neq 0$.

In the thermodynamic limit, relations connecting systems at different $L=M\lambda$ can be rewritten in terms of relations connecting systems at different $\rho$. The change in system size then transfers into a change $\rho^{\prime}=N/M^{\prime}\to\rho=N/M=\rho^{\prime}/(1-\rho^{\prime}m)$ of the density. Specifically, for the relevant steady-state quantities considered below, namely the local density $\varrho_{\rm st}(x;\rho,\sigma)$ and the current $j_{\rm st}(\rho,\sigma)$, we obtain [with $m=\mathrm{int}(\sigma^{\prime}/\lambda)$]

[TABLE]

Equation (25b) can be derived as follows: The Langevin equations (1) remain unchanged under the coordinate transformation $x_{i}^{\prime}\to x_{i}=x_{i}^{\prime}-im\lambda$, which implies equal mean velocities $v_{\rm st}$ in the original and transformed system. Then, using $j_{\rm st}^{\prime}/\rho^{\prime}=v_{\rm st}=j_{\rm st}/\rho$ and Eq. (25a) yields Eq. (25b).

Equations (25) imply a commensurability effect for particle diameters equal to integer multiples of the periodicity length $\lambda$, where the dynamics becomes reducible to that for $\sigma=0$. In this case of hardcore interacting point particles, it is obvious from Eq. (5) that $f^{\rm\scriptscriptstyle int}(x,t)$ vanishes for all $x$, and Eqs. (2) and (3) reduce to the equations for Brownian motion of noninteracting particles subject to the external force $f^{\rm\scriptscriptstyle ext}$. Accordingly, $j_{\rm st}(\rho,m\lambda)$, $m=0,1,\ldots$, is equal to the current $j_{0}(\rho)$ for noninteracting particles. This current $j_{0}(\rho)$ is linearly dependent on the particle density, $j_{0}=v_{0}\rho$, where Ambegaokar and Halperin (1969)

[TABLE]

is the mean velocity of a single particle in the steady state.

An alternative way to reason $j_{\rm st}(\rho,0)=j_{0}(\rho)$ is to resort to the invariance of collective properties under particle exchange for hardcore interacting point particles Ryabov and Chvosta (2011). We thus refer to this behavior as exchange symmetry effect. It can be explained by the consideration of path probabilities: To a path $\mathcal{P}$ in a system $\mathcal{S}$ with hardcore constrains one can assign the set $\{\mathcal{P}^{\prime}\}$ of all paths $\mathcal{P}^{\prime}$ in a system $\mathcal{S}^{\prime}$ of independent particles that result from particle exchanges at all contact points of individual particle trajectories in $\mathcal{P}$. Because the probability for the set $\{\mathcal{P}^{\prime}\}$ in $\mathcal{S}^{\prime}$ is equal to the probability of the path $\mathcal{P}$ in $\mathcal{S}$, it follows that averages of collective quantities, like the current, are equal in $\mathcal{S}^{\prime}$ and $\mathcal{S}$.

IV Current in closed system: Specific example

We now turn to a specific implementation and demonstrate the transport properties and phase transitions in a BASEP with the periodic potential

[TABLE]

As units we choose $\lambda$ for length, $\lambda^{2}/D$ for time, and $k_{\rm\scriptscriptstyle B}T$ for energy (and accordingly $k_{\rm\scriptscriptstyle B}T/\lambda$ for forces). Unless specified otherwise, we set the barrier height $U_{0}=6k_{\rm\scriptscriptstyle B}T$ and the drag force $f=1\,k_{\rm\scriptscriptstyle B}T/\lambda$. The barrier height $U_{0}\gg k_{\rm\scriptscriptstyle B}T$ leads to an effective hopping motion between the potential wells, which resembles the discrete hopping motion in the lattice ASEP, see the trajectories in Fig. 1.

In the simulations of the closed BASEP the system length is set to $L=100\lambda$ and the time step of the simulation scheme to $\Delta t=10^{-4}\lambda^{2}/D$. We checked that our results are not affected by the finite system length and the chosen time step. The current is determined by counting the number of particles per unit time crossing a point of the system in the steady state (and by averaging over many different points to enhance the accuracy). We performed extensive Brownian dynamics simulations for a fine grid of $(\sigma,\rho)$-values covering the area $\{(\sigma,\rho)|\,0\leq\sigma\leq\lambda,0<\rho<\lambda/\sigma\}$. Simulated data presented in the following figures are always shown as quasicontinuous lines. When referring to analytical results, this is explicitly stated.

IV.1 Dependence on particle size and density

As mentioned in the Introduction, we here give a description of the current, which includes all particle sizes and also densities $\rho$ in the range $\mathrm{int}(\sigma+1)<\rho<\lambda/\sigma$, i.e., regimes not considered in our previous work Lips et al. (2018). Representative examples of current-density relations for several $\sigma$ are shown in Fig. 2 for the density ranges (a) $0\leq\rho\leq 1$ and (b) $1\leq\rho\leq 8$.

In the low density limit $\rho\to 0$, particle interactions become negligible and the current-density relations approach the linear relation $j_{0}(\rho)=v_{0}\rho$ of noninteracting particles [solid black line in Fig. 2(a)], where Eq. (26) yields $v_{0}=0.043D/\lambda$ for our parameters. Beyond this common asymptotic limit for all $\sigma$, the form of the current-density relation varies strongly with the particle size. For comparison with the lattice model, the parabolic current-density relations of a corresponding ASEP $j_{\rm\scriptscriptstyle ASEP}(\rho)=v_{0}\rho(1-\rho)$ is shown in Fig. 2(a) also (dashed black line).

The change of the current-density relation with the particle size is due to the interplay of three physical effects Lips et al. (2018): a barrier reduction, a blocking and the exchange-symmetry effect. The barrier reduction effect occurs if a potential well is occupied by more than one particle. It leads to a current enhancement compared to $j_{0}$, because the particles in the well are pushing each other to regions of higher potential energy. Thereby the effective barrier for a transition to the neighboring wells is reduced. In contrast, the blocking effect lowers the current: if a particle attempts a transition to the next potential well, its motion can be blocked by a particle already occupying the neighboring well. The exchange-symmetry effect emerges as a result of the exact invariance of the current under particle exchange for commensurable diameters $\sigma=m\lambda$, as explained Sec. III.4. It leads to a continuous deformation of the curved current-density relation for $\sigma\lesssim m\lambda$ toward the linear dependence for $\sigma=m\lambda$ [see the curve for $\sigma=0.99\lambda$ in Fig. 2(a)]. The exchange symmetry effect thus becomes already notable for $\sigma\lesssim\lambda$.

To understand how these effects influence $j_{\rm st}(\rho,\sigma)$ in Fig. 2(a), we discuss the curves with increasing particle size. For small $\sigma$, $j_{\rm st}(\rho,\sigma)$ is monotonically increasing with $\rho$ and always larger than $j_{0}(\rho)$, see the curve for $\sigma=0.21\lambda$ in Fig. 2(a). The enhancement compared to $j_{0}(\rho)$ is due to the barrier reduction effect, which prevails for small $\sigma$ because of a high multi-occupation probability of potential wells. With increasing $\sigma$, the influence of the blocking effect becomes stronger, which leads to currents smaller than $j_{0}(\rho)$ at intermediate $\sigma$ and not too high $\rho$. In this regime, $j_{\rm st}(\rho,\sigma)$ is still monotonically increasing with $\rho$, see the curve for $\sigma=0.47\lambda$ in Fig. 2 (a). The strong rise of $j_{\rm st}(\rho,\sigma)$ at larger $\rho$ values is caused by double occupancies that are propagating through clusters of single occupied wells in a cascade like manner Lips et al. (2018); Ryabov et al. (2019).

Beyond a certain particle size $\sigma_{\rm\scriptscriptstyle c}$, a local maximum and local minimum appears in $j_{\rm st}(\rho,\sigma)$ at densities $\rho_{\rm\scriptscriptstyle max}=\rho_{\rm\scriptscriptstyle max}(\sigma)$ and $\rho_{\rm\scriptscriptstyle min}=\rho_{\rm\scriptscriptstyle min}(\sigma)$, see the curves for $\sigma=0.61\lambda$ in Fig. 2(a). When further enlarging $\sigma$, the blocking effect dominates the behavior for all $0\leq\rho\leq 1$ and the current-density relations approach $j_{\rm\scriptscriptstyle ASEP}(\rho)$ as a limiting curve with a maximum at $\rho\simeq 0.5$. This occurs in the range $0.74\lambda\lesssim\sigma\lesssim 0.82\lambda$ for our setup. Close to the commensurate diameter $\sigma=\lambda$, the exchange symmetry effect becomes relevant. As a consequence, the position of the maximum in $j_{\rm st}(\rho,\sigma)$ moves to higher densities and the current approaches $j_{0}(\rho)$ from below.

If the number of particles exceeds the number of potential wells, i.e., when $\rho>1$, then the particles cannot all be localized close to a potential minimum and double or multi-occupied wells are permanently present. This leads to a strong increase of the particle current with $\rho$ toward values much larger than those seen in Fig. 2(a) for $\rho<1$ [note the different scales of the axes for the current in Figs. 2(a) and 2(b)]. The upper bound $\mu\rho f/\lambda$ (see Sec. III.3) is shown as the dashed black line in Fig. 2 (b). In the limit $\rho\rightarrow\lambda/\sigma$ of complete filling, the curves approach the upper bound from below.

How the influence of the barrier reduction, blocking and exchange symmetry effects changes with the particle size becomes particularly transparent when plotting the relative change $\Delta j_{\rm st}(\rho,\sigma)=[j_{\rm st}(\rho,\sigma)-j_{0}(\rho)]/j_{0}(\rho)$ of the current with respect to $j_{0}(\rho)$ as a function of $\sigma$ for different fixed $\rho$. Corresponding curves shown in Fig. 3 all display a local maximum at a value $\sigma_{\rm\scriptscriptstyle max}(\rho)$ and show a plateau like behavior in an intermediate $\sigma$ range. For $\sigma$ below the onset of the plateau like regime, the barrier reduction and the blocking effect compete with each other, where the barrier reduction governs the change of $\Delta j_{\rm st}$ for $\sigma<\sigma_{\rm\scriptscriptstyle max}$ and the blocking effect for $\sigma>\sigma_{\rm\scriptscriptstyle max}$. In the plateau like regime, the barrier reduction effect becomes almost negligible. For $\sigma\to\lambda$ the exchange symmetry effect causes the curves to increase back to $j_{0}(\rho)$, which is reached at $\sigma=\lambda$.

Having described the current for all densities $0\leq\rho\leq\lambda/\sigma$ in the range of particle sizes $0\leq\sigma\leq\lambda$, we are able to use the mapping in Eq. (25b) to derive $j_{\rm st}(\rho,\sigma)$ for diameters $\sigma>\lambda$ at arbitrary densities. This is exemplified in Fig. 4, where we show $j_{\rm st}(\rho,\sigma)$ as function of $\sigma$ for a fixed density $\rho=0.2$. The curve (blue line) for $\sigma>\lambda$ was calculated from the simulated data in the range $0\leq\sigma\leq\lambda$ and $0\leq\rho\leq\lambda/\sigma$ by applying Eq. (25b). To demonstrate the validity of this equation, we performed additional simulations for certain $\sigma>\lambda$ for which the current (black circles) is shown in Fig. 4.

The emerging recurrence pattern in the $\sigma$-$\rho$–plane is shown in Fig. 5, where values of the current are represented by a color coding. Note that this pattern is not periodic in $\sigma$ (see also Fig. 4), but has a more complicated structure because of the necessary rescaling of the density.

When comparing the behavior of the steady-state current in the BASEP and ASEP, the blocking effect is present in both models. In contrast, the barrier reduction effect has no analog in the lattice model, because multi-occupation of a site is forbidden in the standard ASEP. The exchange-symmetry effect is also absent in lattice models, even if one introduces a generalized $l$-ASEP, where the particles occupy $l$ lattice sites. This is because the particle size is an integer multiple of the lattice constant and accordingly there is no continuous transition toward a commensurate diameter.

Nevertheless, the reasoning in Sec. III.4 can be taken over to the $l$-ASEP with $l>1$, $N$ particles, $M$ lattice sites with periodic boundary conditions, and hopping rates $\Gamma_{+}$ and $\Gamma_{-}$ in and against bias direction. A transformation $l\to 1$ and $M\to M-N(l-1)$ corresponds to the transformation considered in Sec. III.4 for the BASEP with $m=(l-1)$. Hence from Eq. (25b) and with $j_{\rm\scriptscriptstyle ASEP}(\rho)=(\Gamma_{+}-\Gamma_{-})\rho(1-\rho)$ we obtain

[TABLE]

The reduction from an $l$-ASEP to the standard ($l=1$) ASEP has been used in the literature before to obtain the current-density relation Schönherr and Schütz (2004). In contrast to the equality of the current in the BASEP for commensurate $\sigma=m\lambda$, $m=0,1,2,\ldots$, the currents in the $l$-ASEP change with $l$. This is because $j_{\rm\scriptscriptstyle ASEP}(\rho)$ for the smallest length $l=1$ in the $l$-ASEP is a nonlinear function of $\rho$, while $j_{0}(\rho)$ for the smallest length $\sigma=0$ (point particles) in the BASEP varies linearly with $\rho$.

IV.2 Temperature dependence

With decreasing temperature, the particles become stronger localized at the minima of the potential wells. In the $\sigma$ regime of dominant blocking effect, the current-density relation therefore follows more closely $j_{\rm\scriptscriptstyle ASEP}$. This causes the plateau like regimes discussed in connection with Fig. 3 to extend and to become extremely flat in the zero temperature (low-noise) limit.

However, the stronger localization at the minima of the potential does not mean that the barrier reduction effect disappears for small $\sigma$. In relation to the current $j_{0}$ of a single particle, double occupancies of wells lead to an enhancement at arbitrary low temperatures. As the barrier reduction should become almost independent of temperature at low $T$, $j_{\rm st}/j_{0}$ will even be larger for lower temperatures. Hence, when considering the dependence of $j_{\rm st}/j_{0}$ on temperature for decreasing $T$, we expect a decrease for large $\sigma$ where the blocking effect prevails, and an increase for small $\sigma$ where the barrier reduction dominates. This is indeed the case and demonstrated in Fig. 6, which shows $j_{\rm st}/j_{0}$ as a function of $U_{0}/(k_{\rm\scriptscriptstyle B}T)$ for different $\sigma$ at a fixed density $\rho=0.52$ (and the same ratio $f\lambda/U_{0}=1/6$ as considered before).

IV.3 Analytical approaches in the limit of small driving

Using the exact expression for the steady-state current in Eq. (6), we can calculate an approximation for small drag forces. In an equilibrium system ($f=0$), $j_{\rm st}=0$ and $\overline{f_{\rm st}^{\rm\scriptscriptstyle int}}=0$. In the linear-response regime, we thus obtain from Eq. (6)

[TABLE]

where $\varrho_{\rm\scriptscriptstyle eq}(x)$ is the local density profile in equilibrium. The equilibrium density profile (in the grand-canonical ensemble) can be obtained by minimizing the exact density functional

[TABLE]

for hard rods in one dimension Percus (1976). Here, $\mu_{\rm\scriptscriptstyle ch}$ is the chemical potential and

[TABLE]

Minimizing $\Omega[\varrho(x)]$ in Eq.(30) yields

[TABLE]

We discretized this equation and solved it numerically under periodic boundary conditions [$\varrho_{\rm\scriptscriptstyle eq}(x)=\varrho_{\rm\scriptscriptstyle eq}(x+\lambda)$] to obtain the density profile for a given chemical potential $\mu_{\rm\scriptscriptstyle ch}$ (or density $\rho$). Inserting the solution for $\varrho_{\rm\scriptscriptstyle eq}(x)$ in Eq. (29) and setting $\alpha=0$ in Eq. (29), we obtain a small-driving approximation (SDA) for $j_{\rm st}(\rho,\sigma)$.

Keeping nonzero $\alpha$ requires more advanced approaches. A widely used method is the dynamical density functional theory (DDFT) Marconi and Tarazona (1999, 2000), where the two-particle density at contact in Eq. (5) is related to the single-particle density as in an equilibrium system:

[TABLE]

Here, $\eta(x,t)$ is given by Eq. (31) with $\varrho(y)$ replaced by $\varrho(y,t)$. Combining Eqs. (2), (3), (5) and (33) results in a nonlinear and nonlocal evolution equation for the density:

[TABLE]

To find its stationary solution we used two methods. First, we propagated an initial density profile into the stationary regime with a forward-time central-space scheme. Secondly, we solved the corresponding stationary equation ($\partial\varrho/\partial t=0$) with an iterative scheme. Both methods lead to equal results.

Current-density relations obtained with the SDA, DDFT and Brownian dynamics simulations with the algorithms given in Ref. Behringer and Eichhorn (2012) (BDS1) and Ref. Scala (2012) (BDS2) are shown in Fig. 7 for three particle diameters $\sigma=0.2\lambda$ (blue), $\sigma=0.5\lambda$ (orange), and $\sigma=0.85\lambda$ (yellow) in the small bias regime at $f=0.2k_{\rm\scriptscriptstyle B}T/\lambda$. We see that the DDFT results are almost indistinguishable from the SDA results for all shown diameters, and the results from BDS1 and BDS2 are in very good agreement. Overall the SDA and DDFT capture the qualitative features of the current-density relations.

However, comparing the results of the SDA and DDFT with that of the simulations quantitatively, we observe deviations that become largest for intermediate particle sizes close to $\sigma=0.5\lambda$. The reason is that both the SDA and DDFT underestimate the magnitude of the period-averaged mean interaction force $\overline{f_{\rm st}^{\rm\scriptscriptstyle int}}$, which we determined from BDS1. To get insight into its behavior, we show in the inset of Fig. 7 $\overline{f_{\rm st}^{\rm\scriptscriptstyle int}}$ as a function of $\sigma/\lambda$ for different densities. As can be seen from the data, $\overline{f_{\rm st}^{\rm\scriptscriptstyle int}}$ is always opposite to the bias $f$, in agreement with our discussion in Sec. III.3. Its magnitude is largest close to $\sigma=0.5\lambda$. The corresponding minimum shifts slightly to higher $\sigma$ and becomes more pronounced with increasing $\rho$.

Closer inspection shows that the local mean interaction force $f_{\rm st}^{\rm\scriptscriptstyle int}(x)$ in the steady state can be both parallel and antiparallel to the drag force $f$ and that it is always small close to the local extrema of the external potential, where $f^{\rm\scriptscriptstyle ext}(x)\simeq 0$. This is demonstrated in Fig. 8, where we show representative profiles $f_{\rm st}^{\rm\scriptscriptstyle int}(x)$ for different $\rho$ and $\sigma$. The shape of these profiles changes significantly with the particle diameter. For small $\sigma=0.1\lambda$ [Fig. 8(a)], a region of negative and positive $f_{\rm st}^{\rm\scriptscriptstyle int}(x)$ occurs next to the minimum of the external potential in ($x\lesssim 0.5\lambda$) and against bias ($x\gtrsim 0.5\lambda$) direction, respectively. At an intermediate $\sigma=0.5\lambda$ [Fig. 8(b)], these regions of negative and positive $f_{\rm st}^{\rm\scriptscriptstyle int}(x)$ have shifted to locations close to the potential barriers, and an extended regime of negligible $f_{\rm st}^{\rm\scriptscriptstyle int}(x)\simeq 0$ appears around $x=0.5\lambda$. At large $\sigma=0.9\lambda$ [Fig. 8(c)], the profile from Fig. 8(a) appears to be kind of inverted, with now a region of positive and negative $f_{\rm st}^{\rm\scriptscriptstyle int}(x)$ occurring for $x\lesssim 0.5\lambda$ and $x\gtrsim 0.5\lambda$, respectively. As for the density dependence, it changes the magnitude of $f_{\rm st}^{\rm\scriptscriptstyle int}(x)$ along with a shift of the positions of its local minima and maxima.

These changes of the profiles $f_{\rm st}^{\rm\scriptscriptstyle int}(x)$ can be understood by noting that for small $\sigma$, a particle located at a position left (right) of the potential minimum collides with other particles in multiple-occupied wells, which more frequently are coming from the right (left). This leads to a mean repulsive force that tends to push the particle further away from the potential minimum, i.e., we obtain $f_{\rm st}^{\rm\scriptscriptstyle int}(x)<0$ for $x\lesssim 0.5\lambda$ and $f_{\rm st}^{\rm\scriptscriptstyle int}(x)>0$ for $x\gtrsim 0.5\lambda$. At intermediate $\sigma$, multiple occupancies become unlikely and the particles preferentially occupy positions close to the potential minima. A particle positioned near a potential minimum rarely collides with other particles so that $f_{\rm st}^{\rm\scriptscriptstyle int}(x)\simeq 0$ in a region around the minimum. Particles located close to the potential barriers now interact most strongly with other particles. For large $\sigma$, the blocking effect pushes the particles toward their minima, i.e., we find $f_{\rm st}^{\rm\scriptscriptstyle int}(x)>0$ for $x\lesssim 0.5\lambda$ and $f_{\rm st}^{\rm\scriptscriptstyle int}(x)<0$ for $x\gtrsim 0.5\lambda$.

V Phase transitions in open system

V.1 Extremal current principles

A striking feature of the ASEP is the occurrence of nonequilibrium phase transitions in open systems coupled to particle reservoirs Krug (1991); Schütz and Domany (1993). These phase transitions manifest themselves as discontinuous changes of the bulk density in dependence of control parameters specifying the coupling to the reservoirs, or by jumps in the derivatives of the bulk densities with respect to these control parameters. How theses phases change with experimentally tunable control parameters depends on details of the system reservoir couplings Dierl et al. (2012, 2013). However, all possible phases can be derived from extremal current principles Krug (1991); Antal and Schütz (2000); Dierl et al. (2013). The arguments leading to these principles are quite general for driven diffusive systems and are valid also for driven Brownian motion Maass et al. (2018).

We focus here on an open system with $M$ potential wells $i=1,\ldots,M$ coupled to reservoirs L and R at its left ($i=1$) and right ($i=M$) end. The “filling factor” of the $i$th well in the steady state is given by (note that $\varrho_{\rm st}(x)$ is not a periodic function in the open system)

[TABLE]

and we refer to this as period-averaged density profile.

In the thermodynamic limit ($M\to\infty$), this profile $\rho_{i}$ exhibits a constant bulk value $\rho_{\rm\scriptscriptstyle b}$ in the interior of the system, which defines the order parameter of the phase transitions. For finite large $M$, $\rho_{\rm\scriptscriptstyle b}$ refers to the density of an extended plateau like region in the interior of the system. Using the extremal current principles, $\rho_{\rm\scriptscriptstyle b}$ is given by

[TABLE]

Here $\rho_{\rm\scriptscriptstyle L}$ and $\rho_{\rm\scriptscriptstyle R}$ can be any densities bounding a monotonically varying region encompassing the plateau part from the left and right side. For globally monotonic profiles in particular, it is possible to interpret $\rho_{\rm\scriptscriptstyle L}$ and $\rho_{\rm\scriptscriptstyle R}$ as reservoir densities.

In general, however, interaction effects lead to density oscillations close to the boundaries, and only a specific coupling called “bulk-adapted” ensures a global monotonic behavior and a controlled generation of all possible phases Dierl et al. (2012, 2013). While such bulk-adapted coupling can be realized in a systematic manner in lattice models, its implementation in Brownian dynamics of interacting particles remains challenging. We will take a pragmatic approach and present in Sec. V.3 a simple coupling scheme of particle injection and ejection for the simulation of the phases.

V.2 Phases derived from extremal current principles

Application of the extremal current principles (36) to the simulated current-density relations of the closed system yields the phase diagrams for the open system. The change of the shape $j_{\rm st}(\rho,\sigma)$ with the particle diameter leads to different types of diagrams. Representative examples are shown in Figs. 9(a)-9(d). For $\sigma<\sigma_{\rm c}$, no phase transitions occur, because the current as a function of density exhibits no local extrema. For $\sigma\gtrsim\sigma_{\rm c}$, the maximum number of five phases appears, which we labeled I-V in Fig. 9(a). These phases are colored equally in all other panels. Phases I, III and V are boundary-matching with $\rho_{\rm\scriptscriptstyle b}=\rho_{\rm\scriptscriptstyle L}$ in phases I and V, and $\rho_{\rm\scriptscriptstyle b}=\rho_{\rm\scriptscriptstyle R}$ in phase III. Phases II and IV are maximal and minimal current phases with $\rho_{\rm\scriptscriptstyle b}=\rho_{\rm\scriptscriptstyle max}$ and $\rho_{\rm\scriptscriptstyle b}=\rho_{\rm\scriptscriptstyle min}$, respectively ($\rho_{\rm\scriptscriptstyle max}$ and $\rho_{\rm\scriptscriptstyle min}$ are the densities at which the current-density relation has a local maximum and minimum, see Sec. IV.1). Solid lines mark first-order transitions, where $\rho_{\rm\scriptscriptstyle b}$ changes discontinuously when the line is crossed, and dashed lines mark second-order transition, where $\rho_{\rm\scriptscriptstyle b}$ varies continuously but the gradient of $\rho_{\rm\scriptscriptstyle b}$ with respect to $\rho_{\rm\scriptscriptstyle L}$ and $\rho_{\rm\scriptscriptstyle R}$ changes discontinuously.

With increasing $\sigma>\sigma_{\rm c}$, the phases IV and V shrink [Fig. 9(b)]. This shrinkage continues up to the point where the phase diagram is similar to the one for a corresponding ASEP with three phases [Fig. 9(c)]. Close to the commensurate diameter, the diagram with the three phases becomes asymmetric due to the change of $\rho_{\rm\scriptscriptstyle max}(\sigma)$ [Fig. 9(d)].

V.3 Simulated phases

To verify the phase diagrams derived from the extremal current principles in the Brownian dynamics simulations, we consider particles of size $\sigma=0.58\lambda$, where Fig. 9(a) gives the predicted phase diagram. For the exchange of particles with the reservoirs, we used the following method: If the leftmost (rightmost) potential well is empty, then a particle is injected with a rate $\alpha_{\rm\scriptscriptstyle L}$ ($\alpha_{\rm\scriptscriptstyle R}$). Injected particles are placed at the distance $(\lambda-\sigma)$ away from the boundary. This guarantees that no particle overlaps occur. Ejection of a particle to a reservoir is implemented by removing it from the system once its center position crosses the left or right boundary.

In a simulation with fixed ($\alpha_{\rm\scriptscriptstyle L}$, $\alpha_{\rm\scriptscriptstyle R}$) the system approaches a steady state with density profile $\rho_{i}$ [Eq. (35)] after an initial transient time. Examples of simulated density profiles for the nonequilibrium phases I-IV are shown in Fig. 10(a)-10(d). The bulk density is extracted from an average of $\rho_{i}$ around the center of the system, $\rho_{\rm\scriptscriptstyle b}=\sum_{i=L/2-d}^{L/2+d}\rho_{i}/(2d)$ with $d=5$. These bulk values can be compared with the predicted ones. In all Figs. 10(a)-10(d) we obtain a very good agreement.

When varying the injection rates $\alpha_{\rm\scriptscriptstyle L}$ and $\alpha_{\rm\scriptscriptstyle R}$, the period-averaged boundary densities in the leftmost and rightmost potential well change. Hence, each simulation run with given $(\alpha_{\rm\scriptscriptstyle L},\alpha_{\rm\scriptscriptstyle R})$ results in one set of period-averaged boundary and bulk densities $(\rho_{\rm\scriptscriptstyle L},\rho_{\rm\scriptscriptstyle R},\rho_{\rm\scriptscriptstyle b})$ in the nonequilibrium steady state 222In most simulations, the period-averaged density profile $\rho_{i}$ turned out to be monotonically varying with $i$. Only for very high $\alpha_{\rm\scriptscriptstyle L}$ and $\alpha_{\rm\scriptscriptstyle R}$ small dips appear close to the system boundaries. Then $\rho_{\rm\scriptscriptstyle L}$ and $\rho_{\rm\scriptscriptstyle R}$ were determined from the region of monotonically varying profile.. By performing many simulation runs for different $(\alpha_{\rm\scriptscriptstyle L},\alpha_{\rm\scriptscriptstyle R})$, the various phases can be identified.

Correspondingly simulated data points in a system of length $L=200\lambda$ are shown in Fig. 11 together with the surface $\rho_{\rm\scriptscriptstyle b}=\rho_{\rm\scriptscriptstyle b}(\rho_{\rm\scriptscriptstyle L},\rho_{\rm\scriptscriptstyle R})$ obtained from the extremal current principles (36). As can be seen from the figure, the simulated values for the bulk density (black points) lie very closely to the predicted surface $\rho_{\rm\scriptscriptstyle b}(\rho_{\rm\scriptscriptstyle L},\rho_{\rm\scriptscriptstyle R})$. In particular, the discontinuous first-order transitions from the left-boundary induced phase I to the minimal current phase IV and to the right-boundary induced phase III are clearly visible, as well as the continuous transitions between phase I and the maximal current phase II. Also the continuous transitions between phases III and IV, and between phases II and III can be seen in the simulated data. As for phase V, our simple injection and ejection method did not generate the very high boundary densities $\rho_{\rm\scriptscriptstyle L}$ in this phase.

VI Conclusions

The BASEP represents a wide class of simple models for Brownian single-file transport in periodic structures, designed to explore and understand basic physical mechanisms of nonequilibrium driven motion. We expect that the competition between the barrier reduction, blocking and exchange symmetry effect plays a decisive role in systems with all kinds of short-ranged interactions. The BASEP can thus serve as a reference for more complicated nonequilibrium systems, similar as the hard-sphere fluid became a useful basis in equilibrium liquid theory. One remaining challenge is to develop better analytical approaches for the local mean interaction force acting on a particle. With that at hand, perturbative treatments for interactions beyond hardcore exclusion could be developed.

We have focused in this work on a sinusoidal form of the external potential. In the approximate analytical treatment, arbitrary forms of the periodic potential $U(x)$ can be used by inserting it in the density functional in Eq. (30). Based on our findings for the sinusoidal potential, we conjecture that this method will capture the transport behavior on a qualitative level. A better quantitative agreement requires improved theories for the mean interaction force.

As for the connection of the thermally activated transport in the BASEP to the hopping motion in lattice models, one can think of developing extended jump models. The barrier reduction effect, absent in the standard ASEP, can, for example, be incorporated by allowing for different internal states of the particles that correspond to the different occupancies of the potential wells. Another more obvious approach is to discretize the potential energy landscape in space. However, earlier results indicate that in such models current reversals, absent in the BASEP, become possible Chaudhuri and Dhar (2011); Dierl et al. (2014). Hence, qualitative features can be different in continuum and related lattice models. A deeper understanding of the correspondence of continuum and discrete models in single-file transport should be sought of in future investigations.

A welcome feature of the BASEP is that it describes physics of biased single-file motion generated by laser or magnetic fields in confined geometries, and by flow fields in microfluidic devices. Our results shall help to interpret experimental findings in such systems. Experimental studies in open systems offer the possibility to investigate nonequilibrium phase transitions predicted by theory under well-controlled conditions. A further interesting aspect, both for experimental and theoretical work, is to study the implications of local disturbances in the periodic structure similar as they were investigated in corresponding lattice models Kolomeisky (1998); Brankov et al. (2004).

Acknowledgements.

Financial support by the Czech Science Foundation (Project No. 17-06716S) and the Deutsche Forschungsgemeinschaft (Project No. 397157593) is gratefully acknowledged. A.R. acknowledges financial support from the Portuguese Foundation for Science and Technology (FCT) under Contracts Nos. PTDC/FIS-MAC/28146/2017 (LISBOA-01-0145-FEDER-028146) and UID/FIS/00618/2019. We sincerely thank the members of the DFG Research Unit FOR 2692 for fruitful discussions.