Nonequilibrium Transport and Phase Transitions in Driven Diffusion of Interacting Particles

1 Introduction

Driven diffusive systems of interacting particles constitute an important class of systems to study fundamental aspects of nonequilibrium physics. This holds in particular for one-dimensional models, where exact analytical derivations are possible or reliable approximations are known, for example, when information about exact equilibrium properties can be utilised for the treatment of nonequilibrium states.

A prominent model in the field of driven diffusive systems is the asymmetric simple exclusion process (ASEP), where particles hop between nearest-neighbour sites of a lattice with a bias in one direction and where the sole interaction between particles is a mutual site exclusion, implying that a lattice site cannot be occupied by more than one particle [1], [2]. In the ASEP on a one-dimensional lattice with L sites and periodic boundary conditions, i.e. a ring of L sites, particles jump to vacant nearest-neighbour sites with rates Γ+ and Γ− in clockwise and counterclockwise direction, respectively, where Γ+>Γ− for a bias in clockwise direction. In a corresponding open system with L sites, where the leftmost and rightmost lattice site can exchange particles with reservoirs L and R, respectively, additional rates ΓinL, ΓinR and ΓoutL, ΓoutR specify the corresponding rates for particle injection and ejection. Many properties of the ASEP can be inferred from the even simpler totally asymmetric simple exclusion process (TASEP) with unidirectional transport (Γ−=0).

Stochastic processes in driven lattice gases are described by a master equation for the probabilities of particle configurations, which can be viewed also as the occupation number representation of a Schrödinger equation in imaginary time [3], [4]. This leads to some interesting connections to quantum systems with in general non-Hermitian Hamilton operator H [2], [3], [5], [6]. As an example, we recapitulate in the Appendix the connection of the ASEP with periodic boundary conditions to the XXZ quantum spin chain with non-Hermitian boundary conditions [6]. Spin chains are often used to study fundamental aspects of nonequilibrium quantum physics. Several examples related to current problems, in particular to questions of equilibration in nonintegrable spin chain models, can be found in this special issue on the physics of non-equilibrium systems.

The ASEP has been intensively studied in the past. Let us summarise here some of the most important findings for the ASEP and variants of it:

1. Using the Bethe ansatz for corresponding quantum spin chain models, or a construction in terms of matrix product states, exact results for microstate distributions in nonequilibrium steady states (NESSs) could be derived [1], [2], [7]. Matrix product states in principle exist for driven lattice gases with arbitrary nearest-neighbour interactions [8], although their explicit construction may be difficult.

2. Based on the exact approaches for deriving distribution of microstates in NESS, large deviation functions for fluctuations of time-averaged densities and currents were derived [9], [10], [11]. They have been computed also for coarse-grained descriptions by the macroscopic fluctuation theory [12]. Large deviation functions are argued to play a similar role for time-averaged quantities in NESS as the free energy in equilibrium systems [13]. They can exhibit singularities [14], [15], [16], [17], sometimes referred to as “dynamical phase transitions,” which for certain classes of systems are caused by a violation of an “additivity principle” [18].

3. The Bethe ansatz turned out to be a valuable tool also for deriving microstate distributions of nonsteady states [19], [20]. The propagator for the microstate time evolution in the ASEP was related to integrated Fredholm determinants [21] and led to the derivation of the Tracy–Widom distribution of random matrix theory for the asymptotic behaviour in case of a step initial condition [22]. This result generalised an earlier one derived for the TASEP [23] and proved that the propagation of density fluctuations in the ASEP belongs to the Kardar–Parisi–Zhang (KPZ) universality class [24].

4. In open systems coupled to particle reservoirs, phase transitions between NESS occur [25], [26], [27], [28], where, upon change of control parameters characterising the system-reservoir couplings, the bulk density ρ_b changes discontinuously, or its derivative with respect to the control parameters. Knowing the density dependence of the steady-state bulk current jss(ρ), e.g. from results for a system with periodic boundary conditions, all possible NESS phases with bulk density ρ_b are predicted by the extremal current principles [25], [27], [29]:

   (1)ρb={argminρ−≤ρ≤ρ+{jss(ρ)},ρ−≤ρ+,argmaxρ+≤ρ≤ρ−{jss(ρ)},ρ+≤ρ−.

   Here ρ₋ and ρ₊ can be any densities bounding a monotonically varying region encompassing the plateau part with bulk density ρ_b (which may strictly exist only in the thermodynamic limit of infinite system size). Which of the phases predicted by (1) really occurs for a given control scheme of system-reservoir couplings is given by the dependence of ρ₋ and ρ₊ on respective control parameters.

   The extremal current principles can be reasoned based on the consideration of shock front motions [27], [29], [30] or by resorting to a decomposition of the steady-state current into its drift and diffusive part inside the region of monotonically varying density profile [25]. Because these reasonings do not require specific properties of the ASEP, they are quite generally valid for driven diffusive systems coupled to particle reservoirs. This includes driven lattice gases with interactions other than site exclusion [30], [31], [32], [33], systems with continuous space dynamics, and systems with periodic space structure and/or time-periodic driving, when considering period-averaged densities [34]. For specific system-reservoir couplings termed “bulk-adapted,” it is possible to parameterise the exchange of particles by reservoir densities such that all possible NESS phases must appear. The bulk-adapted couplings can be determined by a general method for driven lattice gases with short-range interactions [33], [34].

5. For random and non-Poissonian hopping rates, Bose–Einstein–type condensations of vacancies can occur in front of the slowest particle with the smallest jump rate [35], [36].

6. Coarse-grained continuum descriptions of the ASEP and of multilane variants [37] give rise to an infinite discrete family of nonequilibrium universality classes in nonlinear hydrodynamics, where density fluctuations spread in time by power laws with exponents given by the Kepler ratios of consecutive Fibonacci numbers [38]. This includes the KPZ class, for which an exact expression for the scaling function was derived [39]. Predictions of the theory of fluctuating nonlinear hydrodynamics were recently confirmed in an exact treatment by considering a two-species exclusion process [40].

As for applications, the ASEP appears as a basic building block in manifold descriptions of biological traffic [41], [42]. In fact, the ASEP was introduced first to describe protein synthesis by ribosomes [43], and it is frequently used in connection with the motion of motor proteins along microtubules or actin tracks [44], [45]. An in vitro study with fluorescently labelled single-headed kinesin motors moving along a microtubule provided experimental evidence for a state of coexisting phases with different motor densities [46]. Other applications concern vehicular traffic [47], [48], diffusion of ions through cell membranes [49] and of molecules through nanopores [50], [51], and electron transport along molecular wires in the incoherent classical limit [52], [53]. However, a direct experimental realisation of the ASEP is difficult, because of its discrete nature. Hence, it is important to see whether the nonequilibrium physics in the ASEP is reflected in models with continuous space dynamics.

For a single particle, it is well known that effective hopping transport emerges from an overdamped Brownian motion in a periodic potential with amplitude much larger than the thermal energy. The particle can be viewed to jump between neighbouring wells on a coarse-grained time scale with a rate determined by the inverse Kramers time [54]. One is thus led to ask whether the driven diffusion of many hardcore-interacting particles in a periodic energy landscape can reflect the driven lattice gas dynamics in the ASEP. To answer this question, we recently introduced a corresponding class of nonequilibrium processes termed Brownian ASEP (BASEP) [55], [56], where hard spheres with diameter σ are driven through a periodic potential with wavelength λ by a constant drag force f. For a sinusoidal external potential, we found that the current–density relation of the ASEP is indeed recaptured in the BASEP, but only for a limited range of particle diameters σ. For other σ, quite different behaviours are obtained.

The nonequilibrium physics of the BASEP should be explorable directly by experiment, for example, in setups utilising advanced techniques of microfluidics and optical and/or magnetic micromanipulation [57], [58], [59], [60], [61]. This includes arrangements where the particles are driven by travelling-wave (TW) potentials [62]. Many of the new collective transport properties seen in the BASEP can be even identified by studying local dynamics of individual transitions between potential wells [63].

In this work, we address the question how the current–density relations found for the BASEP in a sinusoidal external potential are affected when considering different external potentials and short-range interactions other than hardcore exclusions. Our investigation for the different external potentials is carried out based on the small-driving approximation introduced in [55] and [56]. With respect to short-range interactions other than hardcore exclusions, we focus on a Yukawa pair potential. It is shown that the current–density relation for single-well periodic potentials and for the Yukawa interaction has similar features as that of the BASEP. This suggests that the BASEP can serve as a reference model for a wide class of external periodic potentials and pair interactions.

In addition, we extend a former analysis to prove that current reversals cannot occur in systems driven by a constant drag and by travelling waves. These proofs are based on an exact calculation of the total entropy production in corresponding NESS for particles with arbitrary pair interactions. Current reversals refer to steady states, where particle flow is opposite to the external bias. They were reported for lattice models [34], [64], [65], [66], [67] and were recently found experimentally in a rocking Brownian motor [59]. Their absence in TW driven systems was conjectured based on simulation results and a perturbative expansion of the single-particle density in the NESS around its period-averaged value [68].

The article is organised as follows. In Section 2, Current–Density Relations: Analytical Results, we present an analytical treatment of densities and currents for the overdamped one-dimensional Brownian motion of particles with arbitrary pair interactions. This section partly summarises results presented earlier [55], [56] and introduces the small-driving approximation used subsequently for our investigation of hardcore-interacting particles. It also contains our proofs on the absence of current reversals for general pair interactions. In Section 3, Hardcore Interacting Particles in Harmonic Potential, we outline our findings for the BASEP with sinusoidal external potential, and in Section 4, Impact of External Periodic Potential, we contrast them with results for a Kronig–Penney and triple-well periodic potential. In Section 5, Impact of Interactions Other Than Hardcore Exclusions, we discuss our results for the Yukawa interaction. Section 6, Summary and Conclusions, concludes the article with a summary and outlook.

2 Current–Density Relations: Analytical Results

The overdamped single-file Brownian motion of N particles in a periodic potential U(x)=U(x+λ) with pair interaction under a constant drag force f is described by the Langevin equations:

where μ and D=μkBT are the bare mobility and diffusion coefficient, kBT is the thermal energy, and fiint is the interaction force on the i^th particle. The ηi(t) are independent and δ-correlated Gaussian white noise processes with zero mean and unit variance, ⟨ηi⟩=0 and ⟨ηi(t)ηj(t′)⟩=δijδ(t−t′). Unless noted otherwise, we consider closed systems with periodic boundary conditions, which means that the particles are dragged along a ring.

Hardcore interactions imply the boundary conditions |xi−xj|≥σ, i.e. overlaps between neighbouring particles are forbidden. For the BASEP with only hardcore interactions, these boundary conditions must be taken into account, while the interaction force fiint can be set to zero in (2). We define the density as a (dimensionless) filling factor of the potential wells, i.e. by ρ=N/M, where M denotes the total number of periods of U(x). The system length is L=Mλ, and the number density is ρ/λ. For hardcore-interacting particles of size σ, the filling factor ρ has the upper bound λ/σ.

The joint probability function (PDF) of the particle centre coordinates x=(x1,…,xN) evolves in time according to the N-particle Smoluchowski equation^[1],

where the divergence operator acts on the probability current vector J(x,t) with the i^th component given by

The first term describes the drift probability current caused by all forces acting on the i^th particle, and the second term gives the diffusive current. The interaction force is assumed to be conservative and due to pair interactions uint(xi,xj), i.e. fiint(x)=−∂Uint(x)/∂xi with

Additional hardcore interactions are not included in the potential (5) but are incorporated into the dynamics by requiring no-flux (reflecting) boundary conditions

if neighbouring particles hit each other. These boundary conditions ensure conservation of an initial ordering x1<x2<…<xN of the particle positions for all times.

3 Hardcore Interacting Particles in Harmonic Potential

The paradigmatic variant of the BASEP with hardcore-interacting particles diffusing in the external harmonic potential

has been studied thoroughly in our previous works [55], [56], [63]. Here, we review its basic properties that shall serve as a “reference case” for the following analysis.

In all illustrations, we fix units setting λ = 1 (defines units of length), λ2/D=1 (time), kBT=1 (energy); this implies that μ=D/(kBT)=1 also. We assume U0≫1, which leads to a hopping-like motion between potential wells that resembles the dynamics on a lattice.

Four representative shapes of current–density relations are shown in Figure 1. In the low-density limit, all curves collapse to the linear behaviour j0(ρ)=v0ρ with the slope given by the velocity v₀ of a single (noninteracting) particle. This is given by [75]

where β=1/(kBT). Beyond the small-ρ region, the shapes change strongly with the particle size σ. This complex behaviour is caused by three competing collective effects:

Figure 1:

Simulated steady-state current j_ss in the BASEP with cosine potential (36) as a function of the density ρ for different particle sizes σ. The solid black line marks the current of noninteracting particles j0(ρ)=v0ρ; and the dashed line, the current–density relation jASEP(ρ)=v0ρ(1−ρ) of a corresponding ASEP [v₀ = 0.043 from (37)].

Figure 2:

The different periodic externals potentials investigated for comparing current–density relations: (a) cosine (36), (b) Kronig–Penney (39), (c) piecewise linear (42), and (d) triple-well (43).

1. The barrier reduction effect leads to a current increase with ρ. It appears in multioccupied wells, where particles are pushing each other to regions of higher potential energy and thus decrease an effective barrier for a transition to neighbouring wells. The effect is best visible for small σ causing currents to be larger than j0(ρ)=v0ρ (solid black line in Fig. 1). Likewise, for small and moderate σ, the strong current increase at larger ρ is due to the occurrence of double-occupied wells.

2. The blocking effect suppresses the current by reducing the number of transitions between neighbouring wells. It occurs for larger particle sizes: an extended particle is more easily blocked by another one occupying the neighbouring well (compared to smaller σ). To contrast with the most extreme case of blocking, the parabolic current–density relations jASEP(ρ)=v0ρ(1−ρ) of a corresponding ASEP is shown as the dashed line in Figure 1.

3. The exchange symmetry effect causes a deformation of the current–density relation towards the linear behaviour j0(ρ)=v0ρ if the particle size is close to σ=m, m=0, 1, 2,…, that is, a multiple integer of λ. In the commensurate case σ=m, the current of interacting particles becomes equal to that of noninteracting ones. This effect is a consequence of the general relation

   (38)jss(ρ,σ)=(1−mρ)jss(ρ1−mρ,σ−mλ)

   that maps the stationary current in a system with particles of diameter σ and density ρ to that with particles of diameter σ−mλ and density ρ/(1−mρ), where m=int(σ/λ) is the integer part of σ/λ.

4 Impact of External Periodic Potential

As discussed in the Introduction, we consider further external potentials, namely, the Kronig–Penney, a piecewise linear, and a triple-well potential. These potentials are plotted in Figure 2b–d together with the cosine potential of our reference system in Figure 2a.

4.1 Kronig–Penney Potential

The Kronig–Penney potential has the form

(39)U(x)={0,0≤x<λw,U0,λw≤x<λ,

where λ_w is width of the rectangular well, and λb=λ−λw is the width of the rectangular barrier. We are interested in the current–density relation for different λ_w in the limit of large U0≫1. Specifically, we take the same value U₀ = 6 as for the reference BASEP with cosine potential discussed in Section 3. In particular, we aim to clarify whether a current enhancement over that of noninteracting particles still occurs. As all particles dragged from one well to a neighbouring one have to surmount the same barrier height U₀ now, it is not clear whether multiple occupations of wells lead to an effective barrier reduction. The blocking and exchange symmetry effect are expected to influence the current in an analogous manner.

Inserting the Kronig–Penney potential in (37) yields

(40a)v0(λw)=AB

(40b)A=Dλ(βfλ)2(eβfλ−1)

(40c)B=2[eβfλw+eβf(λ−λw)−eβfλ−1]×[1−cosh(βU0)]+βfλ(eβfλ−1)

Figure 3:

Drift velocity of a single particle in dependence of λ_w for the Kronig–Penney potential in (39); the velocity is normalised to f, i.e. its value in a flat (vanishing) external potential. The barrier height and drag force are U₀ = 6 and f = 0.2.

Figure 4:

Current j_ss, normalised with respect to v0(λw) (Fig. 3), for the Kronig–Penney potential in the small-driving approximation as a function of ρ for different σ and (a) λw=0.1, (b) 0.3, (c) 0.5, (d) 0.7, and (e) 0.9. The barrier height is U₀ = 6.

for the single-particle velocity. This result is plotted in Figure 3. As expected, v0(λw) approaches the mean drift velocity μf of a single particle in a flat potential in the limits λw→0 (zero well width) and λw→1 (zero barrier width). With increasing width of the wells (or of the barriers), v0(λw) rapidly decreases. Interestingly, (40c) implies the symmetry v0(λw)=v0(1−λw)=v0(λb), which means the single-particle velocity remains unaltered if the barriers and wells are interchanged.

Current–density relations for hardcore-interacting particles calculated from the SDA (cf. Section 2.2) are shown in Figure 4 for five different values of λ_w. For each λ_w, we plotted jss/v0 vs. ρ for eight rod lengths σ analogous to our representation of current–density curves in Figure 1. As can be seen from the graphs, the shapes of the current–density relation are qualitatively comparable to that in Figure 1 for all λ_w, as well as their overall change with the diameter σ. This means that the interplay of the barrier reduction, blocking, and exchange symmetry is still present. As expected, the overall strengths of the effects in modifying the current of noninteracting particles become weaker with increasing λ_w; for λw→1, the current jss(ρ,λw) indeed approaches j0(ρ,λw→1)=ρv0(λw→1)=μfρ.

The barrier reduction, however, can no longer be associated with a decrease of an effective barrier height, when two or more particles occupy a potential well. For the Kronig–Penney potential in (39), all particles in a multiple-occupied well have zero energy and need to overcome U₀. Nevertheless, we can attribute the enhancement of the current compared to that of noninteracting particles with a barrier reduction. To see this, we analyse the potential of mean force

(41)Umf(x)=kBTlnϱeq(x)+C=Umf(x+λ)

for both the cosine and the Kronig–Penney potential, where the constant C is chosen to give a potential minimum equal to zero. If considering driven Brownian motion of noninteracting particles in the potential Umf(x), the current in the linear response limit would be equal to the many-particle current in the SDA. We therefore can interpret U_mf as an effective barrier in the many-particle system. In Figure 5, we show Umf/U0 for both the cosine and the Kronig–Penney potential. At the maximum of the cosine potential at x = 0, we find Umf/U0<1, which means the barrier height is reduced. In contrast, the barrier at the step of the Kronig–Penney potential equals U₀ (Umf(x=λw)/U0=1 in Figure 5). However, a barrier reduction is now clearly seen in the plateau part of the barrier in the range λw<x<1. We thus can distinguish between two types of barrier reduction, namely, the first type associated with a reduction of the barrier height and the second type associated with a lowering of the barrier plateau.

Figure 5:

Potential of mean force U_mf for the cosine potential (blue dashed line) and the Kronig–Penney potential with λw=0.2 (solid red line). Parameters are σ = 0.2, ρ = 0.5, and U₀ = 6 for both potentials.

Generally, a single-well periodic potential can be characterised roughly by the widths of a valley and barrier part and the flanks in between these parts. A simple representation is given by the piecewise linear potential

(42)U(x)={0,0≤|x|≤λw2,U0λf(|x|−λw2),λw2≤|x|≤λw2+λf,U0,λw2+λf≤|x|≤λ,

shown in Figure 2c, where λ_w, λ_f, and λb=λ−λw−2λf are specifying the widths of the valley, barrier, and flanks. We performed additional calculations in the SDA for this potential. For various fixed λ_w and λ_f, we always found current–density relations with a behaviour similar to that found in Figure 1 for the BASEP with cosine potential. This model may thus be viewed as representative for Brownian single-file transport through single-well periodic potentials with barriers much larger than the thermal energy.

Current–density relations with different characteristics can, however, be obtained for multiple-well periodic potentials, as we discuss next.

4.2 Triple-Well Potential

The triple-well potential shown in Figure 2d is

(43)U(x)=U0∑j=13(j+1)cos(2πjxλ),

where we choose U₀ = 1 here. For this potential, our calculations of jss(ρ) based on the SDA show a very sensitive dependence on σ. We concentrate here on one particle diameter σ = 0.323, where several local extrema occur; see Figure 6. In this figure, currents are shown up to a density (filling factor) N/M=ρ=2.98, corresponding to a coverage ρσ≅96% of the system by the hard rods. If ρ approaches its maximal value 1/σ≅3.10 corresponding to a complete coverage, the numerical calculation of the equilibrium density profile from (19) becomes increasingly difficult. Hence, we refrained to show current data for ρ > 2.98 due to a lack of sufficient numerical accuracy when calculating ϱeq(x). It is important to state in this context that the current for ρ→1/σ is expected to approach that of noninteracting particles in a flat potential [56]; that is, it should hold jss(ρ,σ)∼μfρ for ρ→1/σ (except for the singular point ρ=1/σ, where jss(1/σ,σ)=0). This means that the current in Figure 6 must steeply rise for ρ→1/σ; that is, there must appear a further local minimum for ρ > 2.98. These arguments apply also to the currents shown in Figures 1 and 4.

Figure 6:

Current j_ss, normalised with respect to f, for the triple-well potential in the small-driving approximation as a function of ρ for σ = 0.323. Densities at the local maxima are ρmax, 1=1.14 and ρmax, 2=2.53, and ρmin=1.98 at the local minimum.

To explain the occurrence of the local extrema in Figure 6, we resort to (15) with χ = 0, that is, the SDA. This equation can be interpreted by considering μϱeq(x)dx/λ to be the “local conductivity” of a line segment dx. A serial connection of these segments implies that the “total conductivity” is given by the inverse of the sum of the inverse local conductivities, corresponding to a summation of the respective “local resistivities.” A stronger localisation of ρeq(x) around the minima of the potential leads to a smaller conductivity and hence a smaller current jss(ρ), whereas less localised density profiles lead to larger jss(ρ).

Using this picture, the occurrence of the first maximum in jss(ρ) can be traced back to an increasing occurrence of double-occupied wells for ρ ≳ 1. In double-occupied wells, particle motion is more restricted, leading to a stronger particle localisation at the two deeper minima at about x ≃ 0.2 and x ≃ 0.8; see Figure 2d. Accordingly, jss(ρ) starts to decrease with ρ for ρ ≳ 1 (ρmax, 1≅1.14 in Fig. 6). The decrease of jss(ρ) continues up to a filling factor of about two (ρmin≅1.98 in Fig. 6), above which more than two particles occupy a well on average. With a significant appearance of triple-occupied wells goes along first a stronger spreading of the density, as the minimum of the potential at x=1/2 becomes occupied in wells containing three particles. The spreading of the density causes jss(ρ) to increase for ρ ≳ 2. A counteracting effect, however, is a strong particle localisation at all potential minima in neighbouring triple-occupied wells, where the hardcore constraints force the particles to become strongly localised around the potential minima. For ρ ≳ 2.5 (ρmax, 2≅2.53 in Fig. 6), every second well is occupied by three particles on average, which lets jss(ρ) decrease again with further increasing ρ.

The more complex current–density relation in Figure 6 leads to a richer variety of NESS phases in open systems compared to the reference BASEP, which can exhibit up to five different phases [55], [56]. To identify all possible NESS phases, we consider the particle exchange with two reservoirs L and R at the left and right ends of an open system to be controlled by two parameters ρ_L and ρ_R. As discussed in connection with (1) in the Introduction, these control parameters can be considered as effective densities, or they can be associated with true reservoir densities for specific bulk-adapted couplings of the system to the reservoirs [33], [34].

Applying (1) with ρ−=ρL and ρ+=ρR to the current–density relation in Figure 6 results in the diagram with seven different NESS phases I–VII shown in Figure 7. The colour coding shows the value of the bulk density ρ_b, that is, the order parameter of the phase transitions. Solid lines mark first-order, and dashed lines mark second-order phase transitions, which is reflected in the smooth (continuous) or sudden (jump-like) changes of the colour. The seven phases can be classified in two categories: boundary-matching phases, where ρ_b is equal to either ρ_L or ρ_R, and extremal current phases, where ρ_b is equal to one of the densities at which jss(ρ) has a local extremum in Figure 6. Specifically, phases I and V are left-boundary matching phases with ρb=ρL, phases III and VII are right-boundary matching phases with ρb=ρR, phase II is a maximal current phase with ρb=ρmax, 1, phase VI is a maximal current phase with ρb=ρmax, 2, and phase IV is a minimal current phase with ρb=ρmin. If one takes into account the existence of the further minimum for ρ > 2.98 in the current–density relation (see discussion above), then even more phases are possible. These additional phases, however, must appear in the two stripes 2.98<ρL,R≤1/σ≅3.1, that is, in a very narrow range of one of the two control parameters (marked in black in Fig. 7).

Figure 7:

Phase diagram of NESS for the triple-well potential obtained by applying the extremal current principles to the current–density relation in Figure 6. The colour bar encodes the values of the bulk density ρ_b in an open system coupled to particle reservoirs. Phases I and V are left-boundary matching phases with ρb=ρL, phases III and VII are right-boundary matching phases with ρb=ρR, phases II and VI are maximal current phases with ρb=ρmax, 1 and ρb=ρmax, 2, respectively, and phase IV is a minimal current phase with ρb=ρmin. Solid (dashed) lines mark first-(second-)order phase transitions. The dark black bars at the boundaries mark the two stripes 2.98<ρL,R≤1/σ≅3.1, where additional phases appear (see the discussion in Section 4.2, Triple-Well Potential).

5 Impact of Interactions Other Than Hardcore Exclusions

In this section, we investigate the impact of other particle interactions beyond hardcore exclusion for the cosine external potential in (36). This is done in two different settings. First, we investigate the repulsive Yukawa potential

between particles at distance r for a fixed small amplitude AY=1 and different decay lengths ξ. Second, we combine this Yukawa interaction with hardcore interactions. For obtaining current–density relations, we here employ Brownian dynamics simulations. This is because an exact density functional is not available for the Yukawa interaction, and the SDA with a precise determination of ϱeq(x) cannot be applied. For performing the Brownian dynamics simulations, we used a standard Euler integration scheme of the Langevin equations (2) with a time step Δt=10−4. To deal with the hardcore interactions, the algorithm developed in [76] was applied.

Figure 8:

Simulated current–density relations for the external cosine potential (36) and particle interactions given by the Yukawa potential in (44). The amplitude of the Yukawa potential is AY=1, and ξ values specify different decay lengths. The other simulation parameters are U₀ = 6 and f = 2. The solid line marks the current j0(ρ)=v0ρ for noninteracting particles, and the dashed line indicates the current–density relation jASEP(ρ)=v0ρ(1−ρ) of a corresponding ASEP (same curves as in Fig. 1).

Current–density relations for the Yukawa potential without additional hardcore interactions are shown in Figure 8 for six different values of ξ. These may be viewed to resemble effective particle diameters σ of hardcore-interacting systems. For small ξ, that is, ξ = 0.166 and 0.333 in Figure 8, j_ss shows an enhancement over that of noninteracting particles (solid black line) due to a prevailing barrier reduction effect similar to that in the reference BASEP for small σ. When enlarging ξ, the current is reduced for small ρ compared to that of noninteracting particles, while it rises strongly for large ρ. Again, this behaviour is analogous to that in the reference BASEP for increasing σ. Because the (effective) blocking effect is not so strong for AY=1, the current–density curves do not approach the limiting jASEP(ρ) as closely as for hardcore interactions. Nevertheless, one can say that the change of the current–density relation with varying ξ is reflecting the interplay of a barrier reduction and blocking effect as in the reference BASEP.

Figure 9:

Simulated current–density relations for the cosine external potential and Yukawa interactions as in Figure 8, and additional hardcore interactions for commensurate particle size σ = 1. Parameters for the drag force, as well as for the amplitudes of the cosine and Yukawa potential, are chosen as in Figure 8. The solid line marks the current j0(ρ)=v0ρ for noninteracting particles (same line as in Figs. 1 and 8).

However, one cannot find certain ξ values, where the current–density relation equals that of noninteracting particles for all ρ. The peculiar exchange symmetry effect in the BASEP for commensurate σ=m, m=0, 1, 2…, is caused by the invariance of the stochastic particle dynamics against a specific coordinate transformation containing σ [55], [56]. Such coordinate transformation does not exist for the Yukawa potential.

For the Yukawa potential with additional hardcore interactions, we found changes of current–density curves caused by the barrier reduction and blocking effect as discussed above. But it is interesting to analyse now whether the relation jss(ρ)=j0(ρ)=v0ρ for commensurate σ=m and ξ = 0 is approximately reflected in current–density relations for ξ > 0, where the exchange symmetry effect is no longer strictly valid. One may expect that the hardcore-interacting system should be only weakly perturbed by the Yukawa potential if A_Y is of the order of the thermal energy and ξ not too large compared to σ. This is indeed confirmed by simulation results for σ = 1 shown in Figure 9. The data points for ξ = 0.166 and ξ = 0.333 lie almost directly on the curve j0(ρ) up to the highest simulated density ρ = 0.9. With increasing ξ, deviations from the linear behaviour are seen, which become the more pronounced the larger ρ. But even for ξ = 2, jss(ρ) follows j0(ρ) closely up to ρ = 0.4. We thus conclude that slight deviations from a perfect hardcore interaction, as they are always present in experiments, still allow an identification of the exchange symmetry effect.

6 Summary and Conclusions

To analyse how generic our previous findings are for the nonequilibrium physics of the BASEP in a sinusoidal potential, we have studied the driven Brownian motion of hardcore-interacting particles for other external periodic potentials. Our calculations were carried out based on a small-driving approximation, which refers to the linear response under neglect of a period-averaged mean interaction force. If the external periodic potential exhibits a single-well structure between barriers, that is, if there is only one local minimum per period, our results provide evidence that the various characteristic shapes of bulk current–density relations jss(ρ) for different particle sizes σ are always occurring. There are differences in the exact functional form and at which σ the type of shape is changing. For all single-well periodic potentials, it is the interplay of a barrier reduction, blocking, and exchange symmetry effect that causes a particular shape to appear. Even for a Kronig–Penney potential with alternating rectangular well and barrier parts, where the barrier reduction effect is not so obvious, we showed that an enhancement of the current over that of noninteracting particles occurs. For that potential, this enhancement can be attributed to an effective reduction of the barrier plateau parts. The generic behaviour of the bulk current–density relations implies that for single-well periodic potentials up to five different NESS phases appear in open BASEP systems coupled to particle reservoirs. This can be concluded by applying the extremal current principles [27], [29].

More complex shapes of jss(ρ) can occur in multiple-well periodic potentials. This was demonstrated for a particular triple-well potential, where our calculations yielded a current–density relation with two local maxima for a certain particle size. In that case, the extremal current principles predict more than five different NESS phases in an open system. When neglecting a narrow range of very high effective reservoir densities, which would be very difficult to realise by specific system-reservoir couplings in simulations or experiments, up to seven different NESS phases are possible. We point out that these results were obtained here for demonstration purposes. Systematic investigations of multiple-well external potentials should be performed in the future with a goal to reach a general classification similar as for the BASEP for single-well periodic potentials.

Current–density relations with several local maxima are particularly interesting in the case of “degenerate maxima,” that is, when the current at the maxima has the same value. In such situations, coexisting NESS phases of maximal current can occur in a whole connected region of the space spanned by the parameters controlling the coupling to the environment [77]. Such states of coexisting extremal current phases have not yet been studied in detail in the literature. Preliminary results for driven lattice gases indicate that fluctuations of interfaces separating extremal current phases exhibit an anomalous scaling with time and system length.^[2] This is in contrast to the already well-studied interface fluctuations between the low- and high-density phases in the ASEP, which at long times show a simple random-walk behaviour.

We furthermore performed Brownian dynamics simulations of driven single-file diffusion through a cosine potential for a repulsive particle interaction other than hardcore exclusion. Specifically, we chose a Yukawa interaction with a small interaction amplitude equal to the thermal energy and studied the behaviour for different decay lengths ξ. Current–density relations for this system showed similar shapes as for the BASEP except for the effects implied by the exchange symmetry effect, which is absent for other interactions than hardcore. The change of shapes is solely determined by the interplay of a barrier reduction and effective blocking effect. If the hardcore interaction and the weak Yukawa interaction are combined, the consequences of the exchange symmetry effect can still be seen for particle sizes commensurate with the wavelength of the cosine potential. The current jss(ρ) follows closely that of noninteracting particles up to high densities even for large ξ. This means that deviations from a perfect hardcore interaction in experiments should still allow one to verify the exchange symmetry.