Active gating: rocking diffusion channels
AActive gating: rocking diffusion channels
Tirthankar Banerjee
Instituut voor Theoretische Fysica, KU Leuven, Belgium [email protected]
Christian Maes
Instituut voor Theoretische Fysica, KU Leuven, Belgium [email protected]
Abstract.
When the contacts of an open system flip between different reservoirs,the resulting nonequilibrium shows increased dynamical activity. We investigate suchactive gating for one-dimensional symmetric (SEP) and asymmetric (ASEP) exclusionmodels where the left/right boundary rates for entrance and exit of particles areexchanged at random times. Such rocking makes SEP spatially symmetric and onaverage there is no boundary driving; yet the entropy production increases in therocking rate. For ASEP a non-monotone density profile can be obtained with particlesclustering at the edges. In the totally asymmetric case, there is a bulk transition toa maximal current phase as the rocking exceeds a finite threshold, depending on theboundary rates. We study the resulting density profiles and current as functions of therocking rate. a r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p ctive gating: rocking diffusion channels
1. Introduction
Time-dependence of physical parameters often enters the Hamiltonian or only concernsthe bulk dynamics of a system. The present paper deals with the less common situa-tion where the boundary dynamics is time-dependent, subject in fact to dichotomousnoise [1]. We will see that the noisy gating, which we implement as “rocking the sys-tem,” adds spatial symmetry but also dissipation and the extra activity may change thephase diagram of the original process where the boundary condition is fixed.The resulting systems have a (bulk) particle-conserving dynamics with stochastic bound-ary conditions. To be specific we consider one-dimensional channels with particle hop-ping subject to exclusion, as in boundary-driven lattice gases [2, 3, 5, 4]. The boundarieslet particles in and out, but the entry and exit rates flip between two values at expo-nentially distributed times. Thinking of the edges of the system as gates for flows ofparticles, the models implement active gating . Figure 1 shows a possible scenario ofrocking the channel; its contacts are randomly flipping between two particle-reservoirs.Such modeling appears relevant especially for quasi-one dimensional channels, pores orpolymers sitting or moving at the interface between two chemical reservoirs, as is ubiq-uitous in biological environments. We also think of pores connecting chemical reservoirs,such as the interior and exterior of a biological cell where however the effective chemicalpotentials or kinetic parameters are noisy. The models can also be seen as implementinga version of boundary tumbling, in analogy with active particle processes. Similar mod-els and ideas have appeared in other contexts, e.g. in [7, 8] for mathematical aspects.On the experimental side, the models appear related to the bio- and chemical physicsin e.g. [9, 10].The main results of our work are the following : (i) For symmetric simple exclusionprocesses (SEP), rocking leads to a zero-current nonequilibrium steady state with a flatdensity profile and an associated mean entropy production rate that increases withthe rocking rate. (ii) For totally asymmetric simple exclusion processes (TASEP),rocking modifies the standard phase diagram in terms of the boundary rates. Thereappears a transition as the rocking rate increases, to a maximal current phase. In thethermodynamic limit, bulk density profiles are either linear or flat. (iii) For the moregeneral case of partially asymmetric simple exclusion processes (ASEP), non-monotonedensity profiles appear for small rocking and also the current is non-monotone in therocking rate. The mean entropy production rate for ASEP reflects the nature of thesteady bulk current, and shows nonmonotonicity for moderate bias, while remainingmonotonic in the rocking rate when the bias is either large and very small.The plan of the paper is as follows. We start in the next section by defining therocking dynamics for exclusion models on an open lattice interval. The general questionis to investigate the influence of rocking on entropy production, on the density profileand its role in modifying the phase diagram in terms of particle current. In Sec. 3 weanalyze the active boundary-driven SEP and follow it up with the discussion of the ctive gating: rocking diffusion channels
2. Exclusion models with active gating
We consider standard exclusion processes on the lattice interval of length L withoccupation variables n = ( n i ; i = 1 , , · · · L ); n i = 0 , i th site beingvacant or occupied, respectively. In the bulk, particles hop at rate 1 to the right andrate e − E to the left, where E ≥ σ = ±
1. We define λ in ( σ ) := α + γ σ α − γ (cid:40) α when σ = 1 γ when σ = − λ out ( σ ) := β + δ σ β − δ (cid:40) β when σ = 1 δ when σ = − α and γ and exit frequencies β and δ , non-negative parameters.The variable σ flips between ± r >
0; at exponential times the left andthe right exchange their entrance/exit rates — see Fig. 1. µ (+) = log αβ µ ( − ) = log γδ r r e − E Figure 1.
Rocking an exclusion process with external field E in a two-dimensionalenvironment. The switching of contacts happens at rate r with entrance rate α, γ andexit rates β, δ . The chemical potentials µ (+) and µ ( − ) correspond to the reservoirs,respectively above and below the dashed line. Taking into account the exclusion and the external field E we get for entrance (in)and exit (out) rates, for the left edge: λ leftin ( n , σ ) = λ in ( σ ) (1 − n ) λ leftout ( n , σ ) = λ out ( σ ) e − E n (3)and for the right edge: λ rightin ( n L , σ ) = λ in ( − σ ) e − E (1 − n L ) λ rightout ( n L , σ ) = λ out ( − σ ) n L (4) ctive gating: rocking diffusion channels E .A physically useful parameterization is to write α = a e b / , β = a e − b / , γ = a e b / , δ = a e − b / (5)where the a i are reactivities and the b i are chemical potentials (up to a factor of k B T = 1that we set one from now on) of the two particle reservoirs. Such writing follows thecondition of local detailed balance for the contact with two chemical reservoirs. Thechemical potential of the reservoir which makes contact with the left edge is µ (cid:96) ( σ ) = log λ in ( σ ) λ out ( σ ) = 12 (cid:104) log αγβδ + σ log αδβγ (cid:105) = (cid:40) b for σ = 1 b for σ = − σ . The chemical potential for the right edge is µ r ( σ ) = µ (cid:96) ( − σ ).There is therefore a variable thermodynamic force which equals F ( σ ) := E + 1 L [ µ (cid:96) ( σ ) − µ r ( σ )] = E + 1 L σ log αδβγ (7)In the stationary state (cid:104) σ (cid:105) = 0 , and the average thermodynamic force (cid:104) F ( σ ) (cid:105) = E, while (cid:104) µ (cid:96) ( σ ) (cid:105) = (cid:104) µ r ( σ ) (cid:105) = ( b + b ) / , i.e. , the chemical potentials at the left and theright edges are equal under stationary averaging.The expected instantaneous boundary currents into the environment, at the leftand right edges respectively, equal J (cid:96) ( σ, n ) = λ out ( σ ) e − E n − λ in ( σ )(1 − n ) J r ( σ, n L ) = λ out ( − σ ) n L − λ in ( − σ ) e − E (1 − n L ) (8)given σ, n and n L . There is also the instantaneous bulk current J i which is the expectedparticle flux towards the right between sites i and i + 1: J i ( n ) = [ n i (1 − n i +1 ) − e − E n i +1 (1 − n i )]They enter the expected entropy production rate ˙Σ( n, σ ), given ( n, σ ), [5], as˙Σ( n, σ ) = − µ (cid:96) ( σ ) J (cid:96) ( σ, n ) − µ r ( σ ) J r ( σ, n L ) + E (cid:34) L − (cid:88) i =1 J i ( n ) − J (cid:96) ( σ, n ) + J r ( σ, n L ) (cid:35) (9)The rocking rate r does not appear explicitly in these equations above. The standard,unrocked, exclusion process is recovered by fixing σ ≡ (cid:104)·(cid:105) depending on allparameters is obtained from (9) by using (cid:104) J (cid:96) ( σ, n ) + J r ( σ, n L ) (cid:105) = 0 , (cid:104) J i ( n ) (cid:105) = j ( E, r ), (cid:104) ˙Σ( n, σ ) (cid:105) = −
12 log αδβγ (cid:104) σ [ J (cid:96) ( σ, n ) − J r ( σ, n L )] (cid:105) + ( L + 1) E j ( E, r ) (10)We will see that the rocking leads to a non-zero entropy production even for E = 0 whenthere is left/right symmetry in the steady condition. Also, for E (cid:54) = 0, blocking one ofthe edges, e.g. by putting a = 0 in (5), still allows for positive entropy production. ctive gating: rocking diffusion channels n ( t ) , σ t ) is Markovian. Let us look at the time-evolution of the local density (cid:104) n i (cid:105) t . For i (cid:54) = 1 , L, the density evolves via the usual ASEPequation in the bulk, ∂ t (cid:104) n i (cid:105) = (cid:104) e − E ( n i +1 − n i − n i n i +1 + n i n i − ) + n i − − n i − n i n i − + n i n i +1 (cid:105) t (11)At the boundaries the evolution equations take a different form, ∂ t (cid:104) n (cid:105) t = − (cid:104) J (cid:96) ( σ, n ) (cid:105) t + (cid:104) e − E n (1 − n ) − n (1 − n ) (cid:105) t ∂ t (cid:104) n L (cid:105) t = − (cid:104) J r ( σ, n L ) (cid:105) t + (cid:104) n L − (1 − n L ) − e − E n L (1 − n L − ) (cid:105) t (12)The kinetic equations for (cid:104) σn i (cid:105) t are obtained similarly. At the boundary sites i = 1 , L , ∂ t (cid:104) σn (cid:105) t = (cid:104) λ in ( σ ) σ (1 − n ) (cid:105) t − e − E (cid:104) λ out ( σ ) σn (cid:105) t + e − E (cid:104) σn (1 − n ) (cid:105) t − (cid:104) σn (1 − n ) (cid:105) t − r (cid:104) σn (cid:105) t ∂ t (cid:104) σn L (cid:105) t = e − E (cid:104) λ in ( − σ ) σ (1 − n L ) (cid:105) t − (cid:104) λ out ( − σ ) σn L (cid:105) t (13)+ (cid:104) σn L − (1 − n L ) (cid:105) t − e − E (cid:104) σn L (1 − n L − ) (cid:105) t − r (cid:104) σn L (cid:105) t In the bulk, i.e. , for i = 2 , , · · · , L − ∂ t (cid:104) σn i (cid:105) t = e − E [ (cid:104) σn i +1 (1 − n i ) (cid:105) t − (cid:104) σn i (1 − n i − ) (cid:105) t ]+ (cid:104) σn i − (1 − n i ) (cid:105) t − (cid:104) σn i (1 − n i +1 ) (cid:105) t − r (cid:104) σn i (cid:105) t (14)These equations (11)–(14) are closed for E = 0 but not otherwise. We assume thatthe system starts from an arbitrary density profile and σ = ± E = 0. We discuss it in Section 3. The bulk dynamics is unbiased, and thereis left/right reflection symmetry in the stationary particle process for any r > E = + ∞ , is the subject of Section 4. The dynamics is totally asymmetricwith a particle current from left to right.(iii) fast rocking rate: r ↑ ∞ . When the reservoirs switch at an infinite pace, the flipping σ t decouples from the particle dynamics n ( t ) at all finite times t . The processrestricted to n ( t ) becomes formally equivalent to a standard open (TA)SEP withrates λ leftin = (1 − n ) ( α + γ ) / , λ rightin = e − E (1 − n L )( α + γ ) / λ leftout = n e − E ( β + δ ) / , λ rightout = n L ( β + δ ) / α + γ )( β + δ ) − . Observe however that the limiting (physical) entropyproduction must still be calculated from adding the dissipation into the reservoirs(two chemical and one mechanical, which remain untouched of course) as in (9).(iv) slow rocking rate: r ↓
0. The stationary process is the equal linear combination ofthe two standard (TA)SEPs with respective rates ( α, β ) and ( γ, δ ). We do not havea limiting Markov process for n ( t ). The stationary densities add however. Notethat the process at r = 0 is not defined, except as the large-persistence limit r ↓ ctive gating: rocking diffusion channels E = 0, but even then, it is not an equilibriumSEP.(v) no driving: The model is satisfying detailed balance when the two reservoirs haveequal chemical potential, b = b which means that αδ = βγ , and there is no bulkexternal bias, E = 0. That corresponds to the equilibrium scenario; the stationarysystem is time-reversible. The equilibrium distribution is a product measure withdensity αα + β = γγ + δ . In fact all the thermodynamic observables are independent of r in that case. That does not mean however there is no influence of r ; the dynamicskeeps depending on the rocking (except in the special case when α = γ and β = δ where the rocking does not change the dynamics) and all kinetic observables areexpected to depend on r .(vi) equal chemical potential (but non-zero drive) : we can choose it to be zero, b = b = 0, which corresponds to α = β, γ = δ . That will be the choice inSection 4 for rTASEP.
3. Rocking the boundary driven SEP
For the boundary-driven SEP, in the bulk, particles hop symmetrically to neighbouringvacant sites with rate unity. The boundary rates are determined by (1)–(2). Theenvironment consists of two particle baths, with random switching of contacts witheither left or right boundaries as illustrated in Fig. 2 (with E = 0). α log log γ r µ(−) == δ µ(+) β Figure 2.
Schematic representation of the rocking SEP. Rocking of gating happensat rate r with entrance rate α, γ and exit rates β, δ . The random switching of the gates correlates the two edges of the lattice interval. InFig.3 we see how bulk spatial correlations are vanishingly small, while the edge anti-correlation survives of course, in the thermodynamic limit as well. ctive gating: rocking diffusion channels i/ L C ( i ) i/ L -0.04-0.020 C ( i ) r =10 -3 r =10 -2 r =10 -1 Figure 3.
Plot of the spatial correlation C ( i ) = (cid:104) n n i (cid:105) − ρ vs i/L near the rightboundary i = L for L = 1000. The densities at the two boundaries are negativelycorrelated with the correlations decaying faster for larger r . The inset shows the decayof C ( i ) near i = 1 . Here α = 0 . β = 1 = γ = δ. More explicitly, if the left boundary site is empty, a particle enters with rate λ in ( σ ) = α or = γ depending on σ = 1 or = −
1, respectively (similarly for the right boundary,with entrance rate λ in ( − σ )). A particle can leave the system from the left (respectively,right) boundary site with rate λ out ( σ ) (respectively, λ out ( − σ )). The exit rates switchbetween β and δ . The density varies with the edge-parameters and with the rocking rate according to theequations (8)–(11) for E = 0: for i (cid:54) = 1 , L,∂ t (cid:104) n i (cid:105) t = (cid:104) n i − (cid:105) t + (cid:104) n i +1 (cid:105) t − (cid:104) n i (cid:105) t (15)and at the left and right boundaries, ∂ t (cid:104) n (cid:105) t = − (cid:104) J (cid:96) ( σ, n ) (cid:105) t + (cid:104) n (cid:105) t − (cid:104) n (cid:105) t ∂ t (cid:104) n L (cid:105) t = − (cid:104) J r ( σ, n L ) (cid:105) t + (cid:104) n L − (cid:105) t − (cid:104) n L (cid:105) t (16)with boundary currents into the environment, J (cid:96) ( σ, n ) = λ out ( σ ) n − λ in ( σ )(1 − n ) J r ( σ, n L ) = λ out ( − σ ) n L − λ in ( − σ )(1 − n L ) (17)given σ, n and n L . In the stationary state, the lhs of Eqs. (15)–(16) vanish. Since SEPdynamics corresponds to an infinite temperature and the rocking makes the systemsymmetric under left/right reversal, we expect that the stationary condition has a flatprofile (cid:104) n i (cid:105) = ρ , independent of i. The stationary equations are solved in Appendix A for the thermodynamic limitof the density ρ . We find, with S := α + β + γ + δ, D := β − δ + α − γ , and calling ctive gating: rocking diffusion channels R := r + (cid:112) r ( r + 2), ρ = γβ + αδ + 2 αγ + R ( α + γ )2( α + β )( γ + δ ) + SR = a a [ e ( b − b ) / + e ( b − b ) / + 2 e ( b + b ) / ] + R ( a e b / + a e b / )8 a a cosh b / b / a cosh b / a cosh b / R (18)which gives the density explicitly in terms of the chemical potentials b i and the kineticrates a i . Fig. 4(a) shows the density ρ as a function of r, for different values of thechemical potential b . For b = b = b , we have cancellations giving the equilibriumdensity ρ eq = e b e b . (19)The flat line in Fig. 4(a) corresponds to this equilibrium scenario where the densitybecomes independent of the kinetic parameters r, a , a . In general, however, the densitydepends on these kinetic parameters; see Figure 4(b) where ρ is plotted as a function of b , for different values of a . A special case is when b → −∞ , i.e. , one of the reservoirsonly pumps particle out of the system. In that case,lim b ↓−∞ ρ = γ γ + δ ) + R = 12(1 + e − b ) + R/a which remains nonzero for all finite r, and depends explicitly on the reactivity a . -3 -2 -1 0 1 2 3 b ρ a = 0.1 a = 1 a = 10 r ρ b = -2 b = 0 b = -1 b = 1 b = 2 (a) (b) Figure 4.
Rocking SEP: (a) Density ρ as a function of r for different values of b . Here b = 0 is fixed along with a = 5 and a = 1 . (b) ρ versus b for different valuesof a while r = 1 , a = 1 and b = 0 are fixed. It could be surprising that making the boundary driven SEP more spatially symmetric(by the rocking) can increase the entropy production. It does, because of the increaseddynamical activity. The situation is not unlike the one in [6], where a zero-currentnonequilibrium state is achieved from SEP from dichotomous stochastic resetting. ctive gating: rocking diffusion channels (cid:104) ˙Σ (cid:105) . Forexample, (cid:104) σJ (cid:96) ( σ, n ) (cid:105) = −(cid:104) σJ r ( σ, n L ) (cid:105) = 12 ( γ − α ) + S (cid:104) σ n (cid:105) + Dρ (cid:104) ˙Σ (cid:105) = 12 log βγαδ ( γ − α + S (cid:104) σn (cid:105) + D ρ ) = R (cid:104) σn (cid:105) log αδβγ = RD ( α + γ − Sρ ) log αδβγ Using the exact expression for ρ in (18), we have (cid:104) ˙Σ (cid:105) = ( αδ − βγ ) log αδβγ S + R ( α + β )( γ + δ ) (20)= 2 a a ( b − b ) sinh b − b S + a a R cosh b / b / αδ (cid:54) = βγ , independentof the system size. It is monotone, increasing with r . The first order in r ↓ (cid:104) ˙Σ (cid:105) r ↓ = r ( b − b )2 cosh b / b / b − b b i . The maximal mean entropy productionrate is reached in the limit r ↑ ∞ to give (cid:104) ˙Σ (cid:105) r ↑∞ = 2 a a S ( b − b ) sinh b − b a a /S . Fig. 5(a) shows the dependenceof the mean entropy production rate on the rocking rate r for different values of thechemical potentials and reactivities.It is also interesting to investigate how the entropy production changes with thechemical potential difference ε := b − b . Figure 5(b) shows a plot of (cid:104) ˙Σ (cid:105) as a functionof ε for different values of r and a . For ε = 0 the chemical potentials are equal, thesystem is in equilibrium, and the entropy production vanishes. To quadratic order insmall ε , the mean entropy production rate equals (cid:104) ˙Σ (cid:105) = a a ε (cid:0) a + a + a a R cosh b (cid:1) cosh b (24)and hence keeps kinetic information for all nonzero rocking rates, as clearly visible fromFig. 5(b).
4. Rocking TASEP
Taking infinite external field E = + ∞ in the definitions of Section 2, we get a rockingTASEP (rTASEP) on an open lattice interval. As we also choose to have constant chem-ical potential µ (cid:96) = µ r = 0 in the environment, the rocking dynamics gets characterized ctive gating: rocking diffusion channels -1 -0.5 0 0.5 1 ε 〈Σ. 〉 r =0.1, a =1 r =0.1, a =2 r =1, a =1 r =1, a =2 r 〈Σ. 〉 a =1, b =1 a =1, b =2 a =2, b =1 a =2, b =2 (a) (b) Figure 5.
Rocking SEP: (a) Mean entropy production rate (cid:104) ˙Σ (cid:105) as a function of rockingrate r for different values of a and b . Here a = 1 and b = 0 are fixed. (b) (cid:104) ˙Σ (cid:105) asa function of (cid:15) = b − b for different values of r and a . ε is changed by changing b while keeping b = 1 fixed. We have also taken a = 1. ε = 0 corresponds to equalchemical potentials at the two gates. by only two boundary rates α = β and δ = γ and the rocking rate r >
0. At the leftboundary particles only enter; at the right boundary particles can only exit.We start by highlighting a symmetry. First of all, for every r > σ ↔ − σ . That directly leads to (cid:104) n i (cid:105) = 1 − (cid:104) n L − i +1 (cid:105) . (25)For example, (cid:104) n (cid:105) + (cid:104) n L (cid:105) = 1 always. Similarly, (cid:104) σn i (cid:105) = −(cid:104) σ (1 − n L − i +1 ) (cid:105) implies (cid:104) σn (cid:105) = (cid:104) σn L (cid:105) (26)where all expectations are in the steady state. Note that for all non-zero r , the pure lowdensity and high density phases [12, 11] of standard (unrocked) open TASEP vanish.However, the evolution of the bulk density still follows (11) as for ordinary TASEP( E → ∞ ). At the left and right boundaries we have (12) and ∂ t (cid:104) n (cid:105) t = (cid:104) J in ( σ, n ) (cid:105) t − (cid:104) n (1 − n ) (cid:105) t (27) ∂ t (cid:104) n L (cid:105) t = − (cid:104) J out ( σ, n L ) (cid:105) t + (cid:104) n L − (1 − n L ) (cid:105) t (28)for (incoming) and (outgoing) boundary currents J in and J out , respectively, with J in ( σ, n ) = 12 [( α + δ ) − ( α + δ ) n − ( α − δ ) σn ] (29) J out ( σ, n L ) = 12 [( α + δ ) n L − ( α − δ ) σn L ] (30)In the steady state j = (cid:104) J in ( σ, n ) (cid:105) = (cid:104) J out ( σ, n L ) (cid:105) is a function of r and a symmetricfunction of ( α, δ ). ctive gating: rocking diffusion channels r ↑ ∞ , (cid:104) σn L (cid:105) = 0 = (cid:104) σn (cid:105) . Writing lim r ↑∞ j = j sat , (29) yields j sat = α + δ (cid:104) n L (cid:105) (31)for the saturation current (i.e., for infinitely fast rocking) through the system. When j sat = 1 /
4, then (31) implies (cid:104) n L (cid:105) = 12( α + δ ) = 1 − (cid:104) n (cid:105) , (cid:104) n i (1 − n i +1 ) (cid:105) = 1 / j sat = 1 / (cid:104) n i (cid:105) = 1 /
2. The boundarydensity (cid:104) n (cid:105) = (cid:104) n L (cid:105) = 1 / α + δ = 1. Figure 6 shows how rocking modifies the phase diagram of the usual open TASEP [12].It turns out that the line α + δ = 1 separates two phases in the thermodynamic limit L ↑ ∞ . The first distinction between those two phases follows from the behavior of thebulk current j = j ( r ) which increases as r grows : • Phase A, α + δ <
1: strictly monotone in r , j ( r ) ↑ j sat < /
4. This corresponds toregion I in Fig. 6. • Phase B (regions II and III), α + δ >
1: there exists either a zero or finite r ∗ suchthat j ( r ) = 1 / j sat for all r > r ∗ . We will see that r ∗ = 0 iff both α, δ > / r ∗ is given for the case that some rate, say α < / r ↑ ∞ . Then, the systemfeels an effective rate f = α + δ and connects the left and right edges to densities f and1 − f , respectively, all the while maintaining the relations (25). The average currentthrough the system equals f (1 − f ). Hence for α + δ <
1, one expects a linear profilefor the rocked TASEP that indeed corresponds to moving shocks or delocalized domainwalls [12]. The linear density profiles connecting left boundary density f to the rightboundary density 1 − f are shown by the dotted lines in Fig. 7(a). However, when α + δ >
1, the system goes to its maximum current phase, complemented with a bulkdensity equal to .We now dive into the question of how the density and current in each phase regionbehave as functions of r . In what follows, x refers to i/L , where i labels the lattice siteand L is the system size. We discuss each phase region separately.The phase diagram can be broadly divided into the following three regions :Region I : α + δ < α + δ > { α, δ } < / ctive gating: rocking diffusion channels Max. Current r ∗ > r ∗ > αδ
12 12 r ∗ = 0 j sat < I II IIIII
Figure 6.
Phase diagram for the rocking TASEP. The phase boundary (dotted line) α + δ = 1 separates two regions, characterized by the average steady saturation current j sat being either less than or equal to 1 /
4, respectively in the thermodynamic limit.On the phase boundary itself, the density profile is flat with density ρ = 1 / / / r = r ∗ . In region III, we have r ∗ = 0; see text. Region III : α + δ > α, δ > / • In region I, in the thermodynamic limit, the system shows a linear bulk densityprofile. The results are shown in Fig. 7(a). The density profile always shows apositive slope in the direction of the current. The current is shown in Fig. 7(b).Assuming w = min[ α, δ ], the average current rises from w (1 − w ) for r = 0 to j sat = α + δ (cid:0) − α + δ (cid:1) as r is increased. • The most interesting aspect of rTASEP resides in region II. Here the bulk densityprofile goes from being linear for small r to a constant bulk value 1 /
2; see Fig. 8(a).The average steady current, on the other hand rises monotonically with r to saturateat at a finite value of r = r ∗ . Figure 7(b) also shows a plot of j versus r in regionII. Figure 9 shows the behavior of the average current in region II for various choicesof α and δ . We discuss region II in the next subsection. • In region III, the system always shows a flat density profile in the bulk and thecurrent j ( r ) = 1 / r >
0. Here thesystem basically mimics the so called maximal current phase of standard openboundary TASEP. An intuitive understanding of this region is as follows: Whenboth boundary rates are larger than 0 .
5, rocking essentially gives rise to two newboundary rates which are still larger than 0 .
5. Thus, as expected, the rocked systemin this limit remains in the maximal current phase ( j = 0 . r ;see Fig. 8(b). ctive gating: rocking diffusion channels ρ ( x ) x = i / L r = 10 -4 r = 10 -3 r = 10 -1 r = 10 -3 -2 -1 (b) j r ( α , δ )=(0.1,0.6)( α , δ )=(0.2,0.3)( α , δ )=(0.9,0.2) Figure 7.
Rocking TASEP : (a) Plot of average density ρ ( x ) vs x = iL for systemsize L = 400 for region I : ( α, δ ) = (0 . , . r ↑ ∞ . (b) Plot of average current j vs r inregions I and II. Points represent Monte Carlo results for L = 100. The dashedblack lines refer to the corresponding saturation currents in region I (circles andsquares) j sat = α + δ (cid:0) − α + δ (cid:1) . For each case in region I, j approaches j sat as r increases. The dotted line represents the saturation current in region II (triangles) : j sat = j max = 0 . ρ ( x ) x = i / L r = 10 -3 r = 10 -2 r = 10 -1 r = 10 ρ ( x ) x = i / L r =10 -3 r =1 Figure 8.
Rocking TASEP : Plot of average density ρ ( x ) vs x = iL for systemsize L = 400 (a) in region II: ( α, δ ) = (0 . , . r , the bulk density islinear, while for large enough but finite r , the system goes to its maximal currentphase with average current equal to 1 /
4. (b) Region III: ( α, δ ) = (0 . , . r . The current in the thermodynamic limit is equal to forany r (not shown here). In ordinary TASEP, the system reaches its maximal current phase, with j = , only ifboth the boundary rates are greater than . However, when rocking is introduced, thesystem can show maximum current, even when one of the boundary rates remains lowerthan . Interestingly, that happens with a transition at a finite rocking value r = r ∗ . ctive gating: rocking diffusion channels -2 -1 (a) j r α =0.7, δ =0.4 α =0.9, δ =0.2 α =0.8, δ =0.3 α =0.9, δ =0.3 -2 -1 (b) j r L = 100 L = 400 L = 1000 Figure 9.
Rocking TASEP : (a) Plot of j vs r for various values of α and δ and for L = 100. For each choice of α and δ , such that α + δ > /
2, we find that there is a finite r ∗ where the average current throughthe system reaches 1 /
4. (b) Plot of j vs r for α = 0 . δ = 0 .
9. The value of r atwhich j reaches is independent of the system size L . We extract now the dependence of r ∗ on the boundary rates α and δ in a mean fieldanalysis.We start with the evolution equation of (cid:104) σn i (cid:105) . For the boundary sites i = 1 , L , ∂ t (cid:104) σn (cid:105) t = (cid:104) λ ( σ ) σ (1 − n ) (cid:105) t − (cid:104) n σ (1 − n ) (cid:105) t − r (cid:104) σn (cid:105) t (33)and ∂ t (cid:104) σn L (cid:105) t = −(cid:104) λ ( − σ ) σn L (cid:105) t + (cid:104) n L − σ (1 − n L ) (cid:105) t − r (cid:104) σn L (cid:105) t . (34)For the bulk, the evolution equations for (cid:104) σn i (cid:105) are ∂ t (cid:104) σn i (cid:105) t = (cid:104) n i − σ (1 − n i ) (cid:105) t − (cid:104) σn i (1 − n i +1 ) (cid:105) t − r (cid:104) σn i (cid:105) t (35)From (33) in the steady state,( α − δ ) (cid:104) n L (cid:105) = [ α + δ + 4 r + 2 (cid:104) n L − (cid:105) ] (cid:104) σn (cid:105) , (36)which implies (cid:104) σn (cid:105) = ( α − δ ) (cid:104) n L (cid:105) α + δ + 4 r + 2 (cid:104) n L − (cid:105) , (37)where we have also used (cid:104) σn (cid:105) = (cid:104) σn L (cid:105) . Using either (29) or (30), one can write anequation for the current in terms of the rates α, δ, r and densities (cid:104) n L (cid:105) and (cid:104) n L − (cid:105) , j = 2 αδ + ( α + δ )(2 r + (cid:104) n L − (cid:105) ) α + δ + 4 r + 2 (cid:104) n L − (cid:105) (cid:104) n L (cid:105) . (38)As one would expect, the above relation is symmetric in α and δ . We are interestedin region II of Fig. 6, where α + δ > / r ∗ at which the current j saturates: forsmall r , j ( r ) < / / r ↑ r ∗ . An expression for r ∗ in terms of ctive gating: rocking diffusion channels α and δ ( α + δ >
1, assuming one of these rates to be less than 0 . j = 1 / . α + δ ) (cid:104) n L (cid:105) − ( α − δ ) (cid:104) n L (cid:105) α + δ + 4 r + 2 (cid:104) n L − (cid:105) , (39)implying r ∗ = ( α − δ ) (cid:104) n L (cid:105) α + δ ) (cid:104) n L (cid:105) − − α + δ + 2 (cid:104) n L − (cid:105) r ∗ depends not only on α + δ but also on ( α − δ ) . When the boundaryrates are close to each other, one has to rock the system at a much lower rate toreach the maximal current phase. The smaller boundary rate has a dominant say onthe steady current. Consider, for example, two sets of rates ( α = 0 . , δ = 0 .
3) and( α = 0 . , δ = 0 . α + δ . Clearly from(40), the first set would require a larger value of r ∗ for the system to reach its maximalcurrent phase. Now consider another example when the sets are ( α = 0 . , δ = 0 .
4) and( α = 0 . , δ = 0 .
3) with both choices corresponding to the same value of α − δ . Here thesecond set would require a larger value of r ∗ to reach the maximal current phase. Onecan verify these predictions from Fig. 10) that shows a plot of r ∗ vs δ for fixed values of α . (Note the rapid increase in r ∗ as α + δ → .
5, the smaller r ∗ one requires to attainthe maximal current phase. It must also be stressed that r ∗ corresponds to the bulkthermodynamic limit.Using (32) for (cid:104) n L (cid:105) indeed gives r ∗ → ∞ , thus proving consistency of (40) with theresults of the earlier discussions in the text. Unfortunately our mean-field analysis doesnot give a closed expression for r ∗ in terms of α and δ beyond (40). r* δ α =0.9 α =0.8 α =0.7 Figure 10.
Rocking TASEP : Plot of r ∗ vs δ at fixed α > and for L = 100. r ∗ increases rapidly as α + δ → α + δ , r ∗ increases with α − δ as well, as expected from (40). Also, r ∗ goes towards zero continuously as δ → . ctive gating: rocking diffusion channels r → for a finite size system ρ ( x ) x = i / L r =0.01 r =0.001 r =0.000125 Figure 11.
Rocking TASEP : L = 400 and ( α, δ ) = (0 . , .
9) for different values of r . For r ≈
0, the bulk density goes to , with a negligible slope. At fixed size L , when r →
0, the system mimics the situation where one has twoseparate TASEP lanes, one with entrance and exit rates ( α, δ ) and the other with rates( δ, α ), respectively. Their densities add up. The density in each lane is determined bymin[ α, δ ] till one of the rates is less than 0 .
5, else the system will only show the maximalcurrent phase. Considering α < δ , we expect the density of the system to be given by ρ ( x ) ≈ α +(1 − α )2 = 0 .
5. This relation holds independent of whether α + δ is greater thanor less than 1. Indeed as r ≈
0, the density profile becomes flatter and approaches 0 . even when the bulk density approaches . In fact, as r →
0, the current approachesits value α (1 − α ); see Fig. 7(b). This new feature is in sharp contrast to the unrockedTASEP where a bulk density value around 0 . j ≈ .
5. Rocking ASEP
In the presence of a finite bias field E , we are back to the general set up of Section 2.The density and current are computed in the limit r ↑ ∞ in Appendix B.Here we focus first on a particular interesting aspect of rASEP which occurs for small r for choices of the boundary rates that counter-act the external field. Then, the densityprofile shows non-monotonic behavior with particles heaping up at the boundaries. Thatis similar to the case of run-and-tumble processes in a trap where the particles tend tocluster at the edges for large enough persistence, [14]. An example is shown in Fig. 12(a). The non-monotonicity represents the addition of nonlinear densities; for r ↓
0, thesystem can either be in a standard ASEP maximal current phase or in a low densityphase. ctive gating: rocking diffusion channels j may be non-monotone in r ; see Fig. 12(b). The currentincreases for small r , reaches a maximum, and then decreases to asymptotically approachits predicted value at r ↑ ∞ (computed in Appendix B). The reason is that at small r ,the system is in a low density phase, with the average density being less than , and for r ↑ ∞ , the system reaches a high density (average density greater than ) phase. Atlow densities the current is small due to very few particles being present in the system,while for high densities, the system progressively gets more jammed. In between thesetwo phases, when the average density is around 0 .
5, the current reaches its maximumpossible value for the chosen boundary rates. Note that non-monotone density profilesfor TASEP with particle conservation have been observed in [15] for spatially varyingbut quenched defects. x = i / L ρ ( x ) r =0.01 r =0.02 r =0.1 r =10 (a) -3 -2 -1 r j E =0.5 E =1 E =2 Figure 12. rASEP density and current as functions of r . (a) Plot of ρ ( x ) vs x = i/L for different values of r and fixed E = 1. Non-monotonic density profile is observedfor r ↓
0. The dashed line indicates the flat density profile for r → ∞ in thethermodynamic limit. (b) Plot of j vs r for different values of E. The current increaseswith small r , reaches a maximum and then decays to its value for r → ∞ (indicated byblack dashed lines). The other parameters are a = 0 . , a = 1 . , b = − . , b = 2 . L = 400 . Not shown is the mean entropy production rate as function of E and r . That followshowever the same dependencies as the current, from the dominating effect of the( L + 1) jE term in Eq. 10.
6. Summary
We have studied a class of exclusion models (rSEP, rASEP, rTASEP) in one dimen-sion subject to open and active boundaries. The activity is modeled in terms of adichotomous noise, where the entry and exit rates at the left boundary flip to their rightcounterparts at Poisson rate r . Such models can serve as building blocks for furtherstudies on determining efficiency of transport processes [9], as a function of boundaryactivity. The tuning of chemical potentials in biological environments can be achievedby monitoring pH gradients [10]. ctive gating: rocking diffusion channels r . For rTASEP, the phase diagram is modified:the steady current increases with r and then saturates for r ↑ ∞ to either a value lessthan or equal to 1 /
4. Most interestingly, the system can attain its maximal currentphase even when one of the boundary rates is less than 1 /
2. The role of rocking is reallyto increase the smaller rate. When the sum of the rates exceeds 1, the smaller rate iseffectively pushed beyond 1 / r ↓
0, the systemalways shows a bulk density equal to 1 /
2, but where the current need not be equal to1 /
4. Finally for rASEP, we find that the density profile can be non-monotone for certainboundary rates and for r ↓
0. The associated current can be a non-monotone functionof the rocking rate.The results presented here can be extended in several ways, including more reservoirs ormore types of particles. Some boundary parameters can be kept fixed while others varyrandomly. How that modifies and selects density profiles and current statistics, remainsto be investigated.
Acknowledgements
We thank Urna Basu for many discussions and help throughout the course of the work.TB acknowledges support from Internal Funds KU Leuven.
Appendix A. Particle density in rSEP
From Eq. (17), and using the expressions of the rates from Eqs. (1)-(2), we have, in thestationary state, (cid:104) J (cid:96) ( σ, n ) (cid:105) = 12 [ S (cid:104) n (cid:105) + D (cid:104) σn (cid:105) − ( α + γ )] (A.1) (cid:104) J r ( σ, n L ) (cid:105) = 12 [ S (cid:104) n L (cid:105) − D (cid:104) σn L (cid:105) − ( α + γ )] (A.2)where we denoted S := α + β + γ + δ, D := β − δ + α − γ . The net current to both thereservoirs must also vanish in the stationary state, i.e. , (cid:104) J (cid:96) (cid:105) = (cid:104) J r (cid:105) = 0 and we have S ρ + D (cid:104) σn (cid:105) = α + γ (A.3)for all values of the boundary rates. From symmetry (cid:104) σn (cid:105) = −(cid:104) σn L (cid:105) and both (A.1)and (A.2) give rise to the same relation. In the r ↑ ∞ -limit we have the decoupling (cid:104) σn (cid:105) = (cid:104) σn L (cid:105) = 0 and hence (cid:104) n (cid:105) r ↑∞ = α + γS (A.4)Equation (A.3) is not enough to determine the stationary density as it involves the (cid:104) σn i (cid:105) correlation. We need to use the kinetic equations (13)–(14) at E = 0: at theboundary sites i = 1 , L , ∂ t (cid:104) σn (cid:105) t = (cid:104) λ in ( σ ) σ (1 − n ) (cid:105) t + (cid:104) λ out ( σ ) σn (cid:105) t + (cid:104) σn (cid:105) t − (cid:104) σn (cid:105) − r (cid:104) σn (cid:105) t ctive gating: rocking diffusion channels ∂ t (cid:104) σn L (cid:105) t = (cid:104) λ in ( − σ ) σ (1 − n L ) (cid:105) t + (cid:104) λ out ( − σ ) σn L (cid:105) t − r (cid:104) σn L (cid:105) t (A.5)In the bulk, i.e. , for i = 2 , , · · · , L − ∂ t (cid:104) σn i (cid:105) t = (cid:104) σn i +1 (cid:105) t + (cid:104) σn i − (cid:105) t − r + 1) (cid:104) σn i (cid:105) t (A.6)In the stationary state the time-derivatives vanish, and Eqs (A.5) give, α − γ − ( S + 4 r ) (cid:104) σn (cid:105) − D ρ + 2 (cid:104) σ ( n − n ) (cid:105) = 0 (A.7)Note that (cid:104) σn (cid:105) = −(cid:104) σn L (cid:105) , and the two lines in Eq. (A.5) reduce to the same relation.In the stationary state Eq. (A.6) gives,∆ (cid:104) σn i (cid:105) = 2 r (cid:104) σn i (cid:105) , i = 2 , . . . , L − (cid:104) σn (cid:105) = −(cid:104) σn L (cid:105) . To solve the equation (A.8), we write∆ f j := f j +1 + f j − − f j = 2 r f j , j = 2 , . . . , L − f L + f = 0 and f j ∈ [ − , g j = e kj for arbitrary k. Then, we must verify∆ g j = e kj ( e k + e − k −
2) = 2 rg j which means that k must be such that e k + e − k − r > k > k must be real. Clearly, every linear combination ofsuch g j still solves the eigenequation. Per consequence, f j = B A − [ e − k ( L +1) / e kj − e k ( L +1) / e − kj ] = 2 B A − sinh(( j − L + 12 ) k )solves the boundary condition and the eigenequation (A.9), as long as A = 2 sinh( L − k )and | B | ≤ | f j | ≤
1. In conclusion, we must have for all j = 1 , . . . , L , f j = B sinh(( j − L +12 ) k )sinh( L − k ) , cosh k = r + 1Obviously, f = (cid:104) σn (cid:105) = − B fixes B : f j = −(cid:104) σn (cid:105) sinh(( j − L +12 ) k )sinh( L − k ) , cosh k = r + 1 (A.10)The solution is given by, (cid:104) σn i (cid:105) = (cid:104) σn (cid:105) sinh [( L +12 − i ) k ]sinh( L − k ) , where r = cosh k − (cid:104) σn (cid:105) only which is given by (cid:104) σn (cid:105) = (cid:104) σn (cid:105) sinh (cid:18)(cid:18) L + 12 − (cid:19) k (cid:19) / sinh (cid:18) L − k (cid:19) In the limit of thermodynamic system size L → ∞ , this reduces to, (cid:104) σn (cid:105) = e − k (cid:104) σn (cid:105) (A.12) ctive gating: rocking diffusion channels k = log( r + 1 + √ r + 2 r ) taken positive. Combining (A.7), (A.12) and (A.3), weget two independent equations involving ρ and (cid:104) σn (cid:105) , (2 e − k − S − r − (cid:104) σn (cid:105) − Dρ = γ − αD (cid:104) σn (cid:105) + S ρ = α + γ (A.13)which can be solved immediately to obtain, (cid:104) σn (cid:105) = αδ − βγ α + β )( γ + δ ) + SR (A.14)and ρ = ( α − γ ) D − ( S + 4 r + 2(1 − e − k )( α + γ ) D − ( S + 4 r + 2(1 − e − k )) S (A.15)The final formula (18) follows by using 2 r + 1 − e − k = R = r + (cid:112) r ( r + 2). Appendix B. Density and current in rASEP for r ↑ ∞ For r → ∞ and E >
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