Nonlinear macroscopic transport equations in many-body systems from microscopic exclusion processes
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p Nonlinear macroscopic transport equations inmany-body systems from microscopic exclusionprocesses.
Marta Galanti , , Duccio Fanelli , Francesco Piazza Dipartimento di Fisica and INFN, Universit`a di Firenze, I-50019, Sesto F.no(FI), Italy Centre de Biophysique Mol´eculaire, CNRS-UPR 4301, Universit´e d’Orl´eans,45071 Orl´eans cedex, FranceE-mail:
Abstract.
Describing particle transport at the macroscopic or mesoscopiclevel in non-ideal environments poses fundamental theoretical challenges indomains ranging from inter and intra-cellular transport in biology to diffusionin porous media. Yet, often the nature of the constraints coming from many-body interactions or reflecting a complex and confining environment are betterunderstood and modeled at the microscopic level.In this paper we investigate the subtle link between microscopic exclusionprocesses and the mean-field equations that ensue from them in the continuumlimit. We derive a generalized nonlinear advection diffusion equation suitable fordescribing transport in a inhomogeneous medium in the presence of an externalfield. Furthermore, taking inspiration from a recently introduced exclusion processinvolving agents with non-zero size, we introduce a modified diffusion equationappropriate for describing transport in a non-ideal fluid of d -dimensional hardspheres.We consider applications of our equations to the problem of diffusion toan absorbing sphere in a non-ideal self-crowded fluid and to the problem ofgravitational sedimentation. We show that our formalism allows one to recoverknown results. Moreover, we introduce the notions of point-like and extended crowding, which specify distinct routes for obtaining macroscopic transportequations from microscopic exclusion processes.PACS numbers: 02.50.Ey, 05.10.Gg, 05.60.Cd, 66.10.cg, 61.20.-p Keywords : exclusion processes, transport equations, crowding, non-ideal fluids
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J. Phys. A: Math. Gen. acroscopic transport equations from microscopic exclusion processes.
1. Introduction
Diffusive transport is central in many areas of physics, chemistry, biology and softmatter [1–4]. However, while the mathematics of diffusive processes in dilute andsimple media is fairly well developed and understood [1], many interesting and relevantdiffusive processes take place in strongly non-ideal conditions. These include a wealthof different highly dense media, from non-ideal plasmas [5] to biological membranes [6],media with complex topological structures, including porous media [7–9] and livingcells [10; 11] and strongly confining environments [3; 12–17].Crowding and confinement effects on diffusion-influenced phenomena still posefundamental yet unanswered questions. Concerning molecular mobility, for example,several computational and experimental indications exist of anomalous diffusion in thecell cytoplasm as a result of amount and type of crowding [18; 19], suggesting thatliving cells behave much like fractal or otherwise disordered systems [20; 21]. However,strong evidences also exist in favour of normal (Brownian) diffusion, crowding andconfinement resulting in this scenario in (often nontrivial) modifications of the diffusioncoefficient [10; 22–24]. Another related issue is that of diffusion-limited reactions [25],which are ubiquitous in many domains in biology and chemistry, touching uponproblems such as association, folding and stability of proteins [26; 27] and bimolecularreactions in solution [28–32], including enzyme kinetics [33], but also the dynamics of active agents [34]. Many theoretical studies have tackled these and related problemsunder different angles [16; 26; 30; 33; 35–37]. Nevertheless, a full theoreticalcomprehension of transport in non-ideal media remains an elusive task, Fick’s lawitself and the very notion of effective diffusion coefficient being questionable in adisordered medium [21].In this paper we consider the subtle link between macroscopic transport equations,such as the diffusion equation, and microscopic processes, modeling the stochasticdynamics of some agents. The purpose of our study is two-fold. One the one hand,we wish to understand in greater depth the delicate procedure of obtaining mean-fieldtransport equations from microscopic, agent-based stochastic models, paying specialattention to what exactly is lost in going to the continuum. The idea is that sometimesit may prove simpler or more effective to describe a complex transport process (or asimple one occurring in a complex milieu ) at the microscopic level. On the contrary,it is sometimes better to deal with macroscopic equations. It is thus important toinvestigate how the two levels of description interface with each other. On the otherhand, as a result of our investigation, we will derive new macroscopic equations thatprovide useful analytical tools for exploring particle transport in non-ideal fluids incomplex environments.The paper is organized as follows. In section 2 we discuss the general framework ofsimple exclusion processes (SEPs), which constitute the basic tool of our microscopicdescription, as well as the process of obtaining mean-field equations from SEPs.In section 3, we move a step forward and consider microscopic exclusion processesinvolving agents characterized by a finite size, as opposed to standard SEPs. Weintroduce in particular the notions of point-like and extended crowding and discuss theformalism with reference to the problem of gravitational sedimentation.We then go on in section 4 to propose, based on a simple analogy, a modifieddiffusion equation appropriate for describing transport in non-ideal fluids of d -dimensional hard spheres at high densities. The obtained equation allows one torecover a known result for the basic problem of particle flux at a spherical absorbing acroscopic transport equations from microscopic exclusion processes.
2. From microscopic processes to macroscopic equations
Simple exclusion processes are space-discrete, agent-based stochastic processesmodeling some kind of transport according to specific rules and bound to the constraintthat no two agents can ever occupy the same site. SEPs occupy a central role in non-equilibrium statistical physics [38; 39]. While the first theoretical ideas underlyingsuch processes can be traced back to Boltzmann’s works [40], SEPs were introducedand widely studied in the 70s as simplified models of one-dimensional transport forphenomena like hopping conductivity [41] and kinetics of biopolymerization [5]. Alongthe same lines, the asymmetric exclusion process (ASEP), originally introduced bySpitzer [42], has become a paradigm in non-equilibrium statistical physics [43–45]and has now found many applications, such as the study of molecular motors [46],transport through nano-channels [47] and depolymerization of microtubules [48].The most general SEP in one dimension is described by a stochastic jump processon a 1 D lattice with inequivalent sites in the presence of a field n i ( k + 1) − n i ( k ) = z + i − n i − ( k )[1 − n i ( k )] + z − i +1 n i +1 ( k )[1 − n i ( k )] − z + i n i ( k )[1 − n i +1 ( k )] − z − i n i ( k )[1 − n i − ( k )] (1)Eq. (1) is to be regarded as the update rule for a Monte Carlo process, where n i ( k ) isthe occupancy of site i at time t = k ∆ t , which can be either zero or one. The quantities z ± i are variables which have the value 0 or 1 according to a random number ξ i whichhas a uniform distribution between 0 and 1. By defining the jump probabilities q ± j ( j = i, i ±
1) one can formally write: z + i − = θ ( ξ i ) − θ ( ξ i − q + i − ) z − i +1 = θ ( ξ i − q + i − ) − θ ( ξ i − q + i − − q − i +1 ) z + i = θ ( ξ i − q + i − − q − i +1 ) − θ ( ξ i − q + i − − q − i +1 − q + i ) z − i = θ ( ξ i − q + i − − q − i +1 − q + i ) − θ ( ξ i −
1) (2)where θ ( · ) stands for the Heaviside step function and where we are assuming that q + i − + q − i +1 + q + i + q − i = 1. Note that the ordering of appearance of the q ± j in theabove expressions is arbitrary. Equations (2) entail that h z ± j i = q ± j , where h·i indicatesan average over many values of ξ i , for a given configuration { n i } . The above processis fully determined by the fields q ± i , specifying the probability of jumping from site i to site i + 1 ( q + i ) or to site i − q − i ) in a time interval ∆ t .A (discrete-time) master equation for the above SEP can be obtained by averagingover many Monte Carlo cycles performed according to rule (1) P i ( k + 1) − P i ( k ) = q + i − [ P i − ( k ) − P i,i − ( k )] + q − i +1 [ P i +1 ( k ) − P i,i +1 ( k )] − q + i [ P i ( k ) − P i,i +1 ( k )] − q − i [ P i ( k ) − P i,i − ( k )] (3)where we have defined the one-body and two-body site occupancy probabilities P i ( k ) = hh n i ( k ) ii (4 a ) P i,i ± ( k ) = hh n i ( k ) n i ± ( k ) ii (4 b )Here hh·ii denotes averages performed over many independent Monte Carlo cyclesperformed until time k ∆ t starting from the same initial condition. We emphasize that acroscopic transport equations from microscopic exclusion processes. With the aim of deriving macroscopic transport equations from the microscopicstochastic process described by eqs. (1), it is customary to assume a mean-field (MF)factorization, P i,i ± ( k ) ≡ hh n i ( k ) n i ± ( k ) ii = hh n i ( k ) iihh n i ± ( k ) ii = P i ( k ) P i ± ( k ) (5)With the help of eq. (5), the master equation (3) becomes P i ( k + 1) − P i ( k ) = q + i − P i − ( k )[1 − P i ( k )] + q − i +1 P i +1 ( k )[1 − P i ( k )] − q + i P i ( k )[1 − P i +1 ( k )] − q − i P i ( k )[1 − P i − ( k )] (6)Nonlinear mean-field equations for exclusion process of this type have been used sincethe 70s to investigate one-dimensional transport in solids [49].Let a be the lattice spacing and let us define a reversal probability ǫ i , such that q + i = Q i q − i = Q i − ǫ i (7)The condition (7) (with ǫ i >
0) amounts to considering a field introducing a bias in thepositive x direction. In order to take the continuum limit lim a, ∆ t → P i ( k ) = P ( x, t ),we must require lim a, ∆ t → Q i a ∆ t = D ( x ) (8 a )lim a, ∆ t → ǫ i a ∆ t = v ( x ) (8 b )Eq. (8 a ) defines the position-dependent diffusion coefficient, while eq. (8 b ) defines thefield-induced drift velocity. Note that we are assuming that the reversal probabilityvanishes linearly with a .With the help of eqs. (7), (8 a ) and (8 b ) it is not difficult to see that in thecontinuum limit eq. (6) yield ∂P∂t = (1 − P ) ∇ [ D ( x ) P ] + D ( x ) P ∇ P − ∂∂x [ v ( x ) P (1 − P )] (9)where we have explicitly highlighted the fact that both the diffusion coefficient andthe drift velocity are position-dependent quantities.Eq. (9) is a nonlinear advection-diffusion equation, appropriate for describingthe continuum limit of a microscopic exclusion process occurring on a lattice ofinequivalent sites in the presence of a field. Although it represents the mean-fieldapproximation of a known master equation, we are not aware of any authors statingit in its most general form. It is interesting to note that in the case of equivalentsites, which translates to a constant diffusion coefficient, the diffusive part of eq. (9)becomes linear, i.e. the microscopic exclusion rule is lost in the diffusive part. In thecase of zero field, one then simply recovers the ordinary diffusion equation which, asit is widely known, can be derived from a microscopic jump process with no exclusionrules. This curious observation has been first reported by Huber [49]. If both thediffusion coefficient and the drift velocity are constant, eq. (9) reduces to ∂P∂t = D ∇ P − v ∂∂x [ P (1 − P )] (10) acroscopic transport equations from microscopic exclusion processes. P ( x, t ), which is a numberbetween zero and one. The value P = 1 should correspond to the maximum density ρ M allowed in the system. Thus, a more physical equation can be obtained by introducingthe agent density ρ ( x, t ) ≡ ρ M P ( x, t ) = φ M v ( σ/ P ( x, t ) (11)where v ( r ) = ( π / r ) d Γ(1 + d/
2) (12)is the volume of a d -dimensional sphere ‡ of radius r and φ M is the maximumpacking fraction for systems of d -dimensional hard spheres, φ M = 1 ( d = 1), φ M = π/ √ ≈ .
907 ( d = 2) and φ M = π/ √ ≈ .
740 ( d = 3) [51]. With thesedefinitions, and using a more general vector notation, eq. (9) becomes ∂ρ∂t = (cid:18) − ρρ M (cid:19) ∇ [ D ( x ) ρ ] + D ( x ) (cid:18) ρρ M (cid:19) ∇ ρ − ∇ · (cid:20) v ( x ) ρ (cid:18) − ρρ M (cid:19)(cid:21) (13)This is the first important result of this paper.
3. Point like versus extended crowding in one dimension
The MF diffusion-advection equation (9) has been derived from a master equationcontaining exclusion terms of the type (1 − P i ) in the limit of vanishing lattice spacing.This amounts to considering agents of vanishing size in the continuum limit. We termthis peculiar situation in the macroscopic world point-like crowding . However, onemay argue that considerable microscopic information is lost in going to the continuumlimit with point-like agents. Incidentally, this has to be the reason why the mean-fieldapproximation does lose all the memory of the microscopic exclusion constraint andthe diffusion equation is recovered for equivalent sites in the absence of a field.Interestingly, a generalized exclusion process for agents of extended size in onedimension (hard rods) can be found in the literature, termed the ℓ -ASEP [52]. Theauthors derive a mean-field equation, which, in the absence of a field and for equivalentsites, reads ∂ρ∂t = D ∇ (cid:20) ρ − σρ (cid:21) (14)where σ is the length of the rods. The nonlinear diffusion equation (14) is mostinteresting for a number of reasons. First of all, even if it has been derived through aningenious but complicate change of variables based on a quantitative mapping betweenthe ℓ -ASEP and the zero-range process [52], it turns out that it can be regarded as thelocal-density approximation (LDA) of a simple general property of one-dimensionalexclusion processes. As pointed out in 1967 by Lebowitz and Percus [53] concerningbulk properties[ . . . ] For many purposes, however, adding a finite diameter does not introduceany new complications; it merely requires the replacement in certainexpressions of the actual volume per particle ρ − by the reduced volume ρ − − σ , i.e. ρ → ρ/ (1 − σρ ). ‡ We emphasize that we use the general terminology of d -dimensional hard spheres. Obviously, theseare hard rods in one dimension and hard disks in two. acroscopic transport equations from microscopic exclusion processes. ρ ( x, t ) → ρ ( x, t ) / [1 − σρ ( x, t )], one recovers eq. (14). Extendingthe terminology of the ℓ -ASEP [52], we term this scenario extended-size crowding .Point-like crowding in the mean field approximation corresponds to systems of fullypenetrable spheres, while extended-size crowding yields a transport equation suitablefor systems of totally impenetrable (hard) spheres.It is possible to substantiate and further illustrate the above considerations byshowing that the transport equations for point-like and extended-size crowding forhomogenous systems in a gravitational field allow to recover the appropriate equationsof state (EOS) for fully penetrable spheres (FPSs) and totally impenetrable ( i.e. hard)spheres (TISs), respectively. Let us consider sedimentation in one dimension. Let ̺ p and ̺ s indicate the particle and solvent material densities § , respectively, and m ∗ = ( ̺ p − ̺ s ) v σ the buoyant mass of the particles. The appropriate equation forpoint-like crowding bears in the case of non-zero field the signature of the microscopicexclusion process. Recalling that φ M = 1 in one dimension and thus ρ M = 1 /σ (seeeq. (11)), eq. (13) reduces to d ρdz + 1 ℓ g ddz [ ρ (1 − σρ )] = 0 (15)Here ℓ g = k B T /m ∗ g is the so-called sedimentation (or gravitational) length and wehave taken an upward pointing z -axis. Eq. (15) can be solved by noting that atequilibrium the osmotic current is exactly balanced by the gravitational one, thus J = dρ/dz + ρ (1 − σρ ) /ℓ g = 0, which gives ρ ( z ) = 1 σ (cid:16) Ae z/ℓ g (cid:17) (16)The constant A is to be determined by imposing the boundary conditions. Usually,this is done by introducing the bulk density ρ , such that1 h Z h ρ ( z ) dz = ρ (17)where h is the height of the initially homogenous suspension. This gives ρ ( z ) = σ − " − e − (1 − σρ ) h/ℓ g e σρ ( h/ℓ g ) − ! e z/ℓ g − (18)For h ≫ ℓ g one has that the density at the bottom is exponentially close to themaximum density ρ M = σ − ρ (0) ≃ σ (cid:16) e − σρ ( h/ℓ g ) (cid:17) (19)In practice, particle settling leaves only supernatant solvent at the top of the cell.Hence we can safely take h → ∞ , which shows that the fluid attains maximum packingat the bottom of the cell.The nice thing about this exercise is that we can derive an equation of state (EOS)from the sedimentation profile. In fact, the osmotic pressure Π in the suspension isgiven by Π( z ) k B T = 1 ℓ g Z ∞ z ρ ( z ) dz (20) § For the sake of clarity, we indicate with ̺ mass densities and with ρ number densities, measuredin number of particles per unit volume. acroscopic transport equations from microscopic exclusion processes. h → ∞ . It is a straightforward calculation to show that the generalexpression (16) yields, independently of the chosen boundary conditions,Π k B T = − σ − log[1 − σρ ] (21)The EOS (21) has a straightforward physical interpretation. In a system of fullypenetrable spheres with reduced density σρ , the volume fraction φ p occupied by theparticles is given by φ p = 1 − exp( − σρ ) [51]. Hence, the modified EOS (21) can beobtained from the perfect gas EOS by replacing the reduced density with its expressioncontaining the actual volume fraction. This unveils the meaning of the above modifiedEOS, featuring the true volume fraction in the right-hand side. At the same time, thisalso illustrates the notion of point-like crowding.Summarizing, a point-like microscopic exclusion process such as (6) yields amacroscopic transport equation that describes a fluid of fully penetrable spheres.Therefore, extending this line of reasoning to what we have dubbed the extended-crowding scenario, we expect that the ℓ -ASEP mean-field equation in non-zerofield [52] would yield the EOS of a system of hard rods, also known as the Tonkgas [54] Π k B T = ρ − σρ (22)We shall now prove that this is indeed the case. Inserting the Tonk gas EOS (22) ineq. (20) and differentiating with respect to z , one gets J σ ≡ dρdz + 1 ℓ g ρ (1 − σρ ) = 0 (23)the known LDA equation for sedimentation of hard rods [55]. However, the aboveequation also equals the condition that the total ℓ -ASEP particle flux J σ vanishes atequilibrium. Indeed, the stationary mean-field equation corresponding to the ℓ -ASEPreads [52] d dz (cid:20) ρ − σρ (cid:21) + 1 ℓ g dρdz = 0 (24)which leads to the same expression for the total (constant) particle flux as eq. (23).Hence, the ℓ -ASEP yields a macroscopic transport equation that describes a gas ofhard rods in the local density approximation.The discussion above which leads to the concept of extended crowding appliesto one dimensional systems. Starting from this setting, one can raise the questionwhether similar arguments might be employed to obtain a modified nonlinear equationaccounting for excluded volume effects in the diffusion of hard spheres in two and threedimensions. The following section is devoted to speculating further along this line ofreasoning.
4. Excluded volume interactions of finite-size agents in higher dimensions.
In the preceding section we have shown that the point-like crowding scenario leadsto the mean-field equation (13) from a standard simple exclusion process at the microscopic level. The extended-crowding generalization consists in endowing agents acroscopic transport equations from microscopic exclusion processes. Table 1.
The particle conditional pair distribution function G p for stationaryensembles of d -dimensional spheres of diameter σ [51] d = 1 d = 2 d = 3Fully penetrable spheres 1 1 1Totally impenetrable spheres 11 − σρ a + a (cid:0) σr (cid:1) b + b (cid:0) σr (cid:1) + b (cid:0) σr (cid:1) with a finite size (hard core). The ensuing exclusion process in one dimension hasbeen termed in the literature the ℓ -ASEP [52].We have shown that the nonlinear transformation put forward by Lebowitz andPercus in 1967, under the local density approximation, allows one to recover the ℓ -ASEP mean-field transport equation (14) for the evolution of the density of extendedrod-like agents in one dimension. Interestingly, it is possible to posit a generalizationof such macroscopic transport equation adequate to the mean-field limit of exclusionprocesses involving extended objects in more than one dimension by borrowing generalconcepts in the theory of heterogenous media.With reference to standard definitions of micro-structural descriptors in d dimensions [51], we can identify the aforementioned substitution by Lebowitz andPercus as a mapping between certain statistical properties characterizing systems offully penetrable spheres and totally impenetrable ( i.e. hard) spheres. More precisely,let us introduce the so-called conditional pair distribution function (CPDF) G p ( r ).Let r denote the distance from the center of some reference particle in a system withbulk density ρ . Then, by definition ρ s ( r ) G p ( r ) dr equals the average number ofparticles in the shell of infinitesimal volume s ( r ) dr around the central particle, giventhat the volume v ( r ) of the d -sphere of radius r is empty of other particle centers.Here s ( r ) = dv ( r ) dr = 2 π d/ r d − Γ( d/
2) (25)The CPDF for FPS and TIS systems are reported in Table (1), where one can readilyrecognize the Lebowitz and Percus substitution as a mapping between the FPS andTIS CPDFs in one dimension. The coefficients appearing in the expressions of G p ( r )are given by the following expressions k for d = 2 [51] a = 1 + 0 . φ (1 − φ ) (26 a ) a = − . φ (1 − φ ) (26 b )while for d = 3 one has b = 1 + φ + φ − φ (1 − φ ) (27 a ) b = φ (3 φ − φ − − φ ) (27 b ) b = φ (2 − φ )2(1 − φ ) (27 c ) k These are the expressions appropriate to the liquid phase below the freezing point, i.e. for φ < φ f ( φ f = 0 .
69 for d = 2 and φ f = 0 .
494 for d = 3). acroscopic transport equations from microscopic exclusion processes. φ = v ( σ/ ρ is the packing fraction, σ being the diameter of a d -dimensionalsphere.The above discussion suggests a way to generalize the ℓ -ASEP (14) to describeexcluded-volume effects in more than one dimension in the mean-field approximationin a homogeneous medium and zero field. For a spherically symmetric problem weposit ∂ρ∂t = D ∇ [ ρ G p ( ρ, r )] (28)where we have emphasized that G p depends explicitly on r (for d > G p are given by eqs (26 a ), (26 b ), (27 a ), (27 b ) and (27 c ) and it is understood that,according to the local density approximation, one should replace φ with v σ ρ ( r, t ),where v σ ≡ v ( σ/
2) is the volume of one hard d -dimensional sphere, so that G p depends on r both explicitly and implicitly through ρ ( r ). In order to provide some a posteriori justification for eq. (28), it is instructive toconsider how the classical problem of diffusion to an absorbing sphere is modified ina non-ideal fluid. Let us imagine a fixed sink of radius R s that absorbs hard spheresof radius σ/ ρ . The rate k measuring the number of particlesabsorbed by the sink per unit time equals the total flux into the sink. For ordinaryFickian diffusion, one has the classical result k = k S ≡ πD ( R s + σ/ ρ , known as theSmoluchowski rate [25]. This result is indeed the prediction of a two-body problem, i.e. it amounts to considering the absorption of non-interacting, or equivalentlyfully penetrable, spheres. Thus, it describes the problem in the infinite dilutionlimit. Eq. (28) can now be employed to repeat the same exercise for hard spheres atfinite densities, that is, the Smoluchowski problem with excluded-volume interactionsaccounted for. One should then solve the following boundary-value problem ∇ [ ρ G p ( v σ ρ, r )] = 0 (29 a ) ρ ( r = R s + σ/
2) = 0 (29 b )lim r →∞ ρ ( r ) = ρ (29 c )The rate can be computed readily without really solving the (modified) Laplaceequation. From (29 a ), we have directly ∂∂r [ ρ ( r ) G p ( φ ( r ) , r )] = k πDr (30)where we have defined φ ( r ) = v σ ρ ( r ), so that φ = v σ ρ denotes the bulk packingfraction of the hard spheres. Integrating eq. (30) between R s + σ/ b ) and (29 c ), it is straightforward toobtain kk S = G ∞ p (31)where G p ( ∞ ) ≡ lim r →∞ G p ( φ ( r ) , r ). In three dimensions, one thus has k/k S = b ( φ ),where one can recognize b ( φ ) as the compressibility Z ( φ ) of the hard sphere fluidin the Carnahan-Starling approximation [56]. We see that eq. (28) allows one torecover our previous result k/k S = Z ( φ ), obtained in two different ways, by assuminga density-dependent mobility in the diffusion equation [57] and from a transportequation derived in the local-density approximation [58]. acroscopic transport equations from microscopic exclusion processes.
5. Summary and discussion
In this paper we have discussed a general framework allowing to obtain macroscopictransport equation accounting for excluded volume effects in systems of d -dimensionalhard spheres starting from a microscopic stochastic exclusion processes. The aimof this procedure is to derive mean-field equations suitable for describing transportprocesses in many-body systems in highly non-ideal conditions. We have identifiedtwo strategies for doing so. The first route, termed point-like crowding, leads from astandard simple exclusion process to the mean-field equation (13). For a homogenousmedium in the absence of a field this reduces to a simple diffusion equation, whichis why we have dubbed this scenario point-like crowding. Only for inequivalent sitesand/or in the presence of a field does the microscopic exclusion constraint survives inthe mean-field limit. The second strategy, named extended-crowding , takes inspirationfrom a modified microscopic exclusion process in one dimension involving extendedagents, the so-called ℓ -ASEP [52]. By extending an argument originally put forwardby Lebowitz and Percus in 1967, coupled to a local density approximation, we haveposited a modified nonlinear diffusion equation suitable for studying in an effectivemanner transport processes in dense systems of hard spheres, eq. (32). We haveshown that this equation allows one to recover recent results obtained for the problemof diffusion to an absorbing sphere in a self-crowded medium. Furthermore, wehave brought to the fore an interesting structure underlying the two above strategiesfor obtaining macroscopic equations from microscopic stochastic processes. Whileextended crowding is indeed appropriate for diffusion of hard spheres, the continuumlimit of point-like crowding exclusion processes is only appropriate for systems of fullypenetrable spheres. Therefore, when one takes the continuum limit, all signatures ofthe microscopic exclusion constraints are lost in the diffusive part of the transportequation.We note that eq. (28) provides a sensible description of diffusion in a non-idealfluid in the case where the problem has spherical symmetry. We may ask whether ourline of reasoning may be extended to a problem with different or no symmetry. Guidedby the observation that G ∞ p = Z ( φ ) for a system of d -dimensional hard spheres, wecan speculate that a modified equation could be considered, involving the bulk CPDF G ∞ p ( φ ) in the local density approximation, ∂ρ∂t = D ∇ [ ρ G ∞ p ( ρ )] (32)where again one should understand φ → v σ ρ ( r, t ) in the expression for the coefficients a i ( φ ) and b i ( φ ). Some justification for the above idea can be gathered by recalling theknown equation that one obtains by introducing a density-dependent mobility in thediffusion equation given by the derivative of the osmotic pressure with respect to thedensity [2], D ( ρ ) = D β d Π( ρ ) /dρ = D Z ( ρ ), namely ∂ρ∂t = D ∇ · [ Z ( ρ ) ∇ ρ )] (33)where D is the bare diffusion coefficient (corresponding to the infinite dilution limit)and Z ( ρ ) is the compressibility factor. At zero order, where G p ( ρ ) = G p ( ρ ) = const. and Z ( ρ ) = Z ( ρ ) = const. eq. (32) and eq. (33) are the same, as G ∞ p ( φ ) = Z ( φ ).As a last observation, we see that, if we compare eq. (24) with eq. (15), thepoint-like route to a macroscopic mean-field equation for a homogenous medium bearsthe signature of the microscopic exclusion mechanism in the advection term. On EFERENCES
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