Abstract
We consider a twospecies simple exclusion process on a periodic lattice. We use the method of matched asymptotics to derive evolution equations for the two population densities in the dilute regime, namely a crossdiffusion system of partial differential equations for the two species’ densities. First, our result captures nontrivial interaction terms neglected in the meanfield approach, including a nondiagonal mobility matrix with explicit density dependence. Second, it generalises the rigorous hydrodynamic limit of Quastel (Commun Pure Appl Math 45(6):623–679, 1992), valid for species with equal jump rates and given in terms of a nonexplicit selfdiffusion coefficient, to the case of unequal rates in the dilute regime. In the equalrates case, by combining matched asymptotic approximations in the low and highdensity limits, we obtain a cubic polynomial approximation of the selfdiffusion coefficient that is numerically accurate for all densities. This cubic approximation agrees extremely well with numerical simulations. It also coincides with the Taylor expansion up to the secondorder in the density of the selfdiffusion coefficient obtained using a rigorous recursive method.
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1 Introduction
Stochastic models describing systems of interacting particles are widely used across many disciplines [1,2,3]. A particular class of models concerns excludedvolume or steric interactions, which are local and remain strong regardless of the number of particles in the system. This is in stark contrast with weak or meanfield interactions, which are longrange and weaker as the number of particles increases. Excludedvolume interactions arise from constraints that forbid overlap of particles and, as a result, are particularly relevant in biological applications with crowding, such as transport of tumour cells [4], wound healing [5], and predatorprey systems [6]. Models for diffusive systems with excludedvolume interactions can be broadly split into continuous and discrete depending on the stochastic process employed to describe the motion of particles. The continuous approach considers correlated Brownian motions, with correlations appearing either through soft shortranged interaction potentials or hardcore potentials representing particle shapes. The discrete approach consists of random walks on discrete spaces (e.g. a regular lattice) with exclusion rules that constrain the number of particles allowed in each lattice site. This paper focuses on one such model, namely a simple exclusion process (SEP) whereby only one particle is allowed per site. In particular, we study a SEP with two species of particles on a ddimensional lattice, for \(d \ge 2\).^{Footnote 1} The lattice spacing is h and the total number of particles is N.
We focus on systems with many particles, in a limit where the number of particles \(N\rightarrow \infty \) and the lattice spacing \(h\rightarrow 0\). The central object of interest is the macroscopic density, whose evolution can be described by partial differential equations (PDEs). For a singlespecies SEP, the continuum model for the particle density \(\rho (x,t)\) takes the form [7]
where the diffusivity D and the potential force V are determined from the underlying asymmetric hopping rates of the process. The first approach to obtain the driftdiffusion equation (1) is through a hydrodynamic limit, where the (random) empirical density converges in probability to the (deterministic) solution of the PDE [8,9,10], taking the number of particles \(N\rightarrow \infty \) and the lattice spacing \(h\rightarrow 0\) together, keeping the occupied volume fraction \(\phi = \int \rho \text {d} x\) fixed. An alternative method for deriving macroscopic evolution PDEs such as (1) is to analyse the average density directly [11], thus sidestepping the technical challenges associated with the convergence of random variables. The starting point of this approach is to consider the master equation for the joint density of the N particle system [12], from which a hierarchy of equations for the moments can be obtained (analogously to the BBGKY hierarchy [13]). The macroscopic behaviour is determined by the first moment (mean density) equation, which in general (for interacting processes) depends on the second moment. A standard route to obtain a closed equation is to use a moment closure approximation such as a meanfield closure [14, 15]. Finally, the corresponding PDE description is found by performing a simple perturbation approach (taking the continuous limit assuming the density to be slowly varying [16]).
The case of a single species of SEP (including asymmetric rates) is well understood: the macroscopic limit obtained formally via a meanfield closure, coincides with the rigorous hydrodynamic limit (1) studied in [7]. In [17] and [18] , Quastel derived the hydrodynamic limit and large deviations respectively, for finitely many coloured species undergoing a symmetric SEP (SSEP), corresponding to equal diffusivity D amongst species and setting \(V\equiv 0\) in (1). The hydrodynamic equations in this case take the form of a crossdiffusion system of PDEs for the two species densities \(\rho _r, \rho _b\),
where M is a mobility matrix and \(\delta _{\rho }E\) denotes the functional derivative of a suitable freeenergy functional E. The form (2) can be indentified as a 2Wasserstein gradient flow [19, 20]. The mobility matrix M in [17] is full and depends on the selfdiffusion coefficent \(D_s(\rho )\) of a tagged particle in a SSEP at uniform density \(\rho = \rho _r + \rho _b\). This \(D_s(\rho ) \in C^\infty ([0,1])\) [21], but its dependence on \(\rho \) is not known, although it can be determined from a variational formula [22]. In contrast to the singlespecies SEP, analysis of this hydrodynamic limit is challenging for multispecies systems, because they are not of gradient type [9], this technical distinction will be discussed below. Quastel’s approach has been generalised to study an active SEP (where the orientation of the particle governs the asymmetry in the hopping rates) in [23]. The hydrodynamic limit has also been rigorously derived for similar multispecies systems in [24] and [25], for the zerorange process and interacting Brownian motions in one dimension, respectively. In both of these models the limiting macroscopic PDE is known explicitly.
For the twospecies SEP, the master equation and meanfield closure approach [14, 19] yields a result of the form (2) with an explicit diagonal mobility matrix. Clearly, this result is not consistent with the rigorous hydrodynamic limit whose mobility matrix is full: this situation may be contrasted with the singlespecies case where the result of the meanfield closure is exact.
In this work, we derive a crossdiffusion system of the form (2) from the master equation using the method of matched asymptotics [26] in the lowdensity limit, \(\phi \ll 1\). In contrast to the meanfield closure, our approach has the advantages of being systematic and consistent with the rigorous hydrodynamic limit [17]. In particular, we show that our mobility matrix M agrees up to the expected order in \(\phi \) with the result by Quastel [17]. Moreover, our approach provides an explicit mobility matrix, which makes the analysis of the PDE more amenable, and extends the result of Quastel [17] to handle the general case of different rates amongst species. This is the first result for two species with unequal rates that does not rely on a meanfield approximation to the authors’ knowledge.
The method of matched asymptotics also yields predictions for the motion of a single (tracer) particle in these mixtures via its selfdiffusion constant. For the case where the species have equal diffusivity, this yields the same result as the (rigorous) recursive computation of the selfdiffusion constant of Landim et al. [21]. In this case, we also consider an expansion around the highdensity limit, which we combine with the lowdensity expansion to obtain a cubic approximation of the selfdiffusion coefficient that performs well for the whole range of densities. This leads to an explicit PDE for the density, which agrees well with numerical data.
The remainder of the paper is organised as follows. In Sect. 2 we define the model, discuss existing results for the coloured case, and summarise our main results. The derivation of a crossdiffusion system of the form (2) for the general case via meanfield and the method of matched asymptotic is given in Sect. 3. Then, Sect. 4 concerns the analysis of the selfdiffusion coefficient \(D_s\), and Sect. 5 compares numerical simulations of the PDE systems with stochastic simulations of the microscopic model. We draw together our conclusions in Sect. 6.
2 Model Definitions and Summary of Main Results
2.1 Microscopic Model
Consider N particles on a ddimensional (hyper)cubic lattice \(\Omega \) with \(L^d\) sites and \(d \ge 2\). We embed \(\Omega \) in the unit ddimensional torus \(\mathbb {T}^d\) by setting the lattice spacing \(h=(1/L)\). We impose a simple exclusion constraint so each site contains at most one particle. Hence the lattice spacing, h, can be thought of as the particle diameter, and the lattice sites are points \(x\in \Omega \subset \mathbb {T}^d\) such that \((x/h)\in \{1,2,\dots ,L\}^d\).
We consider a system with two types (species) of particles, where the hopping rates depend on the species. We denote the species as ‘red’ and ‘blue’. The total number of particles is N, of which \(N_r\) are red, and \(N_b\) are blue. Throughout, we use \(\sigma \in \{ r, b\}\) to label a species and \({{\bar{\sigma }}}\) for the opposite species.
The (random) configuration of species \(\sigma \) at time t is denoted by \(\eta ^{\sigma }_t \in \{ 1, 0 \}^\Omega \), where \(\eta ^{\sigma }(x) = 1\) if the site \(x \in \Omega \) is occupied by a \(\sigma \)particle and \(\eta ^{\sigma }(x) = 0\) otherwise. The configuration of the whole system at time t is denoted by \(\eta _t = ( \eta ^{r}_t, \eta ^{b}_t )\in \Sigma \), where \(\Sigma = \{ (0,0), (1,0), (0,1) \}^\Omega \).
The system evolves as a simple exclusion process (SEP), which is a Markov jump process on \({\Sigma }\): a particle of species \(\sigma \) at site x attempts to jump to an adjacent site y with rate \(\lambda _\sigma (x,y)\); if the destination site is empty then the jump is executed; otherwise, the particle remains at x. The microscopic hopping rates \(\lambda _\sigma \) are given by
where \(D_\sigma \) is a diffusion constant and \(V_\sigma \) is a smooth potential, and \(\vert \cdot \vert \) denotes the standard Euclidean norm. Since these rates respect detailed balance, the process is reversible with stationary measure
where the constant of proportionality is fixed by normalisation.
The choice of lattice spacing h and rate of order \(h^{2}\) corresponds to parabolic scaling; this ensures that, when taking \(h\rightarrow 0\), a single \(\sigma \)particle’s motion converges to a driftdiffusion process with drift \(D_\sigma \nabla V_\sigma \).^{Footnote 2} (Note that particle jumps have always \(\vert xy\vert =h\) so \( V_i(x)  V_i(y)\) will be of order h.) For a system with configuration \(\eta = (\eta ^r, \eta ^b)\), a particle at site x jumps to site y with rate
The discrete density of particles of species \(\sigma \) is described by
The value of \({{\mathcal {P}}}_\sigma (x)\) gives the probability that a lattice site x is occupied by a \(\sigma \) particle, so that \(\sum _x {{\mathcal {P}}}_\sigma (x) = N_\sigma \). It is also convenient to define the total density
In the hydrodynamic limit, \(N\rightarrow \infty \) and \(h\rightarrow 0\) at fixed volume fractions \(\phi _\sigma =N_\sigma h^{d}\) we assume in the following that
where \(\rho _\sigma \) and \(\rho \) are the continuous densities in \(\mathbb {T}^d\) that appear in equations such as (2).
2.2 Existing Results for \(D_r=D_b\)
Hydrodynamic limits of simple exclusion processes for mixtures were first analysed by Quastel in [17], with a recent extension by Erignoux in [23]. Both these works consider the case where the diffusivity \(D_\sigma \) is independent of \(\sigma \). We briefly review their main results.
Consider the process described in Sect. 2.1 with \(D_r=D_b\) and \(V_r=0=V_b\), that is, two ‘coloured’ species undergoing a symmetric simple exclusion process (SSEP). Without loss of generality, we choose \(D_r=D_b=1\). The hydrodynamic limit of this process was analysed by [17], who proved convergence of the (random) empirical densities \(\varrho _t^\sigma = h^d \sum _{x \in \Omega } \eta _t^\sigma (x) \delta _x\) to deterministic densities \(\rho _\sigma (x,t)\) solving a crossdiffusion system of PDEs of the form (2). In particular, the free energy is
and the mobility matrix is
where \(D_s(\rho )\) is the selfdiffusion constant of a single tagged particle in a symmetric simple exclusion process at density \(\rho \). (The dependence of \(D_s\) on \(\rho \) is not known in general, but it does have a variational characterisation [21], we return to this object in later sections.)
As discussed in [23] the method can be extended to include a weak speciesdependent drift. The expected generalisation to \(V_r,V_b\ne 0\) is that M remains the same, while E is replaced by [19]
The corresponding thermodynamic force is:
[This thermodynamic force is the object \(\nabla (\delta _{\rho _\sigma } E)\) that appears in (2).]
In contrast to the hydrodynamic limit of a single species SEP, the limit for mixtures [17, 23] is much more challenging. This is because they are nongradient systems, meaning that the instantaneous particle currents along an edge cannot be written as a discrete gradient. The core tool to obtain hydrodynamic limits in gradient systems, namely an integration by parts, is unavailable for nongradient systems. Thus terms of \(O(h^{1})\) need to be controlled by other means [9, 17]. A key advantage of the method of matched asymptotics used here is that it can deal with both gradient and nongradient systems in a relatively straightforward way, including the case of a mixture with unequal and spacedependent rates, which is not covered by the existing hydrodynamic limits literature.
2.3 Summary of Results of This Work
This work uses the method of matched asymptotics to analyse the time evolution of the mean density, including the general case with \(D_r\ne D_b\) and \(V_r, V_b\ne 0\). We apply the method of matched asymptotics in the limit of low but finite volume fraction, corresponding to \(\rho \ll 1\) (but with arbitrary values of \(\rho _r/\rho \) and \(\rho _b/\rho \)). We emphasise that the method yields an equation for the average density, but it does not establish that the (random) empirical density converges to this average value in the hydrodynamic limit.
CrossDiffusion System for \({\varvec{D}}_{{\varvec{r}}}\ne {\varvec{D}}_{{\varvec{b}}}\) in the dilute regime
We derive the following crossdiffusion system for the average densities \(\rho _r,\rho _b\) through a systematic asymptotic expansion as \(\rho = \rho _r + \rho _b \ll 1\), valid up to \(O(\rho ^2)\)
with free energy
and mobility
where
\(\sigma , \sigma ' \in \{ r, b\}\) and \({{\bar{\sigma }}}\) represents the opposite species to \(\sigma \). The constant \(\alpha \) is defined by the relationship
Note that, although not immediately apparent in its form above, \(M^\text {low}\) is symmetric
The value of \(\alpha \) depends on the dimension d of the underlying lattice. One obtains
while \(\alpha \approx 0.265569\) for \(d=3\). From a physical perspective, the coefficient \(\alpha \) accounts for the fact that a particle diffusing in a particular direction tends to acquire an excess of particles in front of it, which acts to slow down its selfdiffusion. A similar effect is observed in the continuous counterpart, namely Brownian hard spheres (see [27] or Eq. (7.9) in [28]).
It is instructive to contrast (12c) with the result one obtains after a simple meanfield closure of the equation of motion [14] (see Subsect. 3.2), which fails to capture some crossdiffusion terms arising from the nondiagonal terms of the mobility matrix,
This corresponds to (12c) with \(\mu _\sigma (\rho )\) replaced by \(\mu _{\sigma }^\text {mf}(\rho ) = 1 \rho \). In Sect. 5 we show numerical results comparing the matched asymptotics and meanfield models, showing that the former significantly outperforms the latter.
The existence of solutions of a PDE system describing macroscopic behaviour such as (2) is guaranteed when it is obtained rigorously as the hydrodynamic limit of a wellposed microscopic process [7, 9, 17, 24, 25]. Conversely, when the PDE is obtained formally via a meanfield approximation or the method of matched asymptotics, existence must be established by other means. The meanfield crossdiffusion system (system (12) with the mobility matrix replaced by (17)) was analysed in [19]. Their globalintime existence of weak solutions relies on the positive definiteness of the mobility matrix \(M^{\text {mf}}\) and exploits the convexity and structure of the energy E (12b), which yields suitable bounds for \(\rho _\sigma \). The method has been coined the boundednessbyentropy method [29] and has since been used to show the existence of a variety of crossdiffusion systems [31]. In [19] they are also able to show the uniqueness of the solution for initial data close to equilibrium. Our system (12) has the same energy (which should ensure that \(\rho _\sigma \ge 0\) and \(\rho < 1\)) and a symmetric mobility matrix \(M^\text {low}\). The matrix is always positive definite for \(d=3\), while for \(d=2\) it is only positive definite if the two diffusion coefficients are not too dissimilar.^{Footnote 3} Provided that one can derive suitable a priori estimates (hypothesis H2’ in [29]), we would therefore expect a similar existence result to [19] for our system (12). To the best of our knowledge, the uniqueness of solutions of the crossdiffusion system (12) in a general sense (far from equilibrium) remains a delicate topic and is still an open problem [30]. This would still be the case if the system was obtained as a hydrodynamic limit.
\({\varvec{Expansion of the Selfdiffusion Coefficient for}}\) \({{\varvec{D}}_{{\varvec{b}}} = {\varvec{D}}_{{\varvec{r}}}}\)
Finally, returning to the case \(D_r=D_b=1\), one sees that (12c) would coincide with the exact result (9b) if \(\mu _{\sigma }(\rho ) \equiv D_s(\rho )\). Using the variational representation of \(D_s\) and applying a recursive approach proposed in [21], in Sect. 4 we obtain the following polynomial expansion
Therefore, comparing (18) and (13), one does indeed have that the matched asymptotics result \(\mu _\sigma \) agrees with the rigorous behaviour of \(D_s\) near \(\rho =0\) to first order in \(\rho \), as expected (since the asymptotic expansion is computed to that order).^{Footnote 4} By contrast, the meanfield result \(\mu _\sigma ^\text {mf}(\rho ) = 1\rho \) already gets the firstorder term in \(\rho \) wrong. We show additionally in Appendix D that extending the matched asymptotic analysis to second order in \(\rho \) yields an improved formula for \(\mu \) that, again, agrees with the expansion of \(D_s(\rho )\) to that order. This suggests that the (formal) method of matched asymptotics recovers the correct asymptotic behaviour of the hydrodynamic equation for \(\rho \ll 1\). Subsect. 4.2 further shows (for \(D_r=D_b\)) that a polynomial ansatz for \(D_s(\rho )\) provides numerically accurate predictions for the behaviour of the oneparticle density in the whole range \(\rho \in [0, 1)\).
3 Analysis by Method of Matched Asymptotics
3.1 Preliminaries
As in [9, 17, 19, 23, 33, 34], we seek a low dimensional PDE to describe the system. Here we derive a set of coupled ODEs for the oneparticle densities, obtained from summation over the master equation of the simple exclusion process. These can then be interpreted as discretising a PDE system defined in \(\mathbb {T}^d\).
3.1.1 Equation of Motion for the OneParticle Probability Density
The model defined in Subsect. 2.1 has generator \(\mathcal {L}\) whose action on a generic function \(f : \Sigma \rightarrow \mathbb {R}\) is
where \(\eta ^{y z}\) denotes the configuration after the states of sites y and z have been swapped, that is
Now take \(f(\eta ) = \eta ^\sigma (x)\) and use the evolution equation, \(\frac{d}{dt} \mathbb {E}[ f ] = \mathbb {E}[\mathcal {L} f ]\), with (6) to obtain
where the dot denotes the time derivative. The only jump rates that contribute to this expectation are those involving the movement of \(\sigma \) particles (because any jump of the other species between sites x, y would require that there is no \(\sigma \) particle on either site). Then by (5):
Consider the marginal probability densities for the position of a single particle and for two particles, described by the functions
and \(P_{\sigma ,{{\bar{\sigma }}}} (x,x,t) = P_{\sigma ,\sigma } (x,x,t) = 0\). The densities are normalised so that \(P_\sigma (x,t),~ P_{\sigma , \sigma '}(x,y,t)\) converge (as \(h\rightarrow 0\)) to probability densities in \(\mathbb {T}^d\) and \(\mathbb {T}^{2d}\) respectively. Note that the volume density (6) is related to the probability density as
The discrete density \(P_\sigma \) is a natural object when considering matched asymptotics because it remains O(1) with respect to both \(h \ll 1\) and \(Nh^d \ll 1\). Combining (23) and (22) implies:
where we have omitted the time variable on the righthand side for ease of presentation. As usual, this equation of motion for the oneparticle (marginal) density involves the twoparticle density, so the system is not closed. We could go back to (26) and obtain equations for the twoparticle densities, but these in turn would depend on the threeparticle densities. This is analogous to the BBGKY hierarchy [13] that one obtains when integrating the Liouville equation for continuous processes. In the following, we present two approaches to close (25), namely the meanfield closure or the method of matched asymptotics to approximate the terms involving the twoparticle densities.
To facilitate the following derivation, it is useful to consider the motion of a single particle, without any interactions. The single \(\sigma \)particle model has a generator, denoted by \({{\mathcal {L}}}_\sigma \) with adjoint
defined for generic functions \(Q : \Omega \rightarrow \mathbb {R}\). Combining (26) and (25), one obtains
where the interaction term \(\mathcal {E}^{\textrm{int}}_\sigma \) includes the effects of the exclusion process and is given by
where \(\mathcal {E}_{\sigma ,\sigma }\) corresponds to the effect of a pairwise interaction between samespecies particles, and \(\mathcal {E}_{\sigma ,{{\bar{\sigma }}}}\) corresponds to the effect on the evolution of \(P_\sigma \) of a pairwise interaction with a particle of the opposite species. Note that the prefactors in these two terms account for the number of such pairwise interactions present in the system. The pairwise interaction terms are given in terms of the twoparticle probability densities,
where \(\sigma , \sigma ' \in \{ r, b\}\). The task in the following will be to estimate the twobody interaction term \(\mathcal {E}^{\textrm{int}}\), to obtain a closed equation for \(P_\sigma \). We first discuss the meanfield approach in Subsect. 3.2 and then consider the method of matched asymptotic expansions in Subsect. 3.3.
3.2 MeanField Approximation
A simple adhoc closure to the onebody equation is obtained by assuming that the occupancies of sites x, y by species \(\sigma ,\sigma '\) are independent, that is,
This is called the meanfield closure [35,36,37] . Substituting this into (28) and using (27a) yields a closed system for \(P_r, P_b\), namely
with
for \(\sigma \in \{ r, b\}\) and \({{\bar{\sigma }}}\) the opposite species.
3.2.1 CrossDiffusion System in Continuous Space
In the next step we assume that the hydrodynamic limit exists, and that \(P_\sigma \) converge to smooth densities \(p_\sigma \) on \(\mathbb {T}^d\),
This is based on the physical intuition that smallscale fluctuations in the particle density are quickly smoothed out by rapid local mixing of the system. Therefore the discrete densities \(P_\sigma (x,t)\) on \(\Omega \subseteq (h\mathbb {Z})^d\) should not vary rapidly across lattice sites. We also expand the rate (3) for \(\vert xy \vert =h\) and small h:
Inserting (32) into (30) and letting \(h \rightarrow 0\) while keeping the volume fractions \(\phi _\sigma \) fixed, one finds
The first divergence on the righthand side corresponds to linear driftdiffusion of a free \(\sigma \)particle, and the second and third terms (premultiplied by \(\phi _\sigma \) and \(\phi _{{{\bar{\sigma }}}}\) respectively) arise from the simple exclusion rule. Multiplying (33) by \(\phi _\sigma \), the average particle density \(\rho _\sigma \) [the continuous analogue of (24)] solves the hydrodynamic PDE
which was obtained previously in [14, 19]. The system of equations (34) for \(\rho _r, \rho _b\) can be written in gradientflow form (2) with mobility (17) and energy (10).
3.3 LowDensity Approximation Via Matched Asymptotics
Deviations from the adhoc meanfield closure (29) are due to particle correlations, which are maximised when particles occupy adjacent sites. Since the interaction term \(\mathcal {E}^\textrm{int}_\sigma \) in (27b) is evaluated exactly at these configurations, one may expect significant correlation effects. Here we approximate the interaction term using the method of matched asymptotics. This is a systematic asymptotic method, wellsuited to study problems with boundary layers governed by a small parameter, and previously used to study systems of interacting Brownian hard spheres [27, 38]. In contrast to the meanfield approach, this systematic procedure does not require a closure assumption and leads to a controlled approximation of \(\mathcal {E}^{\textrm{int}}_\sigma \) as an asymptotic series in \(\phi \) and h. Here we adopt this procedure for the discrete simple exclusion process: consistently with the hydrodynamic limit, the asymptotic expansions at small h take place at fixed \(\phi _\sigma = N_\sigma h^d\). But additionally (in this Section), we assume that the total occupied fraction \(\phi = Nh^d = \phi _r + \phi _b\) is also small, such that we can asymptotically expand the equation for the twoparticle density in powers of \(\phi \).
The interaction term (28) depends on the twoparticle density \(P_{\sigma , \sigma '}\), so in order to approximate it we consider the evolution equation of \(P_{\sigma , \sigma '}\). To this end, we set \(f(\eta ) = \eta ^\sigma (x) \eta ^{\sigma '}(y)\) in (19) and follow a similar calculation to Sect. 3.1 to obtain (expanding asymptotically in \(\phi \))
where the higherorder terms contain particle interactions. This is because, when particles are dilute (\(\phi \ll 1\)), the probability of configurations with three or more particles nearby is much less than that of configurations with only two particles close by. We exploit the fact that, to leadingorder in \(\phi \), (35) is a closed equation for \(P_{\sigma , {{\bar{\sigma }}}}\), so that we can effectively focus on the \(N=2\) problem, while deriving a consistent approximation of \({{\mathcal {E}}}_{\textrm{int}}\). Extension to higher orders in \(\phi \) is discussed in Appendix D.
3.3.1 Inner and Outer Regions
At leading order, (35) can be interpreted as a master equation for a two particle system (of types \(\sigma ,\sigma '\)) where \((x,y) \in \Omega ^2 \setminus \{x=y\}\) are the particle positions. We expect the solution to (35) to display a boundary layer near the excluded diagonal \(x=y\) due to strong correlations arising from the simple exclusion rule. These correlations decay as the separation distance \(\vert xy\vert \) grows. This motivates the use of matched asymptotic expansions, with an outer region in which the two particles are wellseparated (\(\vert xy\vert \gg h\)) and an inner region in which the particles are close to each other (\(\vert xy\vert \sim h\)). Here, in contrast to the standard approach [27, 38], the inner or boundary layer variable will be discrete, and only the outer variable will be continuous. This enables an accurate characterisation of \(P_{\sigma ,\sigma '}\) as \(h\rightarrow 0\); in particular it allows us to keep the exact geometry of the interaction between two particles and accurately evaluate the interaction term \({{\mathcal {E}}}_{\textrm{int}}\). With this in mind, we assume that there exist functions \(P_{\textrm{out}} : \mathbb {T}^2 \times \mathbb {R}_+ \rightarrow \mathbb {R}\) and \(P_{\textrm{in}} : \mathbb {T}\times \mathbb {Z}^d \times \mathbb {R}_+ \rightarrow \mathbb {R}\) such that \(P_{\sigma ,\sigma '}\) can be written for small h as
where the dependence of \(P_{\textrm{out}}, P_{\textrm{in}},\) on \(\sigma , \sigma '\) is left implicit for compactness of notation. We establish asymptotic approximations for \(P_{\textrm{out}}\) and \(P_{\textrm{in}}\), valid in the outer and inner regions respectively. We enforce that they agree in the crossover between the two regions by imposing a matching condition as explained below.
By assumption, the sizeexclusion rule appearing as the condition \(z \ne y\) or \(z \ne x\) in (35) does not appear in the outer region. Recalling \(P_{\sigma ,\sigma '} (x,y,t) = P_\textrm{out}(x,y,t)\) in the outer region, we obtain
using the independent walk or single particle adjoint generator (26) (we write \(\mathcal {L}^*_\sigma P_{\text {out}}(\cdot ,y,t)\) to denote the operator acting on \(P_{\text {out}}(x,y,t)\) as if it was a function of x only for y and t fixed). Therefore, at leading order in \(\phi \), the evolution of \(P_{\text {out}}\) corresponds to two independent \(\sigma \) and \(\sigma '\) particles. That is, \(P_{\text {out}}(x,y,t) = Q_\sigma (x,t) Q_{\sigma '}(y,t)+O(\phi )\) for some functions \(Q_\sigma \) satisfying \(\dot{Q}_\sigma = \mathcal {L}^*_\sigma Q_\sigma \). Using the normalisation condition, \( \sum _{x,y} h^{2d} P_{\sigma ,\sigma '} = 1\), and (31)
The correction term at O(h) comes from approximating discrete densities by continuous densities for \(h \ll 1\). As expected, particles are independent to leading order in the outer region (so the outer solution ‘does not see’ the interaction rule).
In the inner region (\(\vert xy\vert \sim h\)), we introduce inner variables \({{\hat{x}}}, {{\hat{y}}}\) satisfying \(x = {\hat{x}}\) and \(y = {\hat{x}} + h {{\hat{y}}}\). Then the inner density \(P_{\textrm{in}}({{\hat{x}}}, {{\hat{y}}},t)\) is equal to \(P_{\sigma , \sigma '}(x,y,t)\), by (36). As discussed above, the inner density is taken to be continuous with respect to its first argument \({\hat{x}}\), while keeping its second argument \({{\hat{y}}}\in \mathbb {Z}^d\) discrete, to parameterise the boundary layer.
To understand the qualitative dependence of \(P_{\textrm{in}}\) on its second argument \({\hat{y}}\), consider Fig. 1, which displays the auxiliary function \(\psi _1\) that solves (43), below. (It will be shown that the dependence of \(P_{\textrm{in}}\) on \({\hat{y}}\) is similar to that of this function.) The correlation between particles, neglected by the meanfield approximation, is captured at first order by \(\varvec{\psi }\). For \({\hat{y}}\) close to the origin, \(\psi _1({\hat{y}})\) differs significantly between adjacent sites, it describes a boundary layer that is directly affected by the structure of the lattice. For larger \(\hat{y}\), the function decays in modulus, and the relative differences between adjacent sites also decay. This latter property is required because \(P_{\textrm{in}}\) must obey a matching condition with a function \(P_\textrm{out}\) that depends smoothly on its second argument, as \(\vert \hat{y}\vert \rightarrow \infty \), recall (36).
We note that, in the inner region, the summations over the auxiliary variable z in (35) only lead to nonzero terms if z is at a distance h from either x or y. Therefore, introducing the set \(B_{{\hat{y}}}= \{ e \in \mathbb {Z}^d : \vert e \vert =1, e \ne {{\hat{y}}}\}\), the first summation \(\{z \in \Omega , z \ne y\}\) reduces to \(z = {{\hat{x}}}+ h {e}\) and the second summation \(\{z \in \Omega , z \ne x\}\) to \(z = {{\hat{x}}}+ h({{\hat{y}}}e)\), with \({e}\in B_{{\hat{y}}}\). Changing to inner variables in (35) gives, to order \(\phi \),
for \({{\hat{x}}}\in \Omega \subseteq (h\mathbb {Z})^d\) and \({{\hat{y}}}\in {\mathbb {Z}_*^d}\) with \({\mathbb {Z}_*^d}= \mathbb {Z}^d \setminus \{ 0\}\). In the first line above, we have used that \(P_{\text {in}} ({{\hat{x}}}+h{e},{{\hat{y}}}{e},t) = P_{\sigma ,\sigma '}({{\hat{x}}}+h{e}, {{\hat{x}}}+h{e}+ h({{\hat{y}}}{e}), t)\), which is equal to the desired \(P_{\sigma ,\sigma '}(z, y, t)\) converting back to the original variables.
The inner solution \(P_{\text {in}}\) must match with the outer solution \(P_{\text {out}}\) as \(\vert {{\hat{y}}}\vert \rightarrow \infty \). Writing \(P_{\text {out}}\) in (38) in terms of the inner variables and expanding gives
Next, we seek a solution to the inner problem (39) of the form
Using (32) and expanding \(P_{\text {in}}\) with respect to its first argument (which we recall is continuous), the leadingorder inner problem (which comes at order \(h^{2}\)) is
together with the condition at infinity
It is straightforward to see that a function constant in \({{\hat{y}}}\) satisfies (40a). Thus, using (40b) we find that the (trivial) solution for the inner problem (40) is
In what follows we simply write \(p_\sigma \) for \(p_\sigma ({{\hat{x}}},t)\). At \(O(h^{1})\), (39) reads, using (41) and (32),
with
together with the matching condition
where \({\textbf {B}}({{\hat{x}}}, t) = p_{\sigma }({{\hat{x}}}, t) \nabla p_{\sigma '}({{\hat{x}}}, t)\), \({\textbf {C}}( {{\hat{x}}},t) = P_{\text {out}}^{(1)}({{\hat{x}}},{{\hat{x}}},t)\). In order to solve (42), we first define the auxiliary problem
where \(\Delta ^1\) is the standard discrete Laplacian (with unit grid spacing). We remark that for \(d \ge 3\), the auxiliary function \(\psi _j\) can be related to the discrete Green’s function. Its properties are discussed in Appendix B, see also Fig. 1. In particular
where
is a ddependent constant with \(\beta \in (1,0)\) for \(d\ge 2\). It is related to the constant \(\alpha \) of (14,16) as
Therefore, the solution to (42) can be identified as
where
Hence, combining (41) and (48) we arrive at
3.3.2 Systems of Equations for \(P_\sigma \)
The final step of this computation is to evaluate the interaction terms \(\mathcal {E}_{\sigma ,\sigma '}\) in (28), to obtain a closed set of equations for \(P_\sigma \). The summands in (28) are zero unless y is adjacent to x, so in particular, they are zero in the outer region. Therefore we use the inner solution \(P_{\textrm{in}}\) to evaluate them. In inner variables (28) reads,
Combining with (49) yields:
By evaluating \({\textbf {A}}\) and \({\textbf {B}}\) and taking the limit \(h \rightarrow 0\) it follows that
with
noting that \(\gamma _{\sigma ,\sigma } = 1\). Finally, taking the limit \(h\rightarrow 0\) in (27) yields
where the \({{\mathcal {E}}}\) factors are given by (52), resulting in a closed set of equations for the \(p_\sigma \).
Note that we have focused throughout on the limit where \(h\rightarrow 0\) and \(N_\sigma \rightarrow \infty \) at fixed \(\phi _\sigma \), but (54) is still valid if \(N_\sigma =1\) for some species \(\sigma \). In this case \(\phi _\sigma =h^d\) tends to zero in the limit, and the terms corresponding to interactions with this species vanish in (54). (Physically, the single particle of species \(\sigma \) has a negligible effect on the collective motion of the other particles in the system.) Multiplying (54) by \(\phi _\sigma \), the average particle density \(\rho _\sigma \) [the continuous analogue of (24)] solves the hydrodynamic PDE
Upon factorisation this equation takes the form (2): it is a gradient flow with energy (10) and mobility (12c), as anticipated in Sect. 2.1. This is the main result of the matched asymptotic computation.
4 Selfdiffusion Coefficient: Asymptotics and Connection to Rigorous Results
The selfdiffusion coefficient measures the effective diffusion coefficient of a single tagged particle in an interacting environment with many other particles. The standard definition of a selfdiffusion coefficient assumes that the environment is in a homogeneous equilibrium state. In our context, this corresponds to setting \(V_\sigma \equiv 0\) in the microscopic hopping rates (3) and stationary densities \({{\mathcal {P}}}_\sigma (x, t) \equiv \phi _\sigma \) (so that particles evolve according to a SSEP), where we recall that \(\phi _\sigma = N_\sigma h^d\) is the fraction of sites occupied by particles of the \(\sigma \)species.
Because the selfdiffusion coefficient is a macroscopic property of an individual particle, its analysis in the current framework requires tagging a single particle in the populationlevel model. Here we see one big advantage of the matched asymptotic approach: Eq. (54) is still applicable if we take \(\sigma \) to be the ‘species’ of a tagged particle; then the evolution equation for this species yields the selfdiffusion constant. Usually, all the particles in the environment are taken to be identical (same species) [21, 39], and the tagged particle is simply a ‘coloured’ particle, leading to a selfdiffusion coefficient depending on the occupied fraction \(\phi \). However, within our framework, it is possible to consider an environment consisting of a mixture of particles (say red and blue particles), leading to a selfdiffusion coefficient depending on \(\varvec{\phi } = ( \phi _r, \phi _b)\). To this end, we consider the system (54) with three species \(\sigma = \{r,b, g\}\) corresponding to red, blue, and green particles, where \(N_g = 1\) so that the green species is the tagged particle.
The selfdiffusion coefficient of a \(\sigma \)particle is given by the limit^{Footnote 5}
where \(X_t \in (h \mathbb {Z})^d\) for \(d>1\) denotes the position of the tagged particle at time t. The physical effect that controls \(D_{s,\sigma }\) is that when the tagged particle makes a hop in a given direction, it leaves an empty site behind it. For \(\phi >0\), this means that the particle’s next jump is more likely to return to its original location, compared to other adjacent sites. Over many jumps, this generates a ‘density wave’ in front of the tagged particle, which tends to suppress further motion in the same direction. Hence one expects \(D_{s,\sigma }<D_\sigma \). (In one dimension, this is effect is so strong that the tagged particle is subdiffusive, \(D_{s}=0\).)
Note also that the definition (56) applies to the model defined on the infinite lattice, \((h\mathbb {Z})^d\), and in this case, the scaling of the hopping rates \(\lambda _\sigma \) with h means that the righthand side of (56) is independent of h. On the other hand, to estimate \(D_{s,\sigma }\) using a periodic lattice, we approximate this same quantity as \(D^h_{s,\sigma }\), by taking \(X_t  X_0\) as the relative displacement (i.e., the sum of jumps taken up until time t). We expect that \(D^h_{s,\sigma } \rightarrow D_{s,\sigma }\) as \(h \rightarrow 0\)^{Footnote 6}.
4.1 Selfdiffusion for \(D_r \ne D_b\)
The result (54) can be used to compute the selfdiffusion coefficient for small \(\phi \). As described above, we consider three species and take the tagged particle to be the only member of the green species \(g\). Setting \(V_\sigma \equiv 0\) (for all species) and \(N_g= 1\), the generalised form of (54) is
where \({{\mathcal {E}}}_{g,\sigma }\) is given by (52). Combining with that equation yields
where \(\rho _\sigma = \phi _\sigma p_\sigma \) are concentrations. The selfdiffusion coefficent can be identified from the prefactor of \(\nabla p_g\). Using that the environment is stationary (\(\rho _{\sigma } \equiv \phi _\sigma \)), the selfdiffusion of the tagged particle for \(\phi \ll 1\) is
Physically, the selfdiffusion coefficient decreases with increasing excluded volume \(\phi _\sigma \) and increasing diffusivity ratio \(\gamma _{g,\sigma }\). Looking at the expression for \(\gamma _{g,\sigma }\) in (53), this means that diffusion in a slow (or even fixed, if \(D_\sigma \equiv 0\)) environment is harder than in a fast environment.
Choosing the tagged particle to be coloured red (so that it evolves like any other particle in the red species), we define the truncation of the asymptotic series (59) as
Performing the analogous computation on the meanfield discrete PDE (33), we find that the meanfield approximation of the selfdiffusion coefficient, for a tagged red particle is
so it only depends on the total volume fraction \(\phi = \phi _r + \phi _b\).
We perform numerical simulations to test these two approximations for \(D_{s,r}(\varvec{\phi })\). At the beginning of each simulation, we randomly populate the lattice of size \(L \times L\), where \(h = 1/L\), with \(N_r\) red particles and \(N_b\) blue particles such that the probability that a site is occupied by a red particle is \(\phi _r\) and a blue particle is \(\phi _b\). Our stochastic model is the multispecies SSEP with jump rates (5) and \(V_\sigma =0\). A Gillespie algorithm is used to advance the simulation until we reach some predetermined time \(t = T\). Our algorithm first considers all 2Nd possible jumps, including those that would break the particle exclusion rule. Based on these jump rates, a random jump time is generated, and a jump is sampled. The time elapsed, t, is increased to the next jump time. If the chosen jump is to an empty site, the particle jump is executed, and if occupied, no jump is executed. This repeats until \(t \ge T\).
We calculate \(D_{s,r}(\varvec{\phi })\) numerically by averaging \(\vert X_tX_0\vert ^2/(2dt)\) over all the red particles, over \(K=10\) realisations, and over sufficiently large inspection times, \(T \in [250,300]\). We estimate \(D_{s,r}(\varvec{\phi })\) numerically following this procedure for varying total occupied fraction \(\phi \in [0, 0.2]\) and diffusivities ratio \(\gamma _{r,b}\) (whilst keeping \(D_r+D_b = 2\) and \(\phi _r = \phi _b\) fixed). The results are shown as red circles in Fig. 2. The truncation \(D^\text {low}_{s,r}(\varvec{\phi })\), shown as a black line, compares well with the measured values for low volume fractions and performs far better than the meanfield curve \(D^{\text {mf}}_{s,r}\), shown as a black dashedline.
4.2 Selfdiffusion for \(D_r =D_b\)
In contrast to the case \(D_r \ne D_b\), the general behaviour of the selfdiffusion coefficient for the uniform environment (\(D_r = D_b\)) has been widely studied [21, 22]. Without loss of generality, we set \(D_r=D_b=1\) and denote the selfdiffusion in this case simply by \(D_s(\phi )\), dropping the \(\sigma \) index. While \(D_s(\phi ) \in C^\infty ([0,1])\) for \(\phi \in [0,1]\) [21], its dependence on \(\phi \) is not known explicitly, it is given instead by a variational formula [9]. However, Remark 5.3 in [21] provides a method for computing \(D_s(\phi )\) as a Taylor expansion about either \(\phi =0\) or \(\phi =1\). We show in Appendix C that application of this method yields the expansions (18), whose firstorder truncations in \(\phi \) and \(1\phi \) respectively are given by
In order to validate our asymptotic results, we now combine the method of [21] with the hydrodynamic PDE system obtained by Quastel [17], given by the gradientflow structure (2) with mobility and energy (9). In particular, we substitute the lowvolume approximation \(D^\text {low}_{s}(\phi )\) (62) into the mobility matrix (9b). The result is
which agrees with the matched asymptotics result (55) with \(D_r = D_b\). In other words, our asymptotic derivation predicts the correct behaviour of the rigorous hydrodynamic limit [17] up to the expected order. As discussed in Subsect. C.4, the agreement between the two methods can be explained by the connection between the inner problem of the matched asymptotic analysis and the recursive problems arising in the variational characterisation of \(D_s\) [21].
Finally, by combining the low and highvolume asymptotics of \(D_s(\phi )\) (62), we obtain the following (minimal) cubic polynomial approximation \(\tilde{D}_s(\phi )\) that matches \(D^\text {low}_{s}\) and \(D^\text {high}_{s}\) at both ends \(\phi = 0,1\) respectively:
We call this the composite approximation to emphasise that it interpolates between the low and high density asymptotic expansions (62). We numerically estimate \(D_s(\phi )\) using the same procedure as in Subsect. 4.1, now with \(D_r = D_b =1\) and for \(\phi \in [0,1)\). We find that the composite approximation \(\tilde{D}_s\) agrees extremely well with the simulated data (see Fig. 3). This contrasts with the meanfield approximation \(D^{\text {mf}}_{s}\).
5 Numerical Simulations of Collective Dynamics
In this section, we perform numerical simulations of the crossdiffusion PDE system (2) for the concentrations \(\rho _r\) and \(\rho _b\), with the mobility matrix M obtained either via matched asymptotics (12c), meanfield (17) or the composite approximation of (9b) obtained via (64). The microscopic model (simple exclusion process) described in Sect. 2.1 is used to benchmark and test the validity of such PDE models.
As in the previous section, the microscopic model is simulated using the Gillespie algorithm [1], while the PDE models are solved using a discretisation of the gradient flow (2). The scheme is chosen so that in the cases that a discretised equation already exists ((27) and (30a)), the equations will match to \(O(h,\phi ,\Delta t)\). For a small time step \(\Delta t \sim h^2\), we consider the numerical scheme
with periodic boundary conditions and
where the discretised thermodynamic force is,
where \(\rho (x+\tfrac{h}{2}e_k) = [\rho (x+he_k) + \rho (x)]/2\).
We present simulations in two dimensions, with \(\Omega = \{ (h\mathbb {Z})^2 \cap [0,1)^2 \}\) and periodic boundary conditions. Initially red particles are distributed uniformly on \(\{ x \in \Omega : x_1 \le 1/2\}\), \(\rho _r(\cdot ,0) = \phi \cdot \mathbb {1}_{(0,1/2]}\) and blue particles on \(\{ x \in \Omega : x_1 > 1/2\}\), \(\rho _b(\cdot ,0) = \phi \cdot \mathbb {1}_{(1/2,1]}\). We set the potentials to \( D_r V_r = D_b V_b = \sin (2 \pi x) \), and have a lattice spacing \(h =0.01\). Due to the vertical symmetry of the system \(\rho _r\) and \(\rho _b\), are constant in the \(x_2\) direction. At a given time t, we construct the average density profiles, \(\langle \eta _t^r(x) \rangle \) and \(\langle \eta _t^b(x) \rangle \), by averaging over the \(x_2\) coordinates and over K realisations. We plot histograms of \(\langle \eta _t^r(x) \rangle \), \(\langle \eta _t^b(x) \rangle \) with bin width w (red/blue circles).
5.1 Case \(D_r = D_b\)
In the case \(D_r = D_b\) we benchmark two PDE models against the numerical simulations. In the case \(D_r = D_b\) we can still use the mobility \(M^\text {sym}\) from the rigorous hydrodynamic limit. It is noted in [23] that the nongradient method of [17, 23] is compatible with a smooth potential. Hence we obtain a hydrodynamic equation by replacing (9a) with the general hydrodynamic force (67). However, the densitydependent selfdiffusion coefficient \(D_s\) that appears in \(M^\text {sym}\) (9b) is not known explicitly, so it must be approximated to obtain an equation that can be solved numerically. This is achieved by replacing \(D_s\) in the mobility \(M^\text {sym}\) by the composite approximation \(\tilde{D}_s\) (64). We compare the performance of (65) with the mobility \(M^\text {sym}\) and approximate diffusion coefficient with solutions denoted, (\(\tilde{\rho }_r, \tilde{\rho }_b\)), to the solutions with the mean field mobility \(M^\text {mf}\) denoted, (\(\rho _r^\text {mf}, \rho _b^\text {mf} \)).
To test \(\tilde{D}_s\) furthest from its asymptotic derivation, we set \(\phi = 0.5\). We have performed simulations of the twospecies model with \(D_r = D_b =1\). Snapshots of a reduced system (\(h= 0.05\)) are shown in Fig. 4. In Fig. 5 we plot histograms of \(\langle \eta _t^r(x) \rangle \), \(\langle \eta _t^b(x) \rangle \) from \(K=30\) realisations, with bin width \(w = 0.08\), (red/blue circles).
The averaged discrete densities, \(\langle \eta _t^r(x) \rangle \), \(\langle \eta _t^b(x) \rangle \) compare well with the approximate solutions \(\tilde{\rho }_r, \tilde{\rho }_b\) using the composite approximation \(\tilde{D}_s ( \phi ) \) for the selfdiffusion coefficient (Fig. 5). The diffusive nature of the particles causes the initial blocks of red and blue particles to spread out over time, and furthermore the potential \(V_r = \sin (2 \pi x) \) (\(V_b =  \sin (2 \pi x) \)) pushes the red (blue) particles over to the right (left). These factors act together to transport particles and later balance in the steady state. The meanfield equation (34) has a higher mobility than the composite prediction (\(\tilde{D}_s ( \phi ) < D^\text {mf}_s ( \phi )\)) so it appears that the meanfield solution (\(\rho _r^\text {mf}, \rho _b^\text {mf} \)) is significantly ahead of \(\tilde{\rho }_r, \tilde{\rho }_b\) and \(\langle \eta _t^r(x) \rangle \), \(\langle \eta _t^b(x) \rangle \) (\(t= 0.02\) and \(t= 0.08\)). At later times however both solutions, \(\rho _r^\text {mf}, \rho _b^\text {mf} \) and \(\tilde{\rho }_r, \tilde{\rho }_b\) converge to the same steady state (\(t = 0.3\)) because they have the same free energy (10).
The free energy functional E (10) has a unique minimiser \((\rho _r^\infty ,\rho _b^\infty )\) [19], which corresponds to the steady state of the crossdiffusion system (note that all PDE models we discuss share the same steady state as they have the same free energy). We consider the relative free energy
in Fig. 6, with \((\rho _r,\rho _b)\) taken from either the composite approximation model, mean field or stochastic simulations.
Furthermore, the free energy (10) can be calculated using the density profiles. As the concentration of opposite particles decreases, the mobility of a particle increases. Hence in Fig. 6, instead of a straight line, we see a curve with varying gradient. In Fig. 5 we can see that initially, particles are required to move through a high density of the opposing species to reach the steadystate as the red (blue) must travel from left (right) to the right (left). However, transport through the opposing species is reduced for later times when the system is closer to equilibrium. This leads to a steeper slope in Fig. 6 at longer times.
We remark that the estimation of the timedependent free energy (68) from \(\tilde{\rho }_\sigma \) is biased, because fluctuations in \(\tilde{\rho }_\sigma =\langle \eta _t^\sigma (x) \rangle \), lead to a systematic increase in \(\hat{E}\) at \(O( \text {Var} \langle \eta _t^\sigma (x)\rangle )\). However, by ensuring the variance of the free energy is low, we see excellent agreement between the free energy of the asymptotic solution and the free energy evaluated using the average density profiles. On the other hand, the meanfield solution has greater mobility and moves faster towards equilibrium.
5.2 Collective Dynamics: \(D_r \ne D_b\)
In the general setting (\(D_r \ne D_b\)), we lack a rigorous proof of the probabilistic convergence of paths as \(h \rightarrow 0\) and furthermore, lack not only a a global approximation of \(D_{s,\sigma }( \varvec{\phi })\) but also rigorous proof of its existence and regularity. Therefore to model particle density we turn to the asymptotic PDE (54) and use stochastic simulations as a benchmark. We denote the solutions to (65) with asymptotic mobility \(M^\text {low}\) by \(\rho _r^\text {low}, \rho _b^\text {low}\) and solutions with the mean field mobility \(M^\text {mf}\) by \(\rho _r^\text {mf}, \rho _b^\text {mf}\).
Figure 7 shows the results of simulations with \(D_r = 1.5\), \(D_r = 0.5\) and \(\phi _r = \phi _b = 0.05\). We plot histograms of \(\langle \eta _t^r(x) \rangle \), \(\langle \eta _t^b(x) \rangle \) from \(K=60\) realisations, with bin width \(w = 0.08\) (red/blue circles). The averaged discrete densities, \(\langle \eta _t^r(x) \rangle \), \(\langle \eta _t^b(x) \rangle \) compare well with the asymptotic solutions \(\rho _r^\text {low}, \rho _b^\text {low}\) (Fig. 7).
Initially, red particles are placed uniformly on the left half of the periodic lattice and blue on the right. The diffusive nature of the particles causes the block of red (blue) particles to spread out over time, and furthermore the potential \(D_r V_r = \sin (2 \pi x) \) (\(D_b V_b =  \sin (2 \pi x) \)) pushes the red (blue) particles over to the right (left). Initially these factors act together to transport particles and later balance out in the steady state. In the blue particles, we see an additional local minimum form (\(t=0.1,0.2\)) as the potential is strong enough compared to the diffusion constant. In contrast, this bump is spread out by stronger diffusion in the red particles. The mean field equation (34) has a higher mobility than the asymptotic equation (54) so it appears that the meanfield solution \(\rho _r^\text {mf}, \rho _b^\text {mf} \) is ahead of \(\rho _r^\text {low}, \rho _b^\text {low}\) and \(\langle \eta _t^r(x) \rangle \), \(\langle \eta _t^b(x) \rangle \). This difference only appears at intermediate times however as both solutions, \(\rho _r^\text {mf}, \rho _b^\text {mf} \) and \(\rho _r^\text {low}, \rho _b^\text {low}\) converge to the same steady state because they have the same free energy (10).
6 Discussion
We have considered a twospecies SEP on a periodic lattice: this is a discrete model for the diffusion of particle mixtures in the presence of an external potential. Our analysis uses the method of matched asymptotics, which yields a set of coupled ODEs for the (latticebased) probability distribution density of the particle positions \(P_\sigma (x,t)\) for each species. We interpret this as a discrete crossdiffusion PDE system, from which we identify corresponding macroscopic PDEs. These can then be factorised into the gradientflow structure (2). Within this description, a nontrivial mobility matrix M captures the effect of microscopic interactions on the collective dynamics, which manifest as coupled nonlinear crossdiffusion terms.
We compare the predictions of this PDE system with numerical simulations of the discrete stochastic model. Specifically, we consider selfdiffusion and the collective dynamics of the particle densities. For small volume fractions \(\phi \), the predictions of the PDE show excellent agreement with the microscopic model, confirming that it captures the macroscopic behaviour accurately. This quantitative agreement can be attributed to the systematic nature of the method of matched asymptotics, which relies on an expansion for small lattice parameter h and small occupied volume fraction \(\phi \). As \(\phi \rightarrow 0\), particle exclusion interactions are negligible; the expansion over \(\phi \) enables these interactions to be characterised, order by order.
We have focused on the first nontrivial order in this expansion, which yields (55). (The generalisation to higher order is discussed in Appendix D.) The dominant effect of interactions – illustrated by the numerical results of Sect. 5 – is that the presence of the blue species slows down the collective diffusion of the red species (and vice versa). Physically, this is natural because the blue particles act to block the motion of red ones via the exclusion constraint. This effect is underestimated by theories that rely on meanfield closure of the equations of motion [14, 19].
Notably, the results obtained for the SEP latticebased model differ significantly from offlattice models [27], where the collective diffusion is enhanced by interactions between particles of the same species. To understand this, note that the invariant measure of the offlattice system incorporates nontrivial correlations between particles, due to their packing in space. Such effects enter the free energy through the virial expansion [13]. These packing constraints become increasingly important at high volume fractions, increasing the pressure and accelerating diffusion away from highdensity regions. There are no such packing effects in the SEP: the invariant measure has a product structure and the free energy (10) is exact for all \(\phi \). In this case, the interaction terms from the matched asymptotic expansion only appear in the mobility matrix M, which act to suppress diffusive spreading.
Compared with previous results for hydrodynamic limits, our model extends previous work [9, 17, 23] by allowing the twoparticle species to have different diffusion constants, \(D_r\ne D_b\) (as well as different drifts). For the particular case \(D_r=D_b\), our macroscopic PDEs match the ones derived rigorously in [17, 23], up to the expected order in \(\phi \). We also showed (for \(D_r=D_b\)) that the matched asymptotics approach can be used to derive the selfdiffusion constant that agrees with the expansion in \(\phi \) that was derived rigorously in [21]. To the best of our knowledge, our results for multispecies SEP with \(D_r\ne D_b\) are the first to give a consistent picture of the collective dynamics and selfdiffusion.
Rigorous analysis of the hydrodynamic limit is technically challenging in these multispecies systems because they are of nongradient type [9, 17, 23]. In particular, one requires a sharp estimate of the spectral gap to establish the system’s fastmixing properties. We note that recent results for the spectral gap are available for multispecies exclusion processes [42], which might enable a rigorous analysis of the hydrodynamic limit for \(D_r\ne D_s\).
In this context, the role of the matched asymptotic method is to provide a simple and physically intuitive route to the hydrodynamic PDEs: the method avoids the technical difficulties in proving fast mixing, and hence convergence in probability of the particle densities. It also allows latticebased and offlattice systems [27, 34, 38] to be considered on an equal footing. Given this versatility, it will be interesting to explore future applications to other lattice systems of nongradient type, including those for which spectral gap estimates are not available. This might include lattice models that incorporate packing effects, in order to make contact with offlattice results such as [27].
Data availability
The datasets generated during the current study are available from the corresponding author on reasonable request.
Notes
In the case \(d=1\) the system is a reducible Markov chain, as particles cannot jump past one another. This leads to drastically different subdiffusive behaviour [8].
We take the convention that a diffusion process, X, with diffusion coefficient D has the SDE \(\text {d}X_t = \sqrt{2D}\text {d}B_t\) where B is a standard Brownian motion. The probability density for \(X_t\) evolves according to the PDE \(\partial _t p = D \nabla ^2 p\).
The exact condition for (12c) to be semipositive definite for \(d=2\) is that \(\pi 3<D_b/D_r<1/(\pi 3)\). In fact, it is possible to remove the condition on the ratio of diffusivities by setting \(\mu _{\sigma }(\rho ) = (1 \rho )\left[ 1  \alpha (1\rho ) \gamma _{\sigma ,{{\bar{\sigma }}}} \rho \right] \). This only changes \(O(\rho ^3)\) terms, and therefore, the underlying asymptotic equation remains unchanged at \(O(\rho ^2)\).
After submitting this manuscript, we became aware of the work of Nakazato and Kitahara on the selfdiffusion coefficient of the same process under consideration[32]. Their rational approximation of \(D_s\) also agrees with (18) up to the first order at both \(\rho =0\) and \(\rho = 1\).
To understand how \(D^h_{s,\sigma }\) differs from \(D_{s,\sigma }\), note that the density wave that forms in front of the diffusing particle can loop round the periodic boundaries and interact with the other side of the particle. This tends to cause \(D^h_{s,\sigma }>D_{s,\sigma }\). However, for \(h \ll 1\) (i.e \(L = 1/h \gg 1\)) this difference will be small. (For a single species, it is proven that \(D^h_{s,\sigma } \rightarrow D_{s,\sigma }\) [41].)
Equation (C13) corresponds to equation (5.1) of [21]. We note two key differences. The first one is to do with the convention used for the diffusion coefficient: in [21] they define \(D_s\) such that the limiting Brownian motion is \(\sqrt{D_s} {\text {d}} W\) and the jump rates \(\lambda _\sigma \) via a generic symmetric law p(v) with finite range. Instead we have \(\sqrt{2D_s}{\text {d}}W\) and rates \(p(v) = 1_{\vert v \vert =1}\). Second, we note there is a missing factor of two in front of the \(\alpha (1\alpha )\) term in equation (5.1) of [21] (where \(\alpha \) corresponds to our \(\phi \)), cf. equation (2.7) of [41] for the corrected version.
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Acknowledgements
M. Bruna was supported by a Royal Society University Research Fellowship (grant no. URF/R1/180040). J. Mason was supported by the Royal Society Award (RGF/EA/181043) and the Cantab Capital Institute for the Mathematics of Information of the University of Cambridge. The authors thank Tal Agranov, Mike Cates and Jon Chapman for helpful discussions.
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Appendices
Appendix A: Outer Solution
In this appendix we prove the relationship (38). Recall we established \(P_{\text {out}}(x,y,t) = Q_\sigma (x,t) Q_{\sigma '}(y,t)+O(\phi )\) for some functions \(Q_\sigma \) satisfying \(\dot{Q}_\sigma = \mathcal {L}^*_\sigma Q_\sigma \). Hence (40) implies \(P_{\text {in}}({{\hat{x}}},{{\hat{y}}},t) = Q_\sigma ({{\hat{x}}},t) Q_{\sigma '}({{\hat{x}}},t)+O(h,\phi )\).
Although there is no definite boundary between the inner and outer region, here it is useful to fix some radius, \(\varepsilon \), so that \(h \ll \varepsilon \ll 1\) and
By definition, the two particle density must sum to \(h^{2d}\) over its domain. Splitting the sum over y into the inner and outer region yields
where it its implict that \(P_\text {in}\) is evaluated using the respective inner variables. Now we substitute expressions for the inner and outer solutions,
where we use that \(\sum _{y \in \Omega ^x_\text {in}}h^d\) is \(O(\varepsilon ^d)\). Therefore without loss of generality we can take \(\sum _{x \in \Omega }h^{d} Q_{\sigma } = 1 + O(h \varepsilon ^{d})\).
By definition, summing over \(y \in \Omega \) the two particle density must sum to \(h^{d} P_{\sigma }(x,t)\). Splitting the sum over y into the inner and outer region, one obtains
Now we substitute expressions for the inner and outer solutions,
Finally fixing \(\varepsilon = h^\frac{d1}{d}\), it follows that \(P_{\text {out}}(x,y,t) = P_\sigma (x,t) P_{\sigma '}(y,t)+O(\phi )\) as required.
Appendix B: Properties of the Auxiliary Function \(\varvec{\psi }\)
The dvectorial function \(\varvec{\psi }\) has components \(\psi _j\) satisfying (43), which we recall here for convenience
This appendix derives some of its properties. To this end, we consider its semidiscrete Fourier transform \(\tilde{\varvec{\psi }}\), defined for \(\zeta \in [\pi ,\pi ]^d\) as
We recall that the semidiscrete Fourier transform takes an unbounded and discrete spatial domain (\({{\hat{y}}}\in \mathbb {Z}^d\) in our case) to a bounded and continuous frequency domain (\(\zeta \in [\pi ,\pi ]^d\)). Taking the Fourier transform of (B4a) yields
where \(\tilde{{\psi }}_j\) is the jth component of \(\tilde{\varvec{\psi }}\). Then \(\psi _j\) can be recovered from \({\tilde{\psi }}_j\) via the inverse semidiscrete Fourier transform
yielding
We note the following properties:

For \({{\hat{y}}}=0\), the integrand in (B8) is odd in \(\zeta _j\) and so \(\psi _j({{\hat{y}}})=0\).

For \({{\hat{y}}}=e_k\) (a member of the standard basis for \(\mathbb {R}^d\)) the the imaginary part of the integrand in (B8) is odd in \(\zeta _1\), which implies
$$\begin{aligned} {\psi }_j(e_k) = \beta \delta _{jk}, \end{aligned}$$(B9)where \(\beta \) was defined in (45).

For \({{\hat{y}}}= e_k\), we have \({\psi }_j(e_k) = {\psi }_j(e_k) = \beta \delta _{jk}\).
Appendix C: Selfdiffusion Coefficient of SSEP
In this appendix, we review some of the results of [21] and apply them to compute the behaviour of the selfdiffusion coefficient \(D_s(\phi )\) as \(\phi \rightarrow 0\) and \(\phi \rightarrow 1\) in the symmetric and coloured case. In particular, [21] considers the evolution of a single tagged particle in the symmetric simple exclusion process (SSEP) in \(\mathbb {Z}^d\) that is in equilibrium at density \(\phi \). They prove that \(D_s (\phi )\in C^\infty ([0,1])\) and provide a recursive method of compute its Taylor expansion at the boundaries \(\phi = 0,1\). It is this method that is of most relevance to us here.
1.1 C.1: Setup
The microscopic problem studied in [21] is defined similarly to Sect. 2.1 but on an infinite lattice \(\mathbb {Z}^d\) with unit spacing. (It is related to the process in the inner region presented in Subsect. 3.3, as we will see below.) In particular, it corresponds to a twospecies SSEP (\(\lambda _r = \lambda _b = \mathbb {1}_{\vert x y \vert }\)) with a (say) red tagged particle (\(N_r = 1\)) at \(X_t \in \mathbb {Z}^d\) and an enviroment of blue particles (at a density \(\phi _b \approx \phi \)).
The red particle is initialised at the origin, \(X_0 = 0\). The other sites (\({\mathbb {Z}_*^d}= \mathbb {Z}^d \setminus \{ 0\}\)) are initialised with blue particles (with probability \(\phi \)) or vacancies (with probability \(1\phi \)). The selfdiffusion of the red particle (56) coincides with the selfdiffusion coefficient \(D_s(\phi )\) in the singlespecies SSEP at density \(\phi \) [\(D_{s,r}(\varvec{\phi }) \equiv D_s(\phi )\), where \(\varvec{\phi } = (0,\phi )\)]. Note that \(D_s\) can be more generally defined by the tensor [21, 22],
from which it is clear, by the symmetry of the process, that \(\hat{D}^{kl}_{s}(\phi ) = D_s(\phi ) \delta _{kl}\).
It is convenient to define the configuration of the environmental (blue) particles \(\eta ^b_t\) in the reference frame of the tagged particle [21]. The resulting environmental configuration is denoted \(\xi _t \in \{0,1\}^{\mathbb {Z}_*^d}\), and its corresponding site variables are related to those of the original SEP by \(\xi _t(y) = \eta ^b_t(yX_t)\) for \(y \in {\mathbb {Z}_*^d}\). The environmental configuration \(\xi _t\) evolves by a Markov jump process with generator \(\mathcal {L}^{\textrm{t}} = \mathcal {L}_0 +\mathcal {L}_\tau \), where the action of \(\mathcal {L}_0,\mathcal {L}_\tau \) on a generic cylinder function f is
where \({\mathcal {L}}_0\) corresponds to jumps of blue particles, while \({\mathcal {L}}_\tau \) describes the effect of a jump of the red (tagged) particle to the enviroment. Here \(B_0 = \{ {{\hat{y}}}\in {\mathbb {Z}_*^d}: \vert {{\hat{y}}}\vert =1 \}\) and \(\tau _{e} \xi \) stands for the configuration after the tagged particle jumps by e and the enviroment is shifted by \(e\):
(Note that \(\xi (e)=0\) before the jump as, otherwise, the tagged particle would have violated the sizeexclusion rule.) We also introduce \(\mu _\phi \) as a Bernoulli product measure on \(\{0,1\}^{\mathbb {Z}_*^d}\), such that \(\xi (x)=1\) with probability \(\phi \) and zero otherwise. These measures are invariant for the \(\xi \) process, for all \(\phi \in [0,1]\).
1.2 C.2: Variational Formulation of \(D_{\textrm{s}}\)
The selfdiffusion coefficient has a variational characerisation in terms of the \(\xi \) process [22]. In the current notation, it is expressed as
where the infimum is over cylinder functions f. For any two such cylinder functions \(f_1,f_2\), define the inner product
Then a computation shows that^{Footnote 7}
where \(g_k\) is a cylinder function given by
1.3 C.3: A Basis for the Space of Cylinder Functions, and the Corresponding Generator
We consider a basis for these functions to perform the supremum over cylinder functions f. A similar approach was taken [21], but this differs in the details. Noting that, by definition, f depends on \(\xi \) through a finite number of sites, it is possible to represent a generic cylinder function as
where each \(S_n\subset ({\mathbb {Z}_*^d})^n\) is finite, \(a_n:S_n\rightarrow \mathbb {R}\), and every f has a representation with a finite number \(k\ge 0\) of sums on the righthand side. The function f is said to have degree k. [For example, the oneparticle density \(P_\sigma (x)\) is the expectation of a degree one function, while the twoparticle density \(P_{\sigma ,\sigma '}(x,y)\) is the expectation of a degree two function, see (23).]
For each integer \(n \ge 0\), let \({{\mathcal {S}}}_n\) denote all the subsets of \(\mathbb {Z}^d_*\) with n points. Define \({{\mathcal {S}}}= \bigcup _{n\ge 0} {{\mathcal {S}}}_n\). Then for each \(A \in {{\mathcal {S}}}\) define
and \(\Psi _\emptyset =1\). Note that \(\Psi _A(\phi ,\cdot )\) is a cylinder function with the orthonormality relation
where \(\delta _{A,B}=1\) if the sets A and B are equal, and zero otherwise. Moreover, an arbitrary cylinder function can be expressed as
with . This representation is more abstract than (C15) because is a function whose domain is a set of sets, but this abstraction makes it more convenient for later analysis. If f is of degree \(\le k\), then is supported in \(\bigcup _{n=0}^k {{\mathcal {S}}}_n\). For two functions \(f_1,f_2\) represented as (C18), define
so by (C17)
For any given \(\phi \), the functions f and are in onetoone correspondence. It follows that when the generator \(\mathcal{L}^{\textrm{t}}\) acts on f to produce a new cylinder function \(\mathcal{L}^{\textrm{t}}f\), there is a corresponding operator such that
The operator can be constructed as follows. First for a finite subset \(A \subset \mathbb {Z}^d_*\) and \(x,y \in \mathbb {Z}^d_*\) define the set operations
and
where \(Ax\) denotes the set \(\{ y \in {\mathbb {Z}_*^d}: y+x\in A\}\). Then can be written as,
where
in which \(\bar{A}={\mathbb {Z}_*^d}\setminus A\).
1.4 C.4: Dependence of Selfdiffusion Constant on \(\phi \)
So far, our discussion follows [21]. In this subsection we follow the suggestion of Remark 5.3 of that work, to compute the dependence of the selfdiffusion constant on \(\phi \). Define a variable \(\theta \in [0,\pi /2]\) such that
and . Note that, since \(g_k\) in (C14) is a cylinder function, it may be represented as
with
Note that this \(g_k\) is of degree one (because is supported on sets with only one element).
A crucial result of [21] is (see their equation (5.3))
where is defined to be the solution to
Recalling (C13), the behavior of \(D_s\) close to \(\phi =0\) and \(\phi =1\) is available by computing the righthand side of (C26) at \(\theta =0\) and \(\theta =\pi /2\), respectively. In [21] it is shown that the limit \(\lambda \rightarrow 0\) in (C27) is wellbehaved for all \(\theta \), including the end points \(\theta =0,\pi /2\). Therefore, we are left to compute the solution of
The righthand side has degree one. From (C22b) one sees that for \(\theta =0,\pi /2\) then has the same degree as . Hence, in these cases, is of degree one, so there is a function \(\chi _{k,0}:{\mathbb {Z}_*^d}\rightarrow \mathbb {R}\) such that
and similarly for with corresponding function \(\chi _{k,\pi /2}\). To solve (C28), we consider the action of on these functions. Combining (C22) and (C28) for \(\theta =0,\pi /2\) yields
where we recall that \(\Delta ^1\) is the standard Laplacian with unit spacing. Comparing with the inner problem (42), the factor of two corresponds to \(D_\sigma +D_{\sigma '}\) which in this section simplifies to two. As in Subsect. 3.3.1, these problems are solved by a multiple of the auxiliary function \(\psi \) solving (43) and satisfying (44). Recalling from Appendix B that \(\psi _j (0) = 0, \psi _j(\pm e_k) = \pm \beta \delta _{jk}\) and using (46), it is easy to verify
To compute the first order correction to \(D_s\), we evaluate (C13). In particular (C26) corresponds to the contribution missed by the meanfield approximation. Taylor expanding (C26) about \(\theta = 0\) and combining with (C13) yields
and similarly for \(\theta = \pi /2\). Combining with (C30), the expansions (18) follows. The term, corresponds to the part of the interaction term, \( \alpha p_{\sigma '}\nabla p_{\sigma } \), resulting from the \(\psi \) term in the inner solution \(P_\text {in}\).
1.5 C.5: SecondOrder Expansion of \(D_s\)
Recalling (C31), higher order terms in the expansion of \(D_s\) are also available by computing Taylor expansions of (C26) about \(\theta =0\) or \(\theta =\pi /2\). In [21] they prove uniform convergence of and its derivatives, and therefore
By differentiating (C28), we can recursively solve for , and therefore compute \(D_s\) to an arbitrary order.
In order to compare this method with matched asymptotics at \(O(\phi ^2)\), we compute the next coefficient about \(\phi = 0\). To do this we require the order \(\theta ^2 \sim \phi \) terms in the Taylor expansion of about zero. As \(D_s\) is differentiable in \(\phi \), we do not expect an \(O(\theta )\) contribution from , however its computation is still required to solve for . Differentiating (C28) yields
Evaluating and its first and second derivatives and at \(\theta = 0\) using (C22a) yields:
These equations fully determine and but we do not solve for these explicitly. It is notable that increases the degree of a function by one, decreases the degree of a function by one and does not change the degree of a function. Hence will be a degree two function with no degree one component and will be degree three with a degree one component. As is a degree one function , as expected. The next nonzero term in (C31) will be . Collecting \(O(\phi ^2)\) terms in (C31), the next term in the expansion of \(D_s(\phi )\) is . In the following section we show that matched asymptotics gives the same result.
Appendix D: SecondOrder Matched Asymptotics
In this appendix, we show the calculation via matched asymptotics of the order \(\phi ^2\) contribution to equation (54) for \(\phi \ll 1\), and how it is consistent with the secondorder expansion of \(D_s\) in Subsect. C.5 using the rigorous recursive method. This requires evaluating the \(O(\phi )\) terms in equation (35), which as discussed correspond to threeparticle interactions. We show the derivation for the simplest case of equal diffusivities and no drifts, \(D_r = D_b = 1, V_r = V_b \equiv 0\), since this is also the case where the recursive method of Appendix C applies.
We recall the starting point: the oneparticle density \(P_\sigma \) satisfies (27) exactly, with interaction terms \(\mathcal {E}_{\sigma ,\sigma }\) and \(\mathcal {E}_{\sigma ,{{\bar{\sigma }}}}\) in (28) depending on the twoparticle densities \(P_{\sigma ,\sigma }\) and \(P_{\sigma ,{{\bar{\sigma }}}}\), respectively. In Subsect. 3.3 we have computed the leadingorder contributions in \(\mathcal {E}_{\sigma ,\sigma }\) and \(\mathcal {E}_{\sigma ,{{\bar{\sigma }}}}\), which were obtained by considering the equation satisfied by the twoparticle density \(P_{\sigma ,\sigma '}\) (35) to leading order and approximating its solution via inner and outer asymptotic expansions. In order to obtain the next asymptotic term in the equation for \(P_\sigma \), we need to consider (35) to \(O(\phi )\):
where the threeparticle interaction term \(\mathcal {E}^\textrm{int}_{\sigma ,\sigma '}\) is defined analogously to the twoparticle interaction (27b) as
and
Thus we see that the threeparticle interactions in a twospecies mixture involve the threeparticle densities \(P_{\sigma ,\sigma ,\sigma }(\varvec{x},t), P_{\sigma ,\sigma ,{{\bar{\sigma }}}}(\varvec{x},t)\) and \(P_{\sigma ,{{\bar{\sigma }}},{{\bar{\sigma }}}}(\varvec{x},t)\) where \(\varvec{x}= (x,y,z) \in \Omega ^3_h\) and \(\Omega _h^3 = \Omega ^3 \setminus \{x = y, x=z, y=z\}\)
where \(\tilde{N}_{\sigma '} = N_{\sigma '} \mathbb {1}_{\sigma '= \sigma }\) and \(\tilde{N}_{\sigma ''} = N_{\sigma ''} 1 \mathbb {1}_{\sigma ''= \sigma }\).
Our goal in this Appendix is to obtain an asymptotic approximation for small \(\phi \) of (D34c) using matched asymptotic expansions. We proceed with the same method as in Subsect. 3.3, namely, to consider the ‘asymptotically closed’ (for \(\phi \ll 1\)) equation for the threeparticle density \(P_{\sigma ,\sigma ',\sigma ''}(\varvec{x}, t)\) and divide its domain of definition into regions depending on whether three, two or no particles are close to one another.
We make two simplifications specific to the case \(V_r = V_b =0\), but emphasise that the method easily extends to the general case. As \(V_r = V_b =0\), it follows by the symmetry of \(P_{\sigma , \sigma }\) that \({{\mathcal {E}}}_{\sigma , \sigma } = 0\) and therefore we only require \(P_{\sigma , {{\bar{\sigma }}}}\) to evaluate \({{\mathcal {E}}}^{\textrm{int}}\). As a third particle must be either \(\sigma \) or \({{\bar{\sigma }}}\) without loss of generality we solve for \(P_{\sigma , {{\bar{\sigma }}}, {{\bar{\sigma }}}}\), and by permuting species can evaluate \({{\mathcal {E}}}^\textrm{int}_{\sigma , {{\bar{\sigma }}}}\).
1.1 D.1: Matched Asymptotics Expansion for \(P_{\sigma ,{{\bar{\sigma }}},{{\bar{\sigma }}}}\)
Setting \(f(\eta ) = \eta ^\sigma _t(x)\eta ^{\sigma '}_t(y) \eta ^{\sigma ''}_t(z)\) in (19), it follows by similar calculation to Sect. 3.1 that
for \(\varvec{x}\in \Omega _h^3\). We proceed by introducing three regions of \(\Omega _h^3\): an inner region where the three particles are within O(h) of each other; an intermediate region where two particles are close and the third one is far; and an other region where the three particles are far from each other. Consequently, we define
As before, we omit the dependence on \(\sigma ,\sigma ',\sigma ''\) of \(\tilde{P}_\textrm{out}\), \(\tilde{P}_{\textrm{mid}}\) and \(\tilde{P}_{\textrm{in}}\) for ease of notation. Following the procedure in Subsect. 3.3, we keep the boundary layer variables in \({\mathbb {Z}_*^d}\) (discrete) while taking the rest to be continuous.
Outer Region In the outer region, analogously to (37), we have
with solution
Intermediate Region In the intermediate region in which \(\vert x y\vert \sim h\) and \(\vert x z\vert \gg h\), we introduce the intermediate coordinates
and write \(P(\varvec{x},t) = {\tilde{P}}_\text {mid}({{\hat{x}}},{{\hat{y}}},{{\hat{z}}},t)\). By identical calculation to the firstorder inner problem (Subsect. 3.3) and imposing the matching condition to the outer region,
it follows that for \({{\hat{z}}}\in \Omega \)
where \(P_\text {in}({\hat{x}}, {\hat{y}}, t)\) satisfies the twoparticle inner problem (39) and is therefore given by (49). For symmetric and equal rates, \(\tilde{P}_{\text {mid}}\) reduces to
where
and as introduced in Sect. C,
Inner Region As discussed, we now consider the case \(\sigma ',\sigma '' = {{\bar{\sigma }}}\), which is sufficient to deduce \({{\mathcal {E}}}^{\textrm{int}}\) in the case of no potential. This will allow us to solve the inner problem in terms of the function , from Sect. C.4, therefore making comparison simpler. Now consider the inner region where we use the coordinates:
As the second two species are of the same species, the \({{\bar{y}}}, {{\bar{z}}}\) coordinates are interchangeable in \(\tilde{P}_{\text {in}}\). We abuse notation and allow \(\tilde{P}_{\text {in}}({{\bar{x}}}, {{\bar{y}}},{{\bar{z}}}, t)\) to also be denoted \(\tilde{P}_{\text {in}}({{\bar{x}}}, \{{{\bar{y}}},{{\bar{z}}}\}, t)\), to allow the operators (C22b) to act on \(\tilde{P}_{\text {in}}\) in the \(\{{{\bar{y}}},{{\bar{z}}}\}\) argument. In the inner coordinates, (D36) becomes:
The inner problem is complemented by the matching conditions with the intermediate densities (D40)
We look for a solution to (D42) of the form \( \tilde{P}_{\text {in}} ({{\bar{x}}}, {{\bar{y}}}, {{\bar{z}}}, t) \sim \tilde{P}^{(0)}_{\text {in}} ({{\bar{x}}}, {{\bar{y}}}, {{\bar{z}}}, t)+h\tilde{P}^{(1)}_{\text {in}} ({{\bar{x}}}, {{\bar{y}}}, {{\bar{z}}}, t) + \cdots \). At \(O(h^{2})\) we find
where \({\mathbb {Z}_*^{2d}}= \mathbb {Z}^{2d} \setminus \{{{\bar{y}}}= 0, {{\bar{z}}}= 0, {{\bar{y}}}= {{\bar{z}}}\}\) and \(\tilde{P}^{(0)}_{\text {in}}\) are given in (D41). By inspection we have that the solution to (D43) is
The \(O(h^{1})\) of (D42) is, using (D44),
We seek a solution to (D45) of the form:
A simple computation shows that
It follows that
It follows from (C33) that and therefore
ThreeParticle Interaction Term Here we evaluate the three particle interaction term, \(\mathcal {E}_{\sigma ,{{\bar{\sigma }}},{{\bar{\sigma }}}}(x,y,t)\), to leading order in \(\phi \) and h.
Consider the case \(\vert x z \vert \gg h\). In (D34c), \(P_{\sigma , {{\bar{\sigma }}}{{\bar{\sigma }}}}\) is evaluated when \(\varvec{x}\) is in the middle region and therefore we use the middle solution. In middle coordinates (D34c) becomes
where the second term dropped because \(P_{\sigma ,{{\bar{\sigma }}},{{\bar{\sigma }}}}(x,y,z) =P_{\sigma ,{{\bar{\sigma }}},{{\bar{\sigma }}}}(x,z,y)\). Combining with (D40) and (28) yeilds
Now consider \(\vert xy \vert \sim \vert xz \vert \sim h\). In (D34c), \(P_{\sigma , {{\bar{\sigma }}}{{\bar{\sigma }}}}\) is evaluated when \(\varvec{x}\) is in the inner region and therefore we use the inner solution. In inner coordinates (D34c) becomes
where the second term dropped because \(P_{\sigma ,{{\bar{\sigma }}},{{\bar{\sigma }}}}(x,y,z) =P_{\sigma ,{{\bar{\sigma }}},{{\bar{\sigma }}}}(x,z,y)\). Combining with (D49) , (C22b) and (C30) yeilds
Finally noting \(\mathcal {E}_{\sigma ,\sigma ',\sigma ''}(x,y,t) = \mathcal {E}_{\sigma ',\sigma ,\sigma ''}(y,x,t)\) we can deduce \(\mathcal {E}_{\sigma ,{{\bar{\sigma }}},\sigma } \) from (D51) and (D53) using \(\mathcal {E}_{\sigma ,{{\bar{\sigma }}},\sigma }(x,y,t) = \mathcal {E}_{{{\bar{\sigma }}},\sigma ,\sigma }(y,x,t)\). Hence we now have explicit expressions in terms of \(p_\sigma \) and \(p_{{{\bar{\sigma }}}}\) for the three particle interaction term at leading order. In the following section we use this close equation (D34a) to first order in \(\phi \) and solve for \(P_{\sigma ,{{\bar{\sigma }}}}\).
1.2 D.2: Equation for the OneParticle Density
Recall, that to close (27a) we require \(P_{\sigma ,{{\bar{\sigma }}}}\) to evaluate \({{\mathcal {E}}}^\text {int}\). In the calculation in the main text we computed the first nonzero contribution of \(\mathcal {E}_{\sigma ,\sigma '}(x,t)\), which appears at O(1). Here we want to go one order higher. Usually, the terms of interest could be the terms of order \(h\phi \) or \(\phi ^2\) in the equation for \(P_\sigma \). But here, as we take \(h\rightarrow 0\) while keeping \(\phi \) fixed, the former vanishes, and we are after the \(O(\phi ^2)\) contribution, coming from the leadingorder contribution of an inner region with three particles. So the structure of \(\mathcal {E}_{\sigma ,\sigma '}(x,t)\) is as follows
In the main text, we already computed \(\mathcal {E}_{\sigma ,\sigma '}(x,t)\) to the leading order, which remains unchanged. We set
As in the main text, we proceed by introducing two regions: the inner region where two particles are within O(h) and an outer region where the two particles are far apart. And as before,
where we omit the dependence on \(\sigma ,\sigma '\) of \(P_{\textrm{out}}\) and \(P_{\textrm{in}}\) for ease of notation. Following the procedure in Subsect. 3.3, we keep the boundary layer variables in \({\mathbb {Z}_*^d}\) (discrete) while taking the rest to be continuous. We propose asymptotic expansions, but now in both h and \(\phi \)
c to depend on \(\phi \) through \(p_\sigma \). We have already solved the leading order contributions in \(\phi \) and therefore we set \(P^{(n,0)}_\text {out} = P^{(n)}_\text {out}\), and \(P^{(n,0)}_\text {in} = P^{(n)}_\text {in}\) from the main text. In the following, we solve for \(P_{\textrm{in}}\) to \(O(h\phi )\), allowing us to evaluate \(\mathcal {E}_{\sigma ,{{\bar{\sigma }}}}^{(1)}\).
Outer Region In the outer region (D34a) becomes,
using the independent walk or single particle adjoint generator (26) (we write \(\mathcal {L}^*_\sigma P_{\text {out}}(\cdot ,y,t)\) to denote the operator acting on \(P_{\text {out}}(x,y,t)\) as if it was a function of x only for y and t fixed). Therefore, to first order in \(\phi \), the evolution of \(P_{\text {out}}\) corresponds to two independent \(\sigma \) and \({{\bar{\sigma }}}\) particles. That is, \(P_{\text {out}}(x,y,t) = Q_\sigma (x,t) Q_{{{\bar{\sigma }}}}(y,t)+O(\phi ^2, h^d, h\phi )\) for some functions \(Q_\sigma \) satisfying \(\dot{Q}_\sigma = \mathcal {L}^*_\sigma Q_\sigma +\phi _{{{\bar{\sigma }}}} \mathcal {E}_{\sigma ,{{\bar{\sigma }}}}\). Using the normalisation condition, \( \sum _{x,y} h^{2d} P_{\sigma ,\sigma '} = 1\), (27a) and (31),
Therefore the \(\phi \)dependence of \(P^{(0,0)}_{\textrm{out}}(x, y,t) = p_{\sigma } (x,t) p_{{{\bar{\sigma }}}} (y,t)\), contained in \(p_\sigma \) and \(p_{{\bar{\sigma }}}\), is correct to \(O(\phi )\). Hence no correction is needed at \(O(\phi )\) and we set \(P^{(0,1)}_{\textrm{out}} = 0\).
Inner Region In the inner region the interaction term does not contribute at \(O(h^{2})\) and therefore it follows by identical calculation to Sect. 3.3 that at \(O(\phi )\)
In inner coordinates at \(O(h^{1})\) (D34a) reads the same as (42a) but with no contribution from \(P_{\text {out}}^{(0,1)}\) and the addition of the interaction term (D53). The corresponding problem for \(P_{\text {in}}^{(1,1)}\) reads
where we exploit the antisymmetry of to combine \({{\mathcal {E}}}_{\sigma , {{\bar{\sigma }}}, \sigma }\) and \({{\mathcal {E}}}_{\sigma , {{\bar{\sigma }}}, {{\bar{\sigma }}}}\). The corresponding boundary condition is obtained by matching to the outer solution,
The key observation from Sect. C.4 is that the operator \(2\sum _{ e\in B_{{\hat{y}}}}(\tau _{e}+Id)\) acting on a function f, is in fact the same as acting on the degree one function of sets \(\tilde{f}\), where \(\tilde{f}( \{ {{\hat{y}}}\} ) = f({{\hat{y}}})\). With this in hand, and using (C33), it is easily verifiable
where
System of Equations for \({p_\sigma }\)
Evaluating the two particle interaction term (28), using the inner solution to \(O(h \phi )\), and Taylor expanding about x, we find:
where
and \({\bar{\phi }}_\sigma = \phi _\sigma /\phi \) and .
Rescaling to particle density and taking the limit \(h \rightarrow 0\) we find that with this interaction term, equation (27a) has the gradient flow structure (2), with free energy (9a) and mobility (9b) where
This agrees with the expansion of \(D_s(\phi )\) discussed in Appendix C.
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Mason, J., Jack, R.L. & Bruna, M. Macroscopic Behaviour in a TwoSpecies Exclusion Process Via the Method of Matched Asymptotics. J Stat Phys 190, 47 (2023). https://doi.org/10.1007/s10955022030369
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DOI: https://doi.org/10.1007/s10955022030369