1 Introduction

Stochastic models describing systems of interacting particles are widely used across many disciplines [1,2,3]. A particular class of models concerns excluded-volume 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 mean-field interactions, which are long-range and weaker as the number of particles increases. Excluded-volume 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 predator-prey systems [6]. Models for diffusive systems with excluded-volume 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 short-ranged interaction potentials or hard-core 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 d-dimensional 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 single-species SEP, the continuum model for the particle density \(\rho (x,t)\) takes the form [7]

$$\begin{aligned} \partial _t \rho = \nabla \cdot \left[ D \nabla \rho + \rho (1-\rho ) \nabla V \right] , \end{aligned}$$

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 drift-diffusion 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 mean-field 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 mean-field 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 cross-diffusion system of PDEs for the two species densities \(\rho _r, \rho _b\),

$$\begin{aligned} \partial _t \begin{pmatrix} \rho _r \\ \rho _b \end{pmatrix} = \nabla \cdot \left[ M(\rho _r,\rho _b) \nabla \begin{pmatrix} \delta _{\rho _r} E\\ \delta _{\rho _b}E \end{pmatrix} \right] , \end{aligned}$$

where M is a mobility matrix and \(\delta _{\rho }E\) denotes the functional derivative of a suitable free-energy functional E. The form (2) can be indentified as a 2-Wasserstein gradient flow [19, 20]. The mobility matrix M in [17] is full and depends on the self-diffusion 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 single-species SEP, analysis of this hydrodynamic limit is challenging for multi-species 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 multi-species systems in [24] and [25], for the zero-range process and interacting Brownian motions in one dimension, respectively. In both of these models the limiting macroscopic PDE is known explicitly.

For the two-species SEP, the master equation and mean-field 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 single-species case where the result of the mean-field closure is exact.

In this work, we derive a cross-diffusion system of the form (2) from the master equation using the method of matched asymptotics [26] in the low-density limit, \(\phi \ll 1\). In contrast to the mean-field 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 mean-field 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 self-diffusion constant. For the case where the species have equal diffusivity, this yields the same result as the (rigorous) recursive computation of the self-diffusion constant of Landim et al. [21]. In this case, we also consider an expansion around the high-density limit, which we combine with the low-density expansion to obtain a cubic approximation of the self-diffusion 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 cross-diffusion system of the form (2) for the general case via mean-field and the method of matched asymptotic is given in Sect. 3. Then, Sect. 4 concerns the analysis of the self-diffusion 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 d-dimensional (hyper)-cubic lattice \(\Omega \) with \(L^d\) sites and \(d \ge 2\). We embed \(\Omega \) in the unit d-dimensional 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

$$\begin{aligned} \lambda _\sigma (x,y) = \frac{D_\sigma }{h^2}\exp \left( \frac{V_\sigma (x) - V_\sigma (y)}{2} \right) \mathbb {1}_{ \{ \vert x-y\vert =h \} } , \end{aligned}$$

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

$$\begin{aligned} \pi ( \eta ) \propto \exp \Big ( - \sum _{x,\sigma } V_\sigma (x) \eta ^\sigma (x) \Big ), \end{aligned}$$

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 drift-diffusion process with drift \(-D_\sigma \nabla V_\sigma \).Footnote 2 (Note that particle jumps have always \(\vert x-y\vert =h\) 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

$$\begin{aligned} c(\eta , x,y) = \left[ 1-\eta ^r(y) -\eta ^b(y)\right] \left[ \eta ^r(x) \lambda _r(x,y) +\eta ^b(x) \lambda _b(x,y) \right] . \end{aligned}$$

The discrete density of particles of species \(\sigma \) is described by

$$\begin{aligned} {{\mathcal {P}}}_\sigma (x,t) = \mathbb {E}\big [\eta ^\sigma _t (x)\big ]. \end{aligned}$$

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

$$\begin{aligned} {{\mathcal {P}}}= {{\mathcal {P}}}_r+ {{\mathcal {P}}}_b. \end{aligned}$$

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

$$\begin{aligned} {{\mathcal {P}}}_\sigma \rightarrow \rho _\sigma \qquad {{\mathcal {P}}}\rightarrow \rho = \rho _r + \rho _b, \end{aligned}$$

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 cross-diffusion system of PDEs of the form (2). In particular, the free energy is

$$\begin{aligned} E_0[\rho _r,\rho _b] = \int _{\mathbb {T}^d} \left[ \rho _r \log \rho _r + \rho _b \log \rho _b + (1 - \rho )\log (1-\rho ) \right] \text {d}x, \end{aligned}$$

and the mobility matrix is

$$\begin{aligned} M^\text {sym}(\rho _r,\rho _b) = \frac{1-\rho }{\rho } \begin{pmatrix} {\rho _r^2}{} &{}&{} {\rho _r \rho _b}{} \\ {\rho _r\rho _b}{} &{}&{} {\rho _b^2}{} \end{pmatrix} + \frac{\rho _r\rho _b}{\rho } D_s(\rho ) \begin{pmatrix} 1 &{} -1 \\ -1 &{} 1 \end{pmatrix}. \end{aligned}$$

where \(D_s(\rho )\) is the self-diffusion 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 species-dependent drift. The expected generalisation to \(V_r,V_b\ne 0\) is that M remains the same, while E is replaced by [19]

$$\begin{aligned} E[\rho _r,\rho _b] = E_0[\rho _r,\rho _b] + \int _{\mathbb {T}^d} \left[ \rho _r V_r + \rho _b V_b \right] \text {d}x . \end{aligned}$$

The corresponding thermodynamic force is:

$$\begin{aligned} \nabla \frac{\delta E}{\delta \rho _\sigma } = \frac{\nabla \rho _\sigma }{\rho _\sigma }+ \frac{\nabla \rho }{1- \rho } + \nabla V_\sigma . \end{aligned}$$

[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 space-dependent 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.

Cross-Diffusion System for \({\varvec{D}}_{{\varvec{r}}}\ne {\varvec{D}}_{{\varvec{b}}}\) in the dilute regime

We derive the following cross-diffusion 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 \(O(\rho ^2)\)

$$\begin{aligned} \partial _t \begin{pmatrix} \rho _r \\ \rho _b \end{pmatrix} = \nabla \cdot \left[ M^\text {low}(\rho _r,\rho _b) \nabla \begin{pmatrix} \delta _{\rho _r} E\\ \delta _{\rho _b}E \end{pmatrix} \right] , \end{aligned}$$

with free energy

$$\begin{aligned} E = \int _{\mathbb {T}^d} \left[ \rho _r \log \rho _r + \rho _b \log \rho _b + (1 - \rho )\log (1-\rho ) + \rho _r V_r + \rho _b V_b \right] \text {d}x, \end{aligned}$$

and mobility

$$\begin{aligned} M^\text {low} =\begin{pmatrix} D_r &{} 0\\ 0&{} D_b \end{pmatrix} \left[ \frac{1-\rho }{\rho } \begin{pmatrix} {\rho _r^2}{} &{}&{} {\rho _r \rho _b}{} \\ {\rho _r\rho _b}{} &{}&{} {\rho _b^2}{} \end{pmatrix} + \frac{\rho _r\rho _b}{\rho } \begin{pmatrix} \mu _r(\rho ) &{} -\mu _b(\rho ) \\ -\mu _r(\rho ) &{} \mu _b(\rho ) \end{pmatrix} \right] , \end{aligned}$$


$$\begin{aligned} \mu _{\sigma }(\rho ) = (1- \rho ) \left[ 1 - \alpha \gamma _{\sigma ,{{\bar{\sigma }}}} \rho \right] , \qquad \gamma _{\sigma ,\sigma '} = \frac{2D_{\sigma }}{D_\sigma +D_{\sigma '}} , \end{aligned}$$

\(\sigma , \sigma ' \in \{ r, b\}\) and \({{\bar{\sigma }}}\) represents the opposite species to \(\sigma \). The constant \(\alpha \) is defined by the relationship

$$\begin{aligned} \frac{\alpha }{1+\alpha } = \frac{1}{(2\pi )^d} \int _{[-\pi ,\pi ]^d} \frac{\sin ^2\zeta _1}{ 2\sum _k \sin ^2 (\zeta _k/2)} d\zeta . \end{aligned}$$

Note that, although not immediately apparent in its form above, \(M^\text {low}\) is symmetric

$$\begin{aligned} M^\text {low} = (1-\rho ) \begin{pmatrix} D_r\rho _r &{} 0 \\ 0 &{} D_b \rho _b \end{pmatrix} - \frac{2 \alpha (1-\rho ) \rho _r \rho _b}{D_r + D_b}\begin{pmatrix} D_r^2 &{} -D_r D_b\\ -D_r D_b &{} D_b^2 \end{pmatrix}. \end{aligned}$$

The value of \(\alpha \) depends on the dimension d of the underlying lattice. One obtains

$$\begin{aligned} \alpha = (\pi /2) -1 \quad \text {for} \quad d=2, \end{aligned}$$

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 self-diffusion. 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 mean-field closure of the equation of motion [14] (see Subsect. 3.2), which fails to capture some cross-diffusion terms arising from the non-diagonal terms of the mobility matrix,

$$\begin{aligned} M^{\text {mf}}(\rho _r, \rho _b) = (1-\rho ) \begin{pmatrix} D_r\rho _r &{} 0 \\ 0 &{} D_b \rho _b \end{pmatrix} . \end{aligned}$$

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 mean-field 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 well-posed microscopic process [7, 9, 17, 24, 25]. Conversely, when the PDE is obtained formally via a mean-field approximation or the method of matched asymptotics, existence must be established by other means. The mean-field cross-diffusion system (system (12) with the mobility matrix replaced by (17)) was analysed in [19]. Their global-in-time 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 boundedness-by-entropy method [29] and has since been used to show the existence of a variety of cross-diffusion 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 cross-diffusion 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.

\({\varvec{Expansion of the Self-diffusion Coefficient for}}\) \({{\varvec{D}}_{{\varvec{b}}} = {\varvec{D}}_{{\varvec{r}}}}\)

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

$$\begin{aligned} {D_s(\rho )} = {\left\{ \begin{array}{ll} 1-(1+\alpha ) \rho +O(\rho ^2) &{} \text {as }\rho \rightarrow 0,\\ \frac{1}{2\alpha +1}(1-\rho )+O((1-\rho )^2) &{} \text {as }\rho \rightarrow 1. \end{array}\right. } \end{aligned}$$

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 mean-field result \(\mu _\sigma ^\text {mf}(\rho ) = 1-\rho \) already gets the first-order 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 one-particle 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 one-particle 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 One-Particle 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

$$\begin{aligned} \mathcal {L} f (\eta )= \sum _{y,z \in \Omega } c(\eta , y,z) \big [ f(\eta ^{y z})- f(\eta ) \big ] , \end{aligned}$$

where \(\eta ^{y z}\) denotes the configuration after the states of sites y and z have been swapped, that is

$$\begin{aligned} \eta ^{y z}(x) = {\left\{ \begin{array}{ll} \eta (x) &{}\text { if }x\ne y,z, \\ \eta (y) &{}\text { if }x = z, \\ \eta (z) &{}\text { if }x = y. \end{array}\right. } \end{aligned}$$

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

$$\begin{aligned} {\dot{{{\mathcal {P}}}}}_\sigma (x,t)&= \sum _{y \in \Omega } \mathbb {E}\big [ c (\eta _t, y,x)\eta ^\sigma _t(y) - c (\eta _t, x,y) \eta ^\sigma _t(x) \big ], \end{aligned}$$

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 xy would require that there is no \(\sigma \) particle on either site). Then by (5):

$$\begin{aligned} \begin{aligned} {\dot{{{\mathcal {P}}}}}_\sigma (x,t) =&\sum _{ y \in \Omega } \lambda _\sigma (y,x) \mathbb {P}[ \eta ^\sigma _t(y) =1, \eta ^{{{\bar{\sigma }}}}_t(x)= \eta ^\sigma _t(x) =0 ] \\&- \sum _{ y \in \Omega } \lambda _\sigma (x,y) \mathbb {P}[ \eta ^\sigma _t(x) =1, \eta ^{{{\bar{\sigma }}}}_t(y) = \eta ^\sigma _t(y) =0 ]. \end{aligned} \end{aligned}$$

Consider the marginal probability densities for the position of a single particle and for two particles, described by the functions

$$\begin{aligned} P_\sigma (x,t)&= \frac{\mathbb {E}[\eta ^\sigma _t(x)]}{N_\sigma h^d}, \nonumber \\ P_{\sigma ,{{\bar{\sigma }}}} (x,y,t)&= \frac{ \mathbb {E}[\eta ^\sigma _t(x) \eta _t^{{{\bar{\sigma }}}} (y) ] }{N_\sigma N_{{{\bar{\sigma }}}} h^{2d}}, \qquad \text {for } x\ne y ,\\ \nonumber P_{\sigma ,\sigma } (x,y,t)&= \frac{ \mathbb {E}[\eta ^\sigma _t(x) \eta _t^{\sigma }(y) ] }{(N_\sigma -1) N_{\sigma } h^{2d}}, \quad \,\, \text {for } x\ne y , \end{aligned}$$

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

$$\begin{aligned} {{\mathcal {P}}}_\sigma (x,t) = N_\sigma h^d P_\sigma (x,t) . \end{aligned}$$

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:

$$\begin{aligned} \begin{aligned} \dot{P}_\sigma (x,t) =&\sum _{ y \in \Omega } \lambda _\sigma (y,x) \left[ P_\sigma (y) - h^{d}N_{{{\bar{\sigma }}}} P_{\sigma , {{\bar{\sigma }}}}(y,x) - h^{d}(N_\sigma -1) P_{\sigma , \sigma }(y,x) \right] \\&- \sum _{ y \in \Omega } \lambda _\sigma (x,y) \left[ P_\sigma (x) - h^{d}N_{{{\bar{\sigma }}}} P_{\sigma , {{\bar{\sigma }}}}(x,y) - h^{d}(N_\sigma -1) P_{\sigma , \sigma }(x,y) \right] , \end{aligned} \end{aligned}$$

where we have omitted the time variable on the right-hand side for ease of presentation. As usual, this equation of motion for the one-particle (marginal) density involves the two-particle density, so the system is not closed. We could go back to (26) and obtain equations for the two-particle densities, but these in turn would depend on the three-particle 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 mean-field closure or the method of matched asymptotics to approximate the terms involving the two-particle 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

$$\begin{aligned} \mathcal {L}^*_\sigma Q(x)= \sum _{y \in \Omega } \left[ \lambda _\sigma (y,x) Q(y)- \lambda _\sigma (x,y) Q(x) \right] , \end{aligned}$$

defined for generic functions \(Q : \Omega \rightarrow \mathbb {R}\). Combining (26) and (25), one obtains

$$\begin{aligned} \dot{P}_{\sigma } (x,t) = \mathcal {L}^*_\sigma P_\sigma (x,t) + \mathcal {E}^{\textrm{int}}_\sigma (x,t) , \end{aligned}$$

where the interaction term \(\mathcal {E}^{\textrm{int}}_\sigma \) includes the effects of the exclusion process and is given by

$$\begin{aligned} \mathcal {E}^{\textrm{int}}_\sigma (x,t) = ~h^d (N_\sigma -1)\mathcal {E}_{\sigma ,\sigma }(x,t) + h^d N_{{\bar{\sigma }}}\mathcal {E}_{\sigma ,{{\bar{\sigma }}}}(x,t) , \end{aligned}$$

where \(\mathcal {E}_{\sigma ,\sigma }\) corresponds to the effect of a pairwise interaction between same-species 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 two-particle probability densities,

$$\begin{aligned} \mathcal {E}_{\sigma ,\sigma '}(x,t) = ~\sum _{y \in \Omega }\left[ \lambda _\sigma (x,y)P_{\sigma ,\sigma '} (x,y,t)-\lambda _\sigma (y,x)P_{\sigma ,\sigma '} (y,x,t) \right] , \end{aligned}$$

where \(\sigma , \sigma ' \in \{ r, b\}\). The task in the following will be to estimate the two-body interaction term \(\mathcal {E}^{\textrm{int}}\), to obtain a closed equation for \(P_\sigma \). We first discuss the mean-field approach in Subsect. 3.2 and then consider the method of matched asymptotic expansions in Subsect. 3.3.

3.2 Mean-Field Approximation

A simple ad-hoc closure to the one-body equation is obtained by assuming that the occupancies of sites xy by species \(\sigma ,\sigma '\) are independent, that is,

$$\begin{aligned} P_{\sigma ,\sigma '}(x,y,t) = P_\sigma (x,t) P_{\sigma '}(y,t), \qquad x\ne y. \end{aligned}$$

This is called the mean-field closure [35,36,37] . Substituting this into (28) and using (27a) yields a closed system for \(P_r, P_b\), namely

$$\begin{aligned} \dot{P}_\sigma (x,t) = \mathcal {L}^*_\sigma P_\sigma (x,t) + {{\mathcal {E}}}^\textrm{mf}_\sigma (x,t), \end{aligned}$$


$$\begin{aligned} \begin{aligned} \mathcal {E}^{\textrm{mf}}_\sigma (x,t)&=h^d N_{{\bar{\sigma }}}\sum _{y \in \Omega }\left[ \lambda _\sigma (x,y) P_{\sigma } (x,t) P_{{{\bar{\sigma }}}} (y,t)-\lambda _\sigma (y,x)P_{\sigma }(y,t)P_{{{\bar{\sigma }}}} (x,t) \right] \\&\quad +h^d (N_\sigma -1) \sum _{y \in \Omega }\left[ \lambda _\sigma (x,y)P_{\sigma } (x,t)P_{\sigma } (y,t)-\lambda _\sigma (y,x)P_{\sigma }(y,t)P_{\sigma } (x,t) \right] , \end{aligned} \end{aligned}$$

for \(\sigma \in \{ r, b\}\) and \({{\bar{\sigma }}}\) the opposite species.

3.2.1 Cross-Diffusion 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 \(\mathbb {T}^d\),

$$\begin{aligned} P_\sigma (x,t) = p_\sigma (x,t) + O(h), \qquad \text {for } x\in \Omega . \end{aligned}$$

This is based on the physical intuition that small-scale 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 \(\vert x-y \vert =h\) and small h:

$$\begin{aligned} \lambda _\sigma (x,y) = \frac{D_\sigma }{h^2}+\frac{D_\sigma }{h^2}\frac{x-y}{2}\cdot \nabla V_\sigma \Big ( \frac{y+x}{2} \Big )+O(1). \end{aligned}$$

Inserting (32) into (30) and letting \(h \rightarrow 0\) while keeping the volume fractions \(\phi _\sigma \) fixed, one finds

$$\begin{aligned} \begin{aligned} \partial _t p_\sigma (x,t)&= D_\sigma \nabla \cdot \left[ \nabla p_\sigma + p_\sigma \nabla V_\sigma \right] - \phi _\sigma D_\sigma \nabla \cdot \left[ p_\sigma ^2 \nabla V_\sigma \right] \\&\quad + \phi _{{{\bar{\sigma }}}} D_\sigma \nabla \cdot \left[ p_\sigma \nabla p_{{{\bar{\sigma }}}} - p_{{{\bar{\sigma }}}} \nabla p_\sigma - p_\sigma p_{{{\bar{\sigma }}}} \nabla V_\sigma \right] . \end{aligned} \end{aligned}$$

The first divergence on the right-hand side corresponds to linear drift-diffusion 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

$$\begin{aligned} \partial _t \rho _\sigma = D_\sigma \nabla \cdot \Big [ (1-\rho ) \nabla \rho _\sigma + \rho _\sigma \nabla \rho + (1-\rho ) \rho _\sigma \nabla V_\sigma \Big ] , \end{aligned}$$

which was obtained previously in [14, 19]. The system of equations (34) for \(\rho _r, \rho _b\) can be written in gradient-flow form (2) with mobility (17) and energy (10).

3.3 Low-Density Approximation Via Matched Asymptotics

Deviations from the ad-hoc mean-field 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, well-suited 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 mean-field 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 two-particle density in powers of \(\phi \).

The interaction term (28) depends on the two-particle 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 \(\phi \))

$$\begin{aligned} \dot{P}_{\sigma ,\sigma '}(x,y,t) = \sum _{ \begin{array}{c} z \in \Omega \\ z \ne y \end{array} } \left[ \lambda _{\sigma } (z,x) P_{\sigma ,\sigma '}(z,y,t) -\lambda _{\sigma } (x,z) P_{\sigma ,\sigma '}(x,y,t) \right] \nonumber \\ + \sum _{ \begin{array}{c} z \in \Omega \\ z \ne x \end{array} } \left[ \lambda _{\sigma '} (z,y) P_{\sigma ,\sigma '}(x,z,t) -\lambda _{\sigma '} (y,z) P_{\sigma ,\sigma '}(x,y,t) \right] +O(\phi ) , \end{aligned}$$

where the higher-order 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 leading-order 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 \(\vert x-y\vert \) grows. This motivates the use of matched asymptotic expansions, with an outer region in which the two particles are well-separated (\(\vert x-y\vert \gg h\)) and an inner region in which the particles are close to each other (\(\vert x-y\vert \sim h\)). 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

$$\begin{aligned} P_{\sigma ,\sigma '}(x,y,t) = {\left\{ \begin{array}{ll} P_{\textrm{out}}(x,y,t), &{} \vert x-y\vert \gg h, \\ P_{\textrm{in}}\big (x,\frac{y-x}{h},t\big ), &{} \vert x-y\vert \sim h, \end{array}\right. } \end{aligned}$$

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 size-exclusion 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

$$\begin{aligned} \begin{aligned} \dot{P}_{\text {out}}(x,y,t)&= \sum _{ \begin{array}{c} z \in \Omega \end{array} } \left[ \lambda _{\sigma } (z,x) P_{\text {out}}(z,y,t) -\lambda _{\sigma } (x,z) P_{\text {out}}(x,y,t) \right] \\&\quad + \sum _{ \begin{array}{c} z \in \Omega \end{array} } \left[ \lambda _{\sigma '} (z,y) P_{\text {out}}(x,z,t) -\lambda _{\sigma '} (y,z) P_{\text {out}}(x,y,t) \right] +O(\phi ) \\&= \mathcal {L}^*_\sigma P_{\text {out}}(\cdot ,y,t) + \mathcal {L}^*_{\sigma '} P_{\text {out}}(x,\cdot ,t) +O(\phi ) , \end{aligned} \end{aligned}$$

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)

$$\begin{aligned} P_{\text {out}}^{} (x,y,t) = p_{\sigma } (x,t)p_{\sigma '} (y,t) + hP_{\text {out}}^{(1)}(x,y,t)+ O(h^2,\phi ) . \end{aligned}$$

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 (\(\vert x-y\vert \sim h\)), 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.

Fig. 1
figure 1

Solution of the auxiliary function \(\psi _1\) satisfying (43), shown for \(d=2\) and \({\hat{y}} \in \{-10,-9, \dots , 10\}^2\), and evaluated using (B8)

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 mean-field 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 \(\phi \),

$$\begin{aligned} \dot{P}_{\text {in}}({{\hat{x}}},{{\hat{y}}},t)&= \! \! \sum _{ {e}\in B_{{\hat{y}}}} \! \left[ \lambda _{\sigma } ({{\hat{x}}}+h{e},{{\hat{x}}}) P_{\text {in}} ({{\hat{x}}}+h{e},{{\hat{y}}}-{e},t) -\lambda _{\sigma } ({{\hat{x}}},{{\hat{x}}}+h{e}) P_{\text {in}} ({{\hat{x}}},{{\hat{y}}},t)\right] \nonumber \\&\quad + \! \sum _{ {e}\in B_{{\hat{y}}}} \! \big [\lambda _{\sigma '} ({{\hat{x}}}+ h({{\hat{y}}}-{e}),{{\hat{x}}}+ h{{\hat{y}}}) P_{\text {in}} ({{\hat{x}}},{{\hat{y}}}-{e},t) \nonumber \\&\quad - \lambda _{\sigma '} ({{\hat{x}}}+ h{{\hat{y}}}, {{\hat{x}}}+ h({{\hat{y}}}-{e})) P_{\text {in}} ({{\hat{x}}},{{\hat{y}}},t) \big ], \end{aligned}$$

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

$$\begin{aligned} P_{\text {out}}^{}&\sim p_{\sigma } ({{\hat{x}}},t)p_{\sigma '} ({{\hat{x}}}+ h {{\hat{y}}},t) + hP_{\text {out}}^{(1)}({{\hat{x}}},{{\hat{x}}}+ h {{\hat{y}}},t)\\&\sim p_{\sigma } ({{\hat{x}}},t)p_{\sigma '} ({{\hat{x}}},t) + h p_{\sigma } ({{\hat{x}}},t) {{\hat{y}}}\cdot \nabla p_{\sigma '} ({{\hat{x}}},t)+ hP_{\text {out}}^{(1)}({{\hat{x}}},{{\hat{x}}},t) ~~~ \text {as } \vert {{\hat{y}}}\vert \rightarrow \infty . \nonumber \end{aligned}$$

Next, we seek a solution to the inner problem (39) of the form

$$\begin{aligned} P_{\text {in}} ({{\hat{x}}},{{\hat{y}}},t) = P_{\text {in}}^{(0)} ({{\hat{x}}},{{\hat{y}}},t) +h P_{\text {in}}^{(1)} ({{\hat{x}}},{{\hat{y}}},t) + \cdots . \end{aligned}$$

Using (32) and expanding \(P_{\text {in}}\) with respect to its first argument (which we recall is continuous), the leading-order inner problem (which comes at order \(h^{-2}\)) is

$$\begin{aligned} 0 = \sum _{ e \in B_{{\hat{y}}}} ( D_\sigma +D_{\sigma '} )\left[ P_{\text {in}}^{(0)} ({{\hat{x}}}, {{\hat{y}}}-{e},t) - P_{\text {in}}^{(0)} ({{\hat{x}}}, {{\hat{y}}},t) \right] , \qquad {{\hat{y}}}\in {\mathbb {Z}_*^d}, \end{aligned}$$

together with the condition at infinity

$$\begin{aligned} P_{\text {in}}^{(0)} ({{\hat{x}}},{{\hat{y}}},t) \sim p_{\sigma } ({{\hat{x}}},t) p_{\sigma '} ({{\hat{x}}},t) \qquad \text {as} \qquad \vert {{\hat{y}}}\vert \rightarrow \infty . \end{aligned}$$

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

$$\begin{aligned} P_{\text {in}}^{(0)} ({{\hat{x}}},{{\hat{y}}},t) = p_{\sigma } ({{\hat{x}}},t) p_{\sigma '} ({{\hat{x}}},t) . \end{aligned}$$

In what follows we simply write \(p_\sigma \) for \(p_\sigma ({{\hat{x}}},t)\). At \(O(h^{-1})\), (39) reads, using (41) and (32),

$$\begin{aligned} 0 = \sum _{ e\in B_{{\hat{y}}}} \left[ P_{\text {in}}^{(1)} ({{\hat{x}}}, {{\hat{y}}}-{e},t) - P_{\text {in}}^{(1)} ({{\hat{x}}}, {{\hat{y}}},t) \right] - {{\hat{y}}}\cdot \varvec{\kappa }({{\hat{x}}},t) \mathbb {1}_{\{ \vert {{\hat{y}}}\vert =1 \} }, \quad {{\hat{y}}}\in {\mathbb {Z}_*^d}, \end{aligned}$$


$$\begin{aligned} \varvec{\kappa }({{\hat{x}}},t) = \frac{1}{ D_\sigma +D_{\sigma '}} \left[ D_\sigma \nabla \left( p_{\sigma } p_{\sigma '} \right) + (D_\sigma \nabla V_\sigma - D_{\sigma '} \nabla V_{\sigma '} ) p_{\sigma } p_{\sigma '}\right] , \end{aligned}$$

together with the matching condition

$$\begin{aligned} P_{\text {in}}^{(1)} ({{\hat{x}}},{{\hat{y}}},t) \sim {\textbf {B}}({{\hat{x}}},t) \cdot {{\hat{y}}}+ {\textbf {C}}({{\hat{x}}},t) \qquad \text {as } \vert {{\hat{y}}}\vert \rightarrow \infty , \end{aligned}$$

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

$$\begin{aligned} \Delta ^1\psi _j&= {{\hat{y}}}_j \mathbb {1}_{\{ \vert {{\hat{y}}}\vert =1 \}},&{{\hat{y}}}\in {\mathbb {Z}_*^d}, \end{aligned}$$
$$\begin{aligned} \psi _j&\sim 0,&\vert {{\hat{y}}}\vert \rightarrow \infty . \end{aligned}$$

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

$$\begin{aligned} \psi _j (0) = 0, \qquad \psi _j(\pm e_k) = \pm \beta \delta _{jk}, \end{aligned}$$


$$\begin{aligned} \beta = -\frac{1}{(2\pi )^d} \int _{[-\pi ,\pi ]^d} \frac{\sin ^2 \zeta _1}{ 2\sum _k \sin ^2 (\zeta _k/2)} d\zeta , \end{aligned}$$

is a d-dependent constant with \(\beta \in (-1,0)\) for \(d\ge 2\). It is related to the constant \(\alpha \) of (14,16) as

$$\begin{aligned} \alpha = -\frac{\beta }{\beta +1} {\in (0,\infty )}. \end{aligned}$$

Using (43,44), we find that

$$\begin{aligned} \sum _{ e\in B_{{\hat{y}}}} \left[ \psi _j ({{\hat{y}}}-{e}) - \psi _j ({{\hat{y}}}) \right]&= \Delta ^1\psi _j + \left[ \psi _j({{\hat{y}}}) - \psi _j(0)\right] \mathbb {1}_{\{ \vert {{\hat{y}}}\vert =1 \}}\nonumber \\&= (1+ \beta ) {{\hat{y}}}_j \mathbb {1}_{\{ \vert {{\hat{y}}}\vert =1 \}}. \end{aligned}$$

Therefore, the solution to (42) can be identified as

$$\begin{aligned} P_{\text {in}}^{(1)}&= \frac{1}{1+\beta }(\varvec{\kappa }- {\textbf {B}}) \cdot {\varvec{\psi }}({{\hat{y}}}) + {\textbf {B}}\cdot {{\hat{y}}}+{\textbf {C}}\nonumber \\&= {\textbf {A}}\cdot {\varvec{\psi }}({{\hat{y}}}) + {\textbf {B}}\cdot {{\hat{y}}}+{\textbf {C}}, \end{aligned}$$


$$\begin{aligned} {\textbf {A}}({{\hat{x}}},t) = \frac{1}{1+\beta } \bigg [&\left( \frac{D_\sigma }{D_\sigma +D_{\sigma '}} \right) p_{\sigma '} \left( \nabla p_\sigma + p_\sigma \nabla V_\sigma \right) \\&- \left( \frac{D_{\sigma '}}{D_\sigma +D_{\sigma '}} \right) p_{\sigma } \left( \nabla p_{\sigma '} + p_{\sigma '}\nabla V_{\sigma '} \right) \bigg ]. \end{aligned}$$

Hence, combining (41) and (48) we arrive at

$$\begin{aligned} P_{\text {in}} ({{\hat{x}}}, {{\hat{y}}}, t) = p_\sigma ({{\hat{x}}}, t) p_{\sigma '} ({{\hat{x}}}, t) + h \left[ {\textbf {A}}\cdot {\varvec{\psi }}({{\hat{y}}}) + {\textbf {B}}\cdot {{\hat{y}}}+{\textbf {C}}\right] +O(h^2, \phi ). \end{aligned}$$

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,

$$\begin{aligned} \mathcal {E}_{\sigma ,\sigma '}(x,t) = \frac{D_\sigma }{h^2} \sum _{\begin{array}{c} \hat{y} \in \mathbb {Z}^d \\ \vert \hat{y} \vert = 1 \end{array}}\left\{ 1 +\frac{h^2}{8} [\hat{y} \cdot \nabla V_\sigma (\hat{x}) ]^2\right\} \left[ P_\textrm{in}(x , \hat{y}) - P_\textrm{in}(x + h \hat{y} , -\hat{y}) \right] \end{aligned}$$

Combining with (49) yields:

$$\begin{aligned} \mathcal {E}_{\sigma ,\sigma '} = D_\sigma \nabla \cdot \left[ 2\beta {\textbf {A}}+ 2{\textbf {B}}- \nabla ( p_\sigma p_{\sigma '}) - \nabla V_\sigma p_\sigma p_{\sigma '} \right] +O(h, \phi ) . \end{aligned}$$

By evaluating \({\textbf {A}}\) and \({\textbf {B}}\) and taking the limit \(h \rightarrow 0\) it follows that

$$\begin{aligned} \mathcal {E}_{\sigma ,\sigma '}{} & {} = D_\sigma \nabla \cdot \Big [ (1+ \alpha \gamma _{\sigma ',\sigma })p_\sigma \nabla p_{\sigma '} - (1+ \alpha \gamma _{\sigma ,\sigma '} ) p_{\sigma '} \nabla p_\sigma \nonumber \\{} & {} \quad -\, (1+ \alpha \gamma _{\sigma ,\sigma '} ) p_\sigma p_{\sigma '}\nabla V_\sigma + \alpha \gamma _{\sigma ',\sigma } p_\sigma p_{\sigma '}\nabla V_{\sigma '} \Big ] +O(\phi ) , \end{aligned}$$


$$\begin{aligned} \gamma _{\sigma ,\sigma '} = \frac{2D_{\sigma }}{D_\sigma +D_{\sigma '}}, \end{aligned}$$

noting that \(\gamma _{\sigma ,\sigma } = 1\). Finally, taking the limit \(h\rightarrow 0\) in (27) yields

$$\begin{aligned} \partial _t p_\sigma (x,t) = D_\sigma \nabla \cdot \left[ \nabla p_\sigma + p_\sigma \nabla V_\sigma \right] +\phi _\sigma {{\mathcal {E}}}_{\sigma ,\sigma } + \phi _{{\bar{\sigma }}}{{\mathcal {E}}}_{\sigma , {{\bar{\sigma }}}} + O(\phi ^2) . \end{aligned}$$

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

$$\begin{aligned} \partial _t \rho _\sigma{} & {} = D_\sigma \nabla \cdot \Big [ (1-\rho ) \nabla \rho _\sigma + \rho _\sigma \nabla \rho + (1-\rho ) \rho _\sigma \nabla V_\sigma \nonumber \\{} & {} \quad -\, \alpha \gamma _{\sigma ,{{\bar{\sigma }}}} \rho _{{{\bar{\sigma }}}} ( \nabla \rho _\sigma +\rho _\sigma \nabla V_\sigma ) + \alpha \gamma _{{{\bar{\sigma }}},\sigma } \rho _\sigma ( \nabla \rho _{{{\bar{\sigma }}}} + \rho _{{{\bar{\sigma }}}} \nabla V_{{{\bar{\sigma }}}}) \Big ] + O(\rho ^3) . \end{aligned}$$

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 Self-diffusion Coefficient: Asymptotics and Connection to Rigorous Results

The self-diffusion coefficient measures the effective diffusion coefficient of a single tagged particle in an interacting environment with many other particles. The standard definition of a self-diffusion 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 self-diffusion coefficient is a macroscopic property of an individual particle, its analysis in the current framework requires tagging a single particle in the population-level 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 self-diffusion 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 self-diffusion 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 self-diffusion 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 self-diffusion coefficient of a \(\sigma \)-particle is given by the limitFootnote 5

$$\begin{aligned} D_{s,\sigma } (\varvec{\phi } ) = \lim _{t \rightarrow \infty } \frac{1}{t} \mathbb {E}\left( \frac{\vert X_t - X_0 \vert ^2 }{2d} \right) , \end{aligned}$$

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 right-hand 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 Self-diffusion for \(D_r \ne D_b\)

The result (54) can be used to compute the self-diffusion 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

$$\begin{aligned} \partial _t p_g&= D_g\nabla ^2 p_g&\quad +\,\sum _{\sigma ' \in \{r,b\}} \phi _{\sigma }{{\mathcal {E}}}_{g,\sigma } + O(\phi ^2) . \end{aligned}$$

where \({{\mathcal {E}}}_{g,\sigma }\) is given by (52). Combining with that equation yields

$$\begin{aligned} \partial _t p_g{} & {} = D_g\nabla \cdot \left\{ \left[ 1- \sum _{\sigma \in \{r,b\}} (1+\alpha \gamma _{g,\sigma }) \rho _{\sigma }\right] \nabla p_g\right. \nonumber \\{} & {} \quad \left. +\, p_g\sum _{\sigma \in \{r,b\}} (1+\alpha \gamma _{\sigma ,g}) \nabla \rho _{\sigma } \right\} + O(\phi ^2) , \end{aligned}$$

where \(\rho _\sigma = \phi _\sigma p_\sigma \) are concentrations. The self-diffusion coefficent can be identified from the pre-factor of \(\nabla p_g\). Using that the environment is stationary (\(\rho _{\sigma } \equiv \phi _\sigma \)), the self-diffusion of the tagged particle for \(\phi \ll 1\) is

$$\begin{aligned} D_{s,g}({\varvec{\phi }}) = D_g\left[ 1- \sum _{\sigma \in \{r,b\}} (1+\alpha \gamma _{g,\sigma })\phi _{\sigma }\right] + O(\phi ^2) . \end{aligned}$$

Physically, the self-diffusion 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

$$\begin{aligned} D^\text {low}_{s,r}(\varvec{\phi }) = D_r \left[ (1- \sum _{\sigma \in \{r,b\}} (1+\alpha \gamma _{r,\sigma })\phi _{\sigma }\right] . \end{aligned}$$

Performing the analogous computation on the mean-field discrete PDE (33), we find that the mean-field approximation of the self-diffusion coefficient, for a tagged red particle is

$$\begin{aligned} D_{s,r}^\text {mf}( \phi ) = D_r (1- \phi ) , \end{aligned}$$

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 multi-species 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 \(\vert X_t-X_0\vert ^2/(2dt)\) 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 mean-field curve \(D^{\text {mf}}_{s,r}\), shown as a black dashed-line.

Fig. 2
figure 2

Dependence of the self-diffusion coefficient \(D_{s,r}(\phi _r, \phi _b)\) (56) on the particle concentration \(\phi \) and \(\gamma _{r,b}\), up to the concentration \(\phi = 0.2\). Results from stochastic simulation on a \(10^3 \times 10^3\) periodic lattice, \(h = 0.001\), \(\phi _r = \phi _b\), \(D_0 =1\), averaged over \(K= 10\) realisations, with varying N and \(\gamma _{r,b}\). Measured values from stochastic simulations (56) (red circles) and theoretical predictions, \(D^\text {low}_{s,r}\) (60) (solid line) and \(D^\text {mf}_{s,r}\) (61) (dashed line)

4.2 Self-diffusion for \(D_r =D_b\)

In contrast to the case \(D_r \ne D_b\), the general behaviour of the self-diffusion 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 self-diffusion 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 first-order truncations in \(\phi \) and \(1-\phi \) respectively are given by

$$\begin{aligned} D^\text {low}_{s}(\phi ) = 1-(1+\alpha ) \phi ,{} & {} D^\text {high}_{s}(\phi ) = \frac{1}{2\alpha +1}(1-\phi ) . \end{aligned}$$

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 gradient-flow structure (2) with mobility and energy (9). In particular, we substitute the low-volume approximation \(D^\text {low}_{s}(\phi )\) (62) into the mobility matrix (9b). The result is

$$\begin{aligned} \partial _t \rho _\sigma = \nabla \cdot \Big [ (1-\rho ) \nabla \rho _\sigma + \rho _\sigma \nabla \rho + \alpha \rho _{{{\bar{\sigma }}}} \nabla \rho _\sigma - \alpha \rho _\sigma \nabla \rho _{{{\bar{\sigma }}}} \Big ] + O(\rho ^3) . \end{aligned}$$

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 high-volume 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:

$$\begin{aligned} \tilde{D}_s (\phi ) = (1- \phi ) \left[ 1 - \alpha \phi + \dfrac{\alpha (2\alpha -1)}{2\alpha +1} \phi ^2 \right] . \end{aligned}$$

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 mean-field approximation \(D^{\text {mf}}_{s}\).

Fig. 3
figure 3

Dependence of the self-diffusion coefficient \(D_s(\phi )\) (56) on the occupied fraction \(\phi \). Values shown are: simulated values (red circles) measured with (56), composite approximation \(\tilde{D}_s\) (64) (solid line), mean-field approximation \(D^\text {mf}_s\) (61) (dashed line), and low- and high-density approximations \(D^\text {low}_{s}\) and \(D^\text {high}_{s}\) (62) (dot-dashed lines). Stochastic simulations performed on a \(10^3 \times 10^3\) periodic lattice, for \(h = 0.001\) fixed while varying N such that \(Nh^2 = \phi \)

5 Numerical Simulations of Collective Dynamics

In this section, we perform numerical simulations of the cross-diffusion PDE system (2) for the concentrations \(\rho _r\) and \(\rho _b\), with the mobility matrix M obtained either via matched asymptotics (12c), mean-field (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

$$\begin{aligned} \begin{pmatrix} \rho _r(x, t+\Delta t ) \\ \rho _b(x, t+\Delta t ) \end{pmatrix} = \begin{pmatrix} \rho _r(x, t) \\ \rho _b(x, t ) \end{pmatrix}+ \frac{\Delta t}{h^2} \sum _{k} [J_{x+he_k/2}(t)-J_{x-he_k/2}(t)], \end{aligned}$$

with periodic boundary conditions and

$$\begin{aligned} J_{x+he_k/2} = M(\rho _r, \rho _b) \begin{pmatrix} {\bar{F}}_r(x+\tfrac{h}{2}e_k) \\ {{\bar{F}}}_b(x + \tfrac{h}{2}e_k) \end{pmatrix}, \end{aligned}$$

where the discretised thermodynamic force is,

$$\begin{aligned} {{\bar{F}}}_\sigma (x+\tfrac{h}{2}e_k) = \frac{\rho _\sigma (x+h e_k)-\rho _\sigma (x)}{h\rho _\sigma (x+\tfrac{h}{2}e_k)}+ \frac{\rho (x+he_k)-\rho (x)}{h[1- \rho (x+\tfrac{h}{2}e_k)]} + \nabla V_\sigma (x+\tfrac{h}{2}e_k), \end{aligned}$$

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 density-dependent self-diffusion 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 two-species 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).

Fig. 4
figure 4

Visualisation of particle positions, for \(D_r = D_b =1\), \(V_r = -V_b = \sin (2 \pi x) \), \(h =0.05\) and \(\phi _r = \phi _b = 0.25\) (\(N_r = N_b = 100\)). Snapshots of the process are taken at \(t= 0.0,~0.04, ~0.08,~ 0.12\)

Fig. 5
figure 5

Particle densities at times \(t = 0.02, 0.08, 0.3\) with initial data \(\rho _r(\cdot ,0) = \frac{1}{2} \mathbb {1}_{(0,1/2]}\), \(\rho _b(\cdot ,0) = \frac{1}{2}\mathbb {1}_{(1/2,1]}\) and parameters \(D_r = D_b =1\), \(V_r = -V_b = \sin (2 \pi x) \), \(h =0.01\) and \(\phi _r = \phi _b = 0.25\) (\(N_r = N_b = 2500\)). Solutions of (2), \(\tilde{\rho }_r, \tilde{\rho }_b\) (solid line) and \(\rho _r^\text {mf}, \rho _b^\text {mf}\) (dashed line), and histograms of \(\langle \eta _t^r(x) \rangle \), \(\langle \eta _t^b(x) \rangle \) (data points) averaged over \(x_2\) and \(K = 30\) realisations, where the density of each species is shown in its respective colour. Error bars indicate twice the standard error

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 self-diffusion 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 mean-field equation (34) has a higher mobility than the composite prediction (\(\tilde{D}_s ( \phi ) < D^\text {mf}_s ( \phi )\)) so it appears that the mean-field 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).

Fig. 6
figure 6

Relative free energy (10) for \(0.00< t < 0.20\) and parameters \(D_r = D_b =1\), \(V_r = -V_b = \sin (2 \pi x) \), \(h =0.01\) and \(\phi _r = \phi _b = 0.25\) (\(N_r = N_b = 2500\)). Comparison between the free energy evaluated with solutions of (2), \(\tilde{\rho }_r, \tilde{\rho }_b\) (solid line) and \(\rho _r^\text {mf}, \rho _b^\text {mf}\) (dashed line), or via simulated data, \(\langle \eta _t^r(x) \rangle \), \(\langle \eta _t^b(x) \rangle \), averaged over \(x_2\) and K = 60 realisations (data points). Error bars indicate twice the standard error, and are initially covered by the data points

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 cross-diffusion 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

$$\begin{aligned} \hat{E}(t) = E[\rho _r, \rho _b](t) - E[\rho _r, \rho _b](\infty ), \end{aligned}$$

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 steady-state 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 time-dependent 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 mean-field 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 mean-field 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).

Fig. 7
figure 7

Particle densities at times \(t = 0.01, 0.02, 0.04\) with initial data \(\rho _r(\cdot ,0) = 0.1 \mathbb {1}_{(0,1/2]}\), \(\rho _b(\cdot ,0) = 0.1\mathbb {1}_{(1/2,1]}\) and parameters \(D_r = 1.5\), \(D_b=0.5\), \( D_r V_r = - D_b V_b = \sin (2 \pi x) \), \(h =0.01\) and \(\phi _r = \phi _b = 0.05\) (\(N_r = N_b = 500\)). Solution of (54) \(\rho _r^\text {low}, \rho _b^\text {low}\)(solid line), solution of (34) \(\rho _r^\text {mf}, \rho _b^\text {mf}\) (dashed line) and histograms of \(\langle \eta _t^r(x) \rangle \), \(\langle \eta _t^b(x) \rangle \) averaged over \(x_2\) and \(K = 60\) realisations (data points), where the density of each species is shown in its respective colour. Error bars indicate twice the standard error

6 Discussion

We have considered a two-species 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 (lattice-based) probability distribution density of the particle positions \(P_\sigma (x,t)\) for each species. We interpret this as a discrete cross-diffusion PDE system, from which we identify corresponding macroscopic PDEs. These can then be factorised into the gradient-flow structure (2). Within this description, a non-trivial mobility matrix M captures the effect of microscopic interactions on the collective dynamics, which manifest as coupled nonlinear cross-diffusion terms.

We compare the predictions of this PDE system with numerical simulations of the discrete stochastic model. Specifically, we consider self-diffusion 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 non-trivial 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 mean-field closure of the equations of motion [14, 19].

Notably, the results obtained for the SEP lattice-based model differ significantly from off-lattice 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 off-lattice system incorporates non-trivial 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 high-density 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 two-particle 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 self-diffusion constant that agrees with the expansion in \(\phi \) that was derived rigorously in [21]. To the best of our knowledge, our results for multi-species SEP with \(D_r\ne D_b\) are the first to give a consistent picture of the collective dynamics and self-diffusion.

Rigorous analysis of the hydrodynamic limit is technically challenging in these multi-species systems because they are of non-gradient type [9, 17, 23]. In particular, one requires a sharp estimate of the spectral gap to establish the system’s fast-mixing properties. We note that recent results for the spectral gap are available for multi-species 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 lattice-based and off-lattice 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 non-gradient 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 off-lattice results such as [27].