Diffusion of Finite-Size Particles in Confined Geometries

Abstract

The diffusion of finite-size hard-core interacting particles in two- or three-dimensional confined domains is considered in the limit that the confinement dimensions become comparable to the particle’s dimensions. The result is a nonlinear diffusion equation for the one-particle probability density function, with an overall collective diffusion that depends on both the excluded-volume and the narrow confinement. By including both these effects, the equation is able to interpolate between severe confinement (for example, single-file diffusion) and unconfined diffusion. Numerical solutions of both the effective nonlinear diffusion equation and the stochastic particle system are presented and compared. As an application, the case of diffusion under a ratchet potential is considered, and the change in transport properties due to excluded-volume and confinement effects is examined.

Appendix: Derivation of the Narrow-Channel Equation (10) for the (NC2) Case

This appendix is devoted to the derivation of (10) in the two-dimensional channel (NC2) case. The derivation of the three-dimensional cases (NC3) and (PP) or other (simple) geometries follows similarly (see Sect. A.3 for an outline of the conditions/calculations to be carried out).

1.1 A.1 Transformation $\mathcal{T}_{1}$: Reduction from the Individual- to the Population-Level

Our starting point is the Fokker–Planck equation for N hard-disc particles (2a), defined in the high-dimensional (configuration) space $\varOmega_{\epsilon}^{N} \subset\mathbb{R}^{2N}$. Recall that the configuration space has holes that correspond to illegal configurations (with particles’ overlaps) on which the no-flux boundary conditions (2b) hold. The aim of this subsection is to derive the corresponding Fokker–Planck equation for the one-particle density $p({\bf x}_{1}, t) = \int P(\mathbf{x}, t) \,\mathrm {d}{\bf x}_{2} \cdots\,\mathrm {d}{\bf x}_{N}$, where P(x,t) is the joint probability density of the N particles. We first note that the integration of (2a), (2b) over ${\bf x}_{2}, \dots, {\bf x}_{N}$ results in integrals over contact surfaces on which P must be evaluated (Bruna and Chapman 2012b). When the particle volume is small, the dominant contributions to these contact integrals correspond to two-particle interactions, so that we can set N=2 and then extend the result to N arbitrary in a straightforward manner. However, in contrast with the unconfined case studied in Bruna and Chapman (2012b), under confinement conditions the particle–particle–wall interactions (three-body) are not negligible and must be taken into account. For two particles at positions ${\bf x}_{1}$ and ${\bf x}_{2}$, (2a), (2b) reads

(28a)

(28b)

on ${\bf x}_{i} \in\partial\varOmega$ and $\|{\bf x}_{1} - {\bf x}_{2}\| = \epsilon$. Here $\hat{\bf n}_{i} = { {\bf n}}_{i} / \| { {\bf n}}_{i}\|$, where ${ {\bf n}}_{i}$ is the component of the normal vector n corresponding to the ith particle, $\mathbf{n} = ({\bf n}_{1}, {\bf n}_{2})$. We note that $\hat{\bf n}_{1} = 0$ on ${\bf x}_{2} \in\partial \varOmega$, and that $\hat{\bf n}_{1} = -\hat {\bf n }_{2}$ on $\| {\bf x}_{1}-{\bf x}_{2}\| = \epsilon$.

1.1.1 A.1.1 From N Particles to 1 Particle

We denote by $\varOmega({\bf x}_{1}) = \varOmega\setminus B_{\epsilon}({\bf x}_{1})$ the region available to the center of particle 2 when particle 1 is at ${\bf x}_{1}$. Note that when the distance between ${\bf x}_{1}$ and ∂Ω is less than ϵ the area $|\varOmega({\bf x}_{1})|$ increases (see Fig. 16) because the area $\mathcal{U}({\bf x}_{1}) = B_{\epsilon}({\bf x}_{1})\cap\varOmega$ excluded by particle 1 changes with ${\bf x}_{1}$. The points on which the two particles are in contact are given by the collision boundary

$$ \mathcal{C}_{{\bf x}_1} = \bigl\{ {\bf x}_2 \in \varOmega\text{ s.t. } \| {\bf x}_2 - {\bf x}_1\| = \epsilon \bigr\}. $$

(29)

Fig. 16

Sketch of the original channel geometry (solid black lines) and the effective configuration space for the center of a second circular particle, given by the boundary function $\partial \varOmega({{\bf x}_{1}})$ (dash red lines) which depends on the position of the first particle ${\bf x}_{1}$. The collision boundary function $\mathcal{C}_{{\bf x}_{1}}$ forms part of the effective boundary $\partial\varOmega({{\bf x}_{1}})$ (Color figure online)

Integrating Eq. (28a) over $\varOmega({\bf x}_{1})$ using the Reynolds transport theorem (on the moving boundary $\mathcal{C}_{{\bf x}_{1}}$), the divergence theorem, and the boundary condition (28b) yields

(30)

We denote the collision integral above by $\mathcal{I}$. If we now consider the case of N particles we would obtain a collision integral for each pair, so that after some particle relabeling the corresponding equation is

(31)

Equation (31) is halfway through transformation $\mathcal{T}_{1}$ (cf. Fig. 1), since the first half of the equation depends only on $\bf x_{1}$ while the integral $\mathcal{I}$ still depends on the two-particle density P near the collision surface $\mathcal{C}_{{\bf x}_{1}}$.

At this stage, it is common to use a closure approximation such as $P({\bf x}_{1},{\bf x}_{2},t) = p({\bf x}_{1},t) p({\bf x}_{2},t)$ to evaluate $\mathcal{I}$ and obtained a closed equation for p (Rubinstein and Keller 1989). However, the pairwise particle interaction—and, therefore, the correlation between their positions—is exactly localized near the collision surface $\mathcal{C}_{{\bf x}_{1}}$. Instead, in the next section we will use an alternative method based on matched asymptotic expansions to evaluate $\mathcal{I}$ systematically (Bruna and Chapman 2012b).

1.1.2 A.1.2 Matched Asymptotic Expansions

We suppose that when two particles are far apart (|x ₁−x ₂|≫1) they are independent (at leading order), whereas when they are close to each other (|x ₁−x ₂|∼ϵ) they are correlated. We denote these two regions of the configuration space $\varOmega_{\epsilon}^{2}$ the outer region and the inner region, respectively. We use the x-coordinate to distinguish between the two regions because the inner region spans the channel’s cross section. Importantly, this implies that the outer region is disconnected.

Outer Region

In the outer region, we consider the change to the narrow-domain variables (5) and define $\hat{P}(\hat{\bf x}_{1}, \hat{\bf x}_{2},t) = \epsilon^{2} P ({\bf x}_{1}, {\bf x}_{2}, t)$. This scaling is consistent with that introduced for the one-particle density in Sect. 2.2, and is such that P and $\hat{P}$ each integrate to one in their respective domains $\varOmega_{\epsilon}^{2}$ and the narrow-domain variable equivalent $\omega_{\epsilon}^{2}$. Then (28a) becomes

(32a)

for $(\hat{\bf x}_{1}, \hat{\bf x}_{2}) \in\omega_{\epsilon}^{2}$, with

(32b)

(32c)

for i=1,2. The boundary condition on the collision line $\mathcal{C}_{{\bf x}_{1}}$ disappears for $|\hat{x}_{1} - \hat{x}_{2}| > \epsilon$, so that it is “invisible” to the outer region.⁴

In the outer region, we define $P_{\mathrm{out}}(\hat{\bf x}_{1}, \hat{\bf x}_{2},t) = \hat{P}(\hat{\bf x}_{1}, \hat{\bf x}_{2},t)$ and look for an asymptotic solution to (32a)–(32c) by expanding P _out in powers of ϵ. We find at leading order that P _out must be independent of the vertical coordinates $\hat{y}_{1}$ and $\hat{y}_{2}$. By independence in the outer region, we suppose that the leading-order solution is separable, so that it is of the form $q(\hat{x}_{1},t) q(\hat{x}_{2},t)$ for some function q. Solving for the next two orders in ϵ, we find that the solution in the outer region is, to $\mathcal{O}(\epsilon^{2})$,

(33)

where the $\varUpsilon_{i}(\hat{x}_{1}, \hat{x}_{2}, t)$ are arbitrary functions of integration, determined by solvability conditions at higher order. Note that the invariance of P with respect to a switch of particle labels means that in the outer region both particles have the same density function q. From the solvability condition on the O(ϵ ²) terms above, we obtain the following equation for q:

$$ \frac{\partial q}{\partial t} (\hat{x}, t) = \frac{\partial}{\partial \hat{x}} \biggl( \frac{\partial q}{\partial\hat{x}} - f_1(\hat{x},0) q \biggr). $$

(34)

Inner Region

In the inner region, we introduce the inner variables

$$ x_1 = \tilde{x}_1, \qquad y_1= \epsilon\tilde{y}_1,\qquad x_2 = \tilde{x}_1 + \epsilon\tilde{x}, \qquad y_2= \epsilon\tilde{y}_2, $$

(35)

and define $\tilde{P}(\tilde{\bf x}_{1}, \tilde{\bf x}_{2},t) = \epsilon ^{2} P ({\bf x}_{1}, {\bf x}_{2}, t)$. The contact boundary $\mathcal{C}_{{\bf x}_{1}}$ (29) becomes

$$ \tilde{\mathcal{C}}_{\tilde{y}_1} = \bigl\{ (\tilde{x}, \tilde{y}_2) \in \mathbb{R} \times[-h/2, h/2] \ \text{ s.t. } \ \tilde{x}^2 + (\tilde{y}_2 - \tilde{y}_1)^2 = 1 \bigr\}, $$

(36)

and problem (28a), (28b) is transformed to

(37a)

with

(37b)

on $\tilde{\mathcal{C}}_{\tilde{y}_{1}}$ and

(37c)

(37d)

In addition to (37b)–(37d), the inner solution must match with the outer as $\tilde{x} \to\pm\infty$. Expanding the outer solution in terms of the inner variables, which corresponds to replacing $\hat{x}_{1} = \tilde{x}_{1}$, $\hat{y}_{1} = \tilde{y}_{1}$, $\hat{x}_{2} = \tilde{x}_{1} + \epsilon\tilde{x}$ and $\hat{y}_{2} = \tilde{y}_{2}$ in (33), and subsequently expanding in powers of ϵ, we obtain the following matching condition:

(37e)

where $q \equiv q(\tilde{x}_{1}, t)$. Expanding $\tilde{P}$ in powers of ϵ, $\tilde{P}\sim\tilde{P}^{(0)} + \epsilon\tilde{P}^{(1)} + \epsilon^{2} \tilde{P}^{(2)} + \cdots$, we find that the leading- and first-order solutions of (37a)–(37e) are

(38)

(39)

Unlike in the bulk or unconfined problem (Bruna and Chapman 2012a, 2012b), the narrow-channel problem requires computing the second-order inner solution. The solution procedure is rather cumbersome and is omitted here. It involves a further change of variable $\tilde{x} = \sqrt{2} \tilde{s}$ to turn the problem into a Poisson problem, and solving two sub-problems numerically using the commercial finite-element solver COMSOL Multiphysics 4.3. The second-order solution of (37a)–(37e) is (see Appendix C.1 in Bruna 2012 for full details)

(40)

with $\tilde{Q}_{i} (\tilde{x}, \tilde{y}_{1}, \tilde{y}_{2}) = \tilde{v}_{i} (\tilde{x}/\sqrt{2} , \tilde{y}_{1}, \tilde{y}_{2})$, where $\tilde{v}_{1}(\tilde{s}, \tilde{y}_{1}, \tilde{y}_{2})$ and $\tilde{v}_{2} (\tilde{s}, \tilde{y}_{1}, \tilde{y}_{2})$ are given by (numerical solutions of)

$$ \begin{array}{l} \widetilde{\boldsymbol{\nabla}}^2 \tilde{v}_1 = 0,\\[4pt] \widetilde{\boldsymbol{\nabla}} \tilde{v}_1 \cdot \tilde{\boldsymbol {\nu}} = \tilde{s}^2 \quad \text{on} \ \tilde{\mathcal{D}}_{\tilde{y}_1}, \\[7pt] \widetilde{\boldsymbol{\nabla}} \tilde{v}_1 \cdot\tilde{ \boldsymbol {\nu}} = 0 \quad \text{on}\ \tilde{y}_i=\pm\dfrac{h}{2}, \\[7pt] \tilde{v}_1 \sim D_1 |\tilde{s}|\quad \text{as}\ \tilde{s} \to\pm \infty, \end{array} $$

(41)

and

$$ \begin{array}{l} \widetilde{\boldsymbol{\nabla}}^2 \tilde{v}_2 = 0, \\[5pt] \widetilde{\boldsymbol{\nabla}} \tilde{v}_2 \cdot \tilde{\boldsymbol {\nu}} = \tilde{s} (\tilde{y}_1 - \tilde{y}_2), \quad \text{on}\ \tilde {\mathcal{D}}_{\tilde{y}_1}, \\[7pt] \widetilde{\boldsymbol{\nabla}} \tilde{v}_2 \cdot\tilde{ \boldsymbol {\nu}} = 0, \quad \text{on}\ \tilde{y}_i=\pm \dfrac{h}{2}, \\[7pt] \tilde{v}_2 \sim0 ,\quad \text{as}\ \tilde{s} \to\pm\infty. \end{array} $$

(42)

Here, $\widetilde{\boldsymbol{\nabla}}$ stands for the gradient operator with respect to the position vector $(\tilde{s}, \tilde{y}_{1}, \tilde{y}_{2})$, $\tilde{\mathcal{D}}_{\tilde{y}_{1}}$ is the transformed collision boundary $\tilde{\mathcal{C}}_{\tilde{y}_{1}}$ (36), and $\tilde{\boldsymbol{\nu}}$ is the outward unit normal on this mapped boundary, $\tilde{\boldsymbol{\nu}} = - \frac{\sqrt{2}}{2} (2 \tilde{s}, \tilde{y}_{1} - \tilde{y}_{2}, \tilde{y}_{2} - \tilde{y}_{1})$. Finally, the constant field D ₁ at infinity for $\tilde{v}_{1}$ is related to the integration function from the outer ϒ ₁,

$$ D_1 = \frac{\lim_{\hat{x}_2 \to\hat{x}_1} \frac{\partial\varUpsilon _1}{\partial\hat{x}_2} (\hat{x}_1, \hat{x}_2)}{ [ q \frac{\partial ^2 q}{\partial\tilde{x}_1^2} - (\frac{\partial q}{\partial \tilde{x}_1} ) ^2 - \frac{\partial f_1}{\partial x} q^2 ] }. $$

Out of the derivation we also find that $\varUpsilon_{1}(\hat{x}_{1}, \hat{x}_{2}) \equiv\varUpsilon_{1}(|\hat{x}_{1}-\hat{x}_{2}|)$ with ϒ ₁(x) differentiable satisfying ϒ ₁(0)=0. In contrast, the contribution from the other outer function ϒ ₂ is left unknown (it could be determined by matching higher order terms), but we are able to ignore it as it has a zero contribution to the collision integral $\mathcal{I}$ (as we will see in the next section). We note that, in (40), q, f ₁ and f ₂ are functions of the “outer” variable $\tilde{x}_{1}$ only, namely, $q = q(\tilde{x}_{1}, t)$ and $f_{i} = f_{i}(\tilde{x}_{1}, 0)$.

Combining (38), (39), and (40) we have the solution to the inner problem (37a)–(37e) up to $\mathcal{O}(\epsilon^{2})$.

1.1.3 A.1.3 Collision Integral

Now we go back to Eq. (31) and use the asymptotic solution of the previous subsection to turn it into an equation for $p({\bf x}_{1}, t)$ only thus completing transformation $\mathcal{T}_{1}$. Note that, since the integral $\mathcal{I}$ is over the collision boundary $\mathcal{C}_{{\bf x}_{1}}$, it lives in the inner region and we must use $\tilde{P}$ to evaluate it.

In terms of the inner variables, $\mathcal{I}$ is

(43)

where $\tilde{\mathcal{C}}_{\tilde{y}_{1}}$ is given in (36) and $\mathrm {d}\tilde{l}$ is the line integral along this curve (for $\tilde{y}_{1}$ fixed, see Fig. 17). Depending on the channel width h (relative to one, which is the radius of $\tilde{\mathcal{C}}_{\tilde{y}_{1}}$), the integration is over the whole circle or a part of it. Introducing the distances $l_{1} = \max(-h/2-\tilde{y}_{1},-1)$ and $l_{2} = \min(h/2-\tilde{y}_{1}, 1)$, the angles at contact with the lower and upper channel walls are θ ₁=arcsinl ₁ and θ ₂=arcsinl ₂, respectively. (These are equal to ±π/2 for no contact.)

Fig. 17

Domain of integration $\tilde{\mathcal{C}}_{\tilde{y}_{1}}$ (solid green line), distances l ₁ and l ₂ between the vertical coordinate of the first particle and the lower and upper channel walls and corresponding angles θ ₁ and θ ₂ (Color figure online)

Writing $\mathcal{I} = \epsilon^{-2} ( \mathcal{I}^{(0)} + \epsilon \mathcal{I}^{(1)} + \epsilon^{2}\mathcal{I}^{(2)} + \cdots )$ and using (43), we find that

(44)

(45)

(46)

where $f_{i} = f_{i} (\tilde{x}_{1}, 0)$,

(47a)

(47b)

(47c)

and $\mathcal{J}$ is the integral operator $\mathcal{J}[Q](h, \tilde{y}_{1}) = \int_{\tilde{\mathcal{C}}_{\tilde{y}_{1}}} [ Q_{\tilde{y}_{2}} (\tilde{y}_{2}- \tilde{y}_{1}) + Q_{\tilde{x}} \tilde{x} ]\, \mathrm {d}\tilde{l}$. The terms $\mathcal{J}[\tilde{Q}_{1}]$ and $\mathcal{J}[\tilde{Q}_{2}]$ are evaluated numerically with COMSOL (see Appendix C.2 in Bruna 2012 for more details).

1.1.4 A.1.4 Population-Level Fokker–Planck Equation

Combining (45) and (46), we obtain the first two terms of the asymptotic expansion for $\mathcal{I}$, which depends on both the channel width h and the elevation of the first particle $\tilde{y}_{1}$ but is independent of the position of the second particle. Thus, we can drop the first particle label (the subindex 1) for clarity of notation. Inserting this expansion into (31), we obtain an equation for the first particle

(48)

which involves the marginal density $p({\bf x},t)$, the outer density $q(\hat{x},t)$ and the channel width h. This concludes the transformation $\mathcal{T}_{1}$ from N particles to one particle (see Fig. 1).

1.2 A.2 Transformation $\mathcal{T}_{2}$: Reduction of the Number of Geometric Dimensions

Following a similar procedure to that for point particles in Sect. 2.2, we will reduce (48) into a one-dimensional effective equation along the axial direction. First, integrating (28b) over $\varOmega({\bf x})$ for ${\bf x} \in\partial\varOmega$, we obtain the following no-flux boundary condition:

$$ \bigl[ {\boldsymbol{\nabla}}_{{\bf x}} p- {\bf f}({\bf x}) p \bigr] \cdot\hat { \bf n} = 0 \quad\text{on} \ \partial\varOmega. $$

(49)

Analogously to the point-particles case, we use the narrow-domain variables (5) and define $\hat{p}(\hat{\bf x}, t) = \epsilon p({\bf x}, t)$. With this rescaling, Eqs. (48) and (49) become

(50a)

(50b)

(50c)

where $f_{i} \equiv f_{i} (\hat{x}, \epsilon\hat{y})$. There is no need to expand the integral terms $\mathcal{I}^{(i)}$ in terms of the narrow-domain variables since these are written in terms of the inner variables (35), which coincide with the narrow-domain variable for expressions independent of the second particle’s coordinates.

Expanding $\hat{p}$ in powers of ϵ and solving (50a)–(50c) gives, at leading order, that $\hat{p}$ is independent of $\hat{y}$. As before, we introduce the effective one-dimensional densities as $\hat{p}_{e}^{(i)} = \int_{-h/2} ^{h/2} \hat{p}^{(i)}\, \mathrm {d}\hat{y}$. Thus we have that $\hat{p}_{e} ^{(0)} \equiv h \hat{p}_{e}^{(0)}$. At the next order,

(51)

For clarity of notation, in the remaining of this section, we write $f_{i}(\hat{x}, 0) \equiv f_{i}$. Integrating the second order of (50a) over the channel’s cross section and using (50c), gives

(52)

where we have used that $\int_{-h/2}^{h/2} \mathcal{I}^{(1)}\, \mathrm {d}\hat{y} = 0$ [see Eqs. (45) and (47a)]. Note that this equation coincides with the effective equation for point particles (7). It is at the next order that the finite-size effects appear.

Repeating the same procedure of integrating with respect to $\hat{y}$ the $\mathcal{O}(\epsilon^{3})$ of (50a)–(50c) and using (50c) to eliminate $\hat{p}^{(3)}$ yields the following solvability condition:

$$ \frac{\partial\hat{p}^{(1)}_e}{\partial t}(\hat{x}, t) -\frac{\partial }{\partial\hat{x}} \biggl( \frac{\partial\hat{p}^{(1)}_e}{\partial \hat{x}} - f_1 \, \hat{p}^{(1)}_e \biggr) = (N-1) \int_{-h/2}^{h/2} \mathcal{I}^{(2)} \, \mathrm {d}\hat{y}. $$

(53)

Using (46) and (47a)–(47c), the cross-section integral of $\mathcal{I}^{(2)}$ is

(54)

where $M_{1} (h) = \int_{-h/2}^{h/2} \mu_{1}(h, \hat{y})\, \mathrm {d}\hat{y}$ reads

(55)

where Θ(x) is the Heaviside step function, and $\mathcal{M} [Q](h) = \int_{-h/2}^{h/2} \mathcal{J}[Q](h, \hat{y}) \, \mathrm {d}\hat{y}$. Although $\tilde{Q}_{1}$ and $\tilde{Q}_{2}$ are only solved numerically, using information from their respective problems (41) and (42) one can deduce analytical expressions for their integrals $\mathcal{M}[\tilde{Q}_{i}]$, namely that $\mathcal{M}[\tilde{Q}_{1}] = - M_{1}(h)/(2 \sqrt{2})$ and $\mathcal{M}[\tilde{Q}_{2}] = 0$ (see Appendix C.3 in Bruna 2012). Using this, we find that

(56)

Because q is independent of the inner variables ($\tilde{x}, \tilde{y}_{1}, \tilde{y}_{2}$), we can write $q(\tilde{x}_{1}, t) = q_{e} (\hat{x}, t)/h$. Moreover, the normalization condition on $\hat{P}$ gives that $q_{e} (\hat{x}, t) = \hat{p}_{e} (\hat{x}, t) + \mathcal{O}(\epsilon)$. Therefore, the right-hand side of (56) becomes $\frac{ \partial}{ \partial {\hat{x}} } ( \hat{p}_{e} \frac{\partial\hat{p}_{e}}{\partial{\hat{x}} } )/h^{2}$.

Combining (52), (53), and (56) yields

(57)

which coincides with the effective equation (10) for a two-dimensional narrow-channel after writing α _h ≡M ₁(h)/h ²; see (11).

1.3 A.3 Outline of Steps for Other Geometries

In this section, we indicate the key steps to derive the effective continuum Fokker–Planck equation (10) for a general geometry, and in particular for the three-dimensional cases (NC3) and (PP) presented in Sect. 2.3. First, we note below relevant definitions that change with the problem dimension d and the number of confined dimensions k (recall that d _e =d−k):

• (i) Identify the number of confined and effective dimensions: the original position vector is split into two components, ${\bf x}=(\mathbf{x}_{e}, \mathbf{x}_{c})$ (see Table 1). For example, for (NC3) x _e =x and x _c =(y,z), while for (PP) x _e =(x,y) and x _c =z.

  Table 1 Dimension and variables in each of the spaces and subspaces

• (ii) Determine the confinement parameter(s): Next, we must choose an scaling for the confined dimensions relative to the unconfined ones. In all cases considered here, we made that simple by saying that all confined dimensions are of length H relative to the unconfined ones, but there could be of different lengths, too, such as the narrow channel of rectangular cross section mentioned briefly in Sect. 3.3. Suppose that the confinement dimensions are $\mathbf{H} = (H_{1}, \dots, H_{k}) = \mathcal{O}(\epsilon)$. Then the vector of confinement parameters is given by h=(h ₁,…,h _k ), with h _i =H _i /ϵ.

  Note that we are assuming that a confined dimension is always of order ϵ (the particle’s diameter). However, this could also be generalized and introduce an intermediate scaling (between ϵ and one).

• (iii) Narrow-domain variables transformation: the generalization of the change of variables (5) is

  $$ \mathbf{x}_e = \hat{\mathbf{x}}_e, \qquad \mathbf{x}_c = \epsilon\hat{\mathbf{x}}_c. $$

  We write $\hat{\bf x} = (\hat{\mathbf{x}}_{e}, \hat{\mathbf{x}}_{c})$. The one-particle and two-particle densities in the rescaled domain are respectively defined as

  

  The original domain Ω is mapped into the rescaled domain ω.

• (iv) Apply the effective domain transformation: The $\mathcal{T}_{2}$ transformation to reduce the original d-dimensional problem to a d _e -dimensional problem (cf. Sect. A.2) requires the following rescaled densities

  

  where

  $$ \varLambda= \prod_{i=1}^k h_i. $$

  (58)

  This factor is, in fact, equal to the volume of the rescaled domain ω (also |ω|≡|ω _c |). Recall it is introduced so that $\hat{p}_{e}$ and $\hat{P}_{e}$ are defined as densities.

• (v) Evaluate the collision integral $\mathcal{I}$ : The evaluation of the contribution of the two-particle interaction $\mathcal{I}$ reduces to computing one coefficient like M ₁(h) in (57) for each of the unconfined dimensions. Suppose that all the unconfined dimensions are symmetric. In general, it is equal to

  $$ M_1(\mathbf{h}) = \int_{\omega_c} \mu_1(\mathbf{h},\hat {\mathbf{x}}_c) \, \mathrm {d}\hat{\mathbf{x}}_c, $$

  (59)

  where $\hat{\mathbf{x}}_{c}$ are the confined coordinates of the first particle (it was simply $\hat{y}$ in the (NC2) [cf. (55)]. The function $\mu_{1}(\mathbf{h},\hat{\mathbf{x}}_{c})$ is the integral of $\hat{x}^{2}$ over the contact surface between two particles when the first one has coordinates $(\hat{\mathbf{x}}_{e}, \hat{\mathbf{x}}_{c})$. Without loss of generality, we can set $\hat{\mathbf{x}}_{e} \equiv\boldsymbol{0}_{e}$. In the rescaled problem, this surface is a d-dimensional unit sphere centered at $(\boldsymbol{0}_{e}, \hat {\mathbf{x}}_{c})$, and (possibly) intersected with the confinement walls ∂ω _c :

  $$ \mu_1(\mathbf{h},\hat{\mathbf{x}}_c) = \int _{B(\hat {\mathbf{x}}_c) \cap\omega_c} x^2 \, \mathrm {d}S, $$

  (60)

  where $B(\hat{\mathbf{x}}_{c})$ is the unit ball, dS is the surface differential and x is the unconfined dimension of this surface. Once M ₁ is computed, we use it for the nonlinear coefficient of the effective Fokker–Planck equation. The generalization of the coefficient α _h in (10) is given by

  $$ \alpha_{\mathbf{h}} = \frac{1}{\varLambda} M_1({\mathbf{h}}). $$

  (61)