Appendix A: Evaluation of the Collisional Fluxes
The objective of this appendix is to evaluate the collisional contribution to the pressure tensor and heat flux. We will proceed using intuitive arguments, taking into account the collisions that contribute to the flux with their corresponding momentum or energy interchange.
Let us first analyze the pressure tensor case. Let us consider a surface element, \(\varDelta \mathbf {s}\), centered at \({\mathbf {r}}\) and two particles at contact in such a way that the line joining the two centers cross the surface (see Fig. 2). When the collision takes place, the variation of the momentum of particle 2 is
$$\begin{aligned} \varDelta p_{2,i}=m(\varvec{\hat{\sigma }}\cdot {\mathbf {v}}_{12})\hat{\sigma }_i. \end{aligned}$$
(65)
It is assumed that, in order to evaluate the flux, \(\varDelta p_{2,i}\) cross the surface through the intersection of the surface with the line joining the two particles. To calculate the collisional contribution to the flux, we have to consider all the possible collisions of this kind with its corresponding \(\varDelta p_{2,i}\). The surface divides the space in two regions; of course, the centers of the particles must be in different regions. We will consider that particle 2 is in the region pointed by \(\varDelta \mathbf {s}\), as in the Figure. The center of particle 1 can be parameterized by
$$\begin{aligned} {\mathbf {r}}_1(\lambda , \varvec{\hat{\sigma }})={\mathbf {r}}+\lambda \sigma \varvec{\hat{\sigma }}, \end{aligned}$$
(66)
with \(\lambda \in (0,1)\) and \(\varvec{\hat{\sigma }}\) a unitary vector of arbitrary orientation, but compatible with \(\varDelta \mathbf {s}\), i.e. \(\varvec{\hat{\sigma }}\cdot \varDelta \mathbf {s}<0\). In these conditions, particle 2 must be in a solid angle
$$\begin{aligned} \varDelta \hat{\sigma }_2=\frac{|\varvec{\hat{\sigma }}\cdot \varDelta \mathbf {s}|}{(\lambda \sigma )^{d-1}}, \end{aligned}$$
(67)
around
$$\begin{aligned} {\mathbf {r}}_2(\lambda , \varvec{\hat{\sigma }})={\mathbf {r}}-(1-\lambda )\sigma \varvec{\hat{\sigma }}. \end{aligned}$$
(68)
Note that we have used the same notation for the \(\varvec{\hat{\sigma }}\) of the collision in Eq. (65), and for the parameter to specify the position of particle 1 in Eq. (66). This can be done because its difference is of order \(\varDelta \hat{\sigma }_2\).
Let us consider that particle 1 is in the volume element \(\varDelta {\mathbf {r}}_1=(\lambda \sigma )^{d-1}\varDelta (\lambda \sigma )\varDelta \varvec{\hat{\sigma }}\) parameterized by \((\lambda ,\varvec{\hat{\sigma }})\). Hence, if particle 2 collides with particle 1 in the time interval \(\varDelta t\) with \(\varvec{\hat{\sigma }}\), it is in the volume element \(\varDelta {\mathbf {r}}_2=\sigma ^{d-1}\varDelta \hat{\sigma }_2 |{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }}|\varDelta t\). Then, the total number of collisions that contribute to the flux for given \({\mathbf {v}}_1\) and \({\mathbf {v}}_2\) is
$$\begin{aligned}&\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})f_2[{\mathbf {r}}_1(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_1, {\mathbf {r}}_2(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_2]\varDelta {\mathbf {r}}_1\varDelta {\mathbf {v}}_1 \varDelta {\mathbf {r}}_2\varDelta {\mathbf {v}}_2\nonumber \\&=\sigma ^d\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }}) f_2[{\mathbf {r}}_1(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_1, {\mathbf {r}}_2(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_2]|\varvec{\hat{\sigma }}\cdot \varDelta \mathbf {s}| |{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }}|\varDelta {\mathbf {v}}_1\varDelta {\mathbf {v}}_2\varDelta \varvec{\hat{\sigma }}\varDelta \lambda \varDelta t. \end{aligned}$$
(69)
Let us take \(\varDelta \mathbf {s}=\varDelta s\mathbf {e}_j\) where \(\mathbf {e}_j\) is a unit vector in the direction of one of our coordinate axes. The amount of momentum that travels through the surface in the direction of \(\varDelta \mathbf {s}\) per unit time and area due to collisions of particles with velocities \({\mathbf {v}}_1\) and \({\mathbf {v}}_2\) is then
$$\begin{aligned} \varDelta P_{ij}^{(c)}=m\sigma ^d \theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }}) f_2[{\mathbf {r}}_1(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_1, {\mathbf {r}}_2(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_2]\hat{\sigma }_i\hat{\sigma }_j ({\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})^2\varDelta {\mathbf {v}}_1\varDelta {\mathbf {v}}_2\varDelta \varvec{\hat{\sigma }}\varDelta \lambda , \end{aligned}$$
(70)
where we have taken into account that \(|{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }}|=-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }}\) and \(|\hat{\sigma }_j|=-\hat{\sigma }_j\). The net collisional pressure tensor is obtained integrating in Eq. (70) for all the allowed collisions.
Far from the boundary, when there are not geometrical constraints, the result is
$$\begin{aligned} P_{ij}^{(c)}=m\sigma ^d\int d{\mathbf {v}}_1\int d{\mathbf {v}}_2\int _0^1d\lambda \int _{\hat{\sigma }_j<0} d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }}) f_2[{\mathbf {r}}_1(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_1, {\mathbf {r}}_2(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_2]\hat{\sigma }_i\hat{\sigma }_j ({\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})^2. \end{aligned}$$
(71)
The integration can also be done summing for \(\hat{\sigma }_j>0\) but, then, the amount of momentum that crosses the surface is \(\varDelta p_{1,i}=-\varDelta p_{2,i}\), so that
$$\begin{aligned} P_{ij}^{(c)}=m\sigma ^d\int d{\mathbf {v}}_1\int d{\mathbf {v}}_2\int _0^1d\lambda \int _{\hat{\sigma }_j>0} d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }}) f_2[{\mathbf {r}}_1(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_1, {\mathbf {r}}_2(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_2]\hat{\sigma }_i\hat{\sigma }_j ({\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})^2, \end{aligned}$$
(72)
because, in this case, \(|\hat{\sigma }_j|=\hat{\sigma }_j\). Hence, we can re-write Eq. (71) as
$$\begin{aligned} P_{ij}^{(c)}=\frac{m}{2}\sigma ^d \int d{\mathbf {v}}_1\int d{\mathbf {v}}_2\int _0^1d\lambda \int d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }}) f_2[{\mathbf {r}}_1(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_1, {\mathbf {r}}_2(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_2]\hat{\sigma }_i\hat{\sigma }_j ({\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})^2, \end{aligned}$$
(73)
that coincides with the expression derived in [24] for \(d=3\) when the factorization for \(f_2\) given by Eq. (16) is used.
If there are geometrical constraints, we proceed similarly. Integrating in Eq. (70) for the allowed collisions, it is obtained
$$\begin{aligned} P_{ij}^{(c)}=m\sigma ^d\int d{\mathbf {v}}_1\int d{\mathbf {v}}_2\int \int _{\Sigma ^-} d\lambda d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }}) f_2[{\mathbf {r}}_1(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_1, {\mathbf {r}}_2(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_2]\hat{\sigma }_i\hat{\sigma }_j ({\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})^2, \end{aligned}$$
(74)
where the region of integration in \((\lambda , \varvec{\hat{\sigma }})\) is
$$ \begin{aligned} \Sigma ^-=\{(\lambda ,\varvec{\hat{\sigma }})|\varvec{\hat{\sigma }}\in \varOmega _d\quad \text {with}\quad \hat{\sigma }_j<0 \quad \& \quad 0\le \lambda \le 1 \quad \& \quad {\mathbf {r}}_1(\lambda ,\varvec{\hat{\sigma }})\in V \quad \& \quad {\mathbf {r}}_2(\lambda ,\varvec{\hat{\sigma }})\in V \}. \end{aligned}$$
(75)
\(P_{ij}^{(c)}\) can also be calculated summing for \(\hat{\sigma }_j>0\) but, then, the amount of momentum that crosses the surface is \(\varDelta p_{1,i}=-\varDelta p_{2,i}\), so that
$$\begin{aligned} P_{ij}^{(c)}=m\sigma ^d\int d{\mathbf {v}}_1\int d{\mathbf {v}}_2\int \int _{\Sigma ^+} d\lambda d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }}) f_2[{\mathbf {r}}_1(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_1, {\mathbf {r}}_2(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_2]\hat{\sigma }_i\hat{\sigma }_j ({\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})^2, \end{aligned}$$
(76)
where the region of integration in \((\lambda , \varvec{\hat{\sigma }})\) is
$$ \begin{aligned} \Sigma ^+=\{(\lambda ,\varvec{\hat{\sigma }})|\varvec{\hat{\sigma }}\in \varOmega _d\quad \text {with}\quad \hat{\sigma }_j>0 \quad \& \quad 0\le \lambda \le 1 \quad \& \quad {\mathbf {r}}_1(\lambda ,\varvec{\hat{\sigma }})\in V \quad \& \quad {\mathbf {r}}_2(\lambda ,\varvec{\hat{\sigma }})\in V \}. \end{aligned}$$
(77)
Hence, we can re-write Eq. (74) as
$$\begin{aligned} P_{ij}^{(c)}=\frac{m}{2} \sigma ^d\int d{\mathbf {v}}_1\int d{\mathbf {v}}_2\int \int _{\Sigma } d\lambda d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }}) f_2[{\mathbf {r}}_1(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_1, {\mathbf {r}}_2(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_2]\hat{\sigma }_i\hat{\sigma }_j ({\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})^2, \end{aligned}$$
(78)
where the region of integration in \((\lambda , \varvec{\hat{\sigma }})\) is
$$ \begin{aligned} \Sigma =\{(\lambda ,\varvec{\hat{\sigma }})|\varvec{\hat{\sigma }}\in \varOmega _d\quad \& \quad 0\le \lambda \le 1 \quad \& \quad {\mathbf {r}}_1(\lambda ,\varvec{\hat{\sigma }})\in V \quad \& \quad {\mathbf {r}}_2(\lambda ,\varvec{\hat{\sigma }})\in V \}. \end{aligned}$$
(79)
To calculate the collisional contribution to the energy flux, \(J_{E,j}^{(c)}\), the analysis is similar, but taking into account that, when the collision takes place, the variation of the energy of particle 2 is
$$\begin{aligned} \varDelta e_{2,i}=\frac{m}{2}(\varvec{\hat{\sigma }}\cdot {\mathbf {v}}_{12})^2 +m(\varvec{\hat{\sigma }}\cdot {\mathbf {v}}_{12})(\varvec{\hat{\sigma }}\cdot {\mathbf {v}}_{2}). \end{aligned}$$
(80)
Once \(J_{E,j}^{(c)}\) is calculated, the heat flux is expressed as \(q_{j}^{(c)}=J_{E,j}^{(c)}-\sum _{i}u_iP_{ij}^{(c)}\).
Appendix B: Evaluation of the Divergence of the Collisional Fluxes
As in the previous Appendix, we focus on the pressure tensor because the heat flux case is similar. Let us first re-write the collisional pressure tensor given by Eq. (36) in the form
$$\begin{aligned}&P_{ij}^{(c)}({\mathbf {r}},t)=\frac{m}{2}\sigma ^d\int d{\mathbf {v}}_1\int d{\mathbf {v}}_2\int d\varvec{\hat{\sigma }}\nonumber \\&\quad \int _{\lambda _1({\mathbf {r}},\varvec{\hat{\sigma }})}^{\lambda _2({\mathbf {r}},\varvec{\hat{\sigma }})} d\lambda \theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})f_2[{\mathbf {r}}_1(\lambda ,\varvec{\hat{\sigma }}), {\mathbf {v}}_1,{\mathbf {r}}_2(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_2,t] ({\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})^2\hat{\sigma _i}\hat{\sigma _j}, \end{aligned}$$
(81)
where \(\lambda _1({\mathbf {r}},\varvec{\hat{\sigma }})\) and \(\lambda _2({\mathbf {r}},\varvec{\hat{\sigma }})\) are such that \(\sigma \lambda _1({\mathbf {r}},\varvec{\hat{\sigma }})\) and \(\sigma \lambda _2({\mathbf {r}},\varvec{\hat{\sigma }})\) are the minimum and maximum distance from \({\mathbf {r}}\) to \({\mathbf {r}}_1(\lambda ,\varvec{\hat{\sigma }})\) respectively, for a given orientation, \(\varvec{\hat{\sigma }}\). In the bulk of the system, we trivially have \(\lambda _1({\mathbf {r}},\varvec{\hat{\sigma }})=0\) and \(\lambda _2({\mathbf {r}},\varvec{\hat{\sigma }})=1\), for all \(\varvec{\hat{\sigma }}\), but closed to the boundary these functions depend on the geometry of it.
Taking into account Eq. (81), the divergence of \(P_{ij}^{(c)}\) can be expressed as
$$\begin{aligned}&\frac{\partial }{\partial {\mathbf {r}}}\cdot P_{ij}^{(c)}({\mathbf {r}})\nonumber \\&\quad =\frac{m}{2}\sigma ^d\int d{\mathbf {v}}_1\int d{\mathbf {v}}_2\int d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})({\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})^2\varvec{\hat{\sigma }}\varvec{\hat{\sigma }}\cdot \frac{\partial }{\partial {\mathbf {r}}}\lambda _2 f_2[{\mathbf {r}}_1(\lambda _2,\varvec{\hat{\sigma }}), {\mathbf {v}}_1,{\mathbf {r}}_2(\lambda _2,\varvec{\hat{\sigma }}),{\mathbf {v}}_2] \nonumber \\&\qquad -\frac{m}{2}\sigma ^d\int d{\mathbf {v}}_1\int d{\mathbf {v}}_2\int d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})({\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})^2\varvec{\hat{\sigma }}\varvec{\hat{\sigma }}\cdot \frac{\partial }{\partial {\mathbf {r}}}\lambda _1 f_2[{\mathbf {r}}_1(\lambda _1,\varvec{\hat{\sigma }}), {\mathbf {v}}_1,{\mathbf {r}}_2(\lambda _1,\varvec{\hat{\sigma }}),{\mathbf {v}}_2]\nonumber \\&\qquad +\frac{m}{2}\sigma ^d\int d{\mathbf {v}}_1\int d{\mathbf {v}}_2\int d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})({\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})^2\varvec{\hat{\sigma }}\nonumber \\&\int _{\lambda _1({\mathbf {r}},\varvec{\hat{\sigma }})}^{\lambda _2({\mathbf {r}},\varvec{\hat{\sigma }})} d\lambda \varvec{\hat{\sigma }}\cdot \frac{\partial }{\partial {\mathbf {r}}} f_2[{\mathbf {r}}_1(\lambda ,\varvec{\hat{\sigma }}), {\mathbf {v}}_1,{\mathbf {r}}_2(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_2]. \end{aligned}$$
(82)
Taking into account that
$$\begin{aligned} \frac{\partial }{\partial \lambda }f_2[{\mathbf {r}}_1(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_1, {\mathbf {r}}_2(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_2]={\varvec{\sigma }}\cdot \frac{\partial }{\partial {\mathbf {r}}} f_2[{\mathbf {r}}_1(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_1, {\mathbf {r}}_2(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_2], \end{aligned}$$
(83)
the last term of the r.h.s. of Eq. (82) can be written as
$$\begin{aligned}&\frac{m}{2}\sigma ^d\int d{\mathbf {v}}_1\int d{\mathbf {v}}_2\int d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})({\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})^2\varvec{\hat{\sigma }}\int _{\lambda _1({\mathbf {r}},\varvec{\hat{\sigma }})}^{\lambda _2({\mathbf {r}},\varvec{\hat{\sigma }})} d\lambda \varvec{\hat{\sigma }}\nonumber \\&\qquad \cdot \frac{\partial }{\partial {\mathbf {r}}} f_2[{\mathbf {r}}_1(\lambda ,\varvec{\hat{\sigma }}), {\mathbf {v}}_1,{\mathbf {r}}_2(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_2]\nonumber \\&\quad =\frac{m}{2}\sigma ^{d-1}\int d{\mathbf {v}}_1\int d{\mathbf {v}}_2\int d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})({\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})^2\varvec{\hat{\sigma }}f_2[{\mathbf {r}}_1(\lambda _2,\varvec{\hat{\sigma }}), {\mathbf {v}}_1,{\mathbf {r}}_2(\lambda _2,\varvec{\hat{\sigma }}),{\mathbf {v}}_2]\nonumber \\&\qquad -\frac{m}{2}\sigma ^{d-1}\int d{\mathbf {v}}_1\int d{\mathbf {v}}_2\int d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})({\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})^2\varvec{\hat{\sigma }}f_2[{\mathbf {r}}_1(\lambda _1,\varvec{\hat{\sigma }}), {\mathbf {v}}_1,{\mathbf {r}}_2(\lambda _1,\varvec{\hat{\sigma }}),{\mathbf {v}}_2].\nonumber \\ \end{aligned}$$
(84)
Changing variables,
$$\begin{aligned} {\mathbf {v}}_1&\leftrightarrow {\mathbf {v}}_2, \end{aligned}$$
(85)
$$\begin{aligned} \varvec{\hat{\sigma }}&\rightarrow -\varvec{\hat{\sigma }}, \end{aligned}$$
(86)
in the second term of the r.h.s., it is obtained
$$\begin{aligned}&\frac{m}{2}\sigma ^{d-1}\int d{\mathbf {v}}_1\int d{\mathbf {v}}_2\int d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})({\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})^2\varvec{\hat{\sigma }}f_2[{\mathbf {r}}_1(\lambda _1,\varvec{\hat{\sigma }}), {\mathbf {v}}_1,{\mathbf {r}}_2(\lambda _1,\varvec{\hat{\sigma }}),{\mathbf {v}}_2] \nonumber \\&\quad =-\frac{m}{2}\sigma ^{d-1}\int d{\mathbf {v}}_1\int d{\mathbf {v}}_2\int d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})({\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})^2\varvec{\hat{\sigma }}f_2[{\mathbf {r}}_1(\lambda _2,\varvec{\hat{\sigma }}), {\mathbf {v}}_1,{\mathbf {r}}_2(\lambda _2,\varvec{\hat{\sigma }}),{\mathbf {v}}_2],\nonumber \\ \end{aligned}$$
(87)
where it has taken into account that
$$\begin{aligned} {\mathbf {r}}_1[\lambda _1({\mathbf {r}},-\varvec{\hat{\sigma }}),-\varvec{\hat{\sigma }}]= & {} {\mathbf {r}}_2[\lambda _2({\mathbf {r}},\varvec{\hat{\sigma }}),\varvec{\hat{\sigma }}], \end{aligned}$$
(88)
$$\begin{aligned} {\mathbf {r}}_2[\lambda _1({\mathbf {r}},-\varvec{\hat{\sigma }}),-\varvec{\hat{\sigma }}]= & {} {\mathbf {r}}_1[\lambda _2({\mathbf {r}},\varvec{\hat{\sigma }}),\varvec{\hat{\sigma }}]. \end{aligned}$$
(89)
So, we have
$$\begin{aligned}&\frac{m}{2}\sigma ^d\int d{\mathbf {v}}_1\int d{\mathbf {v}}_2\int d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})({\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})^2\varvec{\hat{\sigma }}\int _{\lambda _1({\mathbf {r}},\varvec{\hat{\sigma }})}^{\lambda _2({\mathbf {r}},\varvec{\hat{\sigma }})} d\lambda \varvec{\hat{\sigma }}\nonumber \\&\qquad \cdot \frac{\partial }{\partial {\mathbf {r}}} f_2[{\mathbf {r}}_1(\lambda ,\varvec{\hat{\sigma }}), {\mathbf {v}}_1,{\mathbf {r}}_2(\lambda ,\varvec{\hat{\sigma }}),{\mathbf {v}}_2]\nonumber \\&\quad =m\sigma ^{d-1}\int d{\mathbf {v}}_1\int d{\mathbf {v}}_2\int d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})({\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})^2\varvec{\hat{\sigma }}f_2[{\mathbf {r}}_1(\lambda _2,\varvec{\hat{\sigma }}), {\mathbf {v}}_1,{\mathbf {r}}_2(\lambda _2,\varvec{\hat{\sigma }}),{\mathbf {v}}_2].\nonumber \\ \end{aligned}$$
(90)
Performing the same change of variables in the second term of the r.h.s. of Eq. (82), it is obtained
$$\begin{aligned}&\frac{m}{2}\sigma ^d\int d{\mathbf {v}}_1\int d{\mathbf {v}}_2\int d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})({\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})^2\varvec{\hat{\sigma }}\varvec{\hat{\sigma }}\cdot \frac{\partial }{\partial {\mathbf {r}}}\lambda _1 f_2[{\mathbf {r}}_1(\lambda _1,\varvec{\hat{\sigma }}), {\mathbf {v}}_1,{\mathbf {r}}_2(\lambda _1,\varvec{\hat{\sigma }}),{\mathbf {v}}_2]\nonumber \\&\quad =-\frac{m}{2}\sigma ^d\int d{\mathbf {v}}_1\int d{\mathbf {v}}_2\int d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})({\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})^2\varvec{\hat{\sigma }}\varvec{\hat{\sigma }}\nonumber \\&\qquad \cdot \frac{\partial }{\partial {\mathbf {r}}}\lambda _2 f_2[{\mathbf {r}}_1(\lambda _2,\varvec{\hat{\sigma }}), {\mathbf {v}}_1,{\mathbf {r}}_2(\lambda _2,\varvec{\hat{\sigma }}),{\mathbf {v}}_2]. \end{aligned}$$
(91)
By substituting Eqs. (90) and (91) into Eq. (82), it is obtained
$$\begin{aligned}&\frac{\partial }{\partial {\mathbf {r}}}\cdot P_{ij}^{(c)}({\mathbf {r}})=m\sigma ^{d-1}\int d{\mathbf {v}}_1\int d{\mathbf {v}}_2\int d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})\nonumber \\&\quad ({\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})^2\varvec{\hat{\sigma }}\left[ 1+{\varvec{\sigma }}\cdot \frac{\partial }{\partial {\mathbf {r}}} \lambda _2\right] f_2[{\mathbf {r}}_1(\lambda _2,\varvec{\hat{\sigma }}), {\mathbf {v}}_1,{\mathbf {r}}_2(\lambda _2,\varvec{\hat{\sigma }}),{\mathbf {v}}_2]. \nonumber \\ \end{aligned}$$
(92)
Now, let us analyze the function \(1+{\varvec{\sigma }}\cdot \frac{\partial }{\partial {\mathbf {r}}}\lambda _2\). Let us first consider the simplest case of a plane located at \(z=-\sigma /2\). If \(\varvec{\hat{\sigma }}\in \varOmega ({\mathbf {r}})\), then \(\lambda _2({\mathbf {r}},\varvec{\hat{\sigma }})=1\). Let us define \(\varOmega ^+({\mathbf {r}})\), such that
$$\begin{aligned} \varOmega ({\mathbf {r}})\cup \varOmega ^+({\mathbf {r}}) =\varOmega _d. \end{aligned}$$
(93)
For a given \({\mathbf {r}}\), it is
$$\begin{aligned} z=-\lambda _2({\mathbf {r}},\varvec{\hat{\sigma }})\sigma \hat{\sigma }_z, \quad \text {for}\quad \varvec{\hat{\sigma }}\in \varOmega ^+({\mathbf {r}}), \end{aligned}$$
(94)
so that, for this simple case, we have
$$\begin{aligned} 1+{\varvec{\sigma }}\cdot \frac{\partial }{\partial {\mathbf {r}}} \lambda _2({\mathbf {r}},\varvec{\hat{\sigma }}) =\left\{ \begin{array}{l} 1\quad \text {if }\varvec{\hat{\sigma }}\in \varOmega ({\mathbf {r}})\\ 0\quad \text {if }\varvec{\hat{\sigma }}\in \varOmega ^+({\mathbf {r}}) \end{array} \right. \end{aligned}$$
(95)
In fact, if the plane has a different orientation, the result is the same because the function is a scalar. Moreover, in the general case of an arbitrary wall, the result also holds if the tangent plane is defined at \({\mathbf {r}}+\lambda _2\sigma \varvec{\hat{\sigma }}\).
Hence, by substituting Eq. (95) into Eq. (92), it is finally obtained
$$\begin{aligned}&\frac{\partial }{\partial {\mathbf {r}}}\cdot P_{ij}^{(c)}({\mathbf {r}})\nonumber \\ \nonumber \\&\quad =m\sigma ^{d-1}\int d{\mathbf {v}}_1\int d{\mathbf {v}}_2\int _{\varOmega ({\mathbf {r}})} d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})({\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})^2\varvec{\hat{\sigma }}f_2({\mathbf {r}}+{\varvec{\sigma }}, {\mathbf {v}}_1,{\mathbf {r}},{\mathbf {v}}_2), \end{aligned}$$
(96)
where it has been used that \(\lambda _2({\mathbf {r}},\varvec{\hat{\sigma }})=1\) if \(\varvec{\hat{\sigma }}\in \varOmega ({\mathbf {r}})\).
It still remains to show that \(\frac{\partial }{\partial {\mathbf {r}}}\cdot P_{ij}^{(c)}\) coincides with \(\int d{\mathbf {v}}m{\mathbf {v}}J[f_2]\). By standard manipulations, it can be shown that
$$\begin{aligned}&\int d{\mathbf {v}}\psi ({\mathbf {v}})J[f_2]\nonumber \\&\quad =\sigma ^{d-1}\int d{\mathbf {v}}_1\int d{\mathbf {v}}_2\int _{\varOmega ({\mathbf {r}})} d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }}) |{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }}| f_2({\mathbf {r}}+{\varvec{\sigma }}, {\mathbf {v}}_1,{\mathbf {r}},{\mathbf {v}}_2) (b_{\varvec{\hat{\sigma }}}-1)\psi ({\mathbf {v}}_2).\nonumber \\ \end{aligned}$$
(97)
Taking \(\psi ({\mathbf {v}})={\mathbf {v}}_i\), it is
$$\begin{aligned} \int d{\mathbf {v}}{\mathbf {v}}J[f_2]= -\sigma ^{d-1}\int d{\mathbf {v}}_1\int d{\mathbf {v}}_2\int _{\varOmega ({\mathbf {r}})} d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})({\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }})^2\varvec{\hat{\sigma }}f_2({\mathbf {r}}+{\varvec{\sigma }}, {\mathbf {v}}_1,{\mathbf {r}},{\mathbf {v}}_2). \end{aligned}$$
(98)
Comparing Eq. (98) with Eq. (96), we finally have
$$\begin{aligned} \int d{\mathbf {v}}m{\mathbf {v}}J[f_2]= -\frac{\partial }{\partial {\mathbf {r}}}\cdot P^{(c)}, \end{aligned}$$
(99)
as we wanted to prove.
Appendix C: Evaluation of the Time Derivative of \({\mathcal {H}}\)
Let us first calculate \(\frac{d{\mathcal {H}}^{(k)}}{dt}\). Using standard manipulations and applying the boundary conditions, it is obtained
$$\begin{aligned} \frac{d{\mathcal {H}}^{(k)}}{dt}=\int d{\mathbf {r}}\int d{\mathbf {v}}J_E[f|f] \ln f({\mathbf {r}},{\mathbf {v}},t). \end{aligned}$$
(100)
Eq. (97) reduces in the Enskog case to
$$\begin{aligned} \int d{\mathbf {v}}\psi ({\mathbf {v}})J_E[f|f]= & {} \sigma ^{d-1}\int d{\mathbf {v}}_1\int d{\mathbf {v}}_2 \int _{\varOmega ({\mathbf {r}})} d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }}) |{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }}| g_2({\mathbf {r}}+{\varvec{\sigma }},{\mathbf {r}}|n)\nonumber \\&f({\mathbf {r}}+{\varvec{\sigma }},{\mathbf {v}}_1,t) f({\mathbf {r}},{\mathbf {v}}_2,t) (b_{\varvec{\hat{\sigma }}}-1)\psi ({\mathbf {v}}_2). \end{aligned}$$
(101)
By taking \(\psi =\ln f\), we have
$$\begin{aligned} \frac{d{\mathcal {H}}^{(k)}}{dt}= & {} \sigma ^{d-1}\int d{\mathbf {r}}\int d{\mathbf {v}}_1 \int d{\mathbf {v}}_2\int _{\varOmega ({\mathbf {r}})}d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }}) |{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }}| g_2({\mathbf {r}}+{\varvec{\sigma }},{\mathbf {r}}|n)\nonumber \\&f({\mathbf {r}}+{\varvec{\sigma }},{\mathbf {v}}_1,t) f({\mathbf {r}},{\mathbf {v}}_2,t) \ln \frac{f({\mathbf {r}},{\mathbf {v}}_2',t)}{f({\mathbf {r}},{\mathbf {v}}_2,t)}. \end{aligned}$$
(102)
Changing variables,
$$\begin{aligned} {\mathbf {v}}_1\leftrightarrow&{\mathbf {v}}_2, \end{aligned}$$
(103)
$$\begin{aligned} \varvec{\hat{\sigma }}\rightarrow&-\varvec{\hat{\sigma }}, \end{aligned}$$
(104)
Eq. (102) is transformed into
$$\begin{aligned} \frac{d{\mathcal {H}}^{(k)}}{dt}= & {} \sigma ^{d-1}\int d{\mathbf {r}}\int d{\mathbf {v}}_1 \int d{\mathbf {v}}_2\int _{\widetilde{\varOmega }({\mathbf {r}})}d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }}) |{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }}| g_2({\mathbf {r}}-{\varvec{\sigma }},{\mathbf {r}}|n)\nonumber \\&f({\mathbf {r}}-{\varvec{\sigma }},{\mathbf {v}}_2,t) f({\mathbf {r}},{\mathbf {v}}_1,t) \ln \frac{f({\mathbf {r}},{\mathbf {v}}_1',t)}{f({\mathbf {r}},{\mathbf {v}}_1,t)}, \end{aligned}$$
(105)
where, now, the angular integration is taken over the new region, \(\widetilde{\varOmega }({\mathbf {r}})\), defined in such a way that \(\varvec{\hat{\sigma }}\in \widetilde{\varOmega }({\mathbf {r}})\) if and only if \({\mathbf {r}}-{\varvec{\sigma }}\in V\). Finally, by changing the space variable, \({\mathbf {r}}\rightarrow {\mathbf {r}}+{\varvec{\sigma }}\), it is obtained
$$\begin{aligned} \frac{d{\mathcal {H}}^{(k)}}{dt}= & {} \sigma ^{d-1}\int d{\mathbf {r}}\int d{\mathbf {v}}_1 \int d{\mathbf {v}}_2\int _{\varOmega ({\mathbf {r}})}d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }}) |{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }}| g_2({\mathbf {r}}+{\varvec{\sigma }},{\mathbf {r}}|n)\nonumber \\&f({\mathbf {r}}+{\varvec{\sigma }},{\mathbf {v}}_1,t) f({\mathbf {r}},{\mathbf {v}}_2,t) \ln \frac{f({\mathbf {r}}+{\varvec{\sigma }},{\mathbf {v}}_1',t)}{f({\mathbf {r}}+{\varvec{\sigma }},{\mathbf {v}}_1,t)}, \end{aligned}$$
(106)
where we have taken into account that \(g_2({\mathbf {r}}+{\varvec{\sigma }},{\mathbf {r}}|n) =g_2({\mathbf {r}},{\mathbf {r}}+{\varvec{\sigma }}|n)\). Taking into account Eq. (102) and (106), we have
$$\begin{aligned} \frac{d{\mathcal {H}}^{(k)}}{dt}= & {} \frac{\sigma ^{d-1}}{2} \int d{\mathbf {r}}\int d{\mathbf {v}}_1 \int d{\mathbf {v}}_2\int _{\varOmega ({\mathbf {r}})}d\varvec{\hat{\sigma }}\theta (-{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }}) |{\mathbf {v}}_{12}\cdot \varvec{\hat{\sigma }}| g_2({\mathbf {r}}+{\varvec{\sigma }},{\mathbf {r}}|n)\nonumber \\&f({\mathbf {r}}+{\varvec{\sigma }},{\mathbf {v}}_1,t) f({\mathbf {r}},{\mathbf {v}}_2,t) \ln \frac{f({\mathbf {r}}+{\varvec{\sigma }},{\mathbf {v}}_1',t) f({\mathbf {r}},{\mathbf {v}}_2',t)}{f({\mathbf {r}}+{\varvec{\sigma }},{\mathbf {v}}_1,t) f({\mathbf {r}},{\mathbf {v}}_2,t)}, \end{aligned}$$
(107)
that is the expression of the main text.
Now let us calculate \(\frac{d{\mathcal {H}}^{(c)}}{dt}\). The first contribution is
$$\begin{aligned} \frac{d}{dt}\ln \phi (t)= & {} \frac{1}{\phi (t)}\frac{d}{dt}\int d{\mathbf {r}}_1 w({\mathbf {r}}_1,t)\dots \int d{\mathbf {r}}_Nw({\mathbf {r}}_N,t) \varTheta ({\mathbf {r}}_1,\ldots ,{\mathbf {r}}_N)\nonumber \\= & {} \int d{\mathbf {r}}\frac{n({\mathbf {r}},t)}{w({\mathbf {r}},t)} \frac{\partial }{\partial t}w({\mathbf {r}},t), \end{aligned}$$
(108)
where Eq. (14) has been used. The second contribution is
$$\begin{aligned} \frac{d}{dt}\int d{\mathbf {r}}n({\mathbf {r}},t) \ln \frac{n({\mathbf {r}},t)}{w({\mathbf {r}},t)} =\int d{\mathbf {r}}\left[ \ln \frac{n({\mathbf {r}},t)}{w({\mathbf {r}},t)} \frac{\partial }{\partial t}n({\mathbf {r}},t) -\frac{n({\mathbf {r}},t)}{w({\mathbf {r}},t)} \frac{\partial }{\partial t}w({\mathbf {r}},t)\right] , \end{aligned}$$
(109)
so that
$$\begin{aligned} \frac{d{\mathcal {H}}^{(c)}}{dt}=-\int d{\mathbf {r}} \frac{\partial }{\partial t}n({\mathbf {r}},t) \ln \frac{n({\mathbf {r}},t)}{w({\mathbf {r}},t)}=\int d{\mathbf {r}} \ln \frac{n({\mathbf {r}},t)}{w({\mathbf {r}},t)}\frac{\partial }{\partial {\mathbf {r}}} \cdot [n({\mathbf {r}},t){\mathbf {u}}({\mathbf {r}},t)], \end{aligned}$$
(110)
where the continuity equation, Eq. (31), has been used. As
$$\begin{aligned} \int d{\mathbf {r}} \frac{\partial }{\partial {\mathbf {r}}}\cdot \left[ n({\mathbf {r}},t){\mathbf {u}}({\mathbf {r}},t) \ln \frac{n({\mathbf {r}},t)}{w({\mathbf {r}},t)}\right] = \int _{\partial V} d\mathbf {s}\cdot {\mathbf {u}}({\mathbf {r}},t)n({\mathbf {r}},t) \ln \frac{n({\mathbf {r}},t)}{w({\mathbf {r}},t)}=0, \end{aligned}$$
(111)
because \({\mathbf {u}}({\mathbf {r}},t)\cdot {\mathbf {N}}({\mathbf {r}})=0\) for all \({\mathbf {r}}\in \partial V\), Eq. (110) can be written in the form
$$\begin{aligned} \frac{d{\mathcal {H}}^{(c)}}{dt}=-\int d{\mathbf {r}} n({\mathbf {r}},t){\mathbf {u}}({\mathbf {r}},t)\cdot \frac{\partial }{\partial {\mathbf {r}}}\ln \frac{n({\mathbf {r}},t)}{w({\mathbf {r}},t)}. \end{aligned}$$
(112)
Now, using the property
$$\begin{aligned} \frac{\partial }{\partial {\mathbf {r}}_1} \theta (|{\mathbf {r}}_1-{\mathbf {r}}_2|-\sigma )= \frac{({\mathbf {r}}_1-{\mathbf {r}}_2)}{\sigma } \delta (|{\mathbf {r}}_1-{\mathbf {r}}_2|-\sigma ), \end{aligned}$$
(113)
we get, from the expressions of n and \(n_2\), Eqs. (14) and (18) respectively
$$\begin{aligned} \frac{\partial }{\partial {\mathbf {r}}_1} \left[ \frac{n({\mathbf {r}}_1,t)}{w({\mathbf {r}}_1,t)}\right] = \frac{1}{w({\mathbf {r}}_1,t)}\int d{\mathbf {r}}_2 \frac{({\mathbf {r}}_1-{\mathbf {r}}_2)}{\sigma } \delta (|{\mathbf {r}}_1-{\mathbf {r}}_2|-\sigma ) n_2({\mathbf {r}}_1,{\mathbf {r}}_2,t). \end{aligned}$$
(114)
Performing the pertinent integration to eliminate the delta function, we have
$$\begin{aligned} \frac{\partial }{\partial {\mathbf {r}}} \left[ \frac{n({\mathbf {r}},t)}{w({\mathbf {r}},t)}\right] =-\sigma ^{d-1} \frac{n({\mathbf {r}},t)}{w({\mathbf {r}},t)}\int _{\varOmega ({\mathbf {r}})}d\varvec{\hat{\sigma }}g_2({\mathbf {r}}+{\varvec{\sigma }},{\mathbf {r}}|n) n({\mathbf {r}}+{\varvec{\sigma }},t)\varvec{\hat{\sigma }}. \end{aligned}$$
(115)
This equation is the generalization of Eq. (25) for a generic w in our non equilibrium ensemble given by Eq. (9). By substituting Eq. (115) into (112), we finally obtain
$$\begin{aligned} \frac{d{\mathcal {H}}^{(c)}}{dt}= & {} \sigma ^{d-1}\int d{\mathbf {r}}\int _{\varOmega ({\mathbf {r}})} d\varvec{\hat{\sigma }}n({\mathbf {r}},t) n({\mathbf {r}}+{\varvec{\sigma }},t)\varvec{\hat{\sigma }}\cdot {\mathbf {u}}({\mathbf {r}},t)g_2({\mathbf {r}}+{\varvec{\sigma }},{\mathbf {r}}|n) \nonumber \\= & {} -\sigma ^{d-1}\int d{\mathbf {r}}\int _{\varOmega ({\mathbf {r}})} d\varvec{\hat{\sigma }}n({\mathbf {r}},t) n({\mathbf {r}}+{\varvec{\sigma }},t)\varvec{\hat{\sigma }}\cdot {\mathbf {u}}({\mathbf {r}}+{\varvec{\sigma }},t) g_2({\mathbf {r}}+{\varvec{\sigma }},{\mathbf {r}}|n),\nonumber \\ \end{aligned}$$
(116)
where, in the last step, we have changed \(\varvec{\hat{\sigma }}\rightarrow -\varvec{\hat{\sigma }}\) and \({\mathbf {r}}\rightarrow {\mathbf {r}}+{\varvec{\sigma }}\).
