Continuum mechanics from molecular dynamics via adiabatic time and length scale separation

Abstract

We show how to construct, by exploiting adiabatic time and length scale separation between atomistic and continuum mechanics, a multiscale scheme for continuum dynamics free from macroscopic constitutive modeling. To do so, we introduce a new set of degrees of freedom that simultaneously represent the macroscopic and the microscopic dynamics, based on a space tessellation. In this new formulation, the dynamics of the macroscopic fields steers the microscopic particle dynamics by producing the conditions under which they evolve and, concurrently, the particle dynamics drives the evolution of the macroscopic fields by providing them with atomistically based constitutive information. Under conditions of adiabatic separation both in time and length scales, it is possible to decimate the tessellating cells and let the macroscopic equations of motion be driven by the time average of the terms coupling them with the microscopic degrees of freedom.

Appendices

Appendix A List of selected notations

Pointers between square brackets refer to where each symbol is defined or first used.

$\approx$:

is of the same order of magnitude of   [note 1]

$\gtrapprox$:

is at least of the order of magnitude of   [note 1]

$\lessapprox$:

is at most of the order of magnitude of   [note 1]

$\cong$:

equal to within a preset mesoscopic tolerance   [(2.27)]

$\equiv$:

equivalence of symbols   [note 6]

:

local streaming field at cell    [(2.13)]

$\kappa _i$:

locator index of the i-th particle   [(2.64)]

$\varvec{\lambda }^{\!k}$, $\varvec{\Lambda }^{\!k}$:

Lagrange multipliers affecting particles in the k-th cell   [(3.9)]

:

mass coefficient of the i-th particle in cell    [(2.8)]

$\mu _i^k$:

mass coefficient of the i-th particle in the k-th moving cell   [(2.79)]

:

volume-averaged mass of particles currently in    [(2.35)]

$\varrho$:

macroscopic mass density field   [(2.37)]

:

presence index of the i-th particle in cell    [(2.9)]

$\varphi _i^{k}$:

presence index of the i-th particle in the k-th moving cell   [(2.63)]

$\varvec{\chi }$:

macroscopic (marker) motion   [(2.41)]

:

moving mesoscopic cell initially centered at $\varvec{x}$   [(2.45)]

:

cell mass center ( $\cong$ cell centroid: cf. (2.39b))   [(2.11)]

:

mesoscopic space cell   [(2.5)]

${\textbf {C}}$:

change-of-metric tensor   [(2.46)]

:

central Euler tensor of cell    [(2.17)]

$\varvec{E}^{k}$:

central Euler tensor of the k-th moving cell   [(2.87)]

$\varvec{E}^{k}_{0}$:

scaled central Euler tensor of the k-th moving cell   [(2.88)]

$\textbf{f}_i^{h}$:

force applied to the i-th particle from within the h-th cell   [(3.12)]

$\textbf{f}^{h}$:

total internal force acting on the h-th moving cell   [(3.12)]

$\textbf{f}^{\ne h}$:

total external force acting on the h-th moving cell   [(3.11)]

${\textbf {F}}$:

macroscopic deformation gradient   [(2.42)]

$\varvec{G}^{h}_0$:

initial square gyration tensor of the h-th cell   [(4.6)]

:

central moment of momentum of cell    [(2.16)]

:

total kinetic energy of particles currently in    [(2.22)]

:

linear part of the local streaming field    [(2.13)]

$\textbf{L}$:

gradient of the macroscopic velocity field $\textbf{v}$   [(2.31)]

$L_{\mu }\,$:

microscopic length scale   [sec.2.1]

$L_{\mathsf m}$:

mesoscopic length scale   [(2.5)]

$L_{_{\mathsf M}}$:

macroscopic length scale   [sec. 4]

:

total mass of particles currently in    [(2.12)]

$\mathcal {M}^k$:

mass content of the k-th moving cell   [(2.83)]

$\varvec{M}^h$:

total internal moment acting on the h-th moving cell   [(3.15b)]

$\varvec{M}^{\ne h}$:

total external moment acting on the h-th moving cell   [(3.14)]

 :

central radius vector of the i-th particle currently in    [(2.15)]

:

bounded region of 3D Euclidean space   [sec.2.1]

$\textbf{s}_i$:

‘scaled’ radius vector of the i-th particle    [(2.53)]

$\textbf{S}$:

Piola stress (Piola transform of $\textbf{T}$)   [(4.4)]

$\textbf{t}$:

traction   [(D1)]

$\textbf{T}$:

Cauchy stress   [(D8)]

$T_{\!\mu }$:

  microscopic time scale   [sec.2.1]

$T_\textsf{m}$ :

mesoscopic time scale   [(4.16)]

$T_{_\textsf{M}}$ :

macroscopic time scale   [(4.16)]

$\tau \mapsto \mathbb T_{\tau }$:

moving tessellation   [(2.62)]

$\varvec{\check{\textbf{U}}}$:

Piola transform of the specific virial $\varvec{\check{\varvec{V}}}$    [(4.9)]

:

mass-averaged momentum of particles currently in    [(2.13)]

$\textbf{v}_{\tau }\!\equiv \textbf{v}(\cdot ,\tau )$:

global streaming field at time $\tau$   [(2.3)]

$\varvec{V}^h$:

total virial of the h-th moving cell   [(3.15a)]

$\varvec{\check{\varvec{V}}}$:

specific virial   [(4.7)]

$\varvec{w}_i$ :

global thermal velocity of the i-th particle   [(2.4)]

:

local thermal velocity of the i-th particle currently in    [(2.20)]

Appendix B Least-square minimization

Given a cell and a time $\tau$ (dropped from notations in this appendix), the differential of the convex quadratic function $\mathscr {Q}$ defined in (2.7) is¹⁹

$$\begin{aligned} \begin{aligned} d\mathscr {Q}=&\,2{\textstyle \sum \limits _i}\mu _i\big (\varvec{v}+\varvec{L}\textbf{r}_i-\dot{\varvec{q}}_i\big )\!\cdot \!\big (d\varvec{v}+\varvec{L}d\varvec{o}+(d\varvec{L})\textbf{r}_i\big )\\ =&\,2{\textstyle \sum \limits _i}\mu _i\big (\varvec{v}+\varvec{L}\textbf{r}_i-\dot{\varvec{q}}_i\big )\!\cdot \!d\varvec{v}\, \\&+\,2\varvec{L}^{\!\top }{\textstyle \sum \limits _i}\mu _i\big (\varvec{v}+\varvec{L}\textbf{r}_i-\dot{\varvec{q}}_i\big )\!\cdot \!d\varvec{o}\; \\&+\,2{\textstyle \sum \limits _i}\mu _i\textbf{r}_i\!\otimes \!\big (\varvec{v}+\varvec{L}\textbf{r}_i-\dot{\varvec{q}}_i\big )\!\cdot \!d\varvec{L}\,, \end{aligned} \end{aligned}$$

(B1)

where

$$\begin{aligned} \textbf{r}_i\,\,{:=}\,\,\varvec{q}_i-\varvec{o}. \end{aligned}$$

(B2)

is the radius vector of the i-th particle with respect to the origin $\varvec{o}$. Comparing the second and third lines of (B1) shows that the derivative of $\mathscr {Q}$ with respect to the origin $\varvec{o}$ depends linearly on its derivative with respect to $\varvec{v}$. Hence, the points of stationarity of $\mathscr {Q}$ are the $\infty ^3$ (counting scalar parameters) triplets $(\varvec{o},\varvec{v},\varvec{L})$ that solve the independent linear equations

$$\begin{aligned}&\big ({\textstyle \sum \limits _i}\mu _i\big )\varvec{v}+\varvec{L}\big ({\textstyle \sum \limits _i}\mu _i\textbf{r}_i\big )-{\textstyle \sum \limits _i}\mu _i\dot{\varvec{q}}_i=\varvec{0}, \end{aligned}$$

(B3)

$$\begin{aligned}&\big ({\textstyle \sum \limits _i}\mu _i\textbf{r}_i\big )\!\otimes \!\varvec{v}+\varvec{L}\big ({\textstyle \sum \limits _i}\mu _i\textbf{r}_i\!\otimes \!\textbf{r}_i\big )-{\textstyle \sum \limits _i}\mu _i\textbf{r}_i\!\otimes \!\dot{\varvec{q}}_i=\varvec{0}. \end{aligned}$$

(B4)

All solution triplets share the same $\varvec{L}$, and any two of them, say $(\varvec{o}_1,\varvec{v}_1,\varvec{L})$ and $(\varvec{o}_{2},\varvec{v}_2,\varvec{L})$ satisfy the equality

$$\begin{aligned} \varvec{v}_2-\varvec{v}_1=\varvec{L}(\varvec{o}_{2}-\varvec{o}_1). \end{aligned}$$

(B5)

It is appropriate to choose as origin $\varvec{o}$ the cell mass center

$$\begin{aligned} \varvec{c}\,\,{:=}\,\,\big ({\textstyle \sum \limits _i}\mu _i\big )^{-1}{\textstyle \sum \limits _i}\mu _i\varvec{q}_i, \end{aligned}$$

(B6)

since then

$$\begin{aligned} {\textstyle \sum \limits _i}\mu _i\textbf{r}_i\!=\!\varvec{0}, \end{aligned}$$

(B7)

and equations (B3), (B4) uncouple, yielding straightforwardly the solution

$$\begin{aligned} \begin{aligned} \varvec{v}&=\big ({\textstyle \sum \limits _i}\mu _i\big )^{-1}{\textstyle \sum \limits _i}\mu _i\dot{\varvec{q}}_i, \\ \varvec{L}&=\big ({\textstyle \sum \limits _i}\mu _i\textbf{r}_i\!\otimes \!\textbf{r}_i\big )^{-1}{\textstyle \sum \limits _i}\mu _i\textbf{r}_i\!\otimes \!\dot{\varvec{q}}_i, \end{aligned} \end{aligned}$$

(B8)

written in unabridged notation in (2.12) & (2.14) and (2.16)–(2.18).

Appendix C Evolution equations for the new DOFs

In this Appendix, we detail the derivation of the evolution equations of the dynamical variables, taking for each of them the appropriate variations in the D’Alembert functional (3.5) augmented by the constraints as in (3.9).

In Sects. 1 and 2, for each $h\!\in \!\{1,\dots ,K\}$, we take arbitrary variations of the macroscopic DOFs associated with the h-th cell—either $\varvec{\chi }^{h}$(Sect. 1) or ${\textbf {F}}^{h}$(Sect. 2)—while setting to zero the variations of all other macroscopic DOFs ( $\delta \varvec{\chi }^{k}=\varvec{0}$, $\delta {\textbf {F}}^{k}\!=\varvec{0}$ for $k\ne h$) and of all microscopic DOFs ( $\delta \textbf{s}_i^k=\varvec{0}$ for $i=1,\dots ,N;k=1,\dots ,K$). Accordingly, the integral (3.9) is null, and only one term of the sum over k in (3.5) survives, so that the integral condition to be enforced reads

$$\begin{aligned} \mathop {\int \!\!}\nolimits _{\!\mathscr {I}}\!{\textstyle \sum \limits _i}\varphi _i^{h}(\tau )\big (\textbf{f}_i(\tau )\!\cdot \!\delta \varvec{q}_i(\tau )+m_i\dot{\varvec{q}}_i(\tau )\!\cdot \!\delta \dot{\varvec{q}}_i(\tau )\big )d\tau =0 \end{aligned}$$

(C1)

for all smooth variations $\delta \varvec{q}_i$ ( $i\!=\!1,\dots ,N$) satisfying

$$\begin{aligned} \begin{aligned} \delta \varvec{q}_i(\tau )&=\varphi _i^{h}(\tau )\big (\delta \varvec{\chi }^{h}(\tau )+\delta {\textbf {F}}^{h}(\tau )\textbf{s}_i^h(\tau )\big ), \\ \delta \dot{\varvec{q}}_i(\tau )&=\varphi _i^{h}(\tau )\big (\delta \dot{\varvec{\chi }}^{h}(\tau )+\delta \dot{{\textbf {F}}}^{h}(\tau )\textbf{s}_i^h(\tau )+ \delta {\textbf {F}}^{h}(\tau )\dot{\textbf{s}}_j^h(\tau )\big ). \end{aligned} \end{aligned}$$

(C2)

In Sect. 3, on the contrary, we will take arbitrary variations of the microscopic DOFs of each particle one by one, while setting to zero the variations of all other DOFs, either microscopic or macroscopic.

1.1 Marker dynamics

Substituting (C2) in (C1) after taking $\delta {\textbf {F}}^{h}(\tau )=\delta \dot{{\textbf {F}}}^{h}(\tau )=\varvec{0}$, we get

$$\begin{aligned} \mathop {\int \!\!}\nolimits _{\!\mathscr {I}}\!\Big (\!\big ({\textstyle \sum \limits _i}\varphi _i^{h}(\tau )\textbf{f}_i(\tau )\big )\!\cdot \!\delta \varvec{\chi }^{h}(\tau )+\!\big ({\textstyle \sum \limits _i}\mu _i^h(\tau )\dot{\varvec{q}}_i(\tau )\big )\!\cdot \!\delta \dot{\varvec{\chi }}^{h}(\tau )\!\Big )d\tau =0 \end{aligned}$$

(C3)

for all smooth variations $\delta \varvec{\chi }^{h}$. Substituting (3.8b) in the second sum in (C3) yields

$$\begin{aligned} \begin{aligned}&{\textstyle \sum \limits _i}\mu _i^h(\tau )\dot{\varvec{q}}_i(\tau )=\big ({\textstyle \sum \limits _i}\mu _i^h(\tau )\big )\dot{\varvec{\chi }}^{h}(\tau )\\&\quad =\dot{{\textbf {F}}}^{h}(\tau ){\textbf {F}}^{h}(\tau )^{-1}\big ({\textstyle \sum \limits _i}\mu _i^h(\tau )\textbf{r}_i^h(\tau )\big )+{\textbf {F}}^{h}(\tau )\big ({\textstyle \sum \limits _i}\mu _i^h(\tau )\dot{\textbf{s}}_i^h(\tau )\big ). \end{aligned} \end{aligned}$$

(C4)

Both sums in the second line of (C4) are mesoscopically negligible, as is seen by applying (2.19) to the cell with an obvious adaptation of notations²⁰ for the first sum, and by recalling (2.78a) for the second. Accounting also for (3.13) reduces (C3) to

$$\begin{aligned} \mathop {\int \!\!}\nolimits _{\!\mathscr {I}}\!\big (\textbf{f}^{\ne h}(\tau )\!\cdot \!\delta \varvec{\chi }^{h}(\tau )+\mathcal {M}^h(\tau )\dot{\varvec{\chi }}^{h}(\tau )\!\cdot \!\delta \dot{\varvec{\chi }}^{h}(\tau )\big )d\tau \cong 0, \end{aligned}$$

(C5)

where $\textbf{f}^{\ne h}(\tau )$ is the total external force (3.11) and $\mathcal {M}^h(\tau )\!=\!{\textstyle \sum \limits _i}\mu _i^h(\tau )$ is the mass content of the h-th cell. Then, enforcing condition (C5) and integrating by parts yields the equality

$$\begin{aligned} \mathcal {M}^h(\tau )\,\ddot{\varvec{\chi }}^{h}(\tau )\cong \textbf{f}^{\ne h}(\tau ) \end{aligned}$$

(C6)

at all times $\tau$ when no particle enters/leaves the h-th cell. Instead, at each time t in which a particle, say the j-th one, gets in/out, on any time interval $]t\!-\!\varsigma ,t\!+\!\varsigma [$ in which no other entry/exit occurs condition (C5) reads

$$\begin{aligned} \begin{aligned}&\!\int _{t-\varsigma }^{\,t+\varsigma }\!\!\Big (\textbf{f}^{\ne h}(\tau )\!-\!\big (\mathcal {M}^h\dot{\varvec{\chi }}^{h}\big )^{\!\varvec{\cdot }}\!(\tau )\!\Big )\!\!\cdot \!\delta \varvec{\chi }^{h}(\tau )d\tau \\&\quad =\Big (\!\big (\mathcal {M}^h\dot{\varvec{\chi }}^{h}\big )(t\!-\!\varsigma )\!-\!\big (\mathcal {M}^h\dot{\varvec{\chi }}^{h}\big )(t\!+\!\varsigma )\!\Big )\!\!\cdot \!\delta \varvec{\chi }^{h}(t)\!+\!\mathcal {O}(\varsigma )\cong 0. \end{aligned} \end{aligned}$$

(C7)

Taking the limit for $\varsigma \!\downarrow \!0$ yields

$$\begin{aligned} \dot{\varvec{\chi }}^{h}(t_+)-\dot{\varvec{\chi }}^{h}(t_-)\cong \mp \frac{m_j}{\mathcal {M}^h(t_+)}\dot{\varvec{\chi }}^{h}(t_-). \end{aligned}$$

(C8)

Since the particle mass $m_j\!\!\ll \!\mathcal {M}^h(t_+)$, the velocity jumps (C8) are small. Moreover, as established in (2.83), $\mathcal {M}^h$ is essentially constant. Hence, these small jumps compensate over time, so that their cumulative effect remains mesoscopically negligible. Accordingly, we compute the motion $\varvec{\chi }$ via the ODEs

$$\begin{aligned} \mathcal {M}^h(\tau )\,\ddot{\varvec{\chi }}^{h}(\tau )=\textbf{f}^{\ne h}(\tau )\qquad (h=1,\dots ,K) \end{aligned}$$

(C9)

(rewritten in normal form in (3.10a)) dismissing the distinction between ‘equality’ and ‘equality to within a mesoscopically negligible error’ and ignoring the small self-compensating discontinuities (C8).

1.2 Deformation tensor dynamics

Substituting (C2) in (C1) after taking $\delta \varvec{\chi }^{h}(\tau )=\delta \dot{\varvec{\chi }}^{h}(\tau )=\varvec{0}$, we get

$$\begin{aligned} \begin{aligned}&\mathop {\int \!\!}\nolimits _{\!\mathscr {I}}\!\!\big ({\textstyle \sum \limits _i}\varphi _i^{h}(\tau )\textbf{s}_i^h(\tau )\!\otimes \!\textbf{f}_i(\tau )+ {\textstyle \sum \limits _i}\mu _i^h(\tau )\dot{\textbf{s}}_i^h(\tau )\!\otimes \!\dot{\varvec{q}}_i(\tau )\big )\!\cdot \!\delta {\textbf {F}}^{h}(\tau )d\tau \, \\&\quad \,+\mathop {\int \!\!}\nolimits _{\!\mathscr {I}}\!\!\big ({\textstyle \sum \limits _i}\mu _i^h(\tau )\textbf{s}_i^h(\tau )\!\otimes \!\dot{\varvec{q}}_i(\tau )\big )\!\cdot \!\delta \dot{{\textbf {F}}}^{h}(\tau )d\tau =0 \end{aligned} \end{aligned}$$

(C10)

for all variations $\delta {\textbf {F}}^{h}$. Substituting (3.8b) in the second and third sums over i in (C3) yields²¹

$$\begin{aligned} \begin{aligned}&{\textstyle \sum \limits _i}\mu _i^h(\tau )\dot{\textbf{s}}_i^h(\tau )\!\otimes \!\dot{\varvec{q}}_i(\tau )=\big ({\textstyle \sum \limits _i}\mu _i^h(\tau )\dot{\textbf{s}}_i^h(\tau )\big )\!\otimes \!\dot{\varvec{\chi }}^{h}(\tau )\, \\&\quad +\dot{{\textbf {F}}}^{h}(\tau )\big ({\textstyle \sum \limits _i}\mu _i^h(\tau )\dot{\textbf{s}}_i^h(\tau )\!\otimes \!\textbf{s}_i^h(\tau )\big )+{\textbf {F}}^{h}(\tau )\big ({\textstyle \sum \limits _i}\mu _i^h(\tau )\dot{\textbf{s}}_i^h(\tau )\!\otimes \!\dot{\textbf{s}}_i^h(\tau )\big ), \end{aligned} \end{aligned}$$

(C11a)

$$\begin{aligned} \begin{aligned}&{\textstyle \sum \limits _i}\mu _i^h(\tau )\textbf{s}_i^h(\tau )\!\otimes \!\dot{\varvec{q}}_i(\tau )= {\textbf {F}}^{h}(\tau )^{-1}\big ({\textstyle \sum \limits _i}\mu _i^h(\tau )\textbf{r}_i^h(\tau )\big )\!\otimes \!\dot{\varvec{\chi }}^{h}(\tau )\, \\&\quad +\dot{{\textbf {F}}}^{h}(\tau )\big ({\textstyle \sum \limits _i}\mu _i^h(\tau )\textbf{s}_i^h(\tau )\!\otimes \!\textbf{s}_i^h(\tau )\big )+{\textbf {F}}^{h}(\tau )\big ({\textstyle \sum \limits _i}\mu _i^h(\tau )\textbf{s}_i^h(\tau )\!\otimes \!\dot{\textbf{s}}_i^h(\tau )\big ). \end{aligned} \end{aligned}$$

(C11b)

The first and the second sum in (C11a) are mesoscopically negligible, as is seen by recalling (2.78a) for the first and (2.78b) for the second; in (C11b), the first and the third sum are mesoscopically negligible, as is seen by applying (2.19) to the cell (as in (C4): cf. footnote 20) for the first sum, and by recalling (2.78b) for the second. Hence, (C10) reduces to

$$\begin{aligned} \begin{aligned}&\mathop {\int \!\!}\nolimits _{\!\mathscr {I}}\!\!\big ({\textstyle \sum \limits _i}\varphi _i^{h}(\tau )\textbf{s}_i^h(\tau )\!\otimes \!\textbf{f}_i(\tau )\!+\!{\textbf {F}}^{h}(\tau )\big ({\textstyle \sum \limits _i}\mu _i^h(\tau )\dot{\textbf{s}}_i^h(\tau )\!\otimes \!\dot{\textbf{s}}_i^h(\tau )\big )\big )\!\cdot \!\delta {\textbf {F}}^{h}(\tau )d\tau \, \\&\quad +\mathop {\int \!\!}\nolimits _{\!\mathscr {I}}\!\!\big (\dot{{\textbf {F}}}^{h}(\tau )\varvec{E}^{h}_{0}(\tau )\big )\!\cdot \!\delta \dot{{\textbf {F}}}^{h}(\tau )d\tau \cong 0, \end{aligned} \end{aligned}$$

(C12)

where the only nonnegligible sum in (C11b) appears in the second line, rewritten as the scaled Euler tensor $\varvec{E}^{h}_{0}(\tau )=\!{\textstyle \sum \limits _i}\mu _i^h(\tau )\textbf{s}_i^h(\tau )\otimes \textbf{s}_i^h(\tau )$, defined in (2.88). Recalling the force splitting (3.6) and expressing $\dot{\textbf{s}}_i^h(\tau )$ in terms of the thermal velocity $\varvec{w}_i(\tau )$ via (2.58), we rewrite the two sums in the first line of (C12) in the form

$$\begin{aligned} \begin{aligned} {\textstyle \sum \limits _i}\varphi _i^{h}(\tau )\textbf{s}_i^h(\tau )\!\otimes \!\textbf{f}_i(\tau )=&\,\big ({\textstyle \sum \limits _i}\varphi _i^{h}(\tau )\textbf{r}_i^h(\tau )\!\otimes \!\textbf{f}_i^{\ne h}(\tau )\big ){\textbf {F}}^{h}(\tau )^{-\top } \\&+\,\big ({\textstyle \sum \limits _i}\varphi _i^{h}(\tau )\textbf{r}_i^h(\tau )\!\otimes \!\textbf{f}_i^{h}(\tau )\big ){\textbf {F}}^{h}(\tau )^{-\top }, \end{aligned} \end{aligned}$$

(C13a)

$$\begin{aligned} {\textbf {F}}^{h}(\tau )\big ({\textstyle \sum \limits _i}\mu _i^h(\tau )\dot{\textbf{s}}_i^h(\tau )\!\otimes \!\dot{\textbf{s}}_i^h(\tau )\big )\cong \big ({\textstyle \sum \limits _i}\mu _i^h(\tau )\varvec{w}_i(\tau )\!\otimes \!\varvec{w}_i(\tau )\big ){\textbf {F}}^{h}(\tau )^{-\top }. \end{aligned}$$

(C13b)

Substituting the definitions of the total external moment (3.14) and of the total cell virial (3.15) in (C12) via (C13), we get

$$\begin{aligned} \begin{aligned} \!\!\mathop {\int \!\!}\nolimits _{\!\mathscr {I}}\!\!\left( \!\Big (\!\big (\varvec{M}^{\ne h}(\tau )\!+\!\varvec{V}^h(\tau )\!\big ){\textbf {F}}^{h}(\tau )^{-\top }\Big )\!\cdot \!\delta {\textbf {F}}^{h}(\tau )\!+\!\big (\dot{{\textbf {F}}}^{h}(\tau )\varvec{E}^{h}_{0}(\tau )\!\big )\!\cdot \!\delta \dot{{\textbf {F}}}^{h}(\tau )\!\right) \!d\tau \cong 0. \end{aligned} \end{aligned}$$

(C14)

Then, in exact analogy with the treatment in Sect. 1, we obtain

$$\begin{aligned} \ddot{{\textbf {F}}}^{h}(\tau )\varvec{E}^{h}_{0}(\tau )\cong \big (\varvec{M}^{\ne h}(\tau )\!+\!\varvec{V}^h(\tau )\big ){\textbf {F}}^{h}(\tau )^{-\top } \end{aligned}$$

(C15)

when no particle enters/leaves the h-th cell, and

$$\begin{aligned} \dot{{\textbf {F}}}^{h}(t_+)-\dot{{\textbf {F}}}^{h}(t_-)\cong -\dot{{\textbf {F}}}^{h}(t_-) \big [\!\pm m_j\,\textbf{s}_j^h(t)\!\otimes \!\textbf{s}_j^h(t)\varvec{E}^{h}_{0}(t_+)^{-1}\big ] \end{aligned}$$

(C16)

when the j-th particle gets in/out. Since $\varvec{E}^{h}_{0}(t_+)\approx \mathcal {M}^h(t_+)L_{\mathsf m}^2$,

$$\begin{aligned} \dot{{\textbf {F}}}^{h}(t_+)-\dot{{\textbf {F}}}^{h}(t_-)\approx \mp \frac{m_j}{\mathcal {M}^h(t_+)}\dot{{\textbf {F}}}^{h}(t_-). \end{aligned}$$

(C17)

The term in square bracket is again of the order of $m_j/\mathcal {M}^h(t_+)$ and the scaled Euler tensor $\varvec{E}^{h}_{0}$ is essentially constant (see (2.90)), so that the small jumps (C16) compensate over time and their cumulative effect remains mesoscopically negligible. For the reasons discussed in Sect. 1, we compute the deformation tensor dynamics by integrating the ODEs

$$\begin{aligned} \ddot{{\textbf {F}}}^{h}(\tau )\varvec{E}^{h}_{0}(\tau )=\big (\varvec{M}^{\ne h}(\tau )\!+\!\varvec{V}^h(\tau )\big ){\textbf {F}}^{h}(\tau )^{-\top }\qquad (h=1,\dots ,K), \end{aligned}$$

(C18)

rewritten in normal form in (3.10b).

1.3 Particle dynamics

For each $j\!\in \!\{1,\dots ,N\}$, we take arbitrary variations of the microscopic DOFs of the j-th particle, $\textbf{s}_j^k~(k=1,\dots ,K)$, while setting to zero the variations of all other DOFs, either microscopic or macroscopic (i. e., we let $\delta \varvec{\chi }^{k}=\varvec{0},\delta {\textbf {F}}^{k}=\varvec{0}$, $\delta \textbf{s}_i^k=\varvec{0}$ for $k=1,\dots ,K;i\ne j$). Accordingly, only one term of the sum over i in (3.5) and (3.9) survives, and the integral condition to be enforced reads

$$\begin{aligned} \begin{aligned}&\mathop {\int \!\!}\nolimits _{\!\mathscr {I}}\!{\textstyle \sum \limits _k}\varphi _j^{k}(\tau )\big (m_j^{-1}\textbf{f}_j(\tau )\!\cdot \!\delta \varvec{q}_j(\tau )+\dot{\varvec{q}}_j(\tau )\!\cdot \!\delta \dot{\varvec{q}}_j(\tau )\big )d\tau \\&+\mathop {\int \!\!}\nolimits _{\!\mathscr {I}}\!{\textstyle \sum \limits _k}\varphi _j^{k}(\tau )\big (\varvec{\lambda }^{\!k}(\tau )\!+\!\varvec{\Lambda }^{\!k}(\tau )\textbf{s}_i^k(\tau )\big )\!\cdot \!\delta \textbf{s}_i^k(\tau )d\tau =0 \end{aligned} \end{aligned}$$

(C19)

for all variations $\delta \varvec{q}_j$ satisfying

$$\begin{aligned} \begin{aligned} \delta \varvec{q}_j(\tau )&={\textbf {F}}^{k}(\tau )\delta \textbf{s}_j^k(\tau ), \\ \delta \dot{\varvec{q}}_j(\tau )&=\dot{{\textbf {F}}}^{k}(\tau )\delta \textbf{s}_j^k(\tau )+{\textbf {F}}^{k}(\tau )\delta \dot{\textbf{s}}_j^k(\tau ). \end{aligned} \end{aligned}$$

(C20)

Substituting (C20) in (C19) yields

$$\begin{aligned} \begin{aligned}&\mathop {\int \!\!}\nolimits _{\!\mathscr {I}}\!{\textstyle \sum \limits _k}\varphi _j^{k}(\tau )\big (m_j^{-1}{\textbf {F}}^{k}(\tau )^{\!\top }\textbf{f}_j(\tau )+\varvec{\lambda }^{\!k}(\tau )+\varvec{\Lambda }^{\!k}(\tau )\textbf{s}_j^k(\tau )\big )\!\cdot \!\delta \textbf{s}_j^k(\tau )d\tau \, \\&+\mathop {\int \!\!}\nolimits _{\!\mathscr {I}}\!{\textstyle \sum \limits _k}\varphi _j^{k}(\tau )\Big (\!\big (\dot{{\textbf {F}}}^{k}(\tau )^{\!\top }\dot{\varvec{q}}_j(\tau )\big )\!\cdot \!\delta \textbf{s}_j^k(\tau )\!+\!\big ({\textbf {F}}^{k}(\tau )^{\!\top }\dot{\varvec{q}}_j(\tau )\big )\!\cdot \!\delta \dot{\textbf{s}}_j^k(\tau )\!\Big )d\tau =0, \end{aligned} \end{aligned}$$

(C21)

where $\dot{\varvec{q}}_j$ stands for the sum of terms on the right-hand side of (3.8b). Integration by parts transforms (C21) into

$$\begin{aligned} \begin{aligned}&\mathop {\int \!\!}\nolimits _{\!\mathscr {I}}\!{\textstyle \sum \limits _k}\varphi _j^{k}(\tau ){\textbf {F}}^{k}(\tau )^{\!\top }\!\big (m_j^{-1}\textbf{f}_j(\tau )\!-\!\ddot{\varvec{q}}_j(\tau )\big )\!\cdot \!\delta \textbf{s}_j^k(\tau )d\tau \, \\&+\mathop {\int \!\!}\nolimits _{\!\mathscr {I}}\!{\textstyle \sum \limits _k}\varphi _j^{k}(\tau )\big (\varvec{\lambda }^{\!k}(\tau )\!+\!\varvec{\Lambda }^{\!k}(\tau )\textbf{s}_j^k(\tau )\big )\!\cdot \!\delta \textbf{s}_j^k(\tau )d\tau \, \\&-\mathop {\int \!\!}\nolimits _{\!\mathscr {I}}\!{\textstyle \sum \limits _k}\dot{\varphi }_j^{k}(\tau )\dot{\varvec{q}}_j(\tau )\!\cdot \!\big ({\textbf {F}}^{k}(\tau )\delta \textbf{s}_j^k(\tau )\big ) d\tau =0. \end{aligned} \end{aligned}$$

(C22)

The integrand of the last integral in (C22) vanishes identically on the open time intervals in which the j-th particle moves within a cell. At each time t at which it traverses the boundary between two adjacent cells passing from, say, the h-th to the $\ell$-th cell, $\varphi _j^{h}$ jumps from 1 to 0 and $\varphi _j^{\ell }$ from 0 to 1. Correspondingly, the integral in question receives the contribution

$$\begin{aligned} \dot{\varvec{q}}_j(t)\!\cdot \!\big ({\textbf {F}}^{\ell }(t)\delta \textbf{s}_j^{\ell }(t_+)\!-\!{\textbf {F}}^{h}(t)\delta \textbf{s}_j^h(t_-)\big ). \end{aligned}$$

(C23)

The jump condition (2.70a), rewritten for the j-th particle, reads

$$\begin{aligned} {\textbf {F}}^{\ell }(t)\textbf{s}_j^{\ell }(t_+)-{\textbf {F}}^{h}(t)\textbf{s}_j^h(t_-)=\varvec{\chi }^{h}(t)\!-\varvec{\chi }^{\ell }(t). \end{aligned}$$

(C24)

Taking the variations of only the microscopic DOFs of the j-th particle on both sides of (C24) yields

$$\begin{aligned} {\textbf {F}}^{\ell }(t)\delta \textbf{s}_j^{\ell }(t_+)-{\textbf {F}}^{h}(t)\delta \textbf{s}_j^h(t_-)=\varvec{0}, \end{aligned}$$

(C25)

so that the contribution (C23) is null. In conclusion, expressing $\ddot{\varvec{q}}_j$ in terms of the new DOFs by taking the time derivative on both sides of (3.8b) and enforcing the condition (C22) for all variations $\delta \textbf{s}_j^k$ yields the evolution equation

$$\begin{aligned} \begin{aligned}&{\textbf {F}}^{k}(\tau )\ddot{\textbf{s}}_j^k(\tau )+\ddot{{\textbf {F}}}^{k}(\tau )\textbf{s}_j^k(\tau )+\ddot{\varvec{\chi }}^{k}(\tau ) \\&\quad =m_j^{-1}\textbf{f}_j(\tau )-2\dot{{\textbf {F}}}^{k}(\tau )\dot{\textbf{s}}_j^k(\tau )+{\textbf {F}}^{k}(\tau )^{-\top }\!\big (\varvec{\lambda }^{\!k}(\tau )\!+\!\varvec{\Lambda }^{\!k}(\tau )\textbf{s}_j^k(\tau )\big ), \end{aligned} \end{aligned}$$

(C26)

which applies whenever $\tau$ is in an open time interval in which $\kappa _j(\tau )\!=\!k$. The transition across cell–cell interfaces is ruled by the equalities (2.70)–(2.71), implying that, if at time t the j-th particle leaves, say, the h-th cell to enter the adjacent $\ell$-th cell, then

$$\begin{aligned} \textbf{s}_j^{\ell }(t_+)&={\textbf {F}}^{\ell }(t)^{-1}{\textbf {F}}^{h}(t)\textbf{s}_j^h(t_-)+\varvec{\chi }^{h}(t)\!-\varvec{\chi }^{\ell }(t) \nonumber \\&\cong \textbf{s}_j^h(t_-)+\varvec{\chi }^{h}(t)\!-\varvec{\chi }^{\ell }(t), \end{aligned}$$

(C27a)

$$\begin{aligned} \dot{\textbf{s}}_j^{\ell }(t_+)&\cong {\textbf {F}}^{\ell }(t)^{-1}{\textbf {F}}^{h}(t)\dot{\textbf{s}}_j^h(t_-)\cong \dot{\textbf{s}}_j^h(t_-). \end{aligned}$$

(C27b)

Substituting the values of $\ddot{\varvec{\chi }}^{k}$and $\ddot{{\textbf {F}}}^{k}$ obtained from Eqs. (C9) and (C18) in (C26) yields

$$\begin{aligned} \begin{aligned} \ddot{\textbf{s}}_j^k(\tau )&= {\textbf {C}}^{k}(\tau )^{-1}\big (\varvec{\lambda }^{\!k}(\tau )\!+\!\varvec{\Lambda }^{\!k}(\tau )\textbf{s}_j^k(\tau )\big ) \\&\quad +{\textbf {F}}^{k}(\tau )^{-1}\big (m_j^{-1}\textbf{f}_j(\tau )-\mathcal {M}^k(\tau )^{-1}\textbf{f}^{\ne k}(\tau )-2\dot{{\textbf {F}}}^{k}(\tau )\dot{\textbf{s}}_j^k(\tau )\big ) \\&\quad -{\textbf {F}}^{k}(\tau )^{-1}\big (\varvec{M}^{\ne k}(\tau )\!+\!\varvec{V}^k(\tau )\big ){\textbf {F}}^{k}(\tau )^{-\top }\!\varvec{E}^{k}_{0}(\tau )^{-1}\textbf{s}_j^k(\tau ) \end{aligned} \end{aligned}$$

(C28)

(rewritten as-is in (3.10c) with (cf. (2.46)).

Appendix D Traction and stress

The macroscopic notions of traction and stress are defined here on an atomistic basis, and their fundamental properties are justified in terms of microscopic interactions between particles. The key observation is that, due to the hypothesis (2.2b) of short-range interactions, the force $\textbf{f}_i^{\ne h}(\tau )$ applied to the i-th particle in the h-th cell from outside the cell is significantly nonzero only if the i-th particle is within an $L_{\mu }\,$-thick boundary layer of , whose linear size is . This motivates the assumption that a vector field—the traction (force per unit area) —may be defined [1] on the current cell boundary at each time $\tau$, such that

• (i) the surface integral of the traction is $\cong$-equal to the total external force applied to (the particles in) the h-th cell:

  

  (D1a)

• (ii) the surface integral of the tensor $\textbf{r}(\cdot ,\tau )\!\otimes \!\textbf{t}^h_{\tau }$, where $\textbf{r}(\varvec{z},\tau )\,\,{:=}\,\,\varvec{z}\!-\!\varvec{\chi }^{h}(\tau )$ for , is $\cong$-equal to the total external moment (3.14) applied to (the particles in) the h-th cell:

  

  (D1b)

  with $\textbf{r}_i^h(\tau )\,\,{:=}\,\,\varvec{q}_i(\tau )\!-\!\varvec{\chi }^{h}(\tau )$ (recall from (2.65) that $\varvec{\chi }^{h}(\tau )\equiv \varvec{\chi }_{\tau }(\varvec{x}^h)$ is the center of ).

Statements (i) & (ii) are equivalent to the single assumption that

(D2)

for all affine vector field $\varvec{z}\mapsto \varvec{\alpha }(\varvec{z})=\varvec{v}+\varvec{L}(\varvec{z}\!-\!\varvec{\chi }^{h}(\tau ))$, with arbitrary $\varvec{v}$ and $\varvec{L}$. In fact, the left-hand and right-hand sides of (D2) read, respectively,²²

(D3a)

and

$$\begin{aligned} \begin{aligned} {\textstyle \sum \limits _i}\varphi _i^{h}(\tau )\textbf{f}_i^{\ne h}(\tau )\!\cdot \!\varvec{\alpha }(\varvec{q}_i(\tau ))&=\Big ({\textstyle \sum \limits _i}\varphi _i^{h}(\tau )\textbf{f}_i^{\ne h}(\tau )\Big )\!\cdot \!\varvec{v}\\&\quad +\Big ({\textstyle \sum \limits _i}\varphi _i^{h}(\tau )\textbf{r}_i^h(\tau )\!\otimes \!\textbf{f}_i^{\ne h}(\tau )\!\Big )\!\cdot \!\varvec{L}. \end{aligned} \end{aligned}$$

(D3b)

Recall from Sects. 2.2.1 and 2.3.2 that is well approximated by an open convex polyhedron (in the present context, a parallelepiped). Hence, each integral in (D1) splits into a sum of integrals over the polyhedron faces. Let be the interface separating two adjacent cells, and . Recall from Sects. 2.2.1 and 2.3.2 that is well approximated by a convex polygon having inner and outer diameters $\approx \!L_{\mathsf m}$ and edges whose length $\approx \!L_{\mathsf m}$. Hence, the contribution from the particles in an $L_{\mu }\,$-neighborhood of the edges (or the vertices) of scales as $L_{\mathsf m}L_{\mu }\,^2$ (or as $L_{\mu }\,^3$), while that from those in an $L_{\mu }\,$-neighborhood of scales as $L_{\mathsf m}^2L_{\mu }\,$. For this reason, the contributions from the particles not in nor in but in an $L_{\mu }\,$-neighborhood of the edges and vertices of are negligible, and we may conclude that the external force applied to the particles close to from within the h-th cell is predominantly exerted by the particles close to from within the k-th cell. In the case of binary interactions, we may write

(D4)

Swapping h with k and i with j in (D4) changes the sum on the right-hand side into its opposite, since $\textbf{f}_{ji}\!=\!-\textbf{f}_{ij}$. Hence,

(D5)

We may therefore associate the traction with the individual faces composing the boundary, provided that we endow them with an orientation. Let  ( ) be the face oriented with the unit normal vector ( ) pointing outward of  ( ), with . Then, define

(D6)

Given a point and a unit vector $\varvec{n}$, we could always design a tessellation such that, at a given time $\tau$, the face common to two adjacent cells is centered²³ at $\varvec{y}$ and normal to  $\varvec{n}$. Let us label with k the cell toward which $\varvec{n}$ points and with h the other one. Under the assumption that, for every unit vector $\varvec{n}$, the field $\textbf{t}_{\tau }(\cdot ,\varvec{n})$ is approximately affine on the mesoscopic length scale $L_{\mathsf m}$ (recall definition in Sect. 2.2.2), equality (D5) combined with definitions (D6) yields

$$\begin{aligned} \textbf{t}_{\tau }(\varvec{y},\varvec{n})\cong -\textbf{t}_{\tau }(\varvec{y},-\varvec{n}) \end{aligned}$$

(D7)

for all and all unit vector $\varvec{n}$.

On this basis, the Cauchy stress theorem—fundamental to continuum mechanics—can be proved [1]: for every , the traction $\textbf{t}_{\tau }(\varvec{y},\varvec{n})$ depends linearly on $\varvec{n}$. In other terms, there exists a tensor field, the Cauchy stress field $\textbf{T}_{\tau }\equiv \textbf{T}(\cdot ,\tau )$, smooth and approximately affine on the mesoscopic length scale $L_{\mathsf m}$, such that

$$\begin{aligned} \textbf{t}_{\tau }(\varvec{y},\varvec{n})\cong \textbf{T}_{\tau }(\varvec{y})\varvec{n}. \end{aligned}$$

(D8)

Now, let $\textbf{n}^h_{\tau }$ denote the unit outer normal field on .²⁴ Then,

(D9)

Substituting (D9) in the integral on the left-hand side of (D2) and using the divergence theorem²⁵ yields

(D10)

since $\textbf{T}_{\tau }$ is approximately affine (and hence its divergence approximately constant) on the scale of ; moreover, . Comparing (D10) with (D3a) and (D1), we finally obtain expressions for the total external force $\textbf{f}^{\ne h}(\tau )$ and the total external moment $\varvec{M}^{\ne h}(\tau )$ acting on the h-th cell in terms of the Cauchy stress and its divergence at the cell center:

$$\begin{aligned}&\textbf{f}^{\ne h}(\tau )\cong \mathcal {V}^h(\tau )\text {div}\textbf{T}_{\tau }(\varvec{\chi }^{h}(\tau ))\equiv \mathcal {V}^h(\tau )(\text {div}\textbf{T})^h(\tau ), \end{aligned}$$

(D11a)

$$\begin{aligned}&\varvec{M}^{\ne h}(\tau )\cong \mathcal {V}^h(\tau )\textbf{T}_{\tau }(\varvec{\chi }^{h}(\tau ))\equiv \mathcal {V}^h(\tau )\textbf{T}^{h}(\tau ), \end{aligned}$$

(D11b)

where $\mathcal {V}^h(\tau )$ is the current volume of the h-th cell. Substituting (D11a) and (D11b) in, respectively, (3.10a) and (3.10b), we get

$$\begin{aligned} \mathcal {M}^h(\tau )\,\ddot{\varvec{\chi }}^{h}(\tau )&\cong \mathcal {V}^h(\tau )\,(\text {div}\textbf{T})^h(\tau ), \end{aligned}$$

(D12a)

$$\begin{aligned} \ddot{{\textbf {F}}}^{h}(\tau )\varvec{E}^{h}_{0}(\tau )&\cong \big (\mathcal {V}^h(\tau )\textbf{T}^{h}(\tau )\!+\!\varvec{V}^h(\tau )\big ){\textbf {F}}^{h}(\tau )^{-\top }. \end{aligned}$$

(D12b)

The form of equations (D12b) suggests introducing the field $\textbf{S}_{\tau }\!\!\equiv \!\textbf{S}(\cdot ,\tau )$, defined as the Piola transform of $\textbf{T}_{\tau }$:

$$\begin{aligned} \textbf{S}_{\tau }(\varvec{x}^h)\equiv \textbf{S}^{h}(\tau )\,\,{:=}\,\,\det {\textbf {F}}^{h}(\tau )\textbf{T}^{h}(\tau ){\textbf {F}}^{h}(\tau )^{-\top }, \end{aligned}$$

(D13)

whose divergence enjoys the property²⁶

$$\begin{aligned} \text {div}\textbf{S}_{\tau }(\varvec{x}^h)\equiv (\text {div}\textbf{S})^h(\tau )=\det {\textbf {F}}^{h}(\tau )(\text {div}\textbf{T})^h(\tau ). \end{aligned}$$

(D14)

Substituting (D13) and (D14) in (D11) yields

$$\begin{aligned}&\textbf{f}^{\ne h}(\tau )\cong \mathcal {V}^h(\tau _{_{0}})(\text {div}\textbf{S})^h(\tau ), \end{aligned}$$

(D15a)

$$\begin{aligned}&\varvec{M}^{\ne h}(\tau )\cong \mathcal {V}^h(\tau _{_{0}})\textbf{S}^{h}(\tau ), \end{aligned}$$

(D15b)

i. e., equations (4.4).

Appendix E Running time averages

Definition and general properties Any integrable function f defined on a time interval $\mathscr {I}$ whose duration $\vert \mathscr {I}\vert \gg \!T$ can be split into the sum of its centered running $\theta$-average

$$\begin{aligned} \overline{f}^{\theta }\!(t)\,\,{:=}\,\,\frac{1}{\theta }\int _{t-\theta /2}^{\,t+\theta /2}\!\!\!f(\tau )d\tau \end{aligned}$$

(E1)

with $\theta \approx T$, and its $\theta$-fluctuating component

$$\begin{aligned} f'^{\theta }\!(t)\!\,\,{:=}\,\,f(t)\!-\!\overline{f}^{\theta }\!(t). \end{aligned}$$

(E2)

Of course, $\overline{f'^{\theta }}^{\theta }\!\!=0$. Whenever $\theta$ is clear from the context, we omit the superscript in $\overline{f}^{\theta }$ and $f'^{\theta }$. The bilinear product of two functions of time f, h satisfies the equality

$$\begin{aligned} \overline{fh}=\overline{f}\;\overline{h}+\overline{f'h'}. \end{aligned}$$

(E3)

Since both averaging and differentiating are linear operations, the time rate of the running average equals the running average of the time rate:

$$\begin{aligned} \dot{\overline{f}}=\!\overline{\dot{f}}. \end{aligned}$$

(E4)

Running average of $\,T$-constant functions of time By definition (cf. Sect. 2.2.2), a function of time f is approximately constant on the T time scale (or T-constant) if

$$\begin{aligned} \vert t\!-\!\tau \vert \lessapprox T\;\Longrightarrow \;f(t)\cong f(\tau ). \end{aligned}$$

If f is T-constant, then it is $\cong$-equal to its running $\theta$-average for all $\theta \approx T$ and, equivalently, its $\theta$-fluctuating component is $\cong 0$:

$$\begin{aligned} f\cong \overline{f}^{\theta }\quad \Longleftrightarrow \quad f'^{\theta }\cong 0. \end{aligned}$$

(E5)

Combining (E5) with (E3), we obtain that if at least one of two functions of time $f,h$ is T-constant, then the running $\theta$-average of their product is $\cong$-equal to the product of their running $\theta$-averages, for all $\theta \approx T$:

$$\begin{aligned} h'^{\theta }\cong 0\;\;\Longrightarrow \;\;\overline{fh}^{\theta }\cong \overline{f}^{\theta }\overline{h}^{\theta }\quad \text {for all}\;f. \end{aligned}$$

(E6)

Any time interval $\mathscr {I}$such that $\vert \mathscr {I}\vert \gg T$ can be subdivided into the disjoint union of a large number of subintervals $\mathscr {I}_{_{1}},\dots ,\mathscr {I}_{_{\!S}}$ of duration $\theta \approx T$, i. e.,

$$\begin{aligned} \mathscr {I}\!=\sqcup _\iota \mathscr {I}_{\!\iota }\quad \textrm{with}\quad \vert \mathscr {I}_{\!\iota }\vert =\theta \approx T. \end{aligned}$$

(E7)

Let $t_\iota$ be the middle point of $\mathscr {I}_{\!\iota }$. Then, the integral of f over $\mathscr {I}$ equals the sum over  $\iota$ of the integrals over $\mathscr {I}_{\!\iota }$ of f, each of which equals $\theta \overline{f}(t_\iota )$, due to (E1). If f is T-constant, then

$$\begin{aligned} \overline{f}(t_\iota )\cong f(t_\iota )\cong f(\tau )\cong \overline{f}(\tau )\qquad \mathrm{for~all~}\;\tau \in \!\mathscr {I}_{\!\iota }, \end{aligned}$$

(E8)

on account of (E5). Hence,

$$\begin{aligned} \theta \overline{f}(t_\iota )\cong \!\int _{\!\mathscr {I}_{\!\iota }}\!\overline{f}(\tau )d\tau . \end{aligned}$$

(E9)

Summing over $\iota$, we obtain that the integral over $\mathscr {I}$of f is $\cong$-equal to the integral over $\mathscr {I}$of its running $\theta$-average $\overline{f}$:

$$\begin{aligned} \int _{\!\mathscr {I}}\!f(\tau )d\tau = {\textstyle \sum }_\iota \!\int _{\!\mathscr {I}_{\!\iota }}\!f(\tau )d\tau = {\textstyle \sum }_\iota \theta \overline{f}(t_\iota )\cong {\textstyle \sum }_\iota \!\int _{\!\mathscr {I}_{\!\iota }}\!\overline{f}(\tau )d\tau = \!\int _{\!\mathscr {I}}\!\overline{f}(\tau )d\tau . \end{aligned}$$

(E10)

Combining (E10) with (E6), we obtain that the integral over $\mathscr {I}$of the bilinear product of two functions of time, of which one is T-constant, is $\cong$-equal to the integral of the product of their running $\theta$-averages:

$$\begin{aligned} h'\cong 0\quad \Longrightarrow \quad \int _{\!\mathscr {I}}\!f(\tau )h(\tau )d\tau \,\cong \! \int _{\!\mathscr {I}}\!\overline{f}(\tau )\overline{h}(\tau )d\tau . \end{aligned}$$

(E11)

Properties of essentially constant functions of time By definition (cf. Sect. 2.2.2), a function of time f is essentially constant if

$$\begin{aligned} f(t)\cong f(\tau )\quad \text {for all}\;t,\tau . \end{aligned}$$

(E12)

If f is essentially constant, then it is $\cong$-equal to its running average $\overline{f}$:

$$\begin{aligned} f\cong \overline{f}^{\theta }\qquad \mathrm{for~all~}\;\theta . \end{aligned}$$

(E13)

An essentially constant function of time f need not be continuous, let alone differentiable. However, there exists a time scale $T_{f}$ such that, for all $\theta \gtrapprox T_{f}$, the time derivative of its running $\theta$-average is zero to within a preset tolerance:

$$\begin{aligned} \dot{\overline{f}^{\theta }}\cong 0. \end{aligned}$$

(E14)

Appendix F Computing Lagrange multipliers

The i label being local to each representative cell, we rewrite (4.24) dropping the superscript h from $\textbf{s}_i^h$ and its time derivatives. Premultiplying both sides of the equation by ${\textbf {F}}^{h}(t)^{-1}$ and reordering terms, we obtain

$$\begin{aligned} \begin{aligned} \ddot{\textbf{s}}_{i}(\tau )&={\textbf {F}}^{h}(t)^{-1}\big \{m_i^{-1}\textbf{f}_i(\tau )-2\dot{{\textbf {F}}}^{h}(t)\dot{\textbf{s}}_{i}(\tau ) \\&\quad -\!\big (\ddot{{\textbf {F}}}^{h}(t)\!-\!{\textbf {F}}^{h}(t)^{-\top }\!\varvec{\Lambda }^{\!h}(\tau )\big )\textbf{s}_{i}(\tau ) \\&\quad -\big (\ddot{\varvec{\chi }}^{h}(t)\!-\!{\textbf {F}}^{h}(t)^{-\top }\varvec{\lambda }^{\!h}(\tau )\big )\!\big \}\qquad \;\;\mathrm{for~}\,i=1,\dots ,n^h \end{aligned} \end{aligned}$$

(F1)

(recall that $\textbf{f}_i(\tau )$ is computed as indicated in (4.2)). In each round of time integration of equations (F1), the initial state of the $n^h$ representative particles populating the h-th cell can always be defined in such a way that

$$\begin{aligned} {\textstyle \sum \limits _i}m_i\textbf{s}_i(\tau _{_{0}})=\varvec{0}\end{aligned}$$

(F2)

and that no particle leaves the cell at the initial time $\tau _{_{0}}$. If (and only if) a particle sits initially on the cell boundary, at least one of its images also sits on the boundary. Moreover, if at time $\tau _{_{0}}$ the thermal velocity of that particle points outward, then the thermal velocity of (at least) one of its images points inward. Therefore, it suffices to swap roles between particle and image to fix the initial conditions. Then, the time derivatives of constraints (2.78) at $\tau _{_{0}}$ read

$$\begin{aligned}&{\textstyle \sum \limits _i}m_i\ddot{\textbf{s}}_{i}(\tau _{_{0}})=\varvec{0}, \end{aligned}$$

(F3a)

$$\begin{aligned}&{\textstyle \sum \limits _i}m_i\dot{\textbf{s}}_{i}(\tau _{_{0}})\!\otimes \!\dot{\textbf{s}}_{i}(\tau _{_{0}})+{\textstyle \sum \limits _i}m_i\textbf{s}_i(\tau _{_{0}})\!\otimes \!\ddot{\textbf{s}}_{i}(\tau _{_{0}})=\varvec{0}. \end{aligned}$$

(F3b)

Substituting (F1) in (F3), taking into account (4.3), (4.5a), (F2), (3.14), (3.15), (2.88), (2.53), and (2.58) and using the fact that constraints (2.78) are satisfied at $\tau _{_{0}}$, we get

$$\begin{aligned} \varvec{\lambda }_{(0)}^{h}&={\textbf {F}}^{h}(t)^{\top }\ddot{\varvec{\chi }}^{h}(t), \end{aligned}$$

(F4)

$$\begin{aligned} \varvec{\Lambda }_{(0)}^{h}&={\textbf {F}}^{h}(t)^{\top }\Big (\ddot{{\textbf {F}}}^{h}(t)-\big ({\textstyle \sum \limits _i}\textbf{r}_{i(0)}\!\otimes \!\textbf{f}_{i(0)}\!+\!{\textstyle \sum \limits _i}m_i\varvec{w}_{i(0)}\!\otimes \!\varvec{w}_{i(0)}\big )\big (\varvec{E}_{0}^{h}\big )^{\!-1}\Big ), \end{aligned}$$

(F5)

with

$$\begin{aligned} \textbf{r}_{i(0)}\,\,{:=}\,\,{\textbf {F}}^{h}(t)\textbf{s}_{i(0)},\quad \varvec{w}_{i(0)}\,\,{:=}\,\,{\textbf {F}}^{h}(t)\dot{\textbf{s}}_{i}(0). \end{aligned}$$

(F6)

Then, the solution to (F1) is propagated stepwise by a two-stage velocity Verlet algorithm [30, 31]:

$$\begin{aligned} \textbf{s}_{i(n+1)}&=\textbf{s}_{i(n)}+\epsilon \,\dot{\textbf{s}}_{i(n)}+{\textstyle \frac{1}{2}}\epsilon ^2{\textbf {F}}^{h}(t)^{-1}\big \{m_i^{-1}\textbf{f}_{i(n)}-2\dot{{\textbf {F}}}^{h}(t)\dot{\textbf{s}}_{i(n)}\nonumber \\&-\!\big (\ddot{{\textbf {F}}}^{h}(t)\!-\!{\textbf {F}}^{h}(t)^{-\top }\!\varvec{\Lambda }_{(n)}^{h}\big )\textbf{s}_{i(n)}-\big (\ddot{\varvec{\chi }}^{h}(t)\!-\!{\textbf {F}}^{h}(t)^{-\top }\!\varvec{\lambda }_{(n)}^{h}\big )\!\big \}, \nonumber \\ \dot{\textbf{s}}_{i(n+1)}&=\dot{\textbf{s}}_{i(n)}+{\textstyle \frac{1}{2}}\epsilon {\textbf {F}}^{h}(t)^{-1}\big \{\big [m_i^{-1}\textbf{f}_{i(n)}-2\dot{{\textbf {F}}}^{h}(t)\dot{\textbf{s}}_{i(n)}\end{aligned}$$

(F7)

$$\begin{aligned}&\quad -\!\big (\ddot{{\textbf {F}}}^{h}(t)\!-\!{\textbf {F}}^{h}(t)^{-\top }\!\varvec{\Lambda }_{(n)}^{h}\big )\textbf{s}_{i(n)}-\big (\ddot{\varvec{\chi }}^{h}(t)\!-\!{\textbf {F}}^{h}(t)^{-\top }\!\varvec{\lambda }_{(n)}^{h}\big )\big ] \nonumber \\&\quad +\!\big [m_i^{-1}\textbf{f}_{i(n+1)}-2\dot{{\textbf {F}}}^{h}(t)\dot{\textbf{s}}_{i(n+1)}\nonumber \\&\quad -\!\big (\ddot{{\textbf {F}}}^{h}(t)\!-\!{\textbf {F}}^{h}(t)^{-\top }\!\varvec{\Lambda }_{(n+1)}^{h}\big )\textbf{s}_{i(n+1)}-\big (\ddot{\varvec{\chi }}^{h}(t)\!-\!{\textbf {F}}^{h}(t)^{-\top }\!\varvec{\lambda }_{(n+1)}^{h}\big )\big ]\big \}. \end{aligned}$$

(F8)

where $f_{n}$ stands for $f(\tau _n)$ for each function of time f and each $\tau _0,\tau _1,\dots$. The updated thermal velocity $\dot{\textbf{s}}_{i(n+1)}$ is accepted as is, while the ‘scaled’ radius vector $\textbf{s}_{i(n+1)}$ delivered by (F7) is accepted as final only if

(F9)

i. e., if the i-th particle is still in at time $\tau _{n+1}$. Otherwise, it gets shifted by the unique linear combination of the edges of with coefficients taken in $\{0,\pm 1\}$, denoted $\textbf{u}_{i(n)}$, satisfying the condition

(F10)

Since for $\textbf{u}_{i(n)}\!=\varvec{0}$ (F10) coincides with (F9), in the following we take the (possibly) shifted update $\textbf{s}_{i(n+1)}+\textbf{u}_{i(n)}$ as the scaled radius vector at time $\tau _{n+1}$ of the i-th particle, be it the same particle as at time $\tau _n$ or a new one.²⁷

Then, equations (F8) are solved expressing $\dot{\textbf{s}}_{i(n+1)}$ ( $i\!=\!1,\dots ,n^h$) in terms of $\varvec{\lambda }_{(n+1)}^{h}$ and $\varvec{\Lambda }_{(n+1)}^{h}$, as follows:

$$\begin{aligned} \dot{\textbf{s}}_{i(n+1)}=\textbf{A}^{\!h}(t)\big \{\dot{\textbf{s}}^{\mathsf{{P}}}_{i(n+1)}+{\textstyle \frac{1}{2}}\epsilon {\textbf {C}}^{h}(t)^{\!-1}\big (\varvec{\lambda }_{(n+1)}^{h}\!+\!\varvec{\Lambda }_{(n+1)}^{h}\textbf{s}_{i(n+1)}\big )\!\big \}, \end{aligned}$$

(F11a)

where ${\textbf {C}}^{h}(t)={\textbf {F}}^{h}(t)^{\top }{\textbf {F}}^{h}(t)$, $\textbf{A}^{\!h}(t)\!\,\,{:=}\,\,\big (\textbf{I}+\epsilon \big ({\textbf {F}}^{h}(t)^{-\top }\dot{{\textbf {F}}}^{h}(t)\big )^{\!-1}\!$, and

$$\begin{aligned} \begin{aligned} \dot{\textbf{s}}^{\mathsf{{P}}}_{i(n+1)}\,&\!\,\,{:=}\,\,\dot{\textbf{s}}_{i(n)}+{\textstyle \frac{1}{2}}\epsilon {\textbf {F}}^{h}(t)^{-1}\big \{\big [m_i^{-1}\textbf{f}_{i(n)}-2\dot{{\textbf {F}}}^{h}(t)\dot{\textbf{s}}_{i(n)}\\&\qquad -\!\big (\ddot{{\textbf {F}}}^{h}(t)\!-\!{\textbf {F}}^{h}(t)^{-\top }\!\varvec{\Lambda }_{(n)}^{h}\big )\textbf{s}_{i(n)}-\big (\ddot{\varvec{\chi }}^{h}(t)\!-\!{\textbf {F}}^{h}(t)^{-\top }\!\varvec{\lambda }_{(n)}^{h}\big )\big ] \\&\qquad +\!\big [m_i^{-1}\textbf{f}_{i(n+1)} -\ddot{{\textbf {F}}}^{h}(t)\textbf{s}_{i(n+1)}-\ddot{\varvec{\chi }}^{h}(t)\big ]\big \}. \end{aligned} \end{aligned}$$

(F11b)

According to the SHAKE procedure [32,33,34], substituting (F11a) in the constraint equations (2.78) enforced at time $\tau _{n+1}$, namely

$$\begin{aligned} {\textstyle \sum \limits _i}m_i\dot{\textbf{s}}_{i(n+1)}=\varvec{0},\qquad {\textstyle \sum \limits _i}m_i\big (\textbf{u}_{i(n)}\!+\textbf{s}_{i(n+1)}\big )\!\otimes \!\dot{\textbf{s}}_{i(n+1)}=\varvec{0}, \end{aligned}$$

(F12)

we get

$$\begin{aligned}&{\textstyle \sum \limits _i}m_i\dot{\textbf{s}}^{\mathsf{{P}}}_{i(n+1)}\,+ {\textstyle \frac{1}{2}}\epsilon {\textbf {C}}^{h}(t)^{\!-1}{\textstyle \sum \limits _i}m_i\big (\varvec{\lambda }_{(n+1)}^{h}\!+\!\varvec{\Lambda }_{(n+1)}^{h}\textbf{s}_{i(n+1)}\big )=\varvec{0}, \nonumber \\&{\textstyle \sum \limits _i}m_i\big (\textbf{u}_{i(n)}\!+\textbf{s}_{i(n+1)}\big )\!\otimes \!\dot{\textbf{s}}^{\mathsf{{P}}}_{i(n+1)}\,\end{aligned}$$

(F13a)

$$\begin{aligned}&\quad +{\textstyle \frac{1}{2}}\epsilon {\textbf {C}}^{h}(t)^{\!-1}{\textstyle \sum \limits _i}m_i\big (\textbf{u}_{i(n)}\!+\textbf{s}_{i(n+1)}\big )\!\otimes \!\big (\varvec{\lambda }_{(n+1)}^{h}\!+\!\varvec{\Lambda }_{(n+1)}^{h}\textbf{s}_{i(n+1)}\big )=\varvec{0}. \end{aligned}$$

(F13b)

Under the assumption that

$$\begin{aligned} \begin{aligned}&{\textstyle \sum \limits _i}m_i\big (\textbf{u}_{i(n)}\!+\textbf{s}_{i(n+1)}\big )\cong {\textstyle \sum \limits _i}m_i\textbf{s}_{i(n+1)}\cong \varvec{0}, \\&{\textstyle \sum \limits _i}m_i\big (\textbf{u}_{i(n)}\!+\textbf{s}_{i(n+1)}\big )\!\otimes \!\textbf{s}_{i(n+1)}\cong \\&{\textstyle \sum \limits _i}m_i\big (\textbf{u}_{i(n)}\!+\textbf{s}_{i(n+1)}\big )\!\otimes \!\big (\textbf{u}_{i(n)}\!+\textbf{s}_{i(n+1)}\big )\cong {\textstyle \sum \limits _i}m_i\textbf{s}_{i(n+1)}\!\otimes \!\textbf{s}_{i(n+1)}, \end{aligned} \end{aligned}$$

(F14)

equations (F13) uncouple and deliver (the distinction between ‘equality’ and ‘equality to within a mesoscopically negligible error’ having been dismissed)

$$\begin{aligned} \varvec{\lambda }_{(n+1)}^{h}&=-2\epsilon ^{-1}\mathcal {M}^h(\tau _{_{0}})^{-1}{\textbf {C}}^{h}(t){\textstyle \sum \limits _i}m_i\dot{\textbf{s}}^{\mathsf{{P}}}_{i(n+1)}\,, \end{aligned}$$

(F15)

$$\begin{aligned} \varvec{\Lambda }_{(n+1)}^{h}&=-2\epsilon ^{-1}{\textbf {C}}^{h}(t)\big ({\textstyle \sum \limits _i}m_i\big (\textbf{u}_{i(n)}\!+\textbf{s}_{i(n+1)}\big )\!\otimes \!\dot{\textbf{s}}^{\mathsf{{P}}}_{i(n+1)}\big )\big (\varvec{E}_{0}^{h}\big )^{\!-1}, \end{aligned}$$

(F16)

After substituting (F11b) in (F15), a simple but tedious calculation yields

$$\begin{aligned} \varvec{\lambda }_{(n+1)}^{h}=2{\textbf {F}}^{h}(t)^{\top }\ddot{\varvec{\chi }}^{h}(t)-\varvec{\lambda }_{(n)}^{h}, \end{aligned}$$

(F17)

on account of (4.3). The initial value (F4) entails

$$\begin{aligned} \varvec{\lambda }_{(n)}^{h}=\varvec{\lambda }_{(0)}^{h}={\textbf {F}}^{h}(t)^{\top }\ddot{\varvec{\chi }}^{h}(t)\;\;\mathrm{for~all~}n>0. \end{aligned}$$

(F18)

Then, after substituting (F7) and (F11b) in (F16), a calculation analogous to the one leading to (F17), but rather more tedious, yields

$$\begin{aligned} \varvec{\Lambda }_{(n+1)}^{h}&=2{\textbf {F}}^{h}(t)^{\top }\ddot{{\textbf {F}}}^{h}(t)-\varvec{\Lambda }_{(n)}^{h} \nonumber \\&\quad -2{\textbf {F}}^{h}(t)^{\top }\big ({\textstyle \sum \limits _i}\textbf{r}_{i(n)}\!\otimes \!\textbf{f}_{i(n+\frac{1}{2})}\!+\!{\textstyle \sum \limits _i}m_i\varvec{w}_{i(n)}\!\otimes \!\varvec{w}_{i(n)}\big )\big (\varvec{E}_{0}^{h}\big )^{\!-1} \nonumber \\&\quad -2{\textbf {F}}^{h}(t)^{\top }\Big (\!\big (\epsilon ^{-1}\textbf{I}\!-\!\textbf{L}^h(t)){\textstyle \sum \limits _i}m_i\textbf{u}_{i(n)}\!\otimes \!\varvec{w}_{i(n)}\!+\!{\textstyle \sum \limits _i}\textbf{u}_{i(n)}\!\otimes \!\textbf{f}_{i(n+\frac{1}{2})}\!\Big )\big (\varvec{E}_{0}^{h}\big )^{\!-1} \nonumber \\&\quad +\mathcal {O}(\epsilon ), \end{aligned}$$

(F19)

where

$$\begin{aligned} \begin{aligned} \textbf{r}_{i(n)}\,\,{:=}\,\,{\textbf {F}}^{h}(t)\textbf{s}_{i(n)},\quad \textbf{f}_{i(n+\frac{1}{2})}&\,\,{:=}\,\,{\textstyle \frac{1}{2}}\big (\textbf{f}_{i(n)}\!+\!\textbf{f}_{i(n+1)}\big ), \\ \varvec{w}_{i(n)}\,\,{:=}\,\,{\textbf {F}}^{h}(t)\dot{\textbf{s}}_{i(n)},\quad \textbf{L}^h(t))&\,\,{:=}\,\,\dot{{\textbf {F}}}^{h}(t){\textbf {F}}^{h}(t)^{-1}. \end{aligned} \end{aligned}$$

(F20)

Combining (F19) with the initial value (F5), we see that $\varvec{\Lambda }^{h}$ fluctuates around ${\textbf {F}}^{h}(t)^{\top }\ddot{{\textbf {F}}}^{h}(t)$. We remark that under PBCs, whenever a particle leaves the cell and one of its images enters it, the two simultaneous supplies of momentum compensate each other, while the corresponding supplies of moment of momentum do not, in general. This explains why $\varvec{\Lambda }^{h}$ fluctuates, while $\varvec{\lambda }^{h}$ does not (cf. (F18)).