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Multiscale modeling of thermoelastic bodies

Collaborator: W. Dreyer (FG 7), M. Herrmann (FG 7), J. Sprekels  

(FG 1)

Supported by: DFG: Priority Program ``Analysis, Modellbildung und Simulation von Mehrskalenproblemen'' (Analysis, modeling and simulation of multiscale problems)

Description:

This is a joint project of Dreyer/Sprekels. In the last period two questions were posed and answered:

1.

Is a continuum limit of a microscopic particle system automatically a macroscopic limit?

2.

Is there a rigorous procedure so that the macroscopic limit of time means of microscopic observables can be written as means with macroscopic distribution functions?

Regarding the first question, we have shown that the answer is affirmative only for a special class of microscopic initial conditions, namely those that can be prepared macroscopically.

Regarding the second question, we have studied simple cases, where the microscopic observables only depend on time but not on the particle index. A well-known example is the oscillator motion of the atomic chain, which can be found in [1].

Let $\tau$ be the microscopic time. $f(r(\tau ))$ denotes a microscopic observable that depends continuously on the microscopic variable $r(\tau )$ with $0<r_{-}\leq r(\tau )\leq r_{+}$. The macroscopic time is defined by $t=\epsilon \tau ,$ where $\epsilon$is a small positive parameter, and the macroscopic limit can be established for $\epsilon \rightarrow 0$. To this end we define

$$\bar{f}(\tau )\equiv \int\limits_{0}^{\infty }\chi (\varthet... ... }),\qquad \bar{r}^{\epsilon }(t)\equiv r(\frac{t}{\epsilon }),$$ (1)

where $\chi (\tau )$ denotes a window function with a finite microscopic support. If $r(\tau )$ solves the Newtonian equation of the oscillator motion according to [1], then there holds

$$\lim_{\epsilon \rightarrow 0}\bar{f}^{\epsilon }(t)=\int\limits_{r_{-}}^{r_{+}}f(r)w(r_{-},r_{+},r)dr,$$ (2)

where w is the distribution function of the oscillator motion, and r_-,r₊ are the statistical parameters, see [2] for details.

A second simple example serves to illustrate the suitable mathematics of the macroscopic limit. We consider the following Newtonian system that contains the small parameter $\epsilon$ :

 

$$\frac{d}{ds}{Q}(\tau ) A\Big(Q(\tau )\Big)+{\Big( \begin{array} {c} {0} \ {f(\varepsilon {\tau })}\end{array}\Big)},\quad Q(0)=Q_{0}\,,$$ = (3)

$${\Big( \begin{array} {c} {q} \ {p}\end{array}\Big)},\quad A(Q)={\Big( \begin{array} {c} {p} \ {G(q)}\end{array}\Big)}.$$ Q =

We interprete $\tau ,q,p$ as microscopic time, position, and momentum, respectively, of a microscopic particle, and we assume that an external force f is given, which changes very slowly on the microscopic time scale. For simplicity, we consider the simplest case of a harmonic oscillator with G=-q.

The macroscopic time is defined by $t=\epsilon \tau$, so that the force f varies only on the macroscopic scale, while the variable Q oscillates on the microscopic scale. Fig. 1 shows a typical example: There is a rapid oscillation for small $\epsilon$ from the macroscopic viewpoint.

 

Fig. 1: Numerical example

 

For fixed $\epsilon$ and fixed macroscopic time t_final there exists, within the time interval $0\leq \tau \leq \epsilon ^{-1}t_{final}$, a solution $Q_{\epsilon }(\tau )$ of (3) that we rescale according to

$$\overline{Q}_{\varepsilon }(t):=Q_{\varepsilon }(\varepsilon ^{-1}t),\quad 0\leq {t}\leq {t}_{final}.$$ (4)

Next we consider various observables $\Psi$, which we define as continuous functions of Q and t. It is the objective to study the evolution of $\Psi (Q(t),t)$ on the macroscopic scale, i.e. we are interested in the function $\bar{\Psi}_{\varepsilon }(\overline{Q}_{\varepsilon }(t),t)$ in the limit $\epsilon \rightarrow 0$.

The most important observable is the energy e, which is given by

$$e(Q,\,t)=\frac{1}{2}p^{2}+\frac{1}{2}q^{2}-qf(t).$$ (5)

The rescaled energy is defined by $\bar{e}_{\epsilon }(\,t)=e(\overline{Q}_{\varepsilon }(t),\,t)$ within $0\leq t\leq t_{final}$. We expect that the microscopic and the macroscopic scale decouples in the limit $\epsilon \rightarrow 0$ in a certain sense, and in fact this expectation can be described as follows:

The functions $\overline{Q}_{\varepsilon }$ and $\overline{e}_{\varepsilon }$ converge to Young measures $\mu$and $\nu$, respectively. Precisely we write $\mu =\mu (t,\,Q)$ and $\nu =\nu (t,\,e)$ although $\mu$ and $\nu$ are no functions but Young measures. The Young measures $\mu$ and $\nu$ are completely determined by the following properties

1.

For almost every $0\leq t\leq t_{final}$, the decomposition of $\mu$ at time t is a solution of the following time-dependent but stationary Liouville equation  

$${\mathrm{div}}_{Q}\,\Big(\mu (t,\,Q)\cdot \lbrack A(Q)-{\Big( \begin{array} {c} {0} \ {f(t)}\end{array}\Big)}]\Big)=0.$$ (6)

2.

Equation (6) implies that $\mu$ is completely determined by $\nu$.

3.

The Young measure $\nu$ is a solution to the following transport equation

$$\frac{\partial \,\nu }{\partial \,t}(t,\,e)-\dot{f}(t)f(t)\frac{\partial \,\nu }{\partial \,e}(t,\,e)=0.$$

References:

1.   W. DREYER, M. KUNIK, Cold, thermal and oscillator closure of the atomic chain, J. Phys. A, 33 (2000), pp. 2097-2129.
2.   W. DREYER, M. HERRMANN, Simple but rigorous micro-macro transitions, in preparation.

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