Micro-macro transitions in the atomic chain

[Next]:

Scientific-technical Services

[Up]:

Projects

[Previous]:

Quasilinear nonsmooth evolution systems in L^p-spaces

[Contents]

[Index]

Micro-macro transitions in the atomic chain

Collaborator: W. Dreyer (FG 7), M. Herrmann (FG 7), J. Sprekels (FG 1)

Cooperation with: A. Mielke (Universität Stuttgart)

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

Description: During the period of this report, we have studied micro-macro transitions for the (nonlinear) atomic chain in the context of modulation theory. We have focused on the following problems.

We apply the ideas of classic modulation theory to the nonlinear atomic chain.
We study the thermodynamic properties of the resulting modulation equations including an equation of state and a corresponding Gibbs equation.
We consider special atomic interaction potentials and derive the corresponding modulation equations.

In this project we closely collaborate with the group of A. Mielke, University of Stuttgart.

The atomic chain consists of N identical particles, labeled by $\alpha$ = 1..N, which are located at their positions x(t). The dynamics of the chain is described by Newton's equation

$\displaystyle \ddot{{x}}_{\alpha}^{}$ (t) = $\displaystyle \Phi^{\prime}_{}$ $\displaystyle \Big($ x₊₁(t) - x(t) $\displaystyle \Big)$ - $\displaystyle \Phi^{\prime}_{}$ $\displaystyle \Big($ x(t) - x_-1(t) $\displaystyle \Big)$ ,

(1)

where $\Phi$ denotes a convex interaction potential. In order to pass to the thermodynamic limit N $\rightarrow$ $\infty$ , we introduce a scaling parameter $\varepsilon$ = 1/N and we define the macroscopic time $\overline{{t}}$ and particle index $\overline{{\alpha}}$ by

$\displaystyle \overline{{t}}$ = $\displaystyle \varepsilon$ t, $\displaystyle \overline{{\alpha}}$ = $\displaystyle \varepsilon$ $\displaystyle \alpha$ .

(2)

In contrast to the macroscopic variables, we interpret t and $\alpha$ as the microscopic time and particle index, respectively.

The general strategy of modulation theory is as follows:

We identify a family of special solutions of (1), which are parametrized by a finite number of parameters. In the case at hand, the special solutions are traveling waves.
We modulate the parameters on the macroscopic scale, so that there result at least approximate solutions of (1). There exist modulation restrictions in form of macroscopic PDEs.

A traveling wave for the atomic chain has parameters r, v, k, $\omega$ and can be written as

x $\displaystyle \left(\vphantom{{t}}\right.$ t $\displaystyle \left.\vphantom{{t}}\right)$ = r $\displaystyle \alpha$ + vt + $\displaystyle \mathbb {X}$ $\displaystyle \left(\vphantom{{k\alpha+\omega{t}}}\right.$ k $\displaystyle \alpha$ + $\displaystyle \omega$ t $\displaystyle \left.\vphantom{{k\alpha+\omega{t}}}\right)$ ,

(3)

Here, $\mathbb {X}$ $\left(\vphantom{ {\varphi}}\right.$ $\varphi$ $\left.\vphantom{ {\varphi}}\right)$ is the 1-periodic wave profile, which describes the microscopic oscillations. Since the wave profile depends on r, v, k, and $\omega$ , we write $\mathbb {X}$ (r, v, k, $\omega$ ; $\varphi$ ).

A modulated traveling wave results, if we allow variations of the traveling wave parameters on the macroscopic scale. More precisely, we set

x $\displaystyle \left(\vphantom{{t}}\right.$ t $\displaystyle \left.\vphantom{{t}}\right)$	=	$\displaystyle {\frac{{1}}{{\varepsilon}}}$ X( $\displaystyle \varepsilon$ t, $\displaystyle \varepsilon$ $\displaystyle \alpha$ ) + $\displaystyle \tilde{{\mathbb{X}}}$ ( $\displaystyle \varepsilon$ t, $\displaystyle \varepsilon$ $\displaystyle \alpha$ ; $\displaystyle {\frac{{1}}{{\varepsilon}% }}$ $\displaystyle \Theta$ ( $\displaystyle \varepsilon$ t, $\displaystyle \varepsilon$ $\displaystyle \alpha$ ;)),	(4)
$\displaystyle \tilde{{\mathbb{X}}}$ ( $\displaystyle \varepsilon$ t, $\displaystyle \varepsilon$ $\displaystyle \alpha$ $\displaystyle \varphi$ )	=	$\displaystyle \mathbb {X}$ (r( $\displaystyle \varepsilon$ t, $\displaystyle \varepsilon$ $\displaystyle \alpha$ ), v( $\displaystyle \varepsilon$ t, $\displaystyle \varepsilon$ $\displaystyle \alpha$ ), k( $\displaystyle \varepsilon$ t, $\displaystyle \varepsilon$ $\displaystyle \alpha$ ), $\displaystyle \omega$ ( $\displaystyle \varepsilon$ t, $\displaystyle \varepsilon$ $\displaystyle \alpha$ )),	(5)

where

v = $\displaystyle {\frac{{\partial X}}{{\partial \overline{t}}}}$ , r = $\displaystyle {\frac{{\partial X}}{{\partial \overline{\alpha}}}}$ , $\displaystyle \omega$ = $\displaystyle {\frac{{\partial \Theta}}{{\partial \overline{t}}}}$ , andk = $\displaystyle {\frac{{\partial \Theta}}{{\partial \overline{\alpha}}}}$ .

(6)

The modulation equations govern the macroscopic evolution of the fields r, v, k, and $\omega$ and ensure that the ansatz (4) provides approximate solutions satisfying (1) up to order $\mathcal {O}$ $\left(\vphantom{ {\varepsilon^{2}}}\right.$ $\varepsilon^{{2}}_{}$ $\left.\vphantom{ {\varepsilon^{2}}}\right)$ . The modulation equations read

$\displaystyle {\frac{{\partial}}{{\partial \overline{t}}% }}$ $\displaystyle \begin{pmatrix}% r v k S\end{pmatrix}$ + $\displaystyle {\frac{{\partial}}{{\partial \overline{t}}% }}$ $\displaystyle \begin{pmatrix}% -v +p -\omega +g\end{pmatrix}$ = 0.

(7)

There is an immediate physical interpretation of all quantities: r - specific length, v - mean velocity, p - pressure, k - wave number, $\omega$ - frequency, S - specific entropy density, g - entropy flux. Consequently, the four equations in (7) are the macroscopic conservation laws for mass, momentum, wave number, and entropy. The system is closed by means of a Gibbs equation

dE = v dv + $\displaystyle \omega$ dS - p dr - g dk.

(8)

Here, E abbreviates the specific total energy, which is given by an equation of state E = E(r, v, k, $\omega$ ). Unfortunately, the equation of state is given only implicitly, and thus a complete understanding of the modulation equations needs more insight into the structure of traveling waves. However, there exist some special atomic interaction potentials $\Phi$ , for which the equation of state can be calculated explicitly:

the harmonic potential;
the case of hard sphere collisions;
a model of elastic collision that combines the cases 1 and 2;
the limit of small amplitudes.

References:

W. DREYER, M. KUNIK, Cold, thermal and oscillator closure of the atomic chain, J. Phys. A, Math. Gen., 33 (2000), pp. 2097-2129.
W. DREYER, M. HERRMANN, A. MIELKE, Micro-macro transition for the atomic chain via Whitham's modulation equation, Preprint no. 119 of DFG Priority Program ``Analysis, modelling and simulation of multiscale problems'', Universität Stuttgart, 2004, submitted.
A.-M. FILIP, S. VENAKIDES, Existence and modulation of traveling waves in particle chains, Commun. Pure Appl. Math., 51 (1998), pp. 693-736.
G.B. WHITHAM, Linear and Nonlinear Waves, vol. 1237 of Pure And Applied Mathematics, Wiley Interscience, New York, 1974.