Continuum limits of discrete gases
\textit{J. Greenberg} and \textit{G. Hedstrom} [(*) Arch. Ration. Mech. Anal. 40; 160-165 (1971; Zbl 0214.353)] investigated the limiting behavior of a discrete interacting particle system as the initial particle spacing and masses tend to zero. The model studied assumed a countable collection of particles, each with mass \({\tilde \rho}\)h, moving along the real axis according to \({\tilde \rho}\)hẍ\({}_ k=0\), \(\dot x{}_ k(0)=u_ 0(kh)\), and \(x_ k(0)=kh\), for \(k=0,\pm 1,\pm 2,..\). When particles collide, momenta and energy are conserved. This condition leads to the following interaction rule: if particles \(p,p+1,...,p+q\) collide at some instant and if \(v_ p^->v^-_{p+1}>...>v^-_{p+q}\) are the particle velocities before the collision, then the velocities after the collision, which we denote by \((v^+_ p,v^+_{p+1},...,v^+_{p+q})\), are given by \((v^+_ p,v^+_{p+1},...,v^+_{p+q})=(v^-_{p+q},v^-_{p+q- 1},...,v^-_ p).\) The principal motivation behind the Greenberg and Hedstrom's investigation (*) was to establish the existence of the limiting continuum motion \[ \chi (x,t)=^{def}\lim_{h\to 0,\quad kh=x\quad fixed}x_ k(t) \] and to examine regularity properties of this limit. No limiting equation of motion was obtained for \(\chi\) though it was shown that the limit motion did exhibit shock waves as boundaries of regions where the particle motions were highly oscillatory. In this note we again examine the problem discussed in (*). What is new is the simpler characterization of the limiting field quantities such as density (\({\hat \rho}\)), velocity \((\hat u)\), and pressure \((\hat p)\), as well as a characterization of the equations satisfied by these fields. In certain cases we obtain a closed system of conservation laws for \({\hat \rho}\), \(\hat u,\) and \(\hat p.\)
initial particle spacing and masses tend to zero
scientific article; zbMATH DE number 4135610