Super-Hydrodynamic Limit in Interacting Particle Systems

Abstract

This paper is a follow-up of the work initiated in [3], where it has been investigated the hydrodynamic limit of symmetric independent random walkers with birth at the origin and death at the rightmost occupied site. Here we obtain two further results: first we characterize the stationary states on the hydrodynamic time scale and show that they are given by a family of linear macroscopic profiles whose parameters are determined by the current reservoirs and the system mass. Then we prove the existence of a superhyrdrodynamic time scale, beyond the hydrodynamic one. On this larger time scale the system mass fluctuates and correspondingly the macroscopic profile of the system randomly moves within the family of linear profiles, with the randomness of a Brownian motion

Hydrodynamic limit in a particle system with topological interactions

“…A similar property holds also when x ∈ B (3, * ) which is the set of all x such that x 1 = x 2 , x 3 = x 4 , x 1 and x 4 = x 1 (modulo permutation of labels all x ∈ B (3) are in B (3, * ) ). We write…”

“…The theorem has its continuum analogue which can be proved directly, see Section 4 of [3], but it can also be deduced from Theorem 6.10, as we shall see. …”

“…In [3], we shall study the stationary solutions of (2.11); they are linear functions with slope −2 j. We shall prove that any weak solution (in the sense of barriers) converges as t → ∞ to a linear profile, the one with the same total mass as the initial state.…”

The macroscopic behavior of pantographic sheets depends mainly on their microstructure: experimental evidence and qualitative analysis of damage in metallic specimens

“…However, we remark that the standard procedures used in [44,45] cannot be used when three point interactions at micro-level are not negligible or when relevant bending energy arises in beam lattices. In spite of the relevant and rigorous results obtained in homogenizing microscopically inhomogeneous systems (also when thermal phenomena are involved), for systems which are fluids macroscopically as shown, e.g., in [46][47][48][49], as well as for different classes of biological systems, see, e.g., [50,51]), the problem of homogenizing macroscopically solid behavior remains open if rigorous reasonings of statistical mechanics are to be employed. Therefore, heuristic methods, including mixed micromacro-and ad hoc phenomenological approaches, are expected to play a relevant role in the modeling of the structured media which we consider here.…”

Some Introductory and Historical Remarks on Mechanics of Microstructured Materials

“…Actually while one can obtain relevant and rigorous results in homogenizing microscopically inhomogeneous systems, also including in the picture thermal phenomena for systems behaving macroscopically as fluids (see the e.g. [83][84][85][86][87]) and also for different class of biological systems (see e.g. [88,89]), it is an open problem the determination, via statistical mechanics and rigorous reasonings, of micro properties leading, at macro level, to solid behavior.…”