Multiscaling for Systems with a Broad Continuum of Characteristic Lengths and Times: Structural Transitions in Nanocomposites

摘要

The multiscale approach to N-body systems is generalized to address the broadcontinuum of long time and length scales associated with collective behaviors.A technique is developed based on the concept of an uncountable set of timevariables and of order parameters (OPs) specifying major features of thesystem. We adopt this perspective as a natural extension of the commonly useddiscrete set of timescales and OPs which is practical when only a few,widely-separated scales exist. The existence of a gap in the spectrum oftimescales for such a system (under quasiequilibrium conditions) is used tointroduce a continuous scaling and perform a multiscale analysis of theLiouville equation. A functional-differential Smoluchowski equation is derivedfor the stochastic dynamics of the continuum of Fourier component orderparameters. A continuum of spatially non-local Langevin equations for the OPsis also derived. The theory is demonstrated via the analysis of structuraltransitions in a composite material, as occurs for viral capsids and molecularcircuits.