Abstract
We study the entropy production that is associated with the growing or shrinking of a small granule in, for instance, a colloidal suspension or in an aggregating polymer chain. A granule will fluctuate in size when the energy of binding is comparable to , which is the “quantum” of Brownian energy. Especially for polymers, the conformational energy landscape is often rough and has been commonly modeled as being self-similar in its structure. The subdiffusion that emerges in such a high-dimensional, fractal environment leads to a Fokker–Planck Equation with a fractional time derivative. We set up such a so-called fractional Fokker–Planck Equation for the aggregation into granules. From that Fokker–Planck Equation, we derive an expression for the entropy production of a growing granule.
Highlights
Granules commonly occur in soft materials such as gels and biopolymer complexes
We show below how in a subdiffusive environment, the Fokker–Planck Equation involves fractional derivatives
Fokker–Planck Equation to derive a formula for the entropy production in a subdiffusive environment
Summary
Granules commonly occur in soft materials such as gels and biopolymer complexes. Granules are of the essence in colloidal suspensions. Entropy 2018, 20, 651 diffuses “normally” in its conformational space (i.e., following h x2 (t)i ∝ t, where x is the diffused distance and t is the time), but instead performs so-called subdiffusion [6,7], i.e., h x2 (t)i ∝ tα , where 0 < α < 1. This equation is first order in t and second order in v.
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