Shortest paths on systems with power-law distributed long-range connections

We discuss shortest-path lengths $\ell(r)$ on periodic rings of size L supplemented with an average of pL randomly located long-range links whose lengths are distributed according to P_l \sim l^{-\xpn}. Using rescaling arguments and numerical simulation on systems of up to $10^7$ sites, we show that a characteristic length $\xi$ exists such that $\ell(r) \sim r$ for $r>\xi$. For small p we find that the shortest-path length satisfies the scaling relation \ell(r,\xpn,p)/\xi = f(\xpn,r/\xi). Three regions with different asymptotic behaviors are found, respectively: a) \xpn>2 where $\theta_s=1$, b) 1<\xpn<2 where 0<\theta_s(\xpn)<1/2 and, c) \xpn<1 where $\ell(r)$ behaves logarithmically, i.e. $\theta_s=0$. The characteristic length $\xi$ is of the form $\xi \sim p^{-\nu}$ with \nu=1/(2-\xpn) in region b), but depends on L as well in region c). A directed model of shortest-paths is solved and compared with numerical results.Comment: 10 pages, 10 figures, revtex4. Submitted to PR

How Virtual Machinery Can Bridge The “Explanatory Gap” In Natural and Artificial Systems.

These slides are available in my ‘talks ’ directory

On the influence of interactions between phases on the mechanical stability of retained austenite in transformation-induced plasticity multiphase steels

The mechanical stability of dispersed retained austenite, i.e., the resistance of this austenite to mechanically induced martensitic transformation, was characterized at room temperature on two steels which differed by their silicon content. The steels had been heat treated in such a way that each specimen presented the same initial volume fraction of austenite and the same austenite grain size. Nevertheless, depending on the specimen, the retained austenite contained different amounts of carbon and was surrounded by different phases. Measurements of the variation of the volume fraction of untransformed austenite as a function of uniaxial plastic strain revealed that, besides the carbon content of retained austenite, the strength of the other phases surrounding austenite grains also influences the austenite resistance to martensitic transformation. The presence of thermal martensite together with the silicon solid-solution strengthening of the intercritical ferrite matrix can "shield" austenite from the externally applied load. As a consequence, the increase of the mechanical stability of retained austenite is not solely related to the decrease of the M-s temperature induced by carbon enrichment