Starting from an anomalous monomeric system, where particles interact via a two-scale core-softened potential, we investigate how the system properties evolve inasmuch as particles are put together to form polymers whose chain size varies from 4 up to 32 monomers. We observed that the density and diffusion anomaly regions in the pressure versus temperature phase diagram of the monomeric system is smaller in the monomeric system when compared with the polymers. We also found that the polymers do not fold into themselves to form solid spheres instead they tend to maximize the chain-fluid contact. Also, Rouse and Reptation models can be employed to describe the polymers diffusive behaviour. But, in contrast to results of simulations where mere interacts via Lennard-Jones potentials, our results shown a much shorter entanglement length of at most 8 monomers.
A cosmological extension of the Eisenhart–Duval metric is constructed by incorporating a cosmic scale factor and the energy-momentum tensor into the scheme. The dynamics of the spacetime is governed by the Ermakov–Milne–Pinney equation. Killing isometries include spatial translations and rotations, Newton–Hooke boosts and translation in the null direction. Geodesic motion in Ermakov–Milne–Pinney cosmoi is analyzed. The derivation of the Ermakov–Lewis invariant, the Friedmann equations and the Dmitriev–Zel'dovich equations within the Eisenhart–Duval framework is presented.