During first 18 month of the project we have successfully undertaken the studies of dynamical properties of long-range interacting nanoparticle systems and the results of joint studies have been in the international Journal (J. Phys. A: Math. Theor, v. 50, 12LT02. https://doi.org/10.1088/1751-8121/aa5fcf). In particular, we have investigated how one dimensional model works in order to generalize the results for the much more complicated three dimensional spin system. We have examined the existence and propagation of solitons in a long-range extension of the quartic Fermi–Pasta–Ulam (FPU) chain of anharmonic oscillators where the coupling in the linear term decays as a power-law. We have obtained an analytic perturbative expression of traveling envelope solitons by introducing a non linear Schrödinger equation for the slowly varying amplitude of short wavelength modes. Due to the non analytic properties of the dispersion relation, it is crucial to develop the theory using discrete difference operators. Those properties are also the ultimate reason why kink-solitons may exist but are unstable, at variance with the short-range FPU model. We have successfully compared these approximate analytic results with numerical simulations.

Next we proceed with thermodynamic properties of long-range nanoparticle systems. For this purpose we have performed numerical simulations on three dimensional spin system forming simple cubic lattice and coupled by dipolar forces. We have examined both canonical (system coupled with thermal bath) and microcanonical (isolated system) ensembles and plotted caloric curve. First preliminary results display ensemble inequivalence and existence of temperature jumps and negative specific heat in microcanonical case. It should be especially emphasized that negative specific heat has not been observed up to now in laboratory systems and now we are working to choose specific parameters for the experimental suggestions for observing this exotic effect.