Water-like Anomalies and Breakdown of the Rosenfeld Excess Entropy Scaling Relations for the Core-Softened Systems: Dependence on the Trajectory in Density-Temperature Plane

We show that the existence of the water-like anomalies in kinetic coefficients in the core-softened systems depends on the trajectory in $\rho-T$ plane along which the kinetic coefficients are calculated. In particular, it is shown that the diffusion anomaly does exist along the isotherms, but disappears along the isochores. We analyze the applicability of the Rosenfeld entropy scaling relations to the systems with the core-softened potentials demonstrating the water-like anomalies. It is shown that the validity of the of Rosenfeld scaling relation for the diffusion coefficient also depends on the trajectory in the $\rho-T$ plane along which the kinetic coefficients and the excess entropy are calculated. In particular, it is valid along isochors, but it breaks down along isotherms.