2 In the article [67] an extensive study of another model with bounded potential was reported. This work is con- cerned to the Herzian spheres system which is defined by the interparticle potential of the form by the full excess entropy S$_{ex}$ . It was also shown that Dzugutov formula does not work for silica modelled with an angular dependent potential [50]. It allows to say that Rosenfeld relation is more general. Remember that original idea underlying the Rosen- feld relation is to refer the system under investigation to the hard spheres system. In this respect it is inter- esting whether the Rosenfeld scaling relation is valid for the systems essentially different from the hard spheres. One of the examples of such systems is the system with potentials with negative curvature [19, 20]. It was shown in many publications that the behavior of such systems is very complex [41, 44, 45, 51–57]. In particular such systems can form complicated structures, like cluster liq- uids or different crystal phases. They can demonstrate maximum on the melting line and reentrant melting and many other unusual properties. In particular systems with negative curvature can demonstrate anomalous be- havior [17, 26, 30–32, 39, 45, 58]. The phase diagram of the Herzian spheres system demonstrates very complex behavior, including many crystal phases and reentrant melting. Anomalous dif- fusion is also reported [67]. Φ( r ) = { ε (1 − r/σ ) 5 / $^{2}$, r ≤ σ 0 , r > σ (7) Taking into account that the behavior of the systems with negative curvature potentials and the systems with bounded potentials is rather different from the behavior of hard spheres a question arises if the Rosenfeld scaling relations are applicable for such systems. The purpose of this article is to analyze the validity of the entropy scaling for the systems with anomalous behavior. For our analysis we have chosen the diffusion coefficient since it is the simplest transport coefficient to calculate in simulation. The article is organized as follows. In the second section we describe the models in- vestigated in the present work and the simulation setup. Section III gives the results of the simulations and the discussion of these results. Finally the section IV repre- sents our conclusions. It was suggested that the Rosenfeld relations can hold even in the case of anomalous diffusion [17, 18]. For example, in the paper [18] the dependence of both ex- cess entropy and diffusion coefficient on density are re- ported for the core-softening potential that consists of a combination of a Lennard-Jones potential plus a Gaus- sian well. This potential can represent a whole family of two length scales intermolecular interactions, from a deep double-well potential to a repulsive shoulder. Ac- cordingly to these dependencies both excess entropy and diffusion have non monotonic behavior which allows to preserve exponential dependence of the diffusion coef- ficient on the excess entropy. It means that the ther- modynamically anomalous regions are characterized by anomalous behavior of the excess entropy which induces anomalous diffusion as well. Three systems were studied in the present work: Herzian spheres, Gauss core model and a soft repulsive shoulder system. Another example of systems which can not be ap- proximated by a hard sphere model is the systems with bounded potentials [59–61, 67]. Since these potentials have no singularity in the origin the behavior of such system is strongly different from the behavior of hard spheres. One of the most common model with bounded poten- tial is the Gaussian Core Model (GCM). This system is defined by the potential II. THE SYSTEMS AND METHODS For the simulation of the Herzian spheres we used a system of 1000 particles in a cubic simulation box. NVE MD simulation was carried. Equations of mo- tion were integrated by velocity Verlet algorithm. The time step was set to dt = 0 . 0005. The equilibration run was 5 · 10 5 time steps and the production run 1 . 5 · 10 6 time steps. During the equilibration the velocities were rescaled to keep the temperature constant. The diffu- sions were computed via the Einstein relation for the densities from ρ = 1 . 0 till ρ = 15 . 0 with the step Δ ρ = 0 . 5. Additional simulations were done for com- puting the equation of state for the densities less then unity. Free energy of the liquid was calculated by inte- grating the pressure along an isotherm [68] and the excess entropy was obtained from the relation S$_{ex}$ = U − F ex Nk B T , where U is the internal potential energy of the liquid. The simulations were done for the set of ten isotherms: T = 0 . 01; 0 . 02; 0 . 03; 0 . 05; 0 . 1; 0 . 15; 0 . 2; 0 . 25; 0 . 3; 0 . 5. This potential was introduced by Stillinger [59] for sim- ulation of the plastic crystals system . The phase diagram of the GCM demonstrates two crystal phases - fcc and bcc [62]. Starting from the densities around ρσ 3 ≈ 0 . 25 the melting curve has a negative slope. It was also shown that GCM demonstrates liquid state anomalies: density anomaly [63, 64], diffusion anomaly [64, 65] and struc- tural anomaly [64]. Interestingly Stockes-Einstein rela- tion is also violated in the GCM system [66]. In the case of GCM the system consisted of 2000 particles. The time step was set to dt = 0 . 05. The equilibration and production runs were 4 · 10 5 and 1 · 10 6 time steps respectively. The diffusion U$_{G}$ ( r ) = εe − r $^{2}$/σ$^{2}$. (6)