Diffusion Rosenfeld relation

Diffusion-entropy scaling relation the Rosenfeld relation... 

The Rosenfeld relation between scaled diffusion and excess entropy is defined below [5] ... 

The microscopic origin of the Rosenfeld relation between D and can be understood in the following way. Let us assume k ,isa microscopic relaxation rate between two adjacent microscopic states in the phase space of the system. During the experimental time Tobs, the system makes tobs transitions and therefore visits that many microscopic states. So, we can set tobs proportional to the total number of microscopic states Q so that the states are uniformly distributed. In the next step we assume that each transition results in a displacement a of the particle. So, we have the following relation between diffusion and the rate of transitions as. 

The breakdown of the Rosenfeld relation along isotherms can be seen from the following speculation. The regions of different anomalies do not coincide with each other. In particular, in the case of core-softened, fluids, the diffusion anomaly region is located inside the structural anomaly one. It means that there are some regions where the diffusion is still normal, while the excess entropy is already anomalous. But this kind of behavior cannot be consistent with the Rosenfeld scaling law. 

We discussed the above derivation to bring out the essence of the Rosenfeld scaling relation, which is valid when a system is ergodic with fast transitions between the configurational states of the system, so that the diffusion coefficient increases with entropy. 

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