The correspondence between the Adam-Gibbs and the Rosenfield relations

In this paper, we derive both the Adam-Gibbs and the Rosenfield relations from the microscopic point of view and compare them with the numerical calculation for one and two dimensional systems. The comparison shows there is an excellent agreement between theoretical and numerical calculations for their valid zones (in terms of the thermodynamic temperature) as suggested by experiments. It implies that there may be a transition temperature at which the two relations correspond to each other. We derive a relation to calculate it. Then, we generalize the Rosenfield relation for configurational thermodynamic entropy like quantity(TELQ) and time-dependent Shanon information entropy. At the same time, using a description with a fictitious Hamiltonian, we show that time-dependent configurational Shanon information entropy for a thermodynamic system (of Brownian particles) which is characterized by the absolute temperature, can not be recognized as thermodynamic entropy. At best, it can be identified as a thermodynamic entropy-like quantity. Furthermore, the description based on the fictitious Hamiltonian may lead to the conclusion that the correspondence between the Shanon information entropy and thermodynamic entropy is not a singular feature at equilibrium. It may be a continuation of the correspondence between the information entropy and the thermodynamic entropy-like quantity. Thus, the present study appears to offer important justification for the postulate that the Shannon entropy at steady state may be regarded as a thermodynamic entropy. This postulate holds significant importance in the framework of stochastic thermodynamics.