Rigorous foundation of the Adam-Gibbs relation

The Adam-Gibbs relation which relates a transition rate to a configurational entropy has been used frequently in the analysis of relaxation in non-equilibrium systems, though its foundation is sloppy. In this presentation, I give a rigorous proof of the Adam-Gibbs relation on the basis of the free energy landscape (FEL) approach to non-equilibrium systems [1]. The FEL consists of many basins in a 3N dimensional space, and the elementary process of structural relaxations occurs between two adjacent basins. I define a cooperatively rearranging region (CRR) by the area of atoms forming two adjacent basins and tessellate the entire FEL into a set of CRR's. When the system contains N particles and the number of CRR’s is L, the average size of CRR is given by <??_??RR>=N/L. Denoting the probability of finding the k-th CRR of size ??_?? by ??_??, I find that the configurational entropy is given by ??_??= <−??_??lnΠ_??=1^????_??> =????_??^∗=??_??^∗??∕<??_??RR> with ??_??^∗ =??_??ln2. The transition rate within the k-th CRR can be written as ??_??=??₀exp (−Δ?? ??_??/??_????), where Δ?? is the activation free energy per particle and ??₀ is the attempt frequency. Then, the average transition rate is given by W(T)= (Π_k=1^??W_??)^1/??=??₀exp (−Δ?? Σ_??=1^????_??/??k_????).Therefore, the transition rate is given by W(T)= ??₀exp [−Δ?? ????_??^∗ /??_??????_??(??)], which is the Adam-Gibbs relation. With the proper definition of CRR and the configurational entropy, I assess various estimations of the size of CRR reported before.