A MATHEMATICAL DESCRIPTION OF THE CRITICAL POINT IN PHASE TRANSITIONS AYSE HUMEYRA BILGE * and ONDER PEKCAN † Faculty of Engineering and Natural Sciences Kadir Has University Istanbul, Turkey * ayse.bilge@khas.edu.tr † pekcan@khas.edu.tr Received 30 January 2013 Accepted 31 May 2013 Published 6 August 2013 Let yðxÞ be a smooth sigmoidal curve, y ðnÞ be its nth derivative and fx m;i g and fx a;i g, i ¼ 1; 2; ... , be the set of points where respectively the derivatives of odd and even order reach their extreme values. We argue that if the sigmoidal curve yðxÞ represents a phase transition, then the sequences fx m;i g and fx a;i g are both convergent and they have a common limit x c that we characterize as the critical point of the phase transition. In this study, we examine the logistic growth curve and the Susceptible-Infected-Removed (SIR) epidemic model as typical examples of symmetrical and asymmetrical transition curves. Numerical computations indicate that the critical point of the logistic growth curve that is symmetrical about the point ðx 0 ; y 0 Þ is always the point ðx 0 ; y 0 Þ but the critical point of the asymmetrical SIR model depends on the system parameters. We use the description of the solgel phase transition of polyacrylamide- sodium alginate (SA) composite (with low SA concentrations) in terms of the SIR epidemic model, to compare the location of the critical point as described above with the \gel point" determined by independent experiments. We show that the critical point t c is located in between the zero of the third derivative t a and the in°ection point t m of the transition curve and as the strength of activation (measured by the parameter k= of the SIR model) increases, the phase transition occurs earlier in time and the critical point, t c , moves toward t a . Keywords: Gelation; phase transition; epidemic models. PACS Nos.: 64.60.Bd. 1. Introduction A phase transition is the passage of a physical system from one stable state to another. The abrupt passage from one state to the other is characterized by a \critical point" and a universal behavior near this critical point. Criticality in phase transitions have two aspects: The ¯rst aspect is criticality with respect to a system parameter p: Depending on the values of its structural parameters, a system may or may not undergo a phase transition spontaneously or under a driving force; there is a International Journal of Modern Physics C Vol. 24, No. 10 (2013) 1350065 (19 pages) # . c World Scienti¯c Publishing Company DOI: 10.1142/S0129183113500654 1350065-1