Non-Equilibrium Statistical Mechanics of a Mixed Order Phase Transition in a Dynamical Network Model



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The well-established framework in thermodynamics and statistical mechanics is devoted to systems in equilibrium requiring infinite thermal reservoirs or ideal insulators. Ideal conditions (infinite thermal reservoirs or ideal insulators) to maintain equilibrium are practically impossible to achieve; hence, a majority of physical systems are far from equilibrium. Therefore, developing an analytical platform for non-equilibrium systems is of vital interest for real systems; however, it comes at the cost of losing the Boltzmann-Gibbs framework developed for systems in equilibrium. Qualitative and quantitative arguments, based on the competition between energy and entropy, are described by thermodynamical potentials (free energy). These often fail to describe non-equilibrium systems. Beyond the scope of equilibrium systems, there is limited success, both theoretically and experimentally, in the understanding of non-equilibrium statistical mechanics (NESM). The Generalized Introverts Extroverts (GIE) model and Extreme Introverts Extrovert (XIE) model are among the models that, despite their simplicity, remain non-equilibrium models with the mixed order phase transition (MOT). By decomposing the XIE ensemble into small subensembles with small fluctuations, I have developed a new Mean Field approach, which can capture the exact results of the XIE model at the critical point. Unlike other models have Mixed Order Phase Transition (Inverse Distance Squared Ising model, Poland/Scheraga model), the XIE model lacks spatial coordinates; therefore, the correlation length is not well defined. However, I have found that there are non-trivial pairwise correlations between the elements of the adjacency matrix near criticality. Furthermore, I have shown that the GIE model reaches a non-equilibrium steady state (NESS) when the preferred degrees are non-integer. The similarities between the characteristics of NESS and the hydrodynamics inspired me to measure the probability currents, stream function, and vorticities in the NESS of the GIE model. Analogies between NESS and hydrodynamics facilitate a better understanding of the nature of non-equilibrium steady states. I hope that hydrodynamic analogies can lead to development of a theoretical framework similar to the Boltzmann-Gibbs framework in the paradigm of non-equilibrium steady state.



Statistical mechanics, Non-equilibrium statistical mechanics, Boltzmann-Gibbs framework, Social networks, Mixed order phase transitions, Hydrodynamics


Portions of this document appear in: Zia, R. K. P., Weibin Zhang, Mohammadmehdi Ezzatabadipour, and Kevin E. Bassler. "Exact results for the extreme Thouless effect in a model of network dynamics." EPL (Europhysics Letters) 124, no. 6 (2019): 60008.