How to introduce temperature to the 1D Sznajd model

• Published in: Physica A 2011-06, Vol.390 (11), p.2087-2095
• Main Author: Kondrat, Grzegorz
• Format: Article
• Language: English
• Subjects: Compatibility; Detailed balance; Dynamical systems; Dynamics; Equilibrium; Equivalence; Fluctuation; Lists; Opinion dynamics; Statistical mechanics; Temperature
• ISSN: 0378-4371; 1873-2119
• DOI: 10.1016/j.physa.2011.01.007

We investigate the possibility of introducing temperature to the one dimensional Sznajd model and propose a natural extension of the original model by including other types of interactions. We characterise different kinds of equilibria into which the extended system can evolve. We determine the consequences of fulfilling the detailed balance condition and we prove that in some cases it is equivalent to microscopic reversibility. We found the equivalence of the model to the standard (inflow) model with interactions up to next nearest neighbors. It is shown that under some constraints there exists a Hamiltonian compatible with the dynamics and its form resembles that of the 1D ANNNI model. It appears however, that the standard approach of constructing temperature from the Hamiltonian fails. In this situation we propose a simple definition of the temperature-like quantity that measures the size of fluctuations in the system at equilibrium. The complete list of zero-temperature degenerated cases as well as the list of ground states of the derived Hamiltonian are provided. ► I propose the extension of the 1D Sznajd model for other types of interactions. ► I have found the equivalence between the extended outflow model and 1D ANNNI model with the appropriate Hamiltonian. ► I discuss when the detailed balance condition is obeyed by the system. ► I discuss the impossibility of introducing the temperature to the model in the canonical way, proposing some other possibility.