Exact results for the extreme Thouless effect in a model of network dynamics
| Autor: | Weibin Zhang, Kevin E. Bassler, Royce K.P. Zia, Mohammadmehdi Ezzatabadipour |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
Physics
Phase transition Physics - Physics and Society Conjecture Stationary distribution Statistical Mechanics (cond-mat.stat-mech) Transcendental equation Populations and Evolution (q-bio.PE) FOS: Physical sciences General Physics and Astronomy Order (ring theory) Physics and Society (physics.soc-ph) 01 natural sciences Nonlinear Sciences - Adaptation and Self-Organizing Systems 010305 fluids & plasmas Critical point (thermodynamics) Simple (abstract algebra) FOS: Biological sciences 0103 physical sciences Thermodynamic limit 010306 general physics Quantitative Biology - Populations and Evolution Adaptation and Self-Organizing Systems (nlin.AO) Condensed Matter - Statistical Mechanics Mathematical physics |
| Popis: | If a system undergoing phase transitions exhibits some characteristics of both first and second order, it is said to be of 'mixed order' or to display the Thouless effect. Such a transition is present in a simple model of a dynamic social network, in which $N_{I/E}$ extreme introverts/extroverts always cut/add random links. In particular, simulations showed that $\left\langle f\right\rangle $, the average fraction of cross-links between the two groups (which serves as an 'order parameter' here), jumps dramatically when $\Delta \equiv N_{I}-N_{E}$ crosses the 'critical point' $\Delta _{c}=0$, as in typical first order transitions. Yet, at criticality, there is no phase co-existence, but the fluctuations of $f$ are much larger than in typical second order transitions. Indeed, it was conjectured that, in the thermodynamic limit, both the jump and the fluctuations become maximal, so that the system is said to display an 'extreme Thouless effect.' While earlier theories are partially successful, we provide a mean-field like approach that accounts for all known simulation data and validates the conjecture. Moreover, for the critical system $N_{I}=N_{E}=L$, an analytic expression for the mesa-like stationary distribution, $P\left( f\right) $, shows that it is essentially flat in a range $\left[ f_{0},1-f_{0}\right] $, with $f_0 \ll 1$. Numerical evaluations of $f_{0}$ provides excellent agreement with simulation data for $L\lesssim 2000$. For large $L$, we find $f_{0}\rightarrow \sqrt{\left( \ln L^2 \right) /L}$ , though this behavior begins to set in only for $L>10^{100}$. For accessible values of $L$, we provide a transcendental equation for an approximate $f_{0}$ which is better than $\sim$1% down to $L=100$. We conjecture how this approach might be used to attack other systems displaying an extreme Thouless effect. Comment: 6 pages, 4 figures |
| Databáze: | OpenAIRE |
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