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A modified kinetic Ising model with Metropolis dynamics, so-called $q$-neighbor Ising model, is investigated on random graphs. In this model, each spin interacts only with $q$ spins randomly chosen from its neighborhood. Investigations are performed by means of Monte Carlo (MC) simulations and the analytic pair approximation (PA). The range of parameters such as the size of the $q$-neighborhood and the mean degree of nodes of the random graph is determined for which the model exhibits continuous or discontinuous ferromagnetic (FM) phase transition with decreasing temperature. It is also shown that, in the case of discontinuous transition for large enough and fixed mean degree of nodes, the width of the hysteresis loop oscillates with the parameter $q$, expanding for even and shrinking for odd values of $q$. Predictions of the PA show satisfactory quantitative agreement with results of MC simulations.

Equilibrium properties of Blume–Emery–Griffiths (BEG) model with the arbitrary bilinear (J), biquadratic (K) and crystal field interaction (D) are studied on a body centered cubic lattice by using the pair approximation of the cluster variation method, which is identical to the Bethe approximation. The thermal variation of the stable, metastable and unstable dipole and quadrupolar moment order parameters are found for various values of the two different coupling parameters J/K and D/K. The phase transitions of the stable, metastable and unstable branches of the order parameters are investigated extensively. The critical temperatures in the case of a second-order phase transition are obtained for different values of J/K and D/K by the Hessian determinant. The first-order phase transition temperature is determined by matching the values of the two branches of the free energy values. On the other hand, nonequilibrium behavior of the model is studied by using the path probability method with the pair distribution and the set of nonlinear differential equations, which is also called the dynamic or rate equations, is obtained. The solution of the dynamic equations is expressed by means of flow diagrams. The stable, metastable and unstable solutions are shown and the "overshooting" phenomenon is seen in the flow diagrams, explicitly. The role of the unstable points, as separators between the stable and the metastable points, is described and how a system freezes in a metastable state is also investigated, extensively. Moreover, the dynamic equations are also solved by using the Runge–Kutta method in order to study the relaxation of order parameters. In this investigation, the "flatness" property of metastable states and the "overshooting" phenomenon are seen explicitly.

As a continuation of our previously published work, the metastable phase diagram of the Blume–Emery–Griffiths model with the arbitrary bilinear (J), biquadratic (K) and crystal field interaction (D) is presented in addition to the equilibrium phase diagram in (T/K, J/K) and (T/K, D/K) plane by using the pair approximation of the cluster variation method on a body centered cubic lattice. We also calculate the phase transitions for the unstable branches of order parameters. The calculated first- and second-order phase boundaries of the unstable branches of the order parameters are superimposed on the equilibrium phase diagram and metastable phase diagram. It is found that the metastable phase diagram and the first- and second-order phase boundaries for unstable branches of order parameters always exist at low temperatures, which are consistent with the experimental and theoretical works.

As a continuation of previously published work [A. Erdinç and M. Keskin, Int. J. Mod. Phys. B 18, 1603 (2004), doi:10.1142/S0217979204024823] the hysteretic properties of magnetization ($m$) and quadrupolar moment ($q)$ have been investigated for the spin-1 Blume–Emery–Griffiths model using pair approximation. The magnetic field ($h$) dependence of $m$ is produced for different values of the lattice coordination number ($z$), biquadratic interaction ($k$) and single-ion anisotropy ($d$). The ($m,h$) loops show different and novelty properties when $d<0$. We also observed the hysteresis cycles in the ($q,d)$, ($q,h)$ and ($m,d$) planes. These curves strongly depend on biquadratic interaction $k$. Our results are found to be in good agreement with the other theoretical findings.

We study a three-state stochastic particle system on the square lattice, which extends the contact process. The phase diagram is analyzed by the mean-field approximation, the pair approximation, and numerics. The pair approximation turns out to be better than the meanfield ansatz. We also show that the Harris-FKG type correlation inequalities approximately hold in the present model.

We study the transition from stochasticity to determinism in the three-strategy pair-approximated prisoner's dilemma game. We show that the stochastic solution converges to the deterministic limit cycle attractor as the number of participating players increases. Importantly though, between the stochastic and periodic solutions, we reveal a broad range of population sizes for which the system exhibits deterministic behavior, yet fails to settle onto the limit cycle attractor. We show that these states are characterized by chaos via a rigorous treatment. Results are discussed in view of their sociological importance.

This paper studies asynchronously-updated linear voter cellular automata models, with local and global interactions. In the locally-interacting model, sites copy the states of randomly-chosen sites in the local neighborhood; in the globally-interacting model, sites copy the states of sites chosen at random from the entire lattice. A multiscale model is also considered, mixing the two types of interactions. Such models can be used to model simple genetic drift, or the spread of opinions or ideas. The pair approximation moment-closure method is used to develop systems of differential equations describing the dynamics of the models with k states per site; stochastic spatially explicit simulations are also used. In simulations on large lattices, state frequencies among sites remain relatively constant at their initial values, while autocorrelations between adjacent sites move toward quasiequilibrium distributions and then remain constant with minor stochastic fluctuations. Full analytical solutions for the transient dynamics of the local autocorrelation (clustering) among adjacent sites are obtained with k = 2 states, and compared with simulations. Both show that increasing global interactions decreases spatial autocorrelation, and that even in the absence of long-distance interaction, there will still be mixing among the states. That is, the lattice will not converge to a quasiequilibrium configuration where regions in different states are maximally isolated from each other (completely segregated), but instead sites will have a strictly positive probability of being adjacent to sites in different states.

We discuss the Nucleon Pair Approximation (NPA) of the shell model by a short review, which includes a history survey, its physical foundation, validity, and recent applications, as well as our perspectives.