Two-cycles in spin-systems: sequential versus synchronous updating in multi-state Ising-type ferromagnets

Abstract

Hamiltonians for general multi-state spin-glass systems with Ising symmetry are derived for both sequential and synchronous updating of the spins. The possibly different behaviour caused by the way of updating is studied in detail for the (anti)-ferromagnetic version of the models, which can be solved analytically without any approximation, both thermodynamically via a free-energy calculation and dynamically using the generating functional approach. Phase diagrams are discussed and the appearance of two-cycles in the case of synchronous updating is examined. A comparative study is made for the Q-Ising and the Blume-Emery-Griffiths ferromagnets and some interesting physical differences are found. Numerical simulations confirm the results obtained.

Phase Transition and Hysteresis in an Ensemble of Stochastic Spiking Neurons

“…Sometimes sequential dynamics, where only a reduced number of neurons is updated at every time-step, are used to simulate neural dynamics instead (Herz & Marcus, 1993). It has been shown that sequential and parallel updating can lead to different behavior of the Hopfield model (Fontanari & Köberle, 1988) and multi-state Ising-type ferromagnets (Bolle & Blanco, 2004), which motivates us to investigate the dependence of our results on the type of updating. We will use the most simplest case of sequential updating, where only one neuron is updated at each time-step and modify the model presented in section 2 in the following way:…”

Phase diagram for a zero-temperature Glauber dynamics under partially synchronous updates

“…In the last decade renewed interest in Glauber dynamics [1] has been observed, especially at zero temperature [2][3][4][5][6][7][8][9][10][11][12][13][14]. This is partially caused by recent experiments with so-called single-chain magnets (for a recent review, see [15]) but is also due to the development of the nonequilibrium statistical physics.…”

“…Although Glauber dynamics was originally introduced as a sequential updating process, interesting theoretical results can be obtained also using a synchronous updating mode [4,8,10,12,13,17]. Moreover, clear evidence of a relaxation mechanism which involves the simultaneous reversal of spins has been shown experimentally for magnetic chains at low temperatures [18].…”

Phase Transition between Synchronous and Asynchronous Updating Algorithms

“…The number of attractors in Boolean networks, for example, increases exponentially with the system size [12] for synchronous update, and with a power for critical Boolean networks [13] for asynchronous update. The phase diagrams of the Hopfield neural network model [3,14] and the Blume-Emery-Griffiths model [15] depend on the updating mode as well, while those of the Q-state Ising model [16,17], and the Sherrington-Kirkpatrick spin glass [18] are independent on the used scheme.…”