1980 Preface * 1999 Preface * 1999 Acknowledgements * Introduction * 1 Circular Logic * 2 Phase Singularities (Screwy Results of Circular Logic) * 3 The Rules of the Ring * 4 Ring Populations * 5 Getting Off the Ring * 6 Attracting Cycles and Isochrons * 7 Measuring the Trajectories of a Circadian Clock * 8 Populations of Attractor Cycle Oscillators * 9 Excitable Kinetics and Excitable Media * 10 The Varieties of Phaseless Experience: In Which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways * 11 The Firefly Machine 12 Energy Metabolism in Cells * 13 The Malonic Acid Reagent ('Sodium Geometrate') * 14 Electrical Rhythmicity and Excitability in Cell Membranes * 15 The Aggregation of Slime Mold Amoebae * 16 Numerical Organizing Centers * 17 Electrical Singular Filaments in the Heart Wall * 18 Pattern Formation in the Fungi * 19 Circadian Rhythms in General * 20 The Circadian Clocks of Insect Eclosion * 21 The Flower of Kalanchoe * 22 The Cell Mitotic Cycle * 23 The Female Cycle * References * Index of Names * Index of Subjects

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/596B270197FE27DE5FABB598B6210622/S0025557200150892a.pdf/div-class-title-span-class-italic-the-geometry-of-biological-time-span-by-a-t-winfree-pp-544-dm68-corrected-second-printing-1990-isbn-3-540-52528-9-springer-div.pdf

The geometry of biological time , by A. T. Winfree. Pp 544. DM68. Corrected Second Printing 1990. ISBN 3-540-52528-9 (Springer)

Applied Numerical Analysis. By C.F. Gerald. Pp. xii, 340. £3·60. 1970. (Addison-Wesley.)

H E Stanley Oxford: University Press 1971 pp xx + 308 price ?5 In the past fifteen years or so there has been a lot of experimental and theoretical work on the nature of critical phenomena in the neighbourhood of second order phase transitions. It has not been easy to get a good overall view of this work without digging through the rather complex original literature, although there are some good review articles covering particular aspects of the work.

https://iopscience.iop.org/article/10.1088/0031-9112/23/4/018/pdf

Introduction to Phase Transitions and Critical Phenomena

The Stoner–Wohlfarth (SW) model is the simplest model that describes adequately the physics of fine magnetic grains, the magnetization of which can be used in digital magnetic storage (floppies, hard disks and tapes). Magnetic storage density is presently increasing steadily in almost the same way as electronic device size and circuitry are shrinking, and magnetism in general appears as a new contender for many novel computing applications that were considered traditionally beyond its range. Denser storage leads to finer magnetic grains and smaller size leads to magnetic grains so fine that they contain a single magnetic domain, i.e. a region in the material with a well-defined uniform magnetization best described with the mathematics of the SW model.

https://iopscience.iop.org/article/10.1088/0143-0807/29/3/008/pdf

The Stoner–Wohlfarth model of ferromagnetism

Non-linear and non-local transport processes in heterogeneous media: from long-range correlated percolation to fracture and materials breakdown

When an interacting many-body system, such as a magnet, is driven in time by an external perturbation, such as a magnetic field, the system cannot respond instantaneously due to relaxational delay. The response of such a system under a time-dependent field leads to many novel physical phenomena with intriguing physics and important technological applications. For oscillating fields, one obtains hysteresis that would not occur under quasistatic conditions in the presence of thermal fluctuations. Under some extreme conditions of the driving field, one can also obtain a nonzero average value of the variable undergoing such ``dynamic hysteresis.'' This nonzero value indicates a breaking of the symmetry of the hysteresis loop about the origin. Such a transition to the ``spontaneously broken symmetric phase'' occurs dynamically when the driving frequency of the field increases beyond its threshold value, which depends on the field amplitude and the temperature. Similar dynamic transitions also occur for pulsed and stochastically varying fields. We present an overview of the ongoing research in this not-so-old field of dynamic hysteresis and transitions.

https://arxiv.org/pdf/cond-mat/9811086.pdf

Dynamic transitions and hysteresis

We have studied, using Monte Carlo (MC) simulation for ferromagnetic Ising systems in one to four dimensions and solving numerically the mean-field (MF) equation of motion, the nature of the response magnetization m(t) of an Ising system in the presence of a periodically varying external field [h(t)=${\mathit{h}}_{0}$cos(\ensuremath{\omega}t)]. From these studies, we determine the m-h loop or hysteresis loop area A(=\ensuremath{\oint}mdh) and the dynamic order parameter Q(=\ensuremath{\oint}mdt) and investigate their variations with the frequency (\ensuremath{\omega}) and amplitude (${\mathit{h}}_{0}$) of the applied external magnetic field and the temperature (T) of the system. The variations in A are fitted to a scaling form, assumed to be valid over a wide range of parameter (\ensuremath{\omega},${\mathit{h}}_{0}$,T) values, and the best-fit exponents are obtained in all three dimensions (D=2,3,4). The scaling function is Lorentzian in the MF case and is exponentially decaying, with an initial power law, for the MC cases. The dynamic phase boundary (in the ${\mathit{h}}_{0}$-T plane) is found to be frequency dependent and the transition (from Q\ensuremath{\ne}0 for low T and ${\mathit{h}}_{0}$ to Q=0 for high T and ${\mathit{h}}_{0}$) across the boundary crosses over from a discontinuous to a continuous one at a tricritical point. These boundaries are determined in various cases.We find that the response can be generally expressed as m(t)=P(\ensuremath{\omega}(t-${\mathrm{\ensuremath{\tau}}}_{\mathrm{eff}}$)) where P denotes a periodic function with the same frequency \ensuremath{\omega} of the perturbing field and ${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$(${\mathit{h}}_{0}$,\ensuremath{\omega},T) denotes the effective delay. We established that this effective delay ${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$ of the response is the crucial term and it practically determines all the above observations for A, Q, etc. Investigating the nature of the in-phase (\ensuremath{\chi}\ensuremath{'}) and the out-of-phase (\ensuremath{\chi}\ensuremath{'}\ensuremath{'}) susceptibility, defined as \ensuremath{\chi}\ensuremath{'}=(${\mathit{m}}_{0}$/${\mathit{h}}_{0}$)cos(\ensuremath{\varphi}) and \ensuremath{\chi}\ensuremath{'}\ensuremath{'}=(${\mathit{m}}_{0}$/${\mathit{h}}_{0}$)sin(\ensuremath{\varphi}); \ensuremath{\varphi}=\ensuremath{\omega}${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$ [and ${\mathit{m}}_{0}$ is the amplitude of m(t)], we find that the loop area A is directly given by \ensuremath{\chi}\ensuremath{'}\ensuremath{'} and also the temperature variation of \ensuremath{\chi}\ensuremath{'}\ensuremath{'}, at fixed \ensuremath{\omega} and ${\mathit{h}}_{0}$, gives a prominent peak at the dynamic transition point. We have also studied the behavior of the response magnetization by the application of a short-duration (compared with the relaxation time) pulsed magnetic field.Here, we observed (both in the MC and MF cases) that the width ratio (of the half-width and the width of the response magnetization and of the pulsed field, respectively) and the susceptibility (the ratio of excess magnetization over its equilibrium value and the height of the pulsed field) both show sharp peaks at the order-disorder (ferromagnetic-paramagnetic) transition point. We have also studied similar response behavior of Ising systems in the presence of time-varying longitudinal and transverse fields, solving numerically the mean-field equation of motion. We have again studied here the nature of the dynamic phase transition and the behavior of the ac susceptibility (both longitudinal and transverse) across the dynamic phase boundary. For a short-duration pulse of the transverse field, the width ratio and the pulse susceptibility are again seen to diverge at the order-disorder transition point.

Response of Ising systems to oscillating and pulsed fields: Hysteresis, ac, and pulse susceptibility.

The thermodynamical behaviors of ferromagnetic systems in equilibrium are well studied. However, the ferromagnetic systems far from equilibrium became an interesting field of research in last few d...

Nonequilibrium Phase Transitions in Model Ferromagnets: A Review

The nonequilibrium dynamic phase transition, in the kinetic Ising model in presence of an oscillating magnetic field, has been studied both by Monte Carlo simulation (in two dimensions) and by solving the mean-field dynamical equation of motion for the average magnetization. The temperature variations of hysteretic loss (loop area) and the dynamic correlation have been studied near the transition point. The transition point has been identified as the minimum-correlation point. The hysteretic loss becomes maximum above the transition point. An analytical formulation has been developed to analyze the simulation results. A general relationship among hysteresis loop area, dynamic order parameter, and dynamic correlation has also been developed.

http://arxiv.org/pdf/cond-mat/9712309.pdf

Nonequilibrium phase transition in the kinetic Ising model: Is the transition point the maximum lossy point?

The dynamic phase transition has been studied in the two-dimensional kinetic Ising model in the presence of a time varying (sinusoidal) magnetic field by Monte Carlo simulation. The nature (continuous or discontinuous) of the transition is characterized by studying the distribution of the order parameter and the temperature variation of the fourth-order cumulant. For the higher values of the field amplitude the transition observed is discontinuous and for lower values of the field amplitude it is continuous, indicating the existence of a tricritical point (separating the nature of transition) on the phase boundary. The transition is observed to be a manifestation of stochastic resonance.

Muktish Acharyya

Papers

Dynamic transitions and hysteresis

Response of Ising systems to oscillating and pulsed fields: Hysteresis, ac, and pulse susceptibility.

Nonequilibrium Phase Transitions in Model Ferromagnets: A Review

Nonequilibrium phase transition in the kinetic Ising model: Is the transition point the maximum lossy point?

Nonequilibrium phase transition in the kinetic Ising model: Existence of a tricritical point and stochastic resonance