Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Random hysteresis loops

Gioia Carinci

Full-text: Open access


Dynamical hysteresis is a phenomenon which arises in ferromagnetic systems below the critical temperature as a response to adiabatic variations of the external magnetic field. We study the problem in the context of the mean-field Ising model with Glauber dynamics, proving that for frequencies of the magnetic field oscillations of order $N^{-2/3}$, $N$ the size of the system, the “critical” hysteresis loop becomes random.


L’hystérésis dynamique est un phénomène qu’on observe dans les systèmes ferromagnétiques au-dessous de la temperature critique, en réponse à des variations adiabatiques du champ magnétique extérieur. Nous étudions le problème dans le contexte du modéle d’Ising de champ moyen avec la dynamique de Glauber, en montrant que, pour des fréquences d’oscillations du champ magnétique d’ordre de $N^{-2/3}$, avec $N$ la taille du système, la boucle d’hystérésis « critique » devient aléatoire.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 49, Number 2 (2013), 307-339.

First available in Project Euclid: 16 April 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs

Hysteresis Ising Mean field Glauber dynamics Macroscopic fluctuations


Carinci, Gioia. Random hysteresis loops. Ann. Inst. H. Poincaré Probab. Statist. 49 (2013), no. 2, 307--339. doi:10.1214/11-AIHP461.

Export citation


  • [1] M. Acharyya and B. K. Chakrabarti. Response of Ising systems to oscillating and pulsed fields: Hysteresis, ac, and pulse susceptibility. Phys. Rev. B 52 (1995) 6550–6568.
  • [2] N. Berglund and B. Gentz. Pathwise description of dynamic pitchfork bifurcations with additive noise. Probab. Theory Related Fields 122 (2002) 341–388.
  • [3] N. Berglund and B. Gentz. A sample-paths approach to noise-induced synchronization: Stochastic resonance in a double-well potential. Ann. Appl. Probab. 12 (2002) 1419–1470.
  • [4] N. Berglund and B. Gentz. The effect of additive noise on dynamical hysteresis. Nonlinearity 15 (2002) 605–632.
  • [5] G. Bertotti. Hysteresis in Magnetism. Academic Press, Boston, 1998.
  • [6] G. Bertotti and I. D. Mayergoyz. The Science of Hysteresis, Mathematical Modeling and Applications, Vol. I. Elsevier, Amsterdam, 2006.
  • [7] P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1999.
  • [8] G. Carinci. Stochastic effects in critical regimes. Ph.D. thesis, Università degli Studi dell’Aquila, 2010.
  • [9] P. Jung, G. Gray, R. Roy and P. Mandel. Scaling law for dynamical hysteresis. Phys. Rev. Lett. 65 (1990) 1873–1876.
  • [10] G. Korniss, M. A. Novotny, P. A. Rikvold and C. J. White. Dynamic phase transition, universality, and finite-size scaling in the two-dimensional kinetic Ising model in an oscillating field. Phys. Rev. E 63 (2000) 016120.
  • [11] G. Korniss, M. A. Novotny and P. A. Rikvold. Absence of first-order transition and tricritical point in the dynamic phase diagram of a spatially extended bistable system in an oscillating field. Phys. Rev. E 66 (2002) 056127.
  • [12] H. Landau and L. A. Shepp. On the supremum of a Gaussian process. Sankhya A 32 (1970) 369–378.
  • [13] W. V. Li. Small deviations for Gaussian Markov processes under the sup-norm. J. Theoret. Probab. 12 (1999) 971–984.
  • [14] M. B. Marcus and L. A. Shepp. Sample behaviour of Gaussian processes. In Proceedings of the 6th Berkeley Symposium on Mathematics, Statistic and Probability, Vol. 2 423–441. Univ. California Press, Berkeley, CA, 1972.
  • [15] M. A. Novotny, P. A. Rikvold and S. W. Sides. Stochastic hysteresis and resonance in a kinetic Ising system. Phys. Rev. E 57 (1998) 6512–6533.
  • [16] M. A. Novotny, P. A. Rikvold and S. W. Sides. Kinetic Ising model in an oscillating field: Finite-size scaling at the dynamic phase transition. Phys. Rev. Lett. 81 (1998) 834–837.
  • [17] M. A. Novotny, P. A. Rikvold and S. W. Sides. Kinetic Ising model in an oscillating field: Avrami theory for the hysteretic response and finite-size scaling for the dynamic phase transition. Phys. Rev. E 59 (1999) 2710–2729.
  • [18] D. Pollard. Convergence of Stochastic Processes. Springer, New York, 1984.
  • [19] E. Presutti. Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics. Springer, Berlin, 2009.
  • [20] M. Rao, H. R. Krishnamurthy and R. Pandit. Magnetic hysteresis in two model spin systems. Phys. Rev. B 42-1 (1990) 856–884.
  • [21] T. Tomé and M. J. de Oliveira. Dynamic phase transition in the kinetic Ising model under a time-dependent oscillating field. Phys. Rev. A 41 (1990) 4251–4254.
  • [22] A. Visintin. Differential Models of Hysteresis. Springer, Berlin, 1994.
  • [23] H. Zhu, S. Dong and J. M. Liu. Hysteresis loop area of the Ising model. Phys. Rev. B 70 (2004) 132403.