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Relaxation and overlap-probability function in the spherical and mean-spherical models.

Nicola Fusco1, Marco Zannetti

  • 1Istituto Nazionale per la Fisica della Materia, Unità di Salerno, Università di Salerno, 84081 Baronissi (Salerno), Italy. nicola.fusco@sa.infn.it

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|January 7, 2003
PubMed
Summary

The spherical and mean-spherical models are equivalent only for a short time after a quench. Beyond this crossover time, macroscopic differences emerge, revealing the mean-spherical model

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Area of Science:

  • Statistical Mechanics
  • Condensed Matter Physics

Background:

  • The equivalence of spherical and mean-spherical models is well-established under equilibrium conditions.
  • Understanding the dynamical behavior of these models after a phase transition (quench) is crucial.

Purpose of the Study:

  • To investigate the dynamical equivalence of spherical and mean-spherical models after a quench below the critical temperature.
  • To identify the conditions under which these models diverge during time evolution.

Main Methods:

  • Analysis of time evolution following a quench from above to below the critical temperature.
  • Examination of the crossover time (t*) determining model equivalence.
  • Comparison of off-equilibrium response functions and equilibrium state structures.

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Main Results:

  • A crossover time t* (approximately $\sqrt{2/d}$) was identified, marking the boundary of model equivalence.
  • For times t < t*, the models exhibit equivalent behavior.
  • For times t > t*, macroscopic discrepancies arise between the models.
  • The relationship between off-equilibrium response and equilibrium structure holds for the spherical model but not the mean-spherical model.

Conclusions:

  • The mean-spherical model is not stochastically stable, unlike the spherical model.
  • The dynamical behavior of the mean-spherical model deviates significantly from the spherical model after a critical time.