### Abstract

A finite-size scaling study of the capacity problem for the Hopfield model is presented. Questions of identifying the correct shape of the scaling function, of corrections to finite-size scaling and, in particular, the problem of properly dealing with disorder are carefully addressed. At first-order phase transitions, like the one considered here, relevant physical quantities typically scale exponentially with system size, and it is argued that in disordered systems reliable information about the phase transition can therefore be obtained only by averaging their logarithm rather than by considering the logarithm of their average - an issue reminiscent of the difference between quenched and annealed disorder, but previously ignored in the problem at hand. Our data for the Hopfield model yield α
_{c} = 0.141 ± 0.0015. They are thus closer to the results of a recent one-and two-step replica symmetry breaking (RSB) analysis, and disagree with that of an earlier one-step RSB study, with those of previous simulations, and with that of a recent paper using an infinite-step RSB scheme.

Original language | English |
---|---|

Pages (from-to) | 61-73 |

Number of pages | 13 |

Journal | Physica A: Statistical Mechanics and its Applications |

Volume | 232 |

Issue number | 1-2 |

Publication status | Published - 1996 Oct 15 |

Externally published | Yes |

### Fingerprint

### Keywords

- Disordered systems
- Finite-size scaling
- First-order transitions
- Hopfield model

### ASJC Scopus subject areas

- Mathematical Physics
- Statistical and Nonlinear Physics

### Cite this

*Physica A: Statistical Mechanics and its Applications*,

*232*(1-2), 61-73.

**Averaging and finite-size analysis for disorder : The Hopfield model.** / Stiefvater, Thomas; Muller, Klaus; Kühn, Reimer.

Research output: Contribution to journal › Article

*Physica A: Statistical Mechanics and its Applications*, vol. 232, no. 1-2, pp. 61-73.

}

TY - JOUR

T1 - Averaging and finite-size analysis for disorder

T2 - The Hopfield model

AU - Stiefvater, Thomas

AU - Muller, Klaus

AU - Kühn, Reimer

PY - 1996/10/15

Y1 - 1996/10/15

N2 - A finite-size scaling study of the capacity problem for the Hopfield model is presented. Questions of identifying the correct shape of the scaling function, of corrections to finite-size scaling and, in particular, the problem of properly dealing with disorder are carefully addressed. At first-order phase transitions, like the one considered here, relevant physical quantities typically scale exponentially with system size, and it is argued that in disordered systems reliable information about the phase transition can therefore be obtained only by averaging their logarithm rather than by considering the logarithm of their average - an issue reminiscent of the difference between quenched and annealed disorder, but previously ignored in the problem at hand. Our data for the Hopfield model yield α c = 0.141 ± 0.0015. They are thus closer to the results of a recent one-and two-step replica symmetry breaking (RSB) analysis, and disagree with that of an earlier one-step RSB study, with those of previous simulations, and with that of a recent paper using an infinite-step RSB scheme.

AB - A finite-size scaling study of the capacity problem for the Hopfield model is presented. Questions of identifying the correct shape of the scaling function, of corrections to finite-size scaling and, in particular, the problem of properly dealing with disorder are carefully addressed. At first-order phase transitions, like the one considered here, relevant physical quantities typically scale exponentially with system size, and it is argued that in disordered systems reliable information about the phase transition can therefore be obtained only by averaging their logarithm rather than by considering the logarithm of their average - an issue reminiscent of the difference between quenched and annealed disorder, but previously ignored in the problem at hand. Our data for the Hopfield model yield α c = 0.141 ± 0.0015. They are thus closer to the results of a recent one-and two-step replica symmetry breaking (RSB) analysis, and disagree with that of an earlier one-step RSB study, with those of previous simulations, and with that of a recent paper using an infinite-step RSB scheme.

KW - Disordered systems

KW - Finite-size scaling

KW - First-order transitions

KW - Hopfield model

UR - http://www.scopus.com/inward/record.url?scp=0030258803&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030258803&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0030258803

VL - 232

SP - 61

EP - 73

JO - Physica A: Statistical Mechanics and its Applications

JF - Physica A: Statistical Mechanics and its Applications

SN - 0378-4371

IS - 1-2

ER -