### Abstract

We compute analytically and numerically the four-point correlation function that characterizes nontrivial cooperative dynamics in glassy systems within several models of glasses: elastoplastic deformations, mode-coupling theory (MCT), collectively rearranging regions (CRR's), diffusing defects, and kinetically constrained models (KCM's). Some features of the four-point susceptibility ξ _{4}(t) are expected to be universal: at short times we expect a power-law increase in time as t ^{4} due to ballistic motion (t ^{2} if the dynamics is Brownian) followed by an elastic regime (most relevant deep in the glass phase) characterized by a t or √t growth, depending on whether phonons are propagative or diffusive. We find in both the β and early α regime that ξ _{4}∼t ^{μ}, where μ is directly related to the mechanism responsible for relaxation. This regime ends when a maximum of ξ _{4} is reached at a time t=t* of the order of the relaxation time of the system. This maximum is followed by a fast decay to zero at large times. The height of the maximum also follows a power law ξ _{4}(t*) ∼t* ^{λ} The value of the exponents μ and λ allows one to distinguish between different mechanisms. For example, freely diffusing defects in d=3 lead to μ=2 and λ= 1, whereas the CRR scenario rather predicts either μ,=1 or a logarithmic behavior depending on the nature of the nucleation events and a logarithmic behavior of ξ _{4}(t*). MCT leads to μ=b and λ=1/γ, where b and γ are the standard MCT exponents. We compare our theoretical results with numerical simulations on a Lennard-Jones and a soft-sphere system. Within the limited time scales accessible to numerical simulations, we find that the exponent μ is rather small, μ<1, with a value in reasonable agreement with the MCT predictions, but not with the prediction of simple diffusive defect models, KCM's with noncooperative defects, and CRR's. Experimental and numerical determination of ξ _{4}(t) for longer time scales and lower temperatures would yield highly valuable information on the glass formation mechanism.

Original language | English (US) |
---|---|

Article number | 041505 |

Journal | Physical Review E |

Volume | 71 |

Issue number | 4 |

DOIs | |

State | Published - Apr 2005 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Condensed Matter Physics
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Physical Review E*,

*71*(4), [041505]. https://doi.org/10.1103/PhysRevE.71.041505

**Dynamical susceptibility of glass formers : Contrasting the predictions of theoretical scenarios.** / Toninelli, Cristina; Wyart, Matthieu; Berthier, Ludovic; Biroli, Giulio; Bouchaud, Jean Philippe.

Research output: Contribution to journal › Article

*Physical Review E*, vol. 71, no. 4, 041505. https://doi.org/10.1103/PhysRevE.71.041505

}

TY - JOUR

T1 - Dynamical susceptibility of glass formers

T2 - Contrasting the predictions of theoretical scenarios

AU - Toninelli, Cristina

AU - Wyart, Matthieu

AU - Berthier, Ludovic

AU - Biroli, Giulio

AU - Bouchaud, Jean Philippe

PY - 2005/4

Y1 - 2005/4

N2 - We compute analytically and numerically the four-point correlation function that characterizes nontrivial cooperative dynamics in glassy systems within several models of glasses: elastoplastic deformations, mode-coupling theory (MCT), collectively rearranging regions (CRR's), diffusing defects, and kinetically constrained models (KCM's). Some features of the four-point susceptibility ξ 4(t) are expected to be universal: at short times we expect a power-law increase in time as t 4 due to ballistic motion (t 2 if the dynamics is Brownian) followed by an elastic regime (most relevant deep in the glass phase) characterized by a t or √t growth, depending on whether phonons are propagative or diffusive. We find in both the β and early α regime that ξ 4∼t μ, where μ is directly related to the mechanism responsible for relaxation. This regime ends when a maximum of ξ 4 is reached at a time t=t* of the order of the relaxation time of the system. This maximum is followed by a fast decay to zero at large times. The height of the maximum also follows a power law ξ 4(t*) ∼t* λ The value of the exponents μ and λ allows one to distinguish between different mechanisms. For example, freely diffusing defects in d=3 lead to μ=2 and λ= 1, whereas the CRR scenario rather predicts either μ,=1 or a logarithmic behavior depending on the nature of the nucleation events and a logarithmic behavior of ξ 4(t*). MCT leads to μ=b and λ=1/γ, where b and γ are the standard MCT exponents. We compare our theoretical results with numerical simulations on a Lennard-Jones and a soft-sphere system. Within the limited time scales accessible to numerical simulations, we find that the exponent μ is rather small, μ<1, with a value in reasonable agreement with the MCT predictions, but not with the prediction of simple diffusive defect models, KCM's with noncooperative defects, and CRR's. Experimental and numerical determination of ξ 4(t) for longer time scales and lower temperatures would yield highly valuable information on the glass formation mechanism.

AB - We compute analytically and numerically the four-point correlation function that characterizes nontrivial cooperative dynamics in glassy systems within several models of glasses: elastoplastic deformations, mode-coupling theory (MCT), collectively rearranging regions (CRR's), diffusing defects, and kinetically constrained models (KCM's). Some features of the four-point susceptibility ξ 4(t) are expected to be universal: at short times we expect a power-law increase in time as t 4 due to ballistic motion (t 2 if the dynamics is Brownian) followed by an elastic regime (most relevant deep in the glass phase) characterized by a t or √t growth, depending on whether phonons are propagative or diffusive. We find in both the β and early α regime that ξ 4∼t μ, where μ is directly related to the mechanism responsible for relaxation. This regime ends when a maximum of ξ 4 is reached at a time t=t* of the order of the relaxation time of the system. This maximum is followed by a fast decay to zero at large times. The height of the maximum also follows a power law ξ 4(t*) ∼t* λ The value of the exponents μ and λ allows one to distinguish between different mechanisms. For example, freely diffusing defects in d=3 lead to μ=2 and λ= 1, whereas the CRR scenario rather predicts either μ,=1 or a logarithmic behavior depending on the nature of the nucleation events and a logarithmic behavior of ξ 4(t*). MCT leads to μ=b and λ=1/γ, where b and γ are the standard MCT exponents. We compare our theoretical results with numerical simulations on a Lennard-Jones and a soft-sphere system. Within the limited time scales accessible to numerical simulations, we find that the exponent μ is rather small, μ<1, with a value in reasonable agreement with the MCT predictions, but not with the prediction of simple diffusive defect models, KCM's with noncooperative defects, and CRR's. Experimental and numerical determination of ξ 4(t) for longer time scales and lower temperatures would yield highly valuable information on the glass formation mechanism.

UR - http://www.scopus.com/inward/record.url?scp=41349087334&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=41349087334&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.71.041505

DO - 10.1103/PhysRevE.71.041505

M3 - Article

AN - SCOPUS:41349087334

VL - 71

JO - Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

JF - Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

SN - 1063-651X

IS - 4

M1 - 041505

ER -