^{1}and Francesco Zamponi

^{2}

### Abstract

It has been shown recently that predictions from mode-coupling theory for the glass transition of hard-spheres become increasingly bad when dimensionality increases, whereas replica theory predicts a correct scaling. Nevertheless if one focuses on the regime around the dynamical transition in three dimensions, mode-coupling results are far more convincing than replica theory predictions. It seems thus necessary to reconcile the two theoretic approaches in order to obtain a theory that interpolates between low-dimensional, mode-coupling results, and “mean-field” results from replica theory. Even though quantitative results for the dynamical transition issued from replica theory are not accurate in low dimensions, two different approximation schemes—small cage expansion and replicated hyper-netted-chain (RHNC)—provide the correct qualitative picture for the transition, namely, a discontinuous jump of a static order parameter from zero to a finite value. The purpose of this work is to develop a systematic expansion around the RHNC result in powers of the static order parameter, and to calculate the first correction in this expansion. Interestingly, this correction involves the static three-body correlations of the liquid. More importantly, we separately demonstrate that higher order terms in the expansion are quantitatively relevant at the transition, and that the usual mode-coupling kernel, involving two-body direct correlation functions of the liquid, cannot be recovered from static computations.

H.J.'s Ph.D. work is funded by a CFM - JP Aguilar grant. We acknowledge discussions with Alexei Andreanov, Jean-Louis Barrat, Ludovic Berthier, Giulio Biroli, Patrick Charbonneau, Daniele Coslovich, Silvio Franz, Atsushi Ikeda, Giorgio Parisi, Grzegorz Szamel, Pierfrancesco Urbani, and Frédéric van Wijland.

I. INTRODUCTION

II. STATIC ORDER PARAMETER FOR GLASSES

A. Order parameter with replicas

B. Link with the dynamic order parameter

III. EXPANSION IN POWERS OF THE ORDER PARAMETER

A. Replicated liquid theory

B. Morita and Hiroike functional

C. HNC approximation for the replicated free-energy

D. Improvements over the liquid quantities

E. Systematic expansion in powers of

IV. TREATMENT AT THIRD ORDER IN THE ORDER PARAMETER

A. Third-order Ornstein-Zernicke relation

B. Replica symmetric structure of the theory

C. Final calculation

V. THREE-BODY CORRELATIONS AND NUMERICAL SOLVING

A. Denton and Ashcroft approximation

B. Numerical resolution methodology

C. Results and discussion

VI. HIGHER ORDERS

VII. CONCLUSION AND DISCUSSION

### Key Topics

- Glass transitions
- 44.0
- Correlation functions
- 33.0
- Free energy
- 26.0
- Mean field theory
- 26.0
- Fourier transforms
- 11.0

## Figures

(Filled squares) Non-ergodicity factor of hard spheres at packing fraction 0.599670 (i.e., at the transition), obtain from the replicated hyper-netted-chain approximation equation (46) , combined with the liquid theory HNC approximation equation (45) . (Open circles) The corresponding order parameter .

(Filled squares) Non-ergodicity factor of hard spheres at packing fraction 0.599670 (i.e., at the transition), obtain from the replicated hyper-netted-chain approximation equation (46) , combined with the liquid theory HNC approximation equation (45) . (Open circles) The corresponding order parameter .

Non-ergodicity factor as a function of the wave vector at packing fraction 0.6, for the standard replicated HNC calculation, and from the combination of PY approximation for the diagonal part and RHNC for the off-diagonal correlation.

Non-ergodicity factor as a function of the wave vector at packing fraction 0.6, for the standard replicated HNC calculation, and from the combination of PY approximation for the diagonal part and RHNC for the off-diagonal correlation.

Diagrams that contribute to the free-energy at order . A wiggly line joining two replica indices *a* and *b* is a *h* _{ ab }, with *a* ≠ *b* function, a black dot attached to a zone with replica index *a* is an integration point weighted by a density factor ρ_{ a }.

Diagrams that contribute to the free-energy at order . A wiggly line joining two replica indices *a* and *b* is a *h* _{ ab }, with *a* ≠ *b* function, a black dot attached to a zone with replica index *a* is an integration point weighted by a density factor ρ_{ a }.

Non-ergodicity factor as a function of the wave vector very close to the critical point φ = φ_{ d }, for the PY + HNC calculation, and with inclusion of the three body term.

Non-ergodicity factor as a function of the wave vector very close to the critical point φ = φ_{ d }, for the PY + HNC calculation, and with inclusion of the three body term.

Non-ergodicity factor as a function of the wave vector at fixed packing fraction 0.6, for the PY + RHNC calculation, and with inclusion of the three body term.

Non-ergodicity factor as a function of the wave vector at fixed packing fraction 0.6, for the PY + RHNC calculation, and with inclusion of the three body term.

Diagrams that contribute to the free-energy at order . A wiggly line joining two replica indices *a* and *b* is a *h* _{ ab }, with *a* ≠ *b* function, a black dot attached to a zone with replica index *a* is an integration point weighted by a density factor ρ_{ a }.

*a* and *b* is a *h* _{ ab }, with *a* ≠ *b* function, a black dot attached to a zone with replica index *a* is an integration point weighted by a density factor ρ_{ a }.

*a* and *b* is a *h* _{ ab }, with *a* ≠ *b* function, a black dot attached to a zone with replica index *a* is an integration point weighted by a density factor ρ_{ a }.

*a* and *b* is a *h* _{ ab }, with *a* ≠ *b* function, a black dot attached to a zone with replica index *a* is an integration point weighted by a density factor ρ_{ a }.

Inter-replica pair correlation function at packing fraction φ = 0.6, with and without three-body correction.

Inter-replica pair correlation function at packing fraction φ = 0.6, with and without three-body correction.

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