Equilibrium fluctuations in mean-field disordered models

Folena, Giampaolo; Biroli, Giulio; Hu, Yi; Zamponi, Francesco

doi:10.1103/physreve.106.024605

Cited by 7 publications

(11 citation statements)

References 52 publications

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“…It is readily seen from Equations ( 16 ) and ( 20 ) that the constrained grand potential

is expressed using

The replica trick allows us to have the relation between the constrained grand potential

and the generalized FP potential

which is the first result ( Result 1 ; see Appendix A for the detailed derivation). It is noted that the conventional replica trick proves the necessity of

to consider the quenched type of the FP formalism, though it has been physically motivated to take the limit of

based on the Monasson formalism [ 18 , 22 , 75 ].…”

Section: Main Resultsmentioning

confidence: 99%

“…The overlap

(

) measures the degree of similarity between a pair of replicas using the microscopic density (or the so-called density “operator” [ 79 , 80 , 81 , 82 , 83 , 84 , 85 , 86 , 87 , 88 ]) in replica a ,

. We define that

where a distribution function

specifies the spatial averaging performed over a finite range; for example, we have

using the Heaviside function

and particle diameter a [ 65 , 66 , 67 , 68 , 69 , 70 , 71 , 72 , 73 , 74 , 75 ].…”

Section: Generalized Franz–parisi (Fp) Potentialmentioning

confidence: 99%

“…The FP potential

is obtained in two steps [ 65 , 66 , 67 , 68 , 69 , 70 , 71 , 72 , 73 , 74 , 75 ]. First, we fix a reference configuration

of replica 1, which plays the role of quenched variable in the effective potential

as seen from the following definition:

where

denotes

in the canonical ensemble of replica a for

the inverse thermal energy

, and

the interaction energy of replica a that is the sum of intra-replica interaction energy

and inter-replica one

where

using the intra-replica interaction potential

and the inter-replica one

.…”

Section: Generalized Franz–parisi (Fp) Potentialmentioning

confidence: 99%

“…Equations ( 10 )–( 14 ) reveal that the field-theoretical formulation of the effective potential

can be developed as follows [ 79 , 80 , 81 , 82 , 83 , 84 , 85 , 86 , 87 , 88 ]:

where

In the last line of Equation ( 16 ), we have the ideal gas entropy defined by

The generalized FP potential

is obtained from the grand canonical average of

for the reference configuration as follows:

similar to Equations ( 7 ) and ( 8 ). Equation ( 18 ) clarifies that a given configuration

plays a role of quenched disorder to another replica a [ 65 , 66 , 67 , 68 , 69 , 70 , 71 , 72 , 73 , 74 , 75 ]. Since we consider all possible configurations of

, the statistical weight

is of the Boltzmann form as well as

in Equati...…”

Section: Generalized Franz–parisi (Fp) Potentialmentioning

confidence: 99%

“…The FP potential [ 65 , 66 , 67 , 68 , 69 , 70 , 71 , 72 , 73 , 74 , 75 ] is a function of overlap Q , a weighted average over the system of the two-point correlation function, and plays the same role as the Landau free energy of a global parameter Q that indicates a distance between the two copies in configuration space. Theoretical and simulation studies have demonstrated that the FP potential reproduces the temperature evolution of FELs, just like the Landau free energy [ 65 , 66 , 67 , 68 , 69 , 70 , 71 , 72 , 73 , 74 , 75 ]. With decreasing temperature, the FP potential develops a secondary minimum for

representing a metastable state.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Replica Field Theory for a Generalized Franz–Parisi Potential of Inhomogeneous Glassy Systems: New Closure and the Associated Self-Consistent Equation

Frusawa

2024

Entropy

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On approaching the dynamical transition temperature, supercooled liquids show heterogeneity over space and time. Static replica theory investigates the dynamical crossover in terms of the free energy landscape (FEL). Two kinds of static approaches have provided a self-consistent equation for determining this crossover, similar to the mode coupling theory for glassy dynamics. One uses the Morita–Hiroike formalism of the liquid state theory, whereas the other relies on the density functional theory (DFT). Each of the two approaches has advantages in terms of perturbative field theory. Here, we develop a replica field theory that has the benefits from both formulations. We introduce the generalized Franz–Parisi potential to formulate a correlation functional. Considering fluctuations around an inhomogeneous density determined by the Ramakrishnan–Yussouf DFT, we find a new closure as the stability condition of the correlation functional. The closure leads to the self-consistent equation involving the triplet direct correlation function. The present field theory further helps us study the FEL beyond the mean-field approximation.

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“…It is readily seen from Equations ( 16 ) and ( 20 ) that the constrained grand potential

is expressed using

The replica trick allows us to have the relation between the constrained grand potential

and the generalized FP potential

which is the first result ( Result 1 ; see Appendix A for the detailed derivation). It is noted that the conventional replica trick proves the necessity of

to consider the quenched type of the FP formalism, though it has been physically motivated to take the limit of

based on the Monasson formalism [ 18 , 22 , 75 ].…”

Section: Main Resultsmentioning

confidence: 99%

“…The overlap

(

. We define that

where a distribution function

specifies the spatial averaging performed over a finite range; for example, we have

using the Heaviside function

and particle diameter a [ 65 , 66 , 67 , 68 , 69 , 70 , 71 , 72 , 73 , 74 , 75 ].…”

Section: Generalized Franz–parisi (Fp) Potentialmentioning

confidence: 99%

“…The FP potential

is obtained in two steps [ 65 , 66 , 67 , 68 , 69 , 70 , 71 , 72 , 73 , 74 , 75 ]. First, we fix a reference configuration

of replica 1, which plays the role of quenched variable in the effective potential

as seen from the following definition:

where

denotes

in the canonical ensemble of replica a for

the inverse thermal energy

, and

the interaction energy of replica a that is the sum of intra-replica interaction energy

and inter-replica one

where

using the intra-replica interaction potential

and the inter-replica one

.…”

Section: Generalized Franz–parisi (Fp) Potentialmentioning

confidence: 99%

“…Equations ( 10 )–( 14 ) reveal that the field-theoretical formulation of the effective potential

can be developed as follows [ 79 , 80 , 81 , 82 , 83 , 84 , 85 , 86 , 87 , 88 ]:

where

In the last line of Equation ( 16 ), we have the ideal gas entropy defined by

The generalized FP potential

is obtained from the grand canonical average of

for the reference configuration as follows:

similar to Equations ( 7 ) and ( 8 ). Equation ( 18 ) clarifies that a given configuration

plays a role of quenched disorder to another replica a [ 65 , 66 , 67 , 68 , 69 , 70 , 71 , 72 , 73 , 74 , 75 ]. Since we consider all possible configurations of

, the statistical weight

is of the Boltzmann form as well as

in Equati...…”

Section: Generalized Franz–parisi (Fp) Potentialmentioning

confidence: 99%