T-4: Difference between revisions
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=== RSB and the free- | === RSB and the free-energy landscape === | ||
We have seen an example of mean-field model, the spherical p-spin, in which the low-T phase is glassy, described by a 1-RSB ansatz of the overlap matrix. The thermodynamics in the glassy phase is described by three quantities: the typical overlap <math> q_1^{SP} </math> between configurations belonging to the same pure state, the typical overlap <math> q_0^{SP} </math> between configurations belonging to different pure states, and the probability <math> (1-m^{SP}) </math> that two configurations extracted at equilibrium belong to the same state. It can be shown that the low-T, frozen phase of the REM is also described by this 1-RSB ansatz with <math> q_0^{SP}=0, q_1^{SP}=1</math> and <math> m^{SP}=T/T_c </math>. | We have seen an example of mean-field model, the spherical p-spin, in which the low-T phase is glassy, described by a 1-RSB ansatz of the overlap matrix. The thermodynamics in the glassy phase is described by three quantities: the typical overlap <math> q_1^{SP} </math> between configurations belonging to the same pure state, the typical overlap <math> q_0^{SP} </math> between configurations belonging to different pure states, and the probability <math> (1-m^{SP}) </math> that two configurations extracted at equilibrium belong to the same state. It can be shown that the low-T, frozen phase of the REM is also described by this 1-RSB ansatz with <math> q_0^{SP}=0, q_1^{SP}=1</math> and <math> m^{SP}=T/T_c </math>. Replicas are a way to explore the structure of the free-energy landscape. | ||
The Sherrington-Kirkpatrick model introduced in Lecture 1 also has a low-T phase that is glassy. However, the structure of the free-energy landscape is more complicated the mutual overlaps between equilibrium states are organized in a complicated pattern. To understand it, let us consider a 2-RSB ansatz for the overlap matrix: | |||
Revision as of 17:09, 24 December 2023
Goal of these problems:
Key concepts:
RSB and the free-energy landscape
We have seen an example of mean-field model, the spherical p-spin, in which the low-T phase is glassy, described by a 1-RSB ansatz of the overlap matrix. The thermodynamics in the glassy phase is described by three quantities: the typical overlap between configurations belonging to the same pure state, the typical overlap between configurations belonging to different pure states, and the probability that two configurations extracted at equilibrium belong to the same state. It can be shown that the low-T, frozen phase of the REM is also described by this 1-RSB ansatz with and . Replicas are a way to explore the structure of the free-energy landscape.
The Sherrington-Kirkpatrick model introduced in Lecture 1 also has a low-T phase that is glassy. However, the structure of the free-energy landscape is more complicated the mutual overlaps between equilibrium states are organized in a complicated pattern. To understand it, let us consider a 2-RSB ansatz for the overlap matrix: