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• 13:35, 22 December 2023 Ros talk contribs created page T-3 (Created page with "<strong>Goal of these problems: </strong> In this set of problems, we compute the free energy of the spherical <math>p</math>-spin model in the Replica Symmetric (RS) framework, and in the 1-step Replica Symmetry Broken (1-RSB) framework. <br> <strong>Key concepts: </strong> === The order parameters: overlaps, and their meaning === In the lectures, we have introduced the Edwards-Anderson order parameter <center> <math> q_{EA}=\frac{1}{N}\sum_i \overline{\langle...")
• 12:28, 19 December 2023 Ros talk contribs created page Homework (Created page with " === Homework: freezing as a localization/condensation transition === In this final exercise, we show how the freezing transition can be understood in terms of extreme valued statistics (discussed in the lecture) and localization. We consider the energies of the configurations and define <math> E_\alpha= - N \sqrt{\log 2} + \delta E_\alpha </math>, so that <center><math> {Z} = e^{ \beta N \sqrt{\log 2} }\sum_{\alpha=1}^{2^N} e^{-\beta \delta E_\alpha}= e^{ \beta N \sqrt...")
• 12:21, 19 December 2023 Ros talk contribs created page T-II-3 (Created page with "=== Problem 3: the RS (Replica Symmetric) calculation=== <!-- Remember however that to get the free energy, we are interested in the limit <math>n \to 0</math>. A standard way to proceed is: (i) make an ansatz on the structure of the matrix Q, (ii) compute <math>\mathcal{A}[Q]</math> within this ansatz and expand <math>\mathcal{A}[Q]= n\mathcal{A}_0 + O(n^2)</math>, (iii) perform the saddle-point calculation on <math>\mathcal{A}_0</math>. --> Let us consider the simple...")
• 16:58, 6 December 2023 Ros talk contribs created page T-2 (Created page with "In this set of problems, we use the replica method to study the equilibrium properties of a prototypical toy model of glasses, the spherical <math>p</math>-spin model. === Problem 1: the energy landscape of the REM === In this exercise we study the number <math> \mathcal{N}(E)dE </math> of configurations having energy <math> E_\alpha \in [E, E+dE] </math>. This quantity is a random variable. For large <math> N </math>, we will show that its <em> typical value </em>...")
• 16:59, 24 November 2023 Ros talk contribs created page T-I (Created page with "=== The landscape === To characterize the energy landscape of the REM, we can determine the number <math> \mathcal{N}(E)dE </math> of configurations having energy <math> E_\alpha \in [E, E+dE] </math>. This quantity is a random variable. For large <math> N </math>, its typical value is given by <center><math> \mathcal{N}(E) = e^{N \Sigma\left( \frac{E}{N}\right) + o(N)}, \quad \quad \Sigma(\epsilon) = \begin{cases} \log 2- \epsilon^2 \quad &\text{ if } |\epsilon| \...")
• 13:12, 30 October 2023 Ros talk contribs created page L-1 (Created page with "djdjjnr")
• 13:11, 30 October 2023 Ros talk contribs created page T-I-1 (Created page with "Ciao")
• 10:18, 25 September 2023 User account Ros talk contribs was created by Lptmswikids talk contribs and password was sent by email
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• 10:01, 25 September 2023 User account Zhiqiang talk contribs was created by Lptmswikids talk contribs and password was sent by email