T-2: Difference between revisions
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<li> <em> Entropy contribution.</em> | <li> <em> Entropy contribution.</em> The volume of a sphere of radius <math>\sqrt{N}</math> in dimension <math>N</math> is given by <math>N^{\frac{N}{2}} \pi^{\frac{N}{2}}/(\frac{N}{2})!</math>. Use the large-N asymptotic of this to conclude the calculation of the annealed free energy. </li> | ||
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=== Problem 3: the quenched free energy === | === Problem 3: the quenched free energy === |
Revision as of 23:02, 6 December 2023
In this set of problems, we use the replica method to study the equilibrium properties of a prototypical toy model of glasses, the spherical -spin model.
In the spherical -spin model the configurations that the system can take satisfy the spherical constraint , and the energy associated to each configuration is
where the coupling constants Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_{i_1 \,i_2 \cdots i_p}} are independent random variables with Gaussian distribution with zero mean and variance and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p \geq 3} is an integer.
Problem 1: the annealed free energy
In TD1, we defined the quenched free energy density as the quantity controlling the scaling of the typical value of the partition function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z } . The annealed free energy instead controls the scaling of the average value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z } . It is defined by
Let us compute this quantity.
- Energy correlations. At variance with the REM, in the spherical -spin the energies at different configurations are correlated. Show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{E(\vec{\sigma}) E(\vec{\tau})}= N q(\vec{\sigma}, \vec{\tau})^p/2 + o(1) } , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q(\vec{\sigma}, \vec{\tau})= \frac{1}{N}\sum_{i=1}^N \sigma_i \tau_i } is the overlap between the two configurations. Why an we say that for this model converges with the REM discussed in the previous TD?
- Energy contribution. Show that computing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{Z}} boils down to computing the average Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{e^{-\beta J_{i_1 \, \cdots i_p} \sigma_{i_1} \cdots \sigma_{i_p}}}} . Compute this average. Hint: if X is a centered Gaussian variable with variance , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{e^{\alpha X}}=e^{\frac{\alpha^2 \sigma^2}{2} }} .
- Entropy contribution. The volume of a sphere of radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{N}} in dimension is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N^{\frac{N}{2}} \pi^{\frac{N}{2}}/(\frac{N}{2})!} . Use the large-N asymptotic of this to conclude the calculation of the annealed free energy.
Problem 3: the quenched free energy
- Heavy tails and concentration. ccc
- Inverse participation ratio. cccc