T-I: Difference between revisions
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=== The REM: the energy landscape === | === The REM: the energy landscape === | ||
To characterize the energy landscape of the REM, we | To characterize the energy landscape of the REM, we 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> | <center><math> | ||
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\log 2- \epsilon^2 \quad &\text{ if } |\epsilon| \leq \sqrt{\log 2} \\ | \log 2- \epsilon^2 \quad &\text{ if } |\epsilon| \leq \sqrt{\log 2} \\ | ||
0 \quad &\text{ if } |\epsilon| >\sqrt{\log 2} | 0 \quad &\text{ if } |\epsilon| >\sqrt{\log 2} | ||
\end{cases} | \end{cases}. | ||
</math></center> | </math></center> | ||
The function <math> \Sigma(\epsilon) </math> is the entropy of the model, and it is sketched in Fig. X. The point where the entropy vanishes, <math> \epsilon=- \sqrt{\log 2} </math>, is the energy density of the ground state, consistently with what we obtained with extreme values statistics. The entropy is maximal at <math> \epsilon=0 </math>: the highest number of configurations have vanishing energy density. | |||
*We begin by computing the average <math> \overline{\mathcal{N}(E)} </math>. We set <math> \overline{\mathcal{N}(E)}= e^{N \Sigma^A\left( \frac{E}{N} \right)+ o(N)} </math>, where <math> \Sigma^A </math> is the annealed entropy. Write <math> \mathcal{N}(E)dE= \sum_{\alpha=1}^{2^N} \chi_\alpha(E) dE </math> with <math> \chi_\alpha(E)=1</math> if <math> E_\alpha \in [E, E+dE]</math> and <math> \chi_\alpha(E)=0</math> otherwise. Use this together with <math> p(E)</math> to obtain <math> \Sigma^A </math> : when does this coincide with the entropy? | *We begin by computing the average <math> \overline{\mathcal{N}(E)} </math>. We set <math> \overline{\mathcal{N}(E)}= e^{N \Sigma^A\left( \frac{E}{N} \right)+ o(N)} </math>, where <math> \Sigma^A </math> is the annealed entropy. Write <math> \mathcal{N}(E)dE= \sum_{\alpha=1}^{2^N} \chi_\alpha(E) dE </math> with <math> \chi_\alpha(E)=1</math> if <math> E_\alpha \in [E, E+dE]</math> and <math> \chi_\alpha(E)=0</math> otherwise. Use this together with <math> p(E)</math> to obtain <math> \Sigma^A </math> : when does this coincide with the entropy? |
Revision as of 17:05, 24 November 2023
The REM: the energy landscape
To characterize the energy landscape of the REM, we determine the number of configurations having energy 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 E_\alpha \in [E, E+dE] } . This quantity is a random variable. For large 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 } , its typical value is given by
The 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 \Sigma(\epsilon) } is the entropy of the model, and it is sketched in Fig. X. The point where the entropy vanishes, 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 \epsilon=- \sqrt{\log 2} } , is the energy density of the ground state, consistently with what we obtained with extreme values statistics. The entropy is maximal at : the highest number of configurations have vanishing energy density.
- We begin by 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{\mathcal{N}(E)} } . We set 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{\mathcal{N}(E)}= e^{N \Sigma^A\left( \frac{E}{N} \right)+ o(N)} } , where is the annealed entropy. Write 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 \mathcal{N}(E)dE= \sum_{\alpha=1}^{2^N} \chi_\alpha(E) dE } with 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 \chi_\alpha(E)=1} if 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 \chi_\alpha(E)=0} otherwise. Use this together with 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(E)} to obtain : when does this coincide with the entropy?
- For 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 |\epsilon| \leq \sqrt{\log 2} } the quantity 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 \mathcal{N}(E) } is self-averaging. This means that its distribution concentrates around the average value when 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 \to \infty } . Show that this is the case by computing the second moment 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{\mathcal{N}^2} } and using the central limit theorem. Show that this is no longer true in the region where the annealed entropy is negative.
- For the annealed entropy is negative. This means that configurations with those energy are exponentially rare: the probability to find one is exponentially small in 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 } . Do you have an idea of how to show this, using the expression for 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{\mathcal{N}(E)}} ? Why the entropy is zero in this region? Why the point where the entropy vanishes coincides with the ground state energy of the model?
this will be responsible of the fact that the partition function is not self-averaging in the low-T phase, as we discuss below.
The free energy and the freezing transition
Let us compute the free energy 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 f } of the REM. The partition function reads
We have shown above the behaviour of the typical value of for large 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 } . The typical value of the partition function is
In the limit of large , this integral can be computed with the saddle point method, and one gets
Using the expression of the entropy, we see that the function is stationary at 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 \epsilon^*= -1/2T } , which belongs to the domain of integration whenever . This temperature identifies a transition point: for all values 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 T < T_c } , the stationary point is outside the domain and thus 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 \epsilon^*} has to be chosen at the boundary of the domain, .
The free energy becomes