T-4 - Revision history

http://www.lptms.universite-paris-saclay.fr//wikids/index.php?action=history&feed=atom&title=T-4 T-4 - Revision history 2026-05-17T15:15:50Z Revision history for this page on the wiki MediaWiki 1.43.6 http://www.lptms.universite-paris-saclay.fr//wikids/index.php?title=T-4&diff=4095&oldid=prev Ros: /* Replica Symmetry Breaking: the Parisi scheme */ 2026-02-13T13:18:34Z

<p><span class="autocomment">Replica Symmetry Breaking: the Parisi scheme</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:18, 13 February 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l163">Line 163:</td> <td colspan="2" class="diff-lineno">Line 163:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>There are models for which the glassy phase is described by a more complicated pattern of the mutual overlaps between states, which is obtained iterating the RSB scheme to <math>K</math> levels, ending up with a series of overlaps <math>q_K > q_{K-1} > \cdots >q_0</math>. The underlying picture of the free-energy landscape is as follows: configurations in the same pure state have typical overlap <math>q_K = q_{EA}</math>. They are arranged in clusters such that configurations belonging to different states inside a cluster have overlap <math>q_{K-1}</math>, but such clusters are arranged in other clusters at a higher level, at mutual overlap <math>q_{K-2}</math> and so on.<br></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>There are models for which the glassy phase is described by a more complicated pattern of the mutual overlaps between states, which is obtained iterating the RSB scheme to <math>K</math> levels, ending up with a series of overlaps <math>q_K > q_{K-1} > \cdots >q_0</math>. The underlying picture of the free-energy landscape is as follows: configurations in the same pure state have typical overlap <math>q_K = q_{EA}</math>. They are arranged in clusters such that configurations belonging to different states inside a cluster have overlap <math>q_{K-1}</math>, but such clusters are arranged in other clusters at a higher level, at mutual overlap <math>q_{K-2}</math> and so on.<br></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The exact solution of the <ins>Sherrington-Kirkpatrick model</ins> introduced in Lecture 1 is obtained iterating this procedure an infinite number of times (<math> K \to \infty </math>): this is the <ins>full-RSB</ins> solution. In this limit, the overlap distribution is continuous. The intuition that the free-energy landscape in mean-field spin glasses has this hierarchical structure is due to Giorgio Parisi. Both 1RSB and fullRSB features are found in mean-field models of <ins>structural glasses</ins> <del style="font-weight: bold; text-decoration: none;">[5]</del>. <br></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The exact solution of the <ins>Sherrington-Kirkpatrick model</ins> introduced in Lecture 1 is obtained iterating this procedure an infinite number of times (<math> K \to \infty </math>): this is the <ins>full-RSB</ins> solution. In this limit, the overlap distribution is continuous. The intuition that the free-energy landscape in mean-field spin glasses has this hierarchical structure is due to Giorgio Parisi. Both 1RSB and fullRSB features are found in mean-field models of <ins>structural glasses</ins>. <br></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></div></td></tr> </table>

Ros http://www.lptms.universite-paris-saclay.fr//wikids/index.php?title=T-4&diff=4094&oldid=prev Ros: /* The overlaps, and why they tell us about glassiness */ 2026-02-13T13:16:46Z

<p><span class="autocomment">The overlaps, and why they tell us about glassiness</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:16, 13 February 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l54">Line 54:</td> <td colspan="2" class="diff-lineno">Line 54:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\mathbb{E}[{P}_\beta(q)]:= \lim_{N \to \infty} \mathbb{E}[P_{\beta, N}(q)]=\lim_{n \to 0} \frac{2}{n(n-1)}\sum_{a<b}\delta \left(q- q_{ab}^{*}\right),\quad \quad \quad q_{EA}= \max_{a<b} \left\{ q_{a b}^* \right\}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\mathbb{E}[{P}_\beta(q)]:= \lim_{N \to \infty} \mathbb{E}[P_{\beta, N}(q)]=\lim_{n \to 0} \frac{2}{n(n-1)}\sum_{a<b}\delta \left(q- q_{ab}^{*}\right),\quad \quad \quad q_{EA}= \max_{a<b} \left\{ q_{a b}^* \right\}</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>where <math> q_{ab}^{*}</math> are the saddle point values of the overlap matrix. In the Ising case, a low temperature one has <math> q_{\alpha \alpha}=m^2</math> and <math> q_{\alpha \neq \beta}=-m^2</math>, and thus <math> {P}_\beta(q)</math> has two peaks at <math> \pm m^2</math>. In spin-glass models, the structure of the distribution can be more complicated. </li></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>where <math> q_{ab}^{*}</math> are the saddle point values of the overlap matrix. In the Ising case, a low temperature one has <math> q_{\alpha \alpha}=m^2</math> and <math> q_{\alpha \neq \beta}=-m^2</math>, and thus <math> {P}_\beta(q)</math> has two peaks at <math> \pm m^2</math>. <ins style="font-weight: bold; text-decoration: none;">Moreover, the weights in front of each delta peak is the same, which tells us that there are only two pure states. </ins>In spin-glass models, the structure of the distribution can be more complicated. </li></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></ul></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></ul></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table>

Ros http://www.lptms.universite-paris-saclay.fr//wikids/index.php?title=T-4&diff=4093&oldid=prev Ros: /* The overlaps, and why they tell us about glassiness */ 2026-02-13T13:14:28Z

<p><span class="autocomment">The overlaps, and why they tell us about glassiness</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:14, 13 February 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l43">Line 43:</td> <td colspan="2" class="diff-lineno">Line 43:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>q_{EA}= q_{\alpha \alpha}= \lim_{N \to \infty}\frac{1}{N}\sum_i \langle \sigma_i \rangle_\alpha \langle \sigma_i \rangle_\alpha</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>q_{EA}= q_{\alpha \alpha}= \lim_{N \to \infty}\frac{1}{N}\sum_i \langle \sigma_i \rangle_\alpha \langle \sigma_i \rangle_\alpha</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Notice that this non-zero in a spin-glass phase, but it is also different from zero in unfrustrated ferromagnets: in the low-T phase of an Ising model, it equals to <math>m^2</math>. What is peculiar of glassy systems is that in mean-field glassy models like the spherical <math>p</math>-spin, there are not just two but several different states, not related by symmetry. This shows up in another quantity, the overlap distribution. </li><br></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Notice that this non-zero in a spin-glass phase, but it is also different from zero in unfrustrated ferromagnets: in the low-T phase of an Ising model, it equals to <math>m^2</math>. What is peculiar of glassy systems is that in mean-field glassy models like the spherical <math>p</math>-spin, there are not just two but several different states, not related by symmetry. This shows up in another quantity, the overlap distribution<ins style="font-weight: bold; text-decoration: none;">, that replica calculations produce as a natural output</ins>. </li><br></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table>

Ros http://www.lptms.universite-paris-saclay.fr//wikids/index.php?title=T-4&diff=4092&oldid=prev Ros: /* The overlaps, and why they tell us about glassiness */ 2026-02-13T13:13:42Z

<p><span class="autocomment">The overlaps, and why they tell us about glassiness</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:13, 13 February 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l37">Line 37:</td> <td colspan="2" class="diff-lineno">Line 37:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> q_{EA}= \lim_{t\to \infty} \lim_{N\to \infty} \frac{1}{N}\sum_{i} \sigma_i(0) \sigma_i(t)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> q_{EA}= \lim_{t\to \infty} \lim_{N\to \infty} \frac{1}{N}\sum_{i} \sigma_i(0) \sigma_i(t)</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>This measures the autocorrelation between the configuration of the same spin at <math>t=0</math> and that at infinitely larger time. A non-zero value of <math> q_{EA} </math> is again an indication of ergodicity breaking: if there was not ergodicity breaking, the system would be able to visit dynamically all configurations according to the Boltzmann measure, decorrelating to the initial condition. The fact that <math> q_{EA} >0</math> indicates that the system, even at later times, is constrained to visit configurations that are not too different from the initial ones: this is because it explores only one of the available pure states! <del style="font-weight: bold; text-decoration: none;">The difference with the ferromagnets is that in mean-field glassy </del> <del style="font-weight: bold; text-decoration: none;">models like the spherical <math>p</math>-spin, there are not just two but several different states, not related by symmetry. </del><br></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>This measures the autocorrelation between the configuration of the same spin at <math>t=0</math> and that at infinitely larger time. A non-zero value of <math> q_{EA} </math> is again an indication of ergodicity breaking: if there was not ergodicity breaking, the system would be able to visit dynamically all configurations according to the Boltzmann measure, decorrelating to the initial condition. The fact that <math> q_{EA} >0</math> indicates that the system, even at later times, is constrained to visit configurations that are not too different from the initial ones: this is because it explores only one of the available pure states<ins style="font-weight: bold; text-decoration: none;">, reaching ergodicity only within this pure state</ins>! <br></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The quantity <math>q_{EA}</math> measures the overlap between configurations in the <ins>same state</ins>, that one expects to be the same for all states. In a thermodynamics formalism, it can be written as <sup>[[#Notes|[**] ]]</sup></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The quantity <math>q_{EA}</math> measures the overlap between configurations in the <ins>same state</ins>, that one expects to be the same for all states. In a thermodynamics formalism, it can be written as <sup>[[#Notes|[**] ]]</sup></div></td></tr> <tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l43">Line 43:</td> <td colspan="2" class="diff-lineno">Line 43:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>q_{EA}= q_{\alpha \alpha}= \lim_{N \to \infty}\frac{1}{N}\sum_i \langle \sigma_i \rangle_\alpha \langle \sigma_i \rangle_\alpha</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>q_{EA}= q_{\alpha \alpha}= \lim_{N \to \infty}\frac{1}{N}\sum_i \langle \sigma_i \rangle_\alpha \langle \sigma_i \rangle_\alpha</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Notice that this non-zero in a spin-glass phase, but it is also different from zero in unfrustrated ferromagnets: in the low-T phase of an Ising model, it equals to <math>m^2</math>. </li><br></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Notice that this non-zero in a spin-glass phase, but it is also different from zero in unfrustrated ferromagnets: in the low-T phase of an Ising model, it equals to <math>m^2</math>. <ins style="font-weight: bold; text-decoration: none;">What is peculiar of glassy systems is that in mean-field glassy </ins> <ins style="font-weight: bold; text-decoration: none;">models like the spherical <math>p</math>-spin, there are not just two but several different states, not related by symmetry. This shows up in another quantity, the overlap distribution. </ins></li><br></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table>

Ros http://www.lptms.universite-paris-saclay.fr//wikids/index.php?title=T-4&diff=4091&oldid=prev Ros: /* The overlaps, and why they tell us about glassiness */ 2026-02-13T13:10:31Z

<p><span class="autocomment">The overlaps, and why they tell us about glassiness</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:10, 13 February 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l43">Line 43:</td> <td colspan="2" class="diff-lineno">Line 43:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>q_{EA}= q_{\alpha \alpha}= \lim_{N \to \infty}\frac{1}{N}\sum_i \langle \sigma_i \rangle_\alpha \langle \sigma_i \rangle_\alpha</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>q_{EA}= q_{\alpha \alpha}= \lim_{N \to \infty}\frac{1}{N}\sum_i \langle \sigma_i \rangle_\alpha \langle \sigma_i \rangle_\alpha</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Notice that this non-zero in a spin-glass phase, but it is also different from zero in unfrustrated ferromagnets: in the low-T phase of an Ising model, it equals to <math>m^2</math><del style="font-weight: bold; text-decoration: none;">, where <math>m</math> is the magnetization</del>. </li><br></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Notice that this non-zero in a spin-glass phase, but it is also different from zero in unfrustrated ferromagnets: in the low-T phase of an Ising model, it equals to <math>m^2</math>. <ins style="font-weight: bold; text-decoration: none;"> </ins></li><br></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table>

Ros http://www.lptms.universite-paris-saclay.fr//wikids/index.php?title=T-4&diff=4090&oldid=prev Ros: /* The overlaps, and why they tell us about glassiness */ 2026-02-13T13:08:37Z

<p><span class="autocomment">The overlaps, and why they tell us about glassiness</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:08, 13 February 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l15">Line 15:</td> <td colspan="2" class="diff-lineno">Line 15:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></li><br></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></li><br></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><li> '''Ergodicity breaking, pure states: the ferromagnet.''' The low-T phase of the ferromagnet also corresponds to <ins>ergodicity breaking</ins>. This is a dynamical concept: the system following some equilibrium dynamics (e.g. Langevin) explores only a sub-part of the equilibrium configurations/free-energy shell, which corresponds to the sector of either positive or negative magnetization. Because of this, averages over dynamical trajectories do not converge to Boltzmann averages:</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><li> '''Ergodicity breaking, pure states: the ferromagnet.''' The low-T phase of the ferromagnet also corresponds to <ins>ergodicity breaking</ins>. This is a dynamical concept: the system following some equilibrium dynamics (e.g. Langevin) explores only a sub-part of the equilibrium configurations/free-energy shell, which corresponds to the sector of either positive or negative magnetization. Because of this, averages over dynamical trajectories do not converge to Boltzmann averages <ins style="font-weight: bold; text-decoration: none;"><sup>[[#Notes|[*] ]]</sup></ins>:</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math display="block"></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math display="block"></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>m_\infty:= \lim_{t \to \infty} \lim_{N \to \infty}\; \frac{1}{N}\sum_{i=1}^N \sigma_i(t) = m_{\pm} (\neq m_{\rm eq}=0).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>m_\infty:= \lim_{t \to \infty} \lim_{N \to \infty}\; \frac{1}{N}\sum_{i=1}^N \sigma_i(t) = m_{\pm} (\neq m_{\rm eq}=0).</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l39">Line 39:</td> <td colspan="2" class="diff-lineno">Line 39:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This measures the autocorrelation between the configuration of the same spin at <math>t=0</math> and that at infinitely larger time. A non-zero value of <math> q_{EA} </math> is again an indication of ergodicity breaking: if there was not ergodicity breaking, the system would be able to visit dynamically all configurations according to the Boltzmann measure, decorrelating to the initial condition. The fact that <math> q_{EA} >0</math> indicates that the system, even at later times, is constrained to visit configurations that are not too different from the initial ones: this is because it explores only one of the available pure states! The difference with the ferromagnets is that in mean-field glassy models like the spherical <math>p</math>-spin, there are not just two but several different states, not related by symmetry. <br></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This measures the autocorrelation between the configuration of the same spin at <math>t=0</math> and that at infinitely larger time. A non-zero value of <math> q_{EA} </math> is again an indication of ergodicity breaking: if there was not ergodicity breaking, the system would be able to visit dynamically all configurations according to the Boltzmann measure, decorrelating to the initial condition. The fact that <math> q_{EA} >0</math> indicates that the system, even at later times, is constrained to visit configurations that are not too different from the initial ones: this is because it explores only one of the available pure states! The difference with the ferromagnets is that in mean-field glassy models like the spherical <math>p</math>-spin, there are not just two but several different states, not related by symmetry. <br></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The quantity <math>q_{EA}</math> measures the overlap between configurations in the <ins>same state</ins>, that one expects to be the same for all states. In a thermodynamics formalism, it can be written as <sup>[[#Notes|[*] ]]</sup></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The quantity <math>q_{EA}</math> measures the overlap between configurations in the <ins>same state</ins>, that one expects to be the same for all states. In a thermodynamics formalism, it can be written as <sup>[[#Notes|[<ins style="font-weight: bold; text-decoration: none;">*</ins>*] ]]</sup></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math display="block"></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math display="block"></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>q_{EA}= q_{\alpha \alpha}= \lim_{N \to \infty}\frac{1}{N}\sum_i \langle \sigma_i \rangle_\alpha \langle \sigma_i \rangle_\alpha</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>q_{EA}= q_{\alpha \alpha}= \lim_{N \to \infty}\frac{1}{N}\sum_i \langle \sigma_i \rangle_\alpha \langle \sigma_i \rangle_\alpha</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l59">Line 59:</td> <td colspan="2" class="diff-lineno">Line 59:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><div style="font-size:89%"></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><div style="font-size:89%"></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>: <small>[*]</small> - To be precise, in analogy with the magnetization, we should write <math>q_{EA}=\lim_{\epsilon \to 0}\lim_{N \to \infty}\frac{1}{N}\sum_i \langle \sigma_i^1 \, \sigma_i^2 \rangle_\epsilon </math>, where <math> \vec{\sigma}^1, \vec{\sigma}^2 </math> are two copies of the system, and the average is with respect to a tilted Boltzmann measure which contains a small coupling <math> \epsilon </math> between them, which plays the same role of the infinitesimal magnetic fields in the ferromagnet. The coupling is so weak that it only forces the configurations to fall in the same pure state, but a part from this it leaves them independent.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>: <small>[<ins style="font-weight: bold; text-decoration: none;">*</ins>*]</small> - To be precise, in analogy with the magnetization, we should write <math>q_{EA}=\lim_{\epsilon \to 0}\lim_{N \to \infty}\frac{1}{N}\sum_i \langle \sigma_i^1 \, \sigma_i^2 \rangle_\epsilon </math>, where <math> \vec{\sigma}^1, \vec{\sigma}^2 </math> are two copies of the system, and the average is with respect to a tilted Boltzmann measure which contains a small coupling <math> \epsilon </math> between them, which plays the same role of the infinitesimal magnetic fields in the ferromagnet. The coupling is so weak that it only forces the configurations to fall in the same pure state, but a part from this it leaves them independent.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></div></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></div></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><br></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><br></div></td></tr> </table>

Ros http://www.lptms.universite-paris-saclay.fr//wikids/index.php?title=T-4&diff=4089&oldid=prev Ros: /* The overlaps, and why they tell us about glassiness */ 2026-02-13T13:07:30Z

<p><span class="autocomment">The overlaps, and why they tell us about glassiness</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:07, 13 February 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l27">Line 27:</td> <td colspan="2" class="diff-lineno">Line 27:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>We know how to select each of the two pure states: by adding a small magnetic field <math> h </math> (symmetry breaking). </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>We know how to select each of the two pure states: by adding a small magnetic field <math> h </math> (symmetry breaking). </div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></li><br></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></li><br></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><div style="font-size:89%"></ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">: <small>[*]</small> - Think at Ising mean-field, where magnetization concentrates dynamically and no need to time average.</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></div></ins></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table>

Ros http://www.lptms.universite-paris-saclay.fr//wikids/index.php?title=T-4&diff=4088&oldid=prev Ros: /* The overlaps, and why they tell us about glassiness */ 2026-02-13T13:05:30Z

<p><span class="autocomment">The overlaps, and why they tell us about glassiness</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:05, 13 February 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l17">Line 17:</td> <td colspan="2" class="diff-lineno">Line 17:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><li> '''Ergodicity breaking, pure states: the ferromagnet.''' The low-T phase of the ferromagnet also corresponds to <ins>ergodicity breaking</ins>. This is a dynamical concept: the system following some equilibrium dynamics (e.g. Langevin) explores only a sub-part of the equilibrium configurations/free-energy shell, which corresponds to the sector of either positive or negative magnetization. Because of this, averages over dynamical trajectories do not converge to Boltzmann averages:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><li> '''Ergodicity breaking, pure states: the ferromagnet.''' The low-T phase of the ferromagnet also corresponds to <ins>ergodicity breaking</ins>. This is a dynamical concept: the system following some equilibrium dynamics (e.g. Langevin) explores only a sub-part of the equilibrium configurations/free-energy shell, which corresponds to the sector of either positive or negative magnetization. Because of this, averages over dynamical trajectories do not converge to Boltzmann averages:</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math display="block"></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math display="block"></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>m_\infty:= \lim_{<del style="font-weight: bold; text-decoration: none;">t_0 \to \infty} \lim_{\tau </del>\to \infty}\lim_{N \to \infty}<del style="font-weight: bold; text-decoration: none;">\,\frac{1}{\tau} \int_{t_0}^{t_0+\tau} dt </del>\; \frac{1}{N}\sum_{i=1}^N \sigma_i(t) = m_{\pm} (\neq m_{\rm eq}=0).</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>m_\infty:= \lim_{<ins style="font-weight: bold; text-decoration: none;">t </ins>\to \infty} \lim_{N \to \infty}\; \frac{1}{N}\sum_{i=1}^N \sigma_i(t) = m_{\pm} (\neq m_{\rm eq}=0).</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><br></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><br></div></td></tr> </table>

Ros http://www.lptms.universite-paris-saclay.fr//wikids/index.php?title=T-4&diff=4087&oldid=prev Ros: /* The overlaps, and why they tell us about glassiness */ 2026-02-13T13:03:48Z

<p><span class="autocomment">The overlaps, and why they tell us about glassiness</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:03, 13 February 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l25">Line 25:</td> <td colspan="2" class="diff-lineno">Line 25:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Pure states satisfy the clustering property, which for fully-connected systems reads: <math>\langle \sigma_i \sigma_j \rangle_\alpha=\langle \sigma_i \rangle_\alpha \langle \sigma_j \rangle_\alpha </math>. In the ferromagnet there are two pure states, <math>\alpha= \pm 1 </math>, that correspond to positive and negative magnetization. These two states are related by symmetry. They satisfy the clustering property <math>\langle \sigma_i \sigma_j \rangle_\pm= m^2</math>, while the full Boltzmann measure does not, since <math>\langle \sigma_i \sigma_j \rangle_{\rm eq}= (1/2)\, m^2_{+} + (1/2)\, m^2_{-}= m^2 (\neq \langle \sigma_i \rangle_{\rm eq}\langle \sigma_j \rangle_{\rm eq}=0).</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Pure states satisfy the clustering property, which for fully-connected systems reads: <math>\langle \sigma_i \sigma_j \rangle_\alpha=\langle \sigma_i \rangle_\alpha \langle \sigma_j \rangle_\alpha </math>. In the ferromagnet there are two pure states, <math>\alpha= \pm 1 </math>, that correspond to positive and negative magnetization. These two states are related by symmetry. They satisfy the clustering property <math>\langle \sigma_i \sigma_j \rangle_\pm= m^2</math>, while the full Boltzmann measure does not, since <math>\langle \sigma_i \sigma_j \rangle_{\rm eq}= (1/2)\, m^2_{+} + (1/2)\, m^2_{-}= m^2 (\neq \langle \sigma_i \rangle_{\rm eq}\langle \sigma_j \rangle_{\rm eq}=0).</math></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> A </del>small field <math> h </math> <del style="font-weight: bold; text-decoration: none;">selects one of them </del>(symmetry breaking). </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">We know how to select each of the two pure states: by adding a </ins>small <ins style="font-weight: bold; text-decoration: none;">magnetic </ins>field <math> h </math> (symmetry breaking). </div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></li><br></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></li><br></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table>

Ros http://www.lptms.universite-paris-saclay.fr//wikids/index.php?title=T-4&diff=4086&oldid=prev Ros: /* The overlaps, and why they tell us about glassiness */ 2026-02-13T13:03:01Z

<p><span class="autocomment">The overlaps, and why they tell us about glassiness</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:03, 13 February 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l24">Line 24:</td> <td colspan="2" class="diff-lineno">Line 24:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\langle A \rangle = \sum_{\alpha} \omega_\alpha \langle A \rangle_\alpha, \quad \quad \quad \omega_\alpha= \frac{Z_\alpha}{Z}, \quad \quad \quad Z_\alpha=\int_{\vec{\sigma} \in \text{ state } \alpha} d \vec{\sigma} e^{-\beta E(\vec{\sigma})}= \langle e^{-\beta E (\vec{\sigma})} \rangle_\alpha.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\langle A \rangle = \sum_{\alpha} \omega_\alpha \langle A \rangle_\alpha, \quad \quad \quad \omega_\alpha= \frac{Z_\alpha}{Z}, \quad \quad \quad Z_\alpha=\int_{\vec{\sigma} \in \text{ state } \alpha} d \vec{\sigma} e^{-\beta E(\vec{\sigma})}= \langle e^{-\beta E (\vec{\sigma})} \rangle_\alpha.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Pure states satisfy the clustering property, which for fully-connected systems reads: <math>\langle \sigma_i \sigma_j \rangle_\alpha=\langle \sigma_i \rangle_\alpha \langle \sigma_j \rangle_\alpha </math>. In the ferromagnet there are two pure states, <math>\alpha= \pm 1 </math>, that correspond to positive and negative magnetization. These two states are related by symmetry. They satisfy the clustering property <math>\langle \sigma_i \sigma_j \rangle_\pm= m^2</math>, while the full Boltzmann measure does not, since <math>\langle \sigma_i \sigma_j \rangle_{\rm eq}= (1/2)\, m^2_{+} + (1/2)\, m^2_{-}= m^2 \neq <del style="font-weight: bold; text-decoration: none;">(</del>\langle \sigma_i \rangle_{\rm eq}<del style="font-weight: bold; text-decoration: none;">)^2</del>=0.</math></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Pure states satisfy the clustering property, which for fully-connected systems reads: <math>\langle \sigma_i \sigma_j \rangle_\alpha=\langle \sigma_i \rangle_\alpha \langle \sigma_j \rangle_\alpha </math>. In the ferromagnet there are two pure states, <math>\alpha= \pm 1 </math>, that correspond to positive and negative magnetization. These two states are related by symmetry. They satisfy the clustering property <math>\langle \sigma_i \sigma_j \rangle_\pm= m^2</math>, while the full Boltzmann measure does not, since <math>\langle \sigma_i \sigma_j \rangle_{\rm eq}= (1/2)\, m^2_{+} + (1/2)\, m^2_{-}= m^2 <ins style="font-weight: bold; text-decoration: none;">(</ins>\neq \langle \sigma_i \rangle_{\rm eq}<ins style="font-weight: bold; text-decoration: none;">\langle \sigma_j \rangle_{\rm eq}</ins>=0<ins style="font-weight: bold; text-decoration: none;">)</ins>.</math></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> A small field <math> h </math> selects one of them (symmetry breaking). </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> A small field <math> h </math> selects one of them (symmetry breaking). </div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></li><br></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></li><br></div></td></tr> </table>

Ros