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Dynamic Relaxation - Revision history
2026-09-24T00:39:40Z
Revision history for this page on the wiki
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http://localhost/mediawiki/index.php?title=Dynamic_Relaxation&diff=836&oldid=prev
Jess at 18:27, 17 September 2022
2022-09-17T18:27:34Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 18:27, 17 September 2022</td>
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Jess
http://localhost/mediawiki/index.php?title=Dynamic_Relaxation&diff=825&oldid=prev
Jess at 17:10, 15 September 2022
2022-09-15T17:10:29Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:10, 15 September 2022</td>
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<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>Of course, the assumptions that go into calculating <math>g(\Delta, t)</math> are idealizations of the real field distributions at lattice sites and/or the actual processes involved in muon "hopping" and field "fluctuations". They also omit all effects of ''quantum entanglement'' between the muon spin and the nearby nuclear or electronic spins, which need not remain static during their dipolar or hyperfine interactions with the muon. As has been demonstrated in copper crystals and (more dramatically) in fluoride crystals, such coupled spin systems often evolve together to produce "static" <math>g(\Delta, t)</math> functions bearing little resemblance to that described above. </div></td>
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<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>Of course, the assumptions that go into calculating <math>g(\Delta, t)</math> are idealizations of the real field distributions at lattice sites and/or the actual processes involved in muon "hopping" and field "fluctuations". They also omit all effects of ''quantum entanglement'' between the muon spin and the nearby nuclear or electronic spins, which need not remain static during their dipolar or hyperfine interactions with the muon. As has been demonstrated in copper crystals and (more dramatically) in fluoride crystals, such coupled spin systems often evolve together to produce "static" <math>g(\Delta, t)</math> functions bearing little resemblance to that described above. </div></td>
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<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>
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<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>
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<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>Fortunately, Kehr's recursion relation usually works (numerically) on those "static" functions as well, although of course it is still an approximation.</div></td>
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<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>Fortunately, Kehr's recursion relation usually works (numerically) on those "static" functions as well, although of course it is still an approximation.<ins style="font-weight: bold; text-decoration: none;"> </ins></div></td>
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<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;"><br /></td>
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<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>Note that, for hop or fluctuation rates <math>\nu</math> much larger than the static width <math>\Delta</math>, the dynamic relaxation looks more and more ''exponential'', <math>G^{\rm dyn}(t) [\nu \gg \Delta] \to e^{-\lambda t}</math> where <math>\lambda</math> decreases as <math>\nu</math> increases. This corresponds to the phenomenon of "motional narrowing" in NMR.</div></td>
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Jess
http://localhost/mediawiki/index.php?title=Dynamic_Relaxation&diff=824&oldid=prev
Jess at 17:03, 15 September 2022
2022-09-15T17:03:13Z
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Jess
http://localhost/mediawiki/index.php?title=Dynamic_Relaxation&diff=820&oldid=prev
Jess at 16:50, 15 September 2022
2022-09-15T16:50:27Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:50, 15 September 2022</td>
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<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>Dynamic Gaussian Kubo-Toyabe function for various hop or fluctuation rates. </div></td>
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<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>
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<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;"><br /></td>
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<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>Of course, the assumptions that go into calculating <math>g(\Delta, t)</math> are idealizations of the real field distributions at lattice sites and/or the actual processes involved in muon "hopping" and field "fluctuations". They also omit all effects of ''quantum entanglement'' between the muon spin and the nearby nuclear or electronic spins, which need not remain static during their dipolar or hyperfine interactions with the muon. As has been demonstrated in copper crystals and (more dramatically) in fluoride crystals, such coupled spin systems often evolve together to produce "static" <math>g(\Delta, t)</math> functions bearing little resemblance to that described above. </div></td>
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<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;"><br /></td>
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Jess
http://localhost/mediawiki/index.php?title=Dynamic_Relaxation&diff=817&oldid=prev
Jess at 23:22, 14 September 2022
2022-09-14T23:22:03Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:22, 14 September 2022</td>
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<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>which is sometimes solvable using Laplace transforms. (Numerical methods work too.)</div></td>
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<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>
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<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>In the limit of <math>\nu \to 0</math>, the Gaussian distribution above gives a static function </div></td>
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<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>In the limit of <math>\nu \to 0</math>, the Gaussian distribution above<ins style="font-weight: bold; text-decoration: none;">, centered on <math>B_0 = 0</math>,</ins> gives a static function </div></td>
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<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><center></div></td>
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<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><center></div></td>
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<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>g_G(\Delta, t) = {1\over3} + {2\over3} \exp\left[-{1\over2}\Delta^2 t^2\right] \left(1 - \Delta^2 t^2\right)</math></div></td>
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<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>g_G(\Delta, t) = {1\over3} + {2\over3} \exp\left[-{1\over2}\Delta^2 t^2\right] \left(1 - \Delta^2 t^2\right)</math></div></td>
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Jess
http://localhost/mediawiki/index.php?title=Dynamic_Relaxation&diff=816&oldid=prev
Jess at 23:20, 14 September 2022
2022-09-14T23:20:44Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:20, 14 September 2022</td>
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<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>and the corresponding dynamic relaxation functions look like this: </div></td>
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<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>and the corresponding dynamic relaxation functions look like this: </div></td>
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<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>
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<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;">File</del>:<del style="font-weight: bold; text-decoration: none;"> </del>DGZF.png <del style="font-weight: bold; text-decoration: none;">Dynamic</del> <del style="font-weight: bold; text-decoration: none;">Gaussian</del> <del style="font-weight: bold; text-decoration: none;">ZF</del> <del style="font-weight: bold; text-decoration: none;">Kubo-Toyabe</del> <del style="font-weight: bold; text-decoration: none;">function</del>]</div></td>
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<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;">[Image</ins>:<ins style="font-weight: bold; text-decoration: none;">KT-</ins>DGZF.png<ins style="font-weight: bold; text-decoration: none;">|600px|inline</ins> <ins style="font-weight: bold; text-decoration: none;">image</ins> <ins style="font-weight: bold; text-decoration: none;">(click</ins> <ins style="font-weight: bold; text-decoration: none;">to</ins> <ins style="font-weight: bold; text-decoration: none;">see full</ins> <ins style="font-weight: bold; text-decoration: none;">size)]</ins>]</div></td>
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<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>{| width="80%" </div></td>
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<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>! align="left" |</div></td>
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Jess
http://localhost/mediawiki/index.php?title=Dynamic_Relaxation&diff=813&oldid=prev
Jess at 23:06, 14 September 2022
2022-09-14T23:06:57Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:06, 14 September 2022</td>
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<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>[[Relaxonomy]] --> here</div></td>
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<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>----</div></td>
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<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;"><br /></td>
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<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>So far we have treated only the behavior of the muon polarization at ''fixed locations'' where the local magnetic fields are ''static''. This is often not the case, and situations in which the muons are '''hopping''' between different sites and/or the local fields at those sites are '''fluctuating''' comprise many of the most interesting &micro;SR studies. </div></td>
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<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>So far we have treated only the behavior of the muon polarization at ''fixed locations'' where the local magnetic fields are ''static''. This is often not the case, and situations in which the muons are '''hopping''' between different sites and/or the local fields at those sites are '''fluctuating''' comprise many of the most interesting &micro;SR studies. </div></td>
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Jess
http://localhost/mediawiki/index.php?title=Dynamic_Relaxation&diff=812&oldid=prev
Jess at 23:05, 14 September 2022
2022-09-14T23:05:29Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:05, 14 September 2022</td>
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<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>In the limit of <math>\nu \to 0</math>, the Gaussian distribution above gives a static function </div></td>
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<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>In the limit of <math>\nu \to 0</math>, the Gaussian distribution above gives a static function </div></td>
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<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><center></div></td>
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<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><center></div></td>
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<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><math>g_G(\Delta, t) = {1\over3} + {2\over3} <del style="font-weight: bold; text-decoration: none;">e^{</del>-{1\over2}\Delta^2 t^2<del style="font-weight: bold; text-decoration: none;">}}</del> \left(1 - \Delta^2 t^2\right)</math></div></td>
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<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><math>g_G(\Delta, t) = {1\over3} + {2\over3} <ins style="font-weight: bold; text-decoration: none;">\exp\left[</ins>-{1\over2}\Delta^2 t^2<ins style="font-weight: bold; text-decoration: none;">\right]</ins> \left(1 - \Delta^2 t^2\right)</math></div></td>
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<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></center></div></td>
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<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></center></div></td>
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<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>and the corresponding dynamic relaxation functions look like this: </div></td>
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<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;"><br /></td>
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<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>[File: DGZF.png Dynamic Gaussian ZF Kubo-Toyabe function]</div></td>
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Jess
http://localhost/mediawiki/index.php?title=Dynamic_Relaxation&diff=811&oldid=prev
Jess at 23:00, 14 September 2022
2022-09-14T23:00:37Z
<p></p>
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<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>In the limit of <math>\nu \to 0</math>, the Gaussian distribution above gives a static function </div></td>
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<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>In the limit of <math>\nu \to 0</math>, the Gaussian distribution above gives a static function </div></td>
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<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><center></div></td>
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<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><center></div></td>
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<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><math>g_G(\Delta, t) = {1\over3} + {2\over3} e^{-{1\over2}\Delta^2 t^2}} \left(1 - \Delta^2 t^2\right)</div></td>
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<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><math>g_G(\Delta, t) = {1\over3} + {2\over3} e^{-{1\over2}\Delta^2 t^2}} \left(1 - \Delta^2 t^2\right)<ins style="font-weight: bold; text-decoration: none;"></math></ins></div></td>
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Jess
http://localhost/mediawiki/index.php?title=Dynamic_Relaxation&diff=810&oldid=prev
Jess at 23:00, 14 September 2022
2022-09-14T23:00:08Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:00, 14 September 2022</td>
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<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>One approach, at least in ZF, is to assume that the ''static'' relaxation function <math>G_{zz}^{\rm stat}(t) = g(\Delta, t)</math> simply '''''restarts''''' at every hop/fluctuation, using the muon polarization at the time of the hop as a new starting point. This is called the '''''strong collision model''''' and can be modeled using Kehr's recursion relation: </div></td>
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<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>One approach, at least in ZF, is to assume that the ''static'' relaxation function <math>G_{zz}^{\rm stat}(t) = g(\Delta, t)</math> simply '''''restarts''''' at every hop/fluctuation, using the muon polarization at the time of the hop as a new starting point. This is called the '''''strong collision model''''' and can be modeled using Kehr's recursion relation: </div></td>
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<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><center></div></td>
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<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><math>G^{\rm dyn}(\Delta,\nu,t) = g(\Delta,t) e^{-\nu t} + \nu \int_0^t G^{\rm dyn}(\Delta,\nu, t-\tau) g(\Delta, \tau) e^{-\nu \tau} d\tau </math> ,</div></td>
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<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><math>G^{\rm dyn}(\Delta,\nu,t) = g(\Delta,t)<ins style="font-weight: bold; text-decoration: none;"> \;</ins> e^{-\nu t} + \nu \int_0^t G^{\rm dyn}(\Delta,\nu, t-\tau)<ins style="font-weight: bold; text-decoration: none;"> \;</ins> g(\Delta, \tau)<ins style="font-weight: bold; text-decoration: none;"> \;</ins> e^{-\nu \tau} d\tau </math> ,</div></td>
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<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></center></div></td>
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<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>which is sometimes solvable using Laplace transforms. (Numerical methods work too.)</div></td>
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<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>which is sometimes solvable using Laplace transforms. (Numerical methods work too.)</div></td>
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<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>In the limit of <math>\nu \to 0</math>, the Gaussian distribution above gives a static function </div></td>
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<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><center></div></td>
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<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><math>g_G(\Delta, t) = {1\over3} + {2\over3} e^{-{1\over2}\Delta^2 t^2}} \left(1 - \Delta^2 t^2\right)</div></td>
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Jess