From 8c007b686a12321924a784d12f63a24ca6d47f3a Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Thu, 22 Feb 2018 16:08:26 -0500 Subject: new slides --- aps_mm_2018.html | 41 ++- aps_mm_2018.tex | 2 +- figs/autocorr-scaling-isotherm.png | Bin 0 -> 46733 bytes figs/autocorr-scaling-temp.png | Bin 0 -> 72670 bytes figs/potts.pdf | Bin 0 -> 24447 bytes figs/potts.svg | 683 +++++++++++++++++++++++++++++++++++++ figs/vector-1.pdf | Bin 0 -> 16032 bytes figs/vector-1.png | Bin 0 -> 42681 bytes figs/vector-1.svg | 468 +++++++++++++++++++++++++ figs/vector-2.pdf | Bin 0 -> 15929 bytes figs/vector-2.png | Bin 0 -> 28352 bytes figs/vector-2.svg | 461 +++++++++++++++++++++++++ main.css | 5 + 13 files changed, 1652 insertions(+), 8 deletions(-) create mode 100644 figs/autocorr-scaling-isotherm.png create mode 100644 figs/autocorr-scaling-temp.png create mode 100644 figs/potts.pdf create mode 100644 figs/potts.svg create mode 100644 figs/vector-1.pdf create mode 100644 figs/vector-1.png create mode 100644 figs/vector-1.svg create mode 100644 figs/vector-2.pdf create mode 100644 figs/vector-2.png create mode 100644 figs/vector-2.svg diff --git a/aps_mm_2018.html b/aps_mm_2018.html index 8412808..b9746c1 100644 --- a/aps_mm_2018.html +++ b/aps_mm_2018.html @@ -39,12 +39,12 @@ Described by Hamiltonians $$\mathcal H=-\sum_{\langle ij\rangle}Z(s_i,s_j)-\sum_iH(s_i)$$ -for \\(Z\\) invariant under rotations \\(R\\): \\(Z(R(s),R(t))=Z(s,t)\\). +for \\(Z\\) invariant under rotations \\(r\\): \\(Z(r(s),r(t))=Z(s,t)\\). - + @@ -58,7 +58,7 @@ for \\(Z\\) invariant under rotations \\(R\\): \\(Z(R(s),R(t))=Z(s,t)\\). - +
\(s\)\(R\)\(Z(s_i,s_j)\)\(H(s)\)\(s\)\(r\)\(Z(s_i,s_j)\)\(H(s)\)
Potts model\(\mathbb Z/q\mathbb Z\)addition mod \(q\)\(\delta(s_i,s_j)\)\(\sum_iH_i\delta(i,s)\)
Clock model\(\mathbb Z/q\mathbb Z\)addition mod \(q\)\(\cos(2\pi\frac{s_i-s_j}q)\)\(\sum_iH_i\cos(2\pi\frac{s-i}q)\)Clock modelℤ/qℤaddition mod \(q\)\(\cos(2\pi\frac{s_i-s_j}q)\)\(\sum_iH_i\cos(2\pi\frac{s-i}q)\)
@@ -76,14 +76,14 @@ class: split-40 Standard approach to modelling arbitrary stat mech system: metropolis. 1. Pick random spin. - 2. Pick random rotation \\(R\\). + 2. Pick random rotation \\(r\\). 3. Compute change in energy \\(\Delta\mathcal H\\) resulting from taking \\(s\\) to \\(R(s)\\). 4. Take \\(s\\) to \\(R(s)\\) with probability \\(\max\\{1,e^{-\beta\Delta\mathcal H}\\}\\). Problem: Scales very poorly near phase transitions. -Correlation time `\(\tau\sim L^z\)` at critical point, `\(\tau\sim t^{-z/\nu}\)` +Correlation time *τ* at critical point, *t* – *z/ν* `\(\tau\sim t^{-z/\nu}\)` approaching it. `\(z\)` takes large integer values for Ising, order-`\(n\)`, Potts model critical @@ -104,11 +104,11 @@ class: split-40 1. Pick random spin, add to cluster. 2. Pick random rotation `\(R\)`. 3. For every neighboring spin, add to cluster with probability - `\(\min\{0,1-e^{-\beta(Z(R(s),t)-Z(R(s),R(t)))}\}\)`. + `\(\min\{0,1-e^{-\beta(Z(R(s),R(t))-Z(R(s),t))}\}\)`. 4. Repeat 3 for every spin added to cluster. 5. Transform entire cluster with rotation `\(R\)`. -Relies on symmetry of `\(Z\)` +Relies on symmetry of *Z* Fast near the critical point: early studies thought `\(z\)` was zero, actually 0.1–0.4. @@ -152,6 +152,33 @@ Fast near the critical point: early studies thought `\(z\)` was zero, actually ![scoop details](figs/wolff-scoop_explanation.png) +--- + +# Why is the extended method useful? + +order-n +order-n +  R 
→ + +--- + +# Why is the extended method useful? + +order-n + +--- + +# Correlation time scaling + +Correlation time scales consistently in the whole phase space! + + + + +--- + +# Metastable state direct measurement + diff --git a/aps_mm_2018.tex b/aps_mm_2018.tex index 34e9114..d607d76 100644 --- a/aps_mm_2018.tex +++ b/aps_mm_2018.tex @@ -86,7 +86,7 @@ Problem: Scales very poorly near phase transitions. - Correlation time $\tau\sim L^z$ at critical point, $\tau\sim t^{-z/\nu}$ + Correlation time τ at critical point, $\tau\sim t^{-z/\nu}$ approaching it. $z$ takes large integer values for Ising, order-$n$, Potts model critical diff --git a/figs/autocorr-scaling-isotherm.png b/figs/autocorr-scaling-isotherm.png new file mode 100644 index 0000000..63ff52f Binary files /dev/null and b/figs/autocorr-scaling-isotherm.png differ diff --git a/figs/autocorr-scaling-temp.png b/figs/autocorr-scaling-temp.png new file mode 100644 index 0000000..933a446 Binary files /dev/null and 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a/figs/vector-1.svg b/figs/vector-1.svg new file mode 100644 index 0000000..5e9b9c6 --- /dev/null +++ b/figs/vector-1.svg @@ -0,0 +1,468 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 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