From 301dea6bef491c735ecc0838bfe8b6985d490f60 Mon Sep 17 00:00:00 2001 From: Kriste Krstovski Date: Tue, 8 Nov 2016 09:43:34 -0500 Subject: [PATCH] Update 2016-11-03-ocihd.md --- _posts/2016-11-03-ocihd.md | 28 ++++++++++++++-------------- 1 file changed, 14 insertions(+), 14 deletions(-) diff --git a/_posts/2016-11-03-ocihd.md b/_posts/2016-11-03-ocihd.md index a5a0c38..17a7428 100644 --- a/_posts/2016-11-03-ocihd.md +++ b/_posts/2016-11-03-ocihd.md @@ -42,29 +42,29 @@ The general idea is to solve $$\hat{s}=\text{argmin}(\sum_{N}\rho(\varepsilon_{\mu})+\sum_{P}\sigma(s_{i}))$$ for generic convex $$\rho$$ an $$\sigma$$ by: -1. Defining a physical system augmented with an inverse temperature - $$\beta$$ whose ground state is a solution to the problem as $$\beta\rightarrow\infty$$. - The energy in terms of $$u=s^{0}-s$$ is: +1. Defining a physical system augmented with an inverse temperature + $$\beta$$ whose ground state is a solution to the problem as $$\beta\rightarrow\infty$$. + The energy in terms of $$u=s^{0}-s$$ is: $$E(u)=\sum_{N}\rho(\varepsilon_{\mu}+X_{\mu j}u_{j})+\sum_{P}\sigma(s_{j}^{0}-u_{j})$$ -2. Calculating the free energy of the system using the replica trick - A physical system in the micro-canonical ensemble minimizes $$E$$. A physical system in the canonical ensemble (held at constant temperature $$\frac{1}{\beta}$$) minimizes the free energy +2. Calculating the free energy of the system using the replica trick + A physical system in the micro-canonical ensemble minimizes $$E$$. A physical system in the canonical ensemble (held at constant temperature $$\frac{1}{\beta}$$) minimizes the free energy - $$F=E-TS=\left\langle -\log(Z)\right\rangle =\left\langle -\log(\int \mathrm{d}u e^{-\beta E(u)})\right\rangle.$$ + $$F=E-TS=\left\langle -\log(Z)\right\rangle =\left\langle -\log(\int \mathrm{d}u e^{-\beta E(u)})\right\rangle.$$ -3. Taking the $$\beta\rightarrow\infty$$ limit. +3. Taking the $$\beta\rightarrow\infty$$ limit. - The general idea of the replica trick is to calculate an average over a polynomial of $$Z$$ - instead of an intractable function of $$Z$$. The free energy is calculated using: + The general idea of the replica trick is to calculate an average over a polynomial of $$Z$$ + instead of an intractable function of $$Z$$. The free energy is calculated using: - $$ - F=-\left\langle \log Z\right\rangle = \lim_{n\rightarrow 0}\left\langle \frac{Z^{n}-1}{n}\right\rangle =\lim_{n\rightarrow 0} \left\langle \frac{\partial Z^{n}}{\partial n}\right\rangle - $$ + $$ + F=-\left\langle \log Z\right\rangle = \lim_{n\rightarrow 0}\left\langle \frac{Z^{n}-1}{n}\right\rangle =\lim_{n\rightarrow 0} \left\langle \frac{\partial Z^{n}}{\partial n}\right\rangle + $$ - Where $$Z^{n}$$ is the partition function of n "copies" of the system. The integral over $$X$$, $$s^{0}$$ and $$\varepsilon$$ couples these copies. This interpretation is valid for integer $$n$$, but the formula is taken to hold for continuous $$n$$ + Where $$Z^{n}$$ is the partition function of n "copies" of the system. The integral over $$X$$, $$s^{0}$$ and $$\varepsilon$$ couples these copies. This interpretation is valid for integer $$n$$, but the formula is taken to hold for continuous $$n$$ - We get: + We get: $$ \left\langle Z^{n}\right\rangle = \left\langle \int \prod_{a=1}^{n} \mathrm{d}u^{a} e^{-\sum_a \beta E(u^{a})} \right\rangle