Update 2016-11-03-ocihd.md

_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