Finite-temperature-calculations.md

-W-SLDA Toolkit allows for calculations at finite temperature. Here we describe how the temperature effects are implemented in the toolkit. We emphasize that at conceptual level **some of assumptions may be incorrect**. 
+The W-SLDA Toolkit allows calculations at finite temperature. Here we describe how the temperature effects are implemented in the toolkit. We emphasize that at the conceptual level **some of the assumptions may be incorrect**. 
 
 # Static calculations
-Finite temperature effects are introduced by adding quasiparticle occupation probabilities in form of Fermi-Dirac function:
+Finite temperature effects are introduced by adding quasiparticle occupation probabilities in the form of the Fermi-Dirac function:
 ```math
 f_{\beta}(E_n)=\dfrac{1}{\exp(\beta E_n)+1}
 ```
-to definition of densities (see [here](https://gitlab.fizyka.pw.edu.pl/wtools/wslda/-/wikis/Physical%20quantities#densities) from explicit formulas). Here $`\beta=1/T`$  is inverse of temperature. In zero temperature limit this function reduced to step function. Quasiparticle energies $`E_n`$ are taken from solution of Bogoliubov-de Gennes type equations
+to definition of densities (see [here](Physical-quantities#densities) from explicit formulas). Here $`\beta=1/T`$  is the inverse of temperature. In the zero temperature limit, this function reduces to a step function. Quasiparticle energies $`E_n`$ are taken from the solution of Bogoliubov-de Gennes type equations
 ```math
-H_{\textrm{BdG}} \begin{pmatrix}u_{n\uparrow}(r) \\ v_{n\downarrow}(r)\end{pmatrix}= E_n\begin{pmatrix}u_{n\uparrow}(r) \\ v_{n\downarrow}(r)\end{pmatrix}
+H_{\textrm{BdG}} \begin{pmatrix}u_{n,a}(r) \\ v_{n,b}(r)\end{pmatrix}= E_n\begin{pmatrix}u_{n,a}(r) \\ v_{n,b}(r)\end{pmatrix}
 ```   
-See [here](https://gitlab.fizyka.pw.edu.pl/wtools/wslda/-/wikis/Physical%20quantities#potentials) from explicit form of $`H_{\textrm{BdG}}`$.  
+See [here](Physical-quantities#potentials) from explicit form of $`H_{\textrm{BdG}}`$.  
 
-Presently, **definition of densities is the only place where temperature enters into calculation process**. In general finite temperature DFT allows energy density functional $`\mathcal{E}_{\textrm{edf}}`$ to be temperature dependent, which is not the case neither (A)SLDA or BdG functionals.  
+Presently, the **definition of densities is the only place where temperature enters into the calculation process**. In general, finite temperature DFT allows energy density functional $`\mathcal{E}_{\textrm{edf}}`$ to be temperature dependent, which is not the case for either (A)SLDA or SLDAE functionals.  
 
-Below we provide collection of predictions of ASLDA functional when applied to 3D uniform and spin-symmetric system. Figures are taken from diploma thesis of [Aleksandra Olejak, WUT, 2017
+Below, we present a collection of predictions from ASLDA for functional when applied to 3D uniform and spin-symmetric systems. Figures are taken from the diploma thesis of [Aleksandra Olejak, WUT, 2017
 ](https://apd.usos.pw.edu.pl/diplomas/18015/).  
 
 ![ASLD-finiteT](uploads/4b58c5cd64ad2a853ef30304c861a508/ASLD-finiteT.png)
 
-From these plots it is clearly seen that critical temperature is located around $`T_c\approx0.3\varepsilon_F`$ whereas experimentally measured critical temperature is $`T_c^{(\textrm{exp})}\approx0.15\varepsilon_F`$. It clearly demonstrates that present formulation of ASLDA can only at most simulate finite temperature effects at qualitative level.  
+From these plots, it is clearly seen that the critical temperature is located around $`T_c\approx0.3\varepsilon_F`$, whereas the experimentally measured critical temperature is $`T_c^{(\textrm{exp})}\approx0.15\varepsilon_F`$. It clearly demonstrates that the current formulation of ASLDA can at best simulate finite-temperature effects at a qualitative level.  
 
-In order to activate finite temperature calculations you need to uncomment in input *file*:
+In order to activate finite temperature calculations, you need to uncomment in the input *file*:
 ```bash
 temperature             0.1      # requested temperature in units of eF, default T=0
 ```  
 
-*Recommendation*: For numerical purposes it is convenient  to introduce a very small temperature (much smaller than any other energy scale in the system). It greatly improves convergence properties of algorithm, especially in context of spin-imbalanced systems. 
+*Recommendation*: For numerical purposes, it is convenient to introduce a very small temperature (much smaller than any other energy scale in the system). It greatly improves the convergence properties of the algorithm, especially in spin-imbalanced systems. 
  
-# Time dependent calculations
-**Conceptually correct introduction of temperature effects into time-dependent calculations is still an open question**. For integrity purposes of W-SLDA Toolkit the *temperature effects* are introduced to `td-wslda` codes as follow:  
-1. all densities are calculated with included Fermi-Dirac function, for example:
+# Time-dependent calculations
+**Conceptually correct introduction of temperature effects into time-dependent calculations is still an open question**. For integrity purposes of the W-SLDA Toolkit, the *temperature effects* are introduced to `td-wslda` codes as follows:  
+1. All densities are calculated with the included Fermi-Dirac function, for example:
 ```math
-n_{\uparrow}(\bm{r},t)= \sum_{|E_n|<E_c}|u_{n,\uparrow}(\bm{r},t)|^2 f_{\beta}(E_n)
+n_{a}(\bm{r},t)= \sum_{|E_n|<E_c}|u_{n,a}(\bm{r},t)|^2 f_{\beta}(E_n)
 ```
-2. values of $`E_n`$ are taken from initial (static) solution and keep frozen over entire time evolution.  
+2. values of $`E_n`$ are taken from the initial (static) solution and kept frozen over the entire time evolution.