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**. 
 
 # Static calculations
+Finite temperature effects are introduced by adding quasiparticle occupation probabilities in form of 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/gabrielw/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
+```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}
+```   
+See [here](https://gitlab.fizyka.pw.edu.pl/gabrielw/wslda/-/wikis/Physical%20quantities#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.  
+
+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
+](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.  
+
+In order to activate finite temperature calculations you need to uncomment in 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. 
  
 # 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:
+```math
+n_{\uparrow}(\bm{r},t)= \sum_{|E_n|<E_c}|u_{n,\uparrow}(\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.  
+
+