Functionals.md

@@ -37,19 +37,19 @@ The ASLDA functional has been designed in order to capture properties of **stron
     + [1-\alpha_{\downarrow}(n_{\uparrow},n_{\downarrow})]\dfrac{\bm{j}_{\downarrow}^2}{2n_{\downarrow}}
 \end{aligned}
 ```  
-The functional is fitted to quantum Monte Carlo data, and for spin symmetric and uniform system it provides:
+The functional is fitted to quantum Monte Carlo data, and for spin symmetric and uniform systems it provides:
 ```math
 E/E_{\textrm{ffg}}=\xi=0.40(1),\qquad\Delta/\varepsilon_F=0.504(24).
 ```  
-For more info about fitting procedure and explicit form of functional terms see: [arXiv:1008.3933](https://arxiv.org/abs/1008.3933).
+For more info about fitting procedure and explicit form of functional terms, see: [arXiv:1008.3933](https://arxiv.org/abs/1008.3933).
 
 ## Stabilization of ASLDA functional
-In the case of calculations for trapped system term $`\frac{\bm{j}_{\sigma}^2}{2n_{\sigma}}`$ is source of numerical instabilities. Precisely, for small density regions we have:  
+In the case of calculations for trapped system term $`\frac{\bm{j}_{\sigma}^2}{2n_{\sigma}}`$ is source of numerical instabilities. Precisely, for small-density regions we have:  
 * $`n_{\sigma}\rightarrow 0`$, 
 * $`\bm{j}_{\sigma}\rightarrow 0`$,
 * $`\frac{\bm{j}_{\sigma}^2}{2n_{\sigma}}\rightarrow 0`$.  
 
-However, division of very small numbers is numerically not stable operations. For this reason, we introduce stabilization procedure:  
+However, the division of very small numbers is numerically not stable operation. For this reason, we introduce the stabilization procedure:  
 ```math
 \dfrac{\bm{j}_{\sigma}^2}{2n_{\sigma}}\longrightarrow f_{\textrm{reg.}}(n_{\sigma})\dfrac{\bm{j}_{\sigma}^2}{2n_{\sigma}}
 ```
@@ -61,14 +61,14 @@ Role of $`f_{\textrm{reg.}}`$ is to exclude from computation regions of small de
  * For regions with density smaller than SLDA_STABILIZATION_EXCLUDE_BELOW_DENISTY 
  * contribution from current term j^2/2n is assumed to be zero. 
  * For regions with density above SLDA_STABILIZATION_RETAIN_ABOVE_DENSITY 
- * the contribution is assumed to be intact by stabilization procedure. 
+ * the contribution is assumed to be intact by the stabilization procedure. 
  * */
 #define SLDA_STABILIZATION_RETAIN_ABOVE_DENSITY  1.0e-5
 #define SLDA_STABILIZATION_EXCLUDE_BELOW_DENISTY 1.0e-7
 ```
-There meaning is presented on the figure below. 
+Their meaning is presented in the figure below. 
 ![stabilization](uploads/740d294dbfb30a203cabec0b140fa326/stabilization.png)   
-Smooth transition between densities `SLDA_STABILIZATION_EXCLUDE_BELOW_DENISTY` and `SLDA_STABILIZATION_RETAIN_ABOVE_DENSITY` is introduced in order to avoid discontinuities for quantities, that may lead to divergences when computing derivatives.
+A smooth transition between densities `SLDA_STABILIZATION_EXCLUDE_BELOW_DENISTY` and `SLDA_STABILIZATION_RETAIN_ABOVE_DENSITY` is introduced to avoid discontinuities for quantities, that may lead to divergences when computing derivatives.
 
 ## Chemical potential
 The spin-symmetric and uniform unitary Fermi is expected to be scale-invariant where the following relations for the total energy and the chemical potential satisfy: $`E=\xi E_\textrm{ffg}`$ and $`\mu=\xi \varepsilon_{\textrm{F}}`$, where $`\xi\approx 0.4`$ is the Bertch parameter. However, in numerical realization, the relation for the chemical potential is satisfied only in the limit of very low densities. It is demonstrated in the table below, where results for the uniform solution obtained on lattice `128x128x128`  with `DX=1` is presented, and $`k_{\textrm{F}}=\sqrt{2\varepsilon_{\textrm{F}}}=(6\pi^2 n_{\uparrow})^{1/3}`$.