# Analytically solving (finding a maxima) a system of equations involving PolyLog functions (Fermi-Dirac Integrals)

I have the following system of equations involving PolyLog functions (Fermi Dirac Integrals) where $$d,t\in \mathbb{Z}$$ and $$d,t >0$$ such that

$$J = J_0 \cdot \left[F_{\frac{d-1}{t}}\left(\eta\right)-F_{\frac{d-1}{t}}\left(\eta-v_d\right)\right]\\ n = n_0 \cdot\left[F_{\frac{d-t}{t}}\left(\eta\right)+F_{\frac{d-t}{t}}\left(\eta-v_d\right)\right]$$ Where we denote the Fermi-Dirac integral of order $$j$$ as $$F_{j}\left(\eta\right)= \frac{1}{\Gamma\left(j+1\right)}\int_{0}^{\infty}\frac{u^{j}}{1+e^{u-\eta}}\,\mathrm{d}u$$. Here $$J_0,n_0$$ and $$v_d$$ are constants. Only variable in the above system of equations is $$\eta$$. I want to get an analytical understanding of how $$J/n$$ varies with $$n$$.

I can take two limits of the Fermi-Dirac integral. (Section 4 on Asymptotic Expansions of https://arxiv.org/pdf/0811.0116.pdf)

1. $$\eta \gg 0$$, in which we can approximate $$F_{j}({\eta})\approx \frac{\eta^{j+1}}{\Gamma(j+2)}$$, if we use this we get $$J = Constant \times v_d\times n^{\frac{d-1}{d}}$$. So clearly $$J/n = Constant \times v_d\times n^{-\frac{1}{d}}$$.

2. $$\eta \ll 0$$, in which we can approximate $$F_{j}({\eta})\approx e^{\eta}$$, if we use this we get $$J/n = Constant\times \tanh\left({v_d/2}\right)$$.

So, I can see how the function behaves in the two limits, but what I really want to capture is how $$J/n$$ behaves for all $$\eta$$ (or $$n$$). I solved the two equations numerically for different values of $$d,t$$ and I find that there is always a maxima in $$J/n$$ as a function of $$n$$. If I can in some way capture that analytically, that there would be one maxima and approximately at what value of $$n$$ that maxima would occur, it would be highly useful for me.

• Are $d$ and $t$ linked ? Because, if $d = t= \nu_d = 1$, one can compute an explicit solution and there is no maximum.
– jvc
Apr 15, 2020 at 8:48
• No they are independent, but in your case, I guess the maxima is always near $n \to 0$. Only in $t=1$ case does the maxima occur near $n \to 0$. In all other cases maxima occurs somewhere in middle. Is there any way to show it? Apr 15, 2020 at 17:30
• As a beginning, the case $d=t$, $t>1$ may be more tractable because the denominator, $n$ is explicit.
– jvc
Apr 16, 2020 at 11:42
• Thanks for your help, were you able to deduce anything from the numerical solutions? Apr 16, 2020 at 13:00
• I think we can just study the sign of the derivative $J/n$ according to $\eta$ and prove (with conditions on $d$, $t$, $\nu$) that it is $>0$ when $\eta < 0$ (using the exponential behavior of $F_j$ in $\eta \to -\infty$). Then, I think it is less difficult to prove that it becomes $< 0$ (and maybe get informations on the localization of this maximum) and stay $< 0$ when $\eta > \simeq 0$. The maximum seems to be attained when $\eta \leq \simeq \nu_d$ and it can simplify the study.
– jvc
Apr 17, 2020 at 18:00