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.

  • $\begingroup$ Are $d$ and $t$ linked ? Because, if $d = t= \nu_d = 1$, one can compute an explicit solution and there is no maximum. $\endgroup$
    – jvc
    Apr 15, 2020 at 8:48
  • $\begingroup$ 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? $\endgroup$ Apr 15, 2020 at 17:30
  • $\begingroup$ As a beginning, the case $d=t$, $t>1$ may be more tractable because the denominator, $n$ is explicit. $\endgroup$
    – jvc
    Apr 16, 2020 at 11:42
  • $\begingroup$ Thanks for your help, were you able to deduce anything from the numerical solutions? $\endgroup$ Apr 16, 2020 at 13:00
  • 1
    $\begingroup$ 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. $\endgroup$
    – jvc
    Apr 17, 2020 at 18:00


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