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Help on method of iterationby SadScholar
Tags: iteration 
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#1
Feb2112, 08:32 PM

P: 32

1. The problem statement, all variables and given/known data
Let ##D(\epsilon)## be an arbitrary but wellbehaved (finite at ##\epsilon_F##) density of states, and ##f(\epsilon,\mu,T)## be the Fermi distribution. Assume that ##\mu\approx\epsilon_F## at all temperatures of interest. ##f(\epsilon,\mu,T=0)f(\epsilon,\mu,T)## is an odd function of ##\epsilon\mu## and it is different from zero only about when ##\epsilon\mu\lesssim k_b T##. Use the method of iteration, the Taylor expansion ##D(\epsilon)\approx D(\mu)+D'(\mu)(\epsilon\mu)## and the equation $$ \int_{\epsilon_F}^\mu D(\epsilon)\, \mathrm{d}\epsilon = \int_0^\infty D(\epsilon)[f(\epsilon,\mu,T=0)f(\epsilon,\mu,T)]\, \mathrm{d}\epsilon $$ to find ##\mu## to the leading order in terms of ##D(\epsilon_F)##, ##D'(\epsilon_F)##, ##k_B##, ##T##, and numerical constants. All relevant equations to my knowledge are included above. I'm really not very familiar with the method of iteration. I'm just looking for some hints on how to get started on this. I know that ##\epsilon_f\mu## is going to be a small parameter. I'm not sure if I should use the linear expansion of ##D(\epsilon)## on both sides of the equation or just on one side. I have the feeling that I need to get ##\epsilon_F\mu## as one side of the equation and do some kind of perturbation. I would love to hear any hints and tips or resources on the method of iteration. Where do I begin? 


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