How can we solve the following equation

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lqkhai

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Dear,
I'd like to find the root (Ef) of this equation.

sum_k(1+ln((Ef-E(k))/(kB*T)))=Constant.

sum_k: the sum over k
E(k), kB, T: is already known
Ef: is unknown parameter

Could you please to help me?

Thanks in advance

lqkhai
 

Hi,
This is a usual equation in Physics. And I recommend the Newton-Rapson method as below.
My equation have the form of f(Ef)=0. To find a root of this equation we gave one initial approximation Po and using the iteration

P(k)=P(k-1)-f(P(k))/f(P(k-1)) k=1,2,3...


How a bout your solution?

Please comment more there

Thank in advance

lqkhai
 

Maybe analythic solution

∑ { 1 + ln [ ( Ef - Ek ) / ( kBT ) ] } = C
k + ∑ ln ( Ef - Ek ) - ∑ ln ( kBT ) = C
ln [ Π ( Ef - Ek ) ] = C - k + ∑ ln ( kBT )
Π ( Ef - Ek ) = exp [ C - k + ∑ ln ( kBT ) ]

The last equation is a polynomial equation of degree k. Not all of its roots are acceptable solution (eg. Ef <= Ek ).
 

Thanks man !
I think that approach is difficult to implement numerically. Since there are a large possible roots from your last equation. In my above equation has only one possible root.

cheers,

lqkhai
 

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