Can anyone help me in minimizing the following integral equation and finding the roots ?

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AKANKSHA TRIPATHI 2024-5-15

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https://ww2.mathworks.cn/matlabcentral/answers/2119281-can-anyone-help-me-in-minimizing-the-following-integral-equation-and-finding-the-roots

编辑： Torsten 2024-5-25

Here i am posting an integral equation for which i want the minimization code and methods.

Ω =λ^ 4 (σ 2 + ~π2 − v 2 ) 2 − cσ + νqT∫ (d ^3k)/ (2π) ^3 {ln(1 − nq(T, µ)) + ln(1 − nq¯(T, µ))} equation(1)

we have to minimize the above equation with respect to sigma(σ) and in the scond term,there is integration on the momentum vector k, where integration term also contain the sigma variable in the following ways

nq(T, µ) = 1/ (1 + exp((sqrt(k^2+g^2(σ^2+π^2)) − µ)/T ))

nq¯(T, µ))= 1/ (1 + exp((sqrt(k^2+g^2(σ^2+π^2)) +µ)/T ))

where all the other notation like pi,lambda,vq,T are the some parametrs which we can fix like

lambda=87.6 Mev

T = 500(temperature)

vq=12

g=3.8

mu=100

c=13.6

pi=0

so, we have to find ∂Ω/∂σ and corresponding value of sigma at which above equation (1) is minimized .

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David Goodmanson 2024-5-16

编辑：David Goodmanson 2024-5-16

在 MATLAB Online 中打开

Hello Akanksha,

Is v a separate variable from vq or not, and is the expression

(sigma 2 + ~pi 2 - v 2) 2

actually something like

(sigma^2 + pi^2 - vq^2)^2 ?

The rest of the expression appears to be all right and since the integrand is small for large k, I assume the integral d^3 k is over all of k space. It seems what you will have to do is calculate the integral numerically, assuming it is isotropic, as

Integral {k=0,inf} (4 pi) k^2 dk / (2 pi)^3 * { ln ....... + ln ... }

for a whole lot of values of sigma, which should not be hard to do since it is a fairly simple integral. Then plot it with the rest of the sigma-dependent terms and find the minimum of the derivative with a combination of algebraic and numerical steps. I might try it except that I have many questions about the units. T instead of kT, for example, although the Boltzmann k may have been absorbed somehow. The wave number k seems to have the same units as mu, and if mu were to be a chemical potential then the units do not appear to match up. Also lambda = 87.6 MeV does not seem to fit into this context very well.

AKANKSHA TRIPATHI 2024-5-25

I have tried all these optimizers like fminsearch, fmincon but not getting any result . These are not working on the above equation. Without knowing the σ value we can't calculate the Ω because during the calculation of numerical integration σ is required. Please can you provide some initial steps how can i proceed it in any other ways.

Torsten 2024-5-25

编辑：Torsten 2024-5-25

在 MATLAB Online 中打开

Without knowing the σ value we can't calculate the Ω because during the calculation of numerical integration σ is required.

I repeat for the third time:

The optimizer provides numerical values for σ for which you have to evaluate the function Ω(σ). These σ values will be different test values that the solver uses to move towards the optimal σ_opt.

So if you can evaluate Ω for a given value of σ, you can use any optimizer to solve your minimization problem.

Did you plot the function Ω(σ) in the meantime ?

Do you have a function like

function Omega = fun(sigma)
  Omega = λ^ 4 (σ 2 + ~π2 − v 2 ) 2 − cσ + νqT∫ (d ^3k)/ (2π) ^3 {ln(1 − nq(T, µ)) + ln(1 − nq¯(T, µ))} ;
end

Please can you provide some initial steps how can i proceed it in any other ways.

There is no other way.

Maybe you will understand what I mean if you take a look at the examples provided for the optimizers.

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