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

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 .

[Torsten]

Could you write equation (1) in a readable form ?

I cannot interprete expressions like σ 2 or ~π2 or v 2.

What are g and k ?

You divide by (2π) ^3, but you set pi=0. Or are the two different variables ?

Is σ the only unknown that has to be adjusted to make equation (1) minimal ?

What are the limits of your integration ? What is/are the integration variable(s) ?

The way you present your problem here shows us that you did not yet think about it in sufficient depth.

[David Goodmanson]

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]

Yes, v is separate from vq and it is (sigma^2+pi^2-v^2) and here pi is different from π. pi is some parameter related to pi meson. All the units are in Mev or Gev and here k is the moment vector and mu is chemical potential. I am not getting how can i proceed this and get the proper way for solving it .The limit of integration is from 0 to infinity.Could you please give me a proper direction to implement it in MATLAB.

[David Goodmanson]

Is the first term lambda^4 (sigma^2+pi^2-v^2), as opposed to lambda^4 (sigma^2+pi^2-v^2)^2 ?

(If pions are involved I will withdraw my statement about 87.6 MeV not seeming to fit in).

[AKANKSHA TRIPATHI]

The first term is lambda^4(sigma^2+pi^2-v^2)^2.

[AKANKSHA TRIPATHI]

can somebody help me for the above mentioned problem of optimization in MATLAB coding? I have tried fzero and fsolve but it's not working.

[Torsten]

Given σ, are you able to evaluate Ω in MATLAB ?

If this is the case, you can use any of the MATLAB optimizers, e.g. fminunc, fminsearch, fmincon.

[AKANKSHA TRIPATHI]

yes, for given σ I can calculate the Ω ,but here we can't give any value to σ, first i have to minimize the above equation with respect to sigma and i have to find out the value of sigma at different temperature for which the above equation is minimized and we have to plot the value of sigma at different value of mu and temperature(T).

Actually the problem is that i am not able to minimize the above equation with respect to σ because σ is also involved in intgeration term. Separately i can solve the integration by applying simpson or another method but σ is optimized variable so i am not getting the way how can i proceed it.

help is needed!

[Torsten]

For the optimizers, you only need to supply the value of Ω for a numerical value of σ where the optimizer wants you to evaluate Ω.

You don't need to differentiate Ω with respect to σ and solve dΩ/dσ = 0 to get the optimal σ.

I suggest you first plot the function Ω(σ) to see if your function to compute Ω from σ gives reasonable results.

[AKANKSHA TRIPATHI]

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]

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.