Minimize grand potential functional for density matrix

In summary, the conversation discusses the proof of a functional of the form $$\Omega[\hat \rho] = \text{Tr} \hat \rho \left[ \hat H - \mu \hat N + \frac 1 {\beta} \log \hat \rho \right]$$ and how it relates to the well-known expression $$\Omega[\hat \rho_0] = - \frac 1 {\beta} \log \text{Tr} e^{-\beta (\hat H - \mu \hat N )}$$ The proof involves defining an operator and studying a specific relation to show that the equilibrium value minimizes the functional.
  • #1
dRic2
Gold Member
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TL;DR Summary
##\rho## is the density matrix
##\Omega## is the grand potential
##\text{Tr}## stands for 'trace'
I'd like to show that, by minimizing this functional
$$\Omega[\hat \rho] = \text{Tr} \hat \rho \left[ \hat H - \mu \hat N + \frac 1 {\beta} \log \hat \rho \right]$$
I get the well known expression
$$\Omega[\hat \rho_0] = - \frac 1 {\beta} \log \text{Tr} e^{-\beta (\hat H - \mu \hat N )}$$

I'm familiar with minimizing a functional of the form ##F[g] = \int dx f(g(x))##, but this notations for operators eludes me.

Thanks,
Ric
 
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  • #2
I've found the original paper (PhysRev.137.A1441) where Mermin introduced this functional. You can check the proof in the Appendix (If I'm not mistaken it was first proved by von Neumann).

To sum up the idea behind the proof is to define the operator ##\rho_{\lambda} = \frac {e^{H-\mu N + \lambda \Delta}} {\text{Tr}e^{H-\mu N + \lambda \Delta}}## with ##\Delta = -H + \mu N - \frac 1 {\beta} \log \rho##. You can see that because of ##\text{Tr} \rho =1## for ##\lambda = 1## I get ##\rho_1 = \rho## and for ##\lambda = 0## you get ##\rho = \rho_0## (the equilibrium value). You can then proceed to study the following relation ##\Omega[\rho] - \Omega[\rho_0] = \Omega[\rho_{\lambda = 1}] - \Omega[\rho_{\lambda = 0}] = \int_0^1 \frac {\partial \Omega[\rho_{\lambda}]} {\partial \lambda} d \lambda## and check that is always greater than zero and zero only if ##\lambda = 0## thus proving that ##\rho_0## minimizes the functional
 

1. What is the grand potential functional for density matrix?

The grand potential functional for density matrix is a mathematical expression that describes the energy of a many-particle system in terms of its density matrix. It takes into account both the kinetic energy of the particles and their interactions.

2. What does it mean to minimize the grand potential functional?

Minimizing the grand potential functional means finding the set of parameters that results in the lowest possible value of the functional. In the context of density matrix, this means finding the density matrix that minimizes the energy of the system.

3. Why is minimizing the grand potential functional important?

Minimizing the grand potential functional is important because it allows us to determine the most stable state of a many-particle system. This can provide valuable insights into the behavior and properties of the system, and can also be used to predict and understand phase transitions.

4. How is the grand potential functional minimized?

The grand potential functional is typically minimized using variational methods, which involve finding the set of parameters that minimize the functional through iterative calculations. This can be a complex and computationally intensive process, but it allows for accurate predictions of the behavior of many-particle systems.

5. What are some applications of minimizing the grand potential functional for density matrix?

Minimizing the grand potential functional for density matrix has a wide range of applications in physics and chemistry. It is commonly used in the study of condensed matter systems, such as solids and liquids, as well as in quantum chemistry and materials science. It is also used in the development of new materials and technologies, such as in the design of new electronic devices and energy storage systems.

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