How to get the prefactor for the Planck Distribution

[arunma]

So I have a question on the Planckian (from one of my PhD qualifier practice problems). I'm trying to derive the Planck distribution from the Bose-Einstein distribution, which is,

$$f(\epsilon) = \dfrac{1}{e^{\frac{\epsilon - \mu}{k_B T}} - 1}$$

Now I know that I can set the chemical potential equal to zero, and let $$\epsilon = \hbar \omega = h \nu$$ to get,

$$f(\epsilon) = \dfrac{1}{e^{\frac{h \nu}{k_B T}} - 1}$$

However, this is only the distribution function. There's some sort of prefactor that I need in order to turn it into an energy density. My question is: how do I derive this prefactor? I know it must have something to do with deriving the density of states for a bose gas, since the total occupation is,

$$\int_0 ^\infty g(\epsilon) f(\epsilon) d\epsilon$$

I'm also fairly certain that I need to somehow express energy in terms of the wave vector and convert to spherical coordinates. But how exactly does this work? I'm still not sure how the idea of a density of states applies to a Bose gas, since the Pauli principle doesn't apply. I know I could just look up the Planckian, but I'd like to understand this so I could reproduce it on a qualifying exam (should the need arise). Help would be appreciated, thanks.

 

[eljose79]

I can provide some guidance on how to approach this problem. First, it's important to understand the concept of density of states and how it applies to a Bose gas. The density of states is a measure of the number of energy states available to a system at a given energy level. In the case of a Bose gas, the Pauli exclusion principle does not apply, so there is no limit on the number of particles that can occupy a single energy level. This means that the density of states for a Bose gas is given by,

g(\epsilon) = \dfrac{V}{(2\pi)^3} \dfrac{(2m)^{3/2}}{\hbar^3} \epsilon^{1/2}

Where V is the volume of the system, m is the mass of the particles, and \hbar is the reduced Planck's constant. This expression takes into account the fact that the energy levels in a Bose gas are quantized.

Now, to derive the energy density from the Bose-Einstein distribution, we can use the following equation,

\rho = \int_0 ^\infty g(\epsilon) f(\epsilon) \epsilon d\epsilon

Where \rho is the energy density. Plugging in the expression for the density of states and the Bose-Einstein distribution, we get,

\rho = \dfrac{V}{(2\pi)^3} \dfrac{(2m)^{3/2}}{\hbar^3} \int_0 ^\infty \dfrac{\epsilon^{3/2}}{e^{\frac{\epsilon}{k_B T}} - 1} d\epsilon

To solve this integral, we can use the substitution x = \frac{\epsilon}{k_B T} to get,

\rho = \dfrac{V}{(2\pi)^3} \dfrac{(2m)^{3/2}}{\hbar^3} (k_B T)^{5/2} \int_0 ^\infty \dfrac{x^{3/2}}{e^x - 1} dx

This integral is known as the Riemann zeta function, and its value is approximately 2.612. Plugging this in, we get the final expression for the energy density,

\rho = \dfrac{V}{(2\pi)^3} \

 

[denian]

The prefactor for the Planck distribution can be derived by considering the density of states for a Bose gas. The density of states, denoted by g(\epsilon), represents the number of available energy states per unit energy interval. For a Bose gas, the density of states can be expressed as,

g(\epsilon) = \dfrac{V}{(2\pi)^3} \dfrac{4 \pi k^2}{h^3} \epsilon^{\frac{1}{2}}

where V is the volume of the system, k is the wave vector, and h is the Planck constant.

To obtain the energy density, we need to integrate the product of the density of states and the distribution function over all energy states. This can be written as,

\rho = \int_0 ^\infty g(\epsilon) f(\epsilon) d\epsilon

Substituting the expression for the density of states and the Bose-Einstein distribution, we get,

\rho = \dfrac{V}{(2\pi)^3} \dfrac{4 \pi k^2}{h^3} \int_0 ^\infty \dfrac{\epsilon^{\frac{1}{2}}}{e^{\frac{\epsilon}{k_B T}} - 1} d\epsilon

Integrating this expression using spherical coordinates and making the substitution \epsilon = \hbar \omega = h \nu, we get the final expression for the energy density as,

\rho = \dfrac{V}{(2\pi)^3} \dfrac{4 \pi \hbar^3}{c^3} \dfrac{1}{e^{\frac{\hbar \omega}{k_B T}} - 1}

This is the prefactor for the Planck distribution and is often denoted by B(\omega, T).

In summary, the prefactor for the Planck distribution can be derived by considering the density of states for a Bose gas and integrating it with the Bose-Einstein distribution. The resulting expression involves converting energy to frequency and taking into account the volume of the system. This prefactor is crucial for obtaining the correct energy density and understanding the behavior of a Bose gas at different temperatures.