# Volume of N dimensional phase space

• romeo6
In summary, the conversation is about generalizing a volume integral in 3D phase space to N dimensions. The new integral includes a generalized volume for an N sphere and changes in powers. There is also a mention of multiplying by p^{N-1} to account for the area of the N-1 sphere. The concept of using an N factorial is also discussed.
romeo6
Hi guys,

I have a volume integral in 3D phase space that looks like:

$$\int \frac{4\pi p^2 dp}{h^3}$$

Now, I want to generalize to N dimensions. How does this look:

$$\int \frac{\frac{2\pi^{d/2}}{\Gamma(\frac{d}{2})}p^N dp}{N!h^{3N}}$$

Essentially, I've changed the 4 pi (which I think is the volume of a 2 sphere) into a generalized volume for an N sphere, and made some changes in the powers.

how does this look?

Multiplied by p^{N-1} (with N the dimension of the momentum space) you need to have the area of the N-1 sphere.

What is with the N factorial ?

Daniel.

Do you mean I need to multiply my answer by p^{N-1}?

It should be something like

$$\int_{\Omega} dV =\int_{0}^{\infty} p^{n-1} dp\int_{\partial \Omega} dS_{\Omega}$$

The second integral is the integral giving the area of the "n-1 sphere embedded in R^{n}.

Daniel.

## What is the meaning of "N dimensional phase space"?

N dimensional phase space refers to a mathematical space that describes the possible states of a physical system with N degrees of freedom. Each dimension represents a different variable or coordinate that describes the system's state.

## What is the significance of studying the volume of N dimensional phase space?

The volume of N dimensional phase space is a fundamental concept in statistical mechanics and thermodynamics. It helps us understand the distribution of states of a physical system and make predictions about its behavior.

## How is the volume of N dimensional phase space calculated?

The volume of N dimensional phase space is calculated using a mathematical formula known as the phase space volume element. This formula takes into account the number of degrees of freedom and the ranges of each variable or coordinate.

## What is the relationship between the volume of N dimensional phase space and entropy?

The volume of N dimensional phase space is directly related to the concept of entropy, which is a measure of the disorder or randomness in a system. As the volume of phase space increases, the number of possible states also increases, leading to a higher entropy value.

## How does the volume of N dimensional phase space change with changes in energy or temperature?

The volume of N dimensional phase space is affected by changes in energy or temperature through the Boltzmann factor, which takes into account the energy levels of the system. As the energy or temperature increases, the volume of phase space also increases, leading to a higher number of possible states and a higher entropy value.

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