# Calculating Phase Space Volume for Canonical Ensemble

• mkbh_10
In summary, when calculating entropy in the microcanonical ensemble, we use KlnW where W is the number of accessible microstates to the system. In the canonical ensemble, we calculate the partition function and divide it by a factor of h to make it dimensionless. If asked to calculate the phase space volume, we use the partition function by integrating exp[B(H(q,p))] dqdp without dividing by h. However, it is important to remember that the exponential should have a negative sign. The reason for dividing by h is to make classical calculations work quantum mechanically. Coarse-graining is used to account for the uncertainty in phase space in quantum mechanics. This can be illustrated through various problems, such as solving the harmonic oscillator and a gas

#### mkbh_10

In calculating entropy in micro canonical ensemble we use KlnW where W is the no. of accessible micro states to the system , now when we move on to canonical ensemble ,we calculate the partition function and from there derive the thermodynamics of the system , and divide it by factor of h to make it dimensionless , now if one is asked to calculate phase space volume , it means we calculate the partition function by integration of exp[B(H(q,p))] dqdp and not dividing this by h . Is this correct ??

mkbh_10 said:
In calculating entropy in micro canonical ensemble we use KlnW where W is the no. of accessible micro states to the system , now when we move on to canonical ensemble ,we calculate the partition function and from there derive the thermodynamics of the system , and divide it by factor of h to make it dimensionless , now if one is asked to calculate phase space volume , it means we calculate the partition function by integration of exp[B(H(q,p))] dqdp and not dividing this by h . Is this correct ??

I think that's correct, except the exponential should have a negative sign.

I think you should remember that the whole reason for dividing by "h" is because you are calculating something classically and want it to work quantum mechanically (it's not just a dimension thing). There is no such thing as an integral over dqdp in quantum mechanics (except in the path integral formalism) as you can't know both the position and momentum at the same time. So you coarse grain the phase space by a box of size "h" and hope this uncertainty in phase space gives you the correct classical answer.

there are some good problems in a textbook by Pathria on statistical mechanics that illustrates coarse-graining. a simple one is to solve the harmonic oscillator both classically (where you need to insert "h" by hand) and quantum mechanically (where "h" comes naturally) and compare the two results in the limit of high energy. another one involves solving a gas of particles subject to the condition that if you take a snapshot of the gas, they are all within a n-dimensional regular polyhedra - calculate this classically and compare it quantum mechanically and you'll see that they give more or less the same result.

if you have a good teacher she will go through coarse-graining in class as it's important to know when classical mechanics works and when you need quantum mechanics. unfortunately, I didn't have a good teacher, so I learned from those two problems.

Last edited:

## 1. What is Phase Space Volume in the context of the Canonical Ensemble?

Phase Space Volume refers to the volume in the multi-dimensional space that represents all possible states of a physical system. In the context of the Canonical Ensemble, it represents the volume in the phase space that is accessible to a system at a given temperature, volume, and number of particles.

## 2. What is the significance of calculating Phase Space Volume for the Canonical Ensemble?

Calculating the Phase Space Volume allows us to determine the probability of a system being in a particular state at a given temperature, volume, and number of particles. This is essential for understanding the behavior of systems in thermodynamic equilibrium.

## 3. How is the Phase Space Volume calculated for the Canonical Ensemble?

The Phase Space Volume is calculated by integrating over the positions and momenta of all particles in the system, taking into account the constraints of the given temperature, volume, and number of particles. This can be done analytically or numerically.

## 4. What are the assumptions made when calculating Phase Space Volume for the Canonical Ensemble?

The calculation of Phase Space Volume for the Canonical Ensemble assumes that the system is in thermal equilibrium, meaning that the temperature is constant throughout the system. It also assumes that the particles in the system are indistinguishable and that there are no external forces acting on the system.

## 5. How does the Phase Space Volume change with temperature, volume, and number of particles?

The Phase Space Volume generally increases with increasing temperature, volume, and number of particles. This is because at higher temperatures, there is a greater distribution of energy among the particles, and at larger volumes, there is more space for the particles to move around. However, the specific relationship between the Phase Space Volume and these parameters depends on the specific system and its interactions.