Applying the grand canonical ensemble to a magnetic system

In summary: E>=N. This can be done by taking the derivative of the equation with respect to β and setting it equal to <E>. This will give us the value of β that satisfies the condition and we can then use that to find the mean energy <E>.In summary, to find the chemical potential μ and mean energy <E> of the system, we can use the grand canonical ensemble and the Boltzmann distribution to calculate the average number of particles and average energy of the system. By solving for the value of β that satisfies the conditions <N>=N and <E>=N, we can then find the chemical potential and mean energy of the system.
  • #1
mondeo2015
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Homework Statement


Consider a system with N sites and N particles with magnetic moment m. Each site can be in one of three states: empty with energy 0, occupied by one particle with energy 0 (in the absent of magnetic field) or occupied by two particles with anti parallel moments and energy ε. The system is placed in a magnetic field B, acting on the magnetic moment of the particles. Assume that two particles in the same site are indistinguishable.

a. We are to find the chemical potential of the system μ. Hint: we should use <N>=N as seen in class.

b. We are to find the mean energy of the system <E>.

Homework Equations


The usual grand canonical ensemble. The basics of paramagnetism law of total energy.

The Attempt at a Solution


I figured the idea was to apply the grand canonical ensemble but the technicalities are beyond my ability to approach so I am unsure of whether or not the grand canonical ensemble is the idea here. I truly need help and thank all helpers.
 
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  • #2


Hello, thank you for your post. It seems like you are on the right track in using the grand canonical ensemble to solve this problem. Let's break down the steps to finding the chemical potential and mean energy of the system:

a. To find the chemical potential μ, we can use the fact that <N>=N, where <N> is the average number of particles in the system and N is the total number of particles. In the grand canonical ensemble, the average number of particles is given by:

<N> = ∑n=0,1,2...n∞ nP(n)

Where P(n) is the probability of having n particles in the system. In this case, we can assume that the system is in thermal equilibrium, so the probability of finding a certain number of particles in the system follows the Boltzmann distribution:

P(n) = (1/Z)exp(-βεn)

Where Z is the partition function and β=1/kT. Plugging this into the equation for <N>, we get:

<N> = ∑n=0,1,2...n∞ n(1/Z)exp(-βεn)

We can simplify this further by noting that Z is just a normalization constant and the sum of all probabilities must equal 1. Therefore, we can rewrite the equation as:

<N> = ∑n=0,1,2...n∞ nexp(-βεn)

To find the chemical potential μ, we need to solve for the value of β that satisfies the condition <N>=N. This can be done by taking the derivative of the equation with respect to β and setting it equal to N. This will give us the value of β that satisfies the condition and we can then use that to find the chemical potential μ.

b. To find the mean energy <E> of the system, we can use the same approach as above. However, instead of using the average number of particles <N>, we will use the average energy <E> of the system. This is given by:

<E> = ∑n=0,1,2...n∞ εnP(n)

Using the same steps as above, we can simplify this equation to:

<E> = ∑n=0,1,2...n∞ εnexp(-βεn)

To find the mean energy <E>, we need to solve for the value of β that
 

Related to Applying the grand canonical ensemble to a magnetic system

1. What is the grand canonical ensemble and how is it applied to a magnetic system?

The grand canonical ensemble is a statistical mechanics tool used to describe the thermodynamic properties of a system in equilibrium with a reservoir. In the context of a magnetic system, it takes into account the fluctuations in both the number of particles and the energy of the system. It is applied by considering the system as a whole, rather than individual particles, and using the grand partition function to calculate the probability of a specific energy and number of particles.

2. What are the key assumptions made when applying the grand canonical ensemble to a magnetic system?

The key assumptions made when using the grand canonical ensemble for a magnetic system include: 1) the system is in thermal equilibrium with a reservoir, 2) the particles in the system do not interact with each other, 3) the energy levels in the system are discrete, and 4) the system is in a constant external magnetic field.

3. What are the advantages of using the grand canonical ensemble for a magnetic system?

One advantage of using the grand canonical ensemble for a magnetic system is that it allows for the prediction of the behavior of a large number of particles in equilibrium with a reservoir, which is more realistic than considering individual particles. It also takes into account the fluctuations in both energy and number of particles, providing a more complete understanding of the system's thermodynamic properties.

4. What are some applications of using the grand canonical ensemble for a magnetic system?

The grand canonical ensemble is often used in the study of magnetic materials, such as ferromagnets and antiferromagnets, to predict their behavior at different temperatures and magnetic fields. It is also used in the study of phase transitions and critical phenomena in magnetic systems.

5. Are there any limitations to using the grand canonical ensemble for a magnetic system?

One limitation of using the grand canonical ensemble for a magnetic system is that it assumes that the particles do not interact with each other, which may not be the case in certain systems. It also does not take into account the effects of long-range interactions, which may be significant in some magnetic systems.

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