Kittel Thermal Physics Chapter 2 problem 6

[E-Conservation]

Homework Statement

For the example that two spin systems in thermal contact and attain equilibrium, the product of the original multiplicity function will be the new total multiplicity function, Now, we want to find the probability of the system to be slightly deviated from the equilibrium.

Homework Equations

They are attached.

The Attempt at a Solution

I acknowledge that the multiplicity function is closely related to the probability of the system is found in some state. If I sum over all states of interest, we should obtain the solution. However, no matter how I sum them or integrate them, there is no form of the error function given. I wonder what should I do.

 

[mark726]

Firstly, it is important to note that the multiplicity function is a measure of the number of microstates corresponding to a given macrostate. It is not directly related to the probability of a system being in a particular state, but rather gives us an idea of the relative likelihood of different states.

In order to find the probability of the system being slightly deviated from equilibrium, we can use the concept of entropy. Entropy is a measure of the disorder or randomness in a system, and it is related to the multiplicity function through Boltzmann's equation:

S = k lnΩ

where S is the entropy, k is Boltzmann's constant, and Ω is the multiplicity function.

Using this equation, we can calculate the change in entropy when the system is slightly deviated from equilibrium. This change in entropy can then be used to find the probability of this deviation occurring.

Another approach is to use the concept of statistical mechanics, where we can calculate the probability distribution of the system using the Boltzmann distribution:

P(E) = (1/Z) * e^(-E/kT)

where P(E) is the probability of the system having energy E, Z is the partition function, k is Boltzmann's constant, and T is the temperature.

By calculating the probability distribution, we can then find the probability of the system being in a slightly deviated state from equilibrium.

In summary, while the multiplicity function may not directly give us the probability of a system being in a particular state, we can use other concepts such as entropy and statistical mechanics to calculate the probability of a system being slightly deviated from equilibrium.