Derivation of the Boltzmann Distribution

In summary, the Boltzmann distribution is a mathematical model that describes the probability of finding particles in a given energy state. The equation that is used to find the distribution is a combination of the logarithm and Stirling's series. The extra term -1 comes from Stirling's series. and alpha is found by minimizing the equation.
  • #1
scorpion990
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I'm using McQuarrie's "Statistical Mechanics" for a class, and I'm not quite understanding the the derivation of the Boltzmann Distribution. I'm going to go through it, and then ask a few questions along the way.

All right. You start with a canonical ensemble with N, V, and T fixed. Heat can be exchanged between microsystems, but matter cannot. Each microsystem has a spectrum of energies E1, E2, E3, ... that are repeated for degeneracy as needed. The occupation numbers of each energy state are denoted by a1, a2, a3, etc, and represent the number of microstates that are in energy stated E1, E2, E3, etc.

Imagine that each energy level is a "box" that contains as many particles as its occupation number. A specific distribution of occupation numbers can be achieved in:
[tex]W(a) = \sum^{n}_{i=1}\frac{A!}{\prod^{n}_{i=1}a_{i}!}[/tex]
So that:
[tex]ln(W(a)) = Aln(A) - A - \sum^{n}_{i=1}ln(a_{i}!) =Aln(A) - A - \sum^{n}_{i=1}(a_{i}ln(a_{i})-a_{i}) [/tex]
Subject to the constraints:
[tex]\sum^{n}_{i=1}a_{i}=A[/tex]
and
[tex]\sum^{n}_{i=1}E_{i}a_{i}=\zeta[/tex]

Now.. Here's where many books and I disagree... They say that the set of equations derived by maximizing the above system is:
[tex]-ln(a_{i})-\alpha-1 - \beta E_{i} = 0 [/tex]

However, when I do it out, I get:
[tex]-ln(a_{i})-\alpha - \beta E_{i} = 0 [/tex]

I have no idea where their extra -1 term come from.

In addition, I don't understand most books' method of finding alpha. can somebody guide me through this process more clearly? Thanks!
 
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  • #3
Hi.

scorpion990 said:
Now.. Here's where many books and I disagree... They say that the set of equations derived by maximizing the above system is:
[tex]-ln(a_{i})-\alpha-1 - \beta E_{i} = 0 [/tex]

-1 comes from Stirling's series, the asymptotic expansion of the logarithm.

Regards.
 

FAQ: Derivation of the Boltzmann Distribution

1. What is the Boltzmann Distribution?

The Boltzmann Distribution is a probability distribution that describes the distribution of particles in a system at thermal equilibrium. It is named after the Austrian physicist Ludwig Boltzmann.

2. What is the significance of the Boltzmann Distribution?

The Boltzmann Distribution is significant because it allows us to understand the behavior of particles in a system at thermal equilibrium. It also helps us to predict the most probable state of a system and the average energy of particles in that state.

3. How is the Boltzmann Distribution derived?

The Boltzmann Distribution is derived from the principles of statistical mechanics and the laws of thermodynamics. It takes into account the energy of a particle, the number of particles in a system, and the temperature of the system.

4. What is the formula for the Boltzmann Distribution?

The formula for the Boltzmann Distribution is P(E) = (1/Z) * e^(-E/kT), where P(E) is the probability of a particle having energy E, Z is the partition function, k is the Boltzmann constant, and T is the temperature of the system.

5. How is the Boltzmann Distribution used in real-world applications?

The Boltzmann Distribution is used in a variety of fields, including physics, chemistry, and engineering. It is used to understand the behavior of gases, the distribution of energy in a system, and the equilibrium state of a chemical reaction. It is also used in the development of new materials and technologies.

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