Problem understanding the derivation of the Boltzman distribution

[hideelo]

I am currently reading "Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles" by Robert Eisberg and Robert Resnick (2nd edition). In Appendix C they derive the boltzman distribution and they seem to be saying something that seems to me to be patently false. If you have the book, it's on page C-3 paragraph beginning "Consider a system..."

They describe a closed system in which the total energy of the system is constant. This system is comprised of many individual, distinguishable, identical entities that can interact through the walls separating and are consequentially in thermal equilibrium. They then say that

"Except for the energy conservation constraint, the entities are independent of each other. The presence of one entity in some particular state in no way prohibits or enhances the chance that another identical entity will be in that state."​

(Italics in original)

Now let us take a pause for a moment and analyze what they are saying, they start by saying that these particles are constrained by the conservation of energy, and then they say that having anyone entity in some energy state in no way affects the probabilities of any of the other particles.

This is a contradiction. Let's call the total energy of the system 'E' let us imagine that for some entity, its energy is (3/4)E, given the constraints of energy conservation, the probability of any other entity having an energy greater than (1/4)E is impossible and this is a direct consequence of two things:

1. conservation of energy
2. some entity having an energy of (3/4)E

So we see that having one entity in some energy state does affect the probabilities of the others.Lets continue, they then look at two entities and assert that that since the probability of finding entity 1 in energy state 1 and finding entity 2 in energy state 2 are independent, the probability of finding entity 2 in energy state 2 given that entity 1 is in energy state 1 is just the product of their probabilities.

let us assume, as before that some entity has a nonzero probability of having an energy of (3/4)E, which is allowed since the only constraint on the system was that total energy remain constant. Now since all of the entities are identical they all have the same, nonzero probability of having an energy of (3/4)E. So let's now ask the following, given that some entity has an energy of (3/4)E {which is allowed} what is the probability that some other entity will also have an energy of (3/4)E?

We can answer this in two ways and get two different answers, either by assuming that since all entities are identitical and all have the same probabilities, and given that the energy of one entity does not affect the probability of any other entity, so we just square a nonzero number and get a nonzero probability as the answer.

Or we can say that given our constraint that the energy remain constant, and given that entity 1 has an energy of (3/4)E the probability of any other entity to have an energy greater than (1/4)E is 0.

So here we see this contradiction clearly, we have the same physical question and we get two mutually incompatible answers, this contradiction can and will come up in other ways but I chose this one.

Any help would be appreciated.
Thanks

 

[stevendaryl]

We can answer this in two ways and get two different answers, either by assuming that since all entities are identitical and all have the same probabilities, and given that the energy of one entity does not affect the probability of any other entity, so we just square a nonzero number and get a nonzero probability as the answer.

Or we can say that given our constraint that the energy remain constant, and given that entity 1 has an energy of (3/4)E the probability of any other entity to have an energy greater than (1/4)E is 0.

So here we see this contradiction clearly, we have the same physical question and we get two mutually incompatible answers, this contradiction can and will come up in other ways but I chose this one.

You're right--in an isolated system, with a fixed amount of energy, $E$, the probability of one particle having energy $\epsilon$ is not independent of the probability another particle has that energy. One way around this is to take your given system, system $A$, and put it into thermal equilibrium with another, much larger system, the reservoir, $R$, that has a much higher energy (for example, you put system $A$ in a tub of water at a particular temperature). Then, neither system has a fixed energy, since they are able to exchange energy. But if you adjust the temperature of $R$ then you can make the expected value of the energy for system $A$ be $E$. The energy of system $A$ will fluctuate. If $R$ is much larger than $A$, then the probability distributions for particles in $A$ will be approximately independent. Whether a particle in system $A$ has energy $1/4 E$ or $3/4 E$ makes very little difference to system $R$.

Boltzmann statistics is what you get in the limit as the size of the reservoir goes to infinity.

There is an alternative way to derive Boltzmann statistics that doesn't assume a reservoir, but I think that the reservoir is a good way to picture it.

 

[WannabeNewton]

Any help would be appreciated.
Thanks

Keep reading until the end of that page. Your question will answer itself.

 

[hideelo]

You're right--in an isolated system, with a fixed amount of energy, $E$, the probability of one particle having energy $\epsilon$ is not independent of the probability another particle has that energy. One way around this is to take your given system, system $A$, and put it into thermal equilibrium with another, much larger system, the reservoir, $R$, that has a much higher energy (for example, you put system $A$ in a tub of water at a particular temperature). Then, neither system has a fixed energy, since they are able to exchange energy. But if you adjust the temperature of $R$ then you can make the expected value of the energy for system $A$ be $E$. The energy of system $A$ will fluctuate. If $R$ is much larger than $A$, then the probability distributions for particles in $A$ will be approximately independent. Whether a particle in system $A$ has energy $1/4 E$ or $3/4 E$ makes very little difference to system $R$.

Boltzmann statistics is what you get in the limit as the size of the reservoir goes to infinity.

There is an alternative way to derive Boltzmann statistics that doesn't assume a reservoir, but I think that the reservoir is a good way to picture it.

Thanks, I guess there are two issues here, understanding boltzman, and understanding the book. I think you helped me with the former. As for the latter, I really don't think that the book is describing your case. But thanks again, you helped

 

[hideelo]

Keep reading until the end of that page. Your question will answer itself.

Hi I read until the end of the of the next paragraph (which goes onto the next page) and rather than getting clarification, the problem just got worse.

The author divides the system into two parts, part 1 has the two entities which we will be considering, and part 2 is the rest of the system. He sets it up so that the energy division between the two parts is fixed. The energy in part 1 he describes as ε₁+ε₂ which I will denote by E.

Let us consider for a moment part 1 in which we have our two entities, let us further imagine that entity 1 has some energy which we will call β. Given this and coupled with the fact that that the total energy in this part of the system is fixed, namely the total energy of this part of the system is E, we can therefore immediately conclude that the second entity in this part of the system has energy E-β. In other words we could not ask for a more dependent probability relationship as this relationship between these two entities, since knowing the value of the energy of either of the two entities, unambiguously tells us the energy of the second particle.

All this notwithstanding, the author insists on treating these as independent probabilities, which they cannot be.

Again, any help is appreciated