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
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 any one entity in some energy state in no way affects the probabilities of any of the other particles.
This is a contradiction. Lets 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 lets 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
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 any one entity in some energy state in no way affects the probabilities of any of the other particles.
This is a contradiction. Lets 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 lets 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