Probability in statistical mechanics

[Kashmir]

Suppose we've an isolated box having $N$ classical distinguishable particles in it, the box being hypothetically divided into two parts, left and right with both parts identical.

Its said that the probability of having the configuration of $n$ particles in the left side is given as $P_n=C(n)/2^N$ with $C(n)$ being the total number of ways in which $n$ particles from $N$ can be placed in the left side.

Why should $P_n=C(n)/2^N$ be the probability? It should be true only if each configuration is equiprobable, but I don't think it is equiprobable.

Our initial conditions (position and velocity of particles) will determine the evolution of the states of the system, so it might be that some states occur more frequently than other and some might never occur?

 

[vanhees71]

Maybe they assume equilibrium, which implies that the probability for each particle to be in the one or the other part is 1/2.

 

[Kashmir]

Maybe they assume equilibrium, which implies that the probability for each particle to be in the one or the other part is 1/2.

It's from reif, he says "
Consider then an ideal gas of N molecules confined within a container or box. In order to discuss the simplest possible situation, suppose that this whole system is isolated (i.e., that it does not interact
with any other system) and that it has been left undisturbed for a very
long time".

 

[Kashmir]

[Mentor Note -- two very similar thread starts merged]

Suppose we've an isolated box having $N$ classical distinguishable particles in it, the box being hypothetically divided into two parts, left and right with both parts identical.

Reif says
>...Note that there is only one way of distributing the $N$ molecules so that all $N$ of them are in the left half of the box. It represents only one special configuration of the molecules compared to the $2^{N}$ possible configurations of these molecules. Hence we would expect that, among a very large number of frames of the film, on the average only one out of every $2^{\mathrm{N}}$ frames would show all the molecules to be in the left half. If $P_{N}$ denotes the fraction of frames showing all the $N$ molecules located in the left half of the box, i.e., if $P_{N}$ denotes the relative frequency, or probability, of finding all the $N$ molecules in the left half, then$P_{N}=\frac{1}{2^N}$

Why should we expect that, among a **very large** number of frames of the film, on the average only one out of every $2^{\mathrm{N}}$ frames would show all the molecules to be in the left half?

I have the intuition for it but how can one show this should be the case indeed rigorously?

Why can't the average be more or less than the above value?
Also how large is "very large frames of film" here?

 

[PeroK]

I have the intuition for it but how can one show this should be the case indeed rigorously?

It's really a postulate, relating to a particular statistical model.

That said if you modeled the motion of each particle in some way that it spent half its time in one half and half its time in the other, then it would follow from that. But, that would replace a general statistical assumption with something more specific.

 

[PeroK]

It's from reif, he says "
Consider then an ideal gas of N molecules confined within a container or box. In order to discuss the simplest possible situation, suppose that this whole system is isolated (i.e., that it does not interact
with any other system) and that it has been left undisturbed for a very
long time".

I would take it as a fundamental assumption or postulate. Like, if you shuffle a pack of cards sufficiently, then the probability they are in some given order is $\dfrac{1}{52!}$.

 

[Kashmir]

It's really a postulate, relating to a particular statistical model.

That said if you modeled the motion of each particle in some way that it spent half its time in one half and half its time in the other, then it would follow from that. But, that would replace a general statistical assumption with something more specific.

Thank you. I'm not fully understanding what is the postulate here?

The way the author writes I understand that I should be able to deduce it.

Can you please elaborate a little.
Thank you sir.

 

[vanhees71]

"...been left undisturbed for a very long time." usually means to assume that the system is in thermal equilibrium. Then the probability for each particle to be in either half of the volume is indeed 1/2. The rest is now combinatorics: "How many possibilities do you have to draw $n$ out of $N$ distinguishable balls from an urn."

 

[PeroK]

Thank you. I'm not fully understanding what is the postulate here?

The way the author writes I understand that I should be able to deduce it.

It's an assumption like the assumption about the pack of cards I gave in your other post. You can only deduce it if you make some more fundamental assumptions about what each particle is doing.

There is, however, an unstated assumption that the particles are independent.

 

[Kashmir]

It's an assumption like the assumption about the pack of cards I gave in your other post. You can only deduce it if you make some more fundamental assumptions about what each particle is doing.

There is, however, an unstated assumption that the particles are independent.

So "...we expect that, among a very large number of frames of the film, on the average only one out of every $2^N$ frames would show all the molecules to be in the left half" is a postulate and can't be proven unless we assume other basic postulates exist?

 

[PeroK]

So "...we expect that, among a very large number of frames of the film, on the average only one out of every $2^N$ frames would show all the molecules to be in the left half" is a postulate and can't be proven unless we assume other basic postulates exist?

What do you think? Could you imagine a scenario where, in fact, that's not the case? It wouldn't be true of sheep, as they tend to follow each other around. Or wildebeest on the savannah?

Or human beings on the surface of the Earth? Are we dotted around at random?

 

[Kashmir]

"...been left undisturbed for a very long time." usually means to assume that the system is in thermal equilibrium. Then the probability for each particle to be in either half of the volume is indeed 1/2. The rest is now combinatorics: "How many possibilities do you have to draw $n$ out of $N$ distinguishable balls from an urn."

How does "...been left undisturbed for a very long time." i.e to assume that the system is in thermal equilibrium. Imply that "then the probability for each particle to be in either half of the volume is indeed 1/2" ?

Why can't it be that after long time such state occurs that forbids the existence of some other state?

 

[PeroK]

Why can't it be that after long time such state occurs that forbids the existence of some other state?

That could be the case. Then you'd develop a different theory of statistical mechanics, based on your assumed particle behaviour (e.g. clump together under gravitational attraction).

 

[Kashmir]

That could be the case. Then you'd develop a different theory of statistical mechanics, based on your assumed particle behaviour (e.g. clump together under gravitational attraction).

In the model in question, we assume all states are possible. So basically that's an assumption we make which works well?

 

[Kashmir]

What do you think? Could you imagine a scenario where, in fact, that's not the case? It wouldn't be true of sheep, as they tend to follow each other around. Or wildebeest on the savannah?

Or human beings on the surface of the Earth? Are we dotted around at random?

Yes I can imagine when that's not the case.

 

[vanhees71]

In the model in question, we assume all states are possible. So basically that's an assumption we make which works well?

The point of a physical theory is, whether it's working in the sense that it is in accordance with observations in the real world. The standard assumptions of (quantum) statistical physics lead to a very successful description of all observations.

Here, I guess what's debated is the ideal gas in the canonical ensemble. In thermal equilibrium the only constraints are given by the conservation laws. With the container at rest and neglecting the gravity of the Earth, there's nothing that disinguishes any place within the container from any other. Then by symmetry it follows that in the equilibrium state the density of the gas is constant across the container, and indeed that's also what follows more formally from the maximum-entropy principle: The probability distribution (statistical operator) is determined as that one which maximizes the entropy under the given constraints.

 

[Kashmir]

The point of a physical theory is, whether it's working in the sense that it is in accordance with observations in the real world. The standard assumptions of (quantum) statistical physics lead to a very successful description of all observations.

Here, I guess what's debated is the ideal gas in the canonical ensemble. In thermal equilibrium the only constraints are given by the conservation laws. With the container at rest and neglecting the gravity of the Earth, there's nothing that disinguishes any place within the container from any other. Then by symmetry it follows that in the equilibrium state the density of the gas is constant across the container, and indeed that's also what follows more formally from the maximum-entropy principle: The probability distribution (statistical operator) is determined as that one which maximizes the entropy under the given constraints.

I agree.
The doubt I have is that probably the particles are assumed to be hard spheres that obey Newtons law , also while ignoring gravitation. The author then says that every state has a probability as Cn/2^N , this assumes that every state is indeed possible.
I want to know isn't this an assumption that seems to work or is it a consequence of something else that the author supposes I know but I don't.

 

[PeroK]

The doubt I have is that probably the particles are assumed to be hard spheres that obey Newtons law , also while ignoring gravitation. The author then says that every state has a probability as Cn/2^N , this assumes that every state is indeed possible.
I want to know isn't this an assumption that seems to work or is it a consequence of something else that the author supposes I know but I don't.

There comes a point where the responsibility is on you to say why certain states are not possible. There is something unsatisfactory, perhaps, about demanding that everyone else justifies every last detail, while you offer no justification for your doubts.

I don't think the author, @vanhees71 or I believe we are saying anything profound here. You need to provide some justification for the opposite hypothesis.

 

[vanhees71]

I agree.
The doubt I have is that probably the particles are assumed to be hard spheres that obey Newtons law , also while ignoring gravitation. The author then says that every state has a probability as Cn/2^N , this assumes that every state is indeed possible.
I want to know isn't this an assumption that seems to work or is it a consequence of something else that the author supposes I know but I don't.

If some states are not possible for some reason, of course, you have to take this into account. In statistical physics and thermodynamics it's anyway very important to always be clear about the conditions applied.

 

[PeroK]

I want to know isn't this an assumption that seems to work or is it a consequence of something else that the author supposes I know but I don't.

If, say, it's not possible to have all $N$ particles in one half of the box, then what is the smallest value of $N$ for which this is not possible?

It's definitely possible for $N = 1$. What about $N = 2$? Why is it not possible to have both particles in the same half of the box? What about $N = 3$?

 

[Kashmir]

If some states are not possible for some reason, of course, you have to take this into account. In statistical physics and thermodynamics it's anyway very important to always be clear about the conditions applied.

So suppose we've the situation as I've described earlier. A box with non interacting particles hypothetically divided into two parts. probability of having the configuration of n particles in the left side is given as Pn=C(n)/2N with C(n) being the total number of ways in which n particles from N can be placed in the left side.

The author has hence implicitly assumed that every state is possible. Practically, we can't know which state is or is not possible due to the complexity.

In this case, saying every state is possible , as the author does ,for me is an logical leap which works.

Am I correct?

 

[Kashmir]

There comes a point where the responsibility is on you to say why certain states are not possible. There is something unsatisfactory, perhaps, about demanding that everyone else justifies every last detail, while you offer no justification for your doubts.

I don't think the author, @vanhees71 or I believe we are saying anything profound here. You need to provide some justification for the opposite hypothesis.

There comes a point where the responsibility is on you to say why certain states are not possible. There is something unsatisfactory, perhaps, about demanding that everyone else justifies every last detail, while you offer no justification for your doubts.

I don't think the author, @vanhees71 or I believe we are saying anything profound here. You need to provide some justification for the opposite hypothesis.

I'm just saying some states may be possible while others may not be. This is a true statement in general.

The author assumes that every is possible so the burden proof is on the author.

 

[PeroK]

I'm just saying some states may be possible while others may not be. This is a true statement in general.

The author assumes that every is possible so the burden proof is on the author.

The burden of proof is on you if you want to learn anything about physics. The author already knows the physics. It's not his problem.

You're the one with the problem asking for help on here.

I know from previous posts that you've jumped into textbooks at the deep end without the usual prerequisite knowledge. That's your decision, but you can't expect to be spoon fed every last detail.

Learning physics at this level requires some intellectual effort on your part.

 

[Kashmir]

The burden of proof is on you if you want to learn anything about physics. The author already knows the physics. It's not his problem.

You're the one with the problem asking for help on here.

I know from previous posts that you've jumped into textbooks at the deep end without the usual prerequisite knowledge. That's your decision, but you can't expect to be spoon fed every last detail.

Learning physics at this level requires some intellectual effort on your part.

I think you've misunderstood what I was trying to say. I also agree that my foundations are weak and I also am a student who doesn't have a teacher as is usual in a classroom setting and have almost tried to study everything on my own

What I was trying to ask is that the author was saying a statement which I thought needed justification so I said author needs to prove this by saying" burden of proof is on him".

Still I respect your words since you've helped me so many times, and a like is the only way I can show my respect and gratitude.

Thank you :)