# Definition of an Ensemble

## Homework Statement

Confused about what a statistical ensemble actually means. Why does the ensemble have to have a uniform probability distribution at equilibrium? [If my definition of an ensemble is correct]

## The Attempt at a Solution

This is what I understand so far: [/B]
• For any given macrostate, there is going to be an associated set of microstates ( a region in phase space)
• If we look at a great number of systems started off with the same macrostate, under the same conditions, then if we looked simultaneously at all of them some time later you would have say, five in microstate 1, 2 in microstate 2 etc. etc.
• So in this way you have a probability distribution for the microstates [as a function of time] - is this what an ensemble is?
• But then why is it necessarily true that at equilibrium the probability distribution must be uniform? I.e the ensemble doesn't change with time?
• I can understand the logic for an isolated system - but why does it hold in general?
Thanks!

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Demystifier
I think that that the concept of ensemble in statistical physics creates more confusion than explanation. Most ideas of statistical physics can be more easily understood without it. The central concept one needs to start from is probability, and if one understands what probability is then the concept of ensemble is superfluous. The concept of ensemble is created for those who are confused with the concept of probability, with the motivation to better explain what probability "really is". However, the explanation based on ensemble can easily create even more confusion.

That being said, one can concentrate on your questions which do not mention ensemble. The entropy can be defined in terms of probability (without referring to ensembles) as
$$S=-\sum_x P(x) ln P(x)$$
where $x$ are points in the phase space of physical interest and $P(x)$ is the corresponding probability distribution. Depending on the physical context, the states $x$ can be either (fine-grained) microstates or (coarse-grained) macrostates. By definition, equilibrium probability distribution is the one that maximizes entropy. So it is a matter of relatively straightforward calculation that entropy (defined by equation above) is maximal when probability is uniform.

Or perhaps you wanted to know where does the definition "equilibrium probability distribution is the one that maximizes entropy" comes from?

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• bananabandana and bhobba
A. Neumaier
2019 Award
if one understands what probability is then the concept of ensemble is superfluous.
But to make sense of the concept of probability in the spirit of Kolmogorov one needs the notion of realizations of random variables. The ensemble is just the collection of all conceivable realizations.
why is it necessarily true that at equilibrium [...] the ensemble doesn't change with time?
Equilibrium is defined through stationarity in time. Thus the ensemble is time-independent by definition. If there are changes in time, is is a sure sign of lack of equilibrium.

• bananabandana and vanhees71
A. Neumaier
2019 Award
the ensemble have to have a uniform probability distribution at equilibrium?
Different ensembles have different distributions. The distribution of the canonical ensemble is not uniform.

N88
I think that that the concept of ensemble in statistical physics creates more confusion than explanation. Most ideas of statistical physics can be more easily understood without it. The central concept one needs to start from is probability, and if one understands what probability is then the concept of ensemble is superfluous. The concept of ensemble is created for those who are confused with the concept of probability, with the motivation to better explain what probability "really is". However, the explanation based on ensemble can easily create even more confusion.

That being said, one can concentrate on your questions which do not mention ensemble. The entropy can be defined in terms of probability (without referring to ensembles) as ...
"The central concept one needs to start from is probability, and if one understands what probability is."

Demystifier
It depends on the context, so the question is too general. Can you ask a more specific question?

• bhobba
bhobba
Mentor
"The central concept one needs to start from is probability, and if one understands what probability is." What is your understanding of probability, please?
It often leads to heated argument but my understanding is Kolmogorov's axioms

Like QM itself you can have different interpretations - frequentest, Baysian, Decision theory etc etc.

Interestingly much of QM interpretations is simply an argument about probability
http://math.ucr.edu/home/baez/bayes.html

I hold to the ignorance ensemble interpretation of QM which is ensemble, frequentest in its interpretation of probability etc etc - it all really means the same thing. Copenhagen, Bayesian etc etc (I really cant tell the difference) is Bayesian. Many worlds is decision theory based - at least in modern times.

Chose whatever you like - but for heavens sake don't worry about it - all are equally valid, or not valid - its meaningless really - just a personal preference thing or sometimes a specific choice makes solving a problem easier.

I did a degree in applied math where I had to do mathematical statistics 1a, 1b, 2a, 2b, and as an elective also 3a, 3b. We would seamlessly choose between different views such as frequentest or Baysian purely on the problem. Why not do the same in QM?

Thanks
Bill

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• Demystifier
N88
It depends on the context, so the question is too general. Can you ask a more specific question?
Demystifier, I was asking in the context of your statement: "I think that that the concept of ensemble in statistical physics creates more confusion than explanation. Most ideas of statistical physics can be more easily understood without it. The central concept one needs to start from is probability, and if one understands what probability is then the concept of ensemble is superfluous."

For I take an ensemble to be the starting point for understanding probability.

PS: Thanks Bill; I'm also a fan of Ballentine.

Demystifier
Demystifier, I was asking in the context of your statement: "I think that that the concept of ensemble in statistical physics creates more confusion than explanation. Most ideas of statistical physics can be more easily understood without it. The central concept one needs to start from is probability, and if one understands what probability is then the concept of ensemble is superfluous."

For I take an ensemble to be the starting point for understanding probability.

PS: Thanks Bill; I'm also a fan of Ballentine.
Then let me use an example. Suppose that you flip a coin, but only ones. How would you justify that the probability of getting heads is $p=1/2$? Would you use an ensemble for that?

Then let me use an example. Suppose that you flip a coin, but only ones. How would you justify that the probability of getting heads is $p=1/2$? Would you use an ensemble for that?
It would be premature to try to assign a probability based on one trial ( I hope you don't gamble).

I should say though, that if I knew the phase space of the coin tossing Hamiltonian, I could use that to get a probability.

Demystifier
It would be premature to try to assign a probability based on one trial
How about zero trials? Even without flipping a coin I would predict that probability of getting heads is p=1/2.

How about zero trials? Even without flipping a coin I would predict that probability of getting heads is p=1/2.
You would be making an assumption. Not all coins are 'fair'.

Being serious for a moment - if we have a Hamiltonian we can define an ensemble without reference to probability. Probability is not fundamental - likelihood ( the number of ways something can happen) comes first ( in physics anyway).

Demystifier
Probability is not fundamental - likelihood ( the number of ways something can happen) comes first ( in physics anyway).
I agree with that. But do we necessarily need ensembles for that purpose?

bhobba
Mentor
Probability is not fundamental - likelihood ( the number of ways something can happen) comes first ( in physics anyway).
We are delving into rather obvious circularity here - which is why the axiomatic approach is best for clarity.

Thanks
Bill

stevendaryl
Staff Emeritus
It often leads to heated argument but my understanding is Kolmogorov's axioms
I would take those axioms to define what it means to be a probability function---it's any function on subsets of events such that blah, blah, blah. But it doesn't say what it means to say that something has probability X, because for any set of possibilities, there are infinitely many different probability functions.

• Demystifier
A. Neumaier
2019 Award
We are delving into rather obvious circularity here - which is why the axiomatic approach is best for clarity.
The axiomatic approach only tells one what is permitted to do with probabilities, not what they are.

Various items on the meaning of probability can be found in Chapter A3: Classical probability of my theoretical physics FAQ.

stevendaryl
Staff Emeritus
You would be making an assumption. Not all coins are 'fair'.

Being serious for a moment - if we have a Hamiltonian we can define an ensemble without reference to probability. Probability is not fundamental - likelihood ( the number of ways something can happen) comes first ( in physics anyway).
Well, as Bill says, there is something a little circular about saying that

If there are $N$ possibilities, then each possibility has likelihood $\frac{1}{N}$.

That conclusion assumes that each possibility is equally likely. So you need some notion of likelihood to start with. We might say that "If I throw a pencil, there are three possibilities: It could land on its side, or it could land on its point, or it could land on the eraser." But obviously, those three possibilities don't all have probability 1/3.

It's possible that a lot of probability can be derived from some assumption along the lines of:

If there is a symmetry relating all $N$ possibilities, then they are all equally likely.

bhobba
Mentor
The axiomatic approach only tells one what is permitted to do with probabilities, not what they are.
Hmmmm - yes - but its a bit more complicated

From Feller - An Introduction To Probability Theory And Its Applications page 3
'We shall no more attempt to to explain the true meaning of probability than the modern physicist dwells on the real meaning of mass and energy or the geometer discusses the the nature of a point. Instead we shall prove theorems an show how they are applied'

It is the 'intuition' you build up in seeing how its applied that is its real content. In particular you need to see concrete examples of this thing called event in the axioms.

Thanks
Bill

• Demystifier
A. Neumaier
2019 Award
If I buy a lottery ticket, there are two possibilities: Either I win the jackpot, or I don't. The jackpot is worth many thousand times the lottery ticket. Assigning equal probabilities (or likelihoods) I should buy a dozen tickets and win almost with certainty.
If there is a symmetry relating all N possibilities, then they are all equally likely.
But this principle is almost never applicable. Coins, for example, are not symmetric, neither are dice.

It therefore requires some knowledge of physics to assign correct probabilities to a physical system that is not measured.

A. Neumaier
2019 Award
It is the 'intuition' you build up in seeing how its applied that is its real content. In particular you need to see concrete examples of this thing called event in the axioms.
Yes. This is the analogue of shut-up-and-calculate in applied mathematics.

• bhobba
stevendaryl
Staff Emeritus
I agree with that. But do we necessarily need ensembles for that purpose?
I may have been overly influenced by Bayesianism, but it seemed to me that ensembles (and the associated frequentist probability) doesn't actually help in understanding probabilities. You can understand "A coin toss has 50/50 chance of resulting in heads or tails" in terms of repeated trials as follows:

"A coin toss has a 50/50 chance of heads or tails" means "Tossing a coin 100 times will produce $50 \pm 5$ heads and $50 \mp 5$ tails with probability 99%" (or whatever the number is). But you've just defined the probability for one event (tossing a single coin) in terms of the probability for a different event (tossing 100 coins). You haven't explained anything.

stevendaryl
Staff Emeritus
But this principle is almost never applicable. Coins, for example, are not symmetric, neither are dice
Well, subjective (Bayesian) probability doesn't have this problem, because there is a symmetry in our knowledge about the coins. That is, we don't have any reason to prefer one side of a coin over another.

It therefore requires some knowledge of physics to assign correct probabilities to a physical system that is not measured.
Physics by itself isn't good enough either. Physics allows you to deduce (in principle---in practice, it's often too complicated) probabilities for final states from assumed probability distributions on initial states. But physics alone doesn't tell us the probabilities of the initial states.

Well, as Bill says, there is something a little circular about saying that

If there are $N$ possibilities, then each possibility has likelihood $\frac{1}{N}$.

That conclusion assumes that each possibility is equally likely. So you need some notion of likelihood to start with. We might say that "If I throw a pencil, there are three possibilities: It could land on its side, or it could land on its point, or it could land on the eraser." But obviously, those three possibilities don't all have probability 1/3.

It's possible that a lot of probability can be derived from some assumption along the lines of:

If there is a symmetry relating all $N$ possibilities, then they are all equally likely.
I don't know what you mean. There's no mention of phase space.

I mean something very different by 'likelihood'.

If you throw 2 dice, the various outcomes depend on the number of ways they can happen. And they are not the same. Probability is normalized likelihood and likelihoods are raw phase-space volumes.

bhobba
Mentor
Well, subjective (Bayesian) probability doesn't have this problem, because there is a symmetry in our knowledge about the coins. That is, we don't have any reason to prefer one side of a coin over another.
As I said one flicks between different interpretations depending on the problem. You cant use a frequentest view to assign a reasonable a-priori probability to a coin - but in the Bayesian view its rather trivial then one uses Bayesian inference to update the probabilities.

Thanks
Bill

bhobba
Mentor
If you throw 2 dice, the various outcomes depend on the number of ways they can happen. And they are not the same. Probability is normalized likelihood and likelihoods are raw phase-space volumes.
I think you are introducing a Bayesian reasonable a-priori view here. This stuff is notoriously slippery and circular.

Thanks
Bill