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Definition of an Ensemble

  1. Apr 14, 2016 #1
    1. The problem statement, all variables and given/known data
    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]

    3. The attempt at a solution

    This is what I understand so far:

    • 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!
     
  2. jcsd
  3. Apr 15, 2016 #2

    Demystifier

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    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?
     
    Last edited: Apr 15, 2016
  4. Apr 15, 2016 #3

    A. Neumaier

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    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.
    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.
     
  5. Apr 21, 2016 #4

    A. Neumaier

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    Different ensembles have different distributions. The distribution of the canonical ensemble is not uniform.
     
  6. Apr 21, 2016 #5

    N88

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    "The central concept one needs to start from is probability, and if one understands what probability is."

    What is your understanding of probability, please?
     
  7. Apr 22, 2016 #6

    Demystifier

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    It depends on the context, so the question is too general. Can you ask a more specific question?
     
  8. Apr 22, 2016 #7

    bhobba

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    It often leads to heated argument but my understanding is Kolmogorov's axioms
    http://www.econ.umn.edu/undergrad/math/Kolmogorov's Axioms.pdf

    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
     
    Last edited: Apr 22, 2016
  9. Apr 22, 2016 #8

    N88

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    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.
     
  10. Apr 22, 2016 #9

    Demystifier

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    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?
     
  11. Apr 22, 2016 #10

    Mentz114

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    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.
     
  12. Apr 22, 2016 #11

    Demystifier

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    How about zero trials? Even without flipping a coin I would predict that probability of getting heads is p=1/2.
     
  13. Apr 22, 2016 #12

    Mentz114

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    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).
     
  14. Apr 22, 2016 #13

    Demystifier

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    I agree with that. But do we necessarily need ensembles for that purpose?
     
  15. Apr 22, 2016 #14

    bhobba

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    We are delving into rather obvious circularity here - which is why the axiomatic approach is best for clarity.

    Thanks
    Bill
     
  16. Apr 22, 2016 #15

    stevendaryl

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    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.
     
  17. Apr 22, 2016 #16

    A. Neumaier

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    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.
     
  18. Apr 22, 2016 #17

    stevendaryl

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

    If there are [itex]N[/itex] possibilities, then each possibility has likelihood [itex]\frac{1}{N}[/itex].


    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 [itex]N[/itex] possibilities, then they are all equally likely.
     
  19. Apr 22, 2016 #18

    bhobba

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    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
     
  20. Apr 22, 2016 #19

    A. Neumaier

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    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.
    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.
     
  21. Apr 22, 2016 #20

    A. Neumaier

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    Yes. This is the analogue of shut-up-and-calculate in applied mathematics.
     
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