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Boltzmanns statistics and phase space

  1. Nov 25, 2005 #1
    can anyone handle this one?

    when deriving a distribution function using a purely statistical approach, boltzmann uses some kind of a phase space, that is one with 6N dimentions, 3 for position, 3 for momentum. i see some really short descripions of it but not enough to understand it.

    all i need is an example, like 3 molecules with only 3 units of energy to share. how would this be split up into the possible states in phase space? and what does phase space accomplish in this derivation.

    any brainiacs know how to explain this well???

    thanks for any insight. or links, or books,...
     
  2. jcsd
  3. Nov 25, 2005 #2

    Physics Monkey

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    Start out with a single particle. How many numbers does it take to specify the state of the particle? In three dimensions it takes six numbers: the 3 components of position and the 3 components of momentum (or equivalently velocity). It is important that you remember to tell how fast the particle is going as well as where it is to completely specify the state of the particle. Now imagine a six dimensional space where each point in this space is a list of six numbers. The first 3 numbers are components of the position of the particle and the second three numbers are the components of the momentum of the particle. This six dimensional space is called phase space and any point in phase space corresponds to state of the particle. Now, in statistical mechanics (in the canonical ensemble, say) what you really want to know is the probability that your particle has a certain state i.e. is at a certain point in phase space. According to Boltzmann, this probability distribution is proportional to
    [tex]
    \exp{\left(- \frac{H(\vec{r},\vec{p})}{k T} \right)},
    [/tex]
    where [tex] H [/tex] is the Hamiltonian or energy of your system. For a particle moving in a potential [tex] V [/tex], the energy looks like
    [tex]
    H(\vec{r},\vec{p}) = \frac{p^2}{2 m} + V(\vec{r}).
    [/tex]

    If you now jump to N particles then phase becomes much larger. It now has 6N dimensions, 6 dimensions for each of the N particles. A single point in phase space is a list of 6N numbers that tells you the state (position and momentum) of all N particles in your system. The probability distribution is again proportional to
    [tex]
    \exp{\left(- \frac{H(\vec{r}_1, ...,\vec{r}_N,\vec{p}_1,...,\vec{p}_N)}{k T} \right)},
    [/tex]
    where [tex] H [/tex] is the total energy of the system. This total energy contains the kinetic energy of each of the N particles, the external potential energy of each of the N particles, and the interaction energy that may exist between the particles. Everything you could possibly want to calculate can now be written as integrals over phase space weighted by this probability density. In particular, the partition function is simply the normalization of this probability density.

    That is phase space in five minutes. Of course, I have left out a wealth of details, conditions, caveats, etc. I can try to answer specific questions or clarify anything I brought up, but you would probably do better to consult a book. Any reasonably advanced textbook on classical mechanics will discuss the Hamiltonian approach and phase space as will any reasonable book on statistical mechanics, just go to your library and pick one.
     
  4. Nov 25, 2005 #3
    thank you physics monkey but this is really not helpful. it looks like you are sort of just throwing terms around.

    could someone else just give me a small (but concrete)example. i have the feeling the approach was used in order to cross the border from discrete to continuous energy. but the method and reasoning for "carving up cells" in space needs to be clarified to me.

    thanks to anyone who could help!!
     
  5. Nov 25, 2005 #4

    Physics Monkey

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    I'm sorry you found my reply so unhelpful, 2bootspizza. However, if it all really looks like I'm just throwing words around, then maybe you really need to review some of the things I brought up. Perhaps my presentation was poor, but everything I said was fundamental and extremely important for understanding the role of phase space in statistical mechanics. If you haven't heard of the terms I used, then I strongly suggest you ask your teacher or look in statistical mechanics book. I seriously doubt anyone here will be able to give you an example you can understand, but I hope I'm wrong and I wish you the best.
     
  6. Nov 25, 2005 #5
    ok, i will be a little more specific with your answer.

    i specifically mentioned boltzmanns statistical approach to say, molecules . here we deal with kinetic energy. where do potential energies come into play when deriving the energy distribution function for ideal gases? the hamiltonian is also useless here.

    here is a sentance you throw in:
    In particular, the partition function is simply the normalization of this probability density.

    this comes out of nowhere. do you know what a partition function is? tell me how the number 5 would be partitioned. thats very easy.

    i'm sorry physics monkey, i dont mean to insult you but did you ever take a physics course?

    i will have to get ever more clearer. here is something i read on the net:
    The procedure of dividing μ space into cells is essential here. Indeed, the whole prospect of using combinatorics would disappear if we did not adopt a partition. But the choice to take all cells equal in size in position and momentum variables is not quite self-evident, as Boltzmann himself shows

    this comes from an earlier paragraph:


    We now partition μ into m disjoint cells: μ = ω1∪…∪ωm. These cells are taken to be rectangular in the position and momentum coordinates and of equal size. Further, it is assumed we can characterize each cell in μ with a molecular energy εi.

    For each x, henceforth also called the microstate, we define the macrostate (Boltzmann's term was Komplexion) as Ζ := (n1,…,nm), where ni is the number of particles that have their molecular state in cell ωi. The relation between macro- and microstate is obviously non-unique since many different microstates, e.g., obtained by permuting the molecules, lead to the same macrostate. One may associate with every given macrostate Ζ0 the corresponding set of microstates:

    then later this comment:

    The procedure of dividing μ space into cells is essential here. Indeed, the whole prospect of using combinatorics would disappear if we did not adopt a partition.

    a million hugs to who could answer why this is essential!!!
     
  7. Nov 25, 2005 #6

    Physics Monkey

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    Ok 2bootspizza, I know I should just ignore your personal attacks, but I'm going to respond anyway. This post may be deleted but I will enjoy writing it so here goes.

    First, you don't know what phase space is. When I tell you what phase space is, you tell me I'm "throwing words around". Second, I explained how to obtain the full joint probability distribution for any system you like. You tell me that the Hamiltonian is "useless". Third, how do you suppose molecules hold themselves together? Magic perhaps, little purple monkeys, leprechauns maybe? Try intermolecular forces. If you want to model a monatomic ideal gas then the set the external potential to zero. Wow, that was hard. If you want to model a molecule, you will most likely need to account for some of the rotational, vibrational, etc degrees of freedom. Fourth, do you know what a partition function is? If you had the slightest clue what you were talking about or had ever studied statistical mechanics in a physics course, you would have come across the partition function. http://en.wikipedia.org/wiki/Partition_function_(statistical_mechanics) If you want to talk about the number theoretic partition function then I have some easy advice, learn to count. Fifth, you wouldn't know anything about how the fundamental "discreteness" of phase space utlimately originates from quantum theory. Furthermore, you wouldn't know that a careful analysis of the classical limit of quantum statistical mechanics indicates that the fundamental volume of phase space is set by Planck's constant. Sixth, one is certainly free to discretize phase space more coarsely than the quantum limit depending on the problem at hand but the results are the same. Again, any book on statistical mechanics will discuss all this in detail, see for example "Theoretical Physics" by G. Joos: it's a nice Dover book.

    Do you make a habit of quoting verbatim from your notes without ever trying to understand what you're doing?

    Best wishes!

    P.S. I've had plenty of physics classes, you?
     
    Last edited: Nov 26, 2005
  8. Nov 26, 2005 #7
    here is the number 5 partitioned:
    5
    4:1
    3:2
    3:1:1
    2:2:1
    2:1:1:1
    1:1:1:1:1

    if one knows in principle what a partition function is, the question i asked could have been answered in 20 seconds. and its easier than defining a term with other abstract terms. (instead you refer me to wiki;-)

    further, small examples are the best way to go.

    by the way: 'classical limit' of phase space depends on h?? if the uncertainty principle gives us the limit of a 'point' in phase space, how is that classical?? do you see what i mean by not making sense:)

    i have the feeling you are a high school student which is ok. good luck in college! and i was impolite before, but your answer wouldnt be helpful to any student. i apologize. look for example the way richard feynman writes in his books. he talks so plain. he has alot of confidence in his knowledge and would never throw around abstract terms unless absolutely necessary. he wants to be clear!
     
  9. Nov 26, 2005 #8
    ah yes, to back up what i just said: this is from your favorite site, wiki web on phase space:

    Whilst the classical phase space is a continuum, the introduction of Planck's constant quantisises the space and the movement along the trajectory happens in small "jumps" of size h.

    from http://en.wikipedia.org/wiki/Phase_space

    which brings up a whole other topic which is so interesting. i once read somewhere, where heisenberg was trying to explain why the electron path detected in a cloud chamber really is not a classical path but jumps from cell to cell as described above. that is also very hard for me to understand!! but i'm not here to showcase my intelligence, rather to prove the lackthereof:)
     
  10. Nov 26, 2005 #9

    Physics Monkey

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    And here I thought that you would be gone when instead you came back to tell me that I'm a high school student. How very clever of you.

    Now did you actually read anything on the wiki article about the statistical partition function? The article even has a little paragraph which discusses exactly this issue of coarse graining in phase space. I can see how you are confused in your class if you think that wiki is my favorite website just because I posted one link to it. Maybe you need to work on your logical reasoning skills.

    You also bring up Richard Feynman, but it's obvious you've never read his technical work on statistical mechanics. I found this very funny. Do you know why? It's because the very first thing he writes down is the partition function! It seems even the mighty Feynman thinks the partition function is important, in his own words it is the "summit of statistical mechanics." Go to Amazon.com and look in his book if you don't believe me.

    I'm not going to waste anymore time with you. What's funny is that I can actually do all this stuff, but instead of learning you chose to talk out of your ignorance. Pity.

    Best wishes.
     
  11. Nov 26, 2005 #10
    if you can do this stuff then why can't you partition a simple integer like 5? and why do you state the classical limit of phase space is h? i'm sure you didn't mean that.

    i never stated(re-read above) that the partition function is not important in statistical mechanics. i was referring to the obtuse reference to it.

    i can give a small example with say 3 molecules, and 3 energy units. or something similar. its easy for anyone to come up with the all the possible macrostates and microstates:

    macrostate 1
    3|0|0
    0|3|0
    0|0|3
    macrostate 2
    2|1|0
    0|1|2
    0|2|1
    2|0|1
    1|0|2
    2|1|0
    macrostate 3
    1|1|1

    so we have 3 macrostates and 10 microstates. but this is just a simple discrete example. my guess is that phase space is introduced to as an aid to derive the distribution function, for example when energy is continous and cells are used to partition ranges of energy, or momemtum. but i'm just not sure. my phyiscs is shakey here.

    i'm going to follow this up with something similar.
     
  12. Nov 26, 2005 #11
    as another perspective i have a physics book by arthur beiser called modern physics. as an appendix he derives the maxwell-botzmann energy distribution function.

    as put forth by beiser:

    consider an assembly of N molecules whose energies are limited to e1, e2, e3, ... e(i). can be either discrete or continuous.
    what is the number of ways, W, of in which these molecules arranged in 'phase space'.

    he gives as the product of 2 things:
    the permutations of one microstate which is
    N!/n1!n2!n3!.... (1)
    this is clear to me

    the second term is more confusing. as beiser states -

    if there are g cells with energy e, then the number of ways in which one molecule can have energy e is g. he then says the total number of ways 2 molecules can have eneregy e is g*g. and n molecules would be g*g*g...nth-g or g^n. but this confuses me! i would think the number of ways would be,
    (g + n -1)!/(g - 1)!n!
    for 2 molecules in 3 cells that would be 6 different ways. i know that is correct.
    the only way i can see g^n be correct, would be if g was a probability. like, if 50% of the cells have energy e then g would be 0.5
    then g^n would make sense to me. or what then is phase space here?

    so the starting point for deriving the distribution function is the product of (1) and g^n.

    i just need help understanding g^n here.
     
  13. Nov 27, 2005 #12
    Im not quite sure why 2bootspizza is trying to pick a fight with Physics Monkey because he clearly answered your question. He gave a you a completely valid physical derivation, which is exactly what you asked.

    Not that you understand what you are quoting, but if you are that interested in the statistical approach then you should be roaming a mathematics forum.

    You also clearly needed the defintion of phase space, and if you dont understand Physics Monkey's explanation then you might want to reconsider your chosen profession because it doesnt get any clearer than that.

    Physics Monkey doesn't need anyone to back him up (his answers do that just fine) but I am rather curious how 2bootspizza could come to the conclusion that he had never taken a physics course?
     
  14. Nov 27, 2005 #13

    Gokul43201

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    2boots :

    If you haven't been warned for making personal attacks, consider yourself lucky and take this opportunity to fix your behavior here. Did you think it would be in your best interests to go about insulting people that take time off to help you with your questions ?
     
  15. Nov 27, 2005 #14
    1) 2boots in regards to your last post I think you are mixing up indistinguishable energy units among molecules with distinguishable molecules in phase space. In the case of distinguishable units(gas molecules) you would have 3^3, or 27 permutations whereas with indistinguishable units(photons) there will be (N + M - 1)!/(N - 1)!M! or 10 permutations.

    2) When tallking about statistical mechanics the term 'phase space' often refers to an individual state(microstate) of a system. My honest advice is to get a hold of Boltzmann's original paper(1877) if you want to understand his approach.. He had a habit of being extremely verbose and with that there shouldn't be too much left to the imagination.

    3) Sorry but the above definition would probably only be understandable to a trained physicist with hindsight in this field.
     
  16. Nov 27, 2005 #15
    If you still don't see it I can write out all 27 permutations but I'd rather not.
     
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