Ask A Scientist© Physics Archive

name         Kathryn
status       student
age          13

Question -   What exactly is the Einstein/Boise theory?
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   My best guess is you're talking about Bose-Einstein statistics,
because the ``Bose condensation'' has been in the news, sort of,
lately.
   We begin the (long) answer with a question: how many different
results are there if you flip 1 coin?  The answer is obviously 2: H
(heads) or T (tails).  What if you flip 2 coins?  The answer is 4: HH,
HT, TH, or TT.  We might think of this as follows: for EACH result of
flipping the 1st coin (H or T) we have 2 results for flipping the 2nd.
Hence the total number of results is 2 x 2 = 4.  In the same way if we
had 3 coins we could think: for EACH result of flipping the 1st, we
have 2 choices for flipping the 2nd, and for EACH of those we have 2
choices for fliping the 3rd, which gives 2 x 2 x 2 = 8 results, namely
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.
   Objects that behave this way, where the number of results for N
objects is equal to the number of results for 1 object multiplied by
itself N times, are said to obey Boltzmann statistics.  Practically
every object you know of (people, cars, balls of strings, leaves)
obeys Boltzmann statistics.
   But atoms do not.  Why not?  Let's consider a situation at first
sight identical to our coins.  How many ways are there to put 1 object
into a set of two boxes?  The obvious answer is 2, as follows:

       -----    -----          -----    -----
      | O   |  |     |        |     |  | O   |
       -----    -----          -----    -----

   This seems to be identical to our previous situation if we consider
the object to be a coin, and label the boxes HEADS and TAILS.
   Okay, then, how many ways are there to put 2 exactly identical
objects into a set of two boxes?  From our coin work, we might think,
gee, obviously 2 x 2 = 4.  Not so.  Because the objects are EXACTLY
IDENTICAL, the answer is 3, as follows:

       -----    -----       -----    -----       -----    -----
      | O   |  | O   |     | O O |  |     |     |     |  | O O |
       -----    -----       -----    -----       -----    -----

   It is only if the objects are slightly different that the answer is
4, as follows:

    -----   -----     -----   -----     -----   -----     -----   -----
   | O   | | o   |   | o   | | O   |   | O o | |     |  |     | | O o |
    -----   -----     -----   -----     -----   -----   -----   -----

   We must conclude that only objects that are different obey Boltzmann
statistics.  Objects that are exactly identical behave differently:
they are said to obey Bose-Einstein statistics.
   How do we square this with our results for coins?  Aren't coins
identical?  Why do they obey Boltzmann statistics?  The answer is that
the coins are NOT exactly identical, for a very subtle reason: we see
one being flipped before the other.  The coins are different by virtue
(and only by virtue) of the fact that one is flipped before the other,
and we know which one that is.  That is, it is our knowledge of the
history of the objects that allows us to tell them apart, even if they
look identical.  All objects you see around you follow Bolzmann
statistics, because even if you can't tell them apart by looking at
them, you can always tell them apart by discovering their history,
which will be unique for each object.
   Atoms are different.  Not only are they exactly identical, but you
cannot distinguish their history, even in principle, because quantum
mechanics says atoms do not each have their own individual history.
Hence atoms follow Bose-Einstein statitics.
   Actually, another possibility exists for atoms.  Some atoms are such
that no two are allowed to be in the same situation.  For these atoms
there is only 1 way to put 2 atoms in 2 boxes, since each atom must be
in a different box.  These atoms are said to follow Fermi-Dirac
statistics.  *Why* some atoms follow Fermi-Dirac statistics needs to
be addressed separately, so ask another question if you want to know.
   So what's so special about following Bose-Einstein statitics?  Well,
nothing, really, unless you expect things to always follow Boltzmann
statistics.  Then you can observe stuff that surprises you.
   For example, we see that if we throw 2 atoms into 2 boxes randomly
we'll end up seeing both atoms together in one box more often (1 out
of 3 times) if they follow Bose-Einstein statistics than if they
follow Boltzmann statistics (1 out of 4 times).  If we do the same
experiment with a lot more atoms and a lot more boxes, this difference
turns out to become much larger, as we could prove by writing out all
the possible results, like we did for 2 atoms and 2 boxes above.
   We don't have to think about real boxes, either.  The boxes can be
symbols for any distinguishable situation the atoms might be found in.
We can conclude that if we have a lot of atoms in a lot of possible
situations (``boxes''), and *if* we expect them to follow Boltzmann
statistics, then we will be surprised by how often they end up in one
single situation (box), all doing the same thing.
   The ending-up-in-one-situation is called a ``Bose condensation'',
and it makes the news because it surprises human beings, whose
intuition is to always see Boltzmann statistics.
   There are also a number of interesting things that happen when atoms
do a Bose condensation, such as superfluidity and superconductivity.
Grayce
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