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BoseEinstein Condensation
Note that
is valid only if no state is macroscopically occupied.
So when this inequality is
broken, some state is macroscopically occupied.
Lowest
Temperature
c
TT
when the equality is still satisfied (with
1
z
; actually
1
z
for
all
c
)
2/3
3
2
1
2.612
2
2
2.612
c
c
hn
T
mk
T
mk
Which state is macroscopically occupied?
Bosons can occupy any state macroscopically.
Makes sense
for
0
state though.
Check:
1
1
1
i
i
n
ze
When
0
1
1
0,
as
z
1
1
1
i
z
N
z
z
Also note that
1/2
( ) ~
a
which means
( )
0
a
for
0
and we have not taken the
0
state into
proper account.
So what?
What is one state worth in a continuum?
Problem here is that this state is
macroscopically occupied.
So we better write
0
0
( )
( )
BE
ex
N
a
d
f
N
N
0
3/2
3
1
()
NN
gz
V
When
c
,
0
0
N
, and when
c
,
0
0
N
At
c
T
,
3/2
3/2
~
(1)
c
N
Tg
V
(1)
Below
c
T
,
3/2
0
3/2
~
V
(2)
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 Spring '10
 Chakabarti
 mechanics, Statistical Mechanics

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