Lecture 17 - BE Condensation Revisited in a General Way For...

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BE Condensation Revisited in a General Way For BE condensation, the most important thing to understand is the behavior of the integral 1 0 1 ( ) when 1 1 I a d z ze    (This integral is actually N .) If ( 1) Iz diverges, everything is fine— no condensation. 0 1 ( ( ) 1 I z a d e  Consider ( ) ~ m a 1 11 1 Now if ~ , ~ ( ) ~ ~ ~ nn d d d n n k d k dk d a k dk k dk k k d k ( ) ~ dn n a d d i.e., m n When is large, e dominates hence there is no divergence. Consider small and c is some cut off where this approximation cease to work. 1 00 0 1 ( ~ c Cc m m m I z d d   ( diverges if 0 m ( 0 m logarithmic divergence.) No condensation. For condensation m 0 d -n i.e., 0 n or d > n For non-relativistic particles, 22 ~~ pk i.e., 2 n . That means no BE condensation in d=2 but BE condensation takes place at d=3. For relativistic particles, 1 ~ pc k i.e. 1. n That means BE condensation can take place in both d=2 and d=3.
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Ideal Fermi Gas       / 1 0 1 1 where and 0 1 ( ) 1 i i i i kT i i FD PV n n ze kT N n z e z ze f ad N e         Q 3/2 3/2 1/2 1/2 2 3 3 3/2 1/2 1/2 3
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Lecture 17 - BE Condensation Revisited in a General Way For...

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