# A Bose-Einstein condensate

#### MathematicalPhysicist

Gold Member
In the book A Quantum Approach to Condensed Matter Physics by Taylor and Heinonen they write the following passage on page 87:

Thus there is a temperature $T_c$, defined by $N_0(T_c)=N$, below which the zero-energy state is occupied by a macroscopic number of particles. This phenomenon is known as the Bose-Einstein condensation, and is remarkable in being a phase transition that occurs in the absence of inter-particle forces.
Now I don't understand what does it mean "macroscopic number", how many particles?

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#### DrDu

It means that $N/N_0 = O(N_0^0)$ if N is the number of particles in the condensate and $N_0$ the total number of particles, or, easier, $N =O(N_0)$.

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#### MathematicalPhysicist

Gold Member
@DrDu the total number of particles in where? (in the universe, just outside the condensate, it's not clear to me).

So obviously, $N=aN_0$ where $0<a<1$.

#### DrDu

First, I don't know whether my use of $N$ and $N_0$ coincides with the one from your book.
I consider $N_0$ to be the number of all particles in the substance you are considering while $N$ is the number of particles in the "condensate", i.e. the number of particles in the ground state, or, more generally if the particles are interacting, the lowest eigenvalue of the 1-density matrix.

"Bose-Einstein condensate"

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