What is the significance of a 'macroscopic number' in Bose-Einstein condensates?

[MathematicalPhysicist]

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?

 

[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, $N =O(N_0)$.

 

[MathematicalPhysicist]

@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.

 

[HomogenousCow]

It just means a significant number of the particles in the system are in the groundstate, say half of them.

 

[Laci]

In the book A Quantum Approach to Condensed Matter Physics by Taylor and Heinonen they write the following passage on page 87:
Now I don't understand what does it mean "macroscopic number", how many particles?

The mathematical definition of a macroscopical number of particles is that the ratio Number of particles N to the volume V of the system remains finite as V goes to infinity i.e. the density of particles is finite in the thermodynamic limit.

 

[Demystifier]

Now I don't understand what does it mean "macroscopic number", how many particles?

$\gg 1$

 

[MathematicalPhysicist]

$\gg 1$

2 is enough or we need $\infty$?

 

[DrClaude]

2 is enough or we need $\infty$?

How many grands of sand do you need to make a pile?

Condensates can easily be made with 10⁶-10⁷ atoms.

 

[MathematicalPhysicist]

How many grands of sand do you need to make a pile?

one.

 

[Demystifier]

one.

https://en.wikipedia.org/wiki/Sorites_paradoxhttps://www.physicsforums.com/threads/collection-of-science-jokes-p2.847743/page-53#post-6229594