#### JK423

Gold Member

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Hello guys,

I'm trying to understand bosonic condensation and i would really need your help.

The actual question, before getting into the details is:

"Why Bose-Einstein Condensation doesn't take place in a one-dimensional (1D) space?"

In what follows i'll give you my own (mis)understanding so that you will be able to see for any mistakes in my line of reasoning. I'll try to show, from what i have understood, that a BEC in 1D is possible.

Consider that i have a bosonic gas in a 1D infinite square well, hence with a non-zero ground state. The conservation of the total mean particle number is given by the condition

[itex]N = \sum\limits_i {\frac{1}{{{e^{\left( {{E_i} - \mu (T)} \right)/{k_B}T}}}-1}} \ \ \ (1) [/itex]

from which the chemical potential is determined. For simplicity choose e.g. the mass of the particles suitably so that i can write (1) as

[itex] N = \sum\limits_{n = 1}^\infty {\frac{1}{{{e^{\frac{1}{T}\left( {{n^2} - a(T)} \right)}}}-1}} \ \ \ \ (2)[/itex]

with [itex] - \infty < a(T) = \frac{\mu (T) }{{{k_B}}} \le {E_{ground\_state}}=1 \ \ \ (3)[/itex].

To my understanding, condensation occurs if there is a finite temperature below which the sum of the (mean) number of particles in all the excited states cannot reach [itex]N[/itex], i.e.

[itex] \sum\limits_{n > 1}^\infty {\frac{1}{{{e^{\frac{1}{T}\left( {{n^2} - a\left( T \right)} \right)}}}-1} \le N\,\,{\rm{for}}\,\,\,T \le {T_c}} \ \ \ \ \ (4)[/itex],

and start occupying the ground state.

Well, there is such a critical temperature even for a one-dimensional system! In particular, i did the sum (4) numerically for [itex]N = 10^3 [/itex], and found that for temperatures [itex] {T \le {T_c} = 1397}[/itex] which correspond to [itex]a\left( {{T_c}} \right) \le 1 [/itex] the sum (4) -that excludes the ground state- cannot reach [itex]N[/itex] since [itex]a[/itex] cannot be larger than 1.

Concluding, i gave you an example of 1D bosonic condensation without the need to go to the continuum and work with integrals etc as is usually done. So i have made no reference in the density of states etc, i just used the sum to find the critical temperature numerically.

Now, since i know that 1D condensation is not possible for reasons that i cannot understand, can you please tell at which point the reasoning above is mistaken?

Thank you in advance for your help!

I'm trying to understand bosonic condensation and i would really need your help.

The actual question, before getting into the details is:

"Why Bose-Einstein Condensation doesn't take place in a one-dimensional (1D) space?"

In what follows i'll give you my own (mis)understanding so that you will be able to see for any mistakes in my line of reasoning. I'll try to show, from what i have understood, that a BEC in 1D is possible.

Consider that i have a bosonic gas in a 1D infinite square well, hence with a non-zero ground state. The conservation of the total mean particle number is given by the condition

[itex]N = \sum\limits_i {\frac{1}{{{e^{\left( {{E_i} - \mu (T)} \right)/{k_B}T}}}-1}} \ \ \ (1) [/itex]

from which the chemical potential is determined. For simplicity choose e.g. the mass of the particles suitably so that i can write (1) as

[itex] N = \sum\limits_{n = 1}^\infty {\frac{1}{{{e^{\frac{1}{T}\left( {{n^2} - a(T)} \right)}}}-1}} \ \ \ \ (2)[/itex]

with [itex] - \infty < a(T) = \frac{\mu (T) }{{{k_B}}} \le {E_{ground\_state}}=1 \ \ \ (3)[/itex].

To my understanding, condensation occurs if there is a finite temperature below which the sum of the (mean) number of particles in all the excited states cannot reach [itex]N[/itex], i.e.

[itex] \sum\limits_{n > 1}^\infty {\frac{1}{{{e^{\frac{1}{T}\left( {{n^2} - a\left( T \right)} \right)}}}-1} \le N\,\,{\rm{for}}\,\,\,T \le {T_c}} \ \ \ \ \ (4)[/itex],

and start occupying the ground state.

Well, there is such a critical temperature even for a one-dimensional system! In particular, i did the sum (4) numerically for [itex]N = 10^3 [/itex], and found that for temperatures [itex] {T \le {T_c} = 1397}[/itex] which correspond to [itex]a\left( {{T_c}} \right) \le 1 [/itex] the sum (4) -that excludes the ground state- cannot reach [itex]N[/itex] since [itex]a[/itex] cannot be larger than 1.

Concluding, i gave you an example of 1D bosonic condensation without the need to go to the continuum and work with integrals etc as is usually done. So i have made no reference in the density of states etc, i just used the sum to find the critical temperature numerically.

Now, since i know that 1D condensation is not possible for reasons that i cannot understand, can you please tell at which point the reasoning above is mistaken?

Thank you in advance for your help!

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