Sunday, September 24, 2017

statistical mechanics - Why is chemical potential, μ=0 when calculating critical temperature of BECs?


How do we justify taking the chemical potential, μ as 0 when calculating the critical temperature of Bose-Einstein Condensates (BECs)?


I apologise as I do not how to use LaTeX, for if I did the elegance of mathematics would’ve allowed me to construct my question with ease...


I understand to calculate the total number of particles in a system comprised of non-relativistic bosons of mass m at thermal equilibrium at temperature T. One must simply some over the occupancies for each energy state, the occupancies is given by the bose-einstein distribution...


For some reason during the derivation setting chemical potential to zero within the bose-einstein distribution gives us the largest possible number of particles for a given temperature, can someone explain why this is true?


Edit: Also I know the within the bose-einstein distribution, the energy of the states must always be greater than the chemical potential, this confines the distribution to a range of 0<bose-einstein distribution<+

I can say that the lowest energy state (ground state) has an energy of 0 and thus chemical potential < 0, but if my ground state has an arbitrary non zero energy would the chemical potential = 0?



Answer



To determine the upper limit on chemical potential for a gas of N bosons, look at the form of the Bose distribution in the grand canonical ensemble with N=N. When using the GCE, it's easiest to work at chemical potential μ and to then choose μ(N) so that N(μ)=N. Each state s has average occupancy ns=n0neβn(ϵsμ)n0eβn(ϵsμ)=1Ξs(βμ)Ξs,Ξs=11eβ(ϵsμ),=(βμ)log(1eβϵs+(βμ))=eβμ1eβ(ϵsμ).

This is finite as long as μ<ϵs. In order for N=sns to be finite we need μ<minsϵs=ϵ0. Hence, for any system of bosons where N is conserved we have μ<ϵ0. It is conventional to set ϵ0=0 for simplicity, but you can have systems with ϵ00. As you implied by your final question, in these systems the critical value of μ is ϵ0 in the thermodynamic limit, with N,V,E and μ,p,T held constant. Of course, if the system has no BEC phase then as T0, the chemical potential μ never exceeds some value μmax<ϵ0.


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