### Ideal Bose gas

Introduction

The main motivation for discussing the ideal Bose gas is that it exhibits Bose-Einstein condensation. The original prediction of Bose-Einstein condensation by Satyendra Nath Bose and Albert Einstein in 1924 was considered by some to be a mathematical artifact. In the 1930s Fritz London realized that superfluid liquid helium could be understood in terms of Bose-Einstein condensation. However, the analysis of superfluid liquid helium is complicated by the fact that the helium atoms strongly interact with one another. In 1995 several groups used laser and magnetic traps to create a Bose-Einstein condensate of alkali atoms at approximately 10-6K. In these systems the interaction between the atoms is very weak so that the ideal Bose gas is a good approximation and is no longer only a textbook example.

The behavior of an ideal Bose gas can be understood by calculating N(T,V,μ), the mean number of particles for given values of the temperature T, volume V, and chemical potential μ. In the grand canonical ensemble N is given by

(1)

where g(ε) is the density of states. For a system of particles in three dimensions we have

(2)

and N(T,V,μ) is given by

(3)

Because the integral in Eq. (3) cannot be done analytically, we will evaluate the integral numerically. We introduce the dimensionless variables x = ε/kTc, T* = T/Tc, and μ* = μ/kTc, where Tc is the Bose condensation temperature and satisfies the relation

(4)

The integral on the right-hand side of Eq. (4) can be done numerically, with the result that Tc is given by

(5)

We next express Eq. (3) in terms of the dimensionless variables and obtain

(6)

or

(7)

where we have used Eq. (5).

We will study the behavior of μ* as a function of T*. The idea is to find μ* for a given value of T* such that the right-hand side of Eq. (7) equals unity.

Problems

1. The goal is to find the value of μ* for a given value of T* that makes the right-hand side of Eq. (7) equal to unity. Choose T* = 10 and choose the trial value μ* = -10. Do you have to increase or decrease the value of μ* to make the computed value of the integral closer to 1? Investigate how μ* changes as you decrease the (dimensionless) temperature T*. Choose T* = 5 and find the value of μ*. Does μ* increase or decrease in magnitude? You can generate a plot of μ* versus T* by clicking on the Plot (μ*,T*) button.
2. As T* is decreased, does the magnitude of μ* increase or decrease? What is the implication of this dependence as T* is lowered further?

References

- The 2001 Nobel Prize for Physics was awarded to Eric Cornell, Wolfgang Ketterle, and Carl Wieman for achieving Bose-Einstein condensation in dilute gases of alkali atoms and for early fundamental studies of the properties of the condensate. An excellent resource is at NIST.

Updated 28 December 2009.