Condensate Fraction vs. Temperature Below T_c

1. The Breakdown of the Continuum Approximation

Let's begin by revisiting the core issue we face when $T<T_c$. Our standard method for calculating the total number of particles, $N$, involves converting the sum over discrete quantum states into an integral over a continuous energy spectrum:

In the last lesson, we found that this integral, with the chemical potential at its maximum possible value ($\mu =0$), gives us the total number of particles at the critical temperature, $N_{ex,max}(T_c)=N$.

What happens if we lower the temperature to some $T<T_c$? The number of available slots in the excited states decreases. The maximum number of particles the excited states can hold at this new, lower temperature is:

As we will soon see, this integral evaluates to a value less than $N$. This creates a paradox: where do the "missing" particles, $N-N_{ex,max}(T)$, go?

The resolution lies in a subtlety of the continuum approximation. By replacing the sum ${\sum }_k$ with an integral $\int d^{3}k$, we assign zero weight to any single point in k-space, including the ground state at $\mathbf{k}=0$ ($\epsilon =0$). Above $T_c$, this is justifiable as the occupation of any single state is negligible. Below $T_c$, however, the ground state becomes macroscopically occupied, and failing to account for it is a critical error.

The following notes provide an excellent discussion of this "paradox" and its resolution.

2. Calculating the Condensate Fraction

Having established the need to treat the ground state separately, we can now write the total particle number $N$ as the sum of particles in the ground state, $N_0$, and particles in all excited states, $N_{ex}$:

For any temperature $T\le T_c$, the system adjusts such that the chemical potential is pinned at $\mu \approx 0$ in the thermodynamic limit. This means the excited states are filled to their maximum capacity at that temperature. The number of particles in these excited states is therefore:

We've evaluated this integral form before. In the last lesson, we found that at $T_c$, it gives the total particle number $N$:

The expression for $N_{ex}(T)$ at $T<T_c$ has the exact same structure, just with $T$ instead of $T_c$:

This tells us that the maximum number of particles the excited states can accommodate scales as $T^{3/2}$.

Now, the derivation of the condensate fraction becomes straightforward. We can find the fraction of particles in the excited states by taking the ratio of $N_{ex}(T)$ to $N$:

The remaining particles must be in the ground state. The fraction of particles in the condensate, $N_0/N$, is therefore:

This is our main result. It elegantly describes how the macroscopic ground-state occupation grows from zero at $T_c$ to encompass all particles at $T=0$.

The following video provides a clear, step-by-step walkthrough of this derivation.

For a more mathematically compact and generalized perspective, Nobel laureate Anthony Leggett's notes are highly recommended.

3. Visualizing the Condensate and the Order Parameter

The formula $N_0/N=1-(T/T_c{)}^{3/2}$ represents a continuous phase transition. Below $T_c$, the system is fundamentally different; it has a macroscopic quantum object—the condensate—coexisting with a "normal" gas of thermally excited atoms.

This dramatic change is not just a theoretical construct; it has been stunningly observed in experiments. The image below shows absorption images of a cloud of sodium atoms being cooled through the critical temperature. These images effectively map the momentum distribution of the atoms.

The quantity we derived, the condensate fraction $N_0/N$, is the order parameter for the BEC transition. An order parameter is a quantity that is zero in one phase (the disordered, high-temperature phase) and non-zero in the other (the ordered, low-temperature phase). It quantifies the degree of "order" in the system.

• For $T>T_c$, $N_0/N=0$ (disordered phase, normal gas).
• For $T\le T_c$, $N_0/N=1-(T/T_c{)}^{3/2}>0$ (ordered phase, BEC).

This concept is profoundly important and general. In a ferromagnet, the order parameter is the average magnetization. In a superconductor, it's related to the complex value of the superconducting wavefunction. Your study of BEC provides a clean, calculable introduction to this cornerstone idea of modern condensed matter physics.

Conclusion

In this lesson, we have derived one of the central results in the study of ideal Bose gases: the temperature dependence of the condensate fraction.

Key Takeaways:

• Below the critical temperature $T_c$, the continuum approximation for the density of states fails because it neglects the macroscopic occupation of the $\epsilon =0$ ground state.
• By explicitly separating the ground state population ($N_0$) from the excited state population ($N_{ex}$), we can correctly describe the system for $T\le T_c$.
• The number of particles in excited states scales as $N_{ex}(T)=N(T/T_c{)}^{3/2}$.
• The condensate fraction is given by $\frac{N_0(T)}{N}=1-{\left(\frac{T}{T_c}\right)}^{3/2}$.
• The condensate fraction acts as the order parameter for the BEC phase transition, a key concept for describing phases of matter.

We have now characterized the particle distribution in a Bose gas both above and below $T_c$. The next logical step is to explore the thermodynamic consequences of this phase transition. In our next lesson, we will derive the temperature dependence of the heat capacity, where we will find a distinct signature—a cusp—at $T_c$, providing thermodynamic evidence for the transition we have just described.