Chemical Potential and BEC in 3D Ideal Bose Gas

Welcome to our next lesson on ideal Bose gases. In our previous discussion, we meticulously derived the equation of state for an ideal Bose gas, establishing the crucial separation of the total particle number $N$ into those in the ground state, $N_{0}$ , and those in excited states, $N_{e x}$ . This framework, captured by parametric equations involving the fugacity $z$ , is the foundation for everything that follows.

Today, we will use this machinery to uncover one of the most remarkable phenomena in quantum statistical mechanics. We will investigate the consequences of the equation of state in the low-temperature, high-density regime. This will lead us directly to a quantum phase transition, where the system's macroscopic properties change dramatically. Our objective is to show that a chemical potential $μ > 0$ is forbidden and derive the criterion for Bose-Einstein condensation (BEC) in a 3D ideal Bose gas.

1. The Constraint on the Chemical Potential

We begin by formalizing a constraint that was essential to our previous derivation. The average occupation number of a single-particle state with energy $ε$ is given by the Bose-Einstein distribution:

⟨ n_{ε} ⟩ = \frac{1}{e ^{β (ε - μ)} - 1}

For $⟨ n_{ε} ⟩$ to be a non-negative, physical quantity, the denominator must be positive and real. This requires $e^{β (ε - μ)} > 1$ , which in turn implies $ε - μ > 0$ , or $μ < ε$ .

This condition must hold for every available single-particle energy state $ε$ . Therefore, the chemical potential $μ$ must be strictly less than the energy of the lowest-lying single-particle state, $ε_{min}$ . For a gas of free particles in a three-dimensional box, we conventionally set the ground state energy $ε_{0} = 0$ . The constraint thus becomes $μ < 0$ .

What happens as $μ$ approaches 0 from below? The occupation of the ground state,

⟨ n_{0} ⟩ = \frac{1}{e ^{- β μ} - 1}

diverges as $μ \to 0^{-}$ . This is not a pathology of the theory; rather, it is the signature of a macroscopic number of particles accumulating in the ground state. The full constraint on the chemical potential for an ideal Bose gas is therefore $μ \leq 0$ , which means the fugacity $z = e^{β μ}$ is restricted to the interval $0 < z \leq 1$ .

The following resource provides a concise statement of this fundamental constraint.

The Ideal Bose Gas and Bose-Einstein Condensation

These lecture notes from the University of Rochester offer a very clear and direct path to the BEC criterion. We will use them as our primary guide for this lesson's derivation.

Begin by reading the short derivation on the first page that establishes the constraint on the chemical potential. Focus on the argument starting from the Bose occupation function that leads to the conclusion µ = kBT ln z <= 0.

The behavior of the chemical potential as a function of temperature is a key indicator of the system's state.

This graph shows the chemical potential $\mu$ as a function of temperature $T$ for an ideal Bose gas. At high temperatures, $\mu$ is large and negative, approaching the classical limit. As $T$ decreases, $\mu$ increases towards zero. Below the critical temperature $T_c$, $\mu$ becomes pinned at 0 in the thermodynamic limit.

2. Saturation of Excited States and the BEC Criterion

In our last lesson, we found that the total number of particles $N$ is the sum of those in the ground state, $N_{0}$ , and those in excited states, $N_{e x}$ . For spin-0 bosons, this is:

N = N_{0} + N_{e x} = \frac{z}{1 - z} + \frac{V}{λ ^{3}} g_{3/2} (z)

where $λ = h / 2 π m k_{B} T$ is the thermal de Broglie wavelength and $g_{3/2} (z)$ is a Bose-Einstein function.

Let's focus on the number of particles in the excited states, $N_{e x}$ . For a fixed volume $V$ and temperature $T$ , $N_{e x}$ is a function only of the fugacity $z$ . The function $g_{3/2} (z) = \sum_{k = 1}^{\infty} z^{k} / k^{3/2}$ is monotonically increasing for $z \in [0, 1]$ . This means that for a given temperature, there is a maximum number of particles that can be accommodated in the excited states. This maximum is reached when $z$ takes its maximum possible value, $z = 1$ (i.e., $μ = 0$ ):

N_{e x, max} (T) = \frac{V}{λ ^{3}} g_{3/2} (1)

The value $g_{3/2} (1)$ is simply the Riemann zeta function $ζ (3/2) \approx 2.612$ . Substituting the expression for $λ$ , we find:

N_{e x, max} (T) = V (\frac{2 π m k _{B} T}{h ^{2}})^{3/2} ζ (3/2)

This is a crucial result: the maximum capacity of the excited states is finite and scales as $T^{3/2}$ .

Now, consider a gas with a fixed total number of particles $N$ and a fixed density $n = N / V$ . At high temperatures, $N_{e x, max} (T)$ is very large, and all $N$ particles can easily fit into the excited states with $μ < 0$ ( $z < 1$ ). In this regime, $N_{0}$ is negligible.

However, as we cool the system down, $N_{e x, max} (T)$ decreases. Eventually, we will reach a temperature where the maximum capacity of the excited states is exactly equal to the total number of particles in the system. This temperature is the critical temperature, $T_{c}$ .

The following derivation will walk you through this argument, which is elegantly framed as resolving a paradox.

The Ideal Bose Gas and Bose-Einstein Condensation

Let's return to the Rochester notes to formally derive the BEC criterion and the formula for T_c. The logic is exceptionally clear.

Please read the argument starting from the top of the second page. First, understand the "contradiction" that arises. For a fixed density n, the equation n = g_{3/2}(z)/\lambda^3 cannot be satisfied if T is too low, because the right-hand side has a maximum value that decreases with T. The critical temperature T_c is defined as the temperature at which this contradiction first appears, which is when z=1. Follow the derivation of the formula for T_c. Next, see how the paradox is resolved by explicitly re-introducing the ground state term. The key is that for T < T_c, the fugacity is fixed at z=1, and the "missing" particles populate the ground state. Read the explanation of how the density of particles in the ground state, n_0/V, becomes finite below T_c.

To summarize the derivation, the critical temperature is defined by the condition $n = g_{3/2} (1) / λ_{c}^{3}$ , where $λ_{c}$ is the thermal wavelength at $T_{c}$ . Solving for $T_{c}$ gives:

T_{c} = \frac{h ^{2}}{2 π m k _{B}} (\frac{n}{ζ ( 3/2 )})^{2/3}

This equation provides the critical temperature for a 3D ideal Bose gas of density $n$ .

The criterion for Bose-Einstein condensation can be stated in two equivalent ways:

The system will be in a condensed state if its temperature is at or below the critical temperature, $T \leq T_{c}$ .
Alternatively, using the definition of $T_{c}$ , the criterion is that the dimensionless quantity $n λ^{3}$ must satisfy $n λ^{3} \geq ζ (3/2) \approx 2.612$ .

When this criterion is met ( $T \leq T_{c}$ ), the chemical potential becomes fixed at $μ = 0$ (in the thermodynamic limit), and the number of particles in the ground state becomes a macroscopic fraction of the total:

N_{0} (T) = N - N_{e x, max} (T) = N [1 - (\frac{T}{T _{c}})^{3/2}]

This macroscopic occupation of a single quantum state is the essence of Bose-Einstein condensation.

3. A Physical Picture of Condensation

The mathematical derivation is rigorous, but it's equally important to build a physical intuition for what is happening. The criterion $n λ^{3} \approx 2.612$ provides a powerful clue. It tells us that condensation occurs when the thermal de Broglie wavelength $λ$ becomes comparable to the average interparticle spacing, which is on the order of $n^{- 1/3}$ .

The following video and diagram provide an excellent conceptual visualization of this process.

Bose-Einstein Condensates | Prof B. Prasanna Venkatesh | Lecture 01

This lecture by Prof. B. Prasanna Venkatesh from IIT Gandhinagar offers a wonderful non-mathematical introduction to the physics of BEC, focusing on the de Broglie wavelength argument.

Please watch the segment from the beginning of the "back of the envelope understanding." The lecturer explains how, as a gas is cooled, the de Broglie wavelength of the particles increases. Condensation occurs when these wavelengths become so large that they overlap significantly, causing the individual atoms to lose their identity and merge into a single, coherent "matter wave."

This idea is beautifully illustrated in the following diagram.

The transition to a Bose-Einstein condensate. In the "Incoherent Regime" at high temperatures, particles are far apart and their de Broglie wavelengths ($\lambda$) are small. In the "Transition Regime" near $T_c$, the wavelengths become comparable to the interparticle spacing ($D$). In the "Coherent Regime" below $T_c$, the wavefunctions overlap to form a single macroscopic quantum state.

This physical picture complements the statistical argument we derived earlier. The statistical "saturation" of excited states and the physical "overlapping" of de Broglie waves are two ways of understanding the same underlying quantum phenomenon. Your experience with quantum mechanics provides the context to appreciate that these are not just classical particles crowding into a low-energy state, but matter waves interfering constructively to form a new state of matter.

Conclusion

In this lesson, we have reached the heart of the physics of ideal Bose gases. By analyzing the consequences of the equation of state at low temperatures, we have rigorously derived the conditions for Bose-Einstein condensation.

Key Takeaways:

The chemical potential of an ideal Bose gas is strictly bounded from above: $μ \leq ε_{0} = 0$ . This ensures that occupation numbers remain physical.
For a given temperature, the excited states of a 3D Bose gas have a finite maximum capacity, $N_{e x, max} (T) \propto T^{3/2}$ .
Bose-Einstein condensation occurs when the total number of particles $N$ exceeds this capacity. This happens below a critical temperature $T_{c}$ , given by $T_{c} \propto (n / m)^{2/3}$ .
The criterion for condensation can be expressed as $T \leq T_{c}$ or, equivalently, when the de Broglie wavelength becomes comparable to the interparticle spacing, $n λ^{3} \geq ζ (3/2)$ .
Below $T_{c}$ , the chemical potential is pinned at $μ = 0$ , and all particles in excess of $N_{e x, max} (T)$ occupy the ground state, forming the condensate.

We have now established that a phase transition occurs. In our next lesson, we will build upon this by calculating the fraction of particles in the condensate as a function of temperature and exploring the thermodynamic signatures of this transition.

Recommended resources

1. The Constraint on the Chemical Potential

2. Saturation of Excited States and the BEC Criterion

3. A Physical Picture of Condensation

Conclusion