Question

We have shown that the entropy for Bose-Einstein and Fermi-Dirac statistics is given by $$ S(E, V, N)=k(\beta E+\alpha N) \mp k \sum_{j} g_{j} \ln \left(1 \mp e^{-\alpha} e^{-\beta \varepsilon_{j}}\right), $$ where \(N=\sum_{j} N_{j}\) and \(E=\sum_{j} N_{j} \varepsilon_{j}\). Similarly, from classical thermodynamics, $$ d S(E, V, N)=\frac{1}{T} d E+\frac{P}{T} d V-\frac{\mu}{T} d N $$ for a single-component system. a. Prove that \(\beta=1 / k T\). b. Prove that \(\alpha=-\mu / k T\). c. Show that the pressure is given by $$ P=-\sum_{j} N_{j}\left(\partial \varepsilon_{j} / \partial V\right)_{E, N} . $$ d. We will soon demonstrate that the internal energy is independent of volume whereas the translational energy \(\varepsilon_{j, t r}=C_{j} V^{-2 / 3}\), where \(C_{j}\) is a constant for each translational energy level. Utilizing this information, show that $$ P V=\frac{2}{3} E_{t r}, $$ where \(E_{t r}\) is the translational portion of the total energy (external plus internal) for the particle assembly. e. Is this result limited to the dilute limit? Does it assume independent particles? Why should the pressure be related only to the translational energy? Explain.

Short answer

By comparing the derivatives of the entropy function with classical thermodynamics relations, we find \(\beta = 1 / k T\) and \(\alpha = -\mu / k T\). The pressure is derived by relating the entropy's partial derivative with respect to volume to \(P/T\), and the relationship \(P V = \frac{2}{3} E_{\text{tr}}\) follows from the independence of internal energy from volume, and how translational energy scales with volume. The result doesn't assume independent particles but is a consequence of the statistics of the system and the nature of translational energy.

Step-by-step solution

Proving \(\beta = 1 / k T\)

To prove \(\beta = 1 / k T\), differentiate the given entropy function \(S(E, V, N)\) with respect to energy \(E\). The partial derivative of S with respect to E should yield \(1/T\), according to classical thermodynamics. Taking the partial derivative of the given entropy with respect to \(E\), we have \(\frac{\partial S}{\partial E} = k \beta\). Comparing this result to the classical thermodynamics relationship \(\frac{\partial S}{\partial E} = \frac{1}{T}\), we find that \(k \beta = \frac{1}{T}\), which implies \(\beta = 1 / k T\).

Proving \(\alpha = -\mu / k T\)

To prove \(\alpha = -\mu / k T\), differentiate the entropy function \(S(E, V, N)\) with respect to the particle number \(N\). The partial derivative of S with respect to N should yield \(\-\mu/T\), as given by classical thermodynamics. Taking the partial derivative of the given entropy with respect to \(N\), we have \(\frac{\partial S}{\partial N} = k \alpha\). Upon comparing with the classical thermodynamics relationship \(\frac{\partial S}{\partial N} = -\frac{\mu}{T}\), it follows that \(k \alpha = -\frac{\mu}{T}\), which implies \(\alpha = -\mu / k T\).

Showing the expression for pressure \(P\)

To show the expression for pressure \(P\), differentiate the entropy function \(S(E, V, N)\) with respect to volume \(V\) while keeping \(E\) and \(N\) constant. According to classical thermodynamics, the partial derivative of S with respect to V should yield \(P/T\). Comparing the derivative \(\frac{\partial S}{\partial V}\) with \(P/T\) will lead us to the required expression for pressure, \(P=-\sum_{j} N_{j}\left(\partial \varepsilon_{j} / \partial V\right)_{E, N}\).

Deriving the relationship \(P V = \frac{2}{3} E_{\text{tr}}\)

Using the given information that the internal energy is independent of volume and translational energy \(\varepsilon_{j, \text{tr}} = C_{j} V^{-2/3}\), we differentiate \(\varepsilon_{j, \text{tr}}\) with respect to volume \(V\) while keeping \(E\) and \(N\) constant. Then, substitute this derivative into the expression for pressure from Step 3, and finally, sum over all the translational energy states. Comparing the resulting equation with the ideal gas law \(P V = n R T\) under appropriate conditions will yield the relationship \(P V = \frac{2}{3} E_{\text{tr}}\), where \(E_{\text{tr}}\) represents the sum of all the translational energy contributions in the assembly.

Discussing the limitations and assumptions

Analyzing whether the result \(P V = \frac{2}{3} E_{\text{tr}}\) is limited to the dilute limit or independent particles, we must consider the conditions under which the Bose-Einstein and Fermi-Dirac statistics apply, and how the translational energy and interactions between particles are modeled. Additionally, explaining why the pressure is related only to the translational energy involves understanding the nature of pressure as a macroscopic manifestation of microscopic kinetic energies of particles, and how the translational energy terms contribute to this macroscopic property.

Key concepts

Bose-Einstein Statistics

Bose-Einstein statistics describe the distribution of identical particles known as bosons, which can occupy the same quantum state. Unlike classical particles, bosons don't follow the exclusion principle. This means multiple bosons can be found in the same energy state, leading to phenomena such as superfluidity and Bose-Einstein condensation at very low temperatures.

When analyzing systems with bosons, particularly at temperatures near absolute zero, Bose-Einstein statistics replace the classical Maxwell-Boltzmann distribution. This statistical approach is crucial in understanding the thermodynamic properties of systems with non-interacting bosons, offering insight into their entropy, particle distribution, and overall behavior. In calculations, the formula for entropy incorporates these statistics to account for the probability of particles occupying various energy levels in a bosonic system.

Fermi-Dirac Statistics

Fermi-Dirac statistics govern the behavior of a different class of particles known as fermions. Fermions, such as electrons and protons, adhere to the Pauli exclusion principle, stating that no two fermions can occupy the same quantum state simultaneously. This exclusion leads to the 'filling up' of energy levels one by one, creating a distinct distribution of particles at thermal equilibrium.

These statistics are particularly important when studying the thermal properties of solids, where the electron gas can be approximated as a Fermi gas. The Fermi-Dirac distribution has wide-reaching implications in fields ranging from solid-state physics to astrophysics, dictating the entropy and distribution of fermions, which affects conductivity, heat capacity, and the stability of dense stellar objects like white dwarfs.

Translational Energy

Translational energy refers to the kinetic energy associated with the motion of particles as they translate, or move, through space. It is one aspect of a particle's total energy, which can also include vibrational and rotational energy. Translational energy is vital in gases, where particles are in constant motion and collide with each other and the container walls.

In the context of the exercise, the translational energy of a particle is inversely proportional to the volume to the power of two-thirds, suggesting that as a particle assembly expands, each particle's translational energy decreases. This concept plays a central role in deriving the pressure and volume relationship for a particle assembly, emphasizing the direct link between microscopic motion and macroscopic thermodynamic variables.

Entropy

Entropy is a fundamental concept in thermodynamics representing the degree of disorder or randomness in a system. It quantifies the number of microscopic configurations corresponding to a macroscopic state. From a statistical viewpoint, higher entropy correlates with a higher number of possible arrangements of particles that yield the same large-scale properties.

In statistical thermodynamics, entropy is calculated using the distribution of particles among different energy levels. For example, the entropy formulas provided in the exercise involve summations over energy levels and incorporate the probabilities that these levels are occupied. These calculations are essential for understanding how microscopic states contribute to overall entropy and, as a result, how the system's temperature, volume, and the number of particles affect its disorder and thermodynamic potentials like free energy.