Question

The entropy in the grand ensemble. Write an expres sion for the entropy in the grand canonical ensemble \((T, V, \mu)\).

Short answer

The entropy in the grand canonical ensemble is expressed as \[ S = - \frac{\text{∂Φ}}{\text{∂T}} + \beta (\text{E} - \text{μN}). \]

Step-by-step solution

- Understand the Grand Canonical Ensemble

The grand canonical ensemble is a statistical ensemble that describes a system in thermal equilibrium with a reservoir that can exchange both energy and particles.

- Recall the General Expression for Entropy

The entropy in statistical mechanics is generally given by: \[ S = -k_B \text{Tr}(\rho \text{ln} \rho) \]where \( \rho \) is the density matrix and \( k_B \) is the Boltzmann constant.

- Write the Partition Function

In the grand canonical ensemble, the partition function is: \[ \text{Ξ} = \text{Tr} \bigg( e^{-\beta ( \text{H} - \text{μN} ) } \bigg) \]Here, \( \beta = \frac{1}{k_B T}, H \) is the Hamiltonian, \( \text{N} \) is the number of particles, and \( μ \) is the chemical potential.

- Derive the Gibbs Free Energy

The Gibbs free energy (or grand potential) \( \text{Φ} \) is given by: \[ \text{Φ} = -k_B T \text{ln Ξ} \]

- Express Entropy in Terms of the Grand Potential

Finally, the entropy \( S \) can be expressed in terms of the grand potential as: \[ S = - \frac{\text{∂Φ}}{\text{∂T}} + \beta (\text{E} - \text{μN}) \]where \( \text{E} = - \frac{\text{∂ (lnΞ)}}{\text{∂β}} \) is the average energy.

Key concepts

Statistical Mechanics

Statistical mechanics is a branch of physics that uses probability theory to study the behavior of systems composed of a large number of particles. It connects the microscopic properties of individual atoms and molecules to the macroscopic or bulk properties of materials that can be observed at our scale.
There are three main types of statistical ensembles used in statistical mechanics: microcanonical, canonical, and grand canonical. Each ensemble corresponds to different constraints, like constant energy, volume, and the number of particles for microcanonical, constant temperature and volume for canonical, and constant temperature, volume, and chemical potential for grand canonical.

Partition Function

The partition function is a central quantity in statistical mechanics. It encodes crucial information about a system in thermal equilibrium. For the grand canonical ensemble, the partition function \[\Xi\] is defined as:
\[ \Xi = \text{Tr} \( e^{-\beta ( \text{H} - \mu\text{N} ) } \) \]
Here,

• \( \beta = \frac{1}{k_B T} \)
• \( \text{H} \) is the Hamiltonian (total energy)
• \( \text{N} \) is the number of particles
• \( \mu \) is the chemical potential.

The partition function helps us calculate various thermodynamic properties of a system. In the context of the grand canonical ensemble, it can be used to derive quantities such as the average number of particles, average energy, entropy, and Gibbs free energy.

Gibbs Free Energy

In the grand canonical ensemble, Gibbs free energy, often referred to as the grand potential \( \Phi \), is an essential thermodynamic potential. It provides information about the system's capacity to do useful work at constant temperature. It is given by:
\[ \Phi = -k_B T \text{ln} \Xi \]
This expression connects the grand potential directly to the partition function \( \Xi \). Gibbs free energy or grand potential serves as an anchor for further derivations and calculations, such as finding entropy and other thermodynamic properties.

Density Matrix

The density matrix \( \rho \) is a fundamental concept in quantum statistical mechanics. It helps describe the statistical state of a quantum system. For a system in the grand canonical ensemble, the density matrix is given by:
\[ \rho = \frac{e^{-\beta ( H - \mu N) }}{\Xi} \]
This matrix allows us to calculate the probability of the system being in a particular quantum state, which is essential for determining thermodynamic quantities like entropy. The entropy itself can be expressed using the density matrix as:
\[ S = -k_B \text{Tr}(\rho \text{ln} \rho) \]
This formula shows how the density matrix provides a direct path to determining entropy in thermal equilibrium.

Thermal Equilibrium

Thermal equilibrium is a state in which a system's macroscopic properties do not change over time because it has a uniform temperature throughout. In the context of the grand canonical ensemble, thermal equilibrium implies that the system can exchange both energy and particles with a reservoir, but it maintains a constant temperature, volume, and chemical potential.

• The temperature remains constant throughout the system.
• The volume of the system does not change.
• The chemical potential governs the exchange of particles.

At thermal equilibrium, the system's macroscopic observables reach stable values, allowing us to use the grand canonical ensemble to derive properties such as entropy using the formulas we discussed earlier. The concept of thermal equilibrium ensures that despite any exchanges with the reservoir, the system's overall characteristics remain constant.