Consider a system which has two orbital (single particle)...

Consider a system which has two orbital (single particle) states both of the same energy. When both orbitals are unoccupied, the energy of the system is zero; when one orbital or the other is occupied by one particle, the energy is $\varepsilon$. We suppose that the energy of the system is much higher, say infinitely high, when both orbitals are occupied. Show that the ensemble average number of particles in the level is $$ \langle N\rangle=\frac{2}{2+e^{(\epsilon-\mu) \tau}} . $$

Consider a system which has two orbital (single particle) states both of the same energy. When both orbitals are unoccupied, the energy of the system is zero; when one orbital or the other is occupied by one particle, the energy is $\varepsilon$. We suppose that the energy of the system is much higher, say infinitely high, when both orbitals are occupied. Show that the ensemble average number of particles in the level is
$$
\langle N\rangle=\frac{2}{2+e^{(\epsilon-\mu) \tau}} .
$$

Key Concepts

Grand Canonical Ensemble

The grand canonical ensemble is a statistical framework used to study systems that can exchange both energy and particles with a reservoir. In this ensemble, the system's state is characterized by variables such as energy, particle number, and volume, and the probability of a given configuration is determined by its energy and the chemical potential. This framework is especially useful when the number of particles is not fixed, enabling the calculation of ensemble averages like the average number of particles.

Partition Function

The partition function in the grand canonical ensemble, often called the grand partition function, is a sum over all microstates weighted by the Boltzmann factor e^{-?(energy - ?·number)}. It acts as a generating function from which all thermodynamic properties of the system, including average energy and particle number, can be derived using suitable derivatives. The partition function encapsulates the contributions of various accessible states, accounting for factors like degeneracy and energy constraints.

Fugacity and Chemical Potential

Fugacity is a concept derived from the chemical potential (?) and appears in the exponential factor, often written as e^{??}, and is used as an effective measure of the 'escaping tendency' of particles from the system. The chemical potential plays a crucial role in controlling the number of particles in the ensemble, and together with temperature, it determines the occupancy probabilities of different states. This relationship is key when calculating ensemble averages such as the average number of particles.

Exclusion Principle and Interaction Constraints

When a system has constraints that forbid certain configurations—such as prohibiting double occupancy by assigning it an infinitely high energy—the calculation of the partition function must exclude those configurations. This resembles an exclusion principle where only states that do not violate the imposed constraints contribute to the statistical sum. Such constraints modify the accessible state space and, consequently, affect the computed ensemble averages.

Ensemble Average Calculation

The ensemble average, such as the average number of particles, is calculated by summing over the contributions of each state weighted by their probability, or equivalently, by taking the appropriate derivative of the logarithm of the partition function with respect to the chemical potential. This process systematically accounts for the statistical weights of all allowed configurations and provides a direct link between microscopic state probabilities and macroscopic observables.

Transcript

Please give Ace some feedback