A system consists of \(N\) noninteracting, distinguishable two-level atoms. Each atom can exist in one of two energy states, \(E_{0}=0\) or \(E_{1}=\varepsilon\). The number of atoms in energy level, \(E_{0}\), is \(n_{0}\) and the number of atoms in energy level, \(E_{1}\), is \(n_{1}\). The internal energy of this system is \(U=n_{0} E_{0}+n_{1} E_{1}\). (a) Compute the multiplicity of microscopic states. (b) Compute the entropy of this system as a function of internal energy. (c) Compute the temperature of this system. Under what conditions can it be negative? (d) Compute the heat capacity for a fixed number of atoms, \(N\).

Internal energy is a fundamental concept in the study of statistical physics. For a two-level atomic system, the internal energy is the sum of the energies of all the atoms in the system. Each atom can be in one of two states: a ground state with energy \(E_0=0\) or an excited state with energy \(E_1=\varepsilon\). Therefore, the internal energy \(U\) of the system is calculated as follows:

\[U = n_0 E_0 + n_1 E_1\]

Given that \(E_0 = 0\), the equation simplifies to:

\[U = n_1 \varepsilon\]

This means that the internal energy depends solely on the number of atoms in the excited state and the energy \(\varepsilon\) of that state. Understanding how internal energy works in different systems is crucial for understanding more complex phenomena in physics.

The multiplicity of states, often referred to as the degeneracy, represents the number of ways we can arrange the atoms in the two possible energy levels. For our two-level atomic system, this multiplicity is given by the binomial coefficient:

\[ \Omega(N, n_1) = \binom{N}{n_1} \]

The binomial coefficient can be expanded as:

\[ \Omega(N, n_1) = \frac{N!}{n_1! (N - n_1)!} \]

Here, \(N\) is the total number of atoms, and \(n_1\) is the number of atoms in the excited state. The multiplicity of states is a measure of the number of different ways we can achieve a particular distribution of atoms across the energy levels.

Entropy is a measure of the disorder or randomness of a system. In statistical physics, entropy \(S\) can be related to the multiplicity of states \(\Omega\) through Boltzmann's formula:

\[ S = k_B \ln(\Omega) \]

In our two-level system, the entropy can be written as:

\[ S = k_B \ln \left( \binom{N}{n_1} \right) \]

Using Stirling's approximation for large \(N\), this simplifies to:

\[ S \approx k_B [N \ln(N) - n_1 \ln(n_1) - (N - n_1)\ln(N - n_1)] \]

Thus, the entropy depends on the number of atoms in both the ground and excited states, reflecting the system's multiplicity in a thermodynamic property.

Temperature is a key concept in statistical physics and is linked to entropy and internal energy. The temperature \(T\) of a system can be defined using the relation:

\[ \frac{1}{T} = \frac{\partial S}{\partial U} \]

For a two-level atomic system, this translates to:

\[ \frac{1}{T} = \frac{k_B}{U} \left( \frac{d \ln(\Omega)}{d U} \right) \]

Under certain conditions, such as a higher population of atoms in the excited state compared to the ground state, the temperature can become negative. Negative temperatures indicate populations where higher energy states are more populated than lower ones, which is a non-equilibrium condition in typical thermal systems.

Heat capacity \(C\) is the amount of heat required to change a system's temperature by one unit. It can be defined in terms of internal energy and temperature:

\[ C = \frac{dU}{dT} \]

For our two-level atomic system, we find the heat capacity by differentiating the internal energy with respect to temperature:

\[ U = n_1 \varepsilon \]

Given that \(n_1\) and temperature \(T\) are related through the distribution functions, the heat capacity allows us to understand how the system will respond to changes in temperature and how energy is stored.