A monatomic fluid in equilibrium is contained in a large insulated box of volume \(V\). The fluid is divided (conceptually) into \(m\) cells, each of which has an average number of particles \(N_{0}\), where \(N_{0}\) is large (neglect coupling between cells). Compute the variance in fluctuations of internal energy per particle \(u=U / N,\left\langle\left(\Delta u_{i}\right)^{2}\right\rangle\), in the ith cell. (Hint: Use temperature \(T\) and volume per particle \(v=V / N\) as independent variables.)

When studying statistical mechanics, it's important to understand how internal energy fluctuates in a system in equilibrium. Internal energy is the total energy contained within a system due to the kinetic and potential energies of its particles. In an ideal gas, particularly monatomic, these energies are influenced by temperature and the motion of the particles.

Fluctuations refer to the variations around the average value of a quantity—in this case, the internal energy. These fluctuations are especially relevant when dealing with small systems, where changes are more noticeable. For a larger number of particles, the relative size of these fluctuations decreases, but they still play a crucial role in the thermodynamic behavior of the system.

More formally, the variance in the internal energy fluctuation \textbf{per particle} can be computed to give insights into the stability and thermal properties of the gas. The variance \(\textless(\Delta u_i)^2\textgreater\) provides a measure of the expected deviation of the internal energy per particle from its mean.

A monatomic ideal gas is a theoretical gas composed of identical, single-atom particles that do not interact with each other except through elastic collisions. This idealization helps simplify the complex interactions seen in real gases, making it easier to apply statistical mechanics principles.

The internal energy of a monatomic ideal gas is directly related to its temperature and can be expressed as:\[u = \frac{3}{2} k_B T\] where \(\textsl{k_B}\) is the Boltzmann constant and \(\textsl{T}\) is the temperature in Kelvin. This equation shows that the internal energy per particle is proportional to the temperature, implying that higher temperatures lead to greater internal energies due to increased particle movement.

In the context of our problem, understanding this relationship is critical as it lays the groundwork for calculating fluctuations and other thermodynamic properties.

The variance of a quantity measures how spread out its values are around the mean. In the context of our exercise, we seek to determine the variance in the fluctuations of internal energy per particle. Let’s consider the steps involved:

• The total internal energy \(\textsl{U}\) for \( \textsl{N_0} \) particles in a cell is given by: \[ U = N_0 u = N_0 \frac{3}{2} k_B T \]


• The heat capacity at constant volume for a monatomic ideal gas is: \[ C_V = \frac{3}{2} N_0 k_B \]


• The variance in the fluctuation of total internal energy is then: \[ \langle(\Delta U)^2 \rangle = k_B T^2 C_V \]


• Using the value of \( \textsl{C_V} \): \[ \langle(\Delta U)^2 \rangle = k_B T^2 \frac{3}{2} N_0 k_B = \frac{3}{2} N_0 k_B^2 T^2 \]\


• To find the variance per particle, divide by \( N_0^2 \): \[ \langle(\Delta u)^2 \rangle = \frac{\langle (\Delta U)^2 \rangle}{N_0^2} = \frac{\frac{3}{2} N_0 k_B^2 T^2 }{N_0^2} = \frac{3 k_B^2 T^2}{2 N_0} \]

This final expression tells us the variance in energy fluctuations per particle is inversely proportional to the average number of particles in each cell and directly proportional to the square of the temperature.

Heat capacity is a fundamental concept in thermodynamics and statistical mechanics, often denoted as \( \textsl{C} \), which measures the amount of heat needed to change a system's temperature by a certain amount. For a monatomic ideal gas at constant volume, the heat capacity \( C_V \) is given by: \[ C_V = \frac{3}{2} N_0 k_B \]

This relation reflects the linear relationship between the internal energy and the temperature. It is also essential for calculating the fluctuations in internal energy, linking temperature changes to energy changes.

In our exercise, heat capacity plays a critical role in determining the variance in energy fluctuations. The relationship: \[ \langle (\Delta U)^2 \rangle = k_B T^2 C_V \]

highlights how the heat capacity affects the internal energy's fluctuations. By substituting the value for \( C_V \), we derived the energy variance which directly leads to understanding individual particles' fluctuation behavior.

This illustrates that as the heat capacity increases, either by more particles or higher specific heat, the magnitudes of fluctuations in energy also increase. Such relations are crucial for analyzing thermal systems and predicting their behavior under different conditions.