Question

For a systeFor a system of energy levels, \(\varepsilon_{m}=m^{2} \alpha,\) where \(\alpha\) is a constant with units of energy and \(m=0,1,2, \ldots, \infty,\) What is the internal energy and heat capacity of this system in the high-temperature limit?m of energy levels, \(\varepsilon_{m}=m^{2} \alpha,\) where \(\alpha\) is a constant with units of energy and \(m=0,1,2, \ldots, \infty,\) What is the internal energy and heat capacity of this system in the high-temperature limit?

Short answer

In the high-temperature limit, the internal energy of the system is given by \(U = \frac{3}{2} k_B T\), and the heat capacity is constant and equals \(C_V = \frac{3}{2} k_B\).

Step-by-step solution

Boltzmann Distribution

To find the probabilities of different energy levels in the high-temperature limit, we will use the Boltzmann distribution, which states that the probability of finding the system in a state with energy level \(\varepsilon_m\) is proportional to \(e^{-\frac{\varepsilon_m}{k_BT}}\), where \(k_B\) is the Boltzmann constant and \(T\) is the temperature.

Normalization Factor

To find the normalization factor, we can sum over all probabilities to equal 1: \[Z = \sum_{m=0}^{\infty} e^{-\frac{m^2 \alpha}{k_BT}}\] As we are looking for the high-temperature limit (\(T \to \infty\)), the exponent becomes very small and the sum will approximate an integral: \[Z \approx \int_0^{\infty} e^{-\frac{m^2 \alpha}{k_BT}} dm\] This is a standard Gaussian integral, which has the value: \[\frac{1}{2} \sqrt{\frac{\pi k_B T}{\alpha}}\]

Probability distribution for m

Now, we can find the probability distribution for energy levels: \[P(m) = \frac{1}{Z} e^{-\frac{m^2 \alpha}{k_B T}} = \frac{2}{\sqrt{\pi k_B T / \alpha}} e^{-\frac{m^2 \alpha}{k_B T}}\]

Calculate internal energy

Using the probability distribution, we can find the internal energy of the system, which is the expected value of the energy: \[U = \sum_{m=0}^{\infty} \varepsilon_m P(m) = \frac{2}{\sqrt{\pi k_B T / \alpha}} \sum_{m=0}^{\infty} m^2 \alpha e^{-\frac{m^2 \alpha}{k_B T}}\] Again, let's approximate this sum by an integral: \[U \approx \frac{2}{\sqrt{\pi k_B T / \alpha}} \int_0^{\infty} m^2 \alpha e^{-\frac{m^2 \alpha}{k_B T}} dm\] To solve this integral, we can use the formula for the Gaussian integral: \[\int_0^{\infty} x^ne^{-ax^2}dx = \frac{1}{2} \sqrt{\frac{\pi}{a^{n+1}}} \frac{\Gamma\left(\frac{n+1}{2}\right)}{\Gamma\left(\frac{n}{2}\right)}\] With \(n = 2\) and \(a = \frac{\alpha}{k_B T}\), we get the final result for internal energy: \[U = \frac{3}{2} k_B T\]

Calculate heat capacity

Now we can find the heat capacity at constant volume, which is defined as: \[C_V = \left( \frac{\partial U}{\partial T} \right)_V\] From the above internal energy expression, the heat capacity is: \[C_V = \frac{3}{2} k_B\] In the high-temperature limit, the heat capacity of the system is constant and equals \(\frac{3}{2} k_B\).

Key concepts

Boltzmann Distribution

Understanding the Boltzmann distribution is pivotal for studying statistical mechanics and thermodynamics. It describes the probability of a system to be in a particular energy state at thermal equilibrium. This distribution indicates that at higher temperatures, the particles are more likely to populate higher energy states, while at lower temperatures, lower energy states are more favored.

In practical terms, for a particle in a system with discrete energy levels \( \varepsilon_m \), the probability \( P(m) \) of finding it in a state \( m \) is given by the formula \( P(m) = \frac{e^{-\frac{\varepsilon_m}{k_BT}}}{Z} \) where \( k_B \) is the Boltzmann constant, \( T \) is the absolute temperature, and \( Z \) is the partition function. This relationship emphasizes the influence of temperature on energy distribution within a system.

Partition Function

The partition function, often symbolized by \( Z \) is a central concept in statistical thermodynamics. It plays a critical role by acting as a normalization factor in the Boltzmann distribution to ensure that the probabilities of all possible states sum up to 1.

The partition function is the sum of the Boltzmann factors over all possible energy states of a system and is mathematically expressed as \( Z = \sum_{m=0}^{\infty} e^{-\frac{\varepsilon_m}{k_BT}} \). In high-temperature conditions, many energy states are accessible, and the partition function helps us understand the multitude of possible configurations that a system can adopt. To calculate \( Z \) in these scenarios, one often uses integral approximations, as sums over large numbers of states can become intractable.

Internal Energy

Internal energy, often denoted as \( U \) in thermodynamics, is the total energy contained within a system. It encompasses all forms of energy, including kinetic and potential energies of the particles in the system.

The internal energy can be calculated by taking the expected value of the energy across all possible states, which is the sum of the energy levels weighted by their respective probabilities. For a system following the Boltzmann distribution, this amounts to \( U = \sum_{m=0}^{\infty} \varepsilon_m P(m) \), where \( P(m) \) is the probability of the system being in the state with energy \( \varepsilon_m \). At high temperatures, the internal energy approaches an approximation that simplifies the calculation, reflecting the energy spreading over an increasing number of accessible states.

Gaussian Integral

The Gaussian integral is a fundamental operation in mathematics, often encountered when dealing with normal distributions and statistical applications. It's utilized in various fields of physics and engineering to solve problems involving heat, diffusion, and quantum mechanics.

In a statistical mechanics context, particularly for continuous systems or those at high temperature limits, Gaussian integrals help approximate the sums over states. For example, the integral \( \int_0^{\infty} e^{-ax^2} dx \) calculates the area under the curve of a Gaussian function and has a closed-form solution related to the square root of \( \pi/a \). This analytical solution is a powerful tool when evaluating partition functions and internal energies, as it lets us bypass complex summations which are otherwise very challenging to compute.