Given \(N\) indistinguishable, quasi-independent particles capable of existing in energy levels \(\epsilon_{1}, \epsilon_{2}, \cdots\), with degeneracies \(g_{1}, g_{2}, \cdots\), respectively; in any given macrostate in which there are \(N_{1}\) particles in energy level \(\epsilon_{1}, N_{2}\) particles in energy level \(\epsilon_{2}, \cdots\), assume the thermodynamic probability to be given by the Bose-Einstein expression, $$ \Omega_{\mathrm{BE}}=\frac{\left(g_{1}+N_{1}\right) !\left(g_{2}+N_{2}\right) ! \cdots}{g_{1} ! N_{1} \backslash g_{2} ! N_{2} !} $$ Using Stirling's approximation and the method of Lagrangian multipliers, render \(\ln \Omega_{\mathrm{BE}}\) a maximum, subject to the equations of constraint \(\sum N_{i}=N=\) const. and \(\sum N_{i} \epsilon_{i}=U=\) const., and show that $$ N_{i}=\frac{\boldsymbol{g}_{i}}{\lambda e^{-\beta \mathrm{k}_{4}}-1} $$

Step by step solution

01

- Write the Constraints

First, note the constraints to be utilized:1. \( \sum N_{i} = N = \text{const} \)2. \( \sum N_{i} \epsilon_{i} = U = \text{const} \).

02

- Simplify \( \Omega_{\mathrm{BE}} \) Using Stirling's Approximation

Stirling's approximation is \( \ln x! \approx x \ln x - x \). Apply this to each factorial term in \( \Omega_{\mathrm{BE}} \):\begin{align*} \ln \Omega_{\mathrm{BE}} &\approx \sum_{i} \left( \left(g_{i} + N_{i}\right) \ln\left(g_{i} + N_{i}\right) - \left(g_{i} + N_{i}\right) \right) \& - \sum_{i} \left( g_{i} \ln g_{i} - g_{i} \right) - \sum_{i} \left( N_{i} \ln N_{i} - N_{i} \right) \end{align*}

03

- Form the Lagrangian

Introduce Lagrange multipliers \( \alpha \) and \( \beta \) for the constraints and form the Lagrangian, \( \mathcal{L} \):\begin{align*} \mathcal{L} &= \ln \Omega_{\mathrm{BE}} + \alpha \left( \sum_{i} N_{i} - N \right) + \beta \left( \sum_{i} N_{i} \epsilon_{i} - U \right) \end{align*}

04

- Take Derivatives

Take the partial derivative of \( \mathcal{L} \) with respect to each \( N_{i} \) and set it equal to zero:\begin{align*} \frac{\partial \mathcal{L}}{\partial N_{i}} = \ln\left( 1 + \frac{N_{i}}{g_{i}} \right) - \ln N_{i} + \alpha + \beta \epsilon_{i} = 0 \end{align*}

05

- Solving for \( N_{i} \)

Solve the above equation for \( N_{i} \) to get the distribution:\begin{align*} \ln\left( \frac{1 + \frac{N_{i}}{g_{i}}}{N_{i}} \right) & = \beta \epsilon_{i} - \alpha \ \ \frac{1 + \frac{N_{i}}{g_{i}}}{N_{i}} & = e^{\beta \epsilon_{i} - \alpha} \ \ 1 + \frac{N_{i}}{g_{i}} & = N_{i} e^{\beta \epsilon_{i} - \alpha} \ \ N_{i} \left( e^{\beta \epsilon_{i} - \alpha} - 1 \right) & = g_{i} \ \Rightarrow \ N_{i} = \frac{g_{i}}{e^{\beta \epsilon_{i} - \alpha} - 1} \end{align*}Let \( \lambda = e^{\alpha} \). Then the final expression for the number of particles at each energy level is\begin{align*} N_{i} & = \frac{g_{i}}{\lambda e^{-\beta \epsilon_{i}} - 1} \end{align*}

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Stirling's Approximation

Stirling's approximation is a mathematical method used to approximate the factorial of a large number. It is expressed as: \ \( \ln x! \approx x \ln x - x \). This simplification is useful when dealing with large numbers, as it turns factorials into more manageable polynomial expressions.
In the context of the Bose-Einstein distribution, Stirling's approximation is applied to simplify the expression for thermodynamic probability, making it easier to handle mathematically.

• This method is crucial because factorials grow extremely fast, and handling them directly becomes computationally difficult.
• By converting factorials into logarithmic terms, we can use calculus to further analyze and maximize probabilities.

Lagrange Multipliers

Lagrange multipliers are a technique used in calculus to find the local maxima and minima of a function subject to equality constraints.
In this problem, we use Lagrange multipliers to maximize the logarithm of the thermodynamic probability \( \ln \Omega_{\mathrm{BE}} \).
We introduce two multipliers, \( \alpha \) and \( \beta \), to handle the constraints:

• \( \sum N_i = N = \text{const} \)
• \( \sum N_i \epsilon_i = U = \text{const} \)

The method involves forming a new function called the Lagrangian, which combines the original function and the constraints.
Taking partial derivatives of this Lagrangian with respect to each variable and setting them to zero allows us to find the values that maximize \( \ln \Omega_{\mathrm{BE}} \).

• This method is powerful because it allows us to incorporate constraints directly into the optimization problem.
• It simplifies finding the optimal distribution of particles across different energy levels.

Thermodynamic Probability

Thermodynamic probability (\( \Omega_{\mathrm{BE}} \)) is a measure of the number of ways particles can be distributed among various energy levels while satisfying specific constraints.
In this exercise, thermodynamic probability is given by the Bose-Einstein expression: \[ \Omega_{\mathrm{BE}} = \frac{(g_1 + N_1)! (g_2 + N_2)! \cdots}{g_1! N_1! g_2! N_2! \cdots} \] Here, \( g_i \) represents the degeneracy of the energy level \( \epsilon_i \) (the number of states at that energy level), and \( N_i \) represents the number of particles in that energy level.

• The expression accounts for the indistinguishability and quasi-independence of particles, common in quantum statistics.
• Maximizing \( \Omega_{\mathrm{BE}} \) under given constraints tells us the most probable distribution of particles at equilibrium.

To do so, we simplify it using Stirling's approximation and then apply the method of Lagrange multipliers to find the conditions that maximize \( \Omega_{\mathrm{BE}} \).

Energy Levels

Energy levels \( \epsilon_i \) refer to the specific energies that particles can have in a given system. These energy levels are often quantized, meaning that they can only take on specific, discrete values.
In the Bose-Einstein distribution, we are interested in how particles distribute themselves among these energy levels.

• Each energy level \( \epsilon_i \) can have multiple states, represented by its degeneracy \( g_i \).
• At thermal equilibrium, particles will distribute themselves among the energy levels in a way that maximizes the thermodynamic probability.

The constraints \( \sum N_i = N = \text{const} \) and \( \sum N_i \epsilon_i = U = \text{const} \) ensure that the total number of particles and the total energy of the system remain fixed.
By applying these constraints and optimizing \( \Omega_{\mathrm{BE}} \), we can determine the most probable number of particles \( N_i \) in each energy level \( \epsilon_i \).

Particle Distribution

Particle distribution in the Bose-Einstein context tells us how particles are spread across different energy levels. After applying Stirling's approximation and using Lagrange multipliers, we derive the distribution function: \[ N_i = \frac{g_i}{\lambda e^{-\beta \epsilon_i} - 1} \] Here, \( \lambda \) is related to the chemical potential, and \( \beta \) is inversely proportional to the temperature (\( \beta = \frac{1}{k_B T} \)).

• This formula indicates that the number of particles in each energy level depends on the degeneracy \( g_i \), the energy \( \epsilon_i \), and temperature.
• At higher temperatures, particles distribute more evenly, while at lower temperatures, they tend to occupy lower energy states.

Particle distribution is crucial for understanding various properties of the system, including its entropy, pressure, and specific heat.
By knowing \( N_i \), we can predict the behavior of the system under different conditions and make meaningful connections to physical phenomena.