Question

Classical Maxwell-Boltzmann statistics considers particles to be distinguishable with no limit on the number of particles in each energy state. A physical example is a solid composed of localized atoms at distinguishable lattice sites. The thermodynamic probability in this case is given by $$ W_{M B}=N ! \prod_{j} \frac{g_{j}^{N_{i}}}{N_{j} !} $$ where \(N_{j}\) is the number of particles and \(g_{j}\) is the degeneracy of the \(j\) th energy level. a. Using the methods of statistical thermodynamics, show that the equilibrium particle distribution is $$ N_{j}=g_{j} e^{-\alpha} e^{-\beta \varepsilon_{j}} \text {. } $$ b. Defining the molecular partition function \(Z=\sum_{j} g_{j} e^{-\beta \varepsilon_{j}}\), show that $$ S=k(\beta E+N \ln Z) . $$ c. Using classical thermodynamics, verify that \(\beta=1 / k T\). Hence, the entropy for classical Maxwell-Boltzmann statistics becomes $$ S=\frac{U}{T}+k N \ln Z \text {. } $$ Compare this expression with that for corrected Maxwell-Boltzmann statistics. Explain the difference. d. Show that the probability of a particle being in the \(i\) th energy state is given by $$ P_{i}=\frac{N_{i}}{N}=\frac{e^{-\varepsilon_{i} / k T}}{Z}, $$ where the partition function \(Z=\sum_{i} e^{-\varepsilon_{i} / k T}\). e. Demonstrate that the entropy can be directly related to the probabilities \(P_{i}\) of the various energy states accessible to the system, i.e., $$ S=-k N \sum_{i} P_{i} \ln P_{i} . $$ Discuss the significance of this result.

Short answer

The equilibrium particle distribution is derived by maximizing the multiplicity function with Lagrange multipliers, yielding \( N_{j} = g_{j} e^{-\alpha} e^{-\beta \varepsilon_{j}} \). The entropy expressed using the molecular partition function is \( S=k(\beta E+N \ln Z) \). Verification of \( \beta = 1/kT \) allows the entropy to be expressed as \( S=U/T + kN\ln Z \). The probability of occupying energy state \( i \) is \( P_{i} = e^{-\varepsilon_{i}/kT}/Z \), and the entropy can be related to these probabilities, \( S=-kN\sum_{i} P_{i} \ln P_{i} \), demonstrating a connection to information theory.

Step-by-step solution

Use the Principles of Maximum Multiplicity

The equilibrium distribution of particles is determined by maximizing the thermodynamic probability under constraints (total number of particles and total energy). Apply the method of Lagrange multipliers to the thermodynamic probability to find the condition of maximum multiplicity.

Apply Lagrange Multipliers

Introduce the Lagrange multipliers \( \alpha \) for the constraint on the number of particles and \( \beta \) for the constraint on the energy, and differentiate with respect to \( N_{j} \): \[ \frac{\partial}{\partial N_{j}}(\ln W_{MB} - \alpha (\sum_{j} N_{j} - N) - \beta (\sum_{j} N_{j} \varepsilon_{j} - E)) = 0 \]. Solve for \( N_{j} \) to get the equilibrium particle distribution.

Find the Equilibrium Particle Distribution

After differentiation and solving, you will find \( N_{j} = g_{j} e^{-\alpha} e^{-\beta \varepsilon_{j}} \).

Define the Molecular Partition Function

Define the partition function \( Z = \sum_{j} g_{j} e^{-\beta \varepsilon_{j}} \). The partition function encapsulates the statistical properties of the system and can be used to find various thermodynamic quantities.

Express Entropy Using the Partition Function

Using statistical thermodynamics, express the entropy formula in terms of \( Z \) using \( S = - (\partial F / \partial T)_{V,N} \), where \( F \) is the Helmholtz free energy.

Use Classical Thermodynamics to Verify \( \beta = 1/kT \)

Using the definition of \( \beta \) and relating it to the thermodynamic temperature \( T \), show that \( \beta = 1/kT \), where \( k \) is the Boltzmann constant.

Derive the Maxwell-Boltzmann Entropy

Combine the earlier steps with the relation \( \beta = 1/kT \) to obtain the entropy for classical Maxwell-Boltzmann statistics: \( S = U/T + kN\ln Z \).

Show the Probability of Occupying the \( i \) th Energy State

Using the equilibrium particle distribution and the partition function, show that the probability of a particle being in the \( i \) th energy state is given by \( P_{i} = N_{i}/N = e^{-\varepsilon_{i}/kT}/Z \).

Relate Entropy to Probabilities of Energy States

Use the definition of \( P_{i} \) and the properties of logarithms to express the entropy as \( S = -kN\sum_{i} P_{i} \ln P_{i} \).

Discuss the Significance

This result links statistical mechanics with information theory, where entropy is a measure of the uncertainty or information content of a system. The entropy is maximum when all states are equally probable, reflecting a state of maximum disorder or lack of information about the system's microstate.

Key concepts

Statistical Thermodynamics

Statistical thermodynamics bridges the microscopic behavior of individual atoms and molecules with the macroscopic properties of materials. It utilizes probability theory to derive thermodynamic properties from the statistics of a system's microscopic states. The idea is to count the number of ways particles can be arranged, assuming they obey certain statistics, like the Maxwell-Boltzmann statistics for distinguishable particles.
In the context of the textbook exercise, statistical thermodynamics enables us to calculate the equilibrium distribution of particles among energy levels in a system, using the principle that at equilibrium the distribution will correspond to the most probable arrangement of particles. This explains why higher energy levels, which are less likely to be populated, have fewer particles, a concept manifest in the formula for the equilibrium distribution of particles \( N_j = g_j e^{-\beta \frac{\varepsilon_j}{kT}} \).

Partition Function

The partition function, represented by \( Z \), is a cornerstone of statistical thermodynamics. It incorporates all possible states of the system, with each state's contribution weighted by its probability, which is related to its energy. By summarizing the contributions from all energy levels, the partition function acts as a bridge between microscopic energy levels and macroscopic thermodynamic quantities. In practical terms, it's given by the formula \( Z = \sum_{j} g_{j} e^{-\beta \varepsilon_{j}} \).
Once calculated, the partition function enables the determination of entropy, free energy, internal energy, and other thermodynamic parameters. For instance, in the exercise, the entropy is related to the partition function by \( S = k (\beta E + N \ln Z) \), underlining the partition function's role in connecting statistical properties with thermodynamic descriptions.

Entropy in Thermodynamics

Entropy is a measure of disorder or randomness in a system. In thermodynamics, it's a state function that indicates how energy is distributed within a system and how much of it can be transformed into useful work. Higher entropy typically means higher disorder and less available energy for doing work.
According to the third law of thermodynamics, the entropy of a perfect crystal at zero Kelvin is zero. As temperature increases, so does entropy, reflecting increased molecular disorder. In statistical thermodynamics, entropy is quantitatively related to the number of ways a system's particles can be arranged while maintaining the same energy, captured in the famous Boltzmann's entropy formula \( S = k \ln W \), where \( W \) is the number of microstates. In the given exercise, the relationship \( S = -kN\sum_{i} P_{i} \ln P_{i} \) also connects entropy to the likelihood of a system's microstates, reinforcing the statistical nature of entropy.

Lagrange Multipliers in Physics

The method of Lagrange multipliers is a powerful mathematical tool used to find the local maxima or minima of a function subject to equality constraints. In the context of physics and statistical thermodynamics, it helps to solve problems where a system must be optimized under certain conditions.
The technique involves introducing additional variables, the Lagrange multipliers, which account for the constraints being applied to the system. For example, in optimizing the distribution of particles over different energy levels while conserving the total number of particles and energy, Lagrange multipliers are used to derive the equilibrium distributions. This method forms the core of the textbook exercise, allowing us to find the equilibrium particle distribution, an essential step toward calculating macroscopic properties like entropy.

Boltzmann Constant

The Boltzmann constant (\( k \)) is a fundamental physical constant that appears in various equations in statistical mechanics, linking the microscopic scale with the macroscopic scale. It relates the average kinetic energy of particles in a gas with the temperature of the gas, and it features prominently in Boltzmann's entropy formula.
In numerical terms, it's valued at approximately \( 1.38 \times 10^{-23} \) Joules per Kelvin, a small number reflecting the tiny energy scales at the molecular level. The Boltzmann constant gives the relationship between the system's temperature and the probability of finding a particle in a particular energy state, such as in the exercise where \( \beta = 1/kT \). This underscores its role in translating thermodynamic equations into predictions about statistical behavior.