Derive the statistical expressions of internal energy, entropy, and free energy from the partition function.

Introduction

In statistical mechanics, the partition function, denoted as $ Z $, serves as a fundamental tool for quantifying the thermodynamic properties of a system. Defined as $ Z = \sum_{\text{all microstates}} e^{-\beta E} $, where $ \beta = 1/(kT) $ and $ k $ is Boltzmann’s constant, the partition function encapsulates the statistical distribution of a system’s energy states. This function is central to the canonical ensemble, which describes systems in thermal equilibrium with a reservoir of fixed temperature. By leveraging the partition function, one can derive critical thermodynamic quantities such as internal energy, entropy, and free energy. These expressions provide a bridge between the microscopic behavior of particles and the macroscopic properties of matter, enabling the prediction of phase transitions, equilibrium conditions, and the direction of spontaneous processes.

Internal Energy

The internal energy $ U $ of a system is the total energy associated with its microscopic states. In the canonical ensemble, it is derived from the partition function through the relation:
$$ U = -\frac{\partial}{\partial \beta} \left( \ln Z \right) $$
This expression arises from the definition of the average energy:
$$ U = \frac{1}{Z} \sum_{\text{microstates}} E \cdot e^{-\beta E} $$
Differentiating $ \ln Z $ with respect to $ \beta $ yields:
$$ \frac{\partial}{\partial \beta} \ln Z = -\frac{1}{Z} \sum_{\text{microstates}} E \cdot e^{-\beta E} = -U $$
Thus, the internal energy is given by:
$$ U = -\frac{\partial}{\partial \beta} \left( \ln Z \right) $$
This derivation aligns with the statistical interpretation of energy as a sum over microstates weighted by their probabilities. The result confirms that internal energy is directly proportional to the system’s tendency to minimize its energy, a principle consistent with the second law of thermodynamics.

Entropy

Entropy $ S $ is a measure of the disorder or randomness in a system, and its derivation from the partition function relies on thermodynamic relations. The standard formula for entropy in terms of the partition function is:
$$ S = k \ln Z + \frac{Q}{T} $$
where $ Q $ is the heat capacity at constant volume. This expression is derived from the second law of thermodynamics, which states that entropy increases in isolated systems. The term $ k \ln Z $ accounts for the logarithmic dependence of the partition function on the number of microstates, while $ Q/T $ reflects the system’s ability to exchange energy with its surroundings.

The entropy can also be expressed in terms of the derivative of the Helmholtz free energy $ F $:
$$ S = -\frac{\partial F}{\partial T} $$
Since the Helmholtz free energy is defined as $ F = -k T \ln Z $, differentiating $ F $ with respect to $ T $ yields:
$$ S = k \frac{\partial}{\partial T} \left( \ln Z \right) $$
This formulation highlights the direct relationship between entropy and the rate of change of the partition function with temperature. The entropy’s dependence on the partition function ensures that it captures the system’s ability to explore different microstates, a key factor in determining thermodynamic equilibrium.

Free Energy

The Helmholtz free energy $ F $ is a measure of the maximum work that a system can do on its surroundings at constant temperature and volume. It is derived from the partition function as:
$$ F = -k T \ln Z $$
This expression arises from the definition of the free energy in terms of the system’s energy distribution. The negative sign indicates that the free energy represents the energy available for work, while the logarithmic dependence on the partition function ensures that it accounts for the system’s statistical distribution of energy states.

The relationship between free energy, internal energy, and entropy is encapsulated by the fundamental equation:
$$ F = U - T S $$
Substituting the expressions for $ U $ and $ S $ derived earlier, this equation becomes:
$$ F = -k T \ln Z $$
This equation underscores the interdependence of the system’s energy, entropy, and temperature, and provides a framework for analyzing the stability of thermodynamic states. The free energy’s dependence on temperature and volume allows for the prediction of phase transitions and the conditions under which processes are spontaneous or irreversible.

Conclusion

The derivation of internal energy, entropy, and free energy from the partition function establishes a rigorous foundation for understanding the thermodynamic behavior of physical systems. By linking the microscopic probabilities of energy states to macroscopic properties, these expressions enable the prediction of system behavior under various conditions. The interplay between the partition function and thermodynamic quantities highlights the deep connection between statistical mechanics and the laws of thermodynamics, offering a comprehensive framework for analyzing equilibrium, spontaneity, and the direction of processes. These results are critical in fields ranging from condensed matter physics to chemical thermodynamics, demonstrating the universal applicability of statistical mechanics principles.