Statistical Mechanics Lecture 4

The speaker begins by expressing their intention to delve into the core of statistical mechanics, focusing on the Boltzmann distribution. They aim to illustrate this concept with an example of an ideal gas. A quick review of previously discussed principles for calculating probabilities is also mentioned, emphasizing the importance of understanding occupation numbers and their role in defining the probability distribution of a system's states. The concept of entropy as a function of these probabilities is introduced, and its calculation from a constrained maximization problem is outlined.

The paragraph explains the use of constraints in statistical mechanics and introduces the method of LaGrange multipliers for finding the minimum of a function subject to these constraints. The speaker details the process of adding constraint equations multiplied by Lagrange multipliers to the function to be minimized. Two constraints are identified: the normalization of probabilities and the fixed average energy per subsystem. The role of Lagrange multipliers, denoted as alpha and beta, is discussed in the context of these constraints.

The focus shifts to the process of minimizing the function F, which includes the probabilities and constraints. The speaker outlines the steps for differentiation with respect to the probabilities and setting the derivatives equal to zero to find the minimum. The differentiation process simplifies by ignoring constants and focusing on terms involving the probabilities. The resulting equation is shown to provide insight into the probabilities of the states.

The speaker derives the probability distribution for the states of a system, emphasizing the role of the Lagrange multipliers, particularly beta, in determining the average energy. The relationship between beta and the exponential function of the energy states is explored, revealing that beta is inversely related to temperature. The concept of the partition function Z is introduced, which is a function of beta and is crucial for calculating probabilities.

The paragraph discusses the partition function's role in determining the average energy of a system. The speaker differentiates the partition function with respect to beta to find a formula that relates the average energy to the partition function. The significance of the partition function in statistical mechanics is highlighted, as it encapsulates all the relevant information about the system.

The speaker explores the process of differentiating the partition function to find expressions for energy and entropy. They derive a formula for the average energy per box and relate it to the derivative of the logarithm of the partition function with respect to beta. The connection between the partition function and the system's energy as a function of temperature is established.

The paragraph focuses on establishing the relationship between beta and temperature. The speaker uses the definition of temperature as the change in energy with respect to entropy to derive that the temperature is the inverse of beta. This connection is crucial for understanding the physical significance of beta and its role in statistical mechanics.

The speaker calculates the entropy of the system using the partition function. They derive an expression for entropy in terms of the partition function and show how it can be used to calculate the entropy and, subsequently, the temperature of the system. The importance of the logarithm of the partition function in statistical mechanics is emphasized.

The speaker provides an example of calculating the partition function for an ideal gas, which is a collection of non-interacting point molecules inside a box. They discuss the energy states of the system and how to calculate the partition function using integrals over the positions and momenta of the molecules. The process involves summing over all possible states and integrating over the momentum space.

The speaker explores the implications of considering a gas in a gravitational field, which introduces a potential energy term into the calculations. They discuss how the partition function would be affected and how the energy per particle would include a contribution from the potential energy. The speaker suggests that the problem can be simplified by considering the room to be infinitely high, effectively modeling the atmosphere between the Earth's surface and infinity.

The speaker concludes by mentioning the concept of energy fluctuations and their relation to specific heat. They suggest that fluctuations were not part of the original thermodynamic theory but were incorporated into statistical mechanics by scientists like Einstein and Gibbs. The speaker hints at deriving a beautiful formula that relates energy fluctuations to specific heat in a future session.