# Introduction to Thermodynamics

### Introduction and Summary

The study of thermodynamics is the study of systems that are too large to understand by mechanics alone. For many years, thermodynamics was understood vaguely, and many of the results had been determined only experimentally. Some results posed great theoretical challenges to physicists, who offered many unsuccessful attempts at explaining the origins of the formulas.

With the advent of quantum mechanics came the explanations for the results. The mechanics of individual particles is still too complicated, however. For this reason, statistical physics plays a significant role in the basis of thermodynamics. Instead of worrying about the exact values of properties for each particle in a system, we look at the average values statistically over quantum probabilities. Even fundamental concepts like the energy of a system are derived as averages.

New concepts arise as we talk about large systems, such as entropy and temperature. Defining these carefully from quantum mechanics allows us to make sense of the "3 Laws of Thermodynamics".

There is great symmetry in the structure of thermodynamics. The six variables we look at repeatedly parallel each other in formulations of the energy. We can use a mathematical tool known as the Legendre Transform to posit alternate definitions of energy. This symmetry allows us to derive numerous relationships between the variables, and the multiple definitions of energy greatly simplify problem solving throughout all of thermodynamics.

We can form the Partition Function as a measure of the total weighted probabilities of the various states of a system, and relate this quantum counting result to the energy of a the system. The spectrum of blackbody radiation is derived directly from this counting. For systems in thermal and diffusive contact with a reservoir, the Gibbs Sum replaces the Partition Function.

With the few tools developed up to that point, the entire ideal gas problem can be solved, including the derivation of expressions for all of the interesting variables that describe the gas. In the non-classical regime, an ideal gas behaves quite differently depending on the nature of its constituents . A gas comprised of fermions exhibits a regime of total occupation and a regime of zero occupation, while a gas comprised of bosons can form an Einstein condensate by crowding into the ground orbital of the system.

Heat engines and other devices were the historical motivation for the development of thermodynamics as a science. The devices can be well explained using the framework already developed, and illustrative diagrams can be drawn to make plain the energy and entropy flow involved. Real engines undergo repeated cycles to achieve their purpose. We look at a simplified model known as the Carnot cycle, and discuss different processes and how they relate to the various energies defined.

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