# Introduction

This chapter is a review of the basic concepts of thermodynamics, in particular those that we will use in the following chapters. Therefore it has by no means the intention to be a complete review. For a really complete and rigorous exposition of thermodynamics see H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, Wiley, 1985.

Thermodynamics is the theory that describes the thermodynamic properties of macroscopic systems at equilibrium. It can be generally introduced in two ways, an empirical and an axiomatic one; we will follow the latter. By macroscopic systems we mean systems with many degrees of freedom (like the atoms or molecules that constitute a gas or a fluid), but such that only a few of them are measurable and relevant in order to describe the bulk[1] behaviour of the system itself. These relevant degrees of freedom are the thermodynamic properties of the system; for example, in the case of a liquid or a gas, these variables can be the pressure ${\displaystyle P}$, the volume ${\displaystyle V}$ or the temperature ${\displaystyle T}$, while in the case of magnetic systems we can consider the magnetic filed ${\displaystyle {\vec {H}}}$, the magnetization ${\displaystyle {\vec {M}}}$ (which is the total magnetic dipole moment) and the temperature ${\displaystyle T}$, or finally in electrostatic systems the electrical field ${\displaystyle {\vec {E}}}$, the electric polarization ${\displaystyle {\vec {P}}}$ and the temperature ${\displaystyle T}$. If these thermodynamic variables do not change over time, we say that the system is in equilibrium[2].

The aim of thermodynamics is to find relations between the thermodynamic variables of a system in a given state of equilibrium, so that the value of any other variable of interest can be obtained by the initial ones (and generally one tries to chose experimentally accessible variables). In other words, if we have a system in a known state of equilibrium (namely we know all of its thermodynamic variables) and then change the configuration of the system (for example changing its shape, or giving it a certain amount of energy), in general it will reach another equilibrium state: the scope of thermodynamics is to determine the values of the thermodynamic variables in this new equilibrium.

More in general we can say that the central problem of thermodynamics is to find the final state of equilibrium of a system from a given initial equilibrium state of several thermodynamic systems that can interact.
1. It is in fact impossible to describe the exact behaviour of the system, due to the terribly large number of degrees of freedom (of the order of ${\displaystyle 10^{23}}$ in the case of gases or fluids).
2. This statement should rather be reformulated this way: "If these thermodynamic variables do not change sensibly during the time we take to observe the system, we say that the system is in equilibrium". In fact, the thermodynamic properties of a system can change if we observe it for shorter or longer periods of time. Consider this example: if some boiling water is poured into a tea cup, after some seconds it will reach an unchanging state of rest in which its thermodynamic properties (like volume and temperature) do not change sensibly within some seconds, and thus we can say that during this observation time the cup of water is at equilibrium. However, if we observe it for a longer period of time, the temperature will obviously decrease sensibly and so we cannot say that the system is still in equilibrium; after an hour, say, the temperature of the water will be the same of that of the room, and if the room temperature remains constant for several hours then also the temperature of the water will do so, and therefore if the observation time is within this range of several hours the cup can again be considered in equilibrium. However, if we observe the system for even longer periods, such as a couple of days, then the volume of the water will decrease until it evaporates completely and so within this observation time the system cannot be regarded as being in an equilibrium state. After a few days, then, the cup can be again considered in equilibrium, but strictly speaking this is not a real equilibrium because the molecules of the cup can evaporate, even if it could take years to measure a significant change in the system.