Definitions in thermodynamics: temperature, thermal equilibrium, heat I'm currently reading Fermi's "Thermodynamics" and I'm trying to grasp the (possibly different) right definitions for temperature, 
thermal equilibrium, heat.
To clarify, I'm looking for definitions from a purely thermodynamical point of view, which is also the line followed by the book. 

Let's start with the latter. We can define heat through the 1° principle of Thermodynamics:$$Q=\Delta U + L
,$$that is, “Heat is the quantity of energy that a system absorbs from the ambient in a form that's not mechanical work.”. OK, I see no problem with this, apart from the fact that we need to define the energy $U$ of a thermodynamical system. Let's ignore it.

Now thermal equilibrium. This is the one I'm finding more troubles with, because in all definitions I've come across (maybe not very good ones, or maybe it's my interpretation) there's some reference to temperature, while in the definition of the latter there's reference to the concept of thermal equilibrium, but one has to start somewhere. For example, from (IT) Wikipedia, I read:

Thermal equilibrium: there is no flux of heat, temperature is constant in time and is the same in every point of the system.


The way that temperature is defined in the book is, first of all, the empirical (operational) one:

Temperature can be measured by putting a thermometer in contact with the system, for a sufficient time interval so that thermal equilibrium is established.

Some pages later there's also mention to the gas-thermometer. Finally, in the “Second principle of Thermodynamics” chapter, it's said:

Until now, we've only made use of an empirical scale of temperature. [...]If we put in thermal contact two bodies at a different temperature, heat will flow spontaneously by conduction from one body to the other. Now, by definition, we will say that the body from which the heat flows is the one with the higher temperature.

Now, clearly the definition in the first blockquote requires thermal equilibrium (beetween a body itself and beetween to bodies, I suppose) to be independently defined. Regarding the second, how can one tell what's the direction of the heat flux? Also, the second definition doesn't give a method to measure temperature, but only a way to tell which body is the hotter, right?


As I put them above, those definitions seem to me random pieces of a puzzle, I need to get a clearer picture. So any help is appreciated.
 A: I agree with you that most books do not follow a logical path when defining thermodynamics terms. Even great books such as Fermi's and Pauli's.
The first thing you need to define is the concept of thermodynamic variables.

Thermodynamic variables are macroscopic quantities whose values depend
  only on the current state of thermodynamic equilibrium of the system.

By thermodynamic equilibrium we mean that those variables do not change with time. Their values on the equilibrium cannot depend on the process by which the system achieved the equilibrium. Example of thermodynamic variables are: Volume, pressure, surface tension, magnetization... The equilibrium values of these quantities define the thermodynamic state of a system.
When a thermodynamic system is not isolated, its thermodynamic variables can change under influence of the surrounding. We say the system and the surrounding are in thermal contact. When the system is not in thermal contact with the surrounding we say the system is adiabatically isolated. We can define that,

Two bodies are in thermal equilibrium when they - in thermal contact
  with each other - have constant thermodynamic variables.

Now we are able to define temperature. From a purely thermodynamic point of view this is done through the Zeroth Law. A detailed explanation can be found in this post. Basically,

We say that two bodies have the same temperature if and only if they
  are in thermal equilibrium.

Borrowing the mechanical definition of work one can - by way of experiments - observe that the work needed to achieve a given change in the thermodynamic state of an adiabatically isolated system is always the same. It allows us to define this value as an internal energy change, 
$$W=-\Delta U.$$
By removing the adiabatic isolation we notice that the equation above is no longer valid and we correct it by adding a new term,
$$\Delta U=Q-W,$$
so 

The heat $Q$ is the energy the system exchange with the surrounding in
  a form that is not work.

Notice that I have skipped more basic definitions such as thermodynamic system and isolated system but this can be easily and logically defined in this construction.
