# Why work $W$ and heat $Q$ are different concepts?

You are right. Microscopically, work and heat are just about the same. Both involve molecular collisions transfer energy from one object to the other.

Work involves a kind of "coherent" transfer in a manner of speaking, in which the collisions are predominantly, and to an extreme degree, in one direction. Also, typically the force is applied to one location on the object. And importantly, the boundary of the system is displaced. (E.g. translation or deformation)

On the other hand, transfer of energy by heat is "incoherent", many directions, and typically in all directions. And importantly, the boundary of the system is not displaced.

Finally, everyday phenomena fall into one or the other category, and they differ in their macroscopic behavior. Loosely speaking, when heat is transfered the temperature rises. When work is done, the boundary of the system changes. Of course adding heat generally also results in the boundary moving [say, expanding], and work generally results in a temperature change [Joule's experiment]. I'm trying to motivate the macroscopic separation between work and heat without a lengthy discussion.

To the engineers who first worked all of this out, the very existence of atoms was unknown. To them, the separation between heat and work was very clear. They had little reason to view them as manifestations of the same microscopic process. They thought that heat was a physical fluid. In any event, their remarkable achievements have stood the test of time.

The distinction between heat and work only comes about in statistical physics. The idea is that while work is a transfer of energy through the macroscopic degrees of freedom which are described by macroscopic (i.e. thermodynamic) quantities (like pressure and volume), heat is a transfer of energy through the remaining degrees of freedom which are ignored in the macroscopic description.

In the above sense, the distinction between work and heat is in some way artificial. It is induced by our choice of which degrees of freedom we want to consider in our description of a physical system and which ones we choose to be ignorant about. This also means that temperature and entropy are basically dependent on the convention one chooses to describe the system. This perfectly matches the idea from information theory where the amount of information contained in a message is characterized by the so called Shannon entropy which is exactly the same thing as the physical entropy, except for the factor of $k_B$: a message like *"The OP's name is 21Brunoh."* contains exactly zero information *to you* because you already knew that. That same message may however be considered quite informative by someone unaware of this thread. Thus, information is an observer-dependent concept. Entropy characterizes the *lack of information*.

In thermodynamics however, there is a natural ("canonical") choice of the quantities of a physical system one can know about (like pressure, number of particles) and those which are inaccessible (like the positions and momenta of all the individual particles). The latter lead to a lack of information, characterized by the (physical) entropy. Changes in the observable/macroscopic quantities are called different forms of work, e.g. changes in pressure and volume are referred to as *mechanical work*. Changes in temperature and entropy are referred to as *heat*.

As an additional remark, please note that heat and work are meaningful only as changes of the internal energy. I'd always avoid writing $U=W+Q$ and instead use $\Delta U=W+Q$: changes in the internal energy are due either to work or heat. There is no such thing as the heat content or work content of some object, but one can talk about the energy content. You wouldn't talk about the amount of cash vs. the amount of credit card money you have in your bank account. Both paying in cash and by credit card are ways of changing the balance of your account, though.

It may be useful to have a look at the problem from a more quantitative perspective. Let $E$ denote the total internal energy of the system. For a thermodynamic system, it is defined as an ensemble average: $E = \sum_nE_np_n$, where $E_n$ and $p_n = e^{-E_n/k_BT}/Z$ are the energy and the realisation probability of the $n$th micro-state. Here $Z = \sum_{n}e^{-E_n/k_BT}$ is the partition function. I have used a sum, which may actually represent an integral (depending on the spectrum of the system). Now comes the crunch: apart from the obvious dependence on temperature $T$ via $p_n$, $E$ can also depend on other experimental knobs such as volume, external magnetic field and so on, because $E_n$ can depend on them. Namely, we may put $E_n(x_1,x_2,...)$, where $x_i$ is the $i$th external knob. As a result, $E$ must also depend on these arguments in addition to $T$, $E(T,x_1,...)$. The change in $E$ then reads, $\delta E = \delta Q + \delta W$, where $\delta Q = \delta T \partial E/\partial T$ and $\delta W = \sum_i\delta x_i \partial E/\partial x_i$. The former we call 'heat' while the latter we call 'work'. So, in this respect, heat refers to the energy change due to variation in $p_n$ whereas work to variation in $E_n$.