Entropy: Ask 100 physicists what it is and you'll get 100 different answers. It is quite surprising that this physical quantity, so central for the development of thermodynamics, is still a source of controversy within the scientific community. As a matter of fact there is no universal agreement about what is called entropy in the scientific literature. Different scientists call different physical quantities "entropy". Needless to say this creates a lot of confusion, which in turn may lead into pitfalls.
The problems come when expressing entropy (a quantity that pertains to macroscopic bodies in thermal equilibrium) in terms of the dynamics of its incessantly moving microscopic constituents. For example, when one considers the macroscopic body in thermal insulations, two prescriptions for the calculation of entropy are given in textbooks:
where H(x,p) is the system Hamiltonian and E is its energy. k_B is Boltzmann constant.
Although still nowadays the surface entropy is the most common choice, only few know about the firm theoretical fundations of the volume entropy. Over the last century many physicists have expressed their appreciation for the volume entropy and have recognized its fundamental rationale. Among them are some pillars of statistical mechanics like Helmholtz, Gibbs and Einstein.
Since in most cases the two prescriptions give the same result, whether one should use one or the other has been considered an irrelevant academic questions so far. However, since important exceptions of experimental relevance exists [e.g., systems staying in unstable "negative temperature" states, or small (nano)-system], this fundamental question is far from being an academic one.