@DanielSank nice.thanks...is there any formalization of these concepts? A mathematical definition of thermodynamic/statistical quantities other than the entropy?

@rob Noted.

@ManasDogra Rather than actually formalizing concepts, statistical mechanics describes a system using mechanics (either quantum or classical) on a microscopic level and then we identify certain quantities with macroscopic quantities. As an example , β is the lagrange multiplier of hamiltonian in the canonical ensemble and we identify it with temperature

@DanielSank This is interesting. I had a course about Statistical Mechanics using information theory and as far as I understand, that missing information is not available to us even in principle

depends on what you mean by "in principle" and to which quantum interpretation you subscribe :P

@ACuriousMind all of them

@ACuriousMind I'm not talking about just quantum systems, but also about classical systems

The problem was about storing all the information about a system. To do this you need something bigger than the system, so you can't describe universe as a whole using microscopic mechanics.

There are plenty of places where adiabatic thinking comes into play, in a classical, quantum or statistical context, e.g. time-dependent perturbation theory where a system in a stationary state remains in that stationary state, or deriving adiabatic invariants in the quasi-classical limit, or a thermally isolated system which interacts with it's environment weak enough so that the entropy remains constant under changes of external parameters

@Feynman_00 The mathematical description allows for "truly unavailable information", but Nature doesn't, as far as we know (I think).

The mathematical description is perfectly fine in most cases because even when that "heat information" is there in Nature, we usually have absolutely no hope of recovering it.

Example: I put an atom of gold into a cup of water. That atom very quickly thermalizes with the water, and it's precise location and speed is totally unknown to me. I can only compute things like the probability distribution of the atom's position.

However, if we really could model the position and momentum of every water molecule, we could, in principle, figure out exactly where the gold atom will be. But this is completely hopeless.

In a classical description of a particle in contact with a heat bath, you can add a "noise" term to the particle's motion. For example: $m \ddot{x}(t) = - \gamma \dot{x}(t) + \eta(t)$.

That's the so-called Lengevin equation. It's just Newton's F=ma where the force has a friction term $-\gamma \dot{x}$ and a "noise" term $\eta(t)$.

The function $\eta$ is basically random at each point in time.

There are some interesting ways in which $\gamma$ and $\eta$ have to be related, otherwise the system has unphysical properties.

However, in a quantum description, the relationship between the friction and the noise comes automatically. Basically, because everything in quantum is unitary, you can't just make up a friction or noise term... you have to actually add extra quantum degrees of freedom, and those degrees of freedom give you the friction and the noise together.

very interesting @DanielSank could you please expand on why how what that means 'unitary'