General definition of reversible process without mentioning changes in the environment

Question

I'm looking a general definition of a reversible process for a (closed) system that wouldn't need to say anything about the environment and would be based on the intuitive notion of "reversing".

For adiabatic (no heat exchange) processeses, it can be done:

"An adiabatic process between two equilibrium states consisting of a controlled change in some external variable $V$ is reversible if, when done in reverse, the system goes back to the same internal state".

I wish I could find such a simple definition even in the case of heat exchange, without mentionning changes in the environment. The definition (from here) seems to be fine:

The practical definition of a reversible process is one for which the system (no mass entering or leaving) passes through a continuous sequence of thermodynamic equilibrium states.

With this definition, you don't need to mention "at equilibrium with the environement". The question arises when some heat exchange happens and the environment is not at the same temperature. There is a temperature gradient: either the gradient is outside the system, and then the process is reversible (for the system), or the gradient is inside the system and then the system is not at equilibrium.

Still, if this definition was to be summarized by a single word (linguistically), it would be natural to call it something like "quasi-static". There is no notion of "reversing". I wonder if it is possible to find a definition based on the intuitive notion of "reversing" without mentionning changes in the environment.

(Statistical physics is welcome in answers if required)

Answer 1

Your "practical definition" of a reversible process for a closed system is what Moran et at, Fundamentals of Engineering Thermodynamics call an Internally Reversible Process. Here is a link to Moran et al: http://dl.booktolearn.com/ebooks2/engineering/mechanical/9781119391760_Fundamentals_of_Engineering_Thermodynamics_9th_e857.pdf

In such a process, the system passes through the exact same sequence of states that it would pass through if it were in contact with a surroundings, where both system and surroundings experienced the conventional definition of a reversible process.

Answer 2

In a practical sense, a reversible process means that the state of any part of the system at any instant of the process can be described with a set of continuous functions. More concretely, for any given configuration of interest there is a finite set of variables $X_1, X_2,...X_n$ and functions $f_1, f_2,..., f_m$ such that $$f_1(X_1, X_2,...X_n)=0\\f_2(X_1, X_2,...X_n)=0\\....\\f_m(X_1, X_2,...X_n)=0$$ The emphasis is on the qualifier functions of the variables, thereby we exclude any processes in which rates in time are involved and processes that have memory, for example in a ferromagnetic material.

The variables themselves may include temperature, pressure, polarization, volume, charge density, internal energy, Gibbs energy, etc., the important point is actual the functional relationship among them that is valid at any instant of time. This also includes multi-phase systems as long as each homogeneous phase can be described individually with functions. (I believe all the philosophizing over what quasi-static and/or reversible means is just interpreting the above mathematical definition.)