Where does reversibility collide?

A change of state carried out so slowly and so close to balance that it can be run backward through the same succession of states, returning both the system and its surroundings to where they began with nothing left over. Carnot's 1824 Reflexions sur la puissance motrice du feu introduced the idealized engine whose operations can be exactly retraced, and Clausius's 1865 paper formalized this by showing that such a change generates no net entropy while any real change generates some. The referent here is a limiting ideal of a physical change, judged by whether the total entropy stays fixed, and it is not a reaction that runs both ways, a chain that satisfies balance, or a child's grasp that an action can be undone.[1][2]

Unit

One idealized change of state, scaled by the total entropy it generates (zero at the limit)

Recovery means

Both system and surroundings can be returned exactly to their starting state, which is what defines the ideal

Measured by

Assessed by comparing the entropy change of system plus surroundings, or by how far a real process departs from the quasi-static limit

A property of a random process moving between states in which the flow of probability from state A to state B exactly matches the flow from B to A once the process settles, so a recorded run looks the same played forward or backward in time. Kolmogorov's 1936 Zur Theorie der Markoffschen Ketten gave the condition, now called detailed balance, that makes a Markov chain reversible, and Kelly's 1979 Reversibility and Stochastic Networks built the modern theory on it. The referent is a symmetry in the statistics of a random process, tested by a balance equation on transition rates, and it has nothing to do with an idealized heat process, a two-way chemical change, or a mental undoing.[3][4]

Unit

One stochastic process, holding or failing the detailed-balance condition on its transition rates

Recovery means

Not applicable

Measured by

Checked by verifying that the stationary probabilities and transition rates satisfy detailed balance, or by Kolmogorov's cycle criterion

The capacity of a child in the concrete-operational stage to grasp that a mental action can be run in reverse, so that pouring water into a taller glass or spreading out a row of counters can be mentally undone to recover the starting arrangement. Piaget and Szeminska's 1952 The Child's Conception of Number (first published in French in 1941) treated this undoing as the mark of a true operation and tied it to conservation, and Inhelder and Piaget's 1958 The Growth of Logical Thinking extended it to adolescent reasoning. The referent is a feature of how a mind coordinates operations, inferred from what a child says and does on conservation tasks, and it is unrelated to entropy in a heat process, balance in a random chain, or a reaction running both ways.[5][6]

Unit

One child's competence on tasks that require undoing a transformation, present or absent

Recovery means

Not applicable

Measured by

Inferred from performance on conservation and classification tasks where the correct answer requires reversing a transformation in thought