Irreversibility and the second law

The relation between the second law of thermodynamics and the notion of irreversibility is analysed by distinguishing three different meanings of the latter and studying how they figure in several versions of the second law. A more extensive discussion is given in [1]. 1. THREE CONCEPTS OF (IR)REVERSIBILITY Many physical theories employ a state space Γ containing all possible states s of a system. A process is represented as a parameterised curve: P = {st ∈ Γ : ti ≤ t ≤ t f }. Usually a theory allows only a subclass, say W , of such processes (e.g. the solutions of the equations of motion). Let R be an involution (i.e. R2s = s) that turns state s into its ‘time reversal’ Rs. In classical mechanics, for example, R is the transformation which reverses the sign of all momenta and magnetic fields. In a theory like classical thermodynamics, where the state does not contain velocity-like parameters, one may take R to be the identity. Further, define the time reversal P∗ of process P by: P∗ = {(Rs)−t : −t f ≤ t ≤−ti}. The theory is called time-reversal invariant (TRI) if the class W is closed under time reversal, i.e. iff: P ∈W =⇒ P∗ ∈W . (1) According to this definition the form of the laws (and a choice for R) determines whether a theory is TRI or not. Note that it is irrelevant here whether the processes P∗ actually occur, but only that the theory allows them. Thus, the fact that the sun never rises in the west doesn’t mean that celestial mechanics is non-TRI. Is time-reversal (non)invariance related to the second law? Applying the criterion to thermodynamics is no matter of routine. In contrast to mechanics, thermodynamics does not have equations of motion. Indeed, thermodynamical processes typically occur after external intervention on the system (e.g.: removing a partition, pushing a piston, etc.) and do not reflect its autonomous behaviour. Yet, classical thermodynamics, in the formulation of Clausius, Kelvin or Planck, is concerned with processes, and its second law is clearly not TRI. But in other formulations this is less clear. Now, ‘(ir)reversible’ is an attribute of processes, not theories or laws. But in philosophy of physics, it is closely connected with TRI. Indeed, one calls a process P allowed by a given theory irreversible iff the reversed process P∗ is excluded by this theory. Obviously, such a P exists only if the theory in question is not TRI. Conversely, every non-TRI theory admits irreversible processes in this sense. Therefore, discussions about (ir)reversibility and (non)-TRI in philosophy of physics mostly coincide. However, the thermodynamics literature often uses the term ‘irreversibility’ to denote processes one might also call irrecoverable, ie., when the transition from an initial state si to a final state s f cannot be fully ‘undone’, once the process has taken place. In other words, there is no process which starts off from state s f and restores the initial state si completely. Wear and tear, erosion etc. are the obvious examples. This is the sense of irreversibility that Planck intended, when he called it the essence of the second law. (Ir)recoverability differs from (non)-TRI in at least two respects. First, the only thing that matters here is the retrieval of the initial state si. It is not necessary to find a process P∗ retracing the intermediate stages of the original process in reverse order. Secondly, we are dealing with a complete recovery. This means that all auxiliary systems that may have been used in the original process are also returned to their initial state. A schematic expression of the idea is this. Let s be a state of the system and Z a (formal) state of its environment. Let P be some process that brings about the transition: 〈si,Zi〉 P −→ 〈s f ,Z f 〉 (2) Then P is reversible in Planck’s sense iff there exists another process P ′ that produces 〈s f ,Z f 〉 P ′ −→ 〈si,Zi〉. (3) The term ‘reversible’ is also used in a third sense, to denote processes which proceed so slowly that the system remains in equilibrium ‘up to a negligible error’ during the entire process. This is the meaning embraced by Clausius, and it appears to be the most common usage of the term in the physical-chemical literature; see e.g. [2, 3]. A more apt name for this kind of processes is quasi-static. Of course, the above characterisation is vague, and has to be amended by specifying what ‘errors’ are meant and when they are ‘small’. These criteria invoke a limit procedure so that, strictly speaking, reversibility is here not an attribute of one process but of a series of processes. Quasi-static processes need not be the same as those called reversible in the previous two senses. An ideal harmonic oscillator is reversible in Planck’s sense, but not quasistatic. Conversely, the discharge of a condenser through a large resistance can be made to proceed quasi-statically, but irreversibly in Planck’s sense. Comparison with the notion of TRI is hampered by the fact that ‘quasi-static’ is not strictly a property of a process. Consider a process PN in which a system, originally at temperature θ1 is consecutively put in thermal contact with a sequence of N heat baths, each at a slightly higher temperature than the previous one, until it reaches a temperature θ2. By making N large, and the temperature steps small, such a process becomes quasistatic, and we can represent it by a curve in the space of equilibrium states. However, for any N, the time-reversal of PN is impossible. Nevertheless, many authors call such a curve ‘reversible’, because one may consider other processes QN , in which the system, originally at temperature θ2, is put in contact with a series of heat baths, each slightly colder than the previous. Again, each QN is non-TRI. A forteriori, no QN is the time-reversal of PN . Yet, if we take the quasi-static limit, the state change traverses the same curve in equilibrium space as in the previous case, in opposite direction. The point is, of course, that precisely because this curve is not itself a process, the notion of time reversal does not apply to it.