TIME, AND TIME AGAIN
(IN THERMAL PHYSICS)

1 Overview

The future seems different from the past. Partly, this is down to our memories—we don’t know what the future holds, whereas we remember the past. Delving deeper, there’s an asymmetry between the past and future that physics describes. Certain processes only seem to happen in one direction of time—buildings crumble, ice cubes melt, and wine glasses smash—but only towards the future. It is commonly held, as Reichenbach ([1956]) trenchantly argues, that the key to time asymmetry is to be found in thermal physics.

But there are different types of time asymmetry in thermal physics (see Uffink [2006]). The failure of time-reversal invariance is the most familiar type of time asymmetry, and what I’ll term here ‘irreversibility’. ‘Reversing’ or ‘rewinding’ a sequence of events results in a situation that is forbidden by the theory’s laws. For example, if I were to play you a video of one lane on a busy motorway, you would know immediately whether I am playing the video in rewind or not, since a stream of cars each reversing at high speed along a busy road is not a possibility allowed by the laws of the road. (More formally, a theory is not time-reversal invariant if the time-reversal operator (often t ↦ –t) does not send solutions to solutions.)

There’s a second type of time asymmetry in thermal physics that Uffink terms ‘irrecoverability’, for which the state of the environment is crucial. A recoverable process is one where the system can be taken back to its earlier state without any cost to the environment. The slow compression of a gas with a frictionless piston is a recoverable process, since the gas can be taken back to its larger volume by the slow expansion with the piston. In contrast, an ice cube melting is an example of an irrecoverable process. You can take the ice cube back to its earlier unmelted state, but not without a change in the environment—for example, your energy bill has gone up from using the freezer. Uffink contends that this irrecoverability type of time asymmetry is what Eddington ([1928]) had in mind in his discussion of the ‘running down of the universe’. The ‘running down of the universe’ reflects the impossibility of a perpetual motion machine (of the second kind) and so is closely connected to a form of the second law, but also predates it: Newton discussed how ‘motion is more apt to be lost than got, and always on the decay’, as quoted in (Price [1996], p. 23).

The irrecoverability type of time asymmetry differs from the first example of irreversibility, since the system need not retrace its steps for a process to be recoverable. The car can’t travel in reverse along the motorway from B to A, but there might be another road the car can travel down back to A. The key for (ir)recoverability is whether getting back to A, whatever the route, has an entropic cost to the environment.

At this point, we can look up or down. Looking up, like Callender ([2017]), we can ask: which type of time asymmetry is connected to our experience of time? But I am going to look down towards more fundamental theories and ask: how are each of these types of time asymmetry grounded in more fundamental theories? This is a project in the inter-theoretic relations between three different theories or levels: thermodynamics, statistical mechanics, and the microdynamics.

In thermodynamics, the two types of time asymmetry correspond to two distinct laws. The minus first law says that the system will spontaneously reach a unique equilibrium state, and so embodies the irreversibility type of time asymmetry. One of the myriad statements of the second law is that certain transitions between equilibrium states are forbidden: it is impossible for the system to transition to a state with lower thermodynamic entropy without a compensating entropic change in the environment. Thus, the second law describes the irrecoverability type of time asymmetry. I am going to talk in turn about these laws and their underpinning by lower-level theories, based on my two recent BJPS articles, ‘Asymmetry, Abstraction, and Autonomy: Justifying Coarse-Graining in Statistical Mechanics’ and ‘In Search of the Holy Grail: How to Reduce the Second Law of Thermodynamics’.

2 The Approach to Equilibrium and Irreversibility in Statistical Mechanics

The equations of non-equilibrium statistical mechanics are irreversible (in the familiar non-time-reversal-invariant sense). They quantitively describe, among other things, how a particular system reaches equilibrium, and they provide the underpinning to the minus first law’s assertion that systems will reach equilibrium.

Yet the more fundamental laws underlying statistical mechanics—namely, quantum (or classical) microdynamics—are time-reversal invariant. The time asymmetry in statistical mechanics is the source of a traditional puzzle: how can the fundamental time symmetry be reconciled with the statistical-mechanical asymmetry?

The answer, I believe, lies in a particular formal framework (discussed by Zwanzig ([1960]), Zeh ([2007]), and Wallace ([2015])) that allows us to construct the irreversible statistical-mechanical equations from the underlying reversible microdynamics. This construction requires various assumptions (such as an initial state condition), but provided these assumptions hold, the mystery is resolved.

But this formal framework uses a procedure called coarse-graining, which throws away information about the state of the system. And coarse-graining has been criticized as ‘deceitful’ (Redhead [1996], p. 31), ‘illusory’ (Prigogine [1980]), or ‘subjective’ (Denbigh and Denbigh [1985], p. 53). Grünbaum ([1973]) even wonders if coarse-graining renders statistical mechanics incompatible with scientific realism. If this is so, resolving the mystery is a hollow victory.

The objections to coarse-graining stem from the way that coarse-graining is frequently justified in the literature: due to the imprecision of our measurements, we can’t tell the difference between a fine-grained and a coarse-grained description of the system, and so we should pick a coarse-graining that matches our observational and measurement capacities. On this view, since justifying coarse-graining relies on our inability to tell the difference, it depends sensitively on our perspective on reality, and so the statistical-mechanical time asymmetry is rendered anthropocentric. Worse still, if our observational capacities were to improve—as so often happens throughout the history of science—then we would expect the asymmetry to vanish, making the asymmetry an artefact of our view on reality rather than reality itself. But, I argue, measurement imprecision plays no role in the formal construction of the irreversible equations.

Instead, I offer an alternative justification. Coarse-graining is not a distortion or idealization but is instead is an abstraction; coarse-graining allows us to abstract to a higher level of description. Furthermore, the choice of coarse-graining is determined by whether it uncovers autonomous dynamics—a fact that has little to do with us. To give an analogy: We can abstract from the positions and momenta of each philosopher of science to the centre of mass of all philosophers of science. But if we can’t give a dynamics of how this centre of mass evolves over time without referring back down to the individual level, then we don’t have an autonomous dynamics for this centre of mass variable.

The microdynamical and the statistical-mechanical levels of description are not in conflict. Indeed, the Zwanzig framework—with this justification of coarse-graining in hand—explains how the time asymmetry emerges from the microscopic time-symmetric description it is grounded in (for more on this, see here ).

3 In Search of the Holy Grail: How to Reduce the Second Law

The second type of time asymmetry—irrecoverability—is described by the second law of thermodynamics. But thermodynamics is an odd theory, since it is not ‘dynamical’ in the way that physical theories usually are. One of the central concepts in thermodynamics is an equilibrium state, where the macro-parameters such as temperature and pressure are no longer varying in time. Once a system reaches equilibrium (as mandated by the minus first law), then nothing happens—by the very definition of equilibrium. The system just sits there indefinitely according to thermodynamics. For anything to happen, the system must be prodded by some external intervention: inserting a piston, turning on a magnetic field, and so on. Thermodynamic systems lack spontaneity.

The state-space of thermodynamics is a space of these equilibrium states. Curves through this state-space are usually taken to represent quasi-static processes—those in which the intervention proceeds very slowly and non-dissipatively so that the system is ‘close’ to equilibrium. These quasi-static curves are crucial to the second law, since they are used to define the thermodynamic entropy and are central to its behaviour. The second law legislates two key features of the thermodynamic entropy: for a thermally isolated system, it is (i) constant in quasi-static processes but (ii) increasing in non-quasi-static processes.

Now we turn to reduction. I claim that functionalism is a useful strategy for reduction since it specifies which differences between the two theories can be tolerated. Provided that some statistical-mechanical quantity plays the right role, other differences between the thermodynamic and statistical-mechanical quantity needn’t matter. The key role that the underlying realizer, or reductive basis, must play is to increase in non-quasi-static processes but remain constant in quasi-static processes. And this is contra to the usual reductive project, what Callender (disparagingly) terms the ‘search for the holy grail’: find a non-decreasing statistical-mechanical quantity to call entropy. But this old grail is not enough; the statistical mechanical quantity must increase in the right circumstances too! I show how the Gibbs entropy plays the right role, namely, it increases during non-quasi-static interventions, but remains constant during quasi-static interventions.

However, the Gibbs entropy has been unfairly maligned. Some worry that its ‘ensemble’ nature (namely, that it is a property of a probability distribution) means that it differs from the thermodynamic entropy. But I argue that this doesn’t prevent it from playing the right role. Moreover, in the quantum-statistical mechanical setting, the ‘ensemble problem’ doesn’t arise (for more on this, see ).

4 Summary

To sum up: In this pair of articles, I show how the time asymmetry in thermal physics is to be understood in terms of the underlying theories. In keeping with the practice of physics, but in opposition to much of the philosophical literature, I adopt a broadly Gibbsian approach (although it is disputed whether there is a sharp divide, or whether the Boltzmannian view can be subsumed under the Gibbsian; cf. Wallace [unpublished]). In part, the dispute between Gibbsians and Boltzmannians is over how we should understand probability in statistical mechanics, and in the second law article I discuss how in quantum-statistical mechanics, this is entangled with the quantum measurement problem. So while I hope to have made progress with understanding time (the minus first law) and time again (the second law) in thermal physics, the full picture is only completed by a story about how we should understand probability in statistical mechanics.