Bluff Your Way in the Second Law of Thermodynamics

Abstract

The aim of this article is to analyse the relation between the second law of thermodynamics and the so-called arrow of time. For this purpose, a number of different aspects in this arrow of time are distinguished, in particular those of time-reversal (non-)invariance and of (ir)reversibility. Next I review versions of the second law in the work of Carnot, Clausius, Kelvin, Planck, Gibbs, Carathéodory and Lieb and Yngvason, and investigate their connection with these aspects of the arrow of time. It is shown that this connection varies a great deal along with these formulations of the second law. According to the famous formulation by Planck, the second law expresses the irreversibility of natural processes. But in many other formulations irreversibility or even time-reversal non-invariance plays no role. I therefore argue for the view that the second law has nothing to do with the arrow of time.

Introduction

There is a famous lecture by the British physicist/novelist C. P. Snow about the cultural abyss between two types of intellectuals: those who have been educated in literary arts and those in the exact sciences. This lecture, The Two Cultures (Snow, 1959), characterises the lack of mutual respect between them in a passage:

A good many times I have been present at gatherings of people who, by the standards of the traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists. Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics. The response was cold: it was also negative. Yet I was asking something which is about the equivalent of: have you read a work of Shakespeare?

Snow stands up for the view that exact science is, in its own right, an essential part of civilisation, and should not merely be valued for its technological applications. Anyone who does not know the Second Law of Thermodynamics, and is proud of it too, exposes oneself as a philistine.

Snow's plea will strike a chord with every physicist who has ever attended a birthday party. But his call for cultural recognition creates obligations too. Before one can claim that acquaintance with the Second Law is as indispensable to a cultural education as Macbeth or Hamlet, it should obviously be clear what this law states. This question is surprisingly difficult.

The Second Law made its appearance in physics around 1850, but half a century later it was already surrounded by so much confusion that the British Association for the Advancement of Science decided to appoint a special committee with the task of providing clarity about the meaning of this law. However, its final report (Bryan, 1891) did not settle the issue. Half a century later, the physicist/philosopher Bridgman still complained that there are almost as many formulations of the Second Law as there have been discussions of it (Bridgman, 1941, p. 116). And even today, the Second Law remains so obscure that it continues to attract new efforts at clarification. A recent example is the work of Lieb and Yngvason (1999).

This manifest inability of the physical community to reach consensus about the formulation and meaning of a respectable physical law is truly remarkable. If Snow's question had been: ‘Can you describe the Second Law of Newtonian Mechanics?’, physicists would not have any problem in producing a unanimous answer. The idea of installing a committee for this purpose would be just ridiculous.

A common and preliminary description of the Second Law is that it guarantees that all physical systems in thermal equilibrium can be characterised by a quantity called entropy, and that this entropy cannot decrease in any process in which the system remains adiabatically isolated, i.e. shielded from heat exchange with its environment. But the law has many faces and interpretations; the comparison to a work of Shakespeare is, in this respect, not inappropriate.¹ One of the most frequently discussed aspects of the Second Law is its relation with the ‘arrow of time’. In fact, in many texts in philosophy of physics the Second Law figures as an emblem of this arrow. The idea is, roughly, that typical thermodynamical processes are irreversible, i.e. they can occur in one sense only, and that this is relevant for the distinction between past and future.

At first sight, the Second Law is indeed relevant for this arrow. If the entropy can only increase during a thermodynamical process, then obviously, a reversal of this process is not possible. Many authors believe this is a crucial feature, if not the very essence of the Second Law. Planck, for example, claimed that, were it not for the existence of irreversible processes, ‘the entire edifice of the second law would crumble […] and theoretical work would have to start from the beginning’ (Planck, 1897, § 113), and viewed entropy increase as a ‘universal measure of irreversibility’ (ibid., § 134). A similar view is expressed by Sklar in his recent book on the foundations of statistical mechanics (1993, p. 21): ‘The crucial fact needed to justify the introduction of […] a definite entropy value is the irreversibility of physical processes’.

In this respect, thermodynamics seems to stand in sharp contrast with the rest of classical physics, in particular with mechanics which, at least in Hamilton's formulation, is invariant under time reversal. The problem of reconciling this thermodynamical arrow of time with a mechanical world picture is usually seen as the most profound problem in the foundations of thermal and statistical physics; see Davies (1974), Mackey (1992), Zeh (1992), Sklar (1993) and Price (1996).

However, this is only one of many problems awaiting a student of the Second Law. There are also authors expressing the opposite view. Bridgman writes:

It is almost always emphasized that thermodynamics is concerned with reversible processes and equilibrium states and that it can have nothing to do with irreversible processes or systems out of equilibrium (Bridgman, 1941, p. 133).

It is not easy to square this view—and the fact that Bridgman presents it as prevailing among thermodynamicists—with the idea that irreversibility is essential to the Second Law.

Indeed, one can find other authors maintaining that the Second Law has little to do with irreversibility or the arrow of time; in particular Ehrenfest-Afanassjewa 1925, Ehrenfest-Afanassjewa 1956, Ehrenfest-Afanassjewa 1959, Landsberg (1956) and Jauch 1972, Jauch 1975. For them, the conflict between the irreversibility of thermodynamics and the reversible character of the rest of physics is merely illusory, due to a careless confusion of the meaning of terms. For example, Landsberg remarks that the meaning of the term ‘reversible’ in thermodynamics has nothing to do with the meaning of this term in classical mechanics. However, a fundamental and consistent discussion of the meaning of these concepts is rare.

Another problem is that there are indeed many aspects and formulations of the Second Law, which differ more or less from the preliminary circumscription offered above. For example, consider the so-called ‘approach to equilibrium’. It is a basic assumption of thermodynamics that all systems which are left to themselves, i.e. isolated from all external influences, eventually evolve towards a state of equilibrium, where no further changes occur. One often regards this behaviour as a consequence of the Second Law. This view is also suggested by the well-known fact that equilibrium states can be characterised by an entropy maximum. However, this view is problematic. In thermodynamics, entropy is not defined for arbitrary states out of equilibrium. So how can the assumption that such states evolve towards equilibrium be a consequence of this law?

Even deliberate attempts at careful formulation of the Second Law sometimes end up in a paradox. One sometimes finds a formulation which admits that thermodynamics aims only at the description of systems in equilibrium states, and that, strictly speaking, a system does not always have an entropy during a process. The Second Law, in this view, refers to processes of an isolated system that begin and end in equilibrium states and says that the entropy of the final state is never less than that of the initial state (Sklar, 1974, p. 381). The problem here is that, by definition, states of equilibrium remain unchanged in the course of time, unless the system is acted upon. Thus, an increase of entropy occurs only if the system is disturbed, i.e. when it is not isolated.

It appears then that it is not unanimously established what the Second Law actually says and what kind of relationship it has with the arrow of time. The aim of the present paper is to chart this amazing and confusing multifariousness of the Second Law; if only to help prevent embarrassment when, at a birthday party, the reader is faced with the obvious counter-question by literary companions. Or, if the reader wishes to be counted as a person of literary culture, and guard against arrogant physicists, one can also read this article as a guide to how to bluff your way in the Second Law of Thermodynamics.

The organisation of the article is as follows. In Section 2, I will describe a few general characteristics of thermodynamics, and its status within physics. Section 3 is devoted to the distinction between several meanings one can attribute to the arrow of time. Next, in 4 The Prehistory of Thermodynamics: Carnot, 5 Clausius and Kelvin: The Introduction of the Second Law, 6 From the Steam Engine to the Universe (and Back Again), I will trace the historical development of the orthodox versions of the Second Law, focusing at each stage on its relation to the arrow of time. This historical development finds its climax in the intricate arguments of Planck, which I review in Section 7.

Then I address two less orthodox but perhaps more vital versions of the Second Law, due to Gibbs (Section 8) and Carathéodory (Section 9). I will argue that these versions do not carry implications for an arrow of time (with a slight qualification for Carathéodory). In Section 10, I discuss the debate in the 1920's between Born, Planck and Ehrenfest-Afanassjewa, which was triggered by the work of Carathéodory.

Despite a number of original defects, the approach pioneered by Carathéodory has in recent years turned out to be the most promising route to obtain a clear formulation of the Second Law. Section 11 is devoted to the work of Lieb and Yngvason, which forms the most recent major contribution to this approach. Finally, in Section 12, I will discuss some conclusions. In particular, I will discuss the prospects of giving up the idea that the arrow of time is crucially related to the Second Law.