This paper presents Chapter IV (section 5) from the book THE NATURE OF THE PHYSICAL WORLD by A. S. EDDINGTON. The contents of this book are based on the lectures that Eddington delivered at the University of Edinburgh in January to March 1927.
The paragraphs of original material are accompanied by brief comments in color, based on the present understanding. Feedback on these comments is appreciated.
The heading below links to the original materials.
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Thermodynamical Equilibrium
Progress of time introduces more and more of the random element into the constitution of the world. There is less of chance about the physical universe to-day than there will be to-morrow. It is curious that in this very matter-of-fact branch of physics, developed primarily because of its importance for engineers, we can scarcely avoid expressing ourselves in teleological language. We admit that the world contains both chance and design, or at any rate chance and the antithesis of chance. This antithesis is emphasised by our method of measurement of entropy; we assign to the organisation or non-chance element a measure which is, so to speak, proportional to the strength of our disbelief in a chance origin for it. “A fortuitous concourse of atoms”—that bugbear of the theologian—has a very harmless place in orthodox physics. The physicist is acquainted with it as a much-prized rarity. Its properties are very distinctive, and unlike those of the physical world in general. The scientific name for a fortuitous concourse of atoms is “thermodynamical equilibrium”.
Equilibrium is not a rarity; instead it is the modus operandi of the universe. Progress of time introduces more and more equilibrium into the world through quantization, and not more confusion through random chance. Changes follow the law of equilibrium based on the present condition. The conditions do not change arbitrarily as they arise from the same system. Life including human intelligence is part and parcel of the system.
The system is a system of law. It is the ignorance of laws that makes us believe in a random element. We simply have to discover all the laws, which make the system what it is.
Thermodynamical equilibrium is the other case which we promised to consider in which no increase in the random element can occur, namely, that in which the shuffling is already as thorough as possible. We must isolate a region of the universe, arranging that no energy can enter or leave it, or at least that any boundary effects are precisely compensated. The conditions are ideal, but they can be reproduced with sufficient approximation to make the ideal problem relevant to practical experiment. A region in the deep interior of a star is an almost perfect example of thermodynamical equilibrium. Under these isolated conditions the energy will be shuffled as it is bandied from matter to aether and back again, and very soon the shuffling will be complete.
Thermodynamic equilibrium is essentially the equilibrium between material-substance and its environment formed by the field-substance at atomic level. Equilibrium is not the result of some shuffling. It is the result of the elements of a system coming into balance with each other according to universal laws.
The possibility of the shuffling becoming complete is significant. If after shuffling the pack you tear each card in two, a further shuffling of the half-cards becomes possible. Tear the cards again and again; each time there is further scope for the random element to increase. With infinite divisibility there can be no end to the shuffling. The experimental fact that a definite state of equilibrium is rapidly reached indicates that energy is not infinitely divisible, or at least that it is not infinitely divided in the natural processes of shuffling. Historically this is the result from which the quantum theory first arose. We shall return to it in a later chapter.
Energy of field-substance is proportional to its frequency. The frequency of 1 Hertz can be subdivided infinitely by changing the unit of time. Such a frequency that appears fractional in the units of Hz shows that energy is infinitely divisible.
In such a region we lose time’s arrow. You remember that the arrow points in the direction of increase of the random element. When the random element has reached its limit and become steady the arrow does not know which way to point. It would not be true to say that such a region is timeless; the atoms vibrate as usual like little clocks; by them we can measure speeds and durations. Time is still there and retains its ordinary properties, but it has lost its arrow; like space it extends, but it does not “go on”.
Times’ arrow is a property of how changes follow a sequence, and that such changes endure. It does not necessarily depend on some concept like random element. In equilibrium the changes follow a stable sequence; therefore, time is not passing but simply enduring.
This raises the important question, Is the random element (measured by the criterion of probability already discussed) the only feature of the physical world which can furnish time with an arrow? Up to the present we have concluded that no arrow can be found from the behaviour of isolated individuals, but there is scope for further search among the properties of crowds beyond the property represented by entropy. To give an illustration which is perhaps not quite so fantastic as it sounds, Might not the assemblage become more and more beautiful (according to some agreed aesthetic standard) as time proceeds? (In a kaleidoscope the shuffling is soon complete and all the patterns are equal as regards random element, but they differ greatly in elegance.) The question is answered by another important law of Nature which runs—
Nothing in the statistics of an assemblage can distinguish a direction of time when entropy fails to distinguish one.
I think that although this law was only discovered in the last few years there is no serious doubt as to its truth. It is accepted as fundamental in all modern studies of atoms and radiation and has proved to be one of the most powerful weapons of progress in such researches. It is, of course, one of the secondary laws. It does not seem to be rigorously deducible from the second law of thermodynamics, and presumably must be regarded as an additional secondary law. (The law is so much disguised in the above enunciation that I must explain to the advanced reader that I am referring to “the Principle of Detailed Balancing.” This principle asserts that to every type of process (however minutely particularised) there is a converse process, and in thermodynamical equilibrium direct and converse processes occur with equal frequency. Thus every statistical enumeration of the processes is unaltered by reversing the time-direction, i.e. interchanging direct and converse processes. Hence there can be no statistical criterion for a direction of time when there is thermodynamical equilibrium, i.e. when entropy is steady and ceases to indicate time’s arrow.)
Time’s arrow depends on the sequence of changes. Such changes may appear random from a localized, subjective viewpoint. But from a universal, objective viewpoint certain definite laws may be discerned from them. Entropy determines the direction of equilibrium. Changes in that direction determine time’s arrow. Such an arrow becomes blunted when the equilibrium is reached.
The conclusion is that whereas other statistical characters besides entropy might perhaps be used to discriminate time’s arrow, they can only succeed when it succeeds and they fail when it fails. Therefore they cannot be regarded as independent tests. So far as physics is concerned time’s arrow is a property of entropy alone.
Time’s arrow points in the direction of increasing entropy, which is the direction of equilibrium.
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