As is explained in detail in the article thermodynamics, the laws of thermodynamics make possible the characterization of a given sample of matter—after it has settled down to equilibrium with all parts at the same temperature—by ascribing numerical measures to a small number of properties (pressure, volume, energy, and so forth). One of these is entropy. As the temperature of the body is raised by adding heat, its entropy as well as its energy is increased. On the other hand, when a volume of gas enclosed in an insulated cylinder is compressed by pushing on the piston, the energy in the gas increases while the entropy stays the same or, usually, increases a little. In atomic terms, the total energy is the sum of all the kinetic and potential energies of the atoms, and the entropy, it is commonly asserted, is a measure of the disorderly state of the constituent atoms. The heating of a crystalline solid until it melts and then vaporizes is a progress from a well-ordered, low-entropy state to a disordered, high-entropy state. The principal deduction from the second law of thermodynamics (or, as some prefer, the actual statement of the law) is that, when an isolated system makes a transition from one state to another, its entropy can never decrease. If a beaker of water with a lump of sodium on a shelf above it is sealed in a thermally insulated container and the sodium is then shaken off the shelf, the system, after a period of great agitation, subsides to a new state in which the beaker contains hot sodium hydroxide solution. The entropy of the resulting state is higher than the initial state, as can be demonstrated quantitatively by suitable measurements.

The idea that a system cannot spontaneously become better ordered but can readily become more disordered, even if left to itself, appeals to one’s experience of domestic economy and confers plausibility on the law of increase of entropy. As far as it goes, there is much truth in this naive view of things, but it cannot be pursued beyond this point without a much more precise definition of disorder. Thermodynamic entropy is a numerical measure that can be assigned to a given body by experiment; unless disorder can be defined with equal precision, the relation between the two remains too vague to serve as a basis for deduction. A precise definition is to be found by considering the number, labeled W, of different arrangements that can be taken up by a given collection of atoms, subject to their total energy being fixed. In quantum mechanics, W is the number of different quantum states that are available to the atoms with this total energy (strictly, in a very narrow range of energies). It is so vast for objects of everyday size as to be beyond visualization; for the helium atoms contained in one cubic centimetre of gas at atmospheric pressure and at 0 °C the number of different quantum states can be written as 1 followed by 170 million million million zeroes (written out, the zeroes would fill nearly one trillion sets of the Encyclopædia Britannica).

The science of statistical mechanics, as founded by the aforementioned Ludwig Boltzmann and J. Willard Gibbs, relates the behaviour of a multitude of atoms to the thermal properties of the material they constitute. Boltzmann and Gibbs, along with Max Planck, established that the entropy, S, as derived through the second law of thermodynamics, is related to W by the formula S = k ln W, where k is the Boltzmann constant (1.3806488 × 10⁻²³ joule per kelvin) and ln W is the natural (Naperian) logarithm of W. By means of this and related formulas it is possible in principle, starting with the quantum mechanics of the constituent atoms, to calculate the measurable thermal properties of the material. Unfortunately, there are rather few systems for which the quantum mechanical problems succumb to mathematical analysis, but among these are gases and many solids, enough to validate the theoretical procedures linking laboratory observations to atomic constitution.

When a gas is thermally isolated and slowly compressed, the individual quantum states change their character and become mixed together, but the total number W does not alter. In this change, called adiabatic, entropy remains constant. On the other hand, if a vessel is divided by a partition, one side of which is filled with gas while the other side is evacuated, piercing the partition to allow the gas to spread throughout the vessel greatly increases the number of states available so that W and the entropy rise. The act of piercing requires little effort and may even happen spontaneously through corrosion. To reverse the process, waiting for the gas to accumulate accidentally on one side and then stopping the leak, would mean waiting for a time compared with which the age of the universe would be imperceptibly short. The chance of finding an observable decrease in entropy for an isolated system can be ruled out.

This does not mean that a part of a system may not decrease in entropy at the expense of at least as great an increase in the rest of the system. Such processes are indeed commonplace but only when the system as a whole is not in thermal equilibrium. Whenever the atmosphere becomes supersaturated with water and condenses into a cloud, the entropy per molecule of water in the droplets is less than it was prior to condensation. The remaining atmosphere is slightly warmed and has a higher entropy. The spontaneous appearance of order is especially obvious when the water vapour condenses into snow crystals. A domestic refrigerator lowers the entropy of its contents while increasing that of its surroundings. Most important of all, the state of nonequilibrium of the Earth irradiated by the much hotter Sun provides an environment in which the cells of plants and animals may build order—i.e., lower their local entropy at the expense of their environment. The Sun provides a motive power that is analogous (though much more complex in detailed operation) to the electric cable connected to the refrigerator. There is no evidence pointing to any ability on the part of living matter to run counter to the principle of increasing (overall) disorder as formulated in the second law of thermodynamics.

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The irreversible tendency toward disorder provides a sense of direction for time which is absent from space. One may traverse a path between two points in space without feeling that the reverse journey is forbidden by physical laws. The same is not true for time travel, and yet the equations of motion, whether in Newtonian or quantum mechanics, have no such built-in irreversibility. A motion picture of a large number of particles interacting with one another looks equally plausible whether run forward or backward. To illustrate and resolve this paradox it is convenient to return to the example of a gas enclosed in a vessel divided by a pierced partition. This time, however, only 100 atoms are involved (not 3 × 10¹⁹ as in one cubic centimetre of helium), and the hole is made so small that atoms pass through only rarely and no more than one at a time. This model is easily simulated on a computer, and Figure 13 shows a typical sequence during which there are 500 transfers of atoms across the partition. The number on one side starts at the mean of 50 and fluctuates randomly while not deviating greatly from the mean. Where the fluctuations are larger than usual, as indicated by the arrows, there is no systematic tendency for their growth to the peak to differ in form from the decay from it. This is in accord with the reversibility of the motions when examined in detail.

If one were to follow the fluctuations for a very long time and single out those rare occasions when a particular number occurred that was considerably greater than 50, say 75, one would find that the next number is more likely to be 74 than 76. Such would be the case because, if there are 75 atoms on one side of the partition, there will be only 25 on the other, and it is three times more likely that one atom will leave the 75 than that one will be gained from the 25. Also, since the detailed motions are reversible, it is three times more likely that the 75 was preceded by a 74 rather than by a 76. In other words, if one finds the system in a state that is far from the mean, it is highly probable that the system has just managed to get there and is on the point of falling back. If the system has momentarily fluctuated into a state of lower entropy, the entropy will be found to increase again immediately.

It might be thought that this argument has already conceded the possibility of entropy decreasing. It has indeed, but only for a system on the minute scale of 100 atoms. The same computation carried out for 3 × 10¹⁹ atoms would show that one would have to wait interminably (i.e., enormously longer than the age of the universe) for the number on one side to fluctuate even by as little as one part per million. A physical system as big as the Earth, let alone the entire Galaxy—if set up in thermodynamic equilibrium and given unending time in which to evolve—might eventually have suffered such a huge fluctuation that the condition known today could have come about spontaneously. In that case man would find himself, as he does, in a universe of increasing entropy as the fluctuation recedes. Boltzmann, it seems, was prepared to take this argument seriously on the grounds that sentient creatures could only appear as the aftermath of a large enough fluctuation. What happened during the inconceivably prolonged waiting period is irrelevant. Modern cosmology shows, however, that the universe is ordered on a scale enormously greater than is needed for living creatures to evolve, and Boltzmann’s hypothesis is correspondingly rendered improbable in the highest degree. Whatever started the universe in a state from which it could evolve with an increase of entropy, it was not a simple fluctuation from equilibrium. The sensation of time’s arrow is thus referred back to the creation of the universe, an act that lies beyond the scrutiny of the physical scientist.

It is possible, however, that in the course of time the universe will suffer “heat death,” having attained a condition of maximum entropy, after which tiny fluctuations are all that will happen. If so, these will be reversible, like the graph of Figure 13, and will give no indication of a direction of time. Yet, because this undifferentiated cosmic soup will be devoid of structures necessary for consciousness, the sense of time will in any case have vanished long since.