Imagine a hollow container evacuated except for two lonely molecules of equal mass (a practical impossibility, of course, but a useful idealization for purposes of discussion). Suppose that one of these molecules is moving rapidly, the other slowly. After a time they will collide. In the collision process, energy will be conserved and momentum will be conserved. The precise angles of deflection of the two molecules in the collision cannot be predicted, nor can the energy exchanged between them be predicted. However, a statement about *probability *can be made. The faster molecule is more likely to lose than to gain energy in the collision; the slower molecule is more likely to gain energy than to lose it. This probability implies a tendency toward equalization. In a particular collision, the less energetic molecule might actually lose energy and enhance the discrepancy between the two energies. But after many collisions, the greater probability will win out. Over a long span of time, one molecule will be moving now faster, now slower than the other, in such a way that each one has, on average, the same kinetic energy as the other. This is one example of the equipartition theorem at work. Unequal molecular energies tend to equalize, and once equalized, they remain equal, *on the average*.

For two molecules of identical mass, the symmetry of the situation means that it is unthinkable that anything other than equal average energy (and equal average speed) could result from a series of random collisions. There is no possible reason for one of the equal pair to pull ahead of the other. Therefore the prediction of the equipartition theorem that the total available energy is shared equally by the two molecules is hardly surprising. In its full generality, however, the equipartition theorem goes far beyond this example. The theorem describes energy sharing for any number of molecules, for any mixture of molecules of different kind and different mass, and including all kinds of internal energy, not only kinetic energy.

To understand fully the equipartition theorem, it is necessary to grasp one new idea, the concept of degree of freedom. This is not an easy concept to define straightaway, so I shall first circle around it by discussing how to *count* the number of degrees of freedom. Just as it is easier to count the number of people in a room than it is to say what a person is, it is easier to find out how many degrees of freedom a system possesses than it is to say exactly what one degree of freedom is.

The number of degrees of freedom of a system is the number of coordinates required to specify completely the spatial arrangement of the system. Consider a single bead moving on a fixed straight wire. To simplify matters further, suppose that the wire has a triangular cross section and the hole in the bead is a matching triangle so that the bead can slide along the wire but can’t rotate. This is a system with just one degree of freedom, for only one single coordinate is required to locate the position of the bead. This coordinate could be the distance of the bead from the left end of the wire or from any other point on the wire. There are various possibilities, any one of which suffices. Evidently the fact that a bead on a wire has only one degree of freedom is associated with the fact that the bead can move in only one dimension. This is the second aspect of “oneness” for a bead on a wire. It has only one mode of motion. For eight of these beads strung on the same straight wire, the number of degrees of freedom of the collection is eight, one for each bead. Eight coordinates are required to locate the beads (that is, to specify the exact spatial arrangement of the system).

Consider now another single bead on a straight wire, this one free to rotate. To keep track of its rotation, a stripe is painted on the bead. How many degrees of freedom has this system? To find out, we must find out how many coordinates are required to locate precisely the position and the orientation of the bead. Two suffice to do the job. First is a distance coordinate, specifying the location of the bead on the wire. Second is an angular coordinate, specifying how far the stripe on the bead has rotated from some reference position. Once these two facts are known, the spatial arrangement of the system is fully determined.

Counting the independent coordinates of a system in order to learn the number of degrees of freedom focuses attention on the static aspect of a system, the arrangement of its parts at a particular instant. Exactly correlated with the static aspect is a dynamic aspect. The number of degrees of freedom is also equal to the number of independent modes of motion of the system. This is no coincidence. It is merely a different way of looking at the same thing, for an independent mode of motion is defined as a particular motion (possible in principle but perhaps not realized in practice) in which one coordinate changes while all the rest remain fixed. The bead on the wire can move along the wire without rotating, or it can rotate without moving along the wire. These are two independent modes of motion. There are no more, for any other motion must be compounded of these two. In general, the actual motion of a system is a complex intermixing of the independent modes of motion.

Based on this discussion, we can now define a degree of freedom. It is used in two closely related senses. First, it means the *possibility *of an independent mode of motion. The bead on the wire (of round cross section) has two possibilities for motion. It can slide or it can rotate. These freedoms exist, whether or not they are exercised. On the other hand, the bead is not free to move away from the wire. A man halfway up a flagpole has more degrees of freedom. He can move up and down; he can move around; or, if he is foolhardy, he can leap away from the pole.

Second, degree of freedom is sometimes used to mean the actual independent mode of motion, not merely the possibility of it. Thus we might say of the bead that half of its energy is in its translational degree of freedom and half in its rotational degree of freedom. This means that its sliding motion accounts for half of its energy and its simultaneous turning motion accounts for the other half.

With the concept of degree of freedom in hand, let’s return now to the equipartition theorem. It is literally an equal-division or equal-sharing theorem. It states that after a system comes to equilibrium, its disordered energy is, on the average, equally divided among all of the degrees of freedom of the system. The parcel of energy assigned to each degree of freedom is

(Energy/degree of freedom) = ½*kT *,

where *k* is Boltzmann’s constant^{1} and *T* is the absolute temperature. When the energy is not equally divided, the direction of spontaneous change is toward equal division.

Note the limitation in the statement of this theorem to disordered energy. For a space vehicle hurtling toward the moon, the *ordered *energy is by no means equally divided among the degrees of freedom. A molecule of nitrogen (N_{2}, molecular weight 28) and a molecule of carbon dioxide (CO_{2}, molecular weight 44) in the air within the vehicle are both being carried along at the same velocity toward the moon. The CO_{2 }molecule, being more massive, has more ordered energy. Its degree of freedom associated with motion toward the moon is more richly supplied with energy than is the same degree of freedom of the N_{2 }molecule, and there is no tendency for these to equalize. But superimposed on the headlong flight of these two molecules through space is the random motion of each within the space vehicle. Associated with this random motion is disordered kinetic energy. It is this disordered energy that determines the temperature and that is equally shared by the translational degrees of freedom of the molecules. Therefore in its random motion relative to the space vehicle, the speed of the heavier CO_{2 }molecule is less than the speed of the lighter N_{2 }molecule (on average).

For 13 beads strung on the same straight wire, each one capable of sliding and turning, the number of degrees of freedom of the collection is 26, two for each bead. Consider now 13 nitrogen molecules, and suppose that these occupy an otherwise empty cubical box. How many degrees of freedom has the system? To answer the question, one must add up the number of coordinates required to specify the spatial arrangement of the system precisely. Since the molecules are all alike, it is sufficient to find out how many coordinates are required for each one and multiply that number by 13.

To locate a particular molecule in the box, one must first of all specify three coordinates, which might be the x, y, and *z *coordinates of its center of mass with respect to a corner of the box. Corresponding to these three coordinates, the molecule has three translational modes of motion or three translational degrees of freedom. Having located the spot in the box occupied by the molecule, we need some more coordinates to specify its orientation with respect to the sides of the box, since orientation is part of the total spatial arrangement. A nitrogen molecule, N_{2}, is a dumbbell-shaped object. Three angular coordinates serve to specify its orientation. The first of these is, let’s say, the angle that the molecular axis makes with the *z*-axis, which we choose to be vertical. The second is the angle that the projected molecular axis^{2} makes with the *x*-axis. The third is the angle of rotation of the molecule about its own axis, analogous to the rotation of a bead on a wire. It would appear that there are three more degrees of freedom associated with molecular orientation, to make a total of six. Now we come face to face with the first of two peculiarities added to the degree-of-freedom concept by quantum mechanics. This is that rotation about the molecular symmetry axis does not in fact correspond either to a mode of motion or to a degree of freedom. The molecule actually has only two orientational degrees of freedom.

A real dumbbell, held in the hand, can certainly be rotated about its symmetry axis, and this is undoubtedly an independent mode of motion of the dumbbell. To keep track of this rotation, a thin stripe can be painted along the dumbbell. The nitrogen molecule differs from the dumbbell in one all-important respect. There is no such thing as painting a stripe on it, or in any other way keeping track of this kind of rotation. After rotation through any angle about its axis, the nitrogen molecule is identically and indistinguishably the same as before the rotation. The irrelevance of this angle is a very practical consequence of an almost philosophical question (practical since it changes the number of degrees of freedom): If there is no experimental way to observe a rotation through a particular angle, how can this rotation have any meaning?^{3} The answer provided by quantum mechanics is that no energy can go into this mode of motion. If a mode of motion has no energy and is unable to acquire any, then for practical purposes it does not exist.

The second peculiarity brought to the degree-of-freedom concept by quantum mechanics is the “frozen” degree of freedom. (I address that idea in the next Essay, T4).

We conclude that our 13 nitrogen molecules have 65 degrees of freedom, 5 each. For this and other simple gases, specific heat capacity (the heat required to raise the temperature of 1 kg of the material by 1 kelvin) is simply related to the number of degrees of freedom per molecule. For the collection of nitrogen molecules, three-fifths of any energy added is apportioned to the translational degrees of freedom and therefore contributes to an increase of temperature. The remaining two-fifths of the added energy is apportioned to the rotational degrees of freedom, where it does not influence the temperature. Evidently the larger the number of degrees of freedom, the larger the specific heat capacity, for more degrees of freedom require more energy input to achieve the same temperature rise.

1^{} Ludwig Boltzmann was an Austrian physicist whose work in the latter half of the nineteenth century clarified the links between the submicroscopic and macroscopic worlds.

2^{} Imagine a light shining vertically downward on the molecule. Its shadow in the *x-y *plane is called its projection onto the *x-y *plane. Our second angle is the angle between the *x *axis and the axis of this imaginary shadow.

3^{} This is *not* the same as asking whether a tree falling in a forest makes any noise if there is no one listening. The falling tree produces real physical effects with or without a human listener. The molecule rotating about its own axis has no effects. This is one more difference between the classical and quantum worlds.