The logic of modern physics

by P. W. Bridgman

The general conclusion to which all this discussion leads is that energy is probably not entitled to the fundamental position that physical thought is inclined to give it, but that it is a more or less incidental consequence of more deep-seated properties, and that the character of these deep-seated properties is subject to only the most general restrictions, so that very little about the nature of the details can be inferred from the existence of any energy function.

THE CONCEPTS OF THERMODYNAMICS

We shall not be concerned here with the many technical questions which are the proper subject of treatises on thermodynamics, but shall attempt an examination only of some fundamental concepts.

The most fundamental of these, which sets thermodynamics off apart from the simpler subjects, is probably that of temperature. In origin this concept was without question physiological, in much the same way as the mechanical concept of force was physiological. But just as the force concept was made more precise, so the temperature concept may be more or less divorced from its crude significance in terms of immediate sensation and be given a more precise meaning. This precision may be obtained through the notion of equilibrium states. We have in the first place the fundamental experimental fact that when a small body is placed inside a large system, which we recognize by crude means as comparatively invariable in temperature as time goes on, the small body very soon acquires a steady condition, that is, it comes to equilibrium with its surroundings. We now have the further experimental fact that if the small body A is in equilibrium with its environment, and body B is also in equilibrium with the same environment, there will be no change of condition of A and B when they are brought into contact with each other—that is, A and B are each in equilibrium with the other and also with the environment and therefore, by definition, at the same temperature as the environment. The temperature of the environment is now measured in terms of some of the properties of A and B which crude experience has shown change with the physiological temperature of A and B. The physiological notion of temperature is thus made more precise by being connected with the physical phenomenon of equilibrium.

Now it is at once evident that stated in this way without qualification we have said things that are not true. It is not true in general that, when A is in equilibrium with an environment and B is in equilibrium with the same environment, A will be in equilibrium with B. Suppose, for example, that the environment is a stream of water and A is a tiny water wheel moving freely in its bearings, and that B is a similar wheel with much friction. Then we know that B will become warm, and will not be in equilibrium with A when brought into contact with it. Or we may choose for A a mercury thermometer with bulb covered with putty, and for B a similar thermometer with bulb sheathed in platinum, and we know that the two thermometers will not register the same temperature in the water stream. Or still more simply, we may try to read the temperature of the air in our garden on any bright day with a silvered and with a blackened bulb thermometer; we know that the two thermometers will read different temperatures. It is evident, therefore, that we shall have to specify much more carefully the conditions under which equilibrium holds if we are to give precise significance to the temperature concept.

It seems fairly evident in the first place that we shall have to rule out systems in which there is large scale mechanical motion; the simple notion of temperature does not apply to a system moving with respect to us. Only when the two thermometers A and B move with the same velocity as the stream do we have three-fold equilibrium between the stream, A and B. We may state this in another way by saying that the temperature of a moving body must be measured on a thermometer stationary with respect to the body, but this is only a matter of words, and properly speaking the temperature concept applies only to a certain aspect of the relation between two bodies mutually at rest. We here entirely neglect relativity questions such, for example, as the proper way of correcting for the change of dimensions of moving thermometers.

If now the body whose temperature we are measuring does not move with the same velocity in all its parts, we may still give a meaning to local temperature by dividing the body into parts so small that the velocity of each part is essentially uniform, and measuring the temperature of each part with a thermometer stationary with respect to it. We are now confronted with the question of how far to carry the process of subdivision. Suppose we have a fluid whose motion is completely turbulent when measured with instruments of the ordinary scale of magnitude. For such a fluid the fundamental equilibrium proportions hold between two measuring bodies A and B and the fluid, provided that the bodies A and B are so large that the motion is completely turbulent on their scale of magnitude. We may then define the temperature of the turbulent fluid from the standpoint of these large scale bodies. But we may also define the temperature from the small scale point of view as the average of the temperatures recorded by sufficiently small thermometers, each moving with the velocity of a local bit of the fluid. These two temperatures will in general be different, and we must more or less arbitrarily select one which we define as the true temperature. It would seem that the small scale temperature is the better one to choose, because there is a certain degree of arbitrariness in specifying the scale from which the motion shall be judged completely turbulent But on the other hand, there are difficulties in the small scale definition, because the turbulence may become more and more fine grained, until we end with the motion of the molecules themselves, when the operations certainly fail which give meaning to the temperature concept. In this case of molecular turbulence, we are driven back to the large scale definition, which obviously corresponds to ordinary physical practice.

It appears then that the temperature concept is not a clean cut thing, which can be made to apply to all experience, but that it is more or less arbitrary, involving the scale of our measuring instruments. In any special case, the meaning of the temperature concept must be set by special convention. In practice this does not often make difficulty, because in the majority of cases there is no large scale motion with respect to the thermometers.

Consider now the other aspect of the equilibrium problem suggested by the thermometers with blackened and silvered bulbs in the sunshine. Our common experience tells us how to deal with this situation effectively enough for ordinary purposes. We recognize that the possibility of temperature equilibrium is disturbed by the radiation, and we protect the bulbs of the thermometers from the sun's radiation by appropriate shields. But this only minimizes the difficulty. For the shield is warmed by the sun, and in turn warms to a less degree by its radiation the bulb of the thermometer within. We must recognize that every body, no matter what its temperature, is always emitting radiation, so that the bulb of our thermometer is always in a radiation field. At first this puts us in a serious quandary as to the whole question of equilibrium and the meaning of temperature. The situation is saved by the experimental observation that there is a particular radiation field which affects all thermometers equally, namely, the field inside an infinite body all at the same temperature. Logically this looks like the vicious circle again, for we have not yet defined what we mean by the same temperature. But actually we avoid the circle here, as in so many other physical cases, by a process of asymptotic approximation. The procedure is perhaps something like this: we find that if we experiment with larger and larger bodies, isolated and at great distances from other bodies at approximately the same temperature as judged by crude physiological sensations, two thermometers, identical except that the bulb of one is blackened and that of the other is silvered, record more nearly the same temperature as time goes on and as the thermometers are sunk to greater depths in the body. In actual practice, of course, the radiational opacity of most bodies is so high that these precautions against the effects of external radiation can usually be entirely ignored. At high temperatures, on the other hand, radiation has to be explicitly dealt with.

The conclusion for us from these considerations is that operationally the concept of temperature is tied up with that of radiation—the equilibrium concept of temperature is strictly never exactly applicable; it is only a limiting sort of concept applicable when the radiation field is of a special sort, namely, that of a black body.