Suppose we have a closed room full of air, well sealed and with
well-insulated walls. At one end of this room is a wood stove,
and at the other is a block of ice. We know what happens in
such a system; the ice melts. One way to describe this behavior
is to say that the woodstove makes the ice melt, since
clearly if the woodstove were not in the room, and the walls
were truly well insulated, the ice need not melt. In a coarse
way, this is analogous to the way we describe the effect of the
earth's gravity on objects; it makes them fall in a certain
direction. It does not have to `touch' them directly to be
manifestly the `cause' of the response, as we can see by removing
the cause and seeing the response desist. This kind of a
description is generally called an action at a distance
description of cause and effect. We know, by a process of
exclusion, that the presence of one thing is responsible for the
effect on something else, but the description of this
cause-effect relation may not involve any obvious direct
`contact' between the two.
In the case of the stove and the ice, though, we know there is another, more detailed description possible. A person, placed in the room and free to move about and feel the air, would immediately recognize that it has some property, called temperature, and that the air is hotter near the stove and colder near the ice. We can do many experiments with the behavior of the air temperature. For instance, we might start with all of the air cold, so that the ice does not melt, and with the woodstove off. Then we could fire up the woodstove and have our observer stand still, somewhere between the woodstove and the ice, and tell us what happens over time. We know what he would say. In the first instant that the woodstove is turned on, nothing noticeable changes about the air our observer feels, and the ice also does not begin to melt. After a while, though, the observer begins to notice that the air near him is warmer, and some time later than that, we can note that the ice begins melting. We can easily extend this to experiments including a long string of observers placed between the stove and the ice. It will rapidly become clear from their descriptions that the process of warming of the air takes place sequentially, showing any given effect first near the stove, then further away, and so forth until the ice responds to it.
What we are describing is called a classical field,
in this
case the temperature field. The room is full of a smooth, featureless
stuff called air, and at every place in the room, the air has the
property of being at some temperature. The temperature at a
given place can be measured by someone standing at that place, and
the temperature of all parts of the air clearly does not have to be
the same at any given instant of time. We can describe this
``temperature field'' by making a map of the room (in three
dimensional coordinates, for example), and using the temperature (in
degrees of your choice) to assign a number to any location that
interests us. The set of numbers we assign is clearly just a map of
the temperature field; it represents the results of a large
number of possible measurements. As long as the precision of our
measurements is not too fine, so that we can still regard the air
itself as smooth and featureless, (i.e., we can't see the atomic
bouncing directly) we can make individual
measurements and assign individual temperature values as finely as
we like. In other words, there is nothing intrinsic about the
temperature field that forbids us to think about it as a smooth,
continuously definable quantity.
Unlike our field of numbers, which is just a map, the actual field of temperature itself is a physically real quantity. We know this, first, because it is measureable. In other words, what value the temperature has at a given place always coincides with some particular response of a thermometer put there. A more general way to say this is that we can make good, reliable predictions of what happens to objects once we know the temperature field around them. For instance, the ice does not melt when the air is cold. Even if the stove has just been turned on, as long as the air where the ice is is still cold, the ice does not melt. Only when the air around the ice is warmer does it show the effect of being influenced by the stove, and only to a degree determined by how warm the air is there, independently of how it got that way.
This kind of relation between temperatures and responses allows
us to replace our former `action at a distance' description with
a new, purely local
one. We describe what the ice does
entirely in terms of the value of the temperature field where the ice is. This works because, not only does the
temperature field affect the ice, it affects itself. It
would do no good to say that the ice melts because the air around it
is warm, if we simply followed that by saying that `the stove
makes the air warm where the ice is'. That would be a
paraphrase of our previous description, involving one new
measurement that contributes no new understanding. The experiment
with our string of observers, though, allows us to make a much
more powerful, completely local description.
We start by saying that `the stove makes the air warm where the stove is'. Then we recognized that, when a warm-air region is created next to a cold-air region, the ``contact'' of the two regions makes the cold air warmer. In this way the action of ``being warmed'' propagates, step by step through local contact, from the air where the stove is to the air where the ice is, eventually melting the ice.
Such a description is clearly a lot more complicated, because instead of involving two actors, the stove and the ice, it involves potentially an infinite number of measureable quantities, each of the temperatures at every place in the room. To justify the effort, it had better tell us a lot that the simpler description does not, which indeed it does. In particular, the description in terms of the evolving and uneven temperature field tells us everything that all observers in the room will observe, and not just those standing beside the stove or beside the ice. Thus it enables us to predict what would happen to a third object, say a cup of lukewarm tea, placed somewhere in the room already containing the stove and the ice. Will the tea get colder or hotter? The behavior of the temperature field (where it is) will determine.
The power of this description hinges on the fact that, just as we could write equations to describe the motions of objects, we can write similarly effective equations to predict the changes in the temperature field. The equation that has been found, experimentally, to be the correct description of the temperature field in air is this one:
This is translated into direct experimental predictions as
follows. At any instant of time in our room with the stove and
the ice, we could mark off an imaginary sphere and measure the
temperatures at places all around it. We would expect, for
example, that if the sphere is somewhere between the stove and
the ice, the side nearer the stove will register slightly warmer
air temperatures than the side nearer the ice. What the law
says is that, if we measure the temperature in the center of
that sphere at some instant and then again an instant later, the
air in the center will have gotten warmer or colder in such a
way as to approach the average of the temperatures we measured
around the sphere. How fast it approaches this average
depends on properties of the air (how dense it is, how humid it
is, etc.), and on how large a sphere we used to compute the
average. The law is useful because, as long as we use the same
sphere to compute averages, and air of the same composition, we
can apply this system of measurement, average and prediction,
anyplace in the room, at any time, starting from any
configuration of the air temperature in the room. If we use it
near the stove just after the stove has been turned on, the fact
that the stove keeps making the air on one side of the sphere
warmer and warmer requires that, while the temperature in the
center always moves toward the average, it is chasing a moving
target because of the heating on one side, so it too gets warmer
and warmer. If we then shift the sphere a little toward the ice
(see fig 7.2),
so that the previous edge of the sphere is at the center of the new position, the successive heating of that previous center creates a moving average for the new center, which thus chases it and gets warmer as well. This is how a stove at one place in a room eventually can effect the temperature of air anywhere in the room. Moreover, once we have found the particular rates at which things change, we can correctly predict how long it takes for a given change to be felt somewhere.
Last of all, partly as a sidenote, we mention that an important property of ice is that, by melting, it can keep the air around it cold. Thus, if the stove has been on for a long time, but the ice is not entirely melted, for a while the air temperature at those two places in the room is held fixed at some particular values (freezing temperature where the ice is, and some hot temperature where the stove is). Thus, working from both ends toward the middle, we realize that in this situation, the centers of the spheres do not have to chase a moving average, because working in from either side, first one edge value and then another is held fixed. In other words, there is no change in the values at the centers, or the temperature at the center of every sphere is just equal to the averages of the temperatures around it. The same law that gave us evolution of the temperature field enables us to predict what steady state it can settle into, at least for a while. Both uses of this law for temperatures will be explored in detail in the exercises.