The Second Law of Thermodynamics

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    Imagine that you have come to a railroad on a cold day. After ascertaining that no trains will come by to interrupt your experiment, you light a blowtorch and use it to heat a small spot on the head of one of the rails until that spot glows. Extinguishing the blowtorch and setting it aside, you notice the glow fading and you understand that radiation and air flowing across the rail are taking away heat. But if you touch the rail at a point near the spot you heated, you will discover that it feels warmer than it did before you put heat into the rail. Clearly some of the heat seems to have traveled through the steel and has, thereby, spread away from the hot spot. We want to examine that spreading in greater depth.

    We know that in a given body adding heat increases the amount of scrambled kinetic energy in the body. Even if we had not inferred that fact, we would have known it from the work of Benjamin Thompson, Count Rumford (1753 Mar 26 Ė 1814 Aug 21). In his 1798 work "An Experimental Enquiry Concerning the Source of the Heat which is Excited by Friction" he described how his experiments with boring out cannon showed that heat is a form of motion rather than the substance called caloric. Friction simply converts work or directed kinetic energy (as in a moving body skidding to a stop) into a vast number of minuscule vibrations within the heated material. Because of that nature of the phenomenon, we cannot, strictly speaking, talk about heat moving (because the word energy does not denote a thing-in-itself): we can only properly talk about a redistribution of heat (because we properly conceive energy as a relationship among the parts of a material system). We do, nonetheless, talk about heat spreading or flowing, as if it consisted of a liquid. Remaining aware that we are merely using metaphors, then, we want to ask what kind of rules govern (another metaphor) the flow and spread of heat.

    Assume that we have a solid body, such as a short length of steel rail. Assume further that we increase the amount of heat in one part of the body and diminish the heat in another part of the body by the same amount. If we then isolate the body from all contact with things that could change the energy in it, the first law of thermodynamics tells us that the total amount of heat in the body wonít change. But can we say anything else about the heat in the body, such as, say, its distribution and how it changes? To gain the means to answer that question we need to examine a relevant analogy.

    Consider an ideal gas held inside a rigid container. We had to do a certain amount of work on the gas to get it into the container, so the gas contains that same certain amount of energy of compression, whose density corresponds to pressure in the gas. Feigning ignorance, we ask whether Reality allows us to create an equilibrium state (a stable, unchanging state) in which, in the absence of applied forces, one part of the gas has a higher than average pressure and another part has a lower than average pressure. If we put the packets of gas into rigid shells inside the container, we can create such a situation; but in the absence of such constraints we know that we must answer the question in the negative. Now we want to prove and verify that answer.

    The principle of least action gives us the proof we need. Between two instants, t1 and t2, the energy E contained by the gas enacts S=E(t2-t1) as its action of existence in that interval. That calculation represents the least action that the system enacts (because the energy canít diminish, due to the constraints (i.e. the container) on the system) between the two chosen instants, regardless of how the energy is distributed throughout the gas. But look at how the action itself is distributed within the gas.

    Just as we identify the pressure at a point with the density of the gasís energy at that point, so we can describe the action density enacted by the gas around that point as the product σ=p(t2-t1). In a minuscule volume dV around that point the system enacts a minuscule action dS=σdV=p(t2-t1)dV that contributes as an additional element to the total action enacted by the whole gas. That element, like all of the other elements, comes under the principle of least action by itself.

    We have no constraints within the gas, so the gas can evolve in a way that enables all of its elements to come into conformity with the principle of least action. Thus, each element in the gas acts to diminish its pressure; however, the average pressure within the gas cannot change because when we integrate it over the volume of the container we get the total energy of the gas, which does not change. We infer, then, that the system will evolve by gas flowing out of regions of higher-than-average pressure and flowing into regions of lower-than-average pressure until the pressure in the gas has equalized throughout the entire volume of the container. The principle of least action density leads us to state that, absent constraints, a gas will evolve by flowing from regions of high pressure to regions of low pressure and that a gas will not, of itself, flow from low pressure to high pressure. Classical physics confirms that rule by asserting that pressure exerts a force that acts to move the gas as described in the rule.

    Look again at the solid body, the short segment of steel rail, to one part of which we have added heat and from another part of which we have withdrawn heat. We assume that the body has such a fundamental structure that it can, under the right conditions, redistribute heat from one minuscule part to a neighboring one. Heat denotes scrambled kinetic energy manifested within the structure of the body, so it comes under the principle of least action density. That fact entails, as we saw with pressure, the arousal of forces that will redistribute heat from hotter regions (regions of high energy density) to cooler regions (regions of lower energy density) until the heat density achieves uniformity throughout the body. That analysis explains why the heat that we put into a steel rail seemed to spread out.

    The analysis also entails the statement that heat will not spontaneously get redistributed from regions of lower energy density to regions of higher energy density. And we restate that rule as Rudolf Clausiusís version of the second law of thermodynamics: heat will not, of itself, go from a colder body to a warmer body.


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