THEORY REDUCTION: The Case of the Kinetic Theory of Gases
Soshichi Uchii, Kyoto University, Japan
This is the text-only version of: http://www.bun.kyoto-u.ac.jp/~suchii/reduction1.html
(Acknowledgments: This paper was originally written in 1991 while I was a visiting fellow at the Center for Philosophy of Science, University of Pittsburgh. I wish to thank the Center for its generosity. This paper was read in a lunchtime colloquium at the Center, and also in Jim Joyce's seminar at the Philosophy Department of the University of Michigan; I also wish to express my thanks to the Philosophy Department and Prof. A. W. Burks in particular, for arranging that opportunity.)
1. What is Theory Reduction?
2. What is the problem with the Kinetic Theory?
3. A Quick Review of Thermodynamics
4. Mechanics and Probability: Maxwell's Adventure
5. Mechanics and Irreversibility
6. Can we define Probability in the Kinetic Theory?
7. Ergodic Hypothesis
8. Division between Mechanics and Probability
9. Probability Applicable when we are Ignorant
10. The Case for or against Reductionism?
Bibliography / Appendix
1. What is Theory Reduction?
It is often said that the kinetic theory of gases is one of the best examples of the reduction of one theory into another; that is, the classical theory of thermodynamics [or to be more exact, a significant portion of it] is alleged to be reduced to the kinetic theory, which is based on the Newtonian mechanics and the atomistic view of the matter.
But what is the nature of this alleged "reduction"? If you want to know the right answer to this, the best way is to examine the historical development of the kinetic theory. The kinetic theory is a theoretical attempt to explain the nature of gases and heat processes, in general, in terms of the movements of numerous molecules constituting a gas. Its major advocates were James Clerk Maxwell (1831-79) and Ludwig Boltzmann (1844-1906); in the course of their work on the kinetic theory, they had to struggle with several conceptual problems, as well as with many empirical problems, and these conceptual problems have something to do with our question of theory reduction. And you will see that these problems center on the concept of probability.
2. What is the problem with the Kinetic Theory?
I was led to this question of reduction, and other related questions as well, by reading Dr. Tomonaga's last work, What is Physics? (2 vols., Iwanami, 1979). Dr. Shin'ichiro Tomonaga (1906-1979) was a famous physicist, Nobel Prize Winner; he is known for many excellent works in the field of quantum mechanics; but his last work was on historical and philosophical analysis of physics. [By the way, his father was a professor of philosophy at Kyoto Univertsity, a long time ago.] And I was particularly impressed by his analysis of the kinetic theory of gases, not because this part was written on his death-bed, but because this is undoubtedly the best part of his book. In this part, he begins with Maxwell's adventure of introducing statistical method into physics, then explains Boltzmann's attempt to define entropy in terms of mechanical concepts, and finally comes to the crucial question of irreversibility.
Why is the last question crucial? There are two interrelated problems: (1) First, the basis of the kinetic theory is of course Newtonian mechanics, which is a typical example of deterministic theories; then how can we reconcile this deterministic character with the probabilistic or statistical method, which is also essential in the kinetic theory? To put it crudely, within Newtonian mechanics, there is no room for probability, since every motion of any particle is uniquely determined, given its initial condition, by the laws of motion. Then why and where do we need probability, or is it permissible at all that we introduce probability in order to handle the behavior of a bunch of molecules? Dr. Tomonaga's analysis makes it clear how these questions came to be realized, to be deepened, and to be answered.
(2) Second, the laws of motion are all reversible, in the sense that if a particle moves in a certain way then the reverse motion is also possible according to the same laws. But the second law of thermodynamics is not reversible in this sense; it prohibits, for many thermodynamic processes, the reverse change. In technical words, the entropy of a closed system cannot decrease (if you don't know this, I'll explain it in a moment). Then, the question is, why and how can such an irreversible law derivable, in the kinetic theory, from the mechnical laws which are all reversible? Dr. Tomonaga argues that Boltzmann's attempt to answer this irreversibility question were on the right track, and we can reconstruct it as giving satisfactory answers to both of the preceding two questions.
3. A Quick Review of Thermodynamics
Maybe some words of explanation are in order here, for those of you who are unfamiliar with thermodynamics. Thermodynamics, or the theory of heat, was developed by many people, but probably Clausius is most responsible for its systematization. Roughly speaking, Clausius constructed his theory on two basic laws. The first law is that heat and mechanical work are interchangeable and there is a definite quantitative relation between the two [W=JQ, where J is Joule's constant]. The second law is that heat cannot move from cold body into hot body without some other change; in other words, we need to add some work in order to make such a heat transfer possible. Clausius made this second law more precise and gave it a mathematical formulation in terms of entropy: in any process of a closed system, its entropy either stays the same or increases.
As you already know from your experience, this second law makes many heat processes irreversible. For instance, the hot water in a kettle is bound to cool down, and unless we heat, it can never boil again by itself. In terms of entropy, we can describe it in this way: Suppose that the kettle is in a room and they form a closed system (no heat coming in or going out). Then the initial state of the room with the boiling water has a low entropy; a certain amount of heat is condenced in the kettle. As the time goes, the kettle radiates heat around the room, cooling itself but raising the room temperature a little bit; thus the entropy of the whole system increases. And when its temperature becomes the same as that of the room, the heat process ends, reaching an equilibrium state, which is the state with highest entropy. In other words, the distribution of heat in the room becomes uniform.
You can imagine a similar process in terms of a deck of cards, initially separated into two parts, the one all red and the other all black; as you shuffle the deck, reds and blacks are distributed more and more uniformly. Now, anticipating the kinetic theory, suppose a red card corresponds to a molecule with high velocity (thus with a larger kinetic energy), a black card to a molecule with low velocity. Then the initial state of the deck has a low entropy (energy or temperature unevenly distributed), the uniform distribution a highest entropy.
4. Mechanics and Probability: Maxwell's Adventure
Now getting back to Dr. Tomonaga's analysis, let us tackle the first problem; that is, "Is it all right that we apply probabilistic method on the basis of the deterministic theory of Newtonian mechanics?" Or, maybe we can divide this question into two: (a) Is it consistent to combine probabilistic method with the deterministic theory of Newtonian mechanics? (b) Is the use of probability justified by the Newtonian mechanics?
Now, in order to get better understanding of these problems, we need to know more about the kinetic theory of gases. As I said before, Maxwell began a conceptual adventure of introducing probabilistic method into physics ["Illustrations of the Dynamical Theory of Gases," 1860]. The basic idea of the kinetic theory is this: we can attribute pressure in gases to the random impacts of molecules against the walls of the container; likewise we may be able to explain other important properties of gases, or heat processes in general, by referring to the mechanical movements of a vast number of molecules. In order to make this idea feasible, Maxwell introduced the idea of the statistical distribution of velocities in a gas at uniform pressure (which means, in equilibrium). Of course many molecules are moving with various velocities within the container; but instead of considering each molecule individually, we count the number of molecules within a given range of velocity.
Technically, we can do it this way: let the components of molecular velocity in three axes be x, y, z. Then we can take a small interval for each variable, say x+dx, y+dy, and z+dz. And since the number of all molecules in the container is finite, we can count, in principle, how many molecules have velocities witin this small range. In this way, we can register the number of molecules for each velocity range, and if we review all of these numbers, that gives the statistical distribution of velocities within the given gas.
Maxwell found that this velocity distribution is exactly like the distribution of errors when we make many measurements of any physical quantity. Such distributions are called "normal distribution," and you know that they have a beautiful bell-shape, if you draw them in two-dimensional graphs. And Maxwell was able to determine relevant mean values of mechanical quantities, which are necessary for deriving various properties of a gas. But we need not go into details.
What is important here is that we have a strange mixture of mechanics and statistics. Notice that thermodynamic properties are connected with statistical properties of a set of large number of molecules. In order to derive, in the kinetic theory, the well-known formulas for gases (such as Boyle's or Gay-Lussac's law), we start from Newtonian mechanics for molecules; but somewhere in the derivation, we have to introduce such extra-assumptions as that the gas is in equilibrium, or that molecules are distributed uniformly within the container, or that each molecule has an equal probability for going into any direction, etc. Only with such assumptions we can derive the statistical distribution of velocities within the gas. And you can see, by the way, that the derivation of gas properties cannot be called a reduction in the strict sense, if we need such extra-assumptions in addition to Newtonian mechanics.
Notice that, these assumptions are used not as a mathematical means for calculation but as a factual or physical assumption about gas molecules. Thus the question of consistency or justifiablity becomes really important. The kinetic theory is not only an extended application of Newtonian mechanics to gases, but also an attempt to bring in new concepts and assumptions into Newtonian mechanics. How can we be sure that such an attempt is consistent, or justifiable on the basis of Newtonian mechanics?
One way to answer these questions is to investigate the process through which a gas reaches equilibrium. That is, starting from an arbitrary initial state of a gas, we inquire by what process the gas changes its state; presumably, in most cases the gas finally reaches equilibrium or the state with highest entropy. This way was pursued by Maxwell and, more energetically, by Boltzmann.
5. Mechanics and Irreversibility
Boltzmann generalized Maxwell's approach and succeeded in obtaining a formula which determines the changes in the distribution (of velocities as well as positions) in a gas, resulting from collision between molecules and from external forces. The formula is called "Boltzmann's transport equation" (1872). Moreover, he succeeded in defining the mechanical equivalent of the concept of entropy; and he proved the mechanical equivalent of the second law of thermodynamics. In other words, he showed that a gas in any arbitrary initial state will, as a result of collisions, tend to approach to the state of equilibrium, with highest entropy. This is called, by later people, "Boltzmann's H-Theorem."
But, still, there is something strange about this H-Theorem. For, as we have already seen, the mechanical laws are all reversible. Then by what magic could we derive a law which tells us that certain mechanical procecces are irreversible? But since the H-Theorem translates the second law of thermodynamics into mechanical language, it in effect says that certain mechanical processes are irreversible! [Maxwell was already aware of this strangeness about the second law of thermodynamics which can be "proved" within the kinetic theory; because "Maxwell's demon" introduced in his Theory of Heat (1871) is relevant to this problem. The point of this "demon" is that , if such a demon can exist, then the second law does not hold.] Thus the irreversibility of the second law poses a great difficulty to the kinetic theory.
Josef Loschmidt, a colleague of Boltzmann's, raised this question in his 1876 paper. In order to make it clear that the H-Theorem cannot be proved by mechanics alone, he put forward the following sort of consideration: given a gas in a certain state towards equilibrium, suppose the velocities of all molecules are reversed at some instant; then, clearly, the gas goes from a state of high entropy back to a state of low entropy. That is to say, if the moleculer movements towards equilibrium are possible, the reverse movements must be possible as well, according to the same mechanical laws. And this is against the H-Theorem.
Boltzmann quickly responded to this objection (1877). The essence of his replies is as follows: Indeed we cannot derive the H-Theorem from mechanical laws alone, and we must admit that the Theorem cannot be proved for all initial conditions of a gas; we employed probabilistic assumptions also in order to determine the changes in distribution caused by collisions of molecules. Thus whether or not the H-Theorem holds is a matter of probability. What H-Theorem really means is that it is overwhelmingly probable that the gas tends to approach the state of equilibrium, as time goes on, as the result of collisions of a vast number of molecules. The crucial point is that since there are vastly many more uniform than non-uniform distributions, the number of states which lead to uniform distributions is much greater than the number that lead to non-uniform ones.
Thus, according to Boltzmann, the H-Theorem does not contradict the reversibility of mechanical laws, and at the same time, it can explain the irreversibility of the second law of thermo-dynamics in terms of probability.
6. Can we define Probability in the Kinetic Theory?
However, there is still a smell of question-begging in this reply by Boltzmann. For, in order for this reply to work, he has to show, first of all, that his use of probability in deriving the H-Theorem is indeed consistent with, or justifiable on the basis of, the Newtonian mechanics. But one of the key ideas for this was contained in his 1872 paper [actually, the same idea is already in Boltzmann 1868, 1871]. There, he had already suggested that the probability in the kinetic theory can be defined in terms of the time-average of a gas-state: Since a gas may be in a certain (macroscopic) state, we can measure the length of time during which the gas is in that state. And just as we can define the relative frequency of an event in the long run, we can define the time-average in the long run of each (macroscopic) state of the gas. Namely, the time-average of a state is the relative proportion of time (in the long run) in which the gas has that state. Then we can identify this time-average with the probability of that state.
probability of a gas-state = time-average of the state
= relative proportion of time during which the gas was in that state
In order to see the merit of this definition of probability, let us compare this with another proposal by Boltzmann (1877a) in which he explained the notion of probability in terms of the number of micro-states, or what he called "complexions." This may be explained in the following way: Let us distinguish a macro-state of gas from a micro-state of the molecules constituting the gas. The former corresponds to a thermodynamic state (which we can measure), the latter to a mechanical state (which we cannot usually measure); to be more precise, a micro-state (complexion) is a particular way to distribute a given amount of energy among a specified number of molecules. And a macro-state can be represented in the kinetic theory by a distribution function (i.e., a function determinig the statistical distribution of molecules to all possible values of position and velocity). For any given macro-state, there are many micro-states which correspond to it.
If you think this is too abstract, just imagine a deck of cards. A state of the deck in which all red cards are separated from the black cards is an example of a macro-state. In order to describe such a state, we don't have to refer to individual cards; at most the number of cards and their relative position are relevant (e.g., 26 red cards on the top, 26 black cards on the bottom). But if you refer to each individual card, including its order within the whole, and thereby list a particular combination of 52 cards, this is an example of a micro-state. Of course, there are thousands of such combinations corresponding to the preceding macro-state.
Now, for any given macro-state, we can count, theoretically, the number of micro-states corresponding to it. Or more exactly, we can determine its relative proportion to the whole possible micro-states. Then, Boltzmann's proposal is that, assuming that each micro-state is equally probable, we can measure the probability of a macro-state by this relative proportion. In this way, we can define the probability of a macro-state, and moreover we can show that the state of equilibrium is the most probable one; but clearly, since this presupposes the equiprobability of each micro-state, it fails to give the meaning of probability entirely in terms of physical concepts.
The strength of the former definition of probability in terms of time-average is that it has given the meaning of probability all in terms of mechanical concepts. We start from an actual gas or system of molecules; it goes through many changes by collisions etc., and we measure the time each macro-state assumes in the whole process. Thus the probability in this sense can be defined within the framework of Newtonian mechanics, without bringing in any dubious assumptions--- so it seems.
But still, there is a problem. You can define probability in this way, all right. But how can you assure that we can obtain a unique value of probability in this sense? In other words, how can you assure that a unique time-average exists for each macro-state? That was the crucial problem for Boltzmann.
7. Ergodic Hypothesis
Boltzmann tried to resolve the problem by means of "the Ergodic Hypothesis." The intuitive idea of this hypothesis may be informally explained this way. We have already seen that the assumption of equiprobability of all micro-states leads to the conclusion that the state of equilibrium is most probably realized in any gas, thereby supporting the second law of thermodynamics in a probabilistic way. So if we can justify this equiprobability assumption by mechanical means, Boltzmann's problem will disappear. The Ergodic Hypothesis is supposed to do such a job in a very sophisticated manner.
This hypothesis states that a system of molecules will assume, in the long run, all conceivable micro-states that are compatible with the conservation of energy. And if we can assume this hypothesis, then there is a unique time-average for any macro-state; indeed, we can calculate its value within the Newtonian mechanics. To be more specific, remember that (1) the time-average was a mechanical concept, and that (2) Boltzmann defined probability in terms of this mechanical concept. This means that the mechanical concept of time-average has the same mathematical structure as that of probability. Thus on the assumption of Ergodic Hypothesis, we can calculate the value of probability, within the framework of mechanics. And the H-Theorem should now be understood as saying that the time-average of the equilibrium state or those states which are close to the equilibrium is as large as being almost identical to one. [Actually, the concept of ergodicity has raised many difficult problems; but for simplicity we will ignore them. The philosophical import of the Ergodic Hypothesis can be grasped without getting involved with such problems. If you have a difficulty with the preceding formulation, the following version may help: "A system of molecules will go, in the long run, arbitrarily close to every conceivable micro-state."]
8. Division between Mechanics and Probability
However, there are still problems. First of all, obviously, there are certain initial conditions of gas molecules for which the Ergodic Hypothesis does not hold. Then how should we treat such cases? To this, Boltzmann's answer would be that they are really exceptional cases as far as thermodynamics is concerned; we may confine ourselves to the vast majority of cases in which the hypothesis holds. [Actually, there are a number of difficult problems surrounding the Ergodic Hypothesis, but we will ignore them because they do not seem to be relevant in this context.] The second problem is more essential. Since the concept of time-average is mechanical, even if we can define probability in terms of it, the problem of irreversibility still remains. Granting that Boltzmann is justified in assuming the Ergodic hypothesis, how can he explain the irreversibility of the second law? Does he establish irreversibility on the basis of all reversible laws?
Let us recall Boltzmann's answer to Loschmidt's objection. There, the concept of probability has played an essential role for explaining the irreversibility, consistently with the reversibility of mechanical laws. Thus the irreversibility of the second law is explained in terms of the probabilistic asymmetry between high and low entropy. This asymmetry must now be understood in terms of larger or smaller time-average. Now, the question is, is it sufficient for explaining the irreversibility of thermodynamic processes?
It is here that Dr. Tomonaga's perceptive observation comes in. According to his interpretation of what Boltzmann was doing, the probability as time-average is connected with another concept of probability when we make observations or measurements of the thermodynamic processes of a gas.
You will notice the following: By our paraphrasing, we must have replaced probabilistic terms by mechanical terms; but still, Boltzmann's answer to Loschmidt has revived itself in our explanation. And furthermore, we must notice that its revival is closely connected with such an assertion as this: "if we are to observe the distribution at an arbitrary moment, we must expect there is almost no chance that we come across such a distribution as having the time-average very close to zero."
That we have here such words as "expect" or "chance" means that we have brought in the theory of probability again. And, therefore, Boltzmann's assertion against Loschmidt, that "the H-Theorem is established not by mechanical laws alone but by employing the theory of probability," revives itself again. [Tomonaga 1979, vol. 2, 124-125; my translation]
Now, what does this mean? Dr. Tomonaga is saying that, when we speak of "our expectation" or "chance," we are referring to another concept of probability not reducible to mechanical concepts. We may call this probability "epistemic" or "inductive" (these are not Dr. Tomonaga's words, but mine). Then, Dr. Tomonaga is also saying that we can explain the irreversibility of the second law in terms of this epistemic or inductive probability. However, in this new explanation, the probability theory occupies a completely different place from that in Boltzmann's or Maxwell's early theories. Namely, "it occupies a place where it is impossible to collide with the laws of Newtonian mechanics." Because,
when we say that we, at the moment of our measurement, come across such a distribution as having the time-average nearly equal to zero, we are talking about our own situations in which we make observations, not about the behavior of the given set of molecules. [op. cit., 125]
In short, the set of molecules move according to the deterministic laws of mechanics; but when we speak of our chances of finding such and such distributions within the set, this is a statement outside of, and entirely independent from, the Newtonian mechanics. [If you want to assert that this non-mechanical probability may be defined in terms of frequency or other physical concepts, this does not affect Tomonaga's point. Because, in such a definition, you still need some non-mechanical concepts like "randomness" or "stochastic process," which must therefore be brought in from other sources.]
Thus, according to Dr. Tomonaga, the Ergodic Hypothesis has accomplished three things: First, it has enabled us to use only mechanical concepts when we make "probabilistic" calculations within the kinetic theory. Secondly, by means of that, it has enabled us to separate a genuine probabilistic part from the basic mechanical part. And thirdly and finally, it has enabled us to justify the former by means of the latter.
9. Probability Applicable when we are Ignorant
But with respect to the last point, you may raise a question: What is the nature of such a justification? In order to answer this, Dr. Tomonaga suggests the idea of incomplete knowledge. We often speak of chance when our knowledge is not sufficient for precise predictions (this idea is essentially due to Laplace, and endorced by Maxwell); and this applies here. We cannot solve a vast number of mechanical equations; moreover, we do not even know the exact initial conditions which are indispensable for precise predictions. Thus in the kinetic theory, we have to content ourselves with rougher knowledge of distribution functions, which correspond to thermodynamic quantities observable to us humans.
And, only in this sort of context, the concept of "the time-average in the long run," which is originally a mechanical concept but, functionally, exactly like probability, can be related to probability. That is, their relation is this: The probability that the moment we observe a given set of molecules happens to be within the time in which the set has a certain distribution, is equal to the time-average of that distribution in the long run. [op. cit., 127]
Thus, Dr. Tomonaga asserts that both concepts of probability, one mechanical, the other epistemic or inductive, are necessary in order to give a full picture of the second law of thermodynamics. He is suggesting, in other words, that both the mechanical point of view and the observational point of view are necessary; the latter being, within the context of the kinetic theory, nothing but the statistical point of view. Notice that the condition of our ignorance is essential, in order to combine the two concepts of probability by the preceding relation. For if we had sufficient knowledge about the behavior of molecules, we would be able to predict, with a high probability, when we can find a gas-state with a low entropy, i.e. a state with a very small value of time-average. Thus we can deliberately break the preceding relation of probability with time-average.
Getting back to Tomonaga's interpretation, he emphasizes that the grounds of the Ergodic Hypothesis and its use in deriving a unique value of the time-average are contained in the mathematical structure of the Newtonian mechanics. Without this basis, Tomonaga asserts, it would be impossible to bring in probability in between the behavior of a set of molecules and the humans observing their behavior. In this way, the use of non-mechanical probability and the explanation of irreversibility in terms of it are justified on the mechanical basis, provided that we are ignorant of the behavior of the molecules. This is how I understand Dr. Tomonaga's analysis. [And, whether or not Boltzmann's theory is completely satisfactory, Tomonaga's reconstruction is, it seems to me, one of the most sympathetic readings of Boltzmann's purposes.]
Aside from Dr. Tomonaga's argument, the need for the observational or statistical point of view may be understood in another way. The validity of the second law of thermodynamics of course depends (not exclusively, but essentially) on a vast number of observational data. Empirically, we are quite sure that, without some artificial device (like air-conditioner or refrigerator), heat does not flow from a cold body to a hot body. Now, we do not want to throw away this evidential support of second law when we try to reconstruct thermodynamics in the kinetic theory. Then, we have not only to look for a theoretical device for reconstructing the second law, but also to assure ourselves about its connection with the observational data. But such data are almost all of the statistical nature; that is, our device for measurement is such that it can collect nothing but average values or statistical means of a vast number of mechanical movements. Thus, although we have in the kinetic theory introduced the mechanical or microscopic point of view, this point of view must be supplemented by another point of view, as far as we want to be in touch with observational data. In other words, the need for the observational or statistical point of view is nothing but another way to express the indispensability of genuinely probabilistic concept, not reducible to any mechanical concept.
10. The Case for or against Reductionism?
Now, after this long review of the development of kinetic theory, heavily depending on Dr. Tomonaga's analysis, we can come back to our initial problem of reductionism. As I said at the beginning, the kinetic theory is often referred to as one of the best examples of one theory being reduced to another. But after taking a look at what was going on in the kinetic theory, we have to be more careful. First of all, what does "reduction" mean? Second, what is allleged to be reduced to what?
Let us begin with the first question. I understand that, according to the standard view, one theory T1 is reduced to another theory T2 if and only if the following three conditions hold:
(1) The basic concepts of T1 are all definable in terms of concepts of T2.
(2) All of the basic laws of T1 can be translated, by means of such definitions, into laws of T2, which are of course derivable within T2.
(3) The concepts and laws of T2 are more basic, in some sense, than those of T1.
Now, taking this definition of "reduction" for the moment, what is alleged to be reduced to what?---this is our second question. There may be several alternatives, but simplifying the matter, we will consider only two.
(A) Suppose Thermodynamics is alleged to be reduced to the Newtonian mechanics. Then, clearly, this allegation is false, because the concept of (epistemic or inductive) probability is not contained in the Newtonian mechanics. As Dr. Tomonaga has made clear, we need not only the probability concept as time-average (which may be regarded as a mechanical concept) but another concept independent of the mechanics, in order to do justice to the second law of thermodynamics. Thus even the first condition (1) fails.
(B) Suppose, then, Thermodynamics is alleged to be reduced to the full kinetic theory, which assumes the statistical point of view as well as the mechanical point of view, including all the basic concepts associated with them. Then we may admit that the conditions (1) and (2) can be satisfied. But this time, the condition (3) is dubious, to say the least. Take the concept of irreversibility or entropy from the thermodynamics, and the concept of (epistemic or inductive) probability from the kinetic theory. Can we say that the latter is more basic than the former? Or in what sense is it more basic? Honestly, I am not sure; I really do not know. But certainly not "more basic" in the ontological sense, as some atomist wants to say; because probability does not seem to be an ontological concept.
Without getting involved in this sort of messy questions, I think it is more sensible to say that the word "reduction" is inappropriate here. In the course of the development of the kinetic theory, thermodynamics is integrated within a larger framework, eventually giving birth to a new branch of physics called "statistical mechanics." Whether or not its foundations are more basic, more secure, I really do not know. But one thing seems to be clear. The kinetic theory or statistical mechanics has certainly wider applications, richer in its theoretical content. Thus it seems to me that the word "extension" is more appropriate here. That is to say, Thermodynamics is extended, by adding new concepts and new point of view, into a more powerful theory.
If you are not convinced of this, take an analogous case from mathematics. Almost everyone knows that Frege-Russell's logicism failed. They tried to reduce mathematics into logic. But this program did not work well, primarily because they had to introduce the concept of set when they wanted to go into arithmetic or mathematics. The crucial problem for them turned out to be this: how can we define the concept of set which is rich enough to reproduce all of mathematics, but which is, at the same time, basic or simple enough to be called "logical"? All of their heroic attempts failed, and the upshot is that we can indeed reconstruct mathematics, if we add to logic the concept of set and a number of basic assumptions about sets (axiomatic set theory). That is to say, mathematics is not reducible to logic, but it can be reconstructed as an extension of logic.
I claim that the situation is quite similar in the kinetic theory. Thermodynamics can indeed be reconstructed within the kinetic theory or statistical mechanics; but the latter theory is an extension of the Newtonian mechanics and of the original thermodynamics as well. And, of course, the key concept which has enabled this extension is the concept of probability. It plays a somewhat similar role as that of set in Frege-Russell's logicism.
Thus I wish to conclude that the logicism and the kinetic thery are a typical example in which the original or alleged attempt of reduction has failed and, ironically, ended up with an extension of the original system. However, this conclusion does not mean that "reductionism as a methodology" (or a "research program") is useless; on the contrary, that methodology was quite fuitfull in the case of kinetic theory, giving birth to a new branch of physics, i.e. statistical mechanics. Thus, although I think the kinetic theory failed as a literal reductionism, it has certainly succeeded as a fruitful methodology.
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11. Appendix: Irreversibility still Unexplained?
One of the most perplexing problems for Boltzmann's explanation of irreversibility in terms of probability is this: Given a gas with an initial condition with small time-average, we can predict, with high probability, that its entropy will increase in the future; but we can extend the same reasoning to the past, and tell, with high probability, that its entropy must have been high in the distant past.
This point was already noticed by Boltzmann himself (1877).
I will mention here a peculiar consequence of Loschmidt's theorem, namely that when we follow the state of the world into the infinitely distant past, we are actually just as correct in taking it to be very probable that we would reach a state in which all temperature differences have disappeared, as we would be in the following the state of the world into the distant future. [Brush 1966, 193]
And this makes Boltzmann dubious about the applicability of the kinetic theory to the entire universe (ibid.).
Now, how should we analyze the problem? Maybe we should first distinguish (1) ordinary cases of thermodynamic phenomena from (2) the consideration of the entire universe.
Let us recall that the second law applies to an isolated system. And the preceding consideration applies only to such systems as isolated and having a long or infinite history in the past. Now, how does this affect the two cases?
(1) In the ordinary case, the asssumption that a given gas is an isolatd system is either idealization or approximation. Moreover, it is very hard to imagine that the given gas is literally an isolated system and has an infinite history in the past. Indeed, if we know that this gas has a low entropy state, it must be because we have somehow prepared that state only a short time ago. This means, of course, we cannot apply probability consideration to the past. We already know that the gas had an initial state of low entropy; and we also know that it can have only one-way history into the future. Thus, given this setting, Boltzmann's explanation in terms of probability seems to be all right.
(2) But if we consider the entire universe, it seems that both conditions of isolation and long past history are satisfied. However, is it then probable to conclude that the bulk of the histrory was occupied by high entropy states? Not really; because (a) if the universe has the beginning, its history up to now may occupy only a small proportion to its entire history including the future. On the other hand, (b) if the past history is infinite, the conclusion seems unavoidable. However, aside from the argument that this does not harmonize with our best available theory of the universe, it seems to go far beyond our experience of thermodynamic processes. Thus it seems to me that, in this context, we may not be justified in taking the irreversibility for granted as applicable to the entire universe. If we may take seriously Dr. Tomonaga's condition of insufficient knowledge for combining time-average with probability, Boltzmann's doubt should work in either of the following two ways: Either we should doubt the applicability of the kinetic theory to the entire universe with two-way infinite history, or we should doubt our belief in the irreversibility on such a grand scale.
[For some discussions of irreversibility by philosophers, see the following: L. Sklar, "The Elusive Object of Desire: In pursuit of the Kinetic Equations and the Second Law"; John Earman, "The Problem of Irreversibility," both in PSA 1986, Vol. 2, ed. by A. Fine and P. Machamer, Philosophy of Science Association, 1987. And, of course, Sklar's impressive book, Physics and Chance, Cambridge Univ. Press, 1993, appeared, after the present paper was finished.]
March 1, 1999; last modified March 2, 2001. (c) Soshichi Uchiisuchii@bun.kyoto-u.ac.jp