On the Relation of a General Mechanical Theorem to the Second Law of Thermodynamics(Über die Beziehung eines allgemeine mechanischen Satzes zum zweiten Hauptsätze der Wärmetheorie, Sitzungsberichte Akad. Wiss., Vienna, part II, 75, 67-73 (1877); English trans, Stephen Brush, Kinetic Theory, vol.2, p.188)
SUMMARYLoschmidt has pointed out that according to the laws of mechanics, a system of particles interacting with any force law, which has gone through a sequence of states starting from some specified initial conditions, will go through the same sequence in reverse and return to its initial state if one reverses the velocities of all the particles. This fact seems to cast doubt on the possibility of giving a purely mechanical proof of the second law of thermodynamics, which asserts that for any such sequence of states the entropy must always increase. Since the entropy would decrease as the system goes through this sequence in reverse, we see that the fact that entropy actually increases in all physical processes in our own world cannot be deduced solely from the nature of the forces acting between the particles, but must be a consequence of the initial conditions. Nevertheless, we do not have to assume a special type of initial condition in order to give a mechanical proof of the second law, if we are willing to accept a statistical viewpoint. While any individual non-uniform state (corresponding to low entropy) has the same probability as any individual uniform state (corresponding to high entropy), there are many more uniform states than non-uniform states. Consequently, if the initial state is chosen at random, the system is almost certain to evolve into a uniform state, and entropy is almost certain to increase. In his memoir on the state of thermal equilibrium of a system of bodies with regard to gravity, Loschmidt has stated a theorem that casts doubt on the possibility of a purely mechanical proof of the second law. Since it seems to me to be quite ingenious and of great significance for the correct understanding of the second law, yet in the cited memoir it has appeared in a more philosophical garb, so that many physicists will find it rather difficult to understand, I will try to restate it here. If we wish to give a purely mechanical proof that all natural processes take place in such a way that
∫dQ/T ≤ 0then we must assume the body to be an aggregate of material points. We take the force acting between these points to be a function of the relative positions of the points. When this force is known as a function of these relative positions, we shall say that the law of action of the force is known. In order to calculate the actual motion of the points, and therefore the state variations of the body, we must know also the initial positions and initial velocities of all the points. We say that the initial conditions must be given. If one tries to prove the second law mechanically, he always tries to deduce it from the nature of the law of action of the force without reference to the initial conditions, which are unknown. One therefore seeks to prove that -- whatever may be the initial conditions — the state variations of the body will always take place in such a way that
∫dQ/T ≤ 0We now assume that we are given a certain body as an aggregate of certain material points. The initial conditions at time zero shall be such that the body undergoes state variations for which
∫dQ/T ≤ 0We shall show that then, without changing the law of force, other initial conditions can be found for which conversely
∫dQ/T ≥ 0Consider the positions and velocities of all the points after an arbitrary time t1 has elapsed. We now take, in place of the original initial conditions, the following: all the material points1 shall have the same initial positions at time zero that they had after time t1 with the original initial conditions, and the same velocities but in the opposite directions. For brevity we shall call this state the one opposite to that previously found at time t1. It is clear that the points will pass through the same states as before but in the reverse order. The initial state which they had previously had at time zero, will now be reached after time t1 has elapsed. Whereas previously we found
∫dQ/T ≤ 0this quantity is now ≥ 0. The sign of this integral therefore does not depend on the force law but rather only on the initial conditions2. The fact that this integral is actually 0 for all processes in the world in which we live (as experience shows) is not due to the nature of the forces, but rather to the initial conditions. If, at time zero, the state of all material points in the universe were just the opposite of that which actually occurs at a much later time t1, then the course of all events between times t1 and zero would be reversed, so that
∫dQ/T ≥ 0Thus any attempt to prove from the nature of bodies and of the the force law, without taking account of initial conditions, that
∫dQ/T ≤ 0must necessarily be futile. One sees that this conclusion has great seductiveness and that one must call it an interesting sophism. In order to locate the source of the fallacy in this argument, we shall imagine a system of a finite number of material points which does not interact with the rest of the universe. We imagine a large but not infinite number of absolutely elastic spheres, which move in a closed container whose walls are completely rigid and likewise absolutely elastic. No external forces act on our spheres. Suppose that at time zero the distribution of spheres in the container is not uniform; for example, suppose that the density of spheres is greater on the right than on the left, and that the ones in the upper part move faster than those in the lower, and so forth. The sophism now consists in saying that, without reference to the initial conditions, it cannot be proved that the spheres will become uniformly mixed in the course of time. For the initial conditions which we originally assumed, the spheres will be almost always uniform at time t1, for example. We can then choose in place of the original initial conditions the distribution of states which is just the opposite of the one which would occur (in consequence of the original initial conditions) after time t1 has elapsed. Then the spheres would sort themselves out as time progresses, and at time t1 they would acquire a completely nonuniform distribution of states, even though the initial distribution of states was almost uniform. We must make the following remark: a proof, that after a certain time t1 the spheres must necessarily be mixed uniformly, whatever may be the initial distribution of states, cannot be given. This is in fact a consequence of probability theory, for any nonuniform distribution of states, no matter how improbable it may be, is still not absolutely impossible. Indeed it is clear that any individual uniform distribution, which might arise after a certain time from some particular initial state, is just as improbable as an individual non-uniform distribution; just as in the game of Lotto, any individual set of five numbers is as improbable as the set 1, 2, 3, 4, 5. It is only because there are many more uniform distributions than non-uniform ones that the distribution of states will become uniform in the course of time. One therefore cannot prove that, whatever may be the positions and velocities of the spheres at the beginning, the distribution must become uniform after a long time; rather one can only prove that infinitely many more initial states will lead to a uniform one after a definite length of time than to a non-uniform one. Loschmidt's theorem tells us only about initial states which actually lead to a very non-uniform distribution of states after a certain time t1; but it does not prove that there are not infinitely many more initial conditions that will lead to a uniform distribution after the same time. On the contrary, it follows from the theorem itself that, since there are infinitely many more uniform than non-uniform distributions, the number of states which lead to uniform distributions after a certain time t1 is much greater than the number that leads to non-uniform ones, and the latter are the ones that must be chosen, according to Loschmidt, in order to obtain a non-uniform distribution at t1. One could even calculate, from the relative numbers of the different state distributions, their probabilities, which might lead to an interesting method for the calculation of thermal equilibrium. In just the same way one can treat the second law. It is only in some special cases that it can be proved that, when a system goes over from a non-uniform to a uniform distribution of states, then ∫dQ/T will be negative, whereas it is positive in the opposite case. Since there are infinitely many more uniform than non-uniform distributions of states, the latter case is extraordinarily improbable and can be considered impossible for practical purposes; just as it may be considered impossible that if one starts with oxygen and nitrogen mixed in a container, after a month one will find chemically pure oxygen in the lower half and nitrogen in the upper half, although according to probability theory this is merely very improbable but not impossible. Nevertheless Loschmidt's theorem seems to me to be of the greatest importance, since it shows how intimately connected are the second law and probability theory, whereas the first law is independent of it. In all cases where ∫dQ/T can be negative, there is also an individual very improbable initial condition for which it may be positive; and the proof that it is almost always positive can only be carried out by means of probability theory. It seems to me that for closed paths of the atom, ∫dQ/T must always be zero, which can therefore be proved independently of probability theory. For unclosed paths it can also be negative. 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 following the state of the world into the distant future. This would be similar to the following case: if we know that in a gas at a certain time there is a non-uniform distribution of states, and that the gas has been in the same container without external disturbance for a very long time, then we must conclude that much earlier the distribution of states was uniform and that the rare case occurred that it gradually became non-uniform. In other words: any non-uniform distribution evolves into an almost uniform one after a long time t1. The one opposite to this latter one evolves, after the same time t1, into the initial non-uniform one (more precisely, into the opposite of it). The distribution opposite to the initial one would however, if chosen as an initial distribution, likewise evolve into a uniform distribution after time t1. If perhaps this reduction of the second law to the realm of probability makes its application to the entire universe appear dubious, yet the laws of probability theory are confirmed by all experiments carried out in the laboratory.
See also Boltzmann's responses to Zermelo.
For ScholarsBoltzmann and Statistical Physics
From Part I, Introduction, The Kinetic Theory of Gases, 1895 (1964), pp.27-30 (tr. Stephen G. Brush)
Whence comes the ancient view, that the body does not fill space continuously in the mathematical sense, but rather it consists of discrete molecules, unobservable because of their small size. For this view there are philosophical reasons. An actual continuum must consist of an infinite number of parts; but an infinite number is undefinable. Furthermore, in assuming a continuum one must take the partial differential equations for the properties themselves as initially given. However, it is desirable to distinguish the partial differential equations, which can be subjected to empirical tests, from their mechanical foundations (as Hertz emphasized in particular for the theory of electricity). Thus the mechanical foundations of the partial differential equations, when based on the coming and going of smaller particles, with restricted average values, gain greatly in plausibility; and up to now no other mechanical explanation of natural phenomena except atomism has been successful. A real discontinuity of bodies is moreover established by numerous, and moreover quantitatively agreeing, facts. Atomism is especially indispensable for the clarification of the facts of chemistry and crystallography. The mechanical analogy between the facts of any science and the symmetry relations of discrete particles pertains to those most essential features which will outlast all our changing ideas about them, even though the latter may themselves be regarded as established facts. Thus already today the hypothesis that the stars are huge bodies millions of miles away is similarly viewed only as a mechanical analogy for the representation of the action of the sun and the faint visual perceptions arising from the other heavenly bodies, which could also be criticized on the grounds that it replaces the world of our sense perceptions by a world of imaginary objects, and that anyone could just as well replace this imaginary world by another one without changing the observable facts. I hope to prove in the following that the mechanical analogy between the facts on which the second law of thermodynamics is based, and the statistical laws of motion of gas molecules, is also more than a mere superficial resemblance. The question of the utility of atomistic representations is of course completely unaffected by the fact, emphasized by Kirchhoff, that our theories have the same relation to nature as signs to significates, for example as letters to sounds, or notes to tones. It is likewise unaffected by the question of whether it is not more useful to call theories simply descriptions, in order to remind ourselves of their relation to nature. The question is really whether bare differential equations or atomistic ideas will eventually be established as complete descriptions of phenomena. Once one concedes that the appearance of a continuum is more clearly understood by assuming the presence of a large number of adjacent discrete particles, assumed to obey the laws of mechanics, then he is led to the further assumption that heat is a permanent motion of molecules. Then these must be held in their relative positions by forces, whose origin one can imagine if he wishes. But all forces that act on the visible body but not equally on all the molecules must produce motion of the molecules relative to each other, and because of the indestructibility of kinetic energy these motions cannot stop but must continue indefinitely. In fact, experience teaches that as soon as the force acts equally on all parts of a body — as for example in so-called free fall — all the kinetic energy becomes visible. In all other cases, we have a loss of visible kinetic energy, and hence creation of heat. The view offers itself that there is a resulting motion of molecules among themselves, which we cannot see because we do not see individual molecules, but which however is transmitted to our nerves by contact, and thus creates the sensation of heat. It always moves from bodies whose molecules move rapidly to those whose molecules move more slowly, and because of the indestructibility of kinetic energy it behaves like a substance, as long as it is not transformed into visible kinetic energy or work. We do not know the nature of the force that holds the molecules of a solid body in their relative positions, whether it is action at a distance or is transmitted through a medium, and we do not know how it is affected by thermal motion. Since it resists compression as much as it resists dilatation, we can obviously get it rather rough picture by assuming that in a solid body each molecule has a rest position. If it approaches a neighboring molecule it is repelled by it, but if it moves farther away there is an attraction. Consequently, thermal motion first sets a molecule into pendulum-like oscillations in straight or elliptical paths around its rest position A (in the symbolic Fig. 1, the centers of gravity of the molecules are indicated). If it moves to A', the neighboring molecules B and C repel it, while D and E attract it and hence bring it back to its original rest position. If each molecule vibrates around a fixed rest position, the body will have a fixed form; it is in the solid state of aggregation. The only consequence of the thermal motion is that the rest positions of the molecules will be somewhat pushed apart, and the body will expand somewhat. However, when the thermal motion becomes more rapid, one gets to the point where a molecule can squeeze between its two neighbors and move from A to A" (Fig. 1). It will no longer then be pulled back to its old rest position, but it can instead remain where it is. When this happens to many molecules, they will crawl among each other like earthworms, and the body is molten. Although one may find this description rather crude and childish, it may be modified later and the apparent repulsive force may turn out to be a direct consequence of the motion. In any case, one will allow that when the motions of the molecules increase beyond a definite limit, individual molecules on the surface of the body can be torn off and must fly out freely into space; the body evaporates. If it is in an enclosed vessel, then this will be filled with freely moving molecules, and these can occasionally penetrate into the body again; as soon as the number of recondensing molecules is, on the average, equal to the number of evaporating ones, one says that the vessel is saturated with the vapor of the body in question. A sufficiently large enclosed space, in which only such freely moving molecules are found, provides a picture of a gas. If no external forces act on the molecules, these move most of the time like bullets shot from guns in straight lines with constant velocity. Only when a molecule passes very near to another one, or to the wall of the vessel, does it deviate from its rectilinear path. The pressure of the gas is interpreted as the action of these molecules against the wall of the container.From Part II, Chapter VII, The Kinetic Theory of Gases, 1898 (1964), pp.441-449 (tr. Stephen G. Brush)
§87. Characterization of our assumption about the initial state. When a gas is enclosed in a rigid container, and initially one part of it has a visible motion with respect to the rest, then it soon comes to rest as a consequence of viscosity. When two kinds of gas are initially unmixed, but in contact with each other, then they mix, even if the lighter one was originally on top. In general, when a gas or a system of several kinds of gas has initially some improbable state, then it passes to the most probable state under the given external conditions, and remains there during all observable later times. In order to prove that this is a necessary consequence of the kinetic theory of gases, we used the quantity H defined and discussed in this chapter. We proved that it continually decreases as a result of the motion of the gas molecules among each other. The one-sidedness of this process is clearly not based on the equations of motion of the molecules. For these do not change when the time changes its sign. This one-sidedness rather lies uniquely and solely in the initial conditions. This is not to be understood in the sense that for each experiment one must specially assume just certain initial conditions and not the opposite ones which are likewise possible; rather it is sufficient to have a uniform basic assumption about the initial properties of the mechanical picture of the world, from which it, then follows with logical necessity that, when bodies are always interacting, they must always be found in the correct initial conditions. In particular, our theory does not require that each time when bodies are interacting, the initial state of the system they form must be distinguished by a special property (ordered or improbable) which relatively few states of the same mechanical system would have under the external mechanical conditions in question. Hereby the fact is clarified that this system takes in the course of time states which do not have these properties, and which one calls disordered. Since by far most of the states of the system are disordered, one calls the latter the probable states. The ordered initial states are not related to the disordered ones in the way that a definite state is to the opposite state (arising from the mere reversal of the directions of all velocities), but rather the state opposite to each ordered state is again an ordered state. The self-regulating most probable state — which we call the Maxwell velocity distribution since Maxwell first found its mathematical expression in a special case — is not some kind of special singular state which is contrasted to infinitely many more non-Maxwellian distributions. Rather it is, on the contrary, characterized by the fact that by far the largest number of possible states have the characteristic properties of the Maxwell distribution, and compared to this number, the number of possible velocity distributions which significantly deviate from the Maxwellian is vanishingly small. The criterion of equal possibility or equal probability is provided by Liouville's theorem. In order to explain the fact that the calculations based on this assumption correspond to actually observable processes, one must assume that an enormously complicated mechanical system represents a good picture of the world, and that all or at least most of the parts of it surrounding us are initially in a very ordered — therefore very improbable — state. When this is the case, then whenever two or more small parts of it come into interaction with each other, the system formed by these parts is also initially in an ordered state, and when left to itself it rapidly proceeds to the disordered most probable state. §88. On the return of a system to a former state. We make the following remarks: 1. It is by no means the sign of the time which constitutes the characteristic difference between an ordered and a disordered state. If, in the "initial states" of the mechanical picture of the world, one reverses the directions of all velocities, without changing their magnitudes or the positions of the parts of the system; if, as it were, one follows the states of the system backwards in time, then he would likewise first have an improbable state, and then reach ever more probable states. Only in those periods of time during which the system passes from a very improbable initial state to a more probable later state do the states change in the positive time direction differently than in the negative. 2. The transition from an ordered to a disordered state is only extremely improbable. Also, the reverse transition has a definite calculable (though inconceivably small) probability, which approaches zero only in the limiting case when the number of molecules is infinite. The fact that a closed system of a finite number of molecules, when it is initially in an ordered state and then goes over to a disordered state, finally after an inconceivably long time must again return to the ordered state,* is therefore not a refutation but rather indeed a confirmation of our theory. * H. Poincare, Acta Math. 13, 67 (1890); E. Zermelo, Ann. Phys. [31 57, 485 (1896). One should not however imagine that two gases in a 1/10 liter container, initially unmixed, will mix, then again after a few days, separate, then mix again, and so forth. On the contrary, one finds by the same principles which I used* for a * Ann. Phys.  57, 783 (1896). similar calculation that, not until after a time enormously long compared to 101010 years will there be any noticeable unmixing of the gases. One may recognize that this is practically equivalent to never, if one recalls that in this length of time, according to the laws of probability, there will have been many years in which every inhabitant of a large country committed suicide, purely by accident, on the same day, or every building burned down at the same time — yet the insurance companies get along quite well by ignoring the possibility of such events. If a much smaller probability than this is not practically equivalent to impossibility, then no one can be sure that today will be followed by a night and then a day. * Ann. Phys.  57, 783 (1896). We have looked mainly at processes in gases and have calculated the function H for this case. Yet the laws of probability that govern atomic motion in the solid and liquid states are clearly not qualitatively different in this respect from those for gases, so that the calculation of the function H corresponding to the entropy would not be more difficult in principle, although to be sure it would involve greater mathematical difficulties. §89. Relation to the second law of thermodynamics. If therefore we conceive of the world as an enormously large mechanical system composed of an enormously large number of atoms, which starts from a completely ordered initial state, and even at present is still in a substantially ordered state, then we obtain consequences which actually agree with the observed facts; although this conception involves, from a purely theoretical — I might say philosophical — standpoint, certain new aspects which contradict general thermodynamics based on a purely phenomenological viewpoint. General thermodynamics proceeds from the fact that, as far as we can tell from our experience up to now, all natural processes are irreversible. Hence according to the principles of phenomenology, the general thermodynamics of the second law is formulated in such a way that the unconditional irreversibility of all natural processes is asserted as a so-called axiom, just as general physics based on a purely phenomenological standpoint asserts the unconditional divisibility of matter without limit as an axiom. Just as the differential equations of elasticity theory and hydrodynamics based on this latter axiom will always remain the basis of the phenomenological description of a large group of natural phenomena, since they provide the simplest approximate expression of the facts, so likewise will the formulas of general thermodynamics. No one who has fallen in love with the molecular theory will approve of its being given up completely. But the opposite extreme, the dogma of a self-sufficient phenomenology, is also to be avoided. Just as the differential equations represent simply a mathematical method for calculation, whose clear meaning can only be understood by the use of models which employ a large finite number of elements,1 so likewise general thermodynamics (without prejudice to its unshakable importance) also requires the cultivation of mechanical models representing it, in order to deepen our knowledge of nature — not in spite of, but rather precisely because these models do not always cover the same ground as general thermodynamics, but instead offer a glimpse of a new viewpoint. Thus general thermodynamics holds fast to the invariable irreversibility of all natural processes. It assumes a function (the entropy) whose value can only change in one direction — for example, can only increase — through any occurrence in nature. Thus it distinguishes any later state of the world from any earlier state by its larger value of the entropy. The difference of the entropy from its maximum value — which is the goal [Treibende] of all natural processes — will always decrease. In spite of the invariance of the total energy, its transformability will therefore become ever smaller, natural events will become ever more dull and uninteresting, and any return to a previous value of the entropy is excluded.2 1 Boltzmann, Die Unentbehrlichkeit der Atomistik i.d. Naturwissenschaft. Wien. Ber. 105 (2) 907 (1896); Ann. Phys. [31 60, 231 (1897). Ueber die Frage nach der Existent der Vorgange in der unbelebten Natur, Wien. Ber. 106 (2) 83 (1897).