Who is boltzmann

And the explanation is as follows: if these constants were a little different, life could not exist and we could not be here and now, thinking about why physical constants seem to be fine-tuned for the existence of life on Earth.

It turns out that there are slightly more ordered areas in the Universe, but there is no one nearby who could notice them. Then there is a fluctuation, and a well-organized region of the Universe appears, that leads to the fact that intelligent life is born there, which, in turn, looks around and notices that it lives in an almost impossibly organized world. However, not everything is so simple. Ludwig Boltzmann , an Austrian theoretical physicist of the 19th century, who is often called the genius of entropy, suggested that the brain and other complex ordered objects on Earth were formed as a result of random fluctuations.

But then, why do we see billions of other complex and ordered objects around us? Boltzmann suggested that if random fluctuations could create a brain similar to ours, then brains should fly in space or sit alone in one place on uninhabited planets many light years away. This is the Boltzmann brain. Moreover, these brains should be a more commonplace phenomenon than all the crowds of complex ordered objects that we can see on Earth.

Related: Why do photons disappear when you turn off the light in…. So, we have another paradox. If the only condition for consciousness is a brain similar to what you have in your head, how can you be sure that you yourself are not such a Boltzmann brain?

If you experienced a random consciousness, you would rather be alone in the depths of the cosmos, rather than surrounded by similar consciousnesses. Simple answers seem to require some kind of magic.

Perhaps, consciousness does not arise naturally in the brain — like the brain — but requires metaphysical intervention. Or perhaps, we are not random fluctuations in the thermodynamic broth and were placed here by a reasonable creature? Of course, none of the above answers can be called exhaustive. The basic idea is that the process of natural selection contributes to the development of complex ordered objects, and not just allows them to appear by chance.

As soon as a self-replicating molecule appeared on Earth about 3. The program begins with a string of randomly generated nonsense. The next generation is created from the remaining line in the same way. After many generations, the surviving line will be more and more like a quote. In real life, a similar situation occurs. Objects that are more capable of self-replication and less susceptible to destruction have the ability to reproduce themselves, while others are destroyed.

After many, many, many generations, objects became more and more stable and less often destroyed, before they had the opportunity to reproduce. It turns out that intelligence is a very useful property for an object that can survive and self-replicate.

In short, the solution to the Boltzmann paradox is that it is much more difficult to build one brain than to create an Earth filled with these brains. The random fluctuations needed to start the process of natural selection are much simpler and less accurate than those required to create the Boltzmann brain in the depths of space. So, the next time you feel small and insignificant, remember that you are much more complicated than the 4.

It is nothing of the kind. Boltzmann was involved in various disputes. But this is not to say that he was the innocent victim of hostilities. In many cases he took the initiative by launching a polemic attack on his colleagues.

I will focus below on the most important disputes: with Mach and Ostwald on the reality of atoms; and with colleagues who criticized Boltzmann's own work in the form of the famous reversibility objection Loschmidt and the recurrence objection Zermelo. For a wider sketch of how contemporary scientists took position in the debate on the topics of mechanism and irreversibility I refer to van Strien Ostwald and Mach clearly resisted the atomic view of matter although for different reasons.

Boltzmann certainly defended and promoted this view. But he was not the naive realist or unabashed believer in the existence of atoms that the more popular literature has made of him. Instead, he stressed from the s onwards that the atomic view yielded at best an analogy, or a picture or model of reality cf. In his debate with Mach he advocated c, d this approach as a useful or economical way to understand the thermal behavior of gases.

This means that his views were quite compatible with Mach's views on the goal of science. Boltzmann claimed that no approach in natural science that avoids hypotheses completely could ever succeed.

He argued that those who reject the atomic hypothesis in favor of a continuum view of matter were guilty of adopting hypotheses too. Ultimately, the choice between such views should depend on their fruitfulness, and here Boltzmann had no doubt that the atomic hypothesis would be more successful.

Roughly speaking, energetics presented a conception of nature that took energy as the most fundamental physical entity, and thus represented physical processes as transformations of various forms of energy.

It resisted attempts to comprehend energy, or these transformations in terms of mechanical pictures. But this is surely a great exaggeration. It seems closer to the truth to say that energetics represented a rather small but vocal minority in the physics community, that claimed to put forward a seemingly attractive conception of natural science, and being promoted in the mids by reputed scientists, could no longer be dismissed as the work of amateurs cf.

Deltete Boltzmann, who was member of the programme committee, had already shown interest in the development of energetics in private correspondence with Ostwald. Georg Helm was asked to prepare a report, and at Boltzmann's own suggestion, Ostwald also contributed a lecture. Both Helm and Ostwald, apparently, anticipated that they would have the opportunity to discuss their views on energetics in an open-minded atmosphere.

But at the meeting Boltzmann surprised them with devastating criticism. According to those who were present Boltzmann was the clear winner of the debate. Nevertheless, Boltzmann and Ostwald remained friends, and in Ostwald made a great effort to persuade his home university in Leipzig to appoint Boltzmann cf. Blackmore , 61— Loschmidt was Boltzmann's former teacher and later colleague at the University of Vienna, and a life-long friend.

He had no philosophical reservations against the existence of atoms at all. Indeed, he is best known for his estimate of their size. Rather, his main objection was against the prediction by Maxwell and Boltzmann that a gas column in thermal equilibrium in a gravitational field has the same temperature at all heights.

His now famous reversibility objection arose in his attempts to undermine this prediction. Whether Boltzmann succeeded in refuting the objection or not is still a matter of dispute, as we shall see below section 4.

Zermelo's opposition had a quite different background. When he put forward the recurrence objection in , he was an assistant to Planck in Berlin. And like his mentor, he did not favor the mechanical underpinning of thermal phenomena. Yet his paper Zermelo a is by no means hostile. It presents a careful logical argument that leads him to a dilemma: thermodynamics with its Second Law on the one hand and gas theory in the form as Zermelo understood it on the other cannot both be literally true.

By contrast, it is Boltzmann's b reaction to Zermelo, drenched in sarcasm and bitterness which if anything may have led to hostile feelings between these two authors. In any case, the tone of Zermelo's b is considerably sharper. Still, Zermelo maintained a keen, yet critical, interest in gas theory and statistical physics, and subsequently played an important role in making Gibbs' work known in Germany.

In fact, I think that Boltzmann's rather aggressive reactions to Zermelo and Ostwald should be compared to other polemical exchanges in which he was involved, and sometimes initiated himself e.

It seems to me that Boltzmann enjoyed polemics, and the use of sharp language for rhetorical effect. Certainly, the debates with Ostwald and Zermelo might well have contributed to this personal crisis. But it would be wrong to interpret Boltzmann's plaintive moods as evidence that his critics were, in fact, hostile. Even today, commentators on Boltzmann's works are divided in their opinion.

Some praise them as brilliant and exceptionally clear. Often one finds passages suggesting he possessed all the right answers all along the way — or at least in his later writings, while his critics were simply prejudiced, confused or misguided von Plato, Lebowitz, Kac, Bricmont, Goldstein. Others Ehrenfests, Klein, Truesdell have emphasized that Boltzmann's work is not always clear and that he often failed to indicate crucial assumptions or important changes in his position, while friendly critics helped him in clarifying and developing his views.

Fans and critics of Boltzmann's work alike agree that he pioneered much of the approaches currently used in statistical physics, but also that he did not leave behind a unified coherent theory. His scientific papers, collected in Wissenschaftliche Abhandlungen , contain more than papers on statistical physics alone.

Some of these papers are forbiddingly long, full of tedious calculations and lack a clear coherent structure. Sometimes, vital assumptions, or even a complete change of approach, are stated only somewhere tucked away between the calculations, or at the very last page. Even Maxwell, who might have been in the best position to appreciate Boltzmann's work, expressed his difficulty with Boltzmann's longwindedness in a letter to Tait, August ; see Garber, Brush, and Everett , Boltzmann at his best could be witty, passionate and a delight to read.

He excelled in such qualities in much of his popular work and some of his polemical articles. The foundations of statistical physics may today be characterized as a battlefield between a dozen or so different schools, each firmly dug into their own trenches, e.

Still, many of the protagonists of these schools, regardless of their disagreements, frequently express their debt to ideas first formulated to Boltzmann. Even to those who consider the concept of ensembles as the most important tool of statistical physics, and claim Gibbs rather than Boltzmann as their champion, it has been pointed out that Boltzmann introduced ensembles long before Gibbs.

And those who advocate Boltzmann while rejecting ergodic theory, may similarly be reminded that the latter theory too originated with Boltzmann himself. It appears, therefore, that Boltzmann is the father of many approaches, even if these approaches are presently seen as conflicting with each other. This is due to the fact that during his forty years of work on the subject, Boltzmann pursued many lines of thought. Typically, he would follow a particular train of thought that he regarded promising and fruitful, only to discard it in the next paper for another one, and then pick it up again years later.

This meandering approach is of course not unusual among theoretical physicists but it makes it hard to pin down Boltzmann on a particular set of rock-bottom assumptions, that would reveal his true colors in the modern debate on the foundations of statistical physics. The Ehrenfests in their famous Encyclopedia article, set themselves the task of constructing a more or less coherent framework out of Boltzmann's legacy.

But their presentation of Boltzmann was, as is rather well known, not historically adequate. Without going into a more detailed description of the landscape of the battlefield of the foundations of statistical physics, or a sketch of the various positions occupied, it might be useful to mention only the roughest of distinctions. The first theory aims to explain the properties of gases by assuming that they consist of a very large number of molecules in rapid motion.

During the s probability considerations were imported into this theory. The aim then became to characterize the properties of gases, in particular in thermal equilibrium, in terms of probabilities of various molecular states. Here, molecular states, in particular their velocities, are regarded as stochastic variables, and probabilities are attached to such molecular states of motion.

These probabilities themselves are conceived of as mechanical properties of the state of the total gas system. Either they represent the relative number of molecules with a particular state, or the relative time during which a molecule has that state. In this latter approach, probabilities are not attached to the state of a molecule but of the entire gas system.

Thus, the state of the gas, instead of determining the probability distribution, now itself becomes a stochastic variable. A merit of this latter approach is that interactions between molecules can be taken into account. Indeed, the approach is not restricted to gases, but also applicable to liquids or solids.

The price to be paid, however, is that the probabilities themselves become more abstract. Since probabilities are attributed to the mechanical states of the total system, they are no longer determined by such mechanical states. It is not easy to pinpoint this transition in the course of history, except to say that in Maxwell's work in the s definitely belong to the first category, and Gibbs' book of to the second.

Boltzmann's own works fall somewhere in the middle. His earlier contributions clearly belong to the kinetic theory of gases although his paper already applies probability to an entire gas system , while his work of is usually seen as belonging to statistical mechanics.

However, Boltzmann himself never indicated a clear distinction between these two different theories, and any attempt to draw a demarcation at an exact location in his work seems somewhat arbitrary.

From a conceptual point of view, the transition from kinetic gas theory to statistical mechanics poses two main foundational questions. On what grounds do we choose a particular ensemble, or the probability distribution characterizing the system? Gibbs did not enter into a systematic discussion of this problem, but only discussed special cases of equilibrium ensembles i. A second problem is to relate the ensemble-based probabilities with the probabilities obtained in the earlier kinetic approach for a single gas model.

The Ehrenfests paper was the first to recognize these questions, and to provide a partial answer: Assuming a certain hypothesis of Boltzmann's, which they dubbed the ergodic hypothesis , they pointed out that for an isolated system the micro-canonical distribution is the unique stationary probability distribution.

Hence, if one demands that an ensemble of isolated systems describing thermal equilibrium must be represented by a stationary distribution, the only choice for this purpose is the micro-canonical one.

Similarly, they pointed out that under the ergodic hypothesis infinite time averages and ensemble averages were identical. This, then, would provide a desired link between the probabilities of the older kinetic gas theory and those of statistical mechanics, at least in equilibrium and in the infinite time limit.

Yet the Ehrenfests simultaneously expressed strong doubts about the validity of the ergodic hypothesis. These doubts were soon substantiated when in Rozenthal and Plancherel proved that the hypothesis was untenable for realistic gas models. The Ehrenfests' reconstruction of Boltzmann's work thus gave a prominent role to the ergodic hypothesis, suggesting that it played a fundamental and lasting role in his thinking. Although this view indeed produces a more coherent view of his multifaceted work, it is certainly not historically correct.

Boltzmann himself also had grave doubts about this hypothesis, and expressly avoided it whenever he could, in particular in his two great papers of and b. Since the Ehrenfests, many other authors have presented accounts of Boltzmann's work. Particularly important are Klein and Brush Still, much confusion remains about what exactly his approach to statistical physics was, and how it developed.

For a more elaborate attempt to sketch the general landscape, and Boltzmann's work in particular,I refer to Uffink Roughly speaking, one may divide Boltzmann's work in four periods.

The period — is more or less his formative period. In his first paper , Boltzmann set himself the problem of deriving the full second law from mechanics. The notion of probability does not appear in this paper. The following papers, from and , were written after Boltzmann had read Maxwell's work of and Following Maxwell's example, they deal with the characterization of a gas in thermal equilibrium, in terms of a probability distribution. Even then, he was set on obtaining more general results, and extended the discussion to cases where the gas is subject to a static external force, and might consist of poly-atomic molecules.

He regularly switched between different conceptions of probability: sometimes this referred to a time average, sometimes a particle average or, in an exceptional paper b , it referred to an ensemble average. In some cases Boltzmann also argued it was the unique such state. However, in this period he also presented a completely different method, which did not rely on the SZA but rather on the ergodic hypothesis. In the same period, he also introduced the concept of ensembles, but this concept would not play a prominent role in his thinking until the s.

The paper contained the Boltzmann equation and the H -theorem. Boltzmann claimed that the H -theorem provided the desired theorem from mechanics corresponding to the second law. However, this claim came under a serious objection due to Loschmidt's criticism of The objection was simply that no purely mechanical theorem could ever produce a time-asymmetrical result.

Boltzmann's response to this objection will be summarized later. The result was, however, that Boltzmann rethought the basis of his approach and in b produced a conceptually very different analysis, which might be called the combinatorial argument , of equilibrium and evolutions towards equilibrium, and the role of probability theory.

The distribution function, which formerly represented the probability distribution, was now conceived of as a stochastic variable nowadays called a macrostate subject to a probability distribution. That probability distribution was now determined by the size of the volume in phase space corresponding to all the microstates giving rise to the same macrostate, essentially given by calculating all permutations of the particles in a given macrostate. Equilibrium was now conceived of as the most probable macrostate instead of a stationary macrostate.

The evolution towards equilibrium could then be reformulated as an evolution from less probable to more probable states. Even though all commentators agree on the importance of these two papers, there is still disagreement about what Boltzmann's claims actually were, and whether he succeeded or indeed even attempted in avoiding the reversibility objection in this new combinatorial argument, whether he intended or succeeded to prove that most evolutions go from less probable to more probable states and whether or not he implicitly relied on the ergodic hypothesis in these works.

I shall comment on these issues in due course. See Uffink for a more detailed overview. The third period is taken up by the papers Boltzmann wrote during the 's have attracted much less attention. During this period, he abandoned the combinatorial argument, and went back to an approach that relied on a combination of the ergodic hypothesis and the use of ensembles.

For a while Boltzmann worked on an application of this approach to Helmholtz's concept of monocyclic systems. However, after finding that concept did not always provide the desired thermodynamical analogies, he abandoned this topic again. Next, in the s the reversibility problem resurfaced again, this time in a debate in the columns of Nature. This time Boltzmann chose an entirely different line of counterargument than in his debate with Loschmidt.

A few years later, Zermelo presented another objection, now called the recurrence objection. The same period also saw the publication of the two volumes of his Lectures on Gas Theory.

In this book, he takes the hypothesis of molecular disorder a close relative of the SZA as the basis of his approach.

The combinatorial argument is only discussed as an aside, and the ergodic hypothesis is not mentioned at all. His last paper is an Encyclopedia article with Nabl presenting a survey of kinetic theory. Boltzmann's first paper in statistical physics aimed to reduce the second law to mechanics. Within the next two years he became acquainted with Maxwell's papers on gas theory of and , which introduced probability notions in the description of the gas.

In particular, he had argued that the state of equilibrium is characterized by the so-called Maxwell distribution function:. In , however he replaced these desiderata with the more natural requirement that the equilibrium distribution should be stationary, i. This called for a more elaborate argument, involving a detailed consideration of the collisions between particles. The crucial assumption in this argument is what is now known as the SZA.

For Maxwell, and Boltzmann later, this assumption seemed almost self-evident. One ought to note, however, that by choosing the initial, rather than the final velocities of the collision, the assumption introduced an explicit time-asymmetric element. This, however, was not noticed until Maxwell showed that, under the SZA, the distribution 1 is indeed stationary. He also argued, but much less convincingly, that it should be the only stationary distribution. In his , Boltzmann set out to apply this argument to a variety of other models including gases in a static external force field.

However, Boltzmann started out with a somewhat different interpretation of probability in mind than Maxwell. But, in the same breath, he identifies this with the relative number of particles with this velocity.

This equivocation between different meanings of probability returned again and again in Boltzmann's writing. Yet apart from this striking difference in interpretation, the first section of the paper is a straightforward continuation of the ideas Maxwell had developed in his In particular, the main ingredient is always played by the SZA, or a version of that assumption suitably modified for the case discussed. But in the last section of the paper he suddenly shifts course.

He now focuses on a general Hamiltonian system, i. Assuming now for simplicity that all points in a given energy hypersurface lie on a single trajectory, the probability should be a constant over the energy hypersurface. In other words, the only stationary probability with fixed total energy is the microcanonical distribution. He then showed that this marginal probability distribution tends to the Maxwell distribution when the number of particles tends to infinity.

First, the difference between the approach relying on the ergodic hypothesis and that relying on the SZA is rather striking. This is the first occasion where probability considerations are applied to the state of the mechanical system as whole, instead of its individual particles. If the transition between kinetic gas theory and statistical mechanics may be identified with this caesura, as argued by the Ehrenfests and by Klein it would seem that the transition has already been made right here in , rather than only in But of course, for Boltzmann the transition did not involve a major conceptual move, thanks to his conception of probability as a relative time.

Thus, the probability of a particular state of the total system is still identified with the fraction of time in which that state is occupied by the system.

In other words, he had no need for ensembles or non-mechanical probabilistic assumptions in this paper. There is no route back to infer that this has anything to do with the relative number of particles with this momentum. Second, and more importantly, these results open up a perspective of great generality. It suggests that the probability of the molecular velocities for an isolated system in a stationary state will always assume the Maxwellian form if the number of particles tends to infinity.

Notably, this argument completely dispenses with any particular assumption about collisions, or other details of the mechanical model involved, apart from the assumption that it is Hamiltonian. Indeed it need not even represent a gas. Third, and most importanty, the main weakness of the present result is its assumption that the trajectory actually visits all points on the energy hypersurface.

This is what the Ehrenfests called the ergodic hypothesis. He notes there that exceptions to his theorem might occur, if the microscopic variables would not, in the course of time, take on all values compatible with the conservation of energy. For example this would be the case when the trajectory is periodic. However, Boltzmann observed, such cases would be immediately destroyed by the slightest disturbance from outside, e. He argued that these exceptions would thus only provide cases of unstable equilibrium.

Still, Boltzmann must have felt unsatisfied with his own argument. According to an editorial footnote in the collection of his scientific papers WA I, 96 , Boltzmann's personal copy of the paper contains a hand-written remark in the margin stating that the point was still dubious and that it had not been proven that, even including interaction with an external atom, the trajectory would traverse all points on the energy hypersurface.

However, his doubts were still not laid to rest. His next paper on gas theory a returns to the study of a detailed mechanical gas model, this time consisting of polyatomic molecules, and explicitly avoids any reliance on the ergodic hypothesis. And when he did return to the ergodic hypothesis in b , it was with much more caution.

Indeed, it is here that he actually first described the worrying assumption as an hypothesis , formulated as follows:. Note that Boltzmann formulates this hypothesis for an arbitrary body, i. There is a major confusion among modern commentators about the role and status of the ergodic hypothesis in Boltzmann's thinking. Indeed, the question has often been raised how Boltzmann could ever have believed that a trajectory traverses all points on the energy hypersurface, since, as the Ehrenfests conjectured in , and was shown almost immediately in by Plancherel and Rozenthal, this is mathematically impossible when the energy hypersurface has a dimension larger than 1.

It is a fact that both [WA I, 96] and b [WA I, ] mention external disturbances as an ingredient in the motivation for the ergodic hypothesis. Yet even though Boltzmann clearly expressed the thought that these disturbances might help to motivate the ergodic hypothesis, he never took the idea very seriously. The marginal note in the paper mentioned above indicated that, even if the system is disturbed, there is still no easy proof of the ergodic hypothesis, and all his further investigations concerning this hypothesis assume a system that is either completely isolated from its environment or at most acted upon by a static external force.

Thus, interventionalism did not play a significant role in his thinking. It has also been suggested, in view of Boltzmann's later habit of discretising continuous variables, that he somehow thought of the energy hypersurface as a discrete manifold containing only finitely many discrete cells Gallavotti In this reading, obviously, the mathematical no-go theorems of Rozenthal and Plancherel no longer apply.

Now it is definitely true that Boltzmann developed a preference towards discretizing continuous variables, and would later apply this procedure more and more although usually adding that this was purely for purposes of illustration and more easy understanding. However, there is no evidence in the and b papers that Boltzmann implicitly assumed a discrete structure of mechanical phase space or the energy hypersurface.

Instead, the context of his b makes clear enough how he intended the hypothesis, as has already been argued by Brush See Figure 1 below. See also Cercignani , See Figure Now clearly, in modern language, one should say in the second case that the trajectory lies densely in the surface, but not that it traverses all points. Boltzmann did not possess this language. In fact, he could not have been aware of Cantor's insight that the continuum contains more than a countable infinity of points.

It thus seems eminently plausible, by the fact that this discussion immediately precedes the formulation of the ergodic hypothesis, that the intended reading of the ergodic hypothesis is really what the Ehrenfests dubbed the quasi-ergodic hypothesis , namely, the assumption that the trajectory lies densely i. However, the quasi-ergodic hypothesis does not entail the desired conclusion that the only stationary probability distribution over the energy surface is micro-canonical.

One might then still conjecture that if the system is quasi-ergodic, the only continuous stationary distribution is microcanonical. But even this is fails in general Nemytskii and Stepanov Nevertheless, Boltzmann remained skeptical about the validity of his hypothesis. For this reason, he attempted to explore different routes to his goal of characterizing thermal equilibrium in mechanics. Indeed, both the preceding a and his next paper c present alternative arguments, with the explicit recommendation that they avoid hypotheses.

In fact, he did not return to this hypothesis until the s stimulated by Maxwell's review of the last section of Boltzmann's paper. At that time, perhaps feeling fortified by Maxwell's authority, he would express much more confidence in the ergodic hypothesis see Section 5. So what role did the ergodic hypothesis play? It seems that Boltzmann regarded the ergodic hypothesis as a special dynamical assumption that may or may not be true, depending on the nature of the system, and perhaps also on its initial state.

Note also that the microcanonical distribution immediately implies that the probability of finding the system in any region on the energy hypersurface is proportional to the size of that region as measured by the microcanonical measure. This idea would resurface in his combinatorial argument, although then without the context of characterizing equilibrium thermal equilibrium. The Ehrenfests have suggested that the ergodic hypothesis played a much more fundamental role.

In particular they have pointed out that if the hypothesis is true, averaging over an infinitely long time would be identical to phase averaging with the microcanonical distribution. Thus, they suggested that Boltzmann relied on the ergodic hypothesis in order to equate time averages and phase averages, or in other words, to equate two meaning of probability relative time and relative volume in phase space.

There is however no evidence that Boltzmann ever followed this line of reasoning. He simply never gave any justification for equivocating time and particle averages, or phase averages, at all. Presumably, he thought nothing much depended on this issue and that it was a matter of taste. In Boltzmann published one of his most important papers, its long title often abbreviated as Weitere Studien Further studies.

It was aimed at a something completely new, namely at showing that whatever the initial state of a gas system was, it would always tend to evolve to equilibrium. Thus, this paper is the first work to deal with non-equilibrium theory. The paper contained two celebrated results nowadays known as the Boltzmann equation and the H -theorem. The latter result was the basis of Boltzmann's renewed claim to have obtained a general theorem corresponding to the second law. This paper has been studied and commented upon by numerous authors.

Indeed an integral translation of the text has been provided by Brush Thus, for the present purposes, a succinct summary of the main points might have been sufficient. However, there is still dispute among modern commentators about its actual content. The issue at stake is the question whether the results obtained in this paper are presented as necessary consequences of the mechanical equations of motion, or whether Boltzmann explicitly acknowledged that they would allow for exceptions.

Klein has written. Klein argues that Boltzmann only came to acknowledge the existence of such exceptions thanks to Loschmidt's critique in An opposite opinion is expressed by von Plato Indeed, von Plato states that. The Weitere Studien starts with an appraisal of the role of probability theory in the context of gas theory.

The number of particles in a gas is so enormous, and their movements are so swift that we can observe nothing but average values. The determination of averages is the province of probability calculus. But, Boltzmann says, it would be a mistake to believe that the theory of heat would therefore contain uncertainties. He emphasizes that one should not confuse incompletely proven assertions with rigorously derived theorems of probability theory.

The latter are necessary consequences from their premisses, as in any other theory. They will be confirmed by experience as soon as one has observed a sufficiently large number of cases. This last condition, however, should be no significant problem in the theory of heat because of the enormous number of molecules in macroscopic bodies.

Yet, in this context, one has to make doubly sure that we proceed with the utmost rigor. Thus, the message expressed in the opening pages of this paper seems clear enough: the results Boltzmann is about to derive are advertised as doubly checked and utterly rigorous. Of course, their relationship with experience might be less secure, since any probability statement is only reproduced in observations by sufficiently large numbers of independent data.

Thus, Boltzmann would have allowed for exceptions in the relationship between theory and observation, but not in the relation between premisses and conclusion. He continues by saying what he means by probability, and repeats its equivocation as a fraction of time and the relative number of particles that we have seen earlier in a:.

This equivocation is not vicious however. For most of the paper the intended meaning of probability is always the relative number of molecules with a particular molecular state.

Only at the final stages of his paper WA I, does the time-average interpretation of probability suddenly recur. Boltzmann says that both he and Maxwell had attempted the determination of these probabilities for a gas system but without reaching a complete solution.

Indeed, he announces, he has solved this problem for gases whose molecules consist of an arbitrary number of atoms. His aim is to prove that whatever the initial state in such a system of gas molecules, it must inevitably approach the state characterized by the Maxwell distribution WA I, The next section specializes to the simplest case of monatomic gases and also provides a more complete specification of the problem he aims to solve.

The gas molecules are modelled as hard spheres, contained in a fixed vessel with perfectly elastic walls WA I, There are also a few unstated assumptions that go into the derivation of this equation. The modern procedure to put these requirements in a mathematically precise form is that of taking the so-called Boltzmann-Grad limit. A final ingredient is that all the above assumptions are not only valid at an instant but remain true in the course of time.