## Boltzmann, Ludwig (1844–1906)

## Boltzmann, Ludwig (1844–1906)

# BOLTZMANN, LUDWIG

*(1844–1906)*

Ludwig Boltzmann was born in Vienna, where he received his education. Boltzmann's major contribution to physics and, indirectly, to philosophy, was his profound work in the theory that grounded the phenomenological theory of heat, temperature, and the transformations of internal energy at the macroscopic level—that is to say thermodynamics—in the theoretical description of the underlying mechanical behavior of the basic constituents of a system, such as the molecules of a gas. Boltzmann also contributed directly to the ongoing philosophical discussions about the nature of scientific theories as a member of the group of outstanding physicist-philosophers concerned with such issues in the latter half of the nineteenth century, a group including Pierre Duhem, Ernst Mach, Wilhelm Ostwald, and Heinrich Hertz. During his career he held chairs at Graz, Munich, and Vienna.

After a long career as distinguished researcher and teacher whose influence through popularizing works extended beyond the narrow confines of academic scientists, Boltzmann tragically fell into a terminal depression ending in his suicide.

## Philosophy of Science

It would probably be a mistake to seek for a single, coherent, and fully developed account of the nature of scientific theories in Boltzmann's work. One must extract his views from a large number of short discussions, marginal remarks, and views expressed in correspondence with his colleagues. Nonetheless, certain themes are constant and clear and one can gain some understanding of what Boltzmann was after when one considers the scientific and philosophical context in which his remarks on the nature of theories were made.

Boltzmann's central scientific work posits that a macroscopic piece of matter, such as the volume of gas in a box, is composed of innumerable components—the molecules of the gas—too small to be observed in any direct manner. Following a long development from John Bernoulli, John Herepath, John Waterston, August Krönig, and Rudolf Clausius, and working in parallel with James Clerk Maxwell, Botzmann developed the kinetic theory of heat in which the dynamics of molecules moving more or less independently of one another—except for collisions and short-range interactions with one another and with the walls of a confining box—was used to explain the well-known laws of macroscopic thermodynamics.

It is important to understand just how indirect the evidence was for the genuine existence of molecules at this time. Their existence had been hypothesized in a resurrection of ancient atomic theory by chemists such as John Dalton to explain the combining laws of weight and volume in chemistry. The partial success of kinetic theory also provided indirect evidence of their existence. But the kinds of rich and more direct evidence available now for this particulate view of matter was then nonexistent.

A kind of radical empiricism was popular among the physicist-philosophers with whom Boltzmann associated. Duhem, Mach, and Ostwald shared the view that the aim of science was the production of simple and elegant lawlike regularities among the observable features of matter. They also shared deep skepticism toward any science that hypothesized unobservable entities as real explanatory components of the world. This skepticism included a negative attitude toward any theory positing "unobservable" molecules or atoms. Naturally such a position would be uncongenial to Boltzmann.

Boltzmann sought a view about theories that would legitimate inference to the existence of molecules, but that would not fall prey to empiricist skepticism about any scientific belief that rests upon "mere hypothesis" and that leaps beyond the observable features of the world to the postulation of unobservable entities and properties. Boltzmann's position seems close to that adopted by Hertz.

Theoretical beliefs do, indeed, rest upon hypotheses. New concepts for describing the world arise out of the scientist's imagination and are not all presented to one's direct sensory experience. There is no certainty in theoretical beliefs; they are certainly not derivable by any a priori reasoning, nor can they be established by "induction" from experience. They are hypotheses, guesses, invoked by humans to explain the observable phenomena. Such explanations consist in deductions of the observable phenomena from the hypothesized theory.

Only theories built on such hypotheses and invoking the unobservable will provide truly useful explanations in science. There is no hope of reconstructing science as a set of regularities that range only over the directly observable features of the world. But one must always remember that such hypothesized theories are merely pictures (*Bilder* ) constructed by humans to fit the observable order into a coherent, deductive scheme. And one must always contemplate the possibility that alternative schemes—alternative pictures—may be available. These may present a different picture of the unobservable world, but insofar as they are as empirically adequate as the theories people have adopted, they are equally satisfactory from a scientific point of view.

That the deepest theories rest upon idealization is another reason—in addition to the belief in these theories resting only upon hypothesis—for Boltzmann to retreat from a fully realist position with regard to fundamental physical theories.

Boltzmann's views may perhaps be best understood as a kind of instrumentalism and pragmatism with regard to theories, but with the insistence that physics could not do without such hypothesized theories in its attempts to account for the observable data. Although people must be wary of taking theoretical inferences too realistically, they must not put any of their hopes in a reconstructed physics that eschews the use of concepts and laws invoking the unobservable altogether.

## Theoretical Physics

Boltzmann's great contribution to physics was in kinetic theory and the beginnings of what later was called statistical mechanics. Here his work paralleled that of Maxwell. The two great scientists often came up with similar results independently, but each also found great inspiration in the work of the other.

Maxwell had found a velocity distribution for the molecules of a gas at equilibrium by a curious argument that utilized results from the theory of errors. Boltzmann generalized this distribution to allow for external forces acting on the molecules. In studying the problem of approach to equilibrium, Maxwell derived his so-called "transfer equations." Independently Boltzmann derived his kinetic equation of how the velocity distribution changes with molecular collisions, the famous Boltzmann Equation.

It was easy to show that the Maxwell-Boltzmann equilibrium distribution would be a stationary solution of this equation, hence appropriate for equilibrium that is an unchanging thermodynamic state. To show that this was the only possible such state, Boltzmann invented a quantity "H" as a function of the distribution. He showed that according to his equation this quantity must decrease unless the distribution is the standard equilibrium distribution. Hence the standard distribution is the only one possible for equilibrium.

Boltzmann developed a new method of thinking about the equilibrium as well. Divide a space in which points represent the position and momentum of a single molecule into boxes macroscopically small but in which one expects to find many molecular states. Boltzmann considered all of the ways in which molecules could be permuted among these boxes. He then showed that the combination (number of molecules in specific boxes) corresponding to the largest number of possible ways of permuting the molecules among the boxes (subject to conservation of total energy of the molecules) was that corresponding to the standard equilibrium distribution. One could then think of the numbers of permutations corresponding to a combination as the "probability" of that combination and argue that equilibrium was the overwhelmingly most probable state of the gas. And one could identify thermodynamic entropy as a measure of such probabilities.

Considerations of these results by Maxwell, Boltzmann, and such critics as Samuel Burbury, Edward Culverwell, and later Ernst Zermelo, led Boltzmann to a long process of reinterpretation of his work. Maxwell, considering the possibilities of mechanisms that would molecule-by-molecule subvert the approach to equilibrium (Maxwell's Demon) spoke of the kinetic equation as only describing probable changes in the gas. Considerations of the dynamical reversibility of the system at the molecular level, and of recurrence results for dynamical systems discovered by Henri Poincaré, also forced Boltzmann to modify the initial view of the equation as describing the inevitable behavior of a system.

Reflection revealed that in deriving his equation Boltzmann had used a time-asymmetric hypothesis about the numbers of collisions of molecules of specified kinds that would occur over a given time interval (the *Stosszahlansatz* ). Both Maxwell and Boltzmann began to frequently invoke probabilistic language in their interpretations of their results. What were such "probabilities"? Boltzmann expressed the view that whereas Maxwell thought of them as frequencies with which states would occur in a large collection of similarly prepared systems, he, Boltzmann, thought of them as frequencies with which states would occur over long periods of time for an individual system.

Maxwell and Boltzmann also discovered another approach to calculating equilibrium values, in which these values could be calculated as average values of functions of the microscopic dynamical state of the system in question, where one used (1) a collection of all possible such microscopic states compatible with the macroscopic constraints, and (2) an easily discovered probability distribution over these states, to calculate the mean values. Both Maxwell and Boltzmann introduced dynamical postulates (the Ergodic Hypotheses) to justify this method. The nature of this justification was made much clearer later by the work of Paul and Tatiana Ehrenfest. Although one can show the Ergodic Hypothesis in its Ehrenfest version false, this work led to later, sounder formulations of this approach by means of correct ergodic theorems and important work on the specific dynamics of idealized molecular systems.

Boltzmann, pushed by insightful criticism, realized that invoking probability by itself would not solve all his interpretive problems. His kinetic equation was time asymmetric, but the underlying dynamics was time symmetric. Because for each molecular motion going from nonequilibrium to equilibrium there was one going from equilibrium to nonequilibrium, it was hard to see how one could argue that the equation even characterized "most probable" evolutions of systems. (Although there are current interpretations of the Boltzmann equation that revert to this way of thinking.)

Boltzmann's later interpretation of the whole scheme resorted to cosmological considerations. One thinks of probabilities of states as given by Boltzmann's method. Equilibrium is then the overwhelmingly most probable state. Why is the world in nonequilibrium then? Boltzmann's assistant Dr. Schuetz suggested that maybe the cosmos is in equilibrium overall, but that humans live in a "small" part of it temporarily in a nonequilibrium fluctuational condition. Boltzmann added to this the "anthropic" argument that people must find themselves in such a region because equilibrium regions could not support life-forms. Finally Boltzmann added the argument that what is meant by the "future" direction of time is just the direction of time in which entropy is increasing in this local, nonequilibrium patch of the universe. He draws a deep analogy here with the fact that what people take as "down" is just the local spatial direction of the gravitational force. In equilibrium regions of the cosmos there would be two time directions, but neither could be thought of a "past" or as "future," just as in gravitation-free regions there is no "up" and no "down."

The Ehrenfests later provided a deep interpretation of the kinetic equation and its solutions consonant with this later Boltzmannian interpretation. The solutions to the equation describe neither the inevitable not the most probable behavior of a system, but rather the "concentration curve" that describes the state of most of the systems of a collective of systems started in common nonequilibrium at any later moment of time. But at different times different members of the original collection are making up this majority that is approaching equilibrium.

## Boltzmann's Continued Influence

Boltzmann's methodological thoughts about theories remain provocative and worthy of reflection when one reflect's now on the still problematic status of foundational physical theories. His introduction of probabilistic reasoning into physics was seminal. His work on kinetic theory and statistical mechanics is a rich source of problems for the philosopher of physics interested in probabilistic explanation in physics and in the relationship between phenomenological macroscopic theories and their microscopic, atomistic underpinnings. Boltzmann's invocation of cosmology (still done in current theories of statistical mechanics but within an entirely different cosmological background) also opens up a wide range of important questions for methodologists concerned with how people can construct their fundamental physical explanations. And his views on the "direction of time" remain fundamental for anyone discussing the origin and nature of ideas of the asymmetric nature of past and future.

** See also ** Philosophy of Statistical Mechanics.

## Bibliography

Blackmore, John. *Ludwig Boltzmann His Later Life and Philosophy 1900–1906*. Dordrecht, Netherlands: Kluwer, 1995.

Broda, Englebert. *Ludwig Boltzmann: Man, Physicist, Philosopher*. Woodbridge, CT: Ox Bow Press, 1983.

Brush, Stephen. *The Kind of Motion We Call Heat.* Vol. 2: *Statistical Physics and Irreversible Processes*. Amsterdam: North-Holland, 1976.

Brush, Stephen. *Kinetic Theory.* Vol. 2: *Irreversible Processes*. Oxford: Pergamon Press, 1966.

Brush, Stephen. *Statistical Physics and the Atomic Theory of Matter, from Boyle and Newton to Landau and Onsager*. Princeton, NJ: Princeton University Press, 1983.

Ehrenfest, Paul and Tatiana. *The Conceptual Foundations of the Statistical Approach in Mechanics*. Ithaca, NY: Cornell University Press, 1959.

Sklar, Lawrence. *Physics and Chance: Philosophical Issues in the Foundations of Statistical Mechanics*. Cambridge, U.K.: Cambridge University Press, 1993.

*Lawrence Sklar (2005)*