Ludwig Boltzmann

Ludwig Boltzmann (1902)

Ludwig Eduard Boltzmann (February 20, 1844 – September 5, 1906) was an Austrian physicist and philosopher famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics. He was one of the most important advocates for atomic theory which was still highly controversial.

Quotes

• In my view all salvation for philosophy may be expected to come from Darwin's theory.
  • "Theoretical Physics and Philosophical Problems, Selected Writings", Ludwig Boltzmann, ed. B. McGuinness, 1974, p. 193
• · ... Philosophy gets on my nerves. If we analyse the ultimate ground of everything, then everything finally falls into nothingness. But I have decided to resume my lectures again and look the Hydra of doubt straight in the eye, and it can be quite ominous [verhängnisvoll] if one values one's life. The title [Principles of Natural Philosophy] doesn't tell us anything coherent .... [it is] essentially a joke . . . . I must take care that the lecture is adequate. great difficulties; one doesn't really know what natural philosophy is ...
  • from Boltzmann's notes for his lecture on natural philosophy given on October 24, 1904, as quoted in Blackmore, J. T., ed. Ludwig Boltzmann His Later Life and Philosophy, 1900–1906: Book One: A Documentary History. Vol. 168. Springer Science & Business Media, 1995. p.136.
  • Itself quoting from Ilse M. Fasol-Boltzmann (ed.), Ludwig Boltzmann 'Principien der Naturfilosofi' - Lectures on Natural Philosophy 1903-1906, Springer-Verlag: Berlin etc., 1990. p. 107.
• Let us now tum to the second matter in dispute between us [about the rejection of philosophy in virtually all senses]. That the majority of students don't understand philosophy doesn't bother me. But can any two people understand philosophical [i.e. metaphysical] questions [and agree upon what it is that they understand]? Is there any sense at all in breaking one's head over such questions? Shouldn't the irresistible pressure [Drang] to philosophize be compared with the nausea caused by migraine headaches? As if something could still struggle its strangled way out, even though nothing is actually there at all? My opinion about the high, majestic task of philosophy is to make things clear, in order to finally heal mankind from these terrible migraine headaches. Now, I am one who hopes not to make you angry by my forthrightness, but the first duty of philosophy as love of wisdom is complete frankness. Through my study of Schopenhauer, I am learning Greek ways of thinking again, but piecemeal.
  • letter to Franz Brentano (Vienna, January 4th, 1905), as quoted in Blackmore, J. T., ed. Ludwig Boltzmann His Later Life and Philosophy, 1900–1906: Book One: A Documentary History. Vol. 168. Springer Science & Business Media, 1995. p.125.
• I take my stand before you as a reactionary, a survivor, who is still an enthusiast for the old and the classical as opposed to the modem.
  • a lecture at Munich in 1899. L. Boltzmann, Populare Schriften, Leipzig 1905, p. 205
• Even as a musician can recognize his Mozart, Beethoven, or Schubert after hearing the first few bars, so can a mathematician recognize his Cauchy, Gauss, Jacobi, Helmholtz, or Kirchhoff after the first few pages. The French writers reveal themselves by their extreme formal elegance, while the English, especially Maxwell, by their dramatic sense. Who, for example, is not familiar with Maxwell's memoirs on his dynamic theory of gases? ... The variations of the velocities are, at first, developed majestically; then from one side enter the equations of state; and from the other side, the equations of motion in a central field. Ever higher soars the chaos of formulae. Suddenly, we hear, as from kettle drums, the four beats "put n = 5". The evil spirit V (the relative velocity of the two molecules) vanishes; and, even as in music, a hitherto dominating figure in the bass is suddenly silenced, that which had seemed insuperable has been overcome as if by a stroke of magic. This is not the time to ask why this or that substitution. If you are not swept along with the development, lay aside the paper. Maxwell does not write programme music with explanatory notes. ... One result after another follows in quick succession till at last, as the unexpected climax, we arrive at the conditions for thermal equilibrium together with the expressions for the transport coefficients. The curtain then falls!
  • Quoted in Chandrasekhar, Subrahmanyan. Beauty and the quest for beauty in science. Fermi National Accelerator Laboratory, 1992. The mention of "n = 5" concerns the inverse-fifth power law of force that Maxwell used, as in James Maxwell (1867). "IV. On the dynamical theory of gases". Philosophical Transactions of the Royal Society of London 157: 49–88. ISSN 0261-0523. DOI:10.1098/rstl.1867.0004.

On Certain Questions of the Theory of Gases (February 28, 1895)

Nature Letters to the Editor Vol. 51, No. 1322, Nov, 1894-Apr, 1895, pp. 413-415.

• I propose to answer two questions:—
  (1) Is the Theory of Gases a true physical theory as valuable as any other physical theory?
  (2) What can we demand from any physical theory?

• The first question I answer in the affirmative, but the second belongs not so much to ordinary physics (let us call it orthophysics) as to... metaphysics.

• For a long time the celebrated theory of Boscovich was the ideal of physicists. According to his theory, bodies as well as the ether are aggregates of material points, acting together with forces, which are simple functions of their distances.
  • Ref: Joseph Boscovich, A Theory of Natural Philosophy (1922) from the 1763 text, Venetian edition.

• If this theory were to hold good for all phenomena, we should be still a long way off what Faust's famous famulus [Wagner] hoped to attain, viz., to know everything. But the difficulty of enumerating all the material points of the universe, and of determining the law of mutual force for each pair, would be only a quantitative one; nature would be a difficult problem, but not a mystery for the human mind.

• When Lord Salisbury says that nature is a mystery, he means... that this simple conception of Boscovich is refuted almost in every branch of science, the Theory of Gases not excepted. The assumption that the gas molecules are aggregates of material points, in the sense of Boscovich, does not agree with the facts. But what else are they? And what is the ether through which they move? Let us again hear Lord Salisbury. He says

  "What the atom of each element is, whether it is a movement, or a thing, or a vortex, or a point having inertia, all these questions are surrounded by profound darkness. I dare not use any less pedantic word than entity to designate the ether, for it would be a great exaggeration of our knowledge if I were to speak of it as a body, or even as a substance."

  • Ref: Lord Salisbury, Presidential Address to the British Association meeting at Oxford (Aug, 1894) "Address by the Most Hon. The Marquis of Salisbury, K.G., D.C.L., F.R.S., Chancellor of the University of Oxford, President", Report of the Sixty-fourth Meeting of the British Association for the Advancement of Science held at Oxford (August, 1894) p. 8.

• If this be so—and hardly any physicist will contradict this—then neither the Theory of Gases nor any other physical theory can be quite a congruent account of facts, and I cannot hope with Mr. Burbury, that Mr. Bryan will be able to deduce all the phenomena of spectroscopy from the electromagnetic theory of light. Certainly, therefore, Hertz is right when he says: "The rigour of science requires, that we distinguish well the undraped figure of nature itself from the gay-coloured vesture with which we clothe it at our pleasure." But I think the predilection for nudity would be carried too far if we were to forego every hypothesis. Only we must not demand too much from hypotheses.

Ref: Heinrich Hertz, Untersuchungen über die Ausbreitung der elektrischen Kraft (1892) p. 31: "Aber die Strenge der Wissenschaft erfordert doch, dass wir dies bunte Gewand, welches wir der Theorie überwerfen und dessen Schnitt und Farbe vollständig in unserer Gewalt liegt, wohl unterscheiden von der einfachen und schlichten Gestalt selbst, welche die Natur uns entgegenführt und an deren Formen wir aus unserer Willkür nichts zu ändern vermögen."

• It is curious to see that in Germany, where till lately the theory of action at a distance was much more cultivated than in Newton’s native land itself, where Maxwell’s theory of electricity was not accepted, because it does not start from quite a precise hypothesis, at present every special theory is old-fashioned, while in England interest in the Theory of Gases is still active; vide, ...the excellent papers of Mr. Tait, of whose ingenious results I cannot speak too highly, though I have been forced to oppose them in certain points.

• Every hypothesis must derive indubitable results from mechanically well-defined assumptions by mathematically correct methods. If the results agree with a large series of facts, we must be content, even if the true nature of facts is not revealed in every respect. No one hypothesis has hitherto attained this last end, the Theory of Gases not excepted.

• But this theory agrees in so many respects with the facts, that we can hardly doubt that in gases certain entities, the number and size of which can roughly be determined, fly about pell-mell. Can it be seriously expected that they will behave exactly as aggregates of Newtonian centres of force, or as the rigid bodies of our Mechanics? And how awkward is the human mind in divining the nature of things, when forsaken by the analogy of what we see and touch directly?

• The following assumptions, while not professing to explain the mysteries... nevertheless show that it is possible to explain the spectra of gases while ascribing 5 degrees of freedom to the molecules, and without departing from Boscovich's standpoint.

• Let the molecules of certain gases behave as rigid bodies. The molecules of the gas and of the enclosing vessel move through the ether without loss of energy as rigid bodies, or as Lord Kelvin's vortex rings move through a frictionless liquid in ordinary hydrodynamics. If we were to take a vessel filled with one gram of gas kept during an infinitely long time always at 0° C. and containing always the same portion of ether, every atom of ether and every atom of our gas molecules would reach the same average vis viva. If then we were to raise the temperature to 1° C and to wait till every ponderable and every ether atom was in thermal equilibrium, the total energy would be augmented by what we may call the ideal specific heat.

• But in actually heating one gram of gas, the ether always flows freely through the walls of the vessel. It comes from the universe, and is not at all in thermal equilibrium with the molecules of the gas. It is true that it always carries off energy, if the outside space is colder than the gas; but this energy may be so small as to be quite negligible in comparison with the energy which the gas loses by heat-conduction, and which must be experimentally determined and subtracted in measuring the specific heat. Only certain transverse vibrations of the ether can transfer sensible energy from one ponderable body to another, and therefore a correction for radiant heat must be applied to observations of specific heats.

Heinrich Hertz' first radio transmitter, (1886)

• These transverse vibrations are not produced (as in the older theories of light) by simple atomic vibrations, but their pitch depends on the shape of the hollow space which the molecule forms in the ether, just as Hertzian waves are not caused by vibrations of the ponderable matter of the brass balls, the form of which only determines the pitch.

• The unknown electric action accompanying a chemical process augments these transverse vibrations enormously. The generalised coordinates of the ether, on which these vibrations depend, have not the same vis viva as the coordinates which determine the position of a molecule, because the entire ether has not had time to come into thermal equilibrium with the gas molecules, and has in no respect attained the state which it would have if it were enclosed for an infinitely long time in the same vessel with the molecules of the gas.

• But how can the molecules of a gas behave as rigid bodies? Are they not composed of smaller atoms? Probably they are; but the vis viva of their internal vibrations is transformed into progressive and rotatory motion so slowly, that when a gas is brought to a lower temperature the molecules may retain for days, or even for years, the higher vis viva of their internal vibrations corresponding to the original temperature. This transference of energy... takes place so slowly that it cannot be perceived amid the fluctuations of temperature of the surrounding bodies.

• The possibility of the transference of energy being so gradual cannot be denied, if we also attribute to the ether so little friction that the Earth is not sensibly retarded by moving through it for many hundreds of years.

• If the ether be an external medium which flows freely through the gas, we might find a difficulty in explaining how it is that the source of radiant heat seems to be in the energy of the gas itself. But I still think it possible that the source of energy of the electric vibrations caused by the impact of two gas molecules in the surrounding ether, may be in the progressive and rotatory energy of the molecule. If the electric states of two molecules differ in their motions of approach and separation, the energy of progressive motion may be transformed into electric energy.

• Moreover, it is doubtful whether emission of rays of visible light takes place in simple gases without chemical action. Certainly the light of sodium and that of Gassiot's tubes do not come from gases whose molecules are in thermal equilibrium.

• It may be objected that the above is nothing more than a series of imperfectly proved hypotheses. But granting its improbability, it suffices that this explanation is not impossible. For then I have shown that the problem is not insoluble, and nature will have found a better solution than mine.

• I pointed out in the second part of my paper... that my Minimum Theorem, as well as the so-called Second Law of Thermodynamics, are only theorems of probability. The Second Law can never be proved mathematically by means of the equations of dynamics alone.
  • Ref: Boltzmann, "Bemerkungen über einige Probleme der mechanischen Warmetheorie," Sitz. ber. der k. Wien. Acad. vol. lxxv. 1877

• It can never be proved from the equations of motion alone, that the minimum function H must always decrease. It can only be deduced from the laws of probability, that if the initial state is not specially arranged for a certain purpose, but haphazard governs freely, the probability that H decreases is always greater than that it increases. It is well known that the theory of probability is as exact as any other mathematical theory, if properly understood. If we make 6000 throws with dice, we cannot prove that we shall throw any particular number exactly 1000 times; but we can prove that the ratio of the number of throws in which that number turns up to the whole number of throws, approaches the more to 1/6 the oftener we throw.

• Mr. Culverwell says that my theorem cannot be true because if it were true every atom of the universe would have the same average vis viva, and all energy would be dissipated. I find, on the contrary, that this argument only lends to confirm my theorem, which requires only that in the course of time the universe must tend to a state where the average vis viva of every atom is the same and all energy is dissipated, and that is indeed the case.

• But if we ask why this state is not yet reached, we again come to a "Salisburian mystery."
  I will conclude this paper with an idea of my old assistant, Dr. Schuetz.
  We assume that the whole universe is, and rests for ever, in thermal equilibrium. The probability that one (only one) part of the universe is in a certain state, is the smaller the further this state is from thermal equilibrium; but this probability is greater, the greater is the universe itself. If we assume the universe great enough, we can make the probability of one relatively small part being in any given state (however far from the state of thermal equilibrium), as great as we please. We can also make the probability great that, though the whole universe is in thermal equilibrium, our world is in its present slate. It may be said that the world is so far from thermal equilibrium that we cannot imagine the improbability of such a state. But can we imagine, on the other side, how small a part of the whole universe this world is? Assuming the universe great enough, the probability that such a small part of it as our world should be in its present state, is no longer small.

• If this assumption were correct, our world would return more and more to thermal equilibrium; but because the whole universe is so great, it might be probable that at some future time some other world might deviate as far from thermal equilibrium as our world does at present. Then the aforementioned H-curve would form a representation of what takes place in the universe. The summits of the curve would represent the worlds where visible motion and life exist.

Lectures on Gas Theory (1898)

Published in two volumes as Vorlesungen über Gastheorie. Translated by S.G. Brush, Lectures on Gas Theory, Berkeley: University of California Press, 1964.

• I am conscious of being only an individual struggling weakly against the stream of time. But it still remains in my power to contribute in such a way that, when the theory of gases is again revived, not too much will have to be rediscovered. Thus in this book [this Part] I will now include the parts that are the most difficult and most subject to misunderstanding, and give (at least in outline) the most easily understood exposition of them.
  • Wie ohnmächtig der Einzelne gegen Zeitströmungen bleibt, ist mir bewusst. Um aber doch, was in meinen Kräften steht, dazu beizutragen, dass, wenn man wieder zur Gastheorie zurückgreift, nicht allzuviel noch einmal entdeckt werden muss, nahm ich in das vorliegende Buch nun auch die schwierigsten, dem Missverständnisse am meisten ausgesetzten Theile der Gastheorie auf und versuchte davon wenigstens in den Grundlinien eine möglichst leicht verständliche Darstellung zu geben.
  • Forward
• Who sees the future? Let us have free scope for all directions of research; away with dogmatism, either atomistic or anti-atomistic!
  • p. 26

Translation of Ludwig Boltzmann’s Paper (2015)

"On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium" Sitzungberichte der Kaiserlichen Akademie der Wissenschaften. Mathematisch-Naturwissen Classe. Abt. II, LXXVI 1877, pp 373-435 (Wien. Ber. 1877, 76:373-435). Reprinted in Wiss. Abhandlungen, Vol. II, reprint 42, p. 164-223, Barth, Leipzig, (1909) Tr. Kim Sharp, Franz Matschinsky, Entropy 2015, 17(4), 1971-2009 An MDPI Open Access article. MDPI Open Access Information and Policy.

• It is clear that every single uniform state distribution which establishes itself after a certain time given a defined initial state is equally as probable as every single nonuniform state distribution, comparable to the situation in the game of Lotto where every single quintet is as improbable as the quintet 12345. The higher probability that the state distribution becomes uniform with time arises only because there are far more uniform than nonuniform state distributions... It is even possible to calculate the probabilities from the relationships of the number of different state distributions. This approach would perhaps lead to an interesting method for the calculation of the equilibrium of heat.
  • from "Remarks about several problems of the mechanical theory of heat"

• [I]t is possible to calculate the state of the equilibrium of heat by finding the probability of the different possible states of the system.

• The initial state in most cases is bound to be highly improbable and from it the system will always rapidly approach a more probable state until it finally reaches the most probable state, i.e., that of the heat equilibrium.

• [W]e will be able to identify that quantity which is usually called entropy with the probability of the particular state.

• According to the second fundamental theorem... change has to take place in such a way that the total entropy of the particles increases. This means according to our present interpretation that nothing changes except that the probability of the overall state for all particles will get larger and larger.

• The system of particles always changes from an improbable to a probable state.

• [I]t is our main purpose not to limit our discussion to thermal equilibrium, but to explore the relationship of this probabilistic formulation to the second theorem of the mechanical theory of heat.

• We want first to solve the problem... namely to calculate the probability of state distributions from the number of different distributions. We want first to treat as simple a case as possible, namely a gas of rigid absolutely elastic spherical molecules trapped in a container with absolutely elastic walls. Even in this case, the application of probability theory is not easy. The number of molecules is not infinite... yet the number of velocities each molecule is capable of is effectively infinite... to facilitate understanding, I will... consider a limiting case.

• We assume initially, each molecule is only capable of assuming a finite number of velocities...$0,{\frac {1}{q}},{\frac {2}{q}},{\frac {3}{q}},...{\frac {p}{q}}$where $p$ and $q$ are arbitrary finite numbers. ...but after the collision both molecules still have one of the above velocities ...the actual problem to be solved is re-established by letting p and q go to infinity.

• [W]e will consider the kinetic energy, rather than the velocity of the molecules. Each molecule can have only a finite number of values for its kinetic energy. As a further simplification, we assume that the kinetic energies of each molecule form an arithmetic progression...$0,\epsilon ,2\epsilon ,2\epsilon ,...p\epsilon$We call $P$ the largest possible value of the kinetic energy, $p\epsilon$. ...after the collision, each molecule still has one of the above values of kinetic energy.

Attributed

• Eleganz sei die Sache der Schuster und Schneider
  • Elegance should be left to shoemakers and tailors
    • reported by Arnold Berliner, Curt Thesing (1946). Die Naturwissenschaften. Springer-Verlag. p. 36. 
    • also reported by Albert Einstein, translation by Robert W. Lawson (1921). Relativity. Plain Label Books. p. preface. ISBN 1-603-03164-2. 

• O! immodest mortal! Your destiny is the joy of watching the evershifting battle!
  • S. Rajasekar, N.Athavan, "Ludwig Edward Boltzmann" (7 September 2006), arXiv:physics/0609047v1 [physics.hist-ph]
• Available energy is the main object at stake in the struggle for existence and the evolution of the world.
  • S. Rajasekar, N.Athavan, "Ludwig Edward Boltzmann"
• Which is more remarkable fact about America: that millionaires are idealists or idealists become millionaires.
  • S. Rajasekar, N.Athavan, "Ludwig Edward Boltzmann"
• Bring forward what is true, Write it so that it is clear, Defend it to your last breath!
  • S. Rajasekar, N.Athavan, "Ludwig Edward Boltzmann"
• I am conscious of being only an individual struggling weakly against the stream of time. But it still remains in my power to contribute in such a way that, when the theory of gases is again revived, not too much will have to be rediscovered.
  • S. Rajasekar, N.Athavan, "Ludwig Edward Boltzmann"
• The most ordinary things are to philosophy a source of insoluble puzzles. With infinite ingenuity it constructs a concept of space or time and then finds it absolutely impossible that there be objects in this space or that processes occur during this time.... the source of this kind of logic lies in excessive confidence in the so-called laws of thought.
  • S. Rajasekar, N.Athavan, "Ludwig Edward Boltzmann"

Quotes about Boltzmann

• I adhered scrupulously to the precept of that brilliant theoretical physicist L. Boltzmann, according to whom matters of elegance ought to be left to the tailor and to the cobbler.
  • Albert Einstein, Preface to Relativity: The Special and the General Theory (1916)
• Boltzmann’s Lectures on Gas Theory is an acknowledged masterpiece of theoretical physics... still... [of] considerable scientific value today. It contains a comprehensive exposition of the kinetic theory of gases by a scientist who devoted a large pail of his own career to it, and brought it very nearly to completion as a fundamental part of modern physics. ...Ludwig Boltzmann ...played a leading role in the nineteenth-century movement toward reducing the phenomena of heat, light, electricity, and magnetism to "matter and motion"—in other words, to atomic models based on Newtonian mechanics. His own greatest contribution was to show how... mechanics... previously ...regarded as deterministic and reversible in time, could be used to describe irreversible phenomena in the real world on a statistical basis. His original papers on the statistical interpretation of thermodynamics, the H-theorem, transport theory, thermal equilibrium, the equation of state of gases... occupy about 2,000 pages... [N]ot even the handful of experts on kinetic theory could claim to have read everything he wrote. ...Boltzmann decided to publish his lectures, in which the most important parts of the theory, including his ...contributions, were carefully explained. ...[H]e included his mature reflections and speculations on such questions as the nature of irreversibility and the justification for using statistical methods in physics. His Vorlesungen über Gastheorie was... the standard reference... for advanced researchers, ...[and] a popular textbook ...for the first quarter of the [20th] century ...
  • Stephen G. Brush, Translator's Introduction to Lectures on Gas Theory (1964) a translation of Ludwig Boltzmann's Vorlesungen iiber Gastheorie (1896, 1898)

• The influence of Quetelet's ideas spread throughout the sciences, even to the physical sciences. The two primary founders of the modern kinetic theory of gases, based on considerations of probability, were James Clerk Maxwell and Ludwig Boltzmann. Both acknowledged their debt to Quetelet. ...historians generally consider the influence of the natural sciences on the social sciences, whereas in the case of Maxwell and Boltzmann, there is an influence of the social sciences on the natural sciences, as Theodore Porter has shown.
  • I. Bernard Cohen, The Triumph of Numbers: How Counting Shaped Modern Life (2005)

• Boltzmann summarized most (but not all) of his work in a two volume treatise Vorlesungen Uber Gastheorie [1896, 1898]. This is one of the greatest books in the history of exact sciences and the reader is strongly advised to consult it. It is tough going but the rewards are great.
  • Mark Kac, Probability and Related Topics in Physical Sciences (1959) p. 261.

• In 1902 he returned to his chair in theoretical physics... In addition to... mathematical physics, Boltzmann was given Mach's philosophy course to teach. His philosophy lectures quickly became famous with the audience... too large for the biggest lecture hall available. ...Boltzmann's fame is based on his invention of statistical mechanics... independently of Willard Gibbs. Their theories connected the properties and behaviour of atoms and molecules with... [those of] the substances of which they were the building blocks. ...Boltzmann obtained the Maxwell–Boltzmann distribution in 1871 ...In 1884 ...Boltzmann ...showed how Josef Stefan's empirical T⁴ law for black body radiation ...could be derived from the principles of thermodynamics. ...[T]he Second Law of Thermodynamics...he derived from the principles of mechanics in the 1890s.
  • J.J. O'Conner, E.F. Robertson, "Ludwig Boltzmann" (Sept, 1998)

• In 1895, at a scientific meeting... Wilhelm Ostwald presented a paper... which... stated... The actual irreversibility of natural phenomena thus proves the existence of processes that cannot be described by mechanical equations, and with this the verdict on scientific materialism is settled.
  Sommerfeld... described the resulting battle between Ostwald and Boltzmann. ...Boltzmann was seconded by Felix Klein. The battle resembled the... bull with the supple fighter. However... the bull was victorious... Boltzmann carried the day. We, the young mathematicians of the time, were all on the side of Boltzmann...
  Boltzmann's ideas ...were opposed by many European scientists... not fully grasping the statistical nature of his reasoning.
  • J.J. O'Conner, E.F. Robertson, "Ludwig Boltzmann" (Sept, 1998)

• Maxwell... during the 1860s... showed that when the [molecular] velocities reached the bell-shaped distribution, no further net change was likely. (...Ludwig Boltzmann further elaborated... and strengthened Maxwell's results). Any specific molecule would speed up or slow down, but... other molecules would change in speed to compensate. When a gas reached that state... the gas was at equilibrium. ...[T]his notion of equilibrium is precisely analogous to the Nash equilibrium in game theory. ...[J]ust as the Nash equilibrium is typically a mixed set of strategies, a gas seeks an equilibrium state with a mixed distribution of molecular velocities.
  • Tom Siegfried, A Beautiful Math (2006) Ch. 7: Quetelet's Statistics and Maxwell's Molecules, pp. 139-140.

• Maxwell, and then Boltzmann, and then... J. Willard Gibbs consequently expended enormous intellectual effort in devising... statistical mechanics, or... statistical physics. The uses... extend far beyond gases... describing electric and magnetic interactions, chemical reactions, phase transitions... and all other manner of exchanges of matter and energy.
  The success... has driven the belief among many physicists that it could be applied with similar success to society. ...[E]verything from the flow of funds in the stock market to the flow of traffic on interstate highways ...
  • Tom Siegfried, A Beautiful Math (2006) Ch. 7: Quetelet's Statistics and Maxwell's Molecules, pp. 142-143.

• Boltzmann... felt that all that we were really doing when we stated physical laws was using a series of linguistic representations of reality. To relate force and mass, as Newton had done in his laws of motion, was to relate labels in such a way that we could use the relations for predictive purposes. To read anything more into the terms force and mass was to presume more than we can know.
  • Brian L. Silver, The Ascent of Science (1998)
• Boltzmann was a martyr to his ideas.
  • Ludwig Flamm, Quoted in E. Broda (1980) Ludwig Boltzmann: Man, Physicist, Philosopher, Ox Bow Press, Woodbridge, translated from German by L. Gay., page 33.

Ludwig Boltzmann: the Man who Trusted Atoms (1998)

by Carlo Cercignani

• Boltzmann stands as a link between... James Clerk Maxwell... and Albert Einstein... Maxwell... first found the formula for the probability distribution of velocities of particles in a gas in equilibrium... but... Boltzmann... derived the equation governing the dynamical evolution of the probability distribution, according to which the state of a gas, not necessarily in equilibrium, will... change. Boltzmann's ideas were central to... Planck's... analysis of black body radiation... in which he introduced the quantum of action, thereby... the quantum revolution. In 1905 Einstein... developed it further (...showing that the "atomic hypothesis" applied even to light ..!) but was also influenced by Boltzmann's concepts in two... papers of 1905, one... a method of determining molecular dimensions and the other... explained... Brownian motion... Both gave enormous support to the "atomic hypothesis"...
  • Foreward

• The Boltzmann equation was... the first... describing the time-evolution of a probability. ...[U]nlike ...equations governing the constituent particles ...[it] does not remain unchanged when ...time is reversed. The time-assymetry... arises as an aspect of the second law of thermodynamics... [i.e.,] the entropy of a system out of equilibrium increases with time. The crude meaning of... "entropy" is "disorder"; so... the order of a system is continually... reduced. ...Boltzmann ...give[s] precision to the... notion... identifying it with a... multiple of the logarithm of the volume in phase space defined by... macroscopic parameters specifying the state of the system. Accordingly... the second law could become amenable to precise mathematical treatment.
  • Foreward