# Ludwig Boltzmann

Ludwig Boltzmann (1902)

Ludwig Eduard Boltzmann (born February 20, 1844 in Vienna , † September 5, 1906 in Duino , Austria-Hungary ) was an Austrian physicist and philosopher . He taught at the universities of Vienna , Graz , Munich and Leipzig . His most important achievements are in the field of thermodynamics and statistical mechanics , where he primarily dealt with the question of how the reversible microscopic movements of particles can lead to irreversible macroscopic processes.

As an ardent advocate of atomistics , he defended the real, objective existence of atoms against the attacks of Ernst Mach and Wilhelm Ostwald . He is considered to be one of the masters of classical physics in the 19th century, who himself no longer had any part in the revolutionary innovations in physics at the beginning of the 20th century, such as relativity and quantum theory , but whose methods were in many ways forward-looking.

Seriously ill and suffering from depression, he committed suicide at the age of 62.

## Life

### ancestry

Boltzmann's paternal grandfather, Gottfried Ludwig Boltzmann, was born in Berlin in 1770 and settled in Vienna, where he founded a music box factory. Gottfried's son Georg Ludwig Boltzmann (* 1802, † June 22, 1859) studied Jus and was taxman. In 1837 he married Maria Katharina Pauernfeind in Maria Plain near Salzburg , who came from a respected and wealthy Salzburg merchant family; the mayor of Salzburg, Johann Christian Paurnfeind, was her great-grandfather. Ludwig Georg and Maria Katharina Boltzmann had three children, Ludwig Eduard and his two younger siblings, Albert (* April 22, 1846, † February 14, 1863) and Hedwig (* May 12, 1848, † 1890). Boltzmann's equally gifted brother Albert was the second best student after him at the Academic Gymnasium in Linz, but died at the age of 16 from a lung disease, probably tuberculosis. His sister Hedwig remained unmarried and died mentally deranged.

### Childhood and youth

As an imperial-royal administrative officer, Boltzmann's father was transferred several times, whereupon the family moved to the respective place of employment: from Salzburg to Vienna, where Boltzmann was born, then to Wels and later to Linz . Boltzmann received private lessons up to the age of 10, and in 1855 he entered the Academic High School in Linz . The majority of the teachers belonged to the clergy, 22 of Boltzmann's 29 classmates stated that they wanted to study theology. Boltzmann's upbringing was shaped by the humanistic ideal of education, music was of great importance in the Boltzmann family. The young Anton Bruckner was his piano teacher.

When Boltzmann was 15 years old, his father died. This event weighed heavily on him. The death of his younger brother four years later also shook him badly. Nevertheless, a few months later, in the summer of 1863, he passed the Matura at the academic grammar school with distinction, and shortly afterwards the family moved to Vienna to enable him to study. His wealthy mother was able to finance his studies from her own fortune.

### Studies and doctorate in Vienna (1863–1869)

Ludwig Boltzmann at the age of 24 (1869)

Boltzmann began studying mathematics and physics at the University of Vienna. The Physikalisches Institut was only founded in 1849 by Christian Doppler . It was located at Erdbergstrasse 15 in Wien-Landstrasse and was headed by Doppler's successor, Andreas von Ettingshausen . In addition to Ettinghausen, Boltzmann's teachers were primarily Josef Stefan , the mathematician Josef Petzval and August Kunzek. Boltzmann was initially an "extraordinary pupil" and from the summer semester of 1865 a "regular pupil" at the Physics Institute. In 1866 Stefan succeeded Ettingshausen as director of the Physics Institute, and in October 1866 Boltzmann became his assistant. Before completing his studies, he published two scientific papers. Boltzmann completed his doctoral studies with the three prescribed rigors and was awarded a doctorate in philosophy on December 19, 1866. A dissertation was not required at the time.

In 1866 Josef Loschmidt joined the Physics Institute. Stefan, who was only nine years older than Boltzmann, and the much older Loschmidt were the formative teachers for Boltzmann, with whom he was also friendly. Boltzmann passed the teaching examination for mathematics and physics at upper secondary schools and completed the mandatory probationary year at the academic high school in the academic year 1867/68 . On December 21, 1867, he submitted a request for the venia docendi , which was granted to him on March 19, 1868. Until July 31, 1869 he taught as a private lecturer and gave a lecture on the basic principles of mechanical heat theory .

### First professorship in Graz (1869–1873)

The chair for mathematical physics at the University of Graz had been vacant since the previous professor Ernst Mach was appointed to the University of Prague in 1867 . The newly appointed full professor for general and experimental physics, August Toepler , advocated a new appointment. Boltzmann's application was supported by Stefan, and on July 17, 1869, Emperor Franz Joseph I appointed Boltzmann full professor of mathematical physics at the University of Graz.

Boltzmann was extremely successful in Graz, but missed the contact with international experts. He also accused his esteemed teachers, Stefan and Loschmidt, of being isolated from the professional world: “As far as I know, neither Stefan nor Loschmidt went on a trip outside of the Austrian fatherland. In any case, they never attended a meeting of naturalists, and never entered into closer personal relationships with foreign scholars. I cannot approve of this; I believe that if they had been less secluded they could have done even more. At least they would have made their achievements known more quickly and therefore made them more fruitful. ”In March 1870 he submitted a vacation request and in April and May 1870 undertook the first of his numerous trips abroad, which took him to Heidelberg to see Robert Bunsen , Gustav Kirchhoff and Leo Koenigsberger . In the winter semester of 1871/72 he visited Hermann von Helmholtz at the Friedrich Wilhelms University in Berlin. In 1872 he published one of his most important papers on statistical mechanics, Further Studies on the Heat Equilibrium Among Gas Molecules .

### Full professor of mathematics at the University of Vienna (1873–1876)

In 1873, Boltzmann applied for the chair of mathematics at the University of Vienna, which had become vacant when Franz Moth retired. He was appointed full professor of mathematics on August 30, 1873. As a professor of mathematics, Boltzmann also dealt with physics, he gave a lecture on mechanical heat theory and also worked experimentally in Vienna and Graz. In 1875 he received an offer from the Swiss Federal Polytechnic School in Zurich for a professorship for life, which he turned down, but which helped him to negotiate a higher salary and better conditions with the ministry. He turned down a call to a chair in Freiburg .

### Second professorship in Graz (1876–1890) and failed appointment to Berlin

Boltzmann with his colleagues in Graz 1887. Standing from left: Walther Nernst , Heinrich Streintz , Svante Arrhenius , Richard Hiecke; sitting from left: Eduard Aulinger, Albert von Ettingshausen , Ludwig Boltzmann, Ignaz Klemenčič , Viktor Hausmanninger

In 1875 the Graz full professor for experimental physics, August Toepler, was appointed to the Royal Saxon Polytechnic in Dresden and Boltzmann was appointed as his successor as full professor and head of the Physics Institute at the University of Graz. In the academic year 1878/79 Boltzmann was dean of the Philosophical Faculty and in the academic year 1887/88 rector.

The Graz years were among the happiest and most productive in Boltzmann's life. Four of his children were born here, and he moved into a property on the Platte in Oberkroisbach in the Graz area (today in the Graz district of Mariatrost ), where he felt very comfortable. Nevertheless, this time was overshadowed by health and psychological problems. After the death of his mother on January 23, 1885, Boltzmann went through a severe psychological crisis. When his eldest son Hugo died in 1889, he made serious reproaches because his appendicitis was diagnosed too late.

At the beginning of 1888, Boltzmann was appointed professor of theoretical physics at the Friedrich Wilhelms University in Berlin. This chair became vacant after Gustav Kirchhoff's death in October 1887. Boltzmann accepted and was appointed full professor for theoretical physics on March 19. On June 24, 1888, he surprisingly canceled, referring to medical reports that attested his poor eyesight. On June 27th he sent a telegram to reverse the rejection, which he did not succeed despite further interventions, as the chair had meanwhile been assigned to Max Planck .

### Professor in Munich (1890-1894)

After the Berlin professorship failed, Boltzmann wanted to leave Graz. He finally decided to accept a professorship at the Ludwig Maximilians University in Munich, where he began teaching in the winter semester of 1890/1891. Boltzmann enjoyed the expanded opportunities for scientific exchange compared to Graz and the discussions with numerous specialist colleagues. In Munich, however, the first arguments with Wilhelm Ostwald and “ energetics ” began. In 1892 he attended the 300th anniversary of Trinity College in Dublin and in 1894 Oxford University .

In 1892/93 he refused a call to Vienna. When, after Josef Stefan's death at the beginning of 1893, his chair became vacant, Boltzmann decided to return to Vienna despite the fruitful academic work in Munich.

### First professorship for theoretical physics at the University of Vienna (1894–1900)

Commemorative plaque on Türkenstrasse 3, which housed the Physics Institute of the University of Vienna from 1875 to 1913, where Boltzmann worked.

On September 1, 1894, Boltzmann began his service. The professorship in Vienna was associated with a considerable increase in his income. One of the reasons for Boltzmann's return to Vienna was the provision for old age and the securing of his family's livelihood in the event of his death. In 1895, Boltzmann attended the meeting of naturalists in Lübeck, where there was an argument with the energetic experts Ostwald and Georg Helm . Arnold Sommerfeld compared the dispute to a bullfight in which Boltzmann was like the bull, but "the bull defeated the torero."

On the occasion of an invitation to Clark University in Worcester, Massachusetts , Boltzmann made the first of his three trips to America in 1899. The outward journey on the Kaiser Wilhelm der Große led from Bremerhaven via Cherbourg and Southampton to New York. Boltzmann traveled on to Worcester via Boston and visited Montreal, Niagara Falls, Buffalo, Pittsburgh, Washington, Baltimore and Philadelphia. After four weeks in the USA, he started his journey home on July 25, 1899 on the Trave of North German Lloyd .

After an initial euphoria, however, Boltzmann soon felt uncomfortable again in Vienna and therefore accepted the offer of a professorship in Leipzig. Despite all the scientific contradictions, Ostwald had campaigned intensively for Boltzmann's appointment. For many, he left Vienna as a surprise and without saying goodbye to his colleagues.

### Professorship in Leipzig (1900–1902)

On September 1, 1900, Boltzmann took up his professorship in Leipzig, where he did not feel comfortable from the start. He suffered from the fear that he might lose his memory during the lecture and therefore even temporarily canceled his lectures. In the summer of 1901 he took his son Arthur on an extensive voyage from Hamburg via Gibraltar into the Mediterranean. However, Boltzmann suffered greatly from the heat, and the hoped-for improvement in his health did not occur. Although the Boltzmann and Ostwald families maintained friendly contact - for example, Boltzmann played the piano at music evenings at Ostwald - the scientific disputes with Ostwald strained him. Because of his nervous disorders and suicidal thoughts, he consulted the psychiatrist Paul Flechsig . When an opportunity to return to Vienna presented itself, he seized it without hesitation.

### Second professorship for theoretical physics at the University of Vienna (1902–1906)

Lussinpiccolo (today Mali Lošinj ) around 1900, where Boltzmann stayed for summer vacation in 1903.
Memorial plaque on Boltzmann's house at Haizingergasse 26 in Vienna-Währing

On October 1, 1902, Boltzmann was appointed full professor of theoretical physics at the University of Vienna. In the "summer vacation" by the sea in 1902 and 1903 he was able to recover, and his mental and physical condition improved for some time. He was in the highest esteem: he was received by Emperor Franz Joseph and appointed court counselor . He moved into a new apartment in Haizingergasse in Vienna- Währing and threw himself into his work with zeal. He worked on the second volume of the lectures on the principles of mechanics and visited Göttingen and Kassel, where he had an "extremely lively difference of opinion" with David Hilbert . Further trips took him to England, to the Southport Meeting of the British Association for the Advancement of Science , and to Paris.

Ernst Mach held a chair for philosophy in Vienna, but after a stroke in 1901 could no longer fulfill his teaching duties. In the winter semester of 1903/04, Boltzmann took over his lecture on natural philosophy. He prepared thoroughly for it and wrote to the philosopher Franz Brentano , whom he visited in Florence in 1905. In addition to his intensive preoccupation with philosophy and his teaching activities, his scientific work almost came to a standstill: after 1900 he only published two papers.

From August 21 to October 8, 1904, Boltzmann, accompanied by his son Arthur Ludwig, made his second trip to America to take part in the St. Louis Mathematics Congress. The hardships of the journey put Boltzmann to the test. He found the Belgravia of the Hamburg-America Line to be "very inferior" and uncomfortable. The journey led from New York via Philadelphia, Washington, the Great Lakes with Detroit, Niagara Falls and Chicago to St. Louis. The return journey took place on the Deutschland .

In June 1905, Boltzmann set out on his third and last trip to America, which took him to Berkeley . After a short stay in Leipzig, he embarked in Bremen on the Crown Prince Wilhelm . From New York the express train took him to San Francisco in four days and nights. He was particularly impressed by the Lick Observatory . His lectures at Berkeley were only moderately successful, mainly due to his difficult to understand English. The return trip took place on the Kaiser Wilhelm II. This trip became known through Boltzmann's humorous report Journey of a German professor into the Eldorado . The cheerful, carefree mood of this travel report is seldom clouded, Boltzmann gives little indication of the complaints that tormented him back then. Boltzmann returned in a good mood, but a few months later the collapse occurred: his health deteriorated so much that in the spring of 1906 it became clear that Boltzmann could no longer fulfill his teaching duties.

### Death in Duino

Boltzmann's health had been very bad for a long time. Because of " neurasthenia " (nerve weakness) he received psychiatric treatment several times. He suffered from extreme mood swings, states of highest excitement alternated with deepest dejection. Boltzmann himself said that he was "born on the night between Shrovetide and Ash Wednesday, and this contrast is reflected in his whole life." By 1900 his eyesight had deteriorated so much that he hired a lady to read him scientific literature. He dictated his own work to his wife. He suffered from asthma, nasal polyps, headaches, kidney and bladder problems and various other physical complaints, as evidenced by several worried letters from his wife in 1902 and 1903 to his daughter Ida in Leipzig.

On May 5, 1906, Boltzmann was given leave of absence due to illness due to “severe neurasthenia”, and Stefan Meyer took over his lectures. Boltzmann spent the summer of 1906 with his family on the Adriatic Sea in Duino north of Trieste . On September 5, 1906, the day before the planned trip home, he hanged himself in his hotel room. The Neue Freie Presse reported on September 7th: “Professor Boltzmann had been suffering from neurasthenia for a long time and had come to Duino for his summer stay with a daughter. When the daughter did not see him coming out of his room yesterday morning, she went in and found the father, who had hung himself from an iron bar in the window, dead. ”The next day the Neue Freie Presse published obituaries for Boltzmann by Ernst Mach and Franz-Serafin Exner .

Boltzmann did not leave a suicide note, the immediate cause of his suicide is therefore unknown. Several authors suspect a connection between Boltzmann's depression and the rejection of his atomism. George Jaffé writes: “I do not venture to guess what reasons led a scientist of his caliber to give up his life after what really was a wonderfully successful career. I cannot help feeling, however, that the scientific situation which I have tried to sketch was not altogether disconnected from his resolution. ”(“ I don't dare to guess what made a scientist of his stature to end his life after an extraordinarily successful career I can't help but get the impression that his decision was not entirely unrelated to the scientific circumstances that I tried to outline. ”) It is undisputed that the arguments with the energetic engineers weighed heavily on Boltzmann , however, there is no evidence that these directly contributed to his suicide. Corresponding assumptions are pure speculation. In any case, Boltzmann was by no means an “misunderstood genius” who took his own life out of bitterness for the incomprehension of his surroundings. He was a highly respected scientist and had as many supporters as opponents; he was personally friends with scientific opponents such as Mach and Ostwald.

## Boltzmann's scientific work

Boltzmann worked in almost all areas of 19th century physics. He published a total of 139 original scientific papers, some of which were very extensive, and wrote three textbooks based on material from his lectures.

### Atomism

#### Atomic theory in the 19th century

There were significant advances in chemistry at the beginning of the 19th century. Joseph Louis Proust formulated the law of constant proportions , John Dalton derived from it the existence of atoms as the basic building blocks of chemical compounds and determined the relative atomic weights of several chemical elements . From the observation that atomic weights are approximately integer multiples of the weight of the hydrogen atom, William Prout formulated his hypothesis that the hydrogen atom is the basic building block of matter - it was only discovered in the 20th century that it was next to the hydrogen nucleus, the proton , and the neutron there is another basic building block of the atomic nucleus. By the middle of the 19th century, the existence of atoms was generally recognized among chemists; Mendeleev's formulation of the periodic table of the elements (1869) gave the atomic theory a systematic foundation.

In contrast, the atomic theory was only partially accepted by physicists, also because it was not relevant to many physical questions that were examined in the 19th century. Massive attacks on atomic theory emerged in the 1890s. In addition to the “energetics” Ostwald and Helm, opponents of atomistics were primarily Ernst Mach, whose “ phenomenologicalepistemology rejected the atoms that were not directly accessible to the senses. These currents, however, were locally restricted, mainly to the German-speaking area and partly to France ( Pierre Duhem ); in England these teachings found little support.

#### Boltzmann and atomistics

One of Boltzmann's earliest works from 1867 deals with the number of atoms in gas molecules. The majority of Boltzmann's scientific treatises speak of atoms or molecules, the existence of which is usually taken for granted. It was only in response to the attacks by Mach and the energetics that Boltzmann wrote several essays ( On the Indispensability of Atomistics in Natural Science , Again on Atomistics ) in which he addresses phenomenological arguments: The infinite divisibility of matter is just as inaccessible to the senses as it is Atoms and therefore an equally unproven hypothesis. However, preference should be given to the atomic hypothesis, as it better explains various observable effects:

"The question of whether matter is composed atomistically or continuously is reduced to whether those properties, assuming an extraordinarily large, finite particle number or their limits with an ever-increasing number of particles, represent the observed properties of matter most precisely."

Boltzmann does not start from the indivisibility of atoms, but speculates about the internal structure of the atoms:

“… Everyone is talking about various interesting views on how this structure is built. The word 'atom' must not mislead us, it is taken over from ancient times ... "

#### The triumphant advance of atomic theory after Boltzmann's death

It was only after Boltzmann's death in the first decade of the 20th century that atomic theory was able to prevail, which resulted in revolutionary upheavals in physics. In 1905 Albert Einstein derived from the kinetic theory of heat that the heat movement of molecules in liquids must lead to microscopically visible movements of suspended particles and assumed that these movements were identical to the " Brownian movement ". Smoluchowski came to one independently in 1906 similar result. Shortly afterwards, Jean Perrin was able to experimentally confirm Einstein's predictions:

«… Ainsi, la théorie moléculaire du mouvement brownien peut être regardée comme expérimentalement établie, et, du même coup, il devient assez difficile de nier la réalité objective des molécules.  »

“So the molecular theory of Brownian motion can be considered experimentally confirmed, and it becomes quite difficult to deny the objective existence of molecules. "

After the discovery of X-ray diffraction in crystals, Max von Laue also succeeded in providing direct experimental evidence of the arrangement of atoms in crystals in 1912. The teachings of the energetic were forgotten and were soon little more than a historical curiosity.

### Mechanics and Electrodynamics

Boltzmann called mechanics , which Newton , Euler , Lagrange , Hamilton and others had already brought to a high level of perfection, the "foundation of all natural sciences" and repeatedly dealt with questions of "classical" mechanics, as they differ today to quantum mechanics is called. He also applied his mastery in classical mechanics to other branches of physics. This is especially true for his work on electrodynamics . The theory of electromagnetism, developed in the 1860s by James Clerk Maxwell , was able to find a uniform formulation for the disciplines of electricity, magnetism and optics, which were previously considered separately. Boltzmann played a major role in making Maxwell's work known in continental Europe. In 1895 he translated Maxwell's treatise On Faraday's Lines of Force into German and published it as Über Faraday's Kraftlinien in Ostwald's Classics of Exact Sciences .

In the foreword to the lectures on Maxwell's theory of electricity and light , Boltzmann modestly subordinates himself to Maxwell: “It is therefore no wonder that the Kärrner are now coming to continue the building. I want to be such a carer, whose task it was to pave the way to the building, to clean the façade, perhaps also to add a stone or two to the foundation, and I am proud of it; because if there weren't any Kärrner, how would the kings want to build? ”In these lectures Boltzmann often makes use of mechanical analogies.

### Experimental physics

Boltzmann is mainly known as a theoretical and mathematical physicist, but he also achieved significant achievements in experimental physics. Despite his severe visual impairment, he was considered a skilled experimenter. At the beginning of his stay in Graz he worked with A. Toepler in the field of acoustics. He achieved his most important experimental results in the determination of the dielectric constant (now called permittivity ) of various materials and especially of gases whose dielectric constants differ only slightly from 1 and are therefore difficult to determine, for which Boltzmann had to devise his own methods. Boltzmann's work on the dielectric constant should be seen in connection with his study of Maxwell's electrodynamics. He endeavored to establish the relationship between the refractive index , permittivity number and permeability number, following from Maxwell's theory${\ displaystyle n}$ ${\ displaystyle \ varepsilon _ {r}}$ ${\ displaystyle \ mu _ {r}}$

${\ displaystyle n = {\ sqrt {\ varepsilon _ {r} \ mu _ {r}}}}$

to verify experimentally.

### Kinetic gas theory, thermodynamics and statistical mechanics

Around the middle of the 19th century, Rudolf Clausius had formulated the Second Law of Thermodynamics and coined the term entropy . The increase in entropy in a process is a measure of its reversibility (reversibility): if the entropy remains constant, the process is reversible; however, if it increases (for example when mixing cold with hot water), a reversal is only possible by supplying energy from outside. The laws of classical mechanics that describe the motion of a single particle are, however, invariant to time reversal , i.e. H. For any movement of a particle, movement in the opposite direction is also possible. Boltzmann studied the problem of how the reversible movements of individual particles (e.g. atoms or molecules of a gas) can create an irreversible overall process. This question has preoccupied him all his life and he has taken a number of different approaches. These approaches are based on different explicitly stated or even only implicit assumptions, especially about the properties of the (at that time) not directly observable molecules. In his first work on statistical mechanics from 1866, Boltzmann claimed "to provide a purely analytical, completely general proof of the second law of heat theory, as well as to find the corresponding theorem of mechanics". Boltzmann later moved away from this claim, arguing that such general proof would be impossible.

Boltzmann's application of statistical methods was groundbreaking. Alongside James Clerk Maxwell and Josiah Willard Gibbs, he is considered to be the founder of statistical mechanics. In 1860 Maxwell had determined the distribution of the velocities of the atoms of a gas in thermal equilibrium. Maxwell's results were generalized by Boltzmann and are now known as the Maxwell-Boltzmann distribution . Gibbs succeeded in further generalizing Maxwell and Boltzmann's results, which were mostly restricted to gases, to any system by introducing the term ensemble .

#### H-theorem and Boltzmann equation

In his work from 1872, usually abbreviated as “Further Studies”, Boltzmann looks at a gas in a non-equilibrium state and examines how the impact of the molecules changes the distribution of the “ living forces ” (kinetic energies). With this “impact number approach” he arrives at two important results, which are known today as the H theorem and the Boltzmann equation.

Boltzmann's formulation of the H-theorem reads:

${\ displaystyle E = \ int _ {0} ^ {\ infty} f (x, t) \ left \ {\ log \ left [{\ frac {f (x, t)} {\ sqrt {x}}} \ right] -1 \ right \} \ mathrm {d} x \.}$

Where is the number of particles with kinetic energy at the time . From the impulse number approach, Boltzmann deduces that the size (referred to as in later works , hence the name H-theorem) can never increase. is identical to entropy except for one negative constant factor : With the H-theorem, Boltzmann arrives at a theoretical justification of the Second Law. Boltzmann describes the temporal development of the distribution function with the help of an integro differential equation , which is now called the Boltzmann equation or Boltzmann's transport equation. ${\ displaystyle f (x, t)}$${\ displaystyle x}$${\ displaystyle t}$${\ displaystyle E}$${\ displaystyle H}$${\ displaystyle E}$${\ displaystyle S}$${\ displaystyle f (x, t)}$

#### Entropy and probability

Another high point in Boltzmann's work is his famous work on the relationship between the second law of mechanical heat theory and the calculus of probability or the theorems on thermal equilibrium from 1877. In this work, Boltzmann comes to the conclusion, using probabilistic and combinatory arguments, that the The transition to thermal equilibrium and the associated increase in entropy corresponds to a transition from a more improbable to a more probable state: “The initial state will in most cases be a very improbable one, from which the system will assign ever more probable states until it finally becomes the most probable , d. H. that of thermal equilibrium. If we apply this to the second law, we can identify that quantity, which is usually referred to as entropy, with the probability of the state in question. "

The central result of Boltzmann is often in the form

${\ displaystyle S = k \ log W}$

written. Here is the entropy, the Boltzmann constant , the “thermodynamic probability” and the natural logarithm . The term "probability" is not entirely accurate, rather it is the number of states ("microstates", referred to by Boltzmann as "complex ions"), characterized by the location and momentum of all particles that make up a state of the overall system ("macrostate" , in the case of a gas, characterized by state variables such as pressure, volume and temperature). ${\ displaystyle S}$${\ displaystyle k}$${\ displaystyle W}$${\ displaystyle \ log}$

Boltzmann himself never formulated the formula in this way (it comes from Max Planck), but it is implicit in Boltzmann's much more complex calculations.

Josef Stefan had experimentally determined in 1879 that the radiant power of a black body (the radiant energy emitted per surface area and time) is proportional to the fourth power of the absolute temperature. Based on the laws of thermodynamics and Maxwell's electrodynamics, Boltzmann found a theoretical foundation in 1884, which Lorentz described as the “real pearl of theoretical physics”. The relationship ${\ displaystyle P}$

${\ displaystyle P = \ sigma \ cdot A \ cdot T ^ {4}}$

is called the Stefan-Boltzmann law , where the proportionality factor is called the Stefan-Boltzmann constant , corresponds to the surface of the black body and indicates the absolute temperature. ${\ displaystyle \ sigma}$${\ displaystyle A}$${\ displaystyle T}$

## Boltzmann as a philosopher

Boltzmann's work as a philosopher is overshadowed by his achievements as a physicist. In his inaugural lecture on natural philosophy on October 26, 1903, Boltzmann explained his approach to philosophy from the standpoint of the physicist and practitioner:

“If it is desirable for a professor of medicine or technology that he should continue to practice alongside his teaching activities in order not to ossify, yes, if Moltke was elected to the historical class of the Berlin Academy, not because he was writing history , but because he made history, maybe they chose me too, not because I wrote about logic, but because I belong to a science in which one has the best opportunity for daily practice in the sharpest logic. "

Boltzmann expressed his skepticism towards academic philosophy, especially about its German representatives: “If I only hesitated to follow the call to get involved in philosophy, then the more philosophers mingled with natural science. They came into my enclosure a long time ago. I didn't even understand what they meant, so I wanted to find out more about the basic tenets of philosophy. "

“In order to draw straight from the deepest depths, I reached for Hegel ; but what a vague, thoughtless torrent of words I should find there! My bad star led me from Hegel to Schopenhauer . [...] Yes, even with Kant I could so little understand various things that I almost suspected with his other acumen that he wanted to have the reader for the best or even hypocrites. "

Boltzmann originally wanted to announce a lecture to the Philosophical Society in Vienna on January 21, 1905 as “proof that Schopenhauer was a mindless, ignorant, nonsense smeared his heads through empty verbiage from the ground up and always degenerating philosophaster” Literal quote from Schopenhauer's "About the fourfold root of the principle of sufficient reason", coined by Schopenhauer on Hegel.

On the occasion of his stay at the University of Berkeley, he also spoke about its namesake, George Berkeley , “who is even praised for being the inventor of the greatest folly that a human brain has ever hatched, philosophical idealism, which denies the existence of the material world ".

Boltzmann's materialistic worldview rejects the existence of a "spirit" independent of matter. In the 20th century, Gilbert Ryle coined the term “ghost in the machine” for this “ghost”.

“In my opinion, all salvation for philosophy can be expected from the teaching of Darwin . As long as one believes in a special mind that is able to recognize objects without mechanical means, in a special will that is again capable of wanting what is beneficial for us without mechanical means, one cannot explain the simplest psychological phenomena. "

“Only when one sees that mind and will are not something outside of the body, that they are rather complicated effects of parts of matter whose effectiveness becomes more and more perfect through development, only when one sees that imagination, will and self-consciousness are only the highest The stages of development of those physical-chemical forces of matter through which protoplasmic vesicles were initially enabled to seek regions that are more favorable to them, to avoid those that are unfavorable to them, everything becomes clear to one in psychology. "

## Aftermath

### Boltzmann's influence on his contemporaries

Boltzmann's successor in his chair was his student Friedrich Hasenöhrl , who fell in the First World War in 1915 . His assistant Stefan Meyer turned to radium research and in 1920 became head of the Institute for Radium Research . There was only very limited continuity in Boltzmann's teaching, which is why Boltzmann is considered more the perfecter of classical physics and not so much the pioneer of 20th century physics. On the other hand, however, his student Lise Meitner judges : "... with his thermodynamic research and the introduction of statistical methods he has made a significant contribution to the transition from classical to modern, microphysics."

Two of his Graz students, Svante Arrhenius and Walther Nernst , made significant achievements in the field of physical chemistry and were awarded the Nobel Prize for Chemistry. Boltzmann's student Paul Ehrenfest continued his work in statistical mechanics and, together with his wife Tatjana, wrote the article Conceptual foundations of the statistical conception in mechanics in Klein's Encyclopedia of Mathematical Sciences .

Boltzmann's influence on Albert Einstein should not be underestimated. Einstein carefully studied Boltzmann's work on statistical mechanics and repeatedly criticized it. A number of Einstein's early works deal with thermodynamics and build on the work of Maxwell and Boltzmann.

Max Planck , who was 14 years younger than Boltzmann, rejected his statistical access to physics for many years. In 1900 he derived the energy spectrum of the black body using Boltzmann's statistical methods , which he described as an "act of despair". With this generalization of Stefan-Boltzmann's law, he introduced Planck's quantum of action and thus laid the foundation for modern quantum theory.

The young Erwin Schrödinger began studying physics at the University of Vienna in 1906. Boltzmann's death just before the beginning of the semester shook him badly. However, his teaching was conveyed to him by Hasenöhrl and Franz-Serafin Exner and influenced him decisively.

### Significance for today's science

Numerous formulas, ideas and concepts named after Boltzmann testify to its importance for modern physics and other sciences:

## Personal

### marriage and family

Henriette von Aigentler and Ludwig Boltzmann as fiancé (1875).

In May 1873, Boltzmann met eighteen-year-old Henriette von Aigentler on the occasion of the annual outing of the teacher training institute in Graz. Henriette was born on November 16, 1854 in Stainz . She came from a respected Austrian family, her father Hugo was a lawyer and died in 1864 when Henriette was nine years old; on December 30, 1873, her mother Henrika also died. The orphan was supported by the family of Graz mayor Wilhelm Kienzl, the father of the composer Wilhelm Kienzl .

Henriette attended the Graz Teacher Training Institute and wanted to become a teacher of mathematics and physics, which required attending lectures at the University of Graz. At that time women were not admitted to university. She repeatedly turned to Boltzmann for advice and support in study matters. She succeeded in overcoming all difficulties and as the first student at the University of Graz to attend lectures in mathematics, natural sciences and philosophy as an extraordinary student. On September 27, 1875, Boltzmann made her a marriage proposal in writing, which she accepted by return mail. The wedding took place on July 17, 1876 in the parish church of Graz . Henriette ended her studies with the marriage.

Henriette and Ludwig Boltzmann had five children. The first four were born in Graz, the youngest daughter Elsa in Munich.

• Ludwig Hugo Boltzmann (* 1878; † 1889), Boltzmann's eldest son, died of appendicitis in Graz at the age of eleven.
• Henriette Boltzmann (* May 12, 1880 - March 8, 1945) was a teacher.
• Arthur Ludwig Boltzmann (* May 25, 1881; † November 6, 1952) studied physics, mechanical engineering and electrical engineering. After the First World War he became head of the Federal Office for Metrology and Surveying and Councilor. His wife Pauline Boltzmann was the daughter of the laryngologist Ottokar von Chiari , who had treated Boltzmann. The couple had three children, Ludwig (killed near Smolensk in 1943 ), Ilse-Maria Fasol-Boltzmann (married to Karl Heinz Fasol ; at the same time the academic administrator of her grandfather Ludwig Boltzmann's and also the editor of writings and memorial volumes) and Helga Boltzmann (married with Hans Rodinger). The latter received her doctorate in 1951 at the University of Innsbruck on the subject of "Population geographic studies of the Lungau ".
• Ida Boltzmann (born September 17, 1884 - † April 11, 1910) studied mathematics and physics.
• Elsa Boltzmann (* August 4, 1891, † August 27, 1965) was trained as a therapeutic gymnast in Sweden. On July 12, 1920 she married the physicist Ludwig Flamm , a student of her father. Albert Einstein sent a congratulatory telegram. They had four children, Maria, Werner, Eilhard and Dieter .

Boltzmann's wife survived him by 32 years and died on December 3, 1938 in Vienna.

### Interests

Boltzmann was interested in many things and was highly educated and, in accordance with the humanistic ideal of education, occupied himself extensively with music and literature. He played the piano excellently, in later years he accompanied his son Arthur, who played the violin. Some poems from his hand are also known, the "joke poem" Beethoven in Heaven has been reprinted several times. He was particularly adored by Friedrich Schiller . He dedicated the popular writings to "the manes of Schiller" and wrote in the "foreword":

“The dedicated dedication is not a grind, I thank the works of göthe, whose fist is the greatest of all works of art and from which I took the mottos of my first books, Shakespeare's etc. the highest spiritual elevation; but with schiller it's a little different, thanks to schiller i have become, one in could be a man with the same beard and nose shape as me, but never give me. "

The idiosyncratic spelling is a protest against the spelling reform of 1901 .

He followed with great interest current developments in technology and science that did not affect his immediate area of ​​expertise. He paid particular attention to aviation, about which he gave a lecture in 1894 at the Society of German Natural Scientists and Doctors in Vienna. He corresponded with the aviation pioneers Otto Lilienthal and Wilhelm Kress .

### Boltzmann in the judgment of his contemporaries and students

Boltzmann was considered to be an excellent lecturer. Numerous listeners of his lectures praised the clarity and the "peculiar elegance and transparency of his statements". Lise Meitner wrote "He was an unusually good, spirited, stimulating lecturer, always lively discussing, and was able ... to transfer his own enthusiasm to the audience." In contrast, his often extensive and rambling scientific work was considered difficult to understand, not only because of the sophisticated mathematical derivations. Maxwell's judgment is well known: “By the study of Boltzmann I have been unable to understand him. He could not understand me on account of my shortness, and his length was and is an equal stumbling block to me ”(“ When I was studying Boltzmann, I wasn't able to understand him. He couldn't understand me because of my brevity, and its length was and is equally a stumbling block for me ”). Einstein passed on "the rule of the ingenious theorist L. Boltzmann that elegance should be left to the tailors and shoemakers".

Numerous anecdotes testify to Boltzmann's lack of fluency in society. In a letter to Hans Benndorf in 1944, Stefan Meyer reported: "The invitations to the Boltzmanns are unforgettable because of their unbelievable naivety and awkwardness"; But he judges: “You would get a completely wrong, lopsided judgment about the really first-class tall Boltzmann if you wanted to judge him according to such 'stories'. He was not only a great scholar, but in spite of all his wondrousness a good person inwardly, with a pronounced sense of family and goodwill for others. "Ostwald wrote in his obituary for Boltzmann:" The same man whose mathematical acumen did not escape the slightest scientific discrepancy, was in the daily life of the harmlessness and inexperience of a child. "Wilhelm Kienzl wrote:" He had a high level of general education, which did not add anything to the striking, almost childlike naivety of his nature ... " Gerhard Kowalewski , the Boltzmann during his stay in Leipzig got to know, wrote: “A basic trait of his nature was limitless philanthropy.” Similarly Lise Meitner: “Soft in character, vulnerable and tender-feeling ... full of kindness, belief in ideals and reverence for the miracles of natural law”.

## Honors

A list of honors and awards, especially the numerous memberships in scientific societies and associations, can be found in the section Honors (based on existing certificates) in Ludwig Boltzmann (1844–1906) .

### Honorary doctorates

Boltzmann was awarded an honorary doctorate from three universities. He was personally present at the first two awards.

• Oxford University on August 15, 1894. George Hartley Bryan wrote in his 1906 obituary for Boltzmann that Boltzmann was amused to have been named Doctor of Law . "It was, however, pointed out that as an authority on the laws of thermodynamics the title was a fitting one." ("It was pointed out, however, that this title would be a suitable one for him as an authority on the laws of thermodynamics." )
• On the occasion of the ten-year existence of Clark University in Worcester , he gave four lectures on the basic principles and equations of mechanics and received an honorary doctorate from the university on July 10, 1899.
• Royal Friedrichs University in Christiania on September 6, 1902

Also in 1902 he received an invitation to the 200th anniversary celebration of Yale University , to which Gibbs had invited four physicists: Lord Rayleigh , Poincaré , Boltzmann and Lord Kelvin . Participation would have been linked to the award of an honorary doctorate. Boltzmann canceled due to illness, but thanked Gibbs in a letter "for the award shown" - due to a misunderstanding, Boltzmann considered himself an honorary doctorate from Yale.

### Nominations for the Nobel Prize

The Nobel Prize in Physics was awarded for the first time in 1901. Boltzmann was proposed a total of five times, in 1903 by the chemist Adolf von Baeyer (Nobel Prize 1905), in 1905 and 1906 by Max Planck (Nobel Prize 1918), in 1906 by Philipp Lenard , who had received the Nobel Prize himself the year before, and by the surgeon Vinzenz Czerny .

### Further honors during his lifetime

In 1895 Boltzmann was elected to the Royal Society of Edinburgh and in 1897 to the American Academy of Arts and Sciences , in 1899 to the Russian Academy of Sciences , 1900 to the Académie des Sciences and 1904 to the National Academy of Sciences . On his 60th birthday on February 20, 1904, a large festive event was held in his honor, at which an extensive and splendidly decorated publication was presented. On almost 1000 pages, the Festschrift contains contributions by 117 authors who give an overview of the state of physics at the time, including Max Planck, Johannes van der Waals , Johannes Stark , Ernst Mach, Arnold Sommerfeld . Boltzmann refused to be raised to the nobility - regardless of the aristocratic origin of his wife, whose great-grandfather Joseph Aigentler had been ennobled by Emperor Joseph II in 1783 : “My ancestors, my father, the name Boltzmann was good enough and should mean it to me too To be children and grandchildren ”.

### Posthumous honors

Bust in the arcade courtyard of the University of Vienna.
Honorary grave at the Vienna Central Cemetery

On December 7, 1912, a bust of Boltzmann was unveiled in the arcade courtyard of the University of Vienna . As was customary at the time, the costs were raised by collecting donations; numerous prominent scientists such as Albert Einstein, Loránd Eötvös , Arnold Sommerfeld and Ernest Rutherford can be found in the donation directory.

In 1929, Boltzmann's remains were exhumed from the Döblingen cemetery and transferred to an honorary grave in the Vienna Central Cemetery . A committee of professors "was given the honorable task of providing for the erection of a worthy grave monument", which was then realized in the form of a bust made of Carrara marble . The bust is the work of Gustinus Ambrosi , whose strongly heroic depiction shows little resemblance to Boltzmann. In the background the famous formula is inscribed in the marble. The grave monument was ceremoniously unveiled on July 4, 1933 in the presence of City Councilor Julius Tandler , Wolfgang Pauli , Hans Thirring , Boltzmann's widow and son Arthur and many others. ${\ displaystyle S = k \ log W}$

Since 1953, the Austrian Physical Society has awarded the Ludwig Boltzmann Prize for special achievements in theoretical physics.

In 1960, the Ludwig Boltzmann Society was founded in Vienna as a non-profit association for research funding. In 1965 the first Ludwig Boltzmann Institute, the LBI for Solid State Physics, was established. Further institutes of the Ludwig Boltzmann Society are or were:

Since 1975, the IUPAP has awarded the Boltzmann Medal for services to statistical physics every three years .

A lunar crater was named Boltzmann in 1964 and an asteroid discovered in 1991 was named (24712) Boltzmann .

A number of commemorative events took place on the anniversaries of his birthday and death, for example in Vienna in 1944 during World War II , where Arnold Sommerfeld gave the speech on Boltzmann's 100th birthday. In 1981 the Austrian Post issued a special postage stamp to mark the 75th anniversary of Boltzmann's death. A Boltzmann Memorial Meeting took place on September 4, 2006 in the Duino fort on the 100th anniversary of his death . A plaque was unveiled on the building of the United World College of the Adriatic , which previously housed the Hotel Ples.

### Names of streets

In Austria at least seven streets, in Germany at least five - two of them in Berlin - are named after Boltzmann:

## Fonts

### Original scientific work

The 139 scientific works of Boltzmann were collected after his death by Fritz Hasenöhrl and published in three volumes by Johann Ambrosius Barth in 1909 . They are available online at the University of Vienna:

The copy from the University of Vienna comes from the possession of Stefan Meyers and has a handwritten dedication by Fritz Hasenöhrl. In 2012 the Academic Papers were reprinted by Cambridge University Press .

Boltzmann's last work, written in 1905 together with his assistant Josef Nabl , is not included in the collected treatises and appeared posthumously in 1907:

### Textbooks

The handwritten documents for his lectures on natural philosophy are owned by the family and were published in 1990 by his granddaughter Ilse Fasol-Boltzmann:

• Ilse M. Fasol-Boltzmann (ed.): Ludwig Boltzmann Principien der Naturfilosofi: Lectures on Natural Philosophy 1903-1906. Springer, Berlin 1990, ISBN 3-540-51716-2 .

### Popular fonts

An anthology with various essays from the years 1873 to 1905 was published in 1905 under the name “Popular Writings”:

The popular writings contain, among other things, essays on scientific and philosophical topics (e.g. On Maxwell's theory of electricity , On the indispensability of atomistics in natural science , On a thesis of Schopenhauer ), obituaries on deceased scientists (Gustav Robert Kirchhoff, Josef Stefan, Josef Loschmidt ) and reports on current developments in science and technology ( about aviation , Röntgen's new rays ). The trip of a German professor to the Eldorado , a humorous report on his trip to California in the summer of 1905, was reprinted frequently .

### Letters

Two volumes of Boltzmann's letters have been published:

Erich Zöllner , the husband of Boltzmann's granddaughter Maria Flamm, discovered 125 letters in Boltzmann's apartment in Haizingergasse in Vienna-Währing at an unspecified time, between Boltzmann and his future wife Henriette von Aigentler in the period from 1873 until their wedding changed in July 1876.

• Dieter Flamm (Ed.): Dear Professor! Beloved Louis! . Ludwig Boltzmann, Henriette von Aigentler. Correspondence. Böhlau, Vienna 1995, ISBN 3-205-98266-5 .

### Complete edition

In the 1980s, the publication of a complete edition of Boltzmann's writings began under the direction of Roman Sexl . Three volumes were published by 1982, volume 1 (lectures on gas theory) , volume 2 (lectures on Maxwell's theory of electricity and light) and volume 8 (selected treatises) . After the editor's premature death, however, the complete edition came to a standstill, until 1998 only two more volumes were published, Volume 9 ( Life and Letters , see above) and Volume 10 (Lecture on Experimental Physics in Graz) . The complete edition is still unfinished today (2015).

## literature

• Engelbert Broda : Ludwig Boltzmann: human • physicist • philosopher. Franz Deuticke, Vienna 1955. English translation: Ludwig Boltzmann: Man – Physicist – Philosopher , Ox Bow Press 1983, ISBN 0-918024-24-2 .
• Wolfgang Stiller: Ludwig Boltzmann: Old Masters of Classical Physics. Pioneer of quantum physics and evolutionary theory. Barth, Leipzig 1988.
• Carlo Cercignani : Ludwig Boltzmann: The Man Who Trusted Atoms. Oxford 2006, ISBN 0-19-857064-3 .
• Carlo Cercignani: Boltzmann's Legacy . On the hundredth anniversary of Ludwig Boltzmann's death. In: Physics Journal . tape 05 , no. 07 , 2006, p. 47–51 ( pro-physik.de [PDF]).
• John T. Blackmore (Ed.): Ludwig Boltzmann: His Later Life and Philosophy, 1900-1906. A Documentary History. Kluwer Academic Publishers, Dordrecht 1995, ISBN 978-90-481-4492-1 .
• David Lindley: Boltzmann's Atom: The Great Debate That Launched A Revolution In Physics. Free Press 2001, ISBN 0-684-85186-5 .
• Ilse Maria Fasol-Boltzmann, Gerhard Ludwig Fasol (ed.): Ludwig Boltzmann (1844–1906). On the hundredth anniversary of death . Springer, Vienna / New York, ISBN 978-3-211-33140-8 .
• Stephen G. Brush: Boltzmann, Ludwig . In: Charles Coulston Gillispie (Ed.): Dictionary of Scientific Biography . tape 2 : Hans Berger - Christoph Buys Ballot . Charles Scribner's Sons, New York 1970, p. 260-268 .

Commons : Ludwig Boltzmann  - Collection of images, videos and audio files
Wikisource: Ludwig Boltzmann  - Sources and full texts

## Individual evidence

1. I. Fasol-Boltzmann, G. Fasol (ed.): Ludwig Boltzmann (1844–1906) on the hundredth anniversary of death. Springer, Vienna / New York, ISBN 978-3-211-33140-8 .
2. Ludwig Boltzmann: About the movement of electricity in crooked surfaces . Meeting reports of the Imperial Academy of Sciences 52 Dept. II, pp. 214–221, online in the Google book search
3. a b Ludwig Boltzmann: About the mechanical meaning of the second law of heat theory . Session reports of the Imperial Academy of Sciences 53, pp. 195–220.
4. a b Ludwig Boltzmann: Josef Stefan . In: Popular Writings , pp. 92-103.
5. Ludwig Boltzmann: In memory of Josef Loschmidt . In: Popular Writings , pp. 228-252.
6. ^ A b Ludwig Boltzmann: Further studies on the heat equilibrium among gas molecules. In: Meeting reports of the Imperial Academy of Sciences. 66, pp. 275-370.
7. A monograph is dedicated to Boltzmann's relations to Berlin: Herbert Hörz and Andreas Laaß: Ludwig Boltzmanns Ways to Berlin: A Chapter Austrian-German Scientific Relations . Akademie-Verlag, Berlin 1989, ISBN 3-05-500416-7
8. ^ Wilhelm Ostwald: The overcoming of scientific materialism . Lecture given at the third general meeting of the Assembly of the Society of German Natural Scientists and Doctors in Lübeck on September 20, 1895.
9. David E. Zitarelli: The 1904 St. Louis Congress and Westward expansion of American Mathematics . Notices of the AMS 2001, pp. 1100-1111.
10. ^ A b Ludwig Boltzmann: Journey of a German professor to the Eldorado . In: Popular Writings , pp. 403-435.
11. Boltzmann's statement on the occasion of the celebration of his 60th birthday, handed down by Stefan Meyer to Hans Benndorf. See I. Fasol-Boltzmann, G. Fasol (ed.): Ludwig Boltzmann (1844–1906) on the centenary of death , p. 39.
12. See the foreword to the lectures on gas theory .
13. See e.g. B. Letters No. 52, 60, 65 in: Ludwig Boltzmann: His Later Life and Philosophy, 1900–1906 .
14. ^ Suicide of the court councilor Boltzmann. In:  Neue Freie Presse , September 7, 1906, p. 6 (online at ANNO ).
15. ^ Ernst Mach: Ludwig Boltzmann. In:  Neue Freie Presse , September 8, 1906, p. 1 (online at ANNO ). ; Franz Exner: Ludwig Boltzmann's life's work. In:  Neue Freie Presse , September 8, 1906, p. 1 (online at ANNO ).
16. George Jaffé: Recollections of three great laboratories . Journal of chemical education 05/1952; 29 (5), doi: 10.1021 / ed029p230 .
17. Ludwig Boltzmann: About the number of atoms in gas molecules and the internal work in gases . Vienna. Ber. 56, pp. 682-690 (1867).
18. Ludwig Boltzmann: About the indispensability of atomistics in the natural sciences . In: Popular Writings , pp. 141–157.
19. Ludwig Boltzmann: Again about the atomistics . In: Popular Writings , pp. 158-161.
20. a b Ludwig Boltzmann: About statistical mechanics . In: Popular Writings , pp. 345-363.
21. Albert Einstein: About the motion of particles suspended in liquids at rest, required by the molecular kinetic theory of heat . Annalen der Physik 17, 549 (1905).
22. M. Smoluchowski: On the kinetic theory of Brownian molecular motion and suspensions . In: Annals of Physics . tape 326 , no. 14 , 1906, pp. 756–780 ( digital version ( PDF; 1.4 MB)).
23. Jean Perrin: Mouvement brownien et réalité moléculaire Annales de Chimie et de physique ser. 8, 18 (1909), pp. 5-114.
24. Ludwig Boltzmann: Lectures on the Principles of Mechanics , p. 1.
25. James Clerk Maxwell: On Faraday's Lines of Force . Published by L. Boltzmann, Engelmann, Leipzig 1895.
26. Ludwig Boltzmann: About a new method to analyze the vibrations of sounding columns of air, together with A. Toepler . 'In: Poggendorff's annals . 141, pp. 321–352, 1870. Reprinted in Scientific Treatises by Ludwig Boltzmann , Volume I.
27. Ludwig Boltzmann: Experimental determination of the dielectric constant of some gases. Vienna. Ber. 69, pp. 794-813, 1874; Reprinted in Scientific Treatises by Ludwig Boltzmann. Volume I., pp. 537-555
28. Ludwig Boltzmann: Experimental determination of the dielectric constant of insulators. Vienna. Ber. 67, pp. 17-80, 1873; Reprinted in Scientific Treatises by Ludwig Boltzmann. Volume I., pp. 411-471
29. See Chapter 9.1 Boltzmann's testing of Maxwell's theory of electromagnetism in: Carlo Cercignani: Ludwig Boltzmann: The Man Who Trusted Atoms. Oxford 2006, ISBN 0-19-857064-3 .
30. Jos Uffink: Boltzmann's Work in Statistical Physics.
31. James Clerk Maxwell: Illustrations of the Dynamical Theory of Gases , Philosophical Magazine , January and July 1860. Reprinted in The scientific papers of James Clerk Maxwell , Vol. 1, Cambridge University Press 1890 and Dover Publications 1965.
32. Equation 17 in "Further Studies" , in Scientific Treatises by Ludwig Boltzmann , Volume I., p. 335.
33. See Chapter 4 The Boltzmann equation and 5 Time irreversibility and the H-theorem in: Carlo Cercignani: Ludwig Boltzmann: The Man Who Trusted Atoms. Oxford 2006, ISBN 0-19-857064-3 .
34. Ludwig Boltzmann: About the relationship between the second law of mechanical heat theory and the calculus of probability or the theorems about the equilibrium of heat . In: session area. dk Akad. der Wissenschaften zu Wien II 76, p. 428 (1877). Reprinted in Scientific Discussions by Ludwig Boltzmann , Volume II., Pp. 164–223.
35. a b Max Planck: About the law of energy distribution in the normal spectrum. Ann. Physik 4 (1901) 553-563.
36. Josef Stefan: About the relationship between thermal radiation and temperature. In: Session reports of the mathematical and natural science class of the Imperial Academy of Sciences. Volume 79 (Vienna 1879), pp. 391-428.
37. Ludwig Boltzmann: Derivation of Stefan's law, regarding the dependence of thermal radiation on temperature from the electromagnetic light theory. In: Annals of Physics and Chemistry. Volume 22, 1884, pp. 291-294, doi: 10.1002 / andp.18842580616 .
38. Ludwig Boltzmann: An inaugural lecture on natural philosophy . In: Die Zeit , December 11, 1903, reprinted in: Popular Writings , pp. 338–344.
39. a b from: On a thesis by Schopenhauer. In: Popular Writings , pp. 385–402 (citations on pp. 385 and 396)
40. Arthur Schopenhauer: About the fourfold root of the sentence from sufficient reason , full text on textlog.de
41. Steven Pinker : The Blank Slate: The Modern Denial of Human Nature . Viking 2002, ISBN 978-0-670-03151-1 . German: The blank sheet . Berlin Verlag 2003, ISBN 978-3-8270-0509-0 .
42. Paul and Tatjana Ehrenfest: Conceptual foundations of the statistical conception in mechanics. In: Encyclopedia of Mathematical Sciences. 1909, 1911.
43. Hiroshi Ezawa: Einstein's contribution to statistical mechanics . In: Peter C. Aichelburg and Roman U. Sexl (Ed.): Albert Einstein: His influence on physics, philosophy and politics. Vieweg, Braunschweig 1979, ISBN 3-528-08424-3 , pp. 71-90.
44. ^ Massimiliano Badino: The odd couple: Boltzmann, Planck and the application of statistics to physics (1900-1913) . Annalen der Physik 18, 2–3, pp. 81–101, 2009, doi: 10.1002 / andp.200810336
45. Dieter Flamm: Boltzmann's influence on Schrödinger . In: CW Kilmister (Ed.): Schrödinger. Centenary celebration of a polymath. , Pp. 4-13.
46. ^ Henriettes study and engagement time. In: D. Flamm (Ed.): Dear Professor! Beloved Louis! , Pp. 38-41.
47. ^ Letter No. 15, dated "Wien, VIII Florianigasse 2, September 27, 1875". In: D. Flamm (Ed.): Dear Professor! Beloved Louis! , P. 97.
48. ^ Henriette's reply has not been received, but Boltzmann's reply to Letter No. 16, dated "Vienna, September 30, 1875". In: D. Flamm (Ed.): Dear Professor! Beloved Louis! , P. 98.
49. For Boltzmann's descendants see the Biographical Appendix in: D. Flamm (Ed.): Honored Herr Professor! Beloved Louis! , Pp. 283-304.
50. ↑ Obituary notice Ida Boltzmann. In:  Neue Freie Presse , April 13, 1910, p. 24 (online at ANNO ).
51. Ludwig Boltzmann: About air shipping . In: Popular Writings , pp. 81–91.
52. ^ Sílvio R. Dahmen: Boltzmann and the art of flying. Physics in Perspective 11 (3), 2009, doi: 10.1007 / s00016-008-0395-1
53. ^ Quote from Heinrich Streintz . In: Walter Höflechner (Ed.): Ludwig Boltzmann. Life and Letters , p. I 64.
54. James C. Maxwell to Peter Guthrie Tait (August 1873). See Carlo Cercignani: Ludwig Boltzmann: The Man Who Trusted Atoms , p. 140.
55. In the foreword to: A. Einstein: About the special and general theory of relativity. Vieweg Collection, issue 38, F. Vieweg, Braunschweig 1917; Reprint: Springer, 24th edition 2013, ISBN 3-642-31278-0 .
56. ^ Stefan Meyer to Hans Benndorf in 1944 . In: Walter Höflechner (Ed.): Ludwig Boltzmann. Life and Letters , pp. III 3–9.
57. ^ GH Bryan: Prof. Ludwig Boltzmann . Nature 74, 569 (1927), doi: 10.1038 / 074569a0 .
58. About the basic principles and equations of mechanics. Page 261ff
59. Awarding of an honorary doctorate to Ludwig Boltzmann in: William E. Storey, Louis N. Wilson (eds.): Clark University 1889–1899. Decennial Celebration.
60. ^ Letter from Boltzmann to Gibbs, undated (1901?). No. 573 in: Walter Höflechner (Ed.): Ludwig Boltzmann. Life and Letters , p. II 343.
61. Walter Höflechner (Ed.): Ludwig Boltzmann. Life and Letters , pp. I 218–219.
62. Ludwig Boltzmann in the Nomination Database on nobelprize.org
63. ^ Biographical Index: Former RSE Fellows 1783–2002. Royal Society of Edinburgh, accessed October 10, 2019 .
64. ^ Foreign members of the Russian Academy of Sciences since 1724: Boltzmann, Ludwig. Russian Academy of Sciences, accessed September 22, 2019 (Russian).
65. ^ List of members since 1666: Letter B. Académie des sciences, accessed on September 22, 2019 (French).
66. ^ Festschrift dedicated to Ludwig Boltzmann on the occasion of his 60th birthday, February 20, 1904. Leipzig, Johann Ambrosius Barth 1904.
67. The ancestors of Henriette Aigentler. In: D. Flamm (Ed.): Dear Professor! Beloved Louis! , Pp. 36-37.
68. I. Fasol-Boltzmann, G. Fasol (ed.): Ludwig Boltzmann (1844–1906) on the centenary of death , p. 38.
69. ^ Fritz Hasenöhrl: Report on the erection of a memorial for Ludwig Boltzmann in the arcade courtyard of the University of Vienna. Vienna 1913. Self-published by the reporter.
70. Walter Höflechner (Ed.): Ludwig Boltzmann. Life and Letters , p. I 293.
71. Ludwig Boltzmann 1844–1906. Report on the construction and the ceremonial unveiling of the memorial on July 4, 1933 on the honorary grave donated by the municipality of Vienna in 1929 . Reprint from "Elektrotechnik und Maschinenbau", magazine of the Elektrotechnisches Verein in Vienna, 51st year, issue 53, 1933.
72. ^ The LBG since 1960 ( memento of April 2, 2015 in the Internet Archive ) on the Ludwig Boltzmann Gesellschaft website, accessed on November 2, 2019.
73. Boltzmann in the Gazetteer of Planetary Nomenclature.
74. ^ Carlo Cercignani: Ludwig Boltzmann: The Man Who Trusted Atoms , p. 24.
75. ^ Special post stamp on the 75th anniversary of Ludwig Boltzmann's death in the Austria Forum
76. ^ Ludwig Boltzmann Memorial Meeting in Duino
77. ^ Boltzmanngasse in the Vienna History Wiki of the City of Vienna
78. ^ Wolfgang Volk: Monuments on Mathematicians / Streets in Berlin-Adlershof. In: w-volk.de. April 2013, accessed November 20, 2015 .
79. ^ Ludwig-Boltzmann-Strasse in Adlershof - Treptow-Köpenick. In: streetdir.com. Retrieved September 5, 2016 .
80. Zöllner writes in December 1993 that the find took place “a long time ago”. In: D. Flamm (Ed.): Dear Professor! Beloved Louis! , P. 13.
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