Ludwig Boltzmann

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Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an
Austria The Republic of Austria, commonly just Austria, , bar, Östareich is a country in the southern part of Central Europe, lying in the Eastern Alps. It is a federation of nine States of Austria, states, one of which is the capital, Vienna, ...
n
physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate caus ...
and
philosopher A philosopher is a person who practices or investigates philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Suc ...
. His greatest achievements were the development of
statistical mechanics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scien ...
, and the statistical explanation of the
second law of thermodynamics The second law of thermodynamics is a physical law based on universal experience concerning heat and Energy transformation, energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects ( ...
. In 1877 he provided the current definition of
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
, $S = k_ \ln \Omega \!$, where Ω is the number of microstates whose energy equals the system's energy, interpreted as a measure of statistical disorder of a system.
Max Planck Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a Germans, German theoretical physicist whose discovery of quantum mechanics, energy quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial con ...
named the constant the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...
. Statistical mechanics is one of the pillars of modern
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
. It describes how
macroscopic The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying optical instruments. It is the opposite of microscopic scale, microscopic. Overview When applied to ph ...
observations (such as
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer. Thermometers are calibrated in various Conversion of units of temperature, temp ...
and
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
) are related to
microscopic The microscopic scale () is the scale of objects and events smaller than those that can easily be seen by the naked eye, requiring a lens (optics), lens or microscope to see them clearly. In physics, the microscopic scale is sometimes regarded a ...
parameters that fluctuate around an average. It connects thermodynamic quantities (such as
heat capacity Heat capacity or thermal capacity is a physical quantity, physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The International System of Units, SI unit of heat ca ...
) to microscopic behavior, whereas, in
classical thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
, the only available option would be to measure and tabulate such quantities for various materials.

# Biography

## Childhood and education

Boltzmann was born in Erdberg, a suburb of
Vienna Vienna ( ; german: Wien ; bar, Wean, label=Bavarian language, Austro-Bavarian ) is the Capital city, capital, largest city, and one of States of Austria, nine states of Austria. Vienna is Austria's List of cities and towns in Austria, most populou ...
. His father, Ludwig Georg Boltzmann, was a revenue official. His grandfather, who had moved to Vienna from Berlin, was a clock manufacturer, and Boltzmann's mother, Katharina Pauernfeind, was originally from
Salzburg Salzburg (, ; literally "Salt-Castle"; bar, Soizbuag, label=Bavarian language, Austro-Bavarian) is the List of cities and towns in Austria, fourth-largest city in Austria. In 2020, it had a population of 156,872. The town is on the site of the ...
. He received his primary education at the home of his parents. Boltzmann attended high school in
Linz Linz ( , ; cs, Linec) is the capital of Upper Austria and List of cities and towns in Austria, third-largest city in Austria. In the north of the country, it is on the Danube south of the Czech Republic, Czech border. In 2018, the population ...
,
Upper Austria Upper Austria (german: Oberösterreich ; bar, Obaöstareich) is one of the nine states or of Austria The Republic of Austria, commonly just Austria, , bar, Östareich is a country in the southern part of Central Europe, lying in t ...
. When Boltzmann was 15, his father died. Starting in 1863, Boltzmann studied
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
at the
University of Vienna The University of Vienna (german: Universität Wien) is a public university, public research university located in Vienna, Austria. It was founded by Rudolf IV, Duke of Austria, Duke Rudolph IV in 1365 and is the oldest university in the Geogra ...
. He received his doctorate in 1866 and his
venia legendi Habilitation is the highest academic degree, university degree, or the procedure by which it is achieved, in many European countries. The candidate fulfills a university's set criteria of excellence in research, teaching and further education, us ...
in 1869. Boltzmann worked closely with Josef Stefan, director of the institute of physics. It was Stefan who introduced Boltzmann to
Maxwell's Maxwell's, last known as Maxwell's Tavern, was a bar/restaurant and Music venue, music club in Hoboken, New Jersey. Over several decades the venue attracted a wide variety of acts looking for a change from the New York City concert spaces across t ...
work.

## Academic career

In 1869 at age 25, thanks to a letter of recommendation written by Josef Stefan, Boltzmann was appointed full Professor of
Mathematical Physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
at the
University of Graz The University of Graz (german: link=no, Karl-Franzens-Universität Graz, ), located in Graz, Austria, is the largest and oldest university in Styria, as well as the second-largest and second-oldest university in Austria. History The univers ...
in the province of
Styria Styria (german: Steiermark ; Serbo-Croatian Serbo-Croatian () – also called Serbo-Croat (), Serbo-Croat-Bosnian (SCB), Bosnian-Croatian-Serbian (BCS), and Bosnian-Croatian-Montenegrin-Serbian (BCMS) – is a South Slavic languages, So ...
. In 1869 he spent several months in
Heidelberg Heidelberg (; Palatine German: ''Heidlberg'') is a city A city is a human settlement of notable size.Goodall, B. (1987) ''The Penguin Dictionary of Human Geography''. London: Penguin.Kuper, A. and Kuper, J., eds (1996) ''The Social Science ...
working with
Robert Bunsen Robert Wilhelm Eberhard Bunsen (; 30 March 1811 – 16 August 1899) was a German chemist. He investigated electromagnetic spectroscopy, emission spectra of heated elements, and discovered caesium (in 1860) and rubidium (in 1861) with the physic ...
and Leo Königsberger and in 1871 with
Gustav Kirchhoff Gustav Robert Kirchhoff (; 12 March 1824 – 17 October 1887) was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects. He coine ...
and
Hermann von Helmholtz Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Association ...
in Berlin. In 1873 Boltzmann joined the University of Vienna as Professor of Mathematics and there he stayed until 1876. In 1872, long before women were admitted to Austrian universities, he met Henriette von Aigentler, an aspiring teacher of mathematics and physics in Graz. She was refused permission to audit lectures unofficially. Boltzmann supported her decision to appeal, which was successful. On 17 July 1876 Ludwig Boltzmann married Henriette; they had three daughters: Henriette (1880), Ida (1884) and Else (1891); and a son, Arthur Ludwig (1881). Boltzmann went back to
Graz Graz (; sl, Gradec) is the capital city of the Austria The Republic of Austria, commonly just Austria, , bar, Östareich is a country in the southern part of Central Europe, lying in the Eastern Alps. It is a federation of nine Stat ...
to take up the chair of Experimental Physics. Among his students in Graz were
Svante Arrhenius Svante August Arrhenius ( , ; 19 February 1859 – 2 October 1927) was a Swedes, Swedish scientist. Originally a physicist, but often referred to as a chemist, Arrhenius was one of the founders of the science of physical chemistry. He received ...
and
Walther Nernst Walther Hermann Nernst (; 25 June 1864 – 18 November 1941) was a German chemist known for his work in thermodynamics, physical chemistry, electrochemistry, and solid state physics. His formulation of the Nernst heat theorem helped pave the wa ...
. He spent 14 happy years in Graz and it was there that he developed his statistical concept of nature. Boltzmann was appointed to the Chair of Theoretical Physics at the
University of Munich The Ludwig Maximilian University of Munich (simply University of Munich or LMU; german: Ludwig-Maximilians-Universität München) is a public university, public research university in Munich, Germany. It is Germany's List of universities in Germ ...
in
Bavaria Bavaria ( ; ), officially the Free State of Bavaria (german: Freistaat Bayern, link=no ), is a state in the south-east of Germany Germany,, officially the Federal Republic of Germany, is a country in Central Europe. It is the sec ...
, Germany in 1890. In 1894, Boltzmann succeeded his teacher Joseph Stefan as Professor of Theoretical Physics at the University of Vienna.

## Final years and death

Boltzmann spent a great deal of effort in his final years defending his theories.Cercignani, Carlo (1998) Ludwig Boltzmann: The Man Who Trusted Atoms. Oxford University Press. He did not get along with some of his colleagues in Vienna, particularly
Ernst Mach Ernst Waldfried Josef Wenzel Mach ( , ; 18 February 1838 – 19 February 1916) was a Moravian-born Austrian Empire, Austrian physicist and philosopher, who contributed to the physics of shock waves. The ratio of one's speed to speed of sound, t ...
, who became a professor of philosophy and history of sciences in 1895. That same year Georg Helm and
Wilhelm Ostwald Friedrich Wilhelm Ostwald (; 4 April 1932) was a Baltic German chemist and German philosophy, philosopher. Ostwald is credited with being one of the founders of the field of physical chemistry, with Jacobus Henricus van 't Hoff, Walther Nernst, ...
presented their position on energetics at a meeting in
Lübeck Lübeck (; Low German also ), officially the Hanseatic City of Lübeck (german: Hansestadt Lübeck), is a city in Northern Germany. With around 217,000 inhabitants, Lübeck is the second-largest city on the German Baltic Sea, Baltic coast and ...
. They saw energy, and not matter, as the chief component of the universe. Boltzmann's position carried the day among other physicists who supported his atomic theories in the debate. In 1900, Boltzmann went to the
University of Leipzig Leipzig University (german: Universität Leipzig), in Leipzig in Saxony, Germany, is one of the world's oldest University, universities and the List of universities in Germany#Universities by years of existence, second-oldest university (by conse ...
, on the invitation of
Wilhelm Ostwald Friedrich Wilhelm Ostwald (; 4 April 1932) was a Baltic German chemist and German philosophy, philosopher. Ostwald is credited with being one of the founders of the field of physical chemistry, with Jacobus Henricus van 't Hoff, Walther Nernst, ...
. Ostwald offered Boltzmann the professorial chair in physics, which became vacant when Gustav Heinrich Wiedemann died. After Mach retired due to bad health, Boltzmann returned to Vienna in 1902. In 1903, Boltzmann, together with Gustav von Escherich and
Emil Müller Emil Muller or Emil Müller may refer to: * Emil Müller (mathematician) (1861–1927), Austrian mathematician * Emil Müller (German officer), an officer in the World War I Imperial German Army charged with war crimes at the Leipzig War Crimes Tri ...
, founded the Austrian Mathematical Society. His students included Karl Přibram,
Paul Ehrenfest Paul Ehrenfest (18 January 1880 – 25 September 1933) was an Austrian theoretical physicist, who made major contributions to the field of statistical mechanics and its relations with quantum physics, quantum mechanics, including the theory of ph ...
and
Lise Meitner Elise Meitner ( , ; 7 November 1878 – 27 October 1968) was an Austrian-Swedish physicist who was one of those responsible for the discovery of the element protactinium and nuclear fission. While working at the Kaiser Wilhelm Institute on rad ...
. In Vienna, Boltzmann taught physics and also lectured on philosophy. Boltzmann's lectures on
natural philosophy Natural philosophy or philosophy of nature (from Latin ''philosophia naturalis'') is the philosophy, philosophical study of Physics (Aristotle), physics, that is, nature and the physical universe. It was dominant before the development of mode ...
were very popular and received considerable attention. His first lecture was an enormous success. Even though the largest lecture hall had been chosen for it, the people stood all the way down the staircase. Because of the great successes of Boltzmann's philosophical lectures, the Emperor invited him for a reception at the Palace. In 1906, Boltzmann's deteriorating mental condition forced him to resign his position, and his symptoms indicate he experienced what would today be diagnosed as
bipolar disorder Bipolar disorder, previously known as manic depression, is a mental disorder characterized by periods of Depression (mood), depression and periods of abnormally elevated Mood (psychology), mood that last from days to weeks each. If the elevat ...
. Four months later he died by suicide on 5 September 1906, by
hanging Hanging is the suspension of a person by a noose or ligature strangulation, ligature around the neck.Oxford English Dictionary, 2nd ed. Hanging as method of execution is unknown, as method of suicide from 1325. The ''Oxford English Dictionary' ...
himself while on vacation with his wife and daughter in
Duino Duino ( sl, Devin, german: Tybein) is today a seaside resort A seaside resort is a resort town, town, village, or hotel that serves as a Resort, vacation resort and is located on a coast. Sometimes the concept includes an aspect of official accre ...
, near
Trieste Trieste ( , ; sl, Trst ; german: Triest ) is a city and seaport in northeastern Italy. It is the capital city, and largest city, of the Regions of Italy#Autonomous regions with special statute, autonomous region of Friuli Venezia Giulia, one of ...
(then Austria). He is buried in the Viennese
Zentralfriedhof The Vienna Central Cemetery (german: Wiener Zentralfriedhof) is one of the largest Cemetery, cemeteries in the world by number of interred, and is the most well-known cemetery among Vienna's nearly 50 cemeteries. The cemetery's name is descripti ...
. His tombstone bears the inscription of Boltzmann's entropy formula: $S = k \cdot \log W$.

# Philosophy

Boltzmann's
kinetic theory of gases Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to: * Kinetic theory, describing a gas as particles in random motion * Kinetic energy, the energy of an object that it possesses due to its motion Art and ent ...
seemed to presuppose the reality of
atom Every atom is composed of a atomic nucleus, nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, l ...
s and
molecule A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioche ...
s, but almost all
German philosophers German(s) may refer to: * Germany Germany,, officially the Federal Republic of Germany, is a country in Central Europe. It is the second most populous country in Europe after Russia, and the most populous member state of the European ...
and many scientists like
Ernst Mach Ernst Waldfried Josef Wenzel Mach ( , ; 18 February 1838 – 19 February 1916) was a Moravian-born Austrian Empire, Austrian physicist and philosopher, who contributed to the physics of shock waves. The ratio of one's speed to speed of sound, t ...
and the physical chemist
Wilhelm Ostwald Friedrich Wilhelm Ostwald (; 4 April 1932) was a Baltic German chemist and German philosophy, philosopher. Ostwald is credited with being one of the founders of the field of physical chemistry, with Jacobus Henricus van 't Hoff, Walther Nernst, ...
disbelieved their existence.

# Physics

Boltzmann's most important scientific contributions were in kinetic theory, including for motivating the
Maxwell–Boltzmann distribution In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science ...
as a description of molecular speeds in a gas.
Maxwell–Boltzmann statistics In statistical mechanics, Maxwell–Boltzmann statistics describes the distribution of Classical physics, classical material particles over various energy states in thermal equilibrium. It is applicable when the temperature is high enough or the ...
and the
Boltzmann distribution In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability th ...
remain central in the foundations of classical statistical mechanics. They are also applicable to other
phenomena A phenomenon ( : phenomena) is an observable event. The term came into its modern philosophical Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, Epistemology ...
that do not require quantum statistics and provide insight into the meaning of
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer. Thermometers are calibrated in various Conversion of units of temperature, temp ...
. Most
chemist A chemist (from Greek ''chēm(ía)'' alchemy; replacing ''chymist'' from Medieval Latin ''alchemist'') is a scientist trained in the study of chemistry. Chemists study the composition of matter and its properties. Chemists carefully describe th ...
s, since the discoveries of
John Dalton John Dalton (; 5 or 6 September 1766 – 27 July 1844) was an English chemist, physicist and meteorologist. He is best known for introducing the atomic theory into chemistry, and for his research into Color blindness, colour blindness, which ...
in 1808, and
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and light ...
in Scotland and
Josiah Willard Gibbs Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in t ...
in the United States, shared Boltzmann's belief in
atom Every atom is composed of a atomic nucleus, nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, l ...
s and
molecule A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioche ...
s, but much of the
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
establishment did not share this belief until decades later. Boltzmann had a long-running dispute with the editor of the preeminent German physics journal of his day, who refused to let Boltzmann refer to atoms and molecules as anything other than convenient
theoretical A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or resear ...
constructs. Only a couple of years after Boltzmann's death, Perrin's studies of
colloid A colloid is a mixture in which one substance consisting of microscopically dispersed insoluble particles is suspended throughout another substance. Some definitions specify that the particles must be dispersed in a liquid, while others ext ...
al suspensions (1908–1909), based on Einstein's theoretical studies of 1905, confirmed the values of the Avogadro constant and the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...
, convincing the world that the tiny particles really exist. To quote Planck, "The
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
ic connection between
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
and
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
was first stated by L. Boltzmann in his
kinetic theory of gases Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to: * Kinetic theory, describing a gas as particles in random motion * Kinetic energy, the energy of an object that it possesses due to its motion Art and ent ...
". This famous formula for
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
''S'' is :$S = k_ \ln W$ where ''k''B is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...
, and ln is the
natural logarithm The natural logarithm of a number is its logarithm to the base (exponentiation), base of the mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natur ...
. ''W'' is ''Wahrscheinlichkeit'', a German word meaning the
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
of occurrence of a
macrostate In statistical mechanics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energ ...
or, more precisely, the number of possible
microstates A microstate or ministate is a sovereign state A sovereign state or sovereign country, is a polity, political entity represented by one central government that has supreme legitimate authority over territory. International law defines so ...
corresponding to the macroscopic state of a system — the number of (unobservable) "ways" in the (observable)
thermodynamic Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
state of a system that can be realized by assigning different positions and momenta to the various molecules. Boltzmann's
paradigm In science and philosophy, a paradigm () is a distinct set of concepts or thought patterns, including theories, research methods, postulates, and standards for what constitute legitimate contributions to a field. Etymology ''Paradigm'' comes f ...
was an
ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law The ideal gas law, also called the ge ...
of ''N'' ''identical'' particles, of which ''N''''i'' are in the ''i''th microscopic condition (range) of position and momentum. ''W'' can be counted using the formula for permutations :$W = N! \prod_i \frac$ where ''i'' ranges over all possible molecular conditions, and where $!$ denotes
factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...
. The "correction" in the denominator account for indistinguishable particles in the same condition. Boltzmann could also be considered one of the forerunners of quantum mechanics due to his suggestion in 1877 that the energy levels of a physical system could be discrete.

# Boltzmann equation

The Boltzmann equation was developed to describe the dynamics of an ideal gas. :$\frac+ v \frac+ \frac \frac = \frac\left.\_\mathrm$ where ''ƒ'' represents the distribution function of single-particle position and momentum at a given time (see the
Maxwell–Boltzmann distribution In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science ...
), ''F'' is a force, ''m'' is the mass of a particle, ''t'' is the time and ''v'' is an average velocity of particles. This equation describes the temporal and spatial variation of the probability distribution for the position and momentum of a density distribution of a cloud of points in single-particle
phase space In Dynamical systems theory, dynamical system theory, a phase space is a Space (mathematics), space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. F ...
. (See
Hamiltonian mechanics Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
.) The first term on the left-hand side represents the explicit time variation of the distribution function, while the second term gives the spatial variation, and the third term describes the effect of any force acting on the particles. The right-hand side of the equation represents the effect of collisions. In principle, the above equation completely describes the dynamics of an ensemble of gas particles, given appropriate
boundary conditions In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional Constraint (mathematics), constraints, called the boundary conditions. A solution to a boundary value pr ...
. This first-order
differential equation In mathematics, a differential equation is an functional equation, equation that relates one or more unknown function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the der ...
has a deceptively simple appearance, since ''f'' can represent an arbitrary single-particle distribution function. Also, the
force In physics, a force is an influence that can change the motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to accelerate. Force can ...
acting on the particles depends directly on the velocity distribution function ''f''. The Boltzmann equation is notoriously difficult to integrate.
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
spent years trying to solve it without any real success. The form of the collision term assumed by Boltzmann was approximate. However, for an ideal gas the standard Chapman–Enskog solution of the Boltzmann equation is highly accurate. It is expected to lead to incorrect results for an ideal gas only under
shock wave In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound The speed of sound is the distance travelled per unit of time by a sound wave as it propaga ...
conditions. Boltzmann tried for many years to "prove" the
second law of thermodynamics The second law of thermodynamics is a physical law based on universal experience concerning heat and Energy transformation, energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects ( ...
using his gas-dynamical equation — his famous
H-theorem In classical statistical mechanics, the ''H''-theorem, introduced by Ludwig Boltzmann in 1872, describes the tendency to decrease in the quantity ''H'' (defined below) in a nearly-ideal gas of molecules. L. Boltzmann,Weitere Studien über das Wä ...
. However the key assumption he made in formulating the collision term was "
molecular chaos In the kinetic theory of gases in physics, the molecular chaos hypothesis (also called ''Stosszahlansatz'' in the writings of Paul Ehrenfest) is the assumption that the velocities of colliding particles are uncorrelated, and independent of position ...
", an assumption which breaks
time-reversal symmetry T-symmetry or time reversal symmetry is the theoretical symmetry (physics), symmetry of physical laws under the Transformation (mathematics), transformation of time reversal, : T: t \mapsto -t. Since the second law of thermodynamics states that ...
as is necessary for ''anything'' which could imply the second law. It was from the probabilistic assumption alone that Boltzmann's apparent success emanated, so his long dispute with Loschmidt and others over
Loschmidt's paradox Loschmidt's paradox, also known as the reversibility paradox, irreversibility paradox or ', is the objection that it should not be possible to deduce an irreversible process from time-symmetric dynamics. This puts the time reversal symmetry of (a ...
ultimately ended in his failure. Finally, in the 1970s E.G.D. Cohen and J. R. Dorfman proved that a systematic (power series) extension of the Boltzmann equation to high densities is mathematically impossible. Consequently, nonequilibrium statistical mechanics for dense gases and liquids focuses on the
Green–Kubo relations The Green–Kubo relations (Melville S. Green 1954, Ryogo Kubo 1957) give the exact mathematical expression for transport coefficients \gamma in terms of integrals of Correlation function, time correlation functions: :\gamma = \int_0^\infty \left\l ...
, the
fluctuation theorem The fluctuation theorem (FT), which originated from statistical mechanics, deals with the relative probability that the Entropy (statistical thermodynamics), entropy of a system which is currently away from thermodynamic equilibrium (i.e., maximum ...
, and other approaches instead.

# Second thermodynamics law as a law of disorder

The idea that the
second law of thermodynamics The second law of thermodynamics is a physical law based on universal experience concerning heat and Energy transformation, energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects ( ...
or "entropy law" is a law of disorder (or that dynamically ordered states are "infinitely improbable") is due to Boltzmann's view of the second law of thermodynamics. In particular, it was Boltzmann's attempt to reduce it to a
stochastic Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themsel ...
collision function, or law of probability following from the random collisions of mechanical particles. Following Maxwell, Boltzmann modeled gas molecules as colliding billiard balls in a box, noting that with each collision nonequilibrium velocity distributions (groups of molecules moving at the same speed and in the same direction) would become increasingly disordered leading to a final state of macroscopic uniformity and maximum microscopic disorder or the state of maximum entropy (where the macroscopic uniformity corresponds to the obliteration of all field potentials or gradients). The second law, he argued, was thus simply the result of the fact that in a world of mechanically colliding particles disordered states are the most probable. Because there are so many more possible disordered states than ordered ones, a system will almost always be found either in the state of maximum disorder – the macrostate with the greatest number of accessible microstates such as a gas in a box at equilibrium – or moving towards it. A dynamically ordered state, one with molecules moving "at the same speed and in the same direction", Boltzmann concluded, is thus "the most improbable case conceivable...an infinitely improbable configuration of energy." Boltzmann accomplished the feat of showing that the second law of thermodynamics is only a statistical fact. The gradual disordering of energy is analogous to the disordering of an initially ordered pack of cards under repeated shuffling, and just as the cards will finally return to their original order if shuffled a gigantic number of times, so the entire universe must some-day regain, by pure chance, the state from which it first set out. (This optimistic coda to the idea of the dying universe becomes somewhat muted when one attempts to estimate the timeline which will probably elapse before it spontaneously occurs.) The tendency for entropy increase seems to cause difficulty to beginners in thermodynamics, but is easy to understand from the standpoint of the theory of probability. Consider two ordinary
dice Dice (singular die or dice) are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating Statistical randomness, random values, commonly as part of tabletop games, including List of dice game ...
, with both sixes face up. After the dice are shaken, the chance of finding these two sixes face up is small (1 in 36); thus one can say that the random motion (the agitation) of the dice, like the chaotic collisions of molecules because of thermal energy, causes the less probable state to change to one that is more probable. With millions of dice, like the millions of atoms involved in thermodynamic calculations, the probability of their all being sixes becomes so vanishingly small that the system ''must'' move to one of the more probable states."Collier's Encyclopedia", Volume 22 Sylt to Uruguay, Thermodynamics, by Leo Peters, p. 275

* * * * *

# Awards and honours

In 1885 he became a member of the Imperial
Austrian Academy of Sciences The Austrian Academy of Sciences (german: Österreichische Akademie der Wissenschaften, ÖAW) is a legal entity under the special protection of the Republic of Austria. According to the statutes of the Academy its mission is to promote the scie ...
and in 1887 he became the President of the
University of Graz The University of Graz (german: link=no, Karl-Franzens-Universität Graz, ), located in Graz, Austria, is the largest and oldest university in Styria, as well as the second-largest and second-oldest university in Austria. History The univers ...
. He was elected a member of the
Royal Swedish Academy of Sciences The Royal Swedish Academy of Sciences ( sv, Kungliga Vetenskapsakademien) is one of the royal academies of Sweden. Founded on 2 June 1739, it is an independent, non-governmental scientific organization that takes special responsibility for prom ...
in 1888 and a Foreign Member of the Royal Society (ForMemRS) in 1899. Numerous things are named in his honour.

# See also

*
Thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
* Boltzmann brain

# Further reading

* Roman Sexl & John Blackmore (eds.), "Ludwig Boltzmann – Ausgewahlte Abhandlungen", (Ludwig Boltzmann Gesamtausgabe, Band 8), Vieweg, Braunschweig, 1982. * John Blackmore (ed.), "Ludwig Boltzmann – His Later Life and Philosophy, 1900–1906, Book One: A Documentary History", Kluwer, 1995. * John Blackmore, "Ludwig Boltzmann – His Later Life and Philosophy, 1900–1906, Book Two: The Philosopher", Kluwer, Dordrecht, Netherlands, 1995. * John Blackmore (ed.), "Ludwig Boltzmann – Troubled Genius as Philosopher", in Synthese, Volume 119, Nos. 1 & 2, 1999, pp. 1–232. * * Boltzmann, ''Ludwig Boltzmann – Leben und Briefe'', ed., Walter Hoeflechner, Akademische Druck- u. Verlagsanstalt. Graz, Oesterreich, 1994 * Brush, Stephen G. (ed. & tr.), Boltzmann, ''Lectures on Gas Theory'', Berkeley, California: U. of California Press, 1964 * Brush, Stephen G. (ed.), ''Kinetic Theory'', New York: Pergamon Press, 1965 * * * * * Ehrenfest, P. & Ehrenfest, T. (1911) "Begriffliche Grundlagen der statistischen Auffassung in der Mechanik", in ''Encyklopädie der mathematischen Wissenschaften mit Einschluß ihrer Anwendungen'' Band IV, 2. Teil ( F. Klein and C. Müller (eds.). Leipzig: Teubner, pp. 3–90. Translated as ''The Conceptual Foundations of the Statistical Approach in Mechanics''. New York: Cornell University Press, 1959. * * * * * * * * * English translation by Morton Masius of the 2nd ed. of ''Waermestrahlung''. Reprinted by Dover (1959) & (1991). * Sharp, Kim (2019). ''Entropy and the Tao of Counting: A Brief Introduction to Statistical Mechanics and the Second Law of Thermodynamics'' (SpringerBriefs in Physics). Springer Nature. * Reprinted: Dover (1979).

# External links

* * * Ruth Lewin Sime, ''Lise Meitner: A Life in Physics'
Chapter One: Girlhood in Vienna
gives
Lise Meitner Elise Meitner ( , ; 7 November 1878 – 27 October 1968) was an Austrian-Swedish physicist who was one of those responsible for the discovery of the element protactinium and nuclear fission. While working at the Kaiser Wilhelm Institute on rad ...
's account of Boltzmann's teaching and career. * Eftekhari, Ali,
Ludwig Boltzmann (1844–1906).
Discusses Boltzmann's philosophical opinions, with numerous quotes. * * * {{DEFAULTSORT:Boltzmann, Ludwig 1844 births 1906 suicides Scientists from Vienna 19th-century Austrian physicists Thermodynamicists Fluid dynamicists Burials at the Vienna Central Cemetery University of Vienna alumni Members of the Royal Swedish Academy of Sciences Corresponding members of the Saint Petersburg Academy of Sciences Suicides by hanging in Austria Foreign Members of the Royal Society Foreign associates of the National Academy of Sciences Mathematical physicists Theoretical physicists Rectors of universities in Austria 19th-century Austrian philosophers 20th-century Austrian philosophers Members of the Göttingen Academy of Sciences and Humanities