**Ludwig Eduard Boltzmann** (February 20, 1844 – September 5, 1906) was an

AustriaAustria , officially the Republic of Austria , is a landlocked country of roughly 8.4 million people in Central Europe. It is bordered by the Czech Republic and Germany to the north, Slovakia and Hungary to the east, Slovenia and Italy to the south, and Switzerland and Liechtenstein to the...

n

physicistA physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...

famous for his founding contributions in the fields of

statistical mechanicsStatistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

and statistical thermodynamics. He was one of the most important advocates for

atomic theoryIn chemistry and physics, atomic theory is a theory of the nature of matter, which states that matter is composed of discrete units called atoms, as opposed to the obsolete notion that matter could be divided into any arbitrarily small quantity...

at a time when that scientific model was still highly controversial.

### Childhood and education

Boltzmann was born in

ViennaVienna is the capital and largest city of the Republic of Austria and one of the nine states of Austria. Vienna is Austria's primary city, with a population of about 1.723 million , and is by far the largest city in Austria, as well as its cultural, economic, and political centre...

, the capital of the

Austrian EmpireThe Austrian Empire was a modern era successor empire, which was centered on what is today's Austria and which officially lasted from 1804 to 1867. It was followed by the Empire of Austria-Hungary, whose proclamation was a diplomatic move that elevated Hungary's status within the Austrian Empire...

. His father, Ludwig Georg Boltzmann, was a tax official. His grandfather, who had moved to Vienna from Berlin, was a clock manufacturer, and Boltzmann’s mother, Katharina Pauernfeind, was originally from

Salzburg-Population development:In 1935, the population significantly increased when Salzburg absorbed adjacent municipalities. After World War II, numerous refugees found a new home in the city. New residential space was created for American soldiers of the postwar Occupation, and could be used for...

. He received his primary education from a private tutor at the home of his parents. Boltzmann attended high school in

LinzLinz is the third-largest city of Austria and capital of the state of Upper Austria . It is located in the north centre of Austria, approximately south of the Czech border, on both sides of the river Danube. The population of the city is , and that of the Greater Linz conurbation is about...

,

Upper AustriaUpper Austria is one of the nine states or Bundesländer of Austria. Its capital is Linz. Upper Austria borders on Germany and the Czech Republic, as well as on the other Austrian states of Lower Austria, Styria, and Salzburg...

. At age 15, Boltzmann lost his father.

Boltzmann studied

physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

at the

University of ViennaThe University of Vienna is a public university located in Vienna, Austria. It was founded by Duke Rudolph IV in 1365 and is the oldest university in the German-speaking world...

, starting in 1863. Among his teachers were

Josef LoschmidtJan or Johann Josef Loschmidt , who referred to himself mostly as 'Josef' , was a notable Austrian scientist who performed groundbreaking work in chemistry, physics , and crystal forms.Born in Carlsbad, a town located in the Austrian Empire , Loschmidt...

,

Joseph StefanJoseph Stefan was a physicist, mathematician, and poet of Slovene mother tongue and Austrian citizenship.- Life and work :...

,

Andreas von EttingshausenAndreas Freiherr von Ettingshausen was a German mathematician and physicist.Ettingshausen studied philosophy and jurisprudence in Vienna. In 1817, he joined the University of Vienna and taught mathematics and physics...

and

Jozef PetzvalJoseph Petzval was a Hungarian / Slovak mathematician, inventor, and physicist of German origin, born in Upper Hungary .He is best known for his work in optics...

. Boltzmann received his PhD degree in 1866 working under the supervision of Stefan; his dissertation was on kinetic theory of gases. In 1867 he became a

PrivatdozentPrivatdozent or Private lecturer is a title conferred in some European university systems, especially in German-speaking countries, for someone who pursues an academic career and holds all formal qualifications to become a tenured university professor...

(lecturer). After obtaining his doctorate degree, Boltzmann worked two more years as Stefan’s assistant. It was Stefan who introduced Boltzmann to

Maxwell'sJames Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...

work.

### Academic career

In 1869 at age 25, he was appointed full Professor of Mathematical Physics at the

University of GrazThe University of Graz , a university located in Graz, Austria, is the second-largest and second-oldest university in Austria....

in the province of Styria. In 1869 he spent several months in

Heidelberg-Early history:Between 600,000 and 200,000 years ago, "Heidelberg Man" died at nearby Mauer. His jaw bone was discovered in 1907; with scientific dating, his remains were determined to be the earliest evidence of human life in Europe. In the 5th century BC, a Celtic fortress of refuge and place of...

working with

Robert BunsenRobert Wilhelm Eberhard Bunsen was a German chemist. He investigated emission spectra of heated elements, and discovered caesium and rubidium with Gustav Kirchhoff. Bunsen developed several gas-analytical methods, was a pioneer in photochemistry, and did early work in the field of organoarsenic...

and

Leo KönigsbergerLeo Königsberger was a German mathematician, and historian of science. He is best known for his three-volume biography of Hermann von Helmholtz, which remains the standard reference on the subject.-Biography:...

and then in 1871 he was with

Gustav KirchhoffGustav Robert Kirchhoff was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects...

and

Hermann von HelmholtzHermann Ludwig Ferdinand von Helmholtz was a German physician and physicist who made significant contributions to several widely varied areas of modern science...

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 unofficially audit lectures. Boltzmann advised her to appeal, which she did, successfully. On July 17, 1876 Ludwig Boltzmann married Henriette; they had three daughters and two sons. Boltzmann went back to

GrazThe more recent population figures do not give the whole picture as only people with principal residence status are counted and people with secondary residence status are not. Most of the people with secondary residence status in Graz are students...

to take up the chair of Experimental Physics. Among his students in Graz were

Svante ArrheniusSvante August Arrhenius was a Swedish scientist, originally a physicist, but often referred to as a chemist, and one of the founders of the science of physical chemistry...

and

Walther NernstWalther Hermann Nernst FRS was a German physical chemist and physicist who is known for his theories behind the calculation of chemical affinity as embodied in the third law of thermodynamics, for which he won the 1920 Nobel Prize in chemistry...

. He spent 14 happy years in Graz and it was there that he developed his statistical concept of nature. In 1885 he became a member of the Imperial

Austrian Academy of SciencesThe Austrian Academy of Sciences is a legal entity under the special protection of the Federal Republic of Austria. According to the statutes of the Academy its mission is to promote the sciences and humanities in every respect and in every field, particularly in fundamental research...

and in 1887 he became the President of the

University of GrazThe University of Graz , a university located in Graz, Austria, is the second-largest and second-oldest university in Austria....

. He was elected a member of the

Royal Swedish Academy of SciencesThe Royal Swedish Academy of Sciences or Kungliga Vetenskapsakademien is one of the Royal Academies of Sweden. The Academy is an independent, non-governmental scientific organization which acts to promote the sciences, primarily the natural sciences and mathematics.The Academy was founded on 2...

in 1888.

Boltzmann was appointed to the Chair of Theoretical Physics at the University of Munich in

BavariaBavaria, formally the Free State of Bavaria is a state of Germany, located in the southeast of Germany. With an area of , it is the largest state by area, forming almost 20% of the total land area of Germany...

, Germany in 1890.

In 1893, Boltzmann succeeded his teacher Joseph Stefan as Professor of Theoretical Physics at the University of Vienna.

### Final years

Boltzmann spent a great deal of effort in his final years defending his theories. He did not get along with some of his colleagues in Vienna, particularly

Ernst MachErnst Mach was an Austrian physicist and philosopher, noted for his contributions to physics such as the Mach number and the study of shock waves...

, who became a professor of philosophy and history of sciences in 1895. That same year

Georg HelmGeorg Ferdinand Helm was a German mathematician.Helm graduated from high school from the Annenschule in Dresden in 1867. Thereafter he studied mathematics and natural sciences at the Dresden Polytechnical School, and then at the universities of Leipzig and Berlin...

and

Wilhelm OstwaldFriedrich Wilhelm Ostwald was a Baltic German chemist. He received the Nobel Prize in Chemistry in 1909 for his work on catalysis, chemical equilibria and reaction velocities...

presented their position on Energetics, at a meeting in

LübeckThe Hanseatic City of Lübeck is the second-largest city in Schleswig-Holstein, in northern Germany, and one of the major ports of Germany. It was for several centuries the "capital" of the Hanseatic League and, because of its Brick Gothic architectural heritage, is listed by UNESCO as a World...

in 1895. They saw energy, and not matter, as the chief component of the universe. However, Boltzmann's position carried the day among other physicists who supported his atomic theories in the debate. Thereafter in 1900, Boltzmann went to the

University of LeipzigThe University of Leipzig , located in Leipzig in the Free State of Saxony, Germany, is one of the oldest universities in the world and the second-oldest university in Germany...

, on the invitation of

Wilhelm OstwaldFriedrich Wilhelm Ostwald was a Baltic German chemist. He received the Nobel Prize in Chemistry in 1909 for his work on catalysis, chemical equilibria and reaction velocities...

. After the retirement of Mach due to bad health, Boltzmann came back to Vienna in 1902. In 1903 he founded the

Austrian Mathematical SocietyThe Austrian Mathematical Society is the national mathematical society of Austria and a member society of the European Mathematical Society.-History:...

together with

Gustav von EscherichGustav Ritter von Escherich was an Austrian mathematician.-Biography:Born in Mantua, he studied mathematics and physics at the University of Vienna. From 1876 to 1879 he was professor at the University of Graz...

and

Emil Müller-Biography:Born in Lanškroun, he studied mathematics and physics at the University of Vienna and Vienna University of Technology. In 1898 he defended his dissertation at the University of Königsberg with Wilhelm Franz Meyer. One year later he received his habilitation at the same university...

. His students included

Karl PrzibramKarl Přibram , also known as “Karl Pribram”, was an Austrian-born economist. He is most noted for his work in labor economics, in industrial organization, and in the history of economic thought....

,

Paul EhrenfestPaul Ehrenfest was an Austrian and Dutch physicist, who made major contributions to the field of statistical mechanics and its relations with quantum mechanics, including the theory of phase transition and the Ehrenfest theorem.- Biography :Paul Ehrenfest was born and grew up in Vienna in a Jewish...

and

Lise MeitnerLise Meitner FRS was an Austrian-born, later Swedish, physicist who worked on radioactivity and nuclear physics. Meitner was part of the team that discovered nuclear fission, an achievement for which her colleague Otto Hahn was awarded the Nobel Prize...

.

In Vienna, Boltzmann not only taught physics but also lectured on philosophy. Boltzmann’s lectures on

natural philosophyNatural philosophy or the philosophy of nature , is a term applied to the study of nature and the physical universe that was dominant before the development of modern science...

were very popular, and received a considerable attention at that time. 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.

Boltzmann was subject to rapid alternation of depressed moods with elevated, expansive or irritable moods, likely the symptoms of undiagnosed

bipolar disorderBipolar disorder or bipolar affective disorder, historically known as manic–depressive disorder, is a psychiatric diagnosis that describes a category of mood disorders defined by the presence of one or more episodes of abnormally elevated energy levels, cognition, and mood with or without one or...

. He himself jestingly attributed his rapid swings in temperament to the fact that he was born during the night between

Mardi GrasThe terms "Mardi Gras" , "Mardi Gras season", and "Carnival season", in English, refer to events of the Carnival celebrations, beginning on or after Epiphany and culminating on the day before Ash Wednesday...

and

Ash WednesdayAsh Wednesday, in the calendar of Western Christianity, is the first day of Lent and occurs 46 days before Easter. It is a moveable fast, falling on a different date each year because it is dependent on the date of Easter...

. Meitner relates that those who were close to Boltzmann were aware of his bouts of severe depression and his suicide attempts.

On September 5, 1906, while on a summer vacation in

DuinoDuino is a town at the Adriatic coast in the municipality of Duino-Aurisina, part of the region of Friuli – Venezia Giulia in the province of Trieste, north-eastern Italy....

, near

TriesteTrieste is a city and seaport in northeastern Italy. It is situated towards the end of a narrow strip of land lying between the Adriatic Sea and Italy's border with Slovenia, which lies almost immediately south and east of the city...

, Boltzmann hanged himself during an attack of

depressionMood disorder is the term designating a group of diagnoses in the Diagnostic and Statistical Manual of Mental Disorders classification system where a disturbance in the person's mood is hypothesized to be the main underlying feature...

. He is buried in the Viennese

ZentralfriedhofThe Zentralfriedhof is one of the largest cemeteries in the world, largest by number of interred in Europe and most famous cemetery among Vienna's nearly 50 cemeteries.-Name and location:...

; his tombstone bears the inscription

## Philosophy

Boltzmann's kinetic theory of gases seemed to presuppose the reality of

atomThe atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

s and

moleculeA molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...

s, but almost all German philosophers and many scientists like

Ernst MachErnst Mach was an Austrian physicist and philosopher, noted for his contributions to physics such as the Mach number and the study of shock waves...

and the physical chemist

Wilhelm OstwaldFriedrich Wilhelm Ostwald was a Baltic German chemist. He received the Nobel Prize in Chemistry in 1909 for his work on catalysis, chemical equilibria and reaction velocities...

opposed their existence. During the 1890s Boltzmann attempted to formulate a compromise position which would allow both atomists and anti-atomists to do physics without arguing over atoms. His solution was to use Hertz's theory that atoms were "Bilder", that is, models or pictures. Atomists could think the pictures were the real atoms while the anti-atomists could think of the pictures as representing a useful but unreal model, but this did not fully satisfy either group. Furthermore, Ostwald and many defenders of "pure thermodynamics" were trying hard to refute the kinetic theory of gases and statistical mechanics because of Boltzmann's assumptions about atoms and molecules and especially statistical interpretation of the second law.

Around the turn of the century, Boltzmann's science was being threatened by another philosophical objection. Some physicists, including Mach's student,

Gustav JaumannGustav Jaumann was an Austrian physicist.An assistant to the physicist Ernst Mach, he was gifted in mathematics and an opponent of the reality of small particles like electrons and atoms...

, interpreted Hertz to mean that all electromagnetic behavior is continuous, as if there were no atoms and molecules, and likewise as if all physical behavior were ultimately electromagnetic. This movement around 1900 deeply depressed Boltzmann since it could mean the end of his kinetic theory and statistical interpretation of the second law of thermodynamics.

After Mach's resignation in Vienna in 1901, Boltzmann returned there and decided to become a philosopher himself to refute philosophical objections to his physics, but he soon became discouraged again. In 1904 at a physics conference in St. Louis most physicists seemed to reject atoms and he was not even invited to the physics section. Rather, he was stuck in a section called "applied mathematics," he violently attacked philosophy, especially on allegedly Darwinian grounds but actually in terms of Lamarck's theory of the inheritance of acquired characteristics that people inherited bad philosophy from the past and that it was hard for scientists to overcome such inheritance.

In 1905 Boltzmann corresponded extensively with the Austro-German philosopher

Franz BrentanoFranz Clemens Honoratus Hermann Brentano was an influential German philosopher and psychologist whose influence was felt by other such luminaries as Sigmund Freud, Edmund Husserl, Kazimierz Twardowski and Alexius Meinong, who followed and adapted his views.-Life:Brentano was born at Marienberg am...

in hope of mastering philosophy better apparently so that he could refute its presence in science better, but he became discouraged about this approach as well. In the following year 1906 his mental condition became so bad that he had to resign his position. He committed suicide in September of that same year by hanging himself while on vacation.

## Physics

Boltzmann's most important scientific contributions were in

kinetic theoryThe kinetic theory of gases describes a gas as a large number of small particles , all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container...

, including the

Maxwell–Boltzmann distribution for molecular speeds in a gas. In addition,

Maxwell–Boltzmann statisticsIn statistical mechanics, Maxwell–Boltzmann statistics describes the statistical distribution of material particles over various energy states in thermal equilibrium, when the temperature is high enough and density is low enough to render quantum effects negligible.The expected number of particles...

and the

Boltzmann distributionIn chemistry, physics, and mathematics, the Boltzmann distribution is a certain distribution function or probability measure for the distribution of the states of a system. It underpins the concept of the canonical ensemble, providing its underlying distribution...

over energies remain the foundations of

classicalIn physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

statistical mechanics. They are applicable to the many

phenomenaA phenomenon , plural phenomena, is any observable occurrence. Phenomena are often, but not always, understood as 'appearances' or 'experiences'...

that do not require quantum statistics and provide a remarkable insight into the meaning of

temperatureThermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. Thermodynamic temperature is an "absolute" scale because it is the measure of the fundamental property underlying temperature: its null or zero point, absolute zero, is the...

.

Much of the

physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

establishment did not share his belief in the reality of

atomThe atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

s and

moleculeA molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...

s — a belief shared, however, by

MaxwellJames Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...

in

ScotlandScotland is a country that is part of the United Kingdom. Occupying the northern third of the island of Great Britain, it shares a border with England to the south and is bounded by the North Sea to the east, the Atlantic Ocean to the north and west, and the North Channel and Irish Sea to the...

and

GibbsJosiah Willard Gibbs was an American theoretical physicist, chemist, and mathematician. He devised much of the theoretical foundation for chemical thermodynamics as well as physical chemistry. As a mathematician, he invented vector analysis . Yale University awarded Gibbs the first American Ph.D...

in the United States; and by most

chemistsChemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....

since the discoveries of

John DaltonJohn Dalton FRS was an English chemist, meteorologist and physicist. He is best known for his pioneering work in the development of modern atomic theory, and his research into colour blindness .-Early life:John Dalton was born into a Quaker family at Eaglesfield, near Cockermouth, Cumberland,...

in 1808. He 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 constructs. Only a couple of years after Boltzmann's death,

Perrin'sJean Baptiste Perrin was a French physicist and Nobel laureate.-Early years:Born in Lille, France, Perrin attended the École Normale Supérieure, the elite grande école in Paris. He became an assistant at the school during the period of 1894-97 when he began the study of cathode rays and X-rays...

studies of

colloidA colloid is a substance microscopically dispersed evenly throughout another substance.A colloidal system consists of two separate phases: a dispersed phase and a continuous phase . A colloidal system may be solid, liquid, or gaseous.Many familiar substances are colloids, as shown in the chart below...

al suspensions (1908–1909), based on

Einstein'sAlbert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

theoretical studies of 1905, confirmed the values of

Avogadro's numberIn chemistry and physics, the Avogadro constant is defined as the ratio of the number of constituent particles N in a sample to the amount of substance n through the relationship NA = N/n. Thus, it is the proportionality factor that relates the molar mass of an entity, i.e...

and

Boltzmann's constant, and convinced the world that the tiny particles really exist.

To quote

PlanckMax Karl Ernst Ludwig Planck, ForMemRS, was a German physicist who actualized the quantum physics, initiating a revolution in natural science and philosophy. He is regarded as the founder of the quantum theory, for which he received the Nobel Prize in Physics in 1918.-Life and career:Planck came...

, "The

logarithmThe logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

ic connection between

entropyEntropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

and

probabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

was first stated by L. Boltzmann in his

kinetic theoryThe kinetic theory of gases describes a gas as a large number of small particles , all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container...

of gases". This famous formula for entropy S is

where k = 1.3806505(24) × 10

^{−23} JThe joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...

K^{−1}The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...

is

Boltzmann's constant, and the

logarithmThe logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

is taken to the natural base e. W is the Wahrscheinlichkeit, the

frequencyIn statistics the frequency of an event i is the number ni of times the event occurred in the experiment or the study. These frequencies are often graphically represented in histograms....

of occurrence of a macrostate or, more precisely, the number of possible

microstatesIn statistical mechanics, a microstate is a specific microscopic configuration of a thermodynamic system that the system may occupy with a certain probability in the course of its thermal fluctuations...

corresponding to the macroscopic state of a system — number of (unobservable) "ways" in the (observable)

thermodynamicThermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

state of a system can be realized by assigning different

positionsIn geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...

and

momentaIn classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

to the various molecules. Boltzmann’s paradigm was an

ideal gasAn ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...

of N identical particles, of which N

_{i} are in the ith microscopic condition (range) of position and momentum. W can be counted using the formula for permutations

where i ranges over all possible molecular conditions. (

denotes

factorialIn mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

.) The "correction" in the denominator is because identical particles in the same condition are

indistinguishableIdentical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. Species of identical particles include elementary particles such as electrons, and, with some clauses, composite particles such as atoms and molecules.There are two...

.

Boltzmann was also one of the founders of quantum mechanics due to his suggestion in 1877 that the energy levels of a physical system could be discrete.

The equation for S is engraved on Boltzmann's

tombstoneA headstone, tombstone, or gravestone is a marker, usually stone, that is placed over a grave. In most cases they have the deceased's name, date of birth, and date of death inscribed on them, along with a personal message, or prayer.- Use :...

at the Vienna

ZentralfriedhofThe Zentralfriedhof is one of the largest cemeteries in the world, largest by number of interred in Europe and most famous cemetery among Vienna's nearly 50 cemeteries.-Name and location:...

— his second grave.

## The Boltzmann equation

The Boltzmann equation was developed to describe the dynamics of an ideal gas.

where ƒ represents the distribution function of single-particle position and momentum at a given time (see the

Maxwell–Boltzmann distribution), 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

temporalTime is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

and

spatialSpace is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...

variation of the probability distribution for the position and momentum of a density distribution of a cloud of points in single-particle

phase spaceIn mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...

. (See

Hamiltonian mechanicsHamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...

.) 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. This first-order

differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

has a deceptively simple appearance, since ƒ can represent an arbitrary single-particle distribution function. Also, the

forceIn physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

acting on the particles depends directly on the velocity distribution function ƒ. The Boltzmann equation is notoriously difficult to

integrateIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

.

David HilbertDavid Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

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 waveA shock wave is a type of propagating disturbance. Like an ordinary wave, it carries energy and can propagate through a medium or in some cases in the absence of a material medium, through a field such as the electromagnetic field...

conditions.

Boltzmann tried for many years to "prove" the

second law of thermodynamicsThe second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...

using his gas-dynamical equation — his famous

H-theoremIn Classical Statistical Mechanics, the H-theorem, introduced by Ludwig Boltzmann in 1872, describes the increase in the entropy of an ideal gas in an irreversible process. H-theorem follows from considerations of Boltzmann's equation...

. However the key assumption he made in formulating the collision term was "

molecular chaosIn kinetic theory in physics, molecular chaos is the assumption that the velocities of colliding particles are uncorrelated, and independent of position...

", an assumption which breaks

time-reversal symmetryCPT symmetry is a fundamental symmetry of physical laws under transformations that involve the inversions of charge, parity, and time simultaneously.-History:...

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

LoschmidtJan or Johann Josef Loschmidt , who referred to himself mostly as 'Josef' , was a notable Austrian scientist who performed groundbreaking work in chemistry, physics , and crystal forms.Born in Carlsbad, a town located in the Austrian Empire , Loschmidt...

and others over

Loschmidt's paradoxLoschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from time-symmetric dynamics...

ultimately ended in his failure.

Finally, in the 1970s

E.G.D. CohenE.G.D. Cohen is an American physicist, and is Professor Emeritus at The Rockefeller University. He is widely recognised for his contributions to statistical physics. In 2004 Cohen was awarded the Boltzmann Medal, jointly with Prof. H. Eugene Stanley...

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

liquidLiquid is one of the three classical states of matter . Like a gas, a liquid is able to flow and take the shape of a container. Some liquids resist compression, while others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly...

s focuses on the Green–Kubo relations, the

fluctuation theoremThe fluctuation theorem , which originated from statistical mechanics, deals with the relative probability that the entropy of a system which is currently away from thermodynamic equilibrium will increase or decrease over a given amount of time...

, and other approaches instead.

## The Second Law as a law of disorder

The idea that the

second law of thermodynamicsThe second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...

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. In particular, it was his attempt to reduce it to a

stochasticStochastic refers to systems whose behaviour is intrinsically non-deterministic. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E...

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

codaCoda can denote any concluding event, summation, or section.Coda may also refer to:-Acronyms:* Calgary Olympic Development Association, the former name of the Canadian Winter Sport Institute, a non profit organization...

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

diceA die is a small throwable object with multiple resting positions, used for generating random numbers...

, 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. However, mathematically the odds of all the dice results not being a pair sixes is also as hard as the ones of all of them being sixes, and since statistically the data tend to balance, one in every 36 pairs of dice will tend to be a pair of sixes. And the cards, when shuffled, will sometimes present a certain temporary sequence order even if in its whole they are disordered.

## Energetics of evolution

Boltzmann's views played an essential role in the development of

energeticsEnergetics is the study of energy under transformation. Because energy flows at all scales, from the quantum level to the biosphere and cosmos, energetics is a very broad discipline, encompassing for example thermodynamics, chemistry, biological energetics, biochemistry and ecological energetics...

, the scientific study of energy flows under transformation. In 1922, for example,

Alfred J. LotkaAlfred James Lotka was a US mathematician, physical chemist, and statistician, famous for his work in population dynamics and energetics. An American biophysicist best known for his proposal of the predator-prey model, developed simultaneously but independently of Vito Volterra...

referred to Boltzmann as one of the first proponents of the proposition that available energy can be understood as the fundamental object under contention in the biological, or life-struggle and therefore also in the evolution of the organic world. Lotka interpreted Boltzmann's view to imply that available energy could be the central concept that unified physics and biology as a quantitative physical principle of evolution. In the foreword to Boltzmann's Theoretical Physics and Philosophical Problems, S.R. de Groot noted that

Howard T. OdumHoward Thomas Odum was an American ecologist...

later sought to develop these views when looking at the evolution of ecological systems, and suggested that the

maximum power principleThe maximum power principle has been proposed as the fourth principle of energetics in open system thermodynamics, where an example of an open system is a biological cell. According to Howard T...

was an example of Darwin's law of

natural selectionNatural selection is the nonrandom process by which biologic traits become either more or less common in a population as a function of differential reproduction of their bearers. It is a key mechanism of evolution....

.

## See also

- Boltzmann's Energy Equipartition theorem
In classical statistical mechanics, the equipartition theorem is a general formula that relates the temperature of a system with its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition...

- Boltzmann brain
A Boltzmann brain is a hypothesized self-aware entity which arises due to random fluctuations out of a state of chaos. The idea is named for the physicist Ludwig Boltzmann , who advanced an idea that the known universe arose as a random fluctuation, similar to a process through which Boltzmann...

- Boltzmann machine
A Boltzmann machine is a type of stochastic recurrent neural network invented by Geoffrey Hinton and Terry Sejnowski. Boltzmann machines can be seen as the stochastic, generative counterpart of Hopfield nets...

- History of the molecule
In chemistry, the history of molecular theory traces the origins of the concept or idea of the existence of strong chemical bonds between two or more atoms....

- Lattice Boltzmann methods
Lattice Boltzmann methods is a class of computational fluid dynamics methods for fluid simulation. Instead of solving the Navier–Stokes equations, the discrete Boltzmann equation is solved to simulate the flow of a Newtonian fluid with collision models such as Bhatnagar-Gross-Krook...

, used in computational fluid dynamicsComputational fluid dynamics, usually abbreviated as CFD, is a branch of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with...

- Philosophy of thermal and statistical physics
The philosophy of thermal and statistical physics is that part of the philosophy of physics whose subject matter is classical thermodynamics, statistical mechanics, and related theories...

- Ludwig Boltzmann Gesellschaft
The Ludwig Boltzmann Gesellschaft ' is an Austrian network of specialized research institutes that are not part of a university. It was founded in 1961 and named after physicist Ludwig Boltzmann. As of 1999, the Ludwig Boltzmann Gesellschaft comprised 131 institutes in the fields of medicine,...

- Boltzmann Medal
The Boltzmann Medal is the most important prize awarded to physicists that obtain new results concerning statistical mechanics; it is named after the celebrated physicist Ludwig Boltzmann...

## 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. ISBN 978-0-7923-3231-2
- John Blackmore, "Ludwig Boltzmann – His Later Life and Philosophy, 1900–1906, Book Two: The Philosopher", Kluwer, Dordrecht, Netherlands, 1995. ISBN 978-0-7923-3464-4
- John Blackmore (ed.), "Ludwig Boltzmann – Troubled Genius as Philosopher", in Synthese, Volume 119, Nos. 1 & 2, 1999, pp. 1–232.
- Brush, Stephen G. (ed. & tr.), Boltzmann, Lectures on Gas Theory, Berkeley, CA: U. of California Press, 1964
- Brush, Stephen G. (ed.), Kinetic Theory, New York: Pergamon Press, 1965
- Boltzmann, Ludwig Boltzmann – Leben und Briefe, ed., Walter Hoeflechner, Akademische Druck- u. Verlagsanstalt. Graz, Oesterreich, 1994
- P. Ehrenfest
Paul Ehrenfest was an Austrian and Dutch physicist, who made major contributions to the field of statistical mechanics and its relations with quantum mechanics, including the theory of phase transition and the Ehrenfest theorem.- Biography :Paul Ehrenfest was born and grew up in Vienna in a Jewish...

& T. Ehrenfest (1911) "Begriffliche Grundlagen der statistischen Auffassung in der Mechanik", in Encyklopädie der mathematischen Wissenschaften mit Einschluß ihrer AnwendungenIn mathematics, Klein’s encyclopedia refers to a German mathematical encyclopedia published in six volumes from 1898 to 1933. Felix Klein and Wilhelm Meyer were organizers of the encyclopedia. Its title in English is "Encyclopedia of mathematical sciences including their applications", which is...

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. ISBN 0-486-49504-3 Reprinted: Dover (1979). ISBN 0-486-63896-0 English translation by Morton Masius of the 2nd ed. of Waermestrahlung. Reprinted by Dover (1959) & (1991). ISBN 0-486-66811-8

## External links

- "Ludwig Boltzmann," Universität Wien (German).
- Ruth Lewin Sime, Lise Meitner: A Life in Physics Chapter One: Girlhood in Vienna gives Lise Meitner
Lise Meitner FRS was an Austrian-born, later Swedish, physicist who worked on radioactivity and nuclear physics. Meitner was part of the team that discovered nuclear fission, an achievement for which her colleague Otto Hahn was awarded the Nobel Prize...

's account of Boltzmann's teaching and career.
- E.G.D. Cohen
E.G.D. Cohen is an American physicist, and is Professor Emeritus at The Rockefeller University. He is widely recognised for his contributions to statistical physics. In 2004 Cohen was awarded the Boltzmann Medal, jointly with Prof. H. Eugene Stanley...

, 1996, "Boltzmann and Statistical Mechanics."
- Eftekhari, Ali, "Ludwig Boltzmann (1844–1906)." Discusses Boltzmann's philosophical opinions, with numerous quotes.
- Jacob Bronowski
Jacob Bronowski was a Polish-Jewish British mathematician, biologist, historian of science, theatre author, poet and inventor...

from "The Ascent Of Man"