\right|_\mathrm{collision} </math>

where represents the distribution function of single-particle position and momentum at a given time (see the Maxwell–Boltzmann distribution), is a force, is the mass of a particle, is the time and 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. (See Hamiltonian mechanics.) 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 equation has a deceptively simple appearance, since can represent an arbitrary single-particle distribution function. Also, the force acting on the particles depends directly on the velocity distribution function&nbsp;. The Boltzmann equation is notoriously difficult to integrate. David Hilbert 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 conditions.

Boltzmann tried for many years to "prove" the second law of thermodynamics using his gas-dynamical equation – his famous H-theorem. However the key assumption he made in formulating the collision term was "molecular chaos", an assumption which breaks time-reversal symmetry 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 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 fluctuation theorem, and other approaches instead.

Second thermodynamics law as a law of disorder

thumb|right|Boltzmann's grave in the [[Zentralfriedhof, Vienna, with bust and entropy formula]]

The idea that the second law of thermodynamics 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 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, 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.

Legacy and impact on modern science

Ludwig Boltzmann's contributions to physics and philosophy have left a lasting impact on modern science. His pioneering work in statistical mechanics and thermodynamics laid the foundation for some of the most fundamental concepts in physics. For instance, Max Planck in quantizing resonators in his Black Body theory of radiation used the Boltzmann constant to describe the entropy of the system to arrive at his formula in 1900. However, Boltzmann's work was not always readily accepted during his lifetime, and he faced opposition from some of his contemporaries, particularly in regard to the existence of atoms and molecules. Nevertheless, the validity and importance of his ideas were eventually recognized, and they have since become cornerstones of modern physics. Here, we delve into some aspects of Boltzmann's legacy and his influence on various areas of science.

Atomic theory and the existence of atoms and molecules

Boltzmann's kinetic theory of gases was one of the first attempts to explain macroscopic properties, such as pressure and temperature, in terms of the behaviour of individual atoms and molecules. Although many chemists were already accepting the existence of atoms and molecules, the broader physics community took some time to embrace this view. Boltzmann's long-running dispute with the editor of a prominent German physics journal over the acceptance of atoms and molecules underscores the initial resistance to this idea.

It was only after experiments (including Jean Perrin's studies of colloidal suspensions) confirmed the values of the Avogadro constant and the Boltzmann constant that the existence of atoms and molecules gained wider acceptance. Boltzmann's kinetic theory played a crucial role in demonstrating the reality of atoms and molecules and explaining various phenomena in gases, liquids, and solids.

Statistical mechanics and the Boltzmann constant

Statistical mechanics, which Boltzmann pioneered, connects macroscopic observations with microscopic behaviors. His statistical explanation of the second law of thermodynamics was a significant achievement, and he provided the current definition of entropy (<math>S = k_{\rm B} \ln \Omega</math>), where is the Boltzmann constant and Ω is the number of microstates corresponding to a given macrostate.

Max Planck later named the constant as the Boltzmann constant in honor of Boltzmann's contributions to statistical mechanics. The Boltzmann constant is now a fundamental constant in physics and across many scientific disciplines.

Boltzmann equation and modern uses

Because the Boltzmann equation is practical in solving problems in rarefied or dilute gases, it has been used in many diverse areas of technology. It has been used to calculate Space Shuttle re-entry in the upper atmosphere. It is the basis for neutron transport theory, and ion transport in semiconductors.

Influence on quantum mechanics

Boltzmann's work in statistical mechanics laid the groundwork for understanding the statistical behavior of particles in systems with a large number of degrees of freedom. His 1877 paper on the kinetic theory of heat used discrete energy levels of physical systems as a mathematical device, and went on to show that the same approach could be applied to continuous systems. This might be seen as a forerunner to the development of quantum mechanics. One biographer of Boltzmann says that Boltzmann’s approach “pav[ed] the way for Planck.”

Quantization of energy levels became a fundamental postulate in quantum mechanics, leading to groundbreaking theories like quantum electrodynamics and quantum field theory. Thus, Boltzmann's early insights into the quantization of energy levels had a profound influence on the development of quantum physics.

Awards and honours

In 1885, he became a member of the Imperial Austrian Academy of Sciences and in 1887 he became the President of the University of Graz. He was elected a member of the Royal Swedish Academy of Sciences in 1888 and a Foreign Member of the Royal Society (ForMemRS) in 1899. He was awarded honorary membership of the Manchester Literary and Philosophical Society in 1892. Numerous things are named in his honour.

Works

<gallery>

File:Boltzmann-1.jpg|Volumes I and II of Vorlesungen über Gastheorie (1896-1898)

File:Boltzmann-2.jpg|Title page to volumes I and II of Vorlesungen über Gastheorie (1896-1898)

File:Boltzmann-5.jpg|Table of contents to volumes I and II of Vorlesungen über Gastheorie (1896-1898)

File:Boltzmann-7.jpg|Introduction to volumes I and II of Vorlesungen über Gastheorie (1896-1898)

</gallery>

References

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.&nbsp;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.&nbsp;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).
  • Ludwig Boltzmann - The genius of disorder (Youtube)
  • Ruth Lewin Sime, Lise Meitner: A Life in Physics Chapter One: Girlhood in Vienna gives Lise Meitner's account of Boltzmann's teaching and career.
  • Eftekhari, Ali, "Ludwig Boltzmann (1844–1906)." Discusses Boltzmann's philosophical opinions, with numerous quotes.