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This equation describes the [[time|temporal]] and [[space|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.

This equation describes the [[time|temporal]] and [[space|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.

该方程描述了单粒子相空间中一团点密度分布的位置和动量概率分布的时空变化。(见哈密顿力学。)左边的第一项表示分布函数的显式时间变化，而第二项给出空间变化，第三项描述作用于粒子的任何力响。方程的右边表示碰撞的影响。

该方程描述了单粒子相空间中一团点密度分布的位置和动量概率分布的时空变化(见[[哈密顿力学]])。左边的第一项表示分布函数的显式时间变化，而第二项给出空间变化，第三项描述作用于粒子的任何力响。方程右边表示碰撞的影响。

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 ''ƒ'' can represent an arbitrary single-particle distribution function. Also, the [[force]] acting on the particles depends directly on the velocity distribution function ''ƒ''. The Boltzmann equation is notoriously difficult to [[Integral|integrate]]. [[David Hilbert]] spent years trying to solve it without any real success.

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 ''ƒ'' can represent an arbitrary single-particle distribution function. Also, the [[force]] acting on the particles depends directly on the velocity distribution function ''ƒ''. The Boltzmann equation is notoriously difficult to [[Integral|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 theory|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.

The form of the collision term assumed by Boltzmann was approximate. However, for an ideal gas the standard [[Chapman–Enskog theory|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.

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, [[non-equilibrium statistical mechanics|nonequilibrium statistical mechanics]] for dense gases and liquids focuses on the [[Green–Kubo relations]], the [[fluctuation theorem]], and other approaches instead.

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, [[non-equilibrium statistical mechanics|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==

==Second thermodynamics law as a law of disorder==

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.)<ref>"[[Collier's Encyclopedia]]", Volume 19 Phyfe to Reni, "Physics", by David Park, p. 15</ref> 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.<ref>"Collier's Encyclopedia", Volume 22 Sylt to Uruguay, Thermodynamics, by Leo Peters, p. 275</ref> 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{{Citation needed|date=January 2019}}, 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 the deck was disordered.

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.)<ref>"[[Collier's Encyclopedia]]", Volume 19 Phyfe to Reni, "Physics", by David Park, p. 15</ref> 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.<ref>"Collier's Encyclopedia", Volume 22 Sylt to Uruguay, Thermodynamics, by Leo Peters, p. 275</ref> 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{{Citation needed|date=January 2019}}, 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 the deck was disordered.

==Awards and honours ==

==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 [[List of Fellows of the Royal Society elected in 1899|Foreign Member of the Royal Society (ForMemRS) in 1899]].<ref name="frs">{{cite web|archiveurl=https://web.archive.org/web/20150316060617/https://royalsociety.org/about-us/fellowship/fellows/|archivedate=2015-03-16|url=https://royalsociety.org/about-us/fellowship/fellows/|publisher=[[Royal Society]]|location=London|title=Fellows of the Royal Society}}</ref> [[List of things named after Ludwig Boltzmann|Numerous things]] are named in his honour.

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 [[List of Fellows of the Royal Society elected in 1899|Foreign Member of the Royal Society (ForMemRS) in 1899]].<ref name="frs">{{cite web|archiveurl=https://web.archive.org/web/20150316060617/https://royalsociety.org/about-us/fellowship/fellows/|archivedate=2015-03-16|url=https://royalsociety.org/about-us/fellowship/fellows/|publisher=[[Royal Society]]|location=London|title=Fellows of the Royal Society}}</ref> [[List of things named after Ludwig Boltzmann|Numerous things]] are named in his honour.

*[[Boltzmann brain]]

*[[Boltzmann brain]]

== References==

==References==

{{reflist|30em}}

{{reflist|30em}}

==Further reading==

==Further reading==

*Roman Sexl & John Blackmore (eds.), "Ludwig Boltzmann – Ausgewahlte Abhandlungen", (Ludwig Boltzmann Gesamtausgabe, Band 8), Vieweg, Braunschweig, 1982.

*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 (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, "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.&nbsp;1–232.

*John Blackmore (ed.), "Ludwig Boltzmann – Troubled Genius as Philosopher", in Synthese, Volume 119, Nos. 1 & 2, 1999, pp.&nbsp;1–232.

*{{cite book|last1=Blundell|first1=Stephen|last2=Blundell|first2=Katherine M.|title=Concepts in Thermal Physics|url=https://books.google.com/books?id=vuBHXwAACAAJ|year=2006|publisher=Oxford University Press|isbn=978-0-19-856769-1|page=29}}

*{{cite book|last1=Blundell|first1=Stephen|last2=Blundell|first2=Katherine M.|title=Concepts in Thermal Physics|url=https://books.google.com/books?id=vuBHXwAACAAJ|year=2006|publisher=Oxford University Press|isbn=978-0-19-856769-1|page=29}}

* Boltzmann, ''Ludwig Boltzmann – Leben und Briefe'', ed., Walter Hoeflechner, Akademische Druck- u. Verlagsanstalt. Graz, Oesterreich, 1994

*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. & tr.), Boltzmann, ''Lectures on Gas Theory'', Berkeley, California: U. of California Press, 1964

*Brush, Stephen G. (ed.), ''Kinetic Theory'', New York: Pergamon Press, 1965

*Brush, Stephen G. (ed.), ''Kinetic Theory'', New York: Pergamon Press, 1965