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20:
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Sharp, K.; Matschinsky, F. Translation of Ludwig
Boltzmann’s Paper “On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium” Sitzungberichte der Kaiserlichen Akademie der Wissenschaften.
646:
equally probable—for example, high energy microstates are less probable than low energy microstates for a thermodynamic system kept at a fixed temperature by allowing contact with a heat bath. For thermodynamic systems where microstates of the system may not have equal probabilities, the appropriate
594:
Boltzmann writes: “The first task is to determine the permutation number, previously designated by 𝒫 , for any state distribution. Denoting by J the sum of the permutations 𝒫 for all possible state distributions, the quotient 𝒫 /J is the state distribution’s probability, henceforth denoted by W.
852:
is also sometimes used to indicate entropies calculated based on the approximation that the overall probability can be factored into an identical separate term for each particle—i.e., assuming each particle has an identical independent probability distribution, and ignoring interactions and
366:
A 'microstate' is a state specified in terms of the constituent particles of a body of matter or radiation that has been specified as a macrostate in terms of such variables as internal energy and pressure. A macrostate is experimentally observable, with at least a finite extent in
409:
There are many instantaneous microstates that apply to a given macrostate. Boltzmann considered collections of such microstates. For a given macrostate, he called the collection of all possible instantaneous microstates of a certain kind by the name
625:
Therefore, by making the denominator small, he maximizes the number of states. So to simplify the product of the factorials, he uses their natural logarithm to add them. This is the reason for the natural logarithm in
Boltzmann’s entropy formula.
459:-th microscopic condition (range) of position and momentum. For this case, the probability of each microstate of the system is equal, so it was equivalent for Boltzmann to calculate the number of microstates associated with a macrostate.
371:. A microstate can be instantaneous, or can be a trajectory composed of a temporal progression of instantaneous microstates. In experimental practice, such are scarcely observable. The present account concerns instantaneous microstates.
641:
of interest, plus its surroundings; then the entropy of
Boltzmann's microscopically specified system can be identified with the system entropy in classical thermodynamics. The microstates of such a thermodynamic system are
965:
853:
correlations between the particles. This is exact for an ideal gas of identical particles that move independently apart from instantaneous collisions, and is an approximation, possibly a poor one, for other systems.
1209:
Mathematisch-Naturwissen Classe. Abt. II, LXXVI 1877, pp 373-435 (Wien. Ber. 1877, 76:373-435). Reprinted in Wiss. Abhandlungen, Vol. II, reprint 42, p. 164-223, Barth, Leipzig, 1909. Entropy 2015, 17, 1971-2009.
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as a density in phase space—without mentioning probability—but since this satisfies the axiomatic definition of a probability measure we can retrospectively interpret it as a probability anyway.
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leads to increasingly wrong predictions of entropies and physical behaviours, by ignoring the interactions and correlations between different molecules. Instead one must consider the
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must simultaneously satisfy the two constraints (1) and (2). Since the denominator of 𝒫 is a product, it is easiest to determine the minimum of its logarithm, …”
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In
Boltzmann’s 1877 paper, he clarifies molecular state counting to determine the state distribution number introducing the logarithm to simplify the equation.
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separate identical terms, one term for each particle; and when the summation is taken over each possible state in the 6-dimensional
426:, and this name is widely used today, perhaps partly because Bohr was more interested in the writings of Gibbs than of Boltzmann.
1714:
634:
Boltzmann's formula applies to microstates of a system, each possible microstate of which is presumed to be equally probable.
1350:
614:… values for which 𝒫 is a maximum or since the numerator is a constant, for which the denominator is a minimum. The values w
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99:
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as statistically independent. The probability distribution of the system as a whole then factorises into the product of
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was historically misinterpreted as literally meaning the number of microstates, and that is what it usually means today.
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1067:, rather than single particle states. Gibbs considered several such kinds of ensembles; relevant here is the
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563:. The "correction" in the denominator is due to the fact that identical particles in the same condition are
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Ludwig
Boltzmann (1866). "Über die Mechanische Bedeutung des Zweiten Hauptsatzes der Wärmetheorie".
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is used nowadays. For single particle instantaneous microstates, Boltzmann called the collection an
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We would first like to calculate the permutations đť’« for the state distribution characterized by w
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The
Boltzmann entropy is obtained if one assumes one can treat all the component particles of a
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Interpreted in this way, Boltzmann's formula is the most basic formula for the thermodynamic
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Vorlesungen ĂĽber
Gastheorie, Ludwig Boltzmann (1898) vol. II. J.A. Barth, Leipzig.
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Vorlesungen ĂĽber
Gastheorie, Ludwig Boltzmann (1896) vol. I, J.A. Barth, Leipzig
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960:{\displaystyle S_{\mathrm {G} }=-Nk_{\mathrm {B} }\sum _{i}p_{i}\ln p_{i}}
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formula shows the relationship between entropy and the number of ways the
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This reflects the original statistical entropy function introduced by
876:-dimensional phase space of the system as a whole), the Gibbs entropy
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810:) in his later work and recognized it as more general than equation (
725:{\displaystyle S_{\mathrm {G} }=-k_{\mathrm {B} }\sum p_{i}\ln p_{i}}
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is sometimes called the "thermodynamic probability" since it is an
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43:
1139:. Eric Weisstein's World of Physics (states the year was 1872).
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between 1872 and 1875, but later put into its current form by
308:
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of states of the system as a whole, called by
Boltzmann a
828:)—and not vice versa. In every situation where equation (
801:
gave an explicitly probabilistic interpretation in 1878.
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state of a system can be realized by assigning different
586:
606:“The most likely state distribution will be for those w
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But in thermodynamics, the universe is divided into a
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Boltzmann himself used an expression equivalent to (
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Boltzmann entropy excludes statistical dependencies
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ranges over all possible molecular conditions and "
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1004:
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724:
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378:was originally intended to be proportional to the
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1030:For anything but the most dilute of real gases,
528:{\displaystyle W={\frac {N!}{\prod _{i}N_{i}!}}}
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1304:Ludwig Boltzmann: the Man who Trusted Atoms
1189:Ludwig Boltzmann: the Man who Trusted Atoms
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793:formula as early as 1866. He interpreted
339:The equation was originally formulated by
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42:) is a probability equation relating the
1127:, Vienna, with bust and entropy formula.
603:molecules with kinetic energy ϵ, etc. …
359:was first stated by L. Boltzmann in his
335:, Vienna, with bust and entropy formula.
326:
18:
382:(the German word for probability) of a
192:{\displaystyle S=k_{\mathrm {B} }\ln W}
1707:
1306:, Oxford University Press, Oxford UK,
1191:, Oxford University Press, Oxford UK,
347:in about 1900. To quote Planck, "the
268:) and equal to 1.380649 Ă— 10 J/K, and
1346:
1023:it exactly corresponds to the proper
1019:in 1872. For the special case of an
587:Introduction of the natural logarithm
467:can be counted using the formula for
1328:Introduction to Boltzmann's Equation
1263:Vorlesungen ĂĽber Gastheorie, vol. II
983:simplifies to the Boltzmann entropy
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840:) is valid also—and not vice versa.
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1255:Vorlesungen ĂĽber Gastheorie, vol. I
1176:(1914) The theory of heat radiation
13:
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599:molecules with kinetic energy 0, w
422:. Subsequently, Gibbs called it a
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1211:https://doi.org/10.3390/e17041971
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16:Equation in statistical mechanics
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1052:{\displaystyle S_{\mathrm {B} }}
1005:{\displaystyle S_{\mathrm {B} }}
237:{\displaystyle k_{\mathrm {B} }}
87:{\displaystyle S_{\mathrm {B} }}
1296:
1715:Eponymous equations of physics
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1142:
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822:) is a corollary of equation (
786:{\displaystyle \rho \ln \rho }
1:
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406:to the respective molecules.
1388:Principle of maximum entropy
1283:Gibbs vs Boltzmann entropies
7:
1287:American Journal of Physics
1153:. Oxford University Press.
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146:corresponding to the gas's
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1412:Statistical thermodynamics
1165:(states the year was 1875)
748:This reduces to equation (
577:mathematical probabilities
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300:, as in the image above).
26:—carved on his gravestone.
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1261:Ludwig Boltzmann (1898).
1253:Ludwig Boltzmann (1896).
331:Boltzmann's grave in the
40:Boltzmann–Planck equation
1672:Condensed matter physics
1655:Statistical field theory
1151:A to Z of Thermodynamics
575:greater than one, while
414:, for which Gibbs' term
388:probability distribution
248:(also written as simply
1725:Thermodynamic equations
1530:Mathematical approaches
1519:Lennard-Jones potential
1435:thermodynamic potential
1302:Cercignani, C. (1998).
1187:Cercignani, C. (1998).
1149:Perrot, Pierre (1998).
816:). That is, equation (
754:) if the probabilities
424:microcanonical ensemble
115:{\displaystyle \Omega }
1566:conformal field theory
1265:. J.A. Barth, Leipzig.
1257:. J.A. Barth, Leipzig.
1199:, p. 134, pp. 141–142.
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1374:Statistical mechanics
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281:{\displaystyle \ln }
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36:Boltzmann's equation
24:Boltzmann's equation
1660:elementary particle
1425:partition functions
1178:equation 164, p.119
1101:von Neumann entropy
351:connection between
38:(also known as the
1687:information theory
1594:correlation length
1589:Critical exponents
1576:Critical phenomena
1557:stochastic process
1537:Boltzmann equation
1430:equations of state
1137:Boltzmann equation
1081:History of entropy
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319:can be arranged.
290:natural logarithm
261:{\displaystyle k}
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135:{\displaystyle W}
58:{\displaystyle S}
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977:
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968:
952:
948:
944:
941:
936:
932:
926:
922:
910:
906:
903:
900:
889:
881:
880:
877:
875:
871:
867:
863:
859:
854:
851:
841:
839:
838:
833:
832:
827:
826:
821:
820:
815:
814:
809:
808:
802:
800:
780:
777:
774:
771:
762:
757:
753:
752:
742:
735:
733:
717:
713:
709:
706:
701:
697:
693:
682:
678:
675:
664:
656:
655:
652:
650:
649:Gibbs entropy
645:
640:
635:
627:
623:
604:
592:
584:
582:
578:
574:
566:
562:
545:
538:
536:
519:
514:
510:
504:
500:
494:
491:
485:
482:
475:
474:
471:
470:
447:
440:
436:
432:
427:
425:
421:
417:
413:
407:
405:
401:
397:
396:thermodynamic
393:
389:
385:
381:
374:The value of
372:
370:
364:
362:
358:
354:
350:
346:
342:
334:
329:
320:
318:
314:
310:
306:
301:
299:
295:
292:function (or
291:
275:
255:
247:
223:
209:
202:
200:
186:
183:
180:
169:
165:
162:
155:
154:
151:
149:
145:
129:
101:
97:
73:
52:
45:
41:
37:
33:
25:
21:
1648:Applications
1618:
1599:size scaling
1303:
1298:
1290:
1286:
1262:
1254:
1248:
1239:
1235:
1229:
1213:
1204:
1188:
1183:
1170:
1150:
1144:
1132:
1115:
1068:
1064:
1029:
1014:
982:
971:
873:
869:
861:
855:
849:
847:
835:
829:
823:
817:
811:
805:
803:
763:
755:
749:
747:
736:
643:
636:
633:
624:
605:
593:
590:
550:
539:
469:permutations
445:
428:
423:
419:
415:
411:
408:
390:of possible
379:
373:
365:
338:
302:
214:
203:
100:multiplicity
39:
35:
29:
23:
1639:von Neumann
1509:force field
1501:percolation
1174:Max Planck
866:phase space
579:are always
455:are in the
392:microstates
384:macroscopic
363:of gases".
357:probability
349:logarithmic
144:microstates
1709:Categories
1496:Heisenberg
1242:: 195–220.
1107:References
1091:nat (unit)
559:" denotes
345:Max Planck
148:macrostate
1619:Boltzmann
1542:H-theorem
1420:Ensembles
1314:, p. 134.
1221:CC BY 3.0
1069:canonical
1021:ideal gas
945:
923:∑
904:−
848:The term
781:ρ
778:
772:ρ
710:
694:∑
679:−
561:factorial
501:∏
446:identical
439:ideal gas
400:positions
369:spacetime
313:molecules
305:Boltzmann
184:
110:Ω
96:ideal gas
1629:Tsallis
1293:, 391-8.
1281:(1965).
1223:license.
1075:See also
1061:ensemble
435:paradigm
416:ensemble
94:, of an
1624:Shannon
1611:Entropy
1123:in the
581:numbers
573:integer
437:was an
431:entropy
404:momenta
353:entropy
323:History
288:is the
244:is the
98:to the
44:entropy
1473:Models
1381:Theory
1310:
1195:
1157:
1065:holode
870:single
795:ρ
651:, is:
639:system
551:where
420:ergode
412:monode
215:where
1682:chaos
1634:RĂ©nyi
1491:Potts
1486:Ising
1071:one.
868:of a
799:Gibbs
309:atoms
296:base
1564:and
1308:ISBN
1193:ISBN
1155:ISBN
402:and
355:and
1259:;
644:not
618:, w
610:, w
441:of
311:or
294:log
122:or
30:In
1711::
1437::
1291:33
1289:,
1285:.
1271:^
1240:53
1238:.
1027:.
1012:.
942:ln
775:ln
707:ln
567:.
276:ln
181:ln
150::
34:,
1456:G
1451:F
1446:H
1441:U
1366:e
1359:t
1352:v
1163:.
1044:B
1039:S
997:B
992:S
976:)
974:4
972:(
953:i
949:p
937:i
933:p
927:i
916:B
911:k
907:N
901:=
895:G
890:S
874:N
862:N
837:3
831:1
825:3
819:1
813:1
807:3
759:i
756:p
751:1
741:)
739:3
737:(
718:i
714:p
702:i
698:p
688:B
683:k
676:=
670:G
665:S
620:1
616:0
612:1
608:0
601:1
597:0
569:W
557:!
553:i
544:)
542:2
540:(
520:!
515:i
511:N
505:i
495:!
492:N
486:=
483:W
465:W
461:W
457:i
452:i
450:N
443:N
376:W
298:e
256:k
229:B
224:k
208:)
206:1
204:(
187:W
175:B
170:k
166:=
163:S
130:W
79:B
74:S
53:S
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