E(P,Q)<E 0 where P, Q represent all coordinates in phase space, and E(P,Q) is the total energy as a function of location in phase space.

Transcription

1 Concening the Statistics of Classical Systems by Anulf Schlüte, Max Planc Institute fo Physics, Gottingen (note 1: excepted fom the fist pat of the autho s dissetation) Z. Natufoschg. 3a, (1948) Received on 8 Decembe 1947 [paagaph numbes in bacets fo ease of locating mateial in the oiginal] [1] We fist define the phase volume of a system as the volume contained within the enegy suface in the phase space and define its phase density as its deivative with espective to enegy. Because of the egodic hypothesis, the system phase density can be obtained as the convolution poduct of the phase densities of the individual pats; and these can again, when the enegy of the pats can be sepaated, be depicted as convolution poducts of the enegy contibutions of the assigned phase densities. [2] Cetain, especially impotant pats of these esults can be caied out explicitly. These will equie some distibution poblems to be solved. In paticula, the exact validity of the equipatition theoem will be poved. Out of that and the isotopic invaiance of the phase volume, we conclude that the themodynamic diffeential equations ae stictly adheed to if the entopy is set popotional to the phase volume and the tempeatue is set invesely popotional to its logaithmic deivative. These esults apply fo a system of abitaily few paticles. Phase Space [3] Conside a system obeying the laws of the Classical Mechanics. We imagine that this system is composed of many individual paticles, but we won't yet mae use of this assumption. The instantaneous state (micostate) of the system is defined by as many genealized coodinates, Q, and thei canonically conjugate momenta, P, as the numbe of degees of feedom (note 2: uppe case chaactes indicate vecto quantities epesenting vaiables descibing the whole system). The momenta and coodinates ae theeby intepeted out of Catesian coodinates into a symbolic many dimensional phase space. (Ehenfest s Γ-phase space) [4] Only systems that will not dissociate in time should be investigated. This is assued if we equie that the following integal exists J(E 0 ) =... dpdq, E(P,Q)<E 0 whee P, Q epesent all coodinates in phase space, and E(P,Q) is the total enegy as a function of location in phase space. [5] J(E 0 ) measues the volume of the system phase space with any enegy smalle than E 0 and is called the phase volume. J(E 0 ) is canonically invaiant, meaning it is independent of the choice of genealized coodinates. [6] A mechanical system such as this is called an egodic system if, at constant total enegy, the time aveage of an abitay, not explicitly time-dependent function of P,Q ove adequately long time is equal to the phase space aveage of this function fo this enegy, in the sense, that t 1 F(P(t),Q(t))dt... F(P,Q) dpdq E 0 <E(P,Q)<E 0 +de 0 t F(P,Q) = lim 0 =, Eq. (1) t1 t 1 t 0... dpdq E 0 <E(P,Q)<E 0 +de 0

2 Page 2 whee the last quotient can also be witten as: d de 0... F(P,Q)dPdQ de 0 E(P,Q )<E F(P,Q) = 0 1 d = d... F dpdq, Eq. (1a) de 0 J(E 0 ) G(E 0 ) de 0 E<E 0 de 0 whee the deivative of the phase volume with espect to enegy is abbeviated as G(E), which we will call the phase density of the system being consideed. [7] The dependability of this aveage calculation is as good as the quasiegodic hypothesis in Ehenfest s fomulation. It would be sufficient if the intechangeability equiements fo the aveage wee only equied fo almost all initial conditions. In othe wods, a diffeent elationship could be allowed fo a set of initial conditions of lesse dimensionality (Fo example, fo simple peiodical movements that oiginate though accidental degeneacy). The additional logical consequences would then also be valid only almost always. [8] In such an egodic system, the time aveage of any function of the micostate is a welldefined function of the enegy of the system. This doesn't wo fo othe quantities (it is not tue fo angula momentum, fo example,) and means that the egodic hypothesis, even in ou cautious fomulation, is suely not stong enough. Howeve, we won t go into these difficulties futhe, but athe simply postulate the validity of Eq. (1). [9] If a system consists of multiple pats which ae almost always independent of each othe in the sense that the total enegy of the system can be potayed as the sum of the enegies of the individual pats at any moment in time and the times duing which the pats inteact can be ignoed, then these pats of the system can be said to compise the system. [10] Coespondingly, the state of a subsystem is descibed by a point in its phase space. It possesses in that instant a definite enegy and, theeby, a coesponding definite phase volume and a definite phase density. [11] We conside now a sepaation of the system into two diffeent subsystems (I and II). We inquie, afte a elatively long time, in which the subsystem I has an enegy between E I0 and E I0 + de 0 (note 3. Uppe case lettes with Roman numeal indices epesent lage subsystems). Accoding to the egodic hypothesis, instead of this time aveage, the coesponding aveage in the phase space could be calculated by:... dp I dq I... dp II dq II E W I 0 )de = I 0 <E I (P I,Q I )<E I 0 +de E II 0 <E II (P II,Q II )<E II 0 +de,... dpdq E I 0 +E II 0 <E I (P I,Q I )+E II (P II,Q II )<E I 0 +E II 0 +de Wheeby the numeato oiginates fom the fact that, as long as the subsystem I has the enegy E I, the complementay subsystem II must have the enegy E II, with E I + E II = E being constant. [12] One can bing the phase densities bac in again hee and get (by leaving the Index 0 out): W I ) = G I )G II I ) G + E II ) = G I )G II (E E I ) G(E). Eq. (2)

3 Page 3 [13] The function W I ) is the enegy distibution function (of subsystem I). With its help, the aveage of a function of the enegy of the subsystem I can be immediately specified as: F I )G I ) G II (E E I )de I F I ) = F I ) W I )de I = G(E) The aveage value is (though W I ) a function of the total enegy, E. The Convolution Method [14] Using the definition of the enegy distibution function, its integal must, when extended ove all possible enegies, be nomalized: that means W I )de I =1, G I )G II (E E I )de I = G(E). Eq. (4) [15] Equation (4) clealy states that the phase density of the whole system, G(E), can be obtained though integation ove the enegy of the poduct of the inteelated phase densities of the subsystems. This can be seen as a diect consequence of the egodic hypothesis, though which the equipobability of simila size phase egions consistent with the enegy equiement was postulated. [16] An integal of the fom of Eq. (4) is designated as a convolution integal and the symbol fo it is a sta. G I (E) *G II (E) = G I )G II (E E I )de I = G I (E E II )G II I )de II = G II (E) * G I (E) on which poduct ( convolution poduct ) a special notation is justified, as the opeation is commutative, and also, as a linea opeation, distibutive with espect to addition. One can show (note 4: simply, the Laplace tansfom of a convolution is equal to the usual poduct of the Laplace tansfoms of the sepaate factos, see G. Doetsch, Theoy and Application of the Laplace Tansfom, Belin, 1937, 8, 155ff.) that multiple opeations ae also associative: (G I *G II )*G III = G I *(G II * G III ). [17] Equation (4) is also especially applicable when one consides a system which is composed of only two single paticles with phase densities g 1 (u) and g 2 (u) (note 5: to distinguish them fom whole system vaiables, those descibing individual paticles ae witten in lowe case chaactes with Aabic numbe subscipts), which ae almost always independent of each othe (in the sense defined above fo subsystems): G(E) = g 1 (u)g 2 (E u)du = g 1 (E) * g 2 (E). [18] If the system consists in simila manne of 3 paticles, then 2 paticles can be viewed togethe as one subsystem of the system, whose phase density is geneated though the convolution of the phase density of both paticles. The second convolution with the phase density of the thid paticle poduces the phase density of the entie system. Because of the

4 Page 4 associated laws of convolution, the outcome is independent of the altenative ode of ceation of the subsystems, as equied physically. G(E) = g 1 (E) * g 2 (E) * g 3 (E). So we can natually poceed with additional paticles, so that the following is the phase density G(E) of an egodic system composed of N paticles which ae almost always independent. [with phase densities g i (u)]: G(E) = g 1 (E)*...* g i (E)*...* g N (E), which can be abbeviated: * G(E) = g i (E); i =1,...,N. Eq. (5) (note 6: the index i uns ove all paticles in the system) i [19] The equiements fo the validity of Eq. (5) ae the egodic chaacte of the system and the capability of the total enegy to be sepaated into enegies which ae only dependent on the position and momentum coodinates of each paticle. If enegies exist, fo example because of Coulomb o van de Waals foces between the paticles, which do not depend solely on the positions of the individual pieces, this possibility no longe exists. In the following, only systems should be consideed which (almost always) exist as sepaate, independent pieces so that the convolution method applies. (note 7: concening the numbe of paticles, it is peviously assumed that a system contains at least one paticle. Liewise when a patition of a system into two pats is discussed, it must contain at least two paticles to avoid an empty patition). [20] Often we encounte the situation whee the enegy of individual paticles can also be sepaated into paticle enegies depending on only one position and momentum coodinate. If the enegy of a paticle is completely sepaable, then it can also be witten: u(p,q) = v (q ) + t (p ); =1,..., f (v = components of the potential enegy; t = components of the inetic enegy; f = numbe of degees of feedom, note 8: the index uns ove individual paticle degees of feedom). [21] Evey enegy contibution can then have a phase volume and a phase density assigned to it again, and we can constuct the phase density of the entie system again out of the individual phase densities.

5 Page 5 Figue 1. Phase volume of one enegy component (the index has been suppessed) [22] The meaning of this possibility is that the phase volumes j (v ) o j (t ) can be obtained easily, as long as the enegy functions v (q ) o t (p ) ae nown. The phase volume, j (v 0 ), fo instance, measues the length of the inteval of q, fo which v (q ) < v 0 (see Fig. 1). It is then simply (2) (1) (4) (3) j (v 0 ) = q (v0 ) q (v 0 ) + q (v0 ) q (v0 ) +... Eq. (6) if q (v 0 ) epesent the values of odeed oots of the equation: v (q ) = v 0 In all inteesting cases, this equation has no oots below a minimum enegy, above which it has exactly two oots. (Illustation 1 thus depicts a geneal situation, which doesn't ceate any paticula difficulties.) [23] Diffeentiation with espect to the enegy gives the phase density and, though convolution, of all phase densities (including those which belong to the inetic enegy potions, which can be displayed accodingly), one gets the phase density of all the paticles. (2) dq (u) g(u) = dq (1) (u) du du * dp (2) (u) dp (1) * f (u), =1 du du o witten diffeently as: * f g(u) = u u * u u ; =1 q q (2) (1) p p (2) (1) (1) q = q (1) (u), p = p (1) (u) (2) etc. in lie manne Eq. (7) Eq. (7a)

6 Page 6 In these equations, it is especially impotant to note that the convolution poduct is witten with the esult vaiable, which is a paamete of the integal, as agument instead of using the integation vaiable. [compae equations (4a) and (5)]. Classical Paticles [24] In many models, the individual enegy contibutions fom paticles with sepaable enegy have simila dependence on the associated coodinates. Fo example, the inetic enegy is always a homogeneous quadatic function of the momenta (note 9: this assumes a nonelativistic classical teatment), it can theefoe be potayed as a sum of pats which ae popotional to the squae of the momentum components. Similaly, the potential enegy of hamonic oscillatos is a quadatic function of the position coodinates, while the potential enegy in a homogeneous gavitational field efeenced to a lowe edge (the floo of the containe), at which it can be said to be zeo, is popotional to a position coodinate. The enegy function of a degee of feedom can be bought about in this and also essentially geneal cases though suitable choice of the constants a and c in one of the two foms: 1 v (q ) = 2 Γ(a +1) a q fo < q < and a 1 > 0 and even, Eq. (8a) c o 1 Γ(a +1) a v (q ) = q fo q 0 c ; fo any value of 1 > 0, Eq. (8b) a fo q < 0 whee all a, c > 0, and Γ epesents the gamma function. Fo the inetic enegy the appopiately specialized fom (8a) is sufficient: t (p ) = π p 2 c 2 ; =1,..., f, c > 0. Eq. (9) [25] The use of Eq. (6) immediately gives the phase volume in ageement with (8a) and (8b): c a j (v ) = v Γ(a +1) 0 ; a,c > 0 Eq. (10) and similaly fo the inetic enegy potions (9): ' j (t ) = 2 π c t 1/ 2 0 ; c 3 > 0; =1,..., f ; Γ = π Eq. (11) [26] By this, as also in the following, the pefixed Null (0) means that the function distibution should be eplaced with zeo fo negative aguments: f (x) fo x 0 0 f (x) = 0 fo x < 0. [27] By diffeentiating the phase volumes, one gets the coesponding phase densities. g (v ) = c a v 1 Γ(a ) 0 ; a, c > 0 Eq. (10a)

7 Page 7 g (t ) = c t 1/ 2 ; Γ 1 = π, c π 0 2 > 0 Eq. (11a) [28] Using these foms of the single-paticle phase densities, the convolutions fo the assembly of the phase density of the collection of paticles can be caied out with the help of the convolution ules c 1 Γ(a 1 ) 0 u a 1 1 * c 2 Γ(a 2 ) 0ua 2 1 = c 1 c 2 Γ(a 1 + a 2 ) 0ua 1 +a 2 1 (note 10: This fomula follows by easily veified explicit integation using elementay integals. It is especially elegant, howeve, to pove with help of Laplace tansfomation that c L Γ(a) 0ua 1 = c e uβ Γ(a) 0ua 1 du = cβ a 1 and the tansfomation of fomula (12) simply states c 1 β a 1 c 2β a 2 = c 1c 2 β (a 1 +a ) 2.) [29] The convolution poduct of one-sided powes is thus such that its analytical fom, accoding to ou definition of the constants, depends vey simply on the paametes of the factos. [30] One obtains the oveall density of an entie collection of such paticles with: g(u) = c 0ua 1 Eq. (13) Γ(a) with f f c = c c ; a = a + f (f = numbe of mechanical degees of feedom) =1 =1 2 and though integation of the enegy of the phase volumes: c j(u) = u a Eq. (13a) Γ(a +1) 0 Eq. (12) [31] The fom (13) o (13a) also emains peseved if some o all position coodinates bing no contibution to the potential enegy, but getting fom them instead simply a bounday which depends on the aea available to the paticle. The size of this aea comes in as a facto in the phase density and phase volumes, contibuting thus to the poduct of the c fo the coesponding degees of feedom, while the espective a must be set to null. [32] Paticles with a phase density and phase volume of the fom of equation (13) o (13a) will be called classical paticles hee (note 11: without assuming sepaability of the enegy). We will cay classical in quotes because the applicability of classical mechanics is not sufficient fo the validity of (13) and (13a); we will show, howeve, that most (in the common sense) classical models actually coespond to this type and that the classical fom of phase volume and density is necessay (note 12: see dissetation, section B 12 (compae note 1)) and sufficient fo the equipatition pinciple fo the enegy. [33] The exponent a of the enegy in the phase volume function should be named the chaacteistic numbe of the coesponding paticle, as this chaacteistic numbe is cucial fo the statistical behavio of the paticle. Accoding to Eq. (13), the chaacteistic numbe is equal to the sum of the potential enegy contibution plus half of the numbe of mechanical degees of

8 Page 8 feedom of the paticle. Examples of Classical paticles [34] Ideal gases. In an ideal gas without intenal degees of feedom, the enegy of evey paticle is only dependent on its momentum: u = 1 2m (p 2 x + p 2 y + p 2 z ) [35] This is included in the classical fom (9), if one sets c x = cy = cz = 2mπ [36] The position coodinates simply povide the volume, V, which is occupied by the gas, as a facto, so that one altogethe gets the following as phase density and phase volume of an ideal gas: g(u) = 2π(2m) 3 2 V 0 u 1 2 Eq. (14) j(u) = 4 3 π(2m)3 2 V 0 u 3 2 Eq. (14a) [37] Sedimentation/layeing equilibium. If the ideal gas is in a homogeneous gavitational field, then the enegy of the paticles is given by: u = 1 2m (p 2 x + p 2 y + p 2 z ) + mγz (γ = acceleation of fee fall, z = height, with up being chosen positive). [38] If the containe has height z = 0 as floo and has, fom just above that to infinite height, a constant coss section Q, then the z dependence of the potential enegy has the fom (8b) with a z =1; c z = 1 mγ while the hoizontal coodinates only facto is the coss section Q. This esults in phase density and volume: g(u) = 8 3 π(2m) 1 2 γ 1 Q 0 u 3 2, Eq. (15) j(u) = π(2m) 1 2 γ 1 Q 0 u 5 2. Eq. (15a) The potential enegy thus causes an incease of the chaacteistic numbe a of the paticle fom 3/2 to 5/2. [39] Also in the geneal case whee the coss section of the containe has the fom Q(z) = 0 z b ; b 0 (b not necessaily an intege) depending on the height z, giving the classical phase density of the paticle; we get the following distinct value fo a a = b (note 13: the occasional use of 2a as the numbe of statistical degees of feedom leads to contadictions. The elationship 2a = f + 2Σa give the coect numbe of degees of feedom including potential enegy.)

9 Page 9 [40] Linea hamonic oscillato. The enegy of the one-dimensional hamonic oscillato is u = 1 2m p a2 q 2, If the elastic constant a 2 is used. The dependence of the enegy on momentum and position is of the same ind and gives the phase density g(u) = 2πa 1 m 0 u 0 ( 0 u0 = { 1fo u 0 and 0 fo u 0} ) Eq. (16) and the coesponding phase volume j(u) = 2πa 1 m 0 u Eq. (16a) Statistics of Classical Systems [41] If a system consists of simple classical paticles, we will call it a classical system. Then its phase density can be calculated though epeated use of the convolution theoem (12), and the phase density of the system has the same classical fom: G(E) = C E A 1 ; A,C > 0 Eq. (13') Γ(A) 0 C = Πc i ; A = a i ; i =1,...,N; N >1. i Accodingly, fo the phase volume we get the following: C J(E) = E A. Γ(A +1) 0 i Eq. (13a') [42] These convenient analytical fomulas fo phase volumes and densities pemit the explicit evaluation of vaious distibution poblems. Fo instance, one wants to now the distibution function fo the enegy of a paticle with the phase density g(u) = c Γ(a) 0ua 1 in such a classical system. Equation (2) gives fo this, afte appopiate changes, the expession: w(u) = g(u) G R (E u), G(E) whee G R designates the phase density of the emaining pat of the system without the paticle. In this fomula, g(u) is the a pioi distibution while the quotient gives the influence of the emaining pieces on the enegy distibution. Substitution of the classical expessions gives w(u) = c Γ(a) ua 1 * Γ(A)(E u)a a 1 Γ(A a)ce A 1 ; 0 < u < E. Eq. (17) [43] If the system consists of vey many ( infinitely many) paticles, and the aveage enegy pe paticle is finite, in othe wods when A and E both become infinite at the same ate, this distibution function asymptotically appoached the expession c u A E e lim w(u) = A,E Γ(a) ua 1 c(e / A) ; u > 0. a This is, howeve, the Maxwell-Boltzmann distibution function, if one eplaces E/A with T, which we will justify below.

10 Page 10 [44] The diffeence between the exact distibution (17) and the Boltzmann distibution gives the enlightening conclusion that no paticle can have an enegy lage than the total enegy of the system [w(u) 0 fo u E]. [45] The aveage one-paticle enegy can be obtained though elementay integation (note 14: the simplest evaluation of these is a evese convolution using Eq. 12): u = uw(u)du = uw(u)du to give u = a A E ; u a = E A E 0 = J(E) G(E) (13, 13a). Eq. (18) [46] Equation (18) poves that the aveage enegy of evey paticle is independently popotional to the total enegy, and, in fact, this atio is the same as the elationship of the chaacteistic numbe of the paticle (namely: a) to the system (A = Σa i ). This is the classical equipatition theoem fo enegy, which is also stictly tue, then, in systems with abitay numbes of paticles as long as these ae classical. [47] Equation (18) pesents the coect genealization fo consideing the potential enegy (2a = potential + inetic degees of feedom, compae note 13), contasting to the usual fomulation. [48] Liewise, the analysis of the integal: A(a +1) u 2 = u 2 w(u)du = (A +1)a u a +1 2 A a u 2 gives the aveage squaed enegy, and out of that follows the vaiance: ( u) 2 = u 2 u 2 = A a (A +1)a u 1 2 A a u 2 [49] The most liely enegy of a paticle is the location, u = u ˆ, fo which w(u) is at a maximum, which esults in: A(a 1) u ˆ = u fo a 1 0; A 2 > 0; (A 2)a u ˆ = 0 fo a 1 0; A a 1> 0. u ˆ a 1 u fo a 1 0; A. a [50] Accoding to the deivation, the peceding fomula is only dependent on the analytical fom of the single-paticle phase density and that of the system (both must have the classical fom), but not dependent on the single-paticle phase density belonging to the entie system. The fomulas ae also valid when they ae based on individually sepaable enegy contibutions, such as the potential enegy of a gas paticle in a homogeneous foce field. Equation (17) then gives the distibution of the potential enegy, and that gives the height (and enegy) dependent aveage density of the gas. [51] Liewise these fomulas poduce the distibution, the aveage and so foth, of the enegy of a subsystem consisting of multiple paticles if one sets thei phase density (given by the convolution poduct of the phase density of its paticles) which is its chaacteistic numbe (which is the sum of the chaacteistic numbes of its paticles) in the place of the

11 Page 11 coesponding single-paticle density. Themodynamics of classical systems [52] The enegy of a system in a cetain (mico) state descibed by a point in the phase space of the system, o in othe wods descibed by specifying all P, Q (whee again the pai of vaiables epesent the set), still depends on extenal paametes, which we have peviously pesumed to be constants. Such paametes ae lie the stength of the gavitation field fo sedimentation equilibium o the volume occupied by the paticles of a gas. If the function E = E(P,Q) still depends on the paamete x, then the phase volume and density ae dependent on these, so that at length what must be witten is (note 15: we wite simply x to stand fo all x ): E = E(P,Q;x), J = J(E;x), G = G(E;x) = J(E;x) E [53] The foce, X, which follows fom the paamete, x, depends on the espective values of the P, Q, and all x fom: X (P,Q;x) = E(P,Q;x). x [54] We get the following as enegy balance fo an infinitesimal paamete displacement: E(P,Q;x) 0 = de adiabat + X (P,Q;x)dx = de adiabat dx. x [55] This elation defines when a paamete change may be consideed an adiabatic pocess. This is in contast to a geneal pocess whee the total enegy of the system is changed, not only though wo done by extenal paametes, but also sepaately, pehaps, by combining fo a time the egodic system being consideed with anothe body, in othe wods, that is by binging it into themal equilibium with anothe body. The enegy incease of the system fom this contact is called the heat tansfe, Q (not to be confused with the symbol fo position coodinates), and this givs fo a geneal infinitesimal pocess: dq = de + X dx. Eq. (19) (Fist Law of Themodynamics) [56] The statement of the Fist Law, o the enegy consevation theoem, is because dq fo the system being consideed and that of the body with which the egaded system is bought into contact togethe total to zeo. [57] Especially impotant ae pocesses in which the paamete displacement is done so slowly that, in calculating the enegy incease, ΣX dx, the extenal foce can be eplaced by its time aveage. Such pocesses may be called quasi-static. [58] Because of the egodic hypothesis [Eq. (1a)], the time aveage of an extenal foce can be witten: 1 E(P,Q;x) X (P,Q;x) = X (E 0 ;x) = dpdq G(E 0 ;x) E 0 E(P,Q;x )<E 0 x 1 1 J(E = dpdq = 0 ;x). G(E 0 ;x) x G(E 0 ;x) x E(P,Q;x)<E 0

12 Page 12 [59] Then fo the heat applied duing a quasi-static pocess the following applies: dq quasist = de quasist + X dx = de quasist + 1 J dx. Eq. (19a) G x [60] A pocess which is simultaneously adiabatic and quasi-static, which, in othe wods, only exists in an endlessly slow paamete displacement, is called isentopic. Fo an infinitesimal isentopic pocess the enegy balance is then applicable: 0 = de isentop + 1 J dx. G x (tansl. note: efeed to as Eq. 19b below) [61] This equation is vey valuable because it pemits one to calculate the enegy incease of the system without having to now how the extenal foce paametes depend on the micostate, o the function X (P, Q; x), because only the deivatives of the macoscopic function J(E ; x) ente. [62] Because dj(e;x) = J E de + J dx = G de + 1 J dx Eq. (20) x G x is equivalent to Eq. (19b) with the claim of the isentopic invaiance of the phase volume: dj isentop = 0. (note 16: In the liteatue, usually one speas of the adiabatic invaiance of phase volumes, which is hee eminiscent of apidly completed adiabatic pocesses of macoscopic themodynamics, wheeas hee the infinitely slow pocess is completely essential. Smeal has theefoe poposed simply efeing to paamete invaiance; howeve, this is obviously not substantially bette. Theefoe, we intoduced hee an isentopic pocess though the definition of entopy, analogous to the customay intoduction of an adiabatic pocess o an adiabatic change though the definition of heat.) [63] We can now poceed to define the absolute tempeatue exactly as it is done in macoscopic themodynamics, fist by Helmholtz (note 17: H. v. Helmholtz, J. Math. 97, 111,317 (1884)) and also by Caatheodoy (note 18: Caatheodoy, Math. Ann. 61, 355 (1909)). The absolute tempeatue is defined as the integated denominato of the quasi-static (note 19: Tatyana Ehenfest-Afanasyeva has shown that it is the quasistatic athe than the evesible natue of the heat tansfe upon which this depends (Z. Physi 33, 933 (1925), 34, 638 (1925)) applied heat, which possesses the attibutes of an empiical tempeatue. Thus, an empiical tempeatue is one in which the enegy is dependent monotonically on the state vaiable and is the same fo all bodies in themal equilibium. [64] An integating facto fo the quasi-static added heat is, fo example, G(E), which accoding to (19a) and (20) is: G(E)dQ quasist = dj In geneal, one moe abitay (integable) function, f(j), can be applied to G(E), because f (J)G(E)dQ quasist = f (J)dJ = d f (J)dJ liewise poduces a complete diffeential. (Tansl. note: f(j) = J -1 below) [65] Fo classical systems, which we now etun to the peceding implementations of this section ae valid independent of the special dependence of the phase volume on the enegy we can also immediately define an empiical tempeatue: The aveage enegy pe chaacteistic numbe is the same fo all paticles and also the same fo all subsystems of a classical system,

13 Page 13 and specifically they ae the same as the quotients fom the phase volumes and phase densities (18): u 1 = u 2 =...= E I = E II =... = E a 1 a 2 A I A II A = J(E) G(E). Eq. (18a) paticles subsystems systems Because J/G is also an integable denominato of the quasi-static added heat, J(E) = T Eq. (21) G(E) is the absolute tempeatue, wheeby the initially undefined facto can be identified though an example, fo example an ideal gas, with the Boltzmann constant. [66] Diffeentiation of the enegy with espect to the tempeatue gives the specific heat o espectively the heat capacity of the system: du dt = C v = A, which is, except fo the facto, equal to the chaacteistic numbe of the system. [67] Though the tempeatue, the entopy diffeential is detemined at the same time: ds = dq quasistat 1 = d dj = dlnj, Eq. (22) T J S S 0 = lnj = lnc + Aln E lnγ(a +1); E > 0. If one substitutes the tempeatue as the vaiable instead of the enegy, we get: S S 0 = lnc + AlnT + Aln A lnγ(a +1). Eq. (22a) Using Stiling s appoximation, we find in the limit that A, if one simultaneously eliminates the index A though the heat capacity: S S 0 lnc + C A v lnt + C v ln2πa. Eq. (22b) (tansl. note: befoe the baceted last tem in Eq. (22b) is not in the oiginal) The baceted element is popotionally insignificant compaed to what emains and can be left out. Fo an ideal gas, (22b) now agees with the nown entopy fomula. The emaining themodynamic functions can be constucted out of entopy and tempeatue. [68] Afte ou intoduction of the entopy, one abitay additive constant (S 0 ) emains, which, fo example, could still depend on the paticle numbe. This indeteminacy is totally appopiate, because in macoscopic themodynamics, the entopy constant is fist calculated though the examination of phase o chemical equilibia, which obviously can t be developed though ou model. [69] As a esult of the entopy fomula (22a), a slight entopy coection emeges if two systems of same tempeatue ae bought into themal contact. In the appoximation fomula (22b), this expesses itself in the non-lineaity of the baceted element. The effect disappeas thus with inceased paticle numbe. The entopy incease contains the impossibility of being able to combine systems quasi-statically because that would equie the egodic chaacte of evey system to each othe be maintained. The combination is accodingly also ievesible, because of the sepaation an enegy dispesion occus, as is calculated above (p. 357).

14 Page 14 Conclusion [70] The goal of this study is not to efine the usual statistical fomulas fo systems consisting of few paticles. (Deviations in the deived fomulas, fo example, of the enegy distibution (17) o the entopy fomulas (22, 22a), disappea fo lage numbes of paticles and ae independent of the stict validity of the egodic hypothesis.) It has athe been shown that a sequence of equied postulates fo statistical justification of themodynamics is supefluous: It is in no way fundamentally equied, that systems be egaded as consisting of vey many ( infinitely many ) independent paticles, o that all o some of these must be assumed to be identical in ode to detemine themodynamic quantities. Neithe is it necessay to use complex-valued distibutions of phase volumes, but athe, aleady out of pue classical, constant basic appoaches, we get vey simply both of the Laws of Themodynamics. (the Nenst Theoem can natually not be deived.) Rathe, this shows that the conclusion S S 0 = ln J is in opposition to the Boltzmann-Planc postulate S = lnw, fo one can constue the phase volume J if need be (afte nomalization with the coesponding potential using h) as the total of the ealized possibilities fo all configuations with a smalle enegy than the given E. [71] It is notewothy, that the method of the convolution of canonical invaiant phase densities accomplished hee leads to a staightfowad definition of quantities which fulfill the themodynamic diffeential equations exactly and which, fo a composite system, have a definite value and don't statistically fluctuate as is usually supposed. Tanslation by Joseph Dittle and Randall B. Shits, 2006