Constraints of Compound Systems: Prerequisites for Thermodynamic Modeling Based on Shannon Entropy

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1 Entropy 2014, 16, ; do: /e OPEN ACCESS entropy ISSN Artle Contrant of Compound Sytem: Prerequte for Thermodynam Modelng Baed on Shannon Entropy Martn Pfleger, Thoma Wallek and Andrea Pfenng Inttute of Chemal Engneerng and Envronmental Tehnology, Graz Unverty of Tehnology, NAWI Graz, Inffeldgae 25/C/I, 8010 Graz, Autra * Author to whom orrepondene hould be addreed; E-Mal: martn.georg.pfleger@gmal.om, Tel.: Reeved: 14 Aprl 2014; n reved form: 17 May 2014 / Aepted: 21 May 2014 / Publhed: 26 May 2014 Abtrat: Thermodynam modelng of extenve ytem uually mpltly aume the addtvty of entropy. Furthermore, f th modelng baed on the onept of Shannon entropy, addtvty of the latter funton mut alo be guaranteed. In th ae, the onttuent of a thermodynam ytem are treated a ubytem of a ompound ytem, and the Shannon entropy of the ompound ytem mut be ubeted to ontraned maxmzaton. The ope of th paper to larfy prerequte for applyng the onept of Shannon entropy and the maxmum entropy prnple to thermodynam modelng of extenve ytem. Th aomplhed by nvetgatng how the ontrant of the ompound ytem have to depend on mean value of the ubytem n order to enure addtvty. Two example llutrate the ba dea behnd th approah, omprng the deal ga model and ondened phae latte ytem a lmtng ae of flud phae. The paper the frt tep toward developng a new approah for modelng nteratng ytem ung the onept of Shannon entropy. Keyword: Shannon entropy; ontraned extremalzaton; ompound ytem; addtvty; thermodynam modelng; drete tate; drete modelng

2 Entropy 2014, Introduton In h ba work, Shannon [1] defne a funton H whh meaure the amount of nformaton of a ytem whh an rede n ether of m poble tate by mean of the probablte p of the tate: H(p 1,..., p m ) = K p log p (1) Both, the ontant K and the ba of the logarthm are arbtrary a they ut aount for a alng of H. The et of all p an be wrtten a probablty dtrbuton p: p = {p 1,..., p m }, wth the normalzaton ondton p = 1 (2) In the followng we et K = 1 and hooe the natural logarthm. When buldng the um over all tate, the lmt of the ummaton (1... m) an be omtted and H an formally be wrtten a funton of the probablty dtrbuton: H(p) = p ln p (3) Prevou paper [1 8] worked out that th meaure ha all the properte of thermodynam entropy a ntrodued by tattal phy. Throughout th paper we all H(p) a defned n Equaton (3) the Shannon entropy of the ytem under onderaton, n order to dtnguh t from thermodynam entropy. The range of H(p) gven by 0 H(p) ln m. The zero value for a dtrbuton where one of the p equal 1 and, beaue of the normalzaton ondton (2), all other p are zero. The maxmum value gven for unformly dtrbuted probablte [6]: p = 1 m, = 1... m H(p) = ln m (4a) (4b) 1.1. Compound Sytem We onder a ompound ytem ompoed of N ubytem, eah haraterzed by t ndvdual probablty dtrbuton: {p 1 } = {p 1,1,..., p 1,m1 }. {p } = {p,1,..., p,m }. {p N } = {p N,1,..., p N,mN }

3 Entropy 2014, The tate of the ompound ytem defned by the tate of the ubytem. We therefore wrte the probablty dtrbuton of the ompound ytem a {p } = {p 1,1,...,1,..., p 1,..., N,..., p m1,...,m N }, (5) where p 1,..., N the probablty of the ompound tate where ubytem 1 n the tate 1, ubytem 2 n the tate 2, and o on. It not neeary for the ubytem to have dental probablty dtrbuton. Generally the probablty of the ompound tate A B, omprng the tate A and B of two ubytem gven by p(a B) = p(a/b) p(b) where p(a/b) the probablty of ubytem 1 to be n tate A, gven that ubytem 2 n tate B. If the ubytem are tattally ndependent,.e., p(a/b) = p(a), then t follow (f. [9,10]): p(a B) = p(a) p(b) (6) If all N ubytem omprng the ondered ompound ytem are tattally ndependent, traghtforward applaton of Equaton (6) to the probablty dtrbuton (5) gve: {p} = p 1,..., N = p 1... p N Wth th probablty dtrbuton the Shannon entropy of the ompound ytem : m 1 m 2 H = 1 =1 2 =1 m N N =1 (p 1, 2,..., N ln p 1, 2,..., N ) H = H 1 + H H N (7) Hene the Shannon entropy of ndependent ubytem addtve. In the peal ae of N equal and tattally ndependent ubytem,.e., {p 1 } =... = {p N } {p }, the homogenety of the Shannon entropy of the ompound ytem follow: H = N H (8) Throughout th paper the ndex ued for ngle ytem and for ompound ytem The Brdge to Thermodynam Entropy Conderng a ompound ytem ompoed of N equal and tattally ndependent ubytem, all ubytem are haraterzed by the probablty dtrbuton {p }, and homogenety, Equaton (8), guaranteed. For a large number of ubytem,.e., N >> 1, the probablte p an be expreed a relatve oupaton number N, degnatng the number of ubytem redng n the tate : p N N, (9)

4 Entropy 2014, wth N = the total number of ubytem wthn the ompound ytem, playng the role of the normalzaton ondton (2). Expreng the Shannon entropy of a ngle ubytem wth the oupaton number n Equaton (9) gve N H = N N ln N N, and beaue of homogenety, Equaton (8), the Shannon entropy of the ompound ytem : H = N ln N + N ln N. (10) The rght hand de of Equaton (10) ha the ame form a the logarthm of the thermodynam probablty W known from laal tattal mehan [11]: ln W (N 1, N 2,..., N,...) = N ln N + N ln N (11) The varable N n Equaton (11) gve the number of partle n ell of the µ-pae, and have the very ame meanng a the oupaton number N n Equaton (10),.e., the number of partle redng n the (mehanal) tate. The et (N 1, N 2,..., N,...) alled the oupaton of the µ-pae, and the thermodynam probablty W the number of mrotate realzng the gven oupaton. Hene, the left hand de of Equaton (10) and (11) tand for the ame meaure and an be ombned to: H = ln W. Table 1 ompare the onept behnd the two meaure. Beaue of S = k B ln W where S the thermodynam entropy of the ytem we get the reult: S = k B H Th equaton reveal the equvalene between Shannon entropy and thermodynam entropy related by the Boltzmann ontant.

5 Entropy 2014, Table 1. Comparon between Shannon entropy H and thermodynam probablty W. Shannon entropy H thermodynam probablty W probablty dtrbuton: {p} = {p 1, p 2,..., p,...} oupaton: (N 1, N 2,..., N,...) H = N p ln p W = N! N 1!N 2!...N!... p = 1 N = N aumpton N >> 1 p N N equal ubytem, tattally ndependent N, N 1,..., N >> 1 ln N! N ln N Strlng formula appled to N and to all N H = N ln N + N ln N ln W = N ln N + N ln N 1.3. Addtvty of Shannon Entropy: It Sgnfane for Thermodynam Modelng Homogenety of a ompound ytem, Equaton (8), the tartng pont for thermodynam modelng baed on the tate of t onttuent. On the one hand, a hown n eton 1.2, t buld the brdge between Shannon entropy and the laal thermodynam entropy. On the other hand, the rual property of addtvty enable the alulaton of thermodynam entropy mply by alulatng the Shannon entropy of a ngle ubytem. Subequently, the ompound ytem entropy an be expreed by the um of entrope of the onttutng ubytem. Appled to a ga or flud th mean to derve the Shannon entropy of one atom or moleule baed on ther repetve tate. When peakng of moleular or drete tate we do not neearly onder the quantum-mehanal tate of atom or moleule; the mehanal, ontnuou tate of atom or moleule are alo poble anddate. But when ung the drete formulaton of Shannon entropy, Equaton (1), a dretzaton of the ontnuou tate helpful. When applyng latte model for derbng ondened phae ytem, the goal redued to dervaton of the Shannon entropy of a ngle latte te. However, when dervng the Shannon entropy of a ompound ytem by utlzng homogenety, Equaton (8), we made the followng preaumpton: The frt the aumpton of tattally ndependent ubytem, whh may be plauble for many thermodynam ytem a long a the ubytem (the partle) are not too trongly orrelated n ome nonloal ene [10]. We dd not emphaze the eond aumpton, beaue t eem to be elf-evdent: We ued the probablty dtrbuton a gven ytem varable, a f they were properte of the ubytem, whh - for tattally ndependent ytem - tay ontant. But a known from laal thermodynam, the

6 Entropy 2014, entropy of a ytem depend on ytem varable lke nternal energy, temperature, preure and o on. In addton, entropy not predefned dretly by thee varable, but underle a maxmzaton prnple, tatng that a ytem n thermodynam equlbrum rede n a tate where entropy a maxmum, wth repet to the ontrant gven by the ytem varable. Th mean that we annot deal a pror wth gven probablty dtrbuton, but we have to determne the very probablty dtrbuton whh maxmze entropy wth repet to the ontrant. Hene, f we ak for valdty of addtvty, Equaton (8), we have to nvetgate the maxmzed Shannon entrope of a ompound ytem and t onttutng ngle ytem eparately, a hown n the followng eton. 2. Probablty Dtrbuton wth Maxmum Entropy 2.1. Maxmzaton of Unontraned Sytem A an be een from Equaton (4b), the maxmum value of the Shannon entropy of unontraned ytem,.e., unformly dtrbuted tate, depend olely on the number of ther poble tate. For a ngle ytem wth m poble tate we get H = ln m. In the ae of a ompound ytem ontng of N ngle ytem, eah wth m poble tate, the number of poble tate m N, reultng n a Shannon entropy of H = N ln m,.e., N tme the Shannon entropy of the ngle ytem (f. Equaton (4b)). So for unontraned ytem homogenety, Equaton (8), guaranteed Contraned Maxmzaton of a Sngle Sytem Now a ngle ytem wth m poble tate ondered, eah of them haraterzed by the value f of a random varable F. Let one ontrant be gven by the mean value f of the random varable F : f = f p (12) A lot of probablty dtrbuton p may reult n the ame mean value f. Among thee probablty dtrbuton we are lookng for the one yeldng the maxmum value H for the Shannon entropy. The maxmzng probablty dtrbuton and the reultng Shannon entropy H wll depend on the exat hoe for f, o that both an be expreed a funton of th ontrant: p = p( f ) H = H ( f ) The maxmzng probablty dtrbuton onderng ontrant (12) and the normalzaton ondton (2) an be found by applyng Lagrange method of ontraned extremalzaton [12]. Th method ntrodue the Lagrange funton ) )! L = H λ 0 (1 p λ 1 ( f f p = max., (13)

7 Entropy 2014, where λ 0 and λ 1 are the Lagrangan multpler. L ha to be maxmzed by equatng the dervaton wth repet to all p to zero: p L = 0 Wth H from Equaton (3) and performng all dervaton we get: Inertng Equaton (14) nto the normalzaton ondton (2) yeld: p = e ( 1+λ 0+f λ 1 ) (14) p = 1 = e ( 1+λ 0) Inertng Equaton (14) nto ontrant (12) reult n: f p = f = e ( 1+λ 0) Ung the abbrevaton x e λ 1, ombnaton of Equaton (15) and (16) yeld: f = f e f λ 1 (15) f e f λ 1 (16) Th equaton an be olved numerally for x, whh an now be ued to expre the maxmzng probablty dtrbuton. For that purpoe, Equaton (14) and (15) an be rewrtten a: p = e ( 1+λ 0) 1 = e ( 1+λ 0) Combnng thee equaton yeld the maxmzng probablty dtrbuton (17) p = xf x f. (18) Inertng (18) nto defnton (3) reult n: H = x f ln xf x f The denomnator of the frt fator doe not depend on the ndex of the outer um and an be put n front of the um. The logarthm of the fraton now wrtten a um of two term: ( ) H = 1 ln ln x f x f

8 Entropy 2014, Further rearrangement fnally yeld: H = ln x f + ln (19) Th value repreent the maxmum value of the Shannon entropy among all probablty dtrbuton ompatble wth the normalzaton ondton and the ontrant, Equaton (12) Contraned Maxmzaton of a Cmpound Sytem Gven a ompound ytem ompoed of N ngle ytem a mentoned n the lat eton, eah of whh an rede n ether of m poble tate and wth eah tate beng agned a value f of a drete random varable F, the number of poble tate m N. Wth q k the probablte of the ompound ytem, where k denote the tate, when the ngle ytem 1 n tate 1, ngle ytem 2 n tate 2 and o on: k = 1,..., N, the probablte an be rewrtten a: wth q k = q 1,..., N (20) k = 1... m N 1 = 1... m. N = 1... m Let G be a random varable related to the tate of the ompound ytem, and g k be the aordng value of G n the tate k. g k hoen n uh a way, that t repreent the um of the random varable F of the ngle ytem: g k = g 1,..., N = f f N (21) wth f 1 the value of F of partle 1 n tate 1 and o on. The mean value g of G, whh wll at a ontrant for the ompound ytem, (f. Equaton (12)): m N g = g k q k (22) = k= =1 g 1,..., N q 1,..., N (23) N =1 = f f N (24) The mean value of the ompound ytem the um of the mean value of the ngle ytem. If the ngle ytem are equal, ther mean value the ame: f 1 = f 2 =... = f N f, (25)

9 Entropy 2014, reultng n g = N f. (26) Now we are lookng for the probablty dtrbuton q whh fulfll the normalzaton ondton, guarantee the mean value g gven by the ontrant, and yeld the maxmum value H for the Shannon entropy. The reult wll depend on the ontrant g : m N H = q k ln q k = H ( g ) k=1 q = q( g ) (27) Agan applyng Lagrange method, the oluton (17), (18) and (19) of the ngle ytem an be reued by replang f g, x x, H H and onderng that the number of tate now m N. Wth x beng the oluton of (f. Equaton (17)) g = N N g x g the probablte of the ompound ytem are (f. Equaton (18)) x g (28) q k = xg k N x g and the Shannon entropy of the ompound ytem (f. Equaton (19)): H = N, k = 1... m N, (29) x g ln x g N x g Inertng Equaton (21) nto Equaton (28) reult n: g = N k=1 k=1 m N + ln f k x f k x f k x g (30) a expltly derved n the upplementary materal. Takng nto aount Equaton (26) we get: f = k=1 f k x f k By omparng Equaton (32) wth Equaton (17) one an ee that the oluton x and x fulfll the ame equaton; they are equal and ther ubrpt an therefore be omtted: k=1 x f k (31) (32) x = x = x (33)

10 Entropy 2014, Now we agan ue Equaton (21) and nert t nto Equaton (30). The reult : ln H = N + ln (34) wth ntermedate tep gven n the upplementary materal. The ame expreon an be obtaned when Equaton (20) and (21) are nerted nto the probablte, Equaton (29), reultng n q 1, 2,..., N = xf N ( ) N, (35) and ung th for alulatng Shannon entropy by mean of Equaton (3). Th alternatve dervaton alo nluded n the upplementary materal. Beaue of the equvalene of x and x, Equaton (33), we an rewrte Equaton (19) and (34): ln H = + ln x f x f ln H = N + ln Comparng both thee equaton yeld the reult: H = NH (36) Equaton (36) llutrate that homogenety alo fulflled for ompound ytem underlyng an extremalzaton prnple wth repet to one ontrant. The rual aumpton we made that the ontrant of the ubytem and the ontrant of the ompound ytem obey equaton (21) Sytem Underlyng Several Contrant To be more general we onder ytem under everal ontrant, agan begnnng wth a ngle ytem. Let α be the number of random varable F 1, F 2,... F α a ontrant, and m the number of poble tate. The value of F 1 n the tate f 1,, the aordng value of F 2 f 2, et. The ontrant are gven by the mean value f 1, f 2,..., f α of the random varable, wth f 1 = p f 1, f 2 = p f 2,. f α = p f α,.

11 Entropy 2014, Straghtforward applaton of Lagrange method of ontraned extremalzaton reult [1.0] n (f. Equaton (18)): p = A, (37) A wth the abbrevaton A = X f 1, 1 X f 2, 2... X f α, α (38) and the X 1, X 2,..., X α beng the oluton of the followng ytem of equaton (f. Equaton (17)): f 1, A = f 1 A. f α, A = f α A Calulatng the Shannon entropy wth the probablte gven by Equaton (37) yeld (f. Equaton (19)): A ln A H = + ln A (40) A We onder a ompound ytem ompoed of N of thee ngle ytem, wth α random varable G 1, G 2,..., G α, whh are aoated to the random varable of the ngle ytem n the ame way ndated by Equaton (22), (23), (24), (25) and (26). The value of G 1 n the tate k g 1,k, the aordng value of G 2 g 2,k and o on, and they are related to the random varable of the ngle ytem by (f. Equaton (21)): g 1,k = f 1,1 + f 1, f 1,N (41) The ontrant are gven a the mean value g 1, g 2,..., g α of the random varable, wth (39) g 1 = k g 2 = k. g α = k q k g 1,k = N f 1 q k g 2,k = N f 2 q k g α,k = N f α. (42) Straghtforward applaton of Lagrange method of ontraned extremalzaton reult n (f. Equaton (18) and (37)): q k = B k, (43) B

12 Entropy 2014, wth the abbrevaton B k = Y g 1,k 1 Y g 2,k 2... Y g α,k α (44) and wth the Y 1, Y 2,..., Y α beng the oluton of the followng ytem of equaton (f. Equaton (17) and (39)): g 1,k B k k = g 1 B. g α,k B k k = g α B Calulatng the Shannon entropy wth the probablte gven by Equaton (43) yeld (f. Equaton (19) and (40)): B k ln B k k H = + ln B k (46) B k Aordng to Equaton (42) we replae the rght hand de of the frt equaton of ytem (45) wth N f 1, g 1,k B k k N f 1 =, (47) B and all other equaton of ytem (45) wth the aordng expreon. In the defnton of the B k we replae the exponent g 1,k, g 2,k,..., g α, aordng to Equaton (41) and nert the expreon nto Equaton (47). The evaluaton yeld: f 1 = N f 1,Â. f α = N  f α,â  wth  defned mlarly to A for the ngle ytem, Equaton (38), but now wth the fator Y 1, Y 2,..., Y α :  = Y f 1, 1 Y f 2, 2... Y f α, α The ytem of equaton for X 1,..., X α for the ngle ytem, Equaton (39), the ame a for the Y 1,..., Y α for the ompound ytem, Equaton (48), reultng n: Y 1 = X 1 Y 2 = X 2. Y α = X α, (45) (48)

13 Entropy 2014, Now, replang all Y -fator n the defnton of the B k, (44), wth the aordng X-fator reult n: ( ) ( B k ln B k = N A ln A k ( ) N B k = A k A ) N 1 ln k B k = N ln A Inertng thee expreon nto the Shannon entropy of the ompound ytem, Equaton (46), we get the reult: A ln A H = N + ln A A and omparng wth Equaton (40) we have H = N H It therefore proven that the Shannon entropy of ompound ytem underlyng everal ontrant homogeneou f the ontrant behave aordng to Equaton (41) and (42). Equaton (21) and (26) for ytem underlyng one ontrant a well a Equaton (41) and (42) for ytem underlyng everal ontrant reveal that homogenety of Shannon entropy guaranteed, f the ontrant behave homogeneouly,.e., n a lnear dependene of the number of ubytem: g(n) = N f g(ont. N) = ont. N f = ont. g(n) (49) We an therefore onlude that the aumpton of both ndependent ubytem and of a ompound ytem underlyng a maxmzaton prnple wth repet to addtve ontrant lead to the ame mportant reult: the homogenety of the Shannon entropy of the ompound ytem, hene the homogenety of the modeled thermodynam entropy. Both aumpton alo at a prerequte for homogenety and an be regarded a omplementary vew of the ame property of a ompound ytem; nether of thoe apet preferred to the other. 3. Applaton to Thermodynam Modelng of Flud Phae Amendatory to the prevou eton where onderaton were etablhed for ngle and ompound ytem n general, n the followng the applaton to thermodynam modelng of flud phae hall be dued. Thee model derbe the ytem under onderaton from the vewpont of ther poble tate. For th purpoe we reflet on the lmtng ae of flud phae, the deal ga model and the ondened phae latte ytem.

14 Entropy 2014, Ideal Ga The deal ga an be ondered to be a ompound ytem, ontng of a huge number of deal and equal partle wth no nteraton among them. Thee partle are treated a the ubytem of the ompound ytem. The mehanal tate of a partle n the ene of laal mehan gven by t poton and veloty vetor, o the tate derbed by 6 random varable. The knet tate of the deal partle doe not depend on t poton and ve vera, o the poton and veloty vetor are ndependent random varable. Conequently, a dued n eton 2.4, th lead to two tattally ndependent probablty dtrbuton, and the Shannon entropy plt n a knet and a potonal term: H = H kn + H pot Therefore, maxmzng H an be plt by maxmzng H kn and H pot eparately. We retrt our onderaton to the maxmzaton of the knet term, whh tll ompre three random varable: one for the veloty and two for the dreton of the movement. Aumng otropy for the deal ga mean that the dtrbuton of the knet energy doe not depend on the dreton of the movement. Hene we an agan argue that the veloty ndependent from the two other random varable, and we retrt our onderaton to the knet tate of the partle, defned only by the mean of ther veloty, equvalent to ther knet energy e. The knet energy n fat a ontnuou varable. But n order to ue the drete formulaton of Shannon entropy, Equaton (1), we an dretze t by ntrodung an arbtrarly mall energy quantum ɛ: e = ɛ. The dretzed energy e meant only a a mathematal artfe. In the lmt ɛ 0 all poble ontnuou tate an be repreented. The knet, drete tate k of the deal ga then defned by the knet tate of the partle: k = 1, 2,..., N. Obvouly, the knet energy of the whole ytem n the tate k the um of the knet energe of the ngle partle: E k = E 1,..., N = e e N, f. wth Equaton (21) and (41), and the addtvty of the knet energy atng a ontrant follow mmedatley a E = N e, (50) n aordane wth Equaton (26) and (42), and therefore addtvty of the Shannon entropy of the knet term guaranteed, f. wth Equaton (36): H kn, = NH kn, Th reult the ba for the drete modelng of the deal ga, to be preented n a ubequent paper Condened Phae Latte Sytem Latte ytem are motly appled for trongly nteratng ondened phae where moleular dtane orrepond to the lqud or old tate. Many engneerng model ued n proe mulator

15 Entropy 2014, for hemal-engneerng purpoe uh a atvty oeffent model or equaton of tate are orgnally baed on latte model [13][14][15]. One of the reaon for th that uh model an ealy be verfed by Monte-Carlo mulaton, allevatng model development and verfaton. Therefore, n the followng the peularte of latte ytem hall be dued from the vewpont of Shannon entropy. A latte ytem provde fxed te, eah of whh ouped by one moleule n the mplet ae, eah of whh nterat wth t loet neghbor. In the followng we apply the maxmzaton preented n eton 2 to an exemplary, one-dmenonal latte, omprng moleule of two type. In term of eton 2, n the followng the whole latte ondered a the ompound ytem, ompoed of te whh repreent the ubytem The onept of ubytem appled to a latte ytem The mplet way to defne a ubytem n term of eton 1.1 for a latte to ue a ngle latte te olated from t adaent neghbor. In a lnear latte ytem wth two omponent uh an olated te ha 2 1 = 2 poble drete tate, a llutrated n Fgure 1(a). Fgure 1. All poble drete tate of a ngle latte te a ubytem of a lnear latte ytem when oberved (a) olated from t z nearet neghbor and (b) aoated wth t neghbor. In a lnear latte, onderng only the nearet neghbor, z = (a) (b) However, a onept of an olated latte whh doe not take t nearet, nteratng neghbor nto aount doe not allow for the formulaton of ontrant nludng nteraton energe between te, even though uh ontrant are eental for model development. For th reaon, we ntrodue the onept of a ngle latte te aoated wth t z nearet neghbor a ubytem, z repreentng the oordnaton number. The ubytem ompre (z + 1) te, a llutrated n Fgure 1(b). In th onept, the drete tate of a latte te determned not only by t own moleule type but alo by the type and arrangement of the z nearet neghbor ontrbutng to energet nteraton. Here t aumed that moleular nteraton are onfned to the nearet neghbor of the entral moleule. If moleule beyond the dret neghbor alo ontrbute to nteraton, the onept of aoated te an be extended aordngly The unontraned ytem Shannon entropy of a ubytem: Wthout onderaton of any ontrant, t follow from equaton (4a) that both of the two poble tate hown n Fgure 1(a) have the ame probablty, p = p = 1 2,

16 Entropy 2014, orrepondng to a ytem wth an equal number of blak and whte te. Equaton (4b) for an olated te wth 2 poble tate yeld H, o = ln 2 (51) For an aoated te hown n Fgure 1(b), ontng of (z + 1) te, agan aordng to Equaton (4a), all poble tate are equally probable. The number of poble tate now 2 z+1, yeldng the maxmum Shannon entropy of H, a = ln 2 z+1 = (z + 1) ln 2, (52) whh the (z + 1)-fold of the Shannon entropy of the olated te gven by (51). In a next tep, the poblty of expreng the Shannon entropy of a ompound latte ytem by the Shannon entropy of t onttutng ubytem hall be examned. Shannon entropy of a ompound ytem: When dtrbutng two type of moleule over a latte omprng N te, the number of poble tate gven by 2 N, yeldng the maxmum Shannon entropy of H = ln 2 N = N ln 2 = N H, o, (53) whh the N-fold of the maxmum Shannon entropy of an olated te, f. Equaton (51). When onderng aoated te a ubytem, where a ubytem ont of (z + 1) te, the number of ubytem ˆN = N z + 1 Now the number of tate of the ompound ytem 2 z+1 ˆN, and the aordng Shannon entropy yeld H = ln 2 z+1 ˆN = ˆN ln 2 z+1 = ˆN H, a, (54) whh the ˆN-fold of the maxmum Shannon entropy of an aoated te, f. Equaton (52). Equaton (52)-(54) reveal the homogenety of the Shannon entropy of unontraned latte ytem, f. Equaton (8): H = N H, where H now denote the Shannon entropy of the whole latte a ompound ytem, H the Shannon entropy of the ondered ubytem,.e., olated or aoated te, and N the number of ubytem Sytem onderng ontrant There are baally three type of ontrant to be ondered n a latte ytem: Energy, ompoton and the equvalene of ontat par between moleule of dfferent type. Energy: A mentoned at the begnnng of eton 3.2, the ntended purpoe of latte ytem the onderaton of nteraton energe between adaent latte te. Therefore, the onept of a ngle latte te aoated wth t nearet neghbor wa ntrodued n eton to be ued a ubytem. Baed on th onept, ontrant onderng nteraton energe an be formulated generally n the form u = u p a (55)

17 Entropy 2014, whh n lne wth (12), u degnatng the energy agned to the entral moleule of an aoated te, p a t probablty of redng n tate and u the mean value of energy. Fgure 2 llutrate th nomenlature. Fgure 2. Example for tate and related energe of an aoated latte te a ubytem. ε denote the nteraton energy between two te, where eah te agned the half of t. ubytem tate entral moleule energy u 1 2 (ε + ε ) (ε 2 + ε ) 3 1 (ε 2 + ε ) To formulate the energy agned to a ompound latte ontng of N aoated te a ubytem, we ue the ndex k to denote the tate of the ompound ytem whh determned by the tate of t ubytem: k = 1,..., N. Wth th nomenlature, the energy of a ompound ytem n tate k U k, where U k = U 1,..., N = u u N, whh analogou to (21). Reallng equaton (22) to (25) for the mean value of energe, t follow that U = u u N Beaue all ngle ubytem are of the ame knd and no ngle ubytem preferred to another, ther mean value the ame, reultng n u 1 = u 2 =... = u N u U = N u, (56) whh analogou to (26). Ung (56) a the only ontrant ade from the normalzaton of probablte, maxmzaton of the Shannon entropy analog to (13) requre oluton of the Lagrange funton L = H λ 1 U λ 2 {p } =! max. (57) Applaton of the maxmzaton prnple to (57) n lne wth (27) - (34) fnally reult n H = N H, a (58) a Shannon entropy of the ompound latte ytem, analogouly to Equaton (36). Equaton (58) reveal that the Shannon entropy of a ontraned latte ytem an alo be expreed through the Shannon entropy of ubytem, whereupon ubytem are ngle latte te aoated wth ther repetve nearet neghbor. A hown n eton 2.4, everal funton wth the gener form of equaton (55) an alo be ondered a ontrant n the maxmzaton prnple. Compoton: In the mplet ae of a bnary ytem, there are moleule of two type onttutng the latte. The ontrant for the ompound ytem mply N 1 = N x 1, the total number of 1-moleule.

18 Entropy 2014, x 1 an be nterpreted n two way: a relatve fraton of 1-moleule n the ytem, or a probablty to fnd a 1-moleule at a gven te. Hene, f we onder ompound ytem of dental ompoton, N 1 behave aordng to Equaton (49), fulfllng the prerequte for ontrant that enable homogenety of Shannon entropy. The ame hold for x 2 whh related to x 1 by the normalzaton ondton. Th an ealy be extended to ytem omprng an arbtrary number of omponent. Equvalene of ontat par: Contat par degnate the number of ontat between moleule of dfferent type n the ytem, e.g. N 2 1 (read 2 around 1 ) the number of all 2-moleule around all moleule of type 1, and N 1 2 the number of all 1-moleule around all moleule of type 2. A the number of ontat between moleule of dfferent type mut not depend on the vewpont, the equvalene N 2 1 = N 1 2 ha to be fulflled n latte ytem generally, ndependent of the repetve ze. The aordng ontrant for the maxmzaton prnzple N 2 1 N 1 2 = 0 Th equatng to zero a homogeneou funton n term of Equaton (49). In ummary, all three type of ontrant are homogeneou n term of Equaton (49), enurng that after maxmzaton the Shannon entropy of a ompound ytem alo homogeneou. The omplete Lagrange funton fnally ompre the mean value of energy, ompoton and equvalene of ontat par a ontrant to be ondered n a latte model. Pratal applaton wll be hown n a ubequent paper. 4. Conluon The ope of th paper wa to larfy prerequte for applyng the onept of Shannon entropy and maxmum entropy prnple to thermodynam modelng of extenve ytem. The man rteron for applablty of th knd of modelng the addtvty of the Shannon entropy. It wa hown that th addtvty guaranteed, provded that the addtvty of the ontrant gven. If a thermodynam model ompre addtve ontrant, th prerequte fulflled, and the method expltly applable to ytem of nteratng omponent,.e., real flud. Th wa hown for two lmtng ae of flud phae, the deal ga model and ondened phae latte ytem. The man beneft of thermodynam modelng baed on Shannon entropy that t make the equlbrum dtrbuton of drete tate avalable. Th etablhe new poblte for thermodynam and ma tranport model a t allow onderaton of a more detaled pture of phyal behavor of matter on a moleular ba, beyond the ope of tradtonal modelng method. Th wll be exploted n ubequent paper. Aknowledgement The author gratefully aknowledge upport from NAWI Graz.

19 Entropy 2014, Author Contrbuton M.P. ontrbuted eton 2 and 3.1, T.W. ontrbuted eton 3.2, eton 1 and 4 were ont ontrbuton of M.P. and T.W. The paper wa read and omplemented by A.P. Conflt of Interet The author delare no onflt of nteret. Referene 1. Shannon, C.E. A Mathematal Theory of Communaton. Bell Sy. Tehn. Journ. 1948, 27, , Jayne, E.T. Informaton Theory and Stattal Mehan. Phy. Rev. 1957, 106, Jayne, E.T. Informaton Theory and Stattal Mehan. II. Phy. Rev. 1957, 108, Jayne, E.T. Gbb v Boltzmann Entrope. Am. Journ. Phy. 1965, 33, Wehrl, A. General Properte of Entropy. Rev. Mod. Phy. 1978, 50, Guau, S.; Shentzer, A. The Prnple of Maxmum Entropy. Math. Intell. 1985, 7, Ben-Nam, A. Entropy Demytfed; World Sentf Publhng, Sngapore, Ben-Nam, A. A Farewell to Entropy: Stattal Thermodynam Baed n Informaton; World Sentf Publhng, Sngapore, Curado, E.; Tall, C. Generalzed Stattal Mehan: Conneton wth Thermodynam. J. Phy. A 1991, 24, L69 L Tall, C. Nonaddtve Entropy: The Conept and t Ue. Europ. Phy. Journ. A 2009, 40, Mazek, A. Stattal Thermodynam; Oxford Unverty Pre: Oxford, UK, Wlde, D.J.; Beghtler, C.S. Foundaton of Optmzaton; Prente-Hall: Englewood Clff, NJ, USA, Fowler, R.H.; Kaptza, P.; Mott, N.F.; Bullard, E.C. Mxture - The Theory of the Equlbrum Properte of Some Smple Clae of Mxture, Soluton and Alloy; Oxford at the Clarendon Pre, Abram, D.S.; Prauntz, J.M. Stattal Thermodynam of Lqud Mxture: A New Expreon for the Exe Gbb Energy of Partly or Completely Mble Sytem. AIChE J. 1975, 21, Bronneberg, R.; Pfenng, A. MOQUAC, a New Expreon for the Exe Gbb Energy baed on Moleular Orentaton. Flud Phae Equlb. 2013, 338, by the author; lenee MDPI, Bael, Swtzerland. Th artle an open ae artle dtrbuted under the term and ondton of the Creatve Common Attrbuton lene (