Equation of motion for incompressible mixed fluid driven by evaporation and its application to online rankings

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1 Equaton of moton for ncompressble mxed flud drven by evaporaton and ts applcaton to onlne rankngs Kumko Hattor Department of Mathematcs and Informaton Scences, Tokyo Metropoltan Unversty, Hacho, Tokyo , Japan. emal: Tetsuya Hattor Mathematcal Insttute, Graduate School of Scence, Tohoku Unversty, Senda , Japan. URL: emal: October 10, 2008 ABSTRACT We consder a system of partal dfferental equatons for the denstes of components of one dmensonal ncompressble flud mxture whose moton s drven by evaporaton. We prove exstence and gve explct form of unque global classcal non-negatve soluton to ntal value problem for the system. The soluton s known to be an nfnte partcle lmt of stochastc rankng processes, whch s a smple stochastc model of tme evolutons of Amazon Sales Ranks. As a practcal applcaton, we collected data from the web and performed statstcal fts to our formula. The results suggest that the system of equatons and solutons, though very smple, may have applcatons n the analyss of onlne rankngs. Keywords: evaporaton drven flud; non-lnear wave; stochastc rankng process; long tal; onlne rankng; 2000 Mathematcs Subect Classfcaton: Prmary 35C05; Secondary 35Q35, 82C22 Correspondng author: Tetsuya Hattor, hattor@math.tohoku.ac.p Mathematcal Insttute, Graduate School of Scence, Tohoku Unversty, Senda , Japan tel+fax:

2 2 1 Introducton. Let f 0, =1, 2,, be non-negatve constants, and consder the followng system of non-lnear partal dfferental equatons for the functons u (y, t), =1, 2,, and v(y, t), defned on (y, t) [0, 1) R + : u (y, t) t + (v(y, t) u (y, t)) = f u (y, t), y =1, 2,, (1) u (y, t) =1. (2) We consder ntal value problems for smooth non-negatve ntal data u (y, 0) = u (y); u (y) 0, =1, 2,, u (y) =1, 0 y<1, wth the boundary condtons at y = 0 and y =1: v(1 0,t)=0, (3) for t 0, where, for each, and we assume u (0,t)= f ρ, =1, 2,, (4) f ρ ρ = 0 < 1 0 u (z) dz, (5) f ρ <. (6) Note that addng up (1) over and applyng (2) we have v(y, t) y = f u (y, t), (7) whch, wth (3), determnes v n terms of u. Note also that (2) obvously mples ρ =1. (8) Gven the constants f and the ntal data u (y), the set of equatons (1) (2) (3) (4) defnes the evoluton of our system. The followng arguments and results hold both for fnte components ( =1, 2,,N) and nfnte components. (In fact, we can extend the system and the soluton to a case wth any probablty space Ω, by replacng u (y, t) wth a measure μ(dω, y, t). See [4] for probablty theoretc arguments.) A physcal meanng of the system s as follows. We are consderng a moton of ncompressble flud mxture n an nterval of length 1, where u (y, t) s the densty of -th

3 3 component at space-tme pont (y, t). (2) mples that we normalze total densty of the flud mxture to be 1 at each space-tme pont (y, t), hence u (y, t) represents the rato of -th component at (y, t). We are naturally nterested n the non-negatve solutons u (y, t) 0. v(y, t) s the velocty feld of the flud. (1) s the equaton of contnuty, and the rght-handsde mples that each component evaporates wth rate f per unt tme and unt mass. Note that the set of equatons (7) and (3) s equvalent to v(y, t) = 1 f u (z, t)dz, (9) y whch mples that the velocty feld, or the moton of the flud, s caused solely by fllng the amount of flud whch evaporated from the rght sde of the pont y. In partcular, we have no flux through the boundary y =1(v(1 0,t) = 0). The boundary condton (4) at y = 0 s so tuned by the ntal data that the loss of mass by evaporaton s compensated by the mmedate re-entrance at y = 0 as lqudzed partcles, so that the total mass of each flud component n the nterval [0, 1) s conserved over tme: 1 0 u (z, t) dz = ρ, t 0. (10) In fact, (10) s equvalent to (4), under the condtons (1) (2) (3) (5), f sup f <. (See Appendx A.) Let y C (y, t) =1 1 e f t u (z) dz, 0 y<1, t 0. (11) y For each t 0, y C (,t): [0, 1) [y C (0,t), 1) s a contnuous, strctly ncreasng, onto functon of y, and ts nverse functon ŷ(,t): [y C (0,t), 1) [0, 1) exsts: 1 y = 1 e f t ŷ(y,t) u (z) dz, y C (0,t) y<1, t 0. (12) y C (y, t) denotes the poston of a flud partcle at tme t (on condton that t does not evaporate up to tme t) whose ntal poston s y. ŷ(y, t) denotes the ntal poston of a flud partcle located at y ( y C (0,t)) at tme t. Wth slght abuse of notatons, we wll often wrte y C (t) for y C (0,t): y C (t) =y C (0,t)=1 ρ e f t, t 0. (13) Note that f ρ > 0, as assumed n (6), mples that y C (t) s strctly ncreasng (as well as contnuous) n t, and (8) mples y C (0) = 0. Hence, for each 0 <y<y C (t) there exsts unque t 0 = t 0 (y) (0,t) such that In Secton 2 we prove the followng. y C (t 0 (y)) = y, 0 y<y C (t), t>0. (14)

4 4 Theorem 1 There exsts a unque global (.e., for all t 0) non-negatve classcal soluton to the ntal value problem for the system of partal dfferental equatons defned by (1) (5), whch s explctly gven by (9) and e f t 0 (y) f ρ, t > t 0 (y), e f t 0 (y) f ρ u (y, t) = e ft (15) u (ŷ(y, t)) e ft u (ŷ(y, t)), 0 t<t 0(y), Note that for t>t 0 (y) the soluton s statonary: u t (y, t) =0, t > t 0(y). (16) For t<t 0 (y), effect of ntal condtons exsts n the form of wave propagaton. In natural phenomena where evaporaton s actve, such as producng salt out of sea water, vscosty, surface tenson, and external forces such as gravtatonal forces domnate, and the effect of evaporaton on the moton of flud would be relatvely too small to observe. Thus the equaton and the soluton we consder n ths paper may not have attracted much practcal attenton. However, there are phenomena on the web for whch our formulaton may work as a smplfed mathematcal model, such as the tme evolutons of rankngs of book sales n the onlne booksellers. Such possblty s theoretcally based on a result that (15) appears as an nfnte partcle lmt of the stochastc rankng process [4], whch s a smple model of the tme evolutons of e.g., the number known as the Amazon Sales Rank. (We note that ths number has mathematcally lttle to do wth the perhaps more popular noton of Google Page Ranks.) We collected data of the tme evoluton of the numbers from the web, and performed statstcal fts of the data to (13). Consderng the smplcty of our model and formula, we fnd the fts to be rather good. The results suggest that there s a new applcaton of our results n the analyss of onlne rankngs. The plan of ths paper s as follows. In Secton 2 we gve a proof of Theorem 1. In Secton 3 we recall the stochastc rankng process defned n [4], and relate (15) to the nfnte partcle lmt studed n [4]. In Secton 4 we gve results of fts to (13) of data from the web. The authors would lke to thank Prof. T. Myakawa, Prof. M. Okada, Prof. H. Nawa, and Prof. T. Namk for takng nterest n our work and for nvtng the authors to ther semnars. The research of K. Hattor s supported n part by a Grant-n-Ad for Scentfc Research (C) from the Mnstry of Educaton, Culture, Sports, Scence and Technology, and the research of T. Hattor s supported n part by a Grant-n-Ad for Scentfc Research (B) from the Mnstry of Educaton, Culture, Sports, Scence and Technology. 2 Proof of the man theorem. Put U (y, t) = 1 y u (z, t) dz, =1, 2,, (17)

5 and U (y) = 1 u y (z) dz for the ntal data U (y, 0) = U (y). Wth (17), the the system of equatons (1) (5) s equvalent to the followng: U (y, t) s decreasng n y and U (1 0,t)=0, and U t (y, t)+v(y, t) U y (y, t) = f U (y, t), =1, 2,, (18) U (y, t) =1 y, (19) 5 v(y, t) = f U (y, t), (20) for 0 y<1, t 0 (note (9)), and notng (10), U (0,t)=ρ =1, 2,, t 0. (21) The remander of the proof s a smple applcaton of characterstcs. For each 0 y 0 < 1 let y B = y B (y 0 ; t) be a soluton to an ODE and put Wth (22) and (18) t follows that dy B dt (y 0; t) =v(y B (y 0 ; t),t), t 0, y B (y 0 ;0)=y 0, (22) φ (t) =U (y B (y 0 ; t),t). (23) dφ dt (t) = U t (y B(y 0 ; t),t)+v(y B (y 0 ; t),t) U y (y B(y 0 ; t),t) = f U (y B (y 0 ; t),t)= f φ (t), hence, wth φ (0) = U (y 0, 0) = U (y 0 ), φ s unquely solved as Wth (20) and (22), we then fnd φ (t) =U (y 0 ) e f t. (24) dy B dt (y 0; t) = f U (y 0 ) e f t, hence, usng (19) and (17), we have y B (y 0 ; t) =1 U (y 0 ) e f t =1 1 y 0 u (z) dz e f t = y C (y 0,t), (25) where y C s defned n (11). Wth (12), (23) and (24) we unquely obtan U (y, t) =U (ŷ(y, t)) e f t.

6 6 Dfferentatng by y and usng (17) and (25) u (y, t) = ( ) 1 yb (ŷ(y, t); 0) u (ŷ(y, t)) e ft = u (ŷ(y, t)) e f t y 0 u (ŷ(y, t)) e, f t where y B s the dervatve of y B = y B (y 0 ; t) wth respect to the parameter y 0. Ths proves y 0 (15) for y>y C (t),.e., for 0 t<t 0 (y). Next let t>t 0 (y) and put t 1 = t t 0 (y) (0,t). Let y A be a soluton to an ODE and put y A (s) =v(y A(s),s), s t 1, y A (t 1 )=0, (26) φ (s) =U (y A (s),s), s t 1. (27) Note that (21) mples φ (t 1 )=U (0,t 1 )=ρ, hence, as below (22), φ s unquely solved as Wth (20) and (26), we then fnd φ (s) =ρ e f (s t 1 ). (28) y A(s) = f ρ e f (s t 1 ), Note that (5) and (2) mply ρ = 1. Hence, y A (s) =1 ρ e f (s t 1 ) = y C (s t 1 ),s t 1. (29) where y C s defned n (13). Puttng s = t n (27) and (28), usng (29), and recallng that t 1 = t t 0 (y), we have, wth (14), U (y, t) =U (y C (t 0 (y)),t)=ρ e f t 0 (y). Dfferentatng by y and usng (17), (14) and (13), u (y, t) = ( ) 1 dyc dt (t 0(y)) ρ f e f t 0 (y) = ρ f e f t 0 (y) ρ f e, f t 0 (y) whch proves (15) for t>t 0 (y). Ths completes a proof of Theorem 1.

7 Remark. Observng (18) as a Burgers type system, absence of shock s essental n our proof for unque exstence of global classcal soluton and preservaton of regularty. A suffcent condton for the global absence of shock s (wth other assumed condtons such as statonary boundary condtons and non-negatvty of U s) U y (y 0) 1, (30) whch s satsfed by (19), or the ncompressblty condton (2). In fact, wthout (19), we have, n place of (25), y B (y 0 ; t) =y 0 + U (y 0 )(1 e ft ), whch s strctly ncreasng n y 0 for any t 0, f (30) holds. Hence ŷ(,t), the nverse functon of y B ( ; t) exsts, and consequently, U (y, t) s solved. Before closng ths secton, we gve a couple of examples of the soluton to the equaton of moton. Example 1 (One partcle type). For a pure flud, ρ 1 = 1, hence we have u 1 (y, t) 1. Snce there s only one type of ncompressble flud, the densty s constant. However, there s a flow drven by evaporaton even n ths case, and we actually have y C (t) =1 e f 1t. Example 2 (Two partcle types wth f 2 =0). Consder a mxture of 2 components wth the ratos satsfyng ρ 1 > 0 and ρ 2 =1 ρ 1 > 0. f 2 = 0 means no evaporaton, so the stuaton s a salty sea wth salt densty ρ 2, where evaporaton and flow from a rver balance. We have y C (t) =ρ 1 (1 e f1t ), t 0 (y) = 1 log(1 y ). f 1 ρ 1 The expressons of u (y, t) for y<y C (t) become smple: v(y, t) =f 1 (ρ 1 y), u 1 (y, t) 1, u 2 (y, t) 0. The pure water from the rver comes n up to y<y C (t). (Note that we are consderng a fcttous 1-dmensonal case where no spacal mxng such as turbulence occurs and we have no other dynamcs such as dffuson.) If, furthermore, the ntal dstrbuton s unform on [0, 1): u (y) =ρ, =1, 2, then the expressons for y>y C (t) are also smple: y C (y, t) =1 (1 y)(ρ 1 e f1t 1 y + ρ 2 ), ŷ(y, t) =1, ρ 1 e f 1t + ρ 2 and consequently, ρ 1 e f 1t v(y, t) =(1 y)f 1, u ρ 1 e f 1 (y, t) = ρ 1e f1t ρ 2, u 1t + ρ 2 ρ 1 e f 2 (y, t) =, 1t + ρ 2 ρ 1 e f 1t + ρ 2 for y>y C (t). (In general, the formulas are dependent on ntal data n complex ways, and we do not have explct formula.) 7

8 3 Infnte partcle lmt of stochastc rankng process. In ths secton, we frst recall the stochastc rankng process defned n [4], and relate ts nfnte partcle lmt proved n [4] to (15). Let (Ω, B, P) be a probablty space. A stochastc rankng process of N partcles, {X (N) (t) t 0, =1, 2,,N}, s defned as follows. We assume that the ntal confguraton X (N) (0) = x (N),0, =1, 2,,N, s gven, such that x(n) 1,0,x (N) 2,0,,x (N) N,0 s a permutaton of 1, 2,,N. For each let τ (N),, =0, 1, 2,, be an ncreasng sequence of random ump tmes, such that {τ (N), =0, 1, 2, }, =1, 2,,N, are ndependent (ndependence among partcles), τ (N),0 = 0 and {τ (N),+1 τ (N), =0, 1, 2, } are..d. wth the law of τ (N) = τ (N),1 beng P[ τ (N) t ]=1 e w(n) t,t 0, (31) where, for each, w (N) > 0 s a gven constant (ump rate). (Note that wth probablty 1, τ (N),, =0, 1, 2,, s strctly ncreasng, and that τ (N), τ (N), for any dfferent par of suffces (, ) (, ).) For each =1, 2,,N we defne the tme evoluton of X (N) by, X (N) (t) =x (N),0 + { {1, 2,,N} x (N),0 >x(n),0, τ(n),1 t}, 0 t<τ(n),1, where A denotes the number of elements n the set A, wth = 0, and for each =1, 2, 3, 8 X (N) X (N) (τ (N), )=1, and (t) =1+ { {1, 2,,N} Z + ; τ (N), <τ (N), t}, τ (N), <t<τ (N),+1. Proposton 2 ([4]) Assume that the emprcal dstrbuton of ump rates converges weakly to a probablty dstrbuton λ: λ (N) (dw) = 1 N δ(w w (N) ) dw λ(dw), N. Then N where lm N =1 1 (N) { τ t} = y C (t), n probablty, N y C (t) =1 [0, ) e wt λ(dw), t 0. (32) Proposton 2 (and, the man theorem n [4]) holds for any dstrbuton λ wth fnte mean (see [4] for detals). Here n ths secton, n order to consder the stuaton consstent wth that of Secton 1, we wll restrct ourself to the case where λ s a dscrete measure,.e., s concentrated on a fnte or a countable set of non-negatve numbers {f 1,f 2,f 3, }. Namely, we assume a form λ(dw) = ρ δ(w f )dw, (33) where δ(w f )dw s a unt measure concentrated on f, and ρ are non-negatve constants satsfyng (8) and (6). Then (32) concdes wth (13).

9 9 Let Y (N) (t) = 1 N (X(N) (t) 1), (34) and y (N),0 = 1 N (x(n),0 1) [0, 1) N 1 Z. We consder the N lmt of the emprcal dstrbuton on the product space of ump rate and spacal poston; μ (N) t (dw, dy) = 1 N δ(w w (N) )δ(y Y (N) (t)) dw dy. (35) Accordng to [4], we mpose that the ntal dstrbuton converges weakly to a probablty measure; μ (N) 0 (dw, dy) = 1 δ(w w (N) )δ(y y (N),0 N ) dw dy μ 0(dw, dy), N (weakly.) (36) To meet the notaton n Secton 1, we wrte μ 0 (dw, dy) =μ y,0 (dw) dy = u (y) dy δ(w f )dw, (37) where we assume that u (y) satsfes the assumptons n Secton 1. Note that (5) and (2) respectvely mples that the frst and the second margnal of μ 0 s λ n (33) and the Lebesgue measure, respectvely. Note also that (6) mples (wth (33)) wλ(dw) = f ρ <. (In [4] t has also been assumed that λ({0}) = 0 but the results hold n the present stuaton wth (6).) (37) mples 1 1 y 0 e wt μ 0 (dw, dy) =y C (y, t), where y C s as n (11), hence ts nverse functon ŷ(y, t) ny exsts and s gven by (12). Our notatons are now consstent wth those n [4], wth the correspondence (33) and (37). The man theorem n [4] then mples that the ont emprcal dstrbuton of partcle types and postons at tme t μ (N) t (dw, dy) = 1 N δ(w w (N) )δ(y Y (N) (t)) dw dy (38) converges as N to a dstrbuton μ y,t (dw) dy on R + [0, 1). The lmt dstrbuton s absolutely contnuous wth respect to the Lebesgue measure on [0, 1). The densty μ y,t (dw) wth regard to y s gven by we wt0(y) λ(dw), t > t 0 (y), we wt0(y) λ(dw) μ y,t (dw) = 0 e wt (39) μŷ(y,t),0 (dw), 0 t<t 0 (y). e wt μŷ(y,t),0 (dw) 0

10 10 To see the correspondence between (39) and (15) n Secton 2, let us rewrte (39) n terms of (33) and (37). For y<y C (t), we use (33) n (39) to fnd f e f t 0 (y) ρ δ(w f )dw we wt0(y) λ(dw) = we wt0(y) λ(dw) f e f t 0 (y) ρ 0 whch, accordng to (15), s equal to u (y, t)δ(w f )dw. Smlarly, for y>y C (t), we use (37) n (39) to fnd 0 e wt μŷ(y,t),0 (dw) e wt μŷ(y,t),0 (dw) = e ft u (ŷ(y, t)) δ(w f )dw e ft u (ŷ(y, t)) = u (y, t) δ(w f )dw. In both cases, we have the correspondence μ y,t (dw) = u (y, t) δ(w f )dw, whch relates the soluton (15) of the PDE n Secton 1 to the nfnte partcle lmt of the stochastc rankng process n [4]. 4 Possble applcaton to rankngs on the webs. 4.1 Pareto dstrbuton. The stochastc rankng process may be vewed as a mathematcal model of the tme evoluton of rankngs such as that of books on the onlne bookstores web (e.g., In ths example, N stands for the total number of books, represents a specfc ttle of a book, w (N) s the average rate wth whch the book s sold, x (N),0 s the ntal poston (rankng) of the book, τ (N), s the random tme at whch the book s sold for the -th tme, and X (N) (t) s the rankng of the book at tme t. A partcle (a book, n the case of onlne booksellers) n the rankng umps randomly to the rank 1 (each tme a copy of the book s sold), and ncreases the rankng number by 1 each tme some other partcle of larger rankng number umps to rank 1. Because an ncrease n rankng number s a result of umps of very large number of partcles n the tal sde of the rankng, the partcle effectvely moves on the rankng queue n a determnstc way, even though each ump occurs at a random tme. The ump corresponds to the evaporaton n the flud model. We can therefore predct the tme evolutons of rankngs appearng on the web rankngs based on our model. Here we try to see how a traectory of a partcle (13) could be observed n the web rankngs. In applyng (13) to the actual rankngs, we need to choose the dstrbuton of

11 11 evaporaton rates {(f,ρ ) =1, 2,,N}. In the case of socal or economc studes such as onlne booksellers, ths corresponds to choosng the dstrbuton of actvtes or transactons. In the case of onlne booksellers, we have to choose the dstrbuton of sales rates over books. The Pareto dstrbuton (also called log-lnear dstrbuton n socal studes, or power-law n physcs lteratures) s tradtonally used as a basc model dstrbuton for varous socal rankngs, perhaps a most well-known example s the rankng of ncomes. Let N be the total sze of populaton, and for =1, 2,,N, denote by f the ncome of the -th wealthest person. If ( ) 1/b N f = a,ρ = 1, =1, 2, 3,,N, (40) N holds for some postve constants a and b, then the dstrbuton of ncomes s sad to satsfy the Pareto dstrbuton. (The Pareto dstrbuton assumes all the consttuents to have dstnct f, whch leads to equal weght ρ =1/N n our notaton.) The factor a corresponds to the smallest ncome, and the exponent b reflects a socal equalty of ncomes: n fact the rato of the largest ncome to the smallest s f 1 /f N = N 1/b, whch s close to 1 f b s large (a far socety), whle s large (socety s n monopoly) f b s small. (Our b corresponds to α n a standard textbook on statstcs, θ n [3], and 1/β 2 n [2].) Substtutng (40) n (13) of Secton 2, and approxmatng the summaton by ntegraton, we have, after a change of varable, y C (t) =1 b(at) b Γ( b, at)+o(n 1 ), (41) where Γ(z, p) = p e w w z 1 dw s the ncomplete Gamma functon. rankng normalzed by N, so the tme evoluton of rankng x C (t) s y C (t) s a relatve x C (t) =1+Ny C (t). (42) The O(N 1 ) contrbuton n (41) s (by a careful calculaton) seen to be non-negatve and bounded by 1 N e at 1 N, leadng to a dfference of at most 1 n the rankng x C(t), whch s nsgnfcant for our applcatons below, so we wll gnore t. Note that Γ( b, at) as t 0, for b>0. Ths dvergence s harmless because t s cancelled by t b n (41), but for numercal and asymptotc analyss, t s better to perform a partal ntegraton on the rght-hand sde to fnd whch, wth (42), leads to y C (t) =1 e at +(at) b Γ(1 b, at), (43) x C (t) =N (1 e at +(at) b Γ(1 b, at)) + 1. (44) The constant a, whch denotes the lowest ncome n the Pareto dstrbuton, has a role of a tme constant n (44). In partcular, the short tme behavor of x C (t) for 0 <b<1s x C (t) =ct b + O(t), (45)

12 12 where c = Na b Γ(1 b). (46) For 1 <b<2 we need a partal ntegraton once more for a better expresson; y C (t) =1 e at (1 at b 1 ) (at)b Γ(2 b, at) = ab b 1 a t Γ(2 b) b 1 b b 1 tb + O(t 2 ). (47) Note that for 0 <b<1 the leadng short tme behavor s y C (t) =O(t b ), whch s tangental to the y axs at t = 0, whle for b>1 (the case b 2 can be handled smlarly) the lnear dependence y C (t) =O(t) s domnant for small t. 4.2 Collected web bulletn board thread ndex lstngs. 2ch.net s one of the largest collected web bulletn boards n Japan. Each category ( board ) has an ndex lstng of the ttles of threads or the web pages n the board. The ttles are ordered by the last wrtten thread at the top prncple; f one wrtes an artcle ( response ) to a thread, the ttle of that thread n the ndex lstng umps to the top nstantaneously, and the ttles of other threads whch were orgnally nearer to the top are pushed down by 1 n the lstng accordngly. We can extract the exact tme that a thread umped to rank 1, because the tme of each response n a thread s recorded together wth the response tself. All these features of the 2ch.net ndex lstng match the defnton of the stochastc rankng process, hence 2ch.net s a sutable place for a testng the applcablty of the stochastc rankng process. We note that we can use determnstc (non-stochastc) formula such as (44) f N, the total number of threads n the board s large. In the case of 2ch.net, N s roughly about 700 to 800, so we would expect fluctuaton of a few percent, and up to that accuracy, we expect a tme evoluton predcted by (44). We also note that the tme evolutons are ndependent of whch thread one s lookng at, because the changes n the rankng are caused by the collectve moton of the threads towards the tal; popular threads ump back to the top rankng more frequently than the less popular ones, but as long as the threads reman n the queue (.e., before the next ump), both a popular thread and an unpopular thread should behave n the same way, dependng only on ther poston n the rankng. We collected data of the tme evoluton of an ndex lstng n the 2ch.net, and performed statstcal fts of the data to (13). Fg. 1 s a plot of the threads n a board whch umped to the rank 1 durng actve hours one day and stayed n the queue wthout umps untl mdnght. There were 12 such threads. The rankng s obvously monotone functon of tme between umps, and there are no overtakng, so that the lnes do not cross n the fgure. Fg. 2 s a plot of same data as n Fg. 1, except that, for the horzontal axs the tme s so shfted for each thread that the rankng of the thread s 1 at tme 0. Though each thread starts at rank 1 on dfferent tme of the day, Fg. 2 shows that tme evolutons after rank 1 are on a common curve. N s the total number of threads, whch s N = 795 at the tme of observaton for Fg. 2, and a and b are postve constant parameters to be determned from the data. We performed a least square ft to (44) of n d = 117 data ponts shown n Fg. 2. The best ft for the parameter set (a, b) s(a,b )=( , ) ( χ 2 /n d 1.8).

13 13 rankng 50 12:00 24:00 tme Fg 1: Record of rankng changes n an afternoon for 12 threads n a board of 2ch.net. Ponts from a thread are oned by lne segments to gude the eye Fg 2: Collecton of 12 threads n a board of 2ch.net, same as n Fg. 1. For each thread, tme s shfted so that the rank of the thread s 1 at tme 0. The curve s x C (t) of (44) wth the best ft a = a and b = b to the data. Horzontal and vertcal axes are the hours and rankng, respectvely. In partcular, we see a rather clear behavor close to the orgn that the plotted ponts are on a curve tangental to y axs, ndcatng x C (t) =O(t b ) wth b<1 as n (45). Consderng the smplcty of our model and formula, the fts seem good, suggestng a possblty of new applcaton of flud dynamcs n the analyss of onlne rankngs. 4.3 Amazon.co.p book sales rankngs. We next turn to the rankng n the Amazon.co.p onlne book sales. In ths century of expandng onlne retal busness, the economc mpact of nternet retals has attracted much attenton, and there are studes usng the sales rankngs whch appear on the webs of onlne booksellers such as Amazon.com [2, 3]. We wll study Amazon.co.p, a Japanese counterpart of Amazon.com, whch seems to be not studed (and s easer to access for the authors). Amazon.com and Amazon.co.p are smlar n basc structures of web pages for ndvdual

14 14 books; on a web page for a book there are the ttle, prce and related nformaton such as shppng, bref descrpton of the book, the sales rankng of the book, customer revews and recommendatons. We should note that Amazon.co.p, as well as Amazon.com, does not dsclose exactly how t calculates rankngs of books. In fact, there are observatons [6] that Amazon.com defnes the rankngs for the top sales n a rather nvolved way. Therefore, t would be non-trval and nterestng f we could observe n the data behavors smlar to those of our smple model such as (44). See [5] for economc mplcatons of the rankng numbers n Amazon.co.p. Accordng to observaton, Amazon.co.p, as well as Amazon.com, updates ther rankngs once per hour, n contrast to the 2ch.net where the update procedure s nstantaneous. Ths mples a lmt of short tme observatonal precson of 1 hour. On the other hand, for the long tme observatons, we have to consder a fact that the total number of books N s not constant. It s sad that each year about books are publshed n Japan, or about 5.7 books per hour. Certanly not all of the books are regstered on Amazon.co.p, so the ncrease of N per hour must be less than ths value. Speed of rankng change decreases n the very tal sde of the lstng, and these practcal changes n N wll affect valdty of applyng (44) to the data n the very tal regme of the rankng. Ths gves a practcal lmt to long tme analyss. Fortunately, at rankng as far down as , we stll observe about 200 rankng change per hour, whch makes an ncrease by 5.7 books neglgble, so we expect a chance of applcablty of our theory for long tme data. We wll now summarze our results. The plotted 77 ponts n Fg. 3 show the result of our observaton of a Japanese book rankngs data, taken between the end of May, 2007 and md August, As seen n the fgure, the rankng number falls very rapdly near the top poston (about 200 thousands n 5 days). The sold curve s a least square ft of these ponts to (44). Amazon.co.p announces the total number of Japanese books n ther lst, whch s a few tmes 10 6, but we suspect that ths number ncludes a large number of books whch are regstered but never sell (so that we should dscard n applyng our theory). Therefore n addton to a and b n (44) we nclude N as a parameter to be ft from the data. Also, Amazon does not dsclose the exact pont of sales of each book, unlke 2ch.net where the exact ump tme of a thread s recorded, so that the ump tme to rank 1 of a book s also a parameter. The best ft for the parameter set (N, a, b) s: (N,a,b )=( , , ), ( χ 2 /n d ). Incdentally, we note n Fg. 3 a small ump at about 300 hours. We suspect ths as a result of nventory control such as unregsterng books out of prnt. Obvously, these controls need man-power, so that they appear only occasonally, makng t a knd of unknown tme dependent external source for our analyss. All n all, we thnk t an mpressve dscovery that a smple formula as (44) could explan the data for more than 2 months. Our way of extractng basc socologcal parameters such as the Pareto exponent b from the rankng data on the web has advantages over prevous methods such as n [3], n that because the tme development of the rankng of a book s a result of sales of as large number of books as O(N), the book moves on the rankng queue n a determnstc way though each sale s a stochastc process. The fluctuatons of sales (randomness about who buys what and when) are suppressed through a law-of-large-

15 Fg 3: A long tme sequence of data from Amazon.co.p. The sold curve s a theoretcal ft. Horzontal and vertcal axes are the hours and rankng, respectvely. numbers type mechansm [4]. By lookng at the tme development of the rankng of a sngle book, we are n fact lookng at the total sales of the books on the tal sde of the book. The theory of long-tal economy says [1] that each product mght sell only a lttle, but because of the overwhelmng abundance n the speces of the products the total sales wll be of economc sgnfcance: It s not any specfc sngle book but the total of books on the long-tal that matters. Our analyss on accumulated effect of products each wth random and small sales, s partcularly sutable n analyzng the new and rapdly expandng economc possblty of onlne retals, and moreover, s natural from the long-tal phlosophy pont of vew. Among the parameters to be ft n the Pareto dstrbuton there s an exponent b whch s of mportance n the studes of economy. For example, n the case of dstrbuton of ncomes, whch s usually quanttatvely analyzed by the Pareto dstrbuton, small b means that a few people of hgh ncomes hold most of the wealth (the so called law s a nckname for the Pareto dstrbuton wth b = 1), whle for large b the socety s more equal. In the case of rankng of onlne booksellers, large b means that there are many books (books n the long-tal [1] regme), each of whch does not sell much but the total sales of whch s sgnfcant, further mplyng strong mpacts of onlne retals to economy [3, 2], whle small b favors domnance of tradtonal busness model of greatest hts. Our studes on the 2ch.net bulletn board and the Amazon.co.p onlne bookseller both consstently gve the Pareto exponent b 0.6. Exstng studes on onlne booksellers [3, 2] adopt the value of b =1.2 and b = 1.148, respectvely. These references also quote values from other studes, most of whch satsfy b>1. Note that [6] dscovers, apparently based on extensve observatons, that n

16 16 the long-tal regme the sales are worse than n the head and ntermedate regme, and gves the Pareto exponent b =0.4 n the long-tal regme. Our method gves the total effect of ntermedate and long-tals, so our value b =0.6 could be more or less consstent wth the observaton of [6]. A Proof of equvalence of (10) and (4). Here we prove that, f sup f <, (10) and (4) are equvalent, under the equatons (1) (7) (2) (3) (5), wth postve constants f, ρ, =1, 2,. (The extra condton on boundedness of f s of course rrelevant for the fnte component cases.) Frst assume (10). Then wth (9) we have v(0,t)= f ρ. On the other hand, ntegratng (1) by y from 0 to 1 and usng (10) and (3) we have v(0,t) u (0,t)=f ρ. The two equatons mply (4). Next assume (4) and let Then (9) mples U =(U1,U 2, ); U (t) = v(0,t)= 1 0 u (z, t) dz, =1, 2,. (48) f U (t). (49) Integratng (1) by y from 0 to 1, and usng (3) (4) (49), we have wth d U dt (t) =(A U )(t), t 0, (50) (A U ) (t) = f ρ f U (t) f U (t), =1, 2, 3,, t 0. (51) f ρ The defnton (5) mples U (0) = ρ, hence U (t) =ρ, t 0, s a soluton, mplyng (10). To prove that ths s a unque soluton, let U be a soluton of (50) wth U (0) = ρ := (ρ 1,ρ 2, ). Then d ( U ρ) = A ( U ρ ), hence dt t U (t) ρ A U (s) ρ ds. Note that (51) mples A = sup f <, 0, f ρ f k f kρ k δ f, from whch we have, notng the assumpton t U (t) ρ 2 f U (s) ρ ds 0 t 2 sup f U (s) ρ ds. 0

17 17 Gronwall s nequalty mples U (t) ρ =0,t 0, hence U (t) =ρ, =1, 2,, t 0, s the unque soluton to (50). Ths proves (10). References [1] C. Anderson, The Long Tal: Why the Future of Busness Is Sellng Less of More, Hyperon Books, [2] E. Brynolfsson, Y. Hu, M. D. Smth, Consumer surplus n the dgtal economy: Estmatng the value of ncreased product varety at onlne booksellers, Management Scence (2003) [3] J. Chevaler, A. Goolsbee, Measurng prces and prce competton onlne: Amazon.com and BarnesandNoble.com, Quanttatve Marketng and Economcs, 1 (2) (2003) [4] K. Hattor, T. Hattor, Exstence of an nfnte partcle lmt of stochastc rankng processes, Stochastc Processes and ther Applcatons (2008), to appear. [5] K. Hattor, T. Hattor, Mathematcal analyss of long tal economy usng stochastc rankng processes, preprnt, [6] M. Rosenthal, M. Rosenthal, Prnt-On-Demand Book Publshng: A New Approach to Prntng and Marketng Books for Publshers and Self-Publshng Authors, Foner Books, 2004.