Research Article Exact Partition Function for the Random Walk of an Electrostatic Field

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1 Hndaw Advances n Mathematcal Physcs Volume 2017, Artcle ID , 5 pages Research Artcle Exact Partton Functon for the Random Walk of an Electrostatc Feld Gabrel González 1,2 1 Cátedras CONACYT, Unversdad Autónoma de San Lus Potosí, San Lus Potosí, SLP, Mexco 2 Coordnacón para la Innovacón y la Aplcacón de la Cenca y la Tecnología, Unversdad Autónoma de San Lus Potosí, San Lus Potosí, SLP, Mexco Correspondence should be addressed to Gabrel González; gabrel.gonzalez@uaslp.mx Receved 7 Aprl 2017; Revsed 6 June 2017; Accepted 14 June 2017; Publshed 13 July 2017 Academc Edtor: Jacopo Bellazzn Copyrght 2017 Gabrel González. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. The partton functon for the random walk of an electrostatc feld produced by several statc parallel nfnte charged planes n whch the charge dstrbuton could be ether ±σ s obtaned. We fnd the electrostatc energy of the system and show that t can be analyzed through generalzed Dyck paths. The relaton between the electrostatc feld and generalzed Dyck paths allows us to sum overall possble electrostatc feld confguratons and s used for obtanng the partton functon of the system. We llustrate our results wth one example. 1. Introducton In 1961 Lenard consdered the problem of a system of nfnte chargedplanesfreetomovenonedrectonwthoutany nhbton of free crossng over each other [1]. If all the charged planes carry a surface mass densty of magntude unty and the th one carres an electrc surface charge densty σ, the Hamltonan of Lenard s problem s gven by H (q 1,q 2,...;p 1,p 2,...) = 1 2 p 2 1 σ 2ε σ j 0 q q j, (1) <j where the system s overall neutral. Wth the Hamltonan functon at hand Lenard was able to obtan the exact statstcal mechancs of the system wth the technque of generatng functons [1, 2]. From a physcal pont of vew, Lenard s model can be used n the study of a one-dmensonal Coulomb gas [3]. The onedmensonalcoulombgassastatstcalmechancalproblem where partcles of equal or opposte charges nteract through the Coulomb potental [4]. The model has been extensvely studed n the past and forms one of the classcal exactly soluble problems n one dmenson [5]. Also, the Coulomb gas model has also been used to descrbe the man features of onc lquds [6, 7]. More recently, a very smlar problem was proposed n [8] wth the constrant that each of the charged planes has a fxed poston n space and that the surface charge dstrbuton n each plane could be ether ±σ. Ths modfed model gves rse to a random walk behavor of the electrostatc feld that can be analyzed as a Markovan stochastc process. The purpose of ths paper s to gve a statstcal descrpton of the electrostatc feld generated by several statc parallel nfnte charged planes n whch the surface charge dstrbutoncouldbeether±σ. The key step n our formulaton conssts of the summaton of all possble trajectores of the electrostatc feld for every dfferent charge confguraton. We do ths by showng that there s a one to one correspondence between every electrostatc feld trajectory and a generalzed Dyck path. The artcle s organzed as follows. In Secton 2 we descrbe the model and derve a system of equatons for obtanng the electrostatc energy of the system or Hamltonan.InSecton3wemakeaonetoonecorrespondence between electrostatc feld trajectores and generalzed Dyck paths. In Secton 4 the explct expresson for the partton functon s gven. In Secton 5 a smple numercal analyss s

2 2 Advances n Mathematcal Physcs We can wrte down the boundary condton gven n (3) n matrx form n the followng way: Fgure 1: Possble electrostatc feld confguratons nsde a fourcharged-plane system where the charge dstrbuton s not known. gven to show some features of the partton functon. In the last secton we summarze our conclusons. 2. Model of the System Suppose that we are gven a collecton of 2N nfnte charged planes, half of them have a constant charge dstrbuton σ, and the other half have a constant charge dstrbuton σ; that s, the system s overall neutral. If we randomly place the 2N nfnte charged planes parallel to each other along the zaxs at poston z n =ndwhere n {0,1,...,2N 1},then the electrostatc feld would evolve along the z-axs makng random jumps each tme t crosses an nfnte charged sheet. For example, consder a system of four charged planes. We know that E(z) equals zero at the left and rght sde of the confguraton due to the neutralty of the system [9]. After crossng the frst charged plane, E(z) would ncrease or decrease n an amount of σ/ε 0 dependng on whether the frst charged plane had a postve or negatve surface charge densty. In Fgure 1 we show all the possble electrostatc feld confguratons for the case when the charge densty s not known. Snceweareconcernwththestatstcalpropertesofthe electrostatcfeldwehavefrsttoobtanthehamltonanof the feld or the electrostatc energy of the system. We know from basc electrodynamcs courses that the electrostatc energy s gven by [9] E =δ E ΔV, (2) where δ E = ε 0 E 2 /2 s the electrostatc energy densty and ΔV = Ad sthevolumensdetheregonbetweentwo consecutve parallel charged sheets. Denotng E() as the value of the electrostatc feld n the th regon between two chargedsheetsplacednz = ( 1)d and z=d,thenwehave the followng boundary condton: E () E( 1) = σ ε 0 S, for =1,2,...,2N, (3) where S s a random varable whch takes one of the two possble values ±1 dependng on whether the charged plane at z = ( 1)d s postvely or negatvely charged. z E (0) E (1) S E (2) [. d d. = σ 1 S. ε 2, (4) 0 ] [. ] [. ] [ 0 1 1] [ E (2N)] [ ] S 2N wherewehavencludedthentalcondtonoftheelectrostatc feld; that s, E(0) = 0. Solvng(4)fortheelectrostatc feld we have E () = σ ε 0 j=1 S j, for =1,2...,2N. (5) Substtutng (5) nto (2) we obtan the electrostatc energy of the system whch s gven by E =( ε 0 2 Ad E 2 ())δ 2N =1 S,0, (6) where the Kronecker delta guarantees the neutralty of the system. 3. Electrostatc Feld and Dyck Paths In ths secton we present the relaton between the electrostatc feld and generalzed Dyck paths. Let us consder for the sake of smplcty that all the charged planes carry a surface charge densty of σ=±1/a.letq be the total amount of charge to the left of the th regon; then we can rewrte the electrostatc feld gven n (5) as E () = Q ε 0 A. (7) Note that Q 0 =Q 2N =0and Q Q 1 =1.Theelectrostatc energy n terms of the new varable s gven then by E = 1 2C Q 2, (8) where C=ε 0 A/d. It s convenent to nvestgate the nature of all the possble electrostatc feld confguratons by plottng E().Forthesake of smplcty we wll graph Q versus and connect all the dfferent values of the electrostatc feld n each regon by straght lnes. An example of a graph s shown n Fgure 2. Note that every possble electrostatc feld confguraton conssts of a closed path of length 2N that starts at (0, 0) and ends at (2N, 0) and s unquely determned by a sequence of vertces (Q 0,Q 1,...,Q 2N ),whereq j 1 s connected to Q j by an edge [10, 11]. In the mathematcal lterature ths s known as a closed smple walk of 2N steps on the lattce of ntegers, where the dscrete tme that numbers the steps s measured n the horzontal axs, and the postons of the walker on the lne

3 Advances n Mathematcal Physcs 3 Q that all the electrostatc feld confguratons are the ensembles of generalzed Dyck paths of length 2N that start at (0, 0) and end at (2N, 0). Let us denote by Γ the ensemble of generalzed Dyck paths of length 2N and consder γ Γ; then the partton functon of the electrostatc feld wll be gven by [14] Z = e βe(γ), γ Γ (9) Fgure 2: Generalzed Dyck paths representng a possble electrostatc feld confguraton of 14 parallel statc charged planes. arerecordednthevertcalaxs.snceourwalksstartfromthe orgn, at tme j the walker s on ste s(j) = Q 0 +Q Q j. A Dyck path s a smple closed walk wth the followng constrant s(j) 0 for all j [12]. The constrant on a Dyck path meansthattneverfallsbelowthehorzontalaxs,buttmay touch t at any number of nteger ponts. A more general Dyck path or generalzed Dyck paths are smple closed walks that are allowed to go below the horzontal axs [13]. We note then where β = 1/k B T, k B = J/K s Boltzmann constant, and T sthetemperature.equaton(9)sthesameas calculatng a Feynman ampltude n the frame of generalzed Dyck paths; that s, one sums overall generalzed Dyck paths of length 2N and to each whole path s then attrbuted a weght whch s gven by the Boltzmann factor. 4. Exact Partton Functon In order to calculate the sum of (9) we need to count overall possble generalzed Dyck paths of length 2N.A very powerful tool for countng walks s by means of the transfer matrx method [15, 16]. A transfer matrx method can be used to count all possble generalzed Dyck paths of length 2N.Let us ntroduce a set of (2N + 1) (2N + 1) matrces of the form W l whch represent the number of smple walks n l steps. Snce the paths we are consderng are made of steps ±1,then the transfer matrx for ths case s a bdagonal one gven by 0 b N, N a N+1, N 0 b N+1, N a W= N+2, N+1 0 d d d b N 1,N 1 0, (10) [.. 0 a N 1,N 2 0 b N 1,N ] [ a N,N 1 0 ] where a +1, =e β2 /2C, b,+1 =e β(+1)2 /2C for N N. (11) In ths pcture, the evaluaton of (W l ) j conssts n summng the weghts of all walks of l steps from to j. Therefore, the partton functon for the random walk of an electrostatc feld generated by means of 2N statc parallel randomly charged planes where half of them have a charge surface densty σ = 1/A and the other half has a charge surface densty σ= 1/As gven by Z =[W 2N ;N+1,N+1], (12) where [W 2N ;N+1,N+1]represents the element n row N+1 and column N+1of the transfer matrx W rased to the 2N power. For example, consder the case of 4 parallel statc randomly charged planes wth surface charge dstrbuton σ= ±1/A; forthscasen=2and we can then wrte down the transfer matrx for ths partcular problem whch s 0 e β/2c e 4β/2C W= 0 e β/2c 0 e β/2c 0 ; (13) [ e 4β/2C ] [ e β/2c 0 ]

4 4 Advances n Mathematcal Physcs therefore, the partton functon wll be gven by Z =[W 4 ;3,3]=2e 3β/C +4e β/c. (14) Equaton (14) s gven n the form Z = k g(e k )e βe k,where g(e k ) represents the number of paths that share the same energy E k ;thsmeansthatforthecasewhent all the possble electrostatc feld confguratons wll have the same probablty dstrbuton; that s, lm P(E e βek k)= lm T T Z = (N!)2 (2N!), (15) and ths was the case studed n [8]. We can calculate the expectaton value for the energy of the system when T 0;thats, lm T 0 ln Z E = lm ( T 0 β )= N 2C. (16) Equaton (16) means that, for very low temperatures, only the pathswthmnmumelectrostatcenergywllcontrbuteto the energy of the system. For ths case, the system behaves lke N parallel plate capactors connected n seres. 5. Numercal Analyss We can perform a drect evaluaton of the partton functon gven n (12) for a gven number of charged sheets usng the numercal code gven n the appendx. By usng the numercal code for N = 1, 2,..., we notced that the mnmum energy of the system s gven by E mn =N/2Cand the maxmum energy of the system s gven by E max = N(1 + 2N 2 )/6C.We have also found the entropc factor for the frst two mnmum energes and the maxmum energy, whch nvtes us to wrte the partton functon as Z =2 N e Nβ/2C +2 N 1 (N 1) e (N+4)β/2C + +2e N(1+2N2 )β/6c. (17) From (17) we see that the paths that contrbute the most to theparttonfunctonaregvenbythefrsttwoterms,andwe also see that the second term domnates over the frst term f ( N 1 )e 2β/C >1. (18) 2 Equaton (18) tells us that when N>1+2e 2β/C the paths that contrbutethemosttotheparttonenergybelongtothefrst excted state. 6. Conclusons We have shown that t s possble to obtan the exact partton functon for the electrostatc feld generated by means of several statc parallel nfnte charged planes n whch the surfacechargedstrbutonsnotexplctlyknown.wehave worked out the specal case where the charged planes have a constant surface charge dstrbuton gven by ±1/A and the overall electrostatc system s neutral; that s, there s the same number of postve and negatve charged planes. We use the connecton between generalzed Dyck paths and possble electrostatc feld confguratons to obtan the partton functon of the system as a dscrete Feynman path ntegral. A transfer matrx approach was used to count overall possble generalzed Dyck paths. A smple numercal code s gven to verfy our results. Appendx In ths appendx we present a smple Mathematca code whch evaluates the partton functon gven n (12). The only varable that needs to be specfed s the number of postve or negatve charged sheets N. W=Table [KroneckerDelta [ + 1, j] Exp [ β (+1)2 ] 2 + KroneckerDelta [, j + 1] Exp [ β (j)2 ],{, N, N}, {j, N, N}] ; 2 W 2N = MatrxPower [W, 2N] ; Z=Expand [W 2N [[N+1,N+1]]]. Conflcts of Interest The author declares that they have no conflcts of nterest. Acknowledgments (A.1) Ths work was supported by the program Cátedras CONA- CYT. References [1] A. Lenard, Exact statstcal mechancs of a one-dmensonal system wth Coulomb forces, Mathematcal Physcs, vol. 2, pp , [2] S.F.EdwardsandA.Lenard, Exactstatstcalmechancsofa one-dmensonal system wth Coulomb forces. II. The method of functonal ntegraton, Mathematcal Physcs,vol. 3,pp ,1962. [3] R. J. Baxter, Statstcal mechancs of a one-dmensonal Coulomb system wth a unform charge background, Mathematcal Proceedngs of the Cambrdge Phlosophcal Socety,vol. 59,no.4,pp ,1963. [4] S. Prager, The one-dmensonal plasma, Advances n Chemcal Physcs,2007.

5 Advances n Mathematcal Physcs 5 [5] E.H.Leb,D.C.Matts,andF.J.Dyson,Fnte Mathematcal Physcs n One Dmenson: Exactly Soluble Models of Interactng Partcles, Academc Press, New York, NY, USA, [6] V. Démery, R. Monsarrat, D. S. Dean, and R. Podgornk, Phase dagram of a bulk 1d lattce Coulomb gas, EPL, vol. 113, no. 1, Artcle ID 18008, [7] V. Démery,D.S.Dean,T.C.Hammant,R.R.Horgan,andR. Podgornk, The one dmensonal Coulomb lattce flud capactor, The Chemcal Physcs, vol. 137, no. 6, p , [8] R. Aldana, J. V. Alcalá, and G. González, The random walk of an electrostatc feld usng parallel nfnte charged planes, Revsta Mexcana de Físca, vol. 61, no. 3, pp , [9] D. J. Grffths, Introducton to Electrodynamcs, Prentce-Hall, Upper Saddle Rver, NJ, USA, [10] G. Mohanty, Lattce Path Countng and Applcatons, Academc Press, New York, NY, USA, [11] E. Deutsch, Dyck path enumeraton, Dscrete Mathematcs, vol.204,no.1-3,pp ,1999. [12] R. Chapman, Moments of Dyck paths, Dscrete Mathematcs, vol.204,no.1-3,pp ,1999. [13]J.LabelleandY.N.Yeh, GeneralzedDyckpaths, Dscrete Mathematcs,vol.82,no.1,pp.1 6,1990. [14] G. M. Ccuta, M. Contedn, and L. Molnar, Enumeraton of smple random walks and trdagonal matrces, Physcs. A. Mathematcal and General, vol.35,no.5,pp , [15] L. Shapro, Random walks wth absorbng ponts, Dscrete Appled Mathematcs. The Combnatoral Algorthms, Informatcs and Computatonal Scences,vol.39,no.1,pp.57 67, [16] T. Jonsson and J. F. Wheater, Area dstrbuton for drected random walks, Statstcal Physcs,vol.92,no.3-4,pp , 1998.

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