Joural of Applied Mathematics ad Physics, 26, 4, 6-66 Published Olie August 26 i SciRes. http://www.scirp.org/joural/jamp http://dx.doi.org/.426/jamp.26.487 Iformatio Theory Model for Radiatio Philipp Korreich 9 Wie, Austria, Kig of Prussia 946, USA Received 2 July 26; accepted 22 August 26; published 25 August 26 Copyright 26 by author ad Scietific Research Publishig Ic. This work is licesed uder the Creative Commos Attributio Iteratioal Licese (CC BY). http://creativecommos.org/liceses/by/4./ Abstract Iformatio based models for radiatio emitted by a Black Body which passes through a scatterig medium are aalyzed. I the limit, whe there is o scatterig this model reverts to the Black Body Radiatio Law. The advatage of this mathematical model is that it icludes the effect of the scatterig of the radiatio betwee source ad detector. I the case whe the exact form of the scatterig mechaism is ot kow a model usig a sigle scatterig parameter is derived. A simple versio of this model is derived which is useful for aalyzig large data. Keywords Iformatio Theory, Maximum Iformatio, Etropy, Scatterig, Coditioal Etropy. Itroctio A Iformatio Theory Radiatio Model (ITRM) for radiatio emitted by a black body passig through a scatterig medium is aalyzed. A variatioal method is used to derive three equatios similar to the Black Body Radiatio (BBR) law. But, these equatios iclude the effect of scatterig. I the limit whe there is o scatterig these models revert to the BBR law. The advatage of this mathematical model is that it icludes the effect of the scatterig of the radiatio betwee source ad detector. Three equatios are derived. Oe equatio describes the effect of the scatterig o the radiatio whe the form of the scatterig mechaism is kow. The secod result is a equatio for the case whe the forms of the scatterig mechaisms are ot kow. I this case the model cotais a sigle scatterig parameter. The third equatio is a simplified versio of the case whe the form of the scatterig is ot kow. This is useful for the aalysis of large data. The derivatio of the case whe the form of the scatterig mechaism is ot kow is similar to oe I preseted i a previous publicatio []. It was formulated for the aalysis of the Cosmic Backgroud Radiatio. A variatioal method usig Iformatio Theory was used to derive this model. To obtai the three equatios the Iformatio was maximized, subject to what is kow about the system. The basis of Iformatio Theory was developed by C. E. Shao [2] at Bell Laboratories. The iformatio How to cite this paper: Korreich, P. (26) Iformatio Theory Model for Radiatio. Joural of Applied Mathematics ad Physics, 4, 6-66. http://dx.doi.org/.426/jamp.26.487
P. Korreich theoretical model facilitates the iclusio of the effect of iformatio propagatig through oisy chaels [2]. Here the effect of the thermal radiatio beig scattered o the way from the hot Black Body source to the detector is aalyzed. Oe pheomeo to which oe ca apply this model is to the statistics of light quata. I the Quatum Mechaical model of ature the eergy of the electromagetic radiatio oscillatig with ay give frequecy is divided ito eergy quata or photos. The differet quatum mechaical eergy states are eergy packets cotaiig differet umbers of photos. The iformatio trasmitted at ay frequecy by the hot body is ecoded i the umber of photos radiated. Differet umbers of photos radiated represet differet iformatio, see Figure. Thus, a eergy state represets a amout of iformatio. The amout of iformatio i each photo packet or eergy state is ot equal to the umber of photos, but to a fuctio of the umber of photos. This is illustrated i Figure. The photo packets are represeted by mail bags i Figure. The iformatio i each mail bag is displayed o the tags. The uit of iformatio used i this schematic represetatio is i biary bits. However, the iformatio used i this article is i a uit of Joules per degree Kelvi. The photos travel through space where some are absorbed or scattered. Maybe, the radiatio whe passig through a ioized cloud is eve amplified by stimulated emissio. This ca occur by geeratig additioal photos of the same frequecy as the icomig radiatio eutralizes some of the charges. The model derived here ca be used both whe the details of the scatterig mechaism is kow ad i a simple case whe the details of the scatterig mechaism is ukow. I the case whe the scatterig mechaism is kow, the scatterig mechaism is represeted here by a stochastic model. It is represeted by coditioal probabilities that oe umber of photos were radiated provided aother umber of photos were received. The coditioal probabilities have to be costructed to represet the kow scatterig model. I the case whe the detail forms of the scatterig mechaisms are ot kow, the forms of the coditioal probabilities ca ot be specified. I this case, the iformatio formed by the coditioal probabilities ca be approximated by a fuctio that icludes a sigle average scatterig parameter. Like the BBR, the ITRM depeds o the absolute temperature. I the case whe the radiatio data is kow, the source temperature ad the average scatterig parameter ca be determied by comparig the data to calculated ITRM values. This is the case, for example, for the Cosmic Microwave Backgroud radiatio. Oe result of icludig the effect of scatterig is a blue shift of the distributio of photos, see Figure 2. Ulike i the case of the Doppler effect the wavelegth of the idivial photos do ot chage. However, the distributio chages to more short wave photos. The total umber of photos ca also chage to fewer or more photos. Figure. Schematic represetatio of the ecodig of the iformatio. The iformatio is ecoded i the umber of photos trasmitted. The amout of iformatio i each photo packet is a fuctio of the umber of photos i the packet. The photo packets are represeted by mail bags here. The iformatio i each mail bag is displayed o its tag. 6
P. Korreich Figure 2. Schematic represetatio of a blue shifted photo distributio. Ulike i the case of the Doppler effect, the wavelegth of the idivial photos do ot chage. However, the distributio chages to more short wave photos. 2. Iformatio Model The ITRM for thermal radiatio through a scatterig medium is derived below. Some of the derivatio is similar to the derivatio i my Cosmic Backgroud Radiatio aalysis paper []. The cocepts used here ca be foud i a paper by C. E. Shao [2] ad may Probability texts []-[5]. The iformatio is ecoded i the umber of photos radiated by the hot object, see Figure. The value of the iformatio g i each iformatio packet is a fuctio of the umber of photos. g = kl P (2.) where k is Boltzma costat ad P is the probability that a sigal of photos is beig received. This is similar to the famous Boltzma equatio egraved o his tombstoe. The equatio o the tombstoe is for uiform probabilities P =. The iformatio here is i a uit of Joules per degree Kelvi. Note that the probabilities N P are less or equal to oe. Therefore, the logarithm of the probability is egative ad the iformatio g is positive. The iformatio packets are show schematically as mail bags i Figure. The probabilities P are ormalized. = P = (2.2) The average detected Shao iformatio is equal to the average value H of all the iformatio packets. ) ) a H = gp b H = k Pl P (2.) = = The propagatio of the photos from the hot Black Body source to the detector is modeled by coditioal probabilities P (m photos radiated provided Photos received) that m photos are radiated provided photos were received. Associated with the coditioal probabilities havig the same coditio of photos beig received are coditioal etropies h(s ). Here S is the set of all the differet umbers m of radiated photos ad is the umber of photos that were received [5]. ( S ) k P( m ) l P( m ) h = (2.4) Here k is Boltzma costat ad is the umber of photos that were received. The average value N of the coditioal iformatio is also kow as the oise: = ( ) N = h S P (2.5) 62
P. Korreich The iformatio I is equal to the differece betwee the received Shao iformatio H of Equatio (2.) ad the oise N of Equatio (2.5) I = H N (2.6) The temperature T is the chage of the light eergy U with the iformatio H carried by the photos. For this derivatio the temperature T of the radiatio at the receiver is assumed to be kow. T = U H where the average eergy U of the received photos is give by: = (2.7) ωp U = (2.8) Its value, at this poit, is ot kow. Here is Plak s costat divided by 2π ad ω is the oscillatig frequecy of the radiatio. A variatioal method is used to calculate the values of the Probabilities P. The probabilities P ca be derived by fidig a extremum value of the iformatio I subject to what is kow about the system. I this case the temperature T at the receiver ad the fact that the probabilities are ormalized are kow about the system. However, the Equatio (2.7) for the temperature, is ot i the form of a costrait equatio like Equatios (2.2) ad (2.8). Therefore, it ca ot be used i this process directly. Istead oe has to use the average eergy U first. By multiplyig the two costrait equatios, Equatios (2.2) ad (2.8) by coveiet costats αk ad βk ad addig them to the equatio for the iformatio I oe obtais: ( S ) I = k P l P + P + αp + β ωp + αk + βku (2.9) = k The iformatio I will have a extremum value whe all its derivatives with respect to the probabilities P are equal to zero. By takig the derivative of the iformatio I with respect to oe of the probabilities P, settig the result equal to zero ad solvig for the probability P oe obtais: ( S ) P = exp( α) β ω (2.) The values of the costats α ad β are ot kow at this poit. I order to evaluate the costat α oe substitutes the probability of Equatio (2.) ito the first costrait equatio, which is Equatio (2.2). Oe obtais for exp ( α ) : exp ( α ) = (2.) β ωm The costat α ca be elimiated by substitutig Equatio (2.) ito Equatio (2.). ( S ) β ω k P = (2.2) β ωm The costat β has yet to be evaluated. I order to accomplish this, oe must first calculate the iformatio kl ( P ) of receivig photos by takig the logarithm of the probability P of Equatio (2.2) ad multiplyig the result by mius the Boltzma costat. kl ( P ) = h( S ) + βk ω+ kl β ωm (2.) By substitutig the iformatio associated with receivig photos, Equatio (2.), ito the average Shao iformatio of Equatio (2.) oe obtais: 6
P. Korreich H = h P + βku + k l β ωm (2.4) = where Equatio (2.8) was used for the average eergy U. By solvig Equatio (2.4) for the average eergy U, substitutig the resultig expressio ito Equatio (2.7) ad solvig for β oe obtais the well kow expressio: β = (2.5) The probability P of receivig photos ca ow be completely specified by substitutig Equatio (2.5) for the costat β ito Equatio (2.2). P = ( S ) ω ωm (2.6) The average eergy of a oe dimesioal radiatig system where the radiatio passes through a scatterig medium is derived by substitutig the probabilities of Equatio (2.6) ito the equatio for the average received eergy U, Equatio (2.8). U = where the ormalized frequecy x is give by: ω = ( S ) x x xm (2.7) ω = (2.8) Fially, by multiplyig Equatio (2.7) by a appropriate desity of states costat oe obtais for the chage of the average received eergy desity u with radiatio frequecy ω of a three dimesioal system radiatig through a scatterig medium: ( S ) x exp x 4π = 2 = d ω c h ( S m) xm (2.9) where h(s ) is give by Equatio (2.4) ad where c is the speed of light i free space. This is the first result. It is the Black Body Radiatio law for systems radiatig through a scatterig medium. It ca be used whe the form of the coditioal probabilities that describe the scatterig mechaism are kow. Note that the temperature T is the observed temperature at the receiver. For the case whe there is o scatterig, whe h(s ) is equal to zero, Equatio (2.9) reverts to the stadard Black Body Radiatio law. a ) x exp[ x] = b 2 ) 2 4π 4π x = = c c exp ( x) exp [ xm] (2.2) For the case whe the details of the scatterig models are ot kow the coditioal probabilities ca ot be specified. However, oe ca postulate a simple model for the coditioal iformatio or coditioal etropies [] [4] h(s ). 64
P. Korreich ( ) kρ l for h ( S ) (2.2) for = where ρ is a average scatterig parameter. By substitutig Equatio (2.2) ito Equatio (2.9) oe obtais for the chage of the average received eergy desity u with radiatio frequecy ω for the case whe the scatterig mechaisms are ot kow: ad where ( ) ( ) x exp ρ l x 4π = = 2 c exp ρ l ( m) xm+ (2.22) lim l =. Sice the scatterig parameter ρ is a Etropy Amplitude it must always be positive. This is the secod result. This result was also used i my Cosmic Backgroud Radiatio aalysis paper []. It is applicable whe the form of the scatterig mechaism is ot kow. It describes the scatterig process i terms of a sigle scatterig parameter ρ. This shifts the peak of the distributio to larger values of the ormalized ω frequecy x = see Figure 2. Equatio (2.22) ca be expressed as follows for large values of where ω : ρ x y 2 y 2 ρ + + y + ( ) 4π 2 2 c y y + y + + + ρ ρ 2 ω y = exp (2.2) (2.24) Equatio 2.2 is the third result. This equatio is especially useful whe large data is to be aalyzed. For completeess, the iput iformatio I is calculated by subtractig the oise N from the Shao iformatio H of Equatio (2.). The oise N is calculated i Equatio (2.5). Equatio (2.6) is used for the probabilities P. ( U S m) ( ) h ( ) I = h S P + + kl exp xm h S P (2.25) = T = Note that the first ad last terms of Equatio (2.25) cacel. By makig use of Equatio (2.7) for the average eergy U oe obtais: ( S ) ω x T m k = I = + k l exp xm k exp xm By multiplyig Equatio (2.26) by the same desity of states as was used i Equatio (2.9) oe obtais: ( S ) 2 2 x x m 4π k ω = 2 I = x l exp 2 + m ( m ) xm c c 4π exp xm (2.26) (2.27) Equatio (2.27) is i Joules m Hertz K. The explaatio give here of the shift of the photo distributio to higher frequecies is e to the effect of the scatterig process as discussed above. 65
P. Korreich. Coclusio A Iformatio Theory Radiatio Model (ITRM) for radiatio passig through a scatterig medium radiated by a black body has bee derived. The result of this aalysis is give by Equatios (2.9), (2.22) ad (2.2). Equatio (2.9) is the ITRM for the case whe the coditioal probabilities that describe the scatterig mechaisms are kow. Equatio 2.22 is a approximatio of the ITRM for the case whe the form of the scatterig mechaisms is ot kow. I this case a sigle average scatterig parameter is used to characterize the scatterig process. Equatio (2.2) is a approximatio of Equatio (2.22). It is a equatio for the case whe the form of the scatterig mechaisms are ot kow. It has a simpler form tha Equatio (2.22), but it is oly valid for large ω values of. This equatio is especially useful whe large sets of data are aalyzed. I the limit of o scatter ig the ITRM reverts to the Black Body Radiatio law. Refereces [] Korreich, P. (22) Iformatio Model of Cosmic Backgroud Radiatio. Iteratioal Joural of Astroomy,, 4-2. [2] Shao, C.E. (948) A Mathematical Theory of Commuicatio. The Bell System Techical Joural, 27, 62-656. [] Korreich, P. (28) Mathematical Models of Iformatio ad Stochastic Systems. CRC Press, Taylor ad Fracis Group LLC, Boca Rato, Chapter 8. [4] Papoulis, A. (965) Probability Radom Variables, ad Stochastic Processes. McGraw-Hill Book Compay, New York, Lodo. [5] Rode, M.S. (996) Aalog ad Digital Commuicatio Systems. Pretice-Hall, Upper Saddle River. Submit or recommed ext mauscript to SCIRP ad we will provide best service for you: Acceptig pre-submissio iquiries through Email, Facebook, LikedI, Twitter, etc. A wide selectio of jourals (iclusive of 9 subjects, more tha 2 jourals) Providig 24-hour high-quality service User-friedly olie submissio system Fair ad swift peer-review system Efficiet typesettig ad proofreadig procere Display of the result of dowloads ad visits, as well as the umber of cited articles Maximum dissemiatio of your research work Submit your mauscript at: http://papersubmissio.scirp.org/ 66