Volume 2010, Article ID 504267,12pages doi:10.1155/2010/504267
Research Article
On the Exact Solution of a Generalized Polya Process
Hidetoshi Konno
Department of Risk Engineering, Faculty of Systems and Information Engineering, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan
Correspondence should be addressed to Hidetoshi Konno,[email protected] Received 1 August 2010; Accepted 10 October 2010
Academic Editor: Pierluigi Contucci
Copyrightq2010 Hidetoshi Konno. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
There are two types of master equations in describing nonequilibrium phenomena with memory effect:ithe memory function type andiithe nonstationary type. A generalized Polya process is studied within the framework of a non-stationary type master equation approach. For a transition- rate with an arbitrary time-dependent relaxation function, the exact solution of a generalized Polya process is obtained. The characteristic features of temporal variation of the solution are displayed for some typical time-dependent relaxation functions reflecting memory in the systems.
1. Introduction
The generalized master equation of memory function type1is a useful basis for analyzing non-equilibrium phenomena in open systems as
d
dtPn, t t
0
dτ
j
Knjt−τP j, τ
−Kjnt−τPn, τ
, 1.1
where the kernelKnjtis conventionally assumed to have the product of a memory function φtwith a transition ratewn, jasKnjt φtwn, j. The transition ratewn, jhas the constraint with
jwj, n 1. This generalized master equation approach corresponds to the generalized Langevin equation of the memory function type2,3. One can see many successful applications with long memory along the line of traditional formulation1.
Looking around recent studies in complex open systems, there is an alternative approach based on a generalized non-stationary master equation4as
d
dtPn, t
j
LnjtP j, t
−LjntPn, t
. 1.2
The master equation in this form corresponds to the generalized Langevin equation of the convolutionless type, which is derived with the aid of projection operator method by Tokuyama and Mori 5. The time-dependent coefficient Ljkt may be written in the following form:Lnjt φctwn, j. It is expected from the projection operator method5 that the time-dependent functionφctreflects the memory effect from varying environment in a different way associated with the memory functioncf. also H¨anggi and Talkner 6.
The memory functionMFformalism has been utilized in anomalous diffusion like L´evy type diffusion in atmospheric pollution, diffusion impurities in amorphous materials, and so on. The alternative convolution-less, non-stationary NS formalism gives only a small number of applications. The paper intends to exhibit a potential ability of the NS formalism by taking an arbitrary time-dependent functionφctwhich is representing memory effect.
The paper is organized as follows. Section 2 reviews the non-stationary Poisson process. Section 3shows a generalized Polya process wherein it is involved a generalized non-stationary transition rateλn, t κtαn βwith an arbitrary functionκtof time.
The exact solution and the expression mean and variance are displayed as a function ofκt.
Some important remarks are given for a generalized non-stationary Yule-Furrey process with λn, t κtαn.Section 4discussesithe solvability condition of the generalized Polya model andiithe relation to the memory function approach. The last section is devoted to concluding remarks.
2. Nonstationary Poisson Process
The simplest example of the generalized master equation in the form of 1.2 is a non- stationary Poisson processan inhomogeneous Poisson processdescribed by
d
dtpn, t λtpn−1, t−λtpn, t n≥1, d
dtp0, t −λtp0, t n0,
2.1
whereλtis the time-dependent rate of occurrence of an eventλt≡φctwn−1, n. The functionλtis an arbitrary function of time. The solution is readily obtained, with the aid of the generating function, in the following form:
pn, t Λtn
n! exp−Λt, 2.2
Table 1: Some typical examples ofλtandΛt.
λt ntσn2t Λt constraint i λ0exp−γ0t λ0
γ01−exp−γ0t γ0>0
ii λ0
1 γ0t
λ0
γ0ln1 γ0t γ0>0
iii λ0tγ0−1 λ0
γ0
tγ0 0< γ0<1
0 5 10 15
Numbern 0
0.2 0.4 0.6
(i)Exponential
0 5 10 15
Numbern
(ii)Inverse power 0.10
0.2 0.30.4 0.5
0 5 10 15
Numbern
(iii)Fractional power 0.10
0.20.3 0.4 0.5
Figure 1: Time dependence of pn, t in 2.2 for three relaxation functions of λt in Table 1: i an exponential function,iian inverse power function, andiiia fractional power function; the values of parameters areλ01, γ00.85. The pdf profiles are depicted fort1solid line,t3dotted line,t5 dashed line,t7dash-dotted line, andt9dash-dotted line.
where Λt 0tdτλτ. It is easy to show that the mean and the variance take the same valuentσ2nt Λt. Namely, the Fano factorFσn2t/nttakes 1 for any time- dependent functionλt. The process gives rise to only the PoissonianPstatistics at any time.
Three typical examples of λt are shown in Table 1. All of them are relaxation functionsλt → 0 as the timetgoes to infinity. It is shown in the same table thatΛt ≡
t
0λsdsis the increasing function as the time goes to infinity. The temporal development of the probability densitypn, tin2.2for these three examples is depicted inFigure 1.
In seismology,λt λ0/1 αt Ohmori formula 7is frequently used in analyzing and predicting aftershocks. Many applications are also found in environmental, insurance, and financial problems8. Further, various engineering problems involve many potential applications especially in the probabilistic risk analysis9. However, the applicability of the non-stationary Poisson process is quite limited sincentσ2nt.
3. Generalized Polya Process
3.1. Model EquationNow let us consider a generalized Polya process within the class of generalized birth processes
d
dtpn, t λn−1, tpn−1, t−λn, tpn, t n≥1, 3.1 d
dtp0, t −λ0, tp0, t n0, 3.2
whereλn, ttakes into account then-dependence up to the first order and a memory effect with an arbitrary relaxation functionκtas
λn, t κt αn β
. 3.3
Whenκt 1/1 αtandβ 1, the model reduces to a Polya process10. Whenκt κ0/1 γ0t, the model reduces to an extended Polya process11.
3.2. Exact Solution
The method of characteristic curves is used to get the exact solution under the initial condition pn,0 δn,n0cf., the recursion method with variable transformations11. The generating function is defined bygz, t ∞
n0znpn, t. The equation forgz, tcorresponding to3.1 becomes
d
dtgz, t κtz−1
αz ∂
∂zgz, t βgz, t
. 3.4
From the initial condition, one obtainsgz,0 zn0. To eliminate the second term in the right hand side of3.4, let us assume that
gz, t CzGz, t, 3.5
whenαzd/dzCz βCz 0, one obtainsCz C0z−β/α. Without the loss of generality, C01. So the equation forGz, tbecomes
∂
∂tGz, t κtαzz−1 ∂
∂zGz, t. 3.6
Then, a variable transformation,
ξ 1 α
1
zz−1dz 1 αln
z−1 z
, 3.7
leads3.6to the simple wave equation,
∂
∂tGξ, t κt∂
∂ξGξ, t. 3.8
The solution of the wave equation in3.8is given by
Gξ, t fξ Kt, 3.9
whereKt 0tκτdτ. From the initial conditionpn,0 δn,n0, one obtains
gz,0 z−β/αf 1
αlnz−1 z
zn0. 3.10
Therefore,fxis expressed as
fx
1 1−expαx
β/α n0
. 3.11
Thus, we have
gz, t z−β/αGz, t, 3.12
where
Gz, t
zexp−Λt
1−z
1−exp−Λt β/α n0
, 3.13
and Λt αKt. When n0 0, the exact analytic expression of the probability density functionpn, tis given by
pn, t
exp−Λtβ/α⎛
⎝−β α n
⎞
⎠−1n
1−exp−Λtn
3.14
exp−Λtβ/α⎛
⎝n β α−1 n
⎞
⎠1−exp−Λtn
. 3.15
Table 2: Some examples ofκt, and correspondingΛtand expΛt.
κt Λt F≡expΛt constraint
i κexp−γt ακ
γ 1−exp−γt exp
ακ
γ 1−exp−γt
γ >0
ii κ 1 γt
ακ
γ ln1 γt 1 γtακ/γ γ >0
iii κtγ−1 ακ
γ tγ exp
ακ
γ tγ
0< γ <1
3.3. Mean and Variance
The probability density function in 3.15 is the Pascal distribution fθ the negative binomial distribution
fθ
n r−1 n
θr1−θn, 3.16
with the parametersr, θwithrβ/αandθexp−Λt. Thus, the mean and the variance are obtained in the following form:
nt β α
expΛt−1
, Vt≡σn2t β
αexpΛt
expΛt−1
. 3.17
The variance is generally greater than the mean, that is, the Fano factor is larger than 1 as follows:
Ft Vt
nt expΛt>1. 3.18
It is shown that the generalized Polya process withα /0 andβ /0 is subjected to the super- PoissonianSUPPstatisticsF >1.
Three examples of the relaxation functionκtare given inTable 2. They are decreasing function i.e.,κt → 0as the time goes infinity. In the case of an exponential relaxation i, the Fano factor Ft becomes a double exponential function as shown in the table. In the case of inverse power functionii, the Fano factor takes the form tin the time region t → ∞:a subdiffusion for 2ακ/γ < 1 andb superdiffusion for 2ακ/γ > 1.
On the other hand, in the case of the power relaxationiii, the Fano factorFt becomes the fractional power exponential function of time. To understand the feature of temporal variation, numerical examples are depicted in Figures2aand2bas well as Figures3a and3b.
0 20 40 60 80 100 Timet
0.001 0.01 0.1 1
κ(t)
a κt
0 20 40 60 80 100
Timet 0.1
1 10 100
K(t)
bKt 0tκsds
Figure 2: Time dependence of three relaxation functions forκtinTable 2:ian exponential function solid line,iian inverse power functiondotted line, andiiia fractional power function dashed line; the values of parameters areκ1,γ0.85,α1, andβ1.
3.4. Nonstationary Yule-Furrey Process
Whenβ 0,λ0, t 0 in3.3. So one must omit3.2 i.e., one must redefine the range of variationfor the case of a generalized non-stationary Yule-Furrey process as follows:
d
dtpn, t λn−1, tpn−1, t−λn, tpn, t n≥1. 3.19
The solution ofgz, tunder the initial conditionpn,0 δn,1becomes
gz, t
zexp−Λt
1−z
1−exp−Λt
zexp−Λt∞
n0
1−exp−Λtn zn.
3.20
0 20 40 60 80 100 Timet
10−2 100 102 104 106
<n(t)>
a
0 20 40 60 80 100
Timet 10−2
100 102 104 106
V(t)
b
Figure 3: Time-dependence of the mean and the variance of three relaxation functions forκtinTable 2:i an exponential relaxationsolid line,iian inverse power functiondotted line,iiia fractional power functiondashed line; the values of parameters areκ1,γ0.85,α1, andβ1.
The corresponding probability densitypn, tis obtained as pn, t exp−Λt
1−exp−Λtn−1
. 3.21
This is the geometric distribution
fθ θ1−θn−1, 3.22
with the parameter θ exp−Λt, which is a special case of the Pascal the negative binomialdistribution in3.16. The mean and the variance are obtained as
ntexpΛt, Vt≡σn2t expΛt
expΛt−1
. 3.23
The Fano factor becomes
F Vt
nt expΛt−1>0. 3.24
0 5 10 15 Numbern
0 0.2 0.4 0.6
(i)Exponential
0 5 10 15
Numbern 0
0.2 0.4 0.6
(ii)Inverse power
0 5 10 15
Numbern
(iii)Fractional power 0
0.1 0.2 0.3 0.4
Figure 4: Time dependence ofpn, t in 3.15for three relaxation functions forκtin Table 2: i an exponential function, andiian inverse power function,iiia fractional power function; the values of parameters areα1,κ1, andγ 0.85. The pdf profiles are depicted fort 0.4solid line,t 1.2 dotted line,t2.0dashed line,t2.8dash-dotted lineandt3.6dash-dotted line.
0 5 10 15
Numbern
(i)Exponential 0
0.2 0.4 0.6 0.8
0 5 10 15
Numbern
(ii)Inverse power 0
0.2 0.4 0.6 0.8
0 5 10 15
Numbern
(iii)Fractional power 0
0.2 0.4 0.6 0.8
Figure 5: Time-dependence of pn, tin 3.21 for three relaxation functions forκt in Table 2: i an exponential function, ii an inverse power function, iii a fractional power function; the values of parameters areα1,κ1, andγ 0.85. The pdf profiles are depicted fort 0.4solid line,t 1.2 dotted line,t2.0dashed line,t2.8dash-dotted line, andt3.6dash-dotted line.
This means that the nature of statisticsSub-PoissonianSUBP,F <1, PoissonianP,F 1, and Super-PoissonianSUPP,F >1changes depending on the functional form ofλtand its parameter values involved. It is important to make notice of the fact that the variability of Ftchanges in the two cases forβ /0 andβ0. They are summarized inTable 3.
Table 3: Summary of the generalized Polya and non-stationary Yule-Furrey process.
Parameters Pθ nt Vt F
iα /0, β /0, n00 n r 1
n
θr1−θn rθ−1−1 rθ−1θ−1−1 F >1
iiα /0, β0, n01 θ1−θn−1 θ−1 θ−1θ−1−1 F >0
InTable 3,θ,r, andFare defined by
θexp−Λt, r β
α, Λt α t
0
κτdτ, F Vt
nt. 3.25 The temporal development of the probability densitypn, tfor the generalized Polya process in3.15and the generalized Yule-Furry process in3.21for these three examples is depicted in Figures4and5.
4. Discussions
4.1. Solvability Condition
We have studied the generalized Polya process with the transition rate in3.3. How is the solvability if the transition rate is a more general one than that of3.3as
λn, t αtn βt, 4.1
withαtandβtbeing arbitrary functions with time, how is the solvability. In this case, the exact analytic solution is not obtained. The solvability condition is equivalent to the fact that gz, tis written in the form of3.5;gz, t CzGz, t. For the transition rate in4.1, the time-independent functionCzreduces to
Cz C0z−βt/αt. 4.2
This means thatαtandβtmust have the same time-dependent scaling functionκtwith αt κtαandβt κtβi.e.,λt κtαn βin3.3to get the exact analytic solution.
4.2. Master Equation in Memory Function Formalism
An alternative master equation in the memory functionMFformalism for the generalized Polya process in3.1and3.2may be written as
d
dtpn, t t
0
φt−τ
αn−1 β
pn−1, τdτ− t
0
φt−τ αn β
pn, τdτ n≥1, d
dtp0, t − t
0
φt−τβp0, τdτ n0,
4.3
whereαandβare constantsα /0 andβ /0. The Laplace transform of the memory function is defined byφs 0∞φtexp−stds. Forn≥1, the recursion relation is obtained for the Laplace transformpn, sofpn, tas
s αn β
φs
pn, s φs
αn−1 β
pn−1, s n≥1. 4.4
The general formal solution is given under the initial conditionpn,0 δn,0in terms of the inverse Laplace transform as
pn, t 1 2πi
c i∞
c−i∞
n−1
k0
αk β
φs
n
k0
s αk β
φsexpstds. 4.5
When the memory function φt or the pausing time distribution ψt is given i.e., the Laplace transform ofψtis related toφsasψs φs/s φs, the probability density in4.5can be evaluated numerically. The explicit analytic expressions are obtained only for a few special cases1withα 0. The two formalisms have different features complement with each othercf. Montroll and Shlesinger1, Tokuyama and Mori5, and H¨anggi and Talkner6.
5. Concluding Remarks
In this paper, it is shown that there are two types of generalized master equation: i the memory functionMFformalism in1.1andiithe convolution-less, non-stationaryNS formalism in1.2. Then, we propose a new model in the NS formalism: a generalized Polya process in 3.1 and 3.2with the transition rate λt κtαn β having an arbitrary time-varying function κt in 3.3. Further, we exhibit the exact analytic solutions of the probability densitypn, tand the meanntand varianceσn2tfor an arbitrary function κtof time. For some typical examples ofκt, the temporal variations of the mean and the variance are numerically exhibited.
There are many potential applications of the master equation in the NS formalism to non-equilibrium phenomena. In biological systems, the human EEG response to light flashes 12 i.e., microscopic molecular transport associated with transient visual evoked potential VEP and the transition phenomenon from spiral wave to spiral turbulence in human heart 13can be formulated by the master equation in the NS formalism. In considering a stochastic model of infectious disease like a stochastic SIR model14, the introduction of temporal variation of infection rate on account of various environmental changes leads to the master equation in the NS formalism.
In auditory-nerve spike trains, there are interesting observations 15, 16 that i the Fano factor Ft exhibits temporal variation Ft < 1 in the intermediate time region and ii Ft also shows fractional power dependencet in the time region t → ∞ noninteger number. In the generalized Polya process, time variation of the Fano factorFt i.e., SUBPF < 1, PF 1, and SUPPF < 1changes depending on the choice of the relaxation functionκtand the values ofαandβcf. Tables2and3. The related discussions in detail will be reported elsewhere.
Acknowledgment
This work is partially supported by the JSPS, no. 16500169 and no. 20500251.
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