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\begin{document}

\markboth{    %running head for odd-page (authors' name)
Rosu, Reyes, Valencia 
}{            %running head for even-page (`short' title)
Three-Step Master Equation
}

\centerline{ArXiv: math-ph/0402068 v2}

\bigskip

%..............................................................................
\centerline{%       %You can use \\ for explicit line-break
THE THREE-STEP MASTER EQUATION: CLASS OF PARAMETRIC STATIONARY SOLUTIONS
% Three-Step  Master Equation: Class of Parametric Stationary Solutions
}
%................................................................................

\bigskip

%\subtitle{Subtitle}    %use this when you want a subtitle
%Haret C. Rosu\footnote{e-mail: hcr@ipicyt.edu.mx}, Marco A. Reyes$^*$\footnote{e-mail: marco@fisica.ugto.mx},
%Felipe Valencia\footnote{e-mail: felipe@ipicyt.edu.mx}
\centerline{       %Use \scshape  for the family name
Haret C. \textsc{Rosu\footnote{e-mail: hcr@ipicyt.edu.mx}}$^{a}$, Marco A. \textsc{Reyes} $^{b}$, F. \textsc{Valencia}$^{a}$%
}

\begin{center}
     %Affiliation, neglected when [addenda] or [errata]
$^{a}$ Potosinian Institute of Science and Technology,\\
Apdo Postal 3-74 Tangamanga, 78231 San Luis Potos\'{\i}, Mexico\\

$^{b}$ Institute of Physics, University of Guanajuato, \\ Apdo Postal E-143, Le\'on, Gto, Mexico\\

\end{center}

\bigskip

%\publishedin{%         %Write this ONLY in cases of addenda and errata
%Prog.~Theor.~Phys.\ \textbf{XX} (19YY), page.}

%\recdate{Mmmmm DD, YYYY}%            %Editorial Office will fill in this.

{\small

         %this abstract is neglected when [addenda] or [errata]
We examine the three-step master equation 
from the standpoint of the general solution of the associated discrete Riccati equation. We report by this means stationary master solutions depending 
on a free constant parameter, denoted by $D$, that should be negative 
in order to assure the positivity of the solution. These solutions correspond to different discrete Markov processes characterized by
the value of $D$, which is related to specific renormalizations of the transition rates of the chain of states.}

\bigskip


%\begin{document}

%\maketitle

%\section{INTRODUCTION}
%Start your paper 

In general, the three-step population master equation is used by physicists 
in many studies of diffusion processes of microscopic particles on one-dimensional lattices \cite{j}, but this simple discrete equation has extensive and interesting applications 
in other fields as well, most recently to Hubbell's neutral theory in ecology \cite{ban}. In the following, we shall use a population interpretation. It reads
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation} \label{M1}
\frac{dp_{n}}{dt}=d_{n+1}p_{n+1}- \sigma _{n}p_{n}+b_{n-1}p_{n-1}~, \qquad \qquad   \sigma _{n,}=b_{n}+d_{n}~,
%\eqno(7)
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $b_{n}$ is the transition rate for the birth-type jump $n\rightarrow n+1$  %in the number of individuals of species $k$ 
and $d_{n}$ is the death-type rate for the backward jump $n\rightarrow n-1$,  %in the number of individuals of the same species, 
while
$p_{n}$ is the probability to have $n$ individuals at the instant $t$.
Employing the initial conditions $b_{-1}=d_{0}=0$,
the known stationary solution is, \cite{G}
\begin{equation} \label{M2}
P_{n}^{{\rm st}}= P_{0}
\Bigg(\prod _{j=0}^{n-1} \frac{b_{j}}{d_{j+1}}\Bigg)~,
%\eqno(8)
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $P_{0}$ is a scaling constant that through the probabilistic normalization
condition can be written as $P_{0}=\Bigg(1+\sum _{n=1}^{N}
\prod _{j=0}^{n-1} \frac{b_{j}}{d_{j+1}}\Bigg)^{-1}$ (see, \cite{gil}).


\medskip

%\section*{\bf 2. Riccati Parameter Solutions of the Three-Step Master Equation}

%\medskip

\noindent
We proceed now to show that stationary solutions which are different of Eq.~(\ref{M2}) can be obtained 
that are based on the general solution of the discrete Riccati equation connected to the 
master equation. 
%It has been introduced in the hermitic form by Reyes and Rosu \cite{rr} in the context of one-dimensional lattice models. 
%Here we implement this solution at the classical level that makes a different discussion. 
%The solution depends on a free parameter ($D$) for which we shall give stimulating ecological interpretations in the following.
Indeed, performing the transformation  %\cite{rr}
%...................................................
\begin{equation}\label{M3}
y_{n-1}=\frac{P_{n-1}}{P_{n}}+\frac{1-\sigma _{n}}{b_{n-1}}~, \qquad n\neq 0~,  %{(g_{n}r_{n+1})^{1/2}}
%\eqno(9)
\end{equation}
%.........................................................
in Eq. (\ref{M1}) leads to the following discrete Riccati equation
%..................................................
\begin{equation}\label{M4}
y_{n}=b_{n-1}y_n y_{n-1} -      % -(g_{n}r_{n+1})^{1/2}y_{n+1}y_{n}+
\frac{b_{n-1}}{b_n}(1-\sigma _{n+1})y_{n-1}
+d_{n+1}+\frac{1-\sigma _{n+1}}{b_n}~,
%\eqno(10)
\end{equation}
%.............................................
with the particular solution
%..........................................
\begin{equation}\label{M5}
y_{n}^{0}=\frac{1-b_{n+1}}{b_{n}}~.
%\eqno(11)
\end{equation}
%..........................................
However, it is easy to check that one can write a more general solution of Eq.~(\ref{M4}) as follows   %, as in Eq.~(\ref{A5}) in Appendix B.
%.................................................
\begin{equation}\label{M7}
y_{n}^{1}=y_{n}^{0}+\frac{f_{n}}
{D-\sum _{k=0}^{n}\frac{f_{k}b_{k+1}}{d_{k+2}}
}~,\qquad f_n=\prod _{i=0}^{n}\frac{b_id_{i+2}}{b^{2}_{i+1}}~,
%\eqno(13)
\end{equation}
%.......................................
where $D$ is a real constant.


Using simple discrete algebra, one can obtain the recurrence relationship
%..............................................
\begin{equation}\label{Rec}
P_{n+1}=P_n\left(y_n+\frac{\sigma _{n+1}-1}{b_n}\right)^{-1}
\end{equation}
%...............................................
leading to stationary solutions of the following form
%...........................................                                                  MAIN RESULT
\begin{equation}\label{M8}
P_{n}(D)=\tilde{P_0}\prod _{i=0}^{n-1}\frac{b_i}{d_{i+1}}
%\frac{P_{n}^{{\rm st}}}{\prod _{j=0}^{n-1}
\Bigg(1+\frac{f_i b_i/d_{i+1}}{|D|+\sum _{j=0}^{i}\frac{f_jb_{j+1}}{d_{j+2}}-\frac{f_i b_i}{d_{i+1}}}\Bigg)
\end{equation}
%.................................................
where the normalization constant reads
\begin{equation} \label{M8b}
\tilde{P_0}=\Bigg[1+  \sum _{n=1}^{N} \prod _{i=0}^{n-1} \frac{b_{i}}{d_{i+1}}    \Bigg(1+\frac{f_ib_i/d_{i+1}}{|D|+\sum _{j=0}^{i}\frac{f_jb_{j+1}}{d_{j+2}}-\frac{f_ib_i}{d_{i+1}}}\Bigg)\Bigg]^{-1}~.
\end{equation}

%\subsection*{{\bf 2.1} Discussion of Eq.~(\ref{M8})}

\noindent
Examining Eq.~(\ref{M8}) with normalization (\ref{M8b}), we first notice that for $D\rightarrow -\infty$ we recover the known case of stationary 
master solution with the common normalization. 
In addition, we notice that the factor 
$$
\Bigg(1+\frac{f_i b_i/d_{i+1}}{|D|+\sum _{j=0}^{i}\frac{f_jb_{j+1}}{d_{j+2}}-\frac{f_i b_i}{d_{i+1}}}\Bigg)
$$
looks like a renormalization factor for the transition rates of the original stationary Markov process. 
%Therefore, the solutions with finite $D$ represent a more general class of stationary solutions of the master equation that can be considered 
%more realistic than the $D=-\infty$ case once 
A reasonable interpretation of $D$ depends on the specific application and in 
general is related to initial conditions, boundary conditions, or external applied fields.
In addition, for a physical solution one requires positivity implying
$$
\frac{b_i}{d_{i+1}}\geq \frac{b_{i-1}}{d_{i}}~.
$$
This is a strong condition and in particular cases it could be relaxed.



%\section*{\bf 3. Particular Cases}

\medskip

\noindent
{\bf a}). The most trivial case is $b_k=d_k={\rm const} < 1,\, k=0,..., n$.
This implies $P_{k}^{st}=P_0$; the Riccati solution is $y_k=b^{-1}-1$.
The $D$-dependent solution will be 
%..................................
\begin{equation}\label{pc1}
P_k=\tilde{P}\left(1+\frac{k}{D}\right)
\end{equation}
%......................................
and with the normalization explicitly calculated
%...................................
\begin{equation}\label{pc2}
P_k=\frac{1}{n+1}\left(\frac{1+\frac{k}{D}}{1+\frac{n}{2D}}\right)~.
\end{equation}
%...................................

A plot of $P_k$ for various values of the parameter $D$ is shown in Fig.~(1).

\medskip

\noindent
{\bf b}). For the asymmetric case we take $b_n=\frac{1}{2}(1+\epsilon), \, d_n=\frac{1}{2}(1-\epsilon), \,q=\frac{1-\epsilon}{1+\epsilon}<1,\, n=0,...N$,
and the parametric solution reads
%.........................................
\begin{equation}\label{pc3}
P_n=\tilde{P}_0q^{-n}\prod _{i=0}^{n-1}\frac{(1-q)(D+q^i)+1-q^i}{(1-q)D +1-q^i}~,
\end{equation}
%....................................
and the normalization constant can be easily written down from Eq.~(\ref{M8b}). Plots for this case are displayed in Fig.~(2).

\medskip

\noindent
{\bf c}). Various other cases are presented in Figures~(3) - (5) for exponential parametrizations of the jump rates. 

\medskip

%\section*{\bf 4. Conclusion}

%\medskip

\noindent
In summary, we report one-parameter stationary solutions of the three-step master equation
that are based on the corresponding discrete Riccati general solution.
In the continuous case, the mathematical method we employ here corresponds to
Mielnik's procedure in supersymmetric quantum mechanics \cite{miel}. The parameter of these solutions
could be fixed in applications by initial/boundary conditions or through external perturbations of the underlying birth-death Markov process.  

It is well known that the stationary solution of the master equation of a discrete Markov process is uniquely defined if the 
process contains only one class of ergodic states. In this case, the stationary solution does not depend on the initial condition.
The discrete Riccati mathematical procedure leads to modified transition rates and consequently to different stationary solutions that belongs to different 
master equations.
Modifying the rates at the ends of the chain of states corresponds to a probability current flow through the system, i.e., to a 
driven system.
Thus, the physical interpretation of this class of parametric master solutions is that they are a specific type of current-carrying solutions that are important 
non-equilibrium steady states in many mesoscopic and macroscopic systems, such as Becker-D\"oring nucleation processes \cite{wattis04},  
or superconductivity, where as stated by Geller \cite{geller}, ``it is now understood that supercurrent-carrying states are in fact, metastable {\em non-equilibrium 
states} ... with an extremely long lifetime". The latter states are essential for the tunable supercurrent of Josephson junction technology \cite{baselmans}.  

If, for example, we place us in a population (ecology) context, the difference with respect to the original master equation is already at the level of the $d_0$ rate.
For a finite $D$ the rate $d_0$ is not zero. Thus, for positive $d_0$, one can interpret this rate as an initial immigration rate, while for
negative values as an initial emigration rate.  

%Moreover, out of equilibrium, one can encounter multiple competing metastable states whose relative stability evolves in time and then the state that is selected can depend in a
%well-defined way on the driving force

%We recall here a recent paper by Evans \cite{evans_april04} in which he applies Jaynes' principle of maximum entropy inference to determine constraints on the  transition rates
%of a driven stochastic process exhibiting non-equilibrium steady states with a given flux    

\begin{thebibliography}{99}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Some macros are available for the bibliography:
%  o for general use
%    \JL : general journals                 \andvol : Vol (Year) Page
%  o for individual journal 
%    \AJ   : Astrophys. J.           \NC         : Nuovo Cim.
%    \ANN  : Ann. of Phys.           \NPA, \NPB  : Nucl. Phys. [A,B]
%    \CMP  : Commun. Math. Phys.     \PLA, \PLB  : Phys. Lett. [A,B]
%    \IJMP : Int. J. Mod. Phys.      \PRA - \PRE : Phys. Rev. [A-E]     
%    \JHEP : J. High Energy Phys.    \PRL        : Phys. Rev. Lett.
%    \JMP  : J. Math. Phys.          \PRP        : Phys. Rep.
%    \JP   : J. of Phys.             \PTP        : Prog. Theor. Phys.     
%    \JPSJ : J. Phys. Soc. Jpn.      \PTPS       : Prog. Theor. Phys. Suppl.
% Usage:
%  \PRD{45,1990,345}          ==> Phys.~Rev.\ \textbf{D45} (1990), 345
%  \JL{Nature,418,2002,123}   ==> Nature \textbf{418} (2002), 123
%  \andvol{B123,1995,1020}    ==> \textbf{B123} (1995), 1020
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  
%\bibitem{}

\bibitem{j}
         I. Der\'enyi, C. Lee, and A.L. Barab\'asi, {\em Phys. Rev. Lett.} {\bf 80}, 1473 (1998);
        H.R. Jauslin, 
        %``SUSY partners for random walks"
        {\em Phys. Rev.} A {\bf 41}, 3407 (1990); 
        H.C. Rosu and M. Reyes, 
        %``SUSY time-continuous discrete random walks"
        {\em Phys. Rev.} E {\bf 51}, 5112 (1995);   M.A. Reyes and H.C. Rosu, %General solution of the 3-site master equation and the DRE
        {\em Nuovo Cimento} B {\bf 114}, 717 (1999).
 


\bibitem{ban}
I. Volkov, J.R. Banavar, S.P. Hubbell, and A. Maritan,  %Neutral theory and relative species abundance in ecology
{\em Nature} {\bf 424}, 1035 (2003); A. McKane, D. Alonso, and R.V. Sol\'e, {\em Phys. Rev.} E {\bf 62}, 8466 (2000).
 
%Another analytic study based on an urn model has been proposed by McKane A J, Alonso D, Sol\'e R V,
%Theor. Pop. Biol. {\bf 65}, 67 (2004) %physics/0305022

%\bibitem{hubb} Hubbell S.P., {\em The Unified Neutral Theory of Biodiversity and Biogeography},
%(Monographs in Population Biology, No. 32), 2001. See also, MCGILL B.J., {\em Nature} {\bf 422} (2003) 881.

\bibitem{G}
        C.W. Gardiner, {\em Handbook of stochastic methods}
        (Berlin: Springer-Verlag, 1985). 

\bibitem{gil}
        D.T. Gillespie,
        %``An exact markovian analysis of effusion"
        {\em Am. J. Phys.} {\bf 61}, 595 (1993).

\bibitem{miel}
     B. Mielnik, 
     %``Factorization method and new potentials with the oscillator spectrum"
     {\em J. Math. Phys.} {\bf 25}, 3387 (1984).

\bibitem{wattis04}
       See for example, J.A.D. Wattis, {\em J. Phys. A} {\bf 37}, 7823 (2004) and references therein.

\bibitem{geller}
     M.R. Geller, 
     %Current-carrying ground states in mesoscopic and macroscopic systems
      {\em Phys. Rev.} B {\bf 53}, 9550 (1996).  

\bibitem{baselmans}
     J.J.A. Baselmans, A.F. Morpurgo, B.J. Van Wees, T.M. Klapwijk,
      %Tunable supercurrent in superconductor/normal metal/superconductor Josephson junctions
       {\em Superlattices and Microstructures} {\bf 25}, 973 (1999).
      

\end{thebibliography}


%..........................FIGURE 1
\begin{figure}[htb]
\centerline{
\includegraphics[scale=0.9]{grafica1.eps}}
\caption{$P_k(-1000)$, $P_k(-2000)$ and $P_k(-\infty)$ for $b_i=d_i={\rm const}<1$ and $n=20$.
} \label{f1M}  
\end{figure}
%........................

%.......................FIGURE 2
\begin{figure}[htb]
\centerline{
\includegraphics[scale=0.7]{graph_5.eps}}
\caption{$P_n(-4)$ - black straight line; $P_n(-40)$ - blue dotted line; $P_n(-\infty)$ - red-dotted line,
all of them for $\epsilon=0.02$.
} \label{f2M}  
\end{figure}
%.....................

%........................FIGURE 3
\begin{figure}[htb]
\centerline{
\includegraphics[scale=0.7]{graph_1.eps}}
\caption{Black straight line - $P_n(-4)$; blue-dotted line is $P_n(-4000)$; red-dotted-dashed line is $P_n(-\infty)$- for $b_i=0.1+ \exp[-0.12\,i]$ and $d_i=0.1+ \exp[-0.15\,i]$.
} \label{f3M}  
\end{figure}
%.........................


%........................FIGURE 4
\begin{figure}[htb]
\centerline{
\includegraphics[scale=0.7]{graph_2.eps}}
\caption{$P_n(-4)$ - black straight line; $P_n(-4000)$ - blue dotted line; $P_n(-\infty)$ - red-dotted line, for $b_i= \exp[-0.12\,i^{1/2}]$ and $d_i=\exp[-0.15\,i^{1/2}]$.
} \label{f4M}  
\end{figure}
%......................


%.....................FIGURE 5
\begin{figure}[htb]
\centerline{
\includegraphics[scale=0.7]{graph_3.eps}}
\caption{$P_n(-4)$ - black straight line; $P_n(-4000)$ - blue dotted line; $P_n(-\infty)$ - red-dotted line,
for $b_i= 0.01+\exp[-0.15\,i]$ and $d_i=0.01+\exp[-0.12\,i]$.
} \label{f5M}  
\end{figure}
%...........................





%\section*{Acknowledgements}
%We would like to thank ...........


%\appendix
%\section{First Appendix} %Empty argument \section{} yields `Appendix'. 
%
%\section{Second Appendix}



\end{document}


























math-ph0402068.extracted_bib
\begin{thebibliography}{99}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Some macros are available for the bibliography:
%  o for general use
%    \JL : general journals                 \andvol : Vol (Year) Page
%  o for individual journal 
%    \AJ   : Astrophys. J.           \NC         : Nuovo Cim.
%    \ANN  : Ann. of Phys.           \NPA, \NPB  : Nucl. Phys. [A,B]
%    \CMP  : Commun. Math. Phys.     \PLA, \PLB  : Phys. Lett. [A,B]
%    \IJMP : Int. J. Mod. Phys.      \PRA - \PRE : Phys. Rev. [A-E]     
%    \JHEP : J. High Energy Phys.    \PRL        : Phys. Rev. Lett.
%    \JMP  : J. Math. Phys.          \PRP        : Phys. Rep.
%    \JP   : J. of Phys.             \PTP        : Prog. Theor. Phys.     
%    \JPSJ : J. Phys. Soc. Jpn.      \PTPS       : Prog. Theor. Phys. Suppl.
% Usage:
%  \PRD{45,1990,345}          ==> Phys.~Rev.\ \textbf{D45} (1990), 345
%  \JL{Nature,418,2002,123}   ==> Nature \textbf{418} (2002), 123
%  \andvol{B123,1995,1020}    ==> \textbf{B123} (1995), 1020
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  
%\bibitem{}

\bibitem{j}
         I. Der\'enyi, C. Lee, and A.L. Barab\'asi, {\em Phys. Rev. Lett.} {\bf 80}, 1473 (1998);
        H.R. Jauslin, 
        %``SUSY partners for random walks"
        {\em Phys. Rev.} A {\bf 41}, 3407 (1990); 
        H.C. Rosu and M. Reyes, 
        %``SUSY time-continuous discrete random walks"
        {\em Phys. Rev.} E {\bf 51}, 5112 (1995);   M.A. Reyes and H.C. Rosu, %General solution of the 3-site master equation and the DRE
        {\em Nuovo Cimento} B {\bf 114}, 717 (1999).
 


\bibitem{ban}
I. Volkov, J.R. Banavar, S.P. Hubbell, and A. Maritan,  %Neutral theory and relative species abundance in ecology
{\em Nature} {\bf 424}, 1035 (2003); A. McKane, D. Alonso, and R.V. Sol\'e, {\em Phys. Rev.} E {\bf 62}, 8466 (2000).
 
%Another analytic study based on an urn model has been proposed by McKane A J, Alonso D, Sol\'e R V,
%Theor. Pop. Biol. {\bf 65}, 67 (2004) %physics/0305022

%\bibitem{hubb} Hubbell S.P., {\em The Unified Neutral Theory of Biodiversity and Biogeography},
%(Monographs in Population Biology, No. 32), 2001. See also, MCGILL B.J., {\em Nature} {\bf 422} (2003) 881.

\bibitem{G}
        C.W. Gardiner, {\em Handbook of stochastic methods}
        (Berlin: Springer-Verlag, 1985). 

\bibitem{gil}
        D.T. Gillespie,
        %``An exact markovian analysis of effusion"
        {\em Am. J. Phys.} {\bf 61}, 595 (1993).

\bibitem{miel}
     B. Mielnik, 
     %``Factorization method and new potentials with the oscillator spectrum"
     {\em J. Math. Phys.} {\bf 25}, 3387 (1984).

\bibitem{wattis04}
       See for example, J.A.D. Wattis, {\em J. Phys. A} {\bf 37}, 7823 (2004) and references therein.

\bibitem{geller}
     M.R. Geller, 
     %Current-carrying ground states in mesoscopic and macroscopic systems
      {\em Phys. Rev.} B {\bf 53}, 9550 (1996).  

\bibitem{baselmans}
     J.J.A. Baselmans, A.F. Morpurgo, B.J. Van Wees, T.M. Klapwijk,
      %Tunable supercurrent in superconductor/normal metal/superconductor Josephson junctions
       {\em Superlattices and Microstructures} {\bf 25}, 973 (1999).
      

\end{thebibliography}