text:
\section{Introduction}

Understanding charge and energy transfer processes in soft and disordered materials such as thin films made from
organic polymers or large biological molecules is central to the development of
electronic and energy solar-harvesting applications \cite{VMay}.
To model such systems efficiently, a coarse-grained model of the surrounding environment is often assumed.
In turn, this approach allows the application of kinetic, Markovian master equations for
following the dynamics of averaged variables.

Mean first-passage quantities are useful for characterizing stochastic processes.
Specifically, the mean first-passage time defines a timescale
to visit a specified target (or a threshold value) for the first time \cite{VKampen}.
This measure finds numerous applications in physics, chemistry and biology,
e.g. for quantifying reaction rates,
transmission of particles in channels, molecular processes such as receptor binding and adhesion,
and cellular processes such as cell division \cite{Zilman}.
Here, we focus on (charge or energy) transfer processes in flexible molecular systems
and use the mean first-passage time to report on the transfer process from a certain initial site to a final location \cite{Klafter98}.

The capacity of molecules to transfer electric charge can be measured in different setups.
In transient absorption spectroscopy experiments, an initial state is carefully prepared and monitored in time,
see e.g. Refs. \cite{LewisJacs,LewisAcc,Majima}.
In contrast, in electrochemical experiments a molecule of interest is attached to an electrode,
and the rate of charge transfer from the electrode to redox groups in the molecule is monitored,
see e.g. Ref. \cite{Waldeck}.
Alternatively, in molecular conductance experiments a molecule links
two voltage-biased electrodes,  e.g., an STM tip and a conducting substrate,
and the molecule is characterized by its electrical conductance \cite{BJ}.
To thoroughly understand charge transfer processes in a single molecule or ensemble of molecules,
it is critical to resolve the connection between measured observables
in such different experiments.

Several theoretical studies predicted a linear relationship between the intramolecular charge transfer rate
in a donor-bridge-acceptor system and the low bias conductance in a metal-molecule-metal junction
\cite{NitzanR1,NitzanR2,Berlin,Traub,BeratanP}.
Nevertheless, experiments revealed a more complex behavior \cite{Zhou,Waldeck,Bueno}.
For example, in Ref. \cite{Waldeck} the electrochemical rate constant
and the molecular conductance were examined in alkane chains and peptide nucleic acid oligomers,
generally manifesting nonlinear correlations between the transfer rate and the conductance.
These deviations from linearity are rationalized by noting that, for the same molecule, transient,
electrochemical, and conductance measurements
are performed at undoubtedly different settings, considering ensemble vs. single molecule,
using different solvents, and experiencing different environmental conditions.
As a result, the energetics of the molecule and its dephasing and relaxation processes are different in the
these distinct types of experiments.

In this work, we revisit the following basic question:
Considering carrier transfer processes in one-dimensional hopping systems,
what is the relationship between transient measures and steady state observables,
particularly going beyond donor-bridge-acceptor systems?
We leave aside the challenging, practical problem discussed above, that in different experimental
settings the molecule and its local environment are modified, and for simplicity assume that the
structures are identical in the different types of experiments.

Using classical rate equations, we compare transient and steady state measures for the transfer process,
namely, the trapping mean first-passage time (MFPT) and the inverse of the steady state rate,
referred to as the steady state transfer time (SSTT).
Based on these two measures, we study the roles of structural motifs
on carrier transfer in different networks, including
linear or branched chains, uniform, bridge-mediated, energy-biased, single and multi-component networks.
We derive a simple, intuitive relationship between the two timescales, MFPT and the SSTT, and
explain under what conditions they agree.
Generally, we find that these measures scale differently with system size and energy,
and that they can disclose distinct properties of the system.
We further discuss the enhancement of the transfer speed by optimizing the
energy profile of the system, achieved e.g. by chemical modifications or gating.

\section{Model and Measures}
\label{Model}

We study the problem of a random walk across a finite system.
The model includes $n+2$ sites, with a donor D (site 0), acceptor A (site $n+1$), and $n$ intermediate sites.
In addition, the model includes a trap $T$;  transitions from A to the trap are irreversible
with a trapping rate constant $\Gamma_A$.
Variants of this model have been adopted to describe charge transfer and excitation energy transport in
polymers, biomolecules, and amorphous systems.  A model with multiple acceptors is discussed in the Appendix.
At this point, we do not specify the connectivity nor the energy profile of the system,
to be described in examples. As well, we do not include lossy processes within the system,
such as exciton recombination.

The Pauli master equation describes the time evolution of sites populations, $p_{i}$, $i=0,1,..,n+1$,
\begin{eqnarray}
\dot {\bf p}(t) = {\bf M p}(t),
\label{eq:psol}
\end{eqnarray}
with the vector of population {\bf p} and rate constant matrix ${\bf M}$.
Note that this system of equations does not include the equation of motion for the trap, which
fulfills $\dot p_T= \Gamma_A p_A$.
In such a coarse-grained random walk model, the properties of the environment,
such as its temperature, determine the hopping rates;
to make a transition up (down) in energy the walker absorbs (dissipates) heat from (to) the
surrounding thermal bath.
The coupled kinetic rate equations (\ref{eq:psol}) are linear and first-order, and therefore can be readily solved.
We particularly mention that in many instances,
experiments of charge transfer in DNA are well described by the Pauli rate equations, see e.g.
\cite{Bixon,Giese,Berlin01,Burin,Porath,Wagner}.

The success rate of a transfer process in a kinetic network can be quantified
using different measures, such as the transfer time, yield, flux.
Before we introduce the different observables,
we present time dependent and steady state experiments.
In the first case, realized e.g. with transient absorption spectroscopy,
one prepares a well-defined initial condition and follows the time evolution of the system towards
its long-time state. For example, in studies of charge transfer in DNA, an excess charge is prepared on a
donor site using a laser excitation. Once injected into the DNA, the excess charge migrates until
it arrives at an acceptor molecule attached to the DNA  \cite{LewisJacs,LewisAcc,Majima}.
Mathematically, transient dynamics is revealed by solving the time dependent,
first order differential equation
(\ref{eq:psol}). The formal solution is ${\bf p}(t) =\exp ({\bf M}t) {\bf p_0}$, with the vector of the initial condition
${\bf p_0}$.

Alternatively, one may study the same system---albeit in a steady state situation---by defining
a boundary condition rather than an initial condition, identifying a source and a sink
(possible more than one).
For example, the donor population can be maintained fixed by continuously feeding in particles;
population leaves the acceptor at the same rate.
In this scenario, the relevant measure for the transfer process is the flux of outgoing particles,
from the acceptor---outside.
As was discussed in e.g. Refs. \cite{NitzanR1,NitzanR2,Berlin,Traub,BeratanP}, in a donor-bridge-acceptor
configuration and under some simplifying assumptions this steady state setup
can be furthermore related to the low-voltage molecular conductances.

In what follows, we consider time dependent and steady state settings,
examine transfer measures, and discuss their relationships.
We emphasize that we focus here on systems that include a single donor and a single acceptor,
and that so far we do not consider losses within the network.

\subsection{Mean first-passage time}
\label{secMFPT}

The mean first-passage time defines a timescale for a random event to occur for the first time.
A trapping timescale  describes the mean time it takes for
population to leave the system, that is, to transfer from the donor to the trap.
Following Ref. \cite{BeratanIs}, we define the trapping mean first-passage time as
\begin{eqnarray}
\tau_m \equiv \int_0^{\infty} t \dot p_T(t) dt.
\label{eq:MFPTdef}
\end{eqnarray}
Here, $p_T(t)$ is the trap population and
$\dot p_T(t)$ is the probability density of the passage time.
Since the probability of not having transitioned to the
trap is equal to the probability of being in the other $n+2$ sites of the system,
$\dot p_{T}(t) = -\sum_{j=0}^{n+1} \dot p_j(t)$.
Integrating by parts we get
\begin{eqnarray}
\tau_m &=&\sum_{j=0}^{n+1} \int_0 ^{\infty}p_j(t) dt
\nonumber\\
&=& \sum_{j=0}^{n+1}r_j,
\label{eq:MFPT1}
\end{eqnarray}
where the residence time is defined as
\begin{eqnarray}
r_j\equiv \left[\int_0^{\infty} \exp({\bf M} t) {\bf p_0} dt\right]_j = \left[-{\bf M}^{-1} {\bf p_0}\right]_j.
\label{eq:residencet}
\end{eqnarray}
Note that we count rows starting from 0 to $n+1$.
Alternatively, the MFPT can be calculated from the accumulated population on the acceptor; the rate equation describing the kinetics of the trap fulfills
$\dot p_T(t) = \Gamma_Ap_A(t)$, therefore
\begin{eqnarray}
\tau_m = \Gamma_A \int_0^{\infty} t p_A(t) dt.
\label{eq:MFPT2}
\end{eqnarray}
Eqs.\@ (\ref{eq:MFPT1}) and (\ref{eq:MFPT2}) are equivalent; computationally, the former is more convenient. We emphasize that the MFPT, as well as other
observables considered below, are calculated without the explicit consideration of the trap state population.

Another method to calculate the MFPT is to replace the trap and the donor states
with a combined donor/trap state. The steady state for this configuration gives the MFPT by
\begin{eqnarray}
\frac{1}{\tau_m} = \Gamma_A  p_{D/T},
\end{eqnarray}
where $p_{D/T}$ is the steady state population of the combined donor/trap state.
This scheme allows calculation of MFPT for systems with a backflow from the trap state to the acceptor \cite{BeratanFDisc}.
In the rest of the paper, we calculate the MFPT based on Eq. (\ref{eq:MFPT1}).

\subsection{Mean transfer time}

Another measure that has been commonly introduced in the literature is the average transfer time to
reach the acceptor, written as
\begin{eqnarray}
\tau_t\equiv \frac{\int_0^{\infty} t p_A(t) dt }{\int_0^{\infty}p_A(t)dt }.
\label{eq:Mt}
\end{eqnarray}
Nevertheless, since   $\int_0^\infty \Gamma_A p_A dt =  \int_0^\infty  \dot p_T dt=1$,
Eq.\@ (\ref{eq:Mt})  is identical to Eq.\@ (\ref{eq:MFPT2}), concluding that the MFPT and the mean transfer time are in fact the same observable here.
Note that these considerations hold even if the underlying dynamics involves nonthermal effects and/or quantum effects
comprising the time evolution of coherences.

\subsection{Steady state flux}
\label{secSSTT}

In a steady state experiment we maintain the population of the donor state fixed, $p_D(t)=1$, and
measure the outgoing particle flux from the acceptor towards the trap.
In the long time limit, transient effects die out, $\dot {\bf p}(t)=0$,
and the problem reduces to an algebraic equation,
\begin{eqnarray}
 \widetilde {\bf M} {\bf p}_{ss}= {\bf v}.
\end{eqnarray}
Here, $\widetilde {\bf M}$ corresponds to the matrix ${\bf M}$ of Eq. (\ref{eq:psol}) by replacing its first row by
the condition $p_D^{ss}=1$, which is the donor state population in steady state. The inhomogeneous column ${\bf v}$ has a single nonzero value (unity) in its first element.
The steady state flux is defined as
\begin{eqnarray}
k_{ss}&\equiv&\Gamma_A p_A^{ss}/p_D^{ss}
\nonumber\\
&=& \Gamma_A\left[\widetilde {\bf M}^{-1}{\bf v} \right]_{n+1}.
\label{eq:kss}
\end{eqnarray}
The inverse of the flux defined a timescale
\begin{eqnarray}
\tau_{ss}\equiv k_{ss}^{-1},
\label{eq:SSTT}
\end{eqnarray}
which is referred to as the ``steady state transfer time''.

\subsection{On the relation between the MFPT and the steady state flux}
\label{Sec-relation}

Since different protocols (transient and steady state) may be used to interrogate the same system,
it is critical to resolve the relationship between the MFPT and the SSTT.
Do these measures necessarily convey the same information?
If not, when do these timescales agree?
This question will be discussed throughout the paper: We provide examples in which the
two measures, (\ref{eq:MFPT1}) and (\ref{eq:SSTT}) converge, and situations in which they report on
different properties of the system.

Intuitively, as was explained in e.g. Ref. \cite{nitzancar},
the MFPT and the SSTT agree when the dynamics is dominated by a single decaying exponent.
Considering the eigenvalues of the kinetic matrix ${\bf M}$, this situation manifests itself when
$|(\lambda_1-\lambda_0)/\lambda_0|\gg1$. Here,
 $\lambda_0$  is the smallest (in magnitude) eigenvalue of ${\bf M}$, and $\lambda_1$ is the next eigenvalue in order
of increasing magnitude.
While mathematically, this condition is clear, its physical implications are not transparent. In what follows,
we derive a relationship between the MFPT and the SSTT, which can be readily rationalized.

First, we note that in the time dependent protocol, as explained in Sec. \ref{secMFPT},
the total population of the system  ($n+2$ sites and trap) adds up to one.
In contrast, in a steady state experiment as described in Sec. \ref{secSSTT},
the population of the system  exceeds unity.
Let us now define the rate constant
$k_{ss} = \Gamma_A p_A^{ss}$ [we ignore thus far the denominator of Eq.\@ (\ref{eq:kss})].
Since the outgoing flux is equal to the incoming flux, we can write down
\begin{eqnarray} {\bf M} {\bf p}^{ss} = \begin{pmatrix} - k_{ss} & 0 & 0 & \dots \end{pmatrix}^T = - k_{ss}{\bf p}_0,
\label{eq:popss}
\end{eqnarray}
with ${\bf p}^{ss}$  the vector of population in steady state,
which can be obtained by inverting ${\bf M}$.
Using Eq.\@ (\ref{eq:residencet}), the solution to Eq. (\ref{eq:popss}) is given in terms of the residence times,
\begin{eqnarray}
{\bf  p}^{ss} =k_{ss} {\bf r}.
\end{eqnarray}
Summing up the rows and identifying the MFPT by the sum of the residence times we get
\begin{eqnarray}
\sum_{j=0}^{n+1} [ p^{ss}]_j =   k_{ss} \tau_m.
\end{eqnarray}
We now define the total steady state population of the system as $p^{ss}_{sys}\equiv \sum_{j=0}^{n+1} [ p^{ss}]_j$,
and arrive at
\begin{eqnarray}
\tau_{ss} p_{sys}^{ss}= \tau_m.
\end{eqnarray}
This simple  expression directly relates the MFPT to the SSTT, and it hands over a physical meaning to the eigenvalue argument
mentioned above. It is important to note that this relation is general and that it does not rely on the details of the network
i.e., the model may include transitions beyond nearest neighbors, loops, etc.

We now consider different normalizations for the steady state populations:
(i) A condition which is not straightforward to experimentally implement is
$\sum_{j=0}^{n+1} p_{j}^{ss}=1$,  that is, the total population of the system is maintained at unity
in the steady state limit. In this (unnatural) case, the SSTT and the MFPT precisely coincide,
$\tau_{ss} = \tau_m$.
(ii) Alternatively,  we can set the steady state
by invoking the more plausible choice, $p_D^{ss}=1$, which is the common case discussed
in Sec. \ref{secSSTT}.  We then find that
the total population, $\sum_{j=0}^{n+1} p_{j}^{ss}$, exceeds unity and therefore the
two timescales diverge.
Nevertheless, as long as $\sum p_{j\neq D}^{ss} \ll 1$, the total
population of the system remains close to 1,  and therefore the SSTT approaches the MFPT.
This scenario precisely arises in a donor-bridge-acceptor model when the energy of the bridge is
placed high relative to the energy of the donor  \cite{Segal00}.

\section{One-dimensional chains with arbitrary rates: Analytic results and examples}
\label{sec:1D}

In this section, we derive analytical expressions for the MFPT and the SSTT in a
nearest-neighbor one-dimensional (1D) model.
We then exemplify these measures in problems that are
interesting in the context of charge and excitation transfer in
molecular chains and junctions such as organic polymers and DNA.

The model includes  a single donor, $n$ intermediate sites on a chain, an acceptor, and a trap.  We calculate the MFPT and the SSTT,
discuss their relationship, and exemplify that they may convey distinct, complementary information.
Our analytical expression for the MFPT agrees with a previous work \cite{Klafter98}. Nevertheless,
our simple derivation and the associated analysis of the SSTT
provide new physical insights.
The population equations of motion satisfy  ($i=1,2...,n$)
\begin{eqnarray}
&&\dot p_0(t) =-\kr{1}p_0(t) + \kl{0} p_1(t),
\nonumber\\
&&\dot p_i(t) =-\left( \kl{i+1}+\kr{i-1} \right) p_i(t) + \kl{i} p_{i+1}(t) +\kr{i}p_{i-1}(t),
\nonumber\\
&&\dot p_{n+1}(t) = -\left( \kl{n} + \Gamma_A \right)  p_{n+1}(t)
+ \kr{n+1} p_{n}(t),
\label{eq:kin}
\end{eqnarray}
with $\kl{i}$ ($\kr{i}$) as the rate constant to make a transition towards site $i$
which is at the left (right) of the current site.
For example, assuming an equilibrium bath at temperature $T$,
the rate constants satisfy the detailed balance relation,
$\frac{\kr{i+1}}{\kl{i}}=e^{-(E_{i+1}-E_i)/T}$;
we set the Boltzmann constant as $k_B\equiv 1$.
Nevertheless, our general results in this Section do not rely on this particular choice for the rates.

\subsection{MFPT}
\label{3MFPT}

The rate matrix (\ref{eq:kin}) of dimension $(n+2)\times(n+2)$ is tridiagonal (all missing entries are zero),
\begin{eqnarray} \label{eq:arbratematrix}
\textbf{M} = \begin{pmatrix}
-\kr{1}          & \kl{0}          &\tikzmark{c}      &             &                   &  \\
\kr{1}           &-\kl{0}-\kr{2}   & \kl{1}           &             & \mbox{\Huge 0}                  &                   \\
\tikzmark{a}     & \kr{2}          &-\kl{1}-\kr{3}    &\ddots       &                   &                   \\
                 &                 &\kr{3}            &\ddots       & \kl{n-1}          &\tikzmark{d}       \\
                 &\mbox{\Huge 0}&                &\ddots       & -\kl{n-1}-\kr{n+1}& \kl{n}            \\
 &                &                  & & \tikzmark{b} \kr{n+1}         &-\kl{n} - \Gamma_A \\
\end{pmatrix}
\end{eqnarray}
Our objective is to find the residence times using
Eq.\@~(\ref{eq:residencet}), $-\textbf{M}\left({\bf r}\right) = {\bf p}_0$.
Since  ${\bf M}$ is tridiagonal, by following the Thomas algorithm \cite{tridiagonal}
we reduce the above equation to
 \begin{eqnarray}
 \begin{pmatrix}
 1 & -\frac{\kl{0}}{\kr{1}} &\tikzmark{g}            &       & \mbox{\Huge 0} \\
\tikzmark{e}   & 1          & -\frac{\kl{1}}{\kr{2}} &       &  \\
   &                        & 1                      &\ddots &\tikzmark{h}  \\
   &                        &                        &\ddots &-\frac{\kl{n}}{\kr{n+1}}  \\
\mbox{\Huge 0}&                        &                        &\tikzmark{f}       &  1   \\
\end{pmatrix}
\left( {\bf r} \right) =
  \begin{pmatrix}
\kr{1}^{-1} \\\kr{2}^{-1} \\\kr{3}^{-1} \\ \vdots \\ \kr{n+1}^{-1} \\ \Gamma_A^{-1}
\end{pmatrix}
 \end{eqnarray}
\MyLine[thick]{e}{f}
\MyLine[thick]{g}{h}
The transfer times $r_j$ are obtained by progressing recursively from the last row to the first row,
\begin{eqnarray}
{\bf r} =
\begin{pmatrix}
\frac{1}{\kr{1}} \left( 1 + \kl{0}  r_1 \right) \\
\frac{1}{\kr{2}} \left( 1 + \kl{1}  r_2 \right) \\
\frac{1}{\kr{3}} \left( 1 + \kl{2}  r_3 \right) \\
\vdots \\
\frac{1}{\kr{n+1}} \left( 1 + \kl{n}  r_{n+1} \right)  \\
\frac{1}{\Gamma_A}
\end{pmatrix}
\label{eq:arbres}
 \end{eqnarray}
or explicitly,
\begin{eqnarray}
{\bf r} =
 \begin{pmatrix}
\frac{1}{\kr{1}} \left(1 + \frac{\kl{0}}{\kr{2}}\left( 1 + \frac{\kl{1}}{\kr{3}} + \dots \right) \right)\\
\frac{1}{\kr{2}} \left(1 + \frac{\kl{1}}{\kr{3}}\left( 1 + \frac{\kl{2}}{\kr{4}} + \dots \right) \right)\\
\frac{1}{\kr{3}} \left(1 + \frac{\kl{2}}{\kr{4}}\left( 1 + \frac{\kl{3}}{\kr{5}} + \dots \right) \right)\\
\vdots \\
\frac{1}{\kr{n+1}} + \frac{\kl{n}}{\kr{n+1}\Gamma_A} \\
\frac{1}{\Gamma_A}
\end{pmatrix}
\end{eqnarray}
The recursive form has a clear interpretation: $\kr{i+1}^{-1}$
is the residence time on site $i$ before moving forward once;
$r_{i+1}\kl{i}/ \kr{i+1} $ is the contribution to the $i$th residence time from a particle
that jumps to site $i$ from $i+1$, before moving forward again.
The mean first-passage time is the sum of the residence times, given by
\begin{eqnarray}
\tau_m =
\frac{1}{\kr{1}} \left(1 + \frac{\kl{0} + \kr{1}}{\kr{2}}\left( 1 + \frac{\kl{1} + \kr{2}}{\kr{3}} \left( \dots \right) \right) \right)
\label{eq:arbmftt}
\end{eqnarray}
The  recurrence relation for the residence time can also be seen from the flux condition:
The population must leave to the right
one more time than it enters from the left,
$ \kr{i} r_{i-1} - \kl{i-1} r_{i} = 1$.

\subsection{SSTT}
\label{3SSTT}

Following the discussion of Sec. \ref{secSSTT}, our objective is to solve
the algebraic equation $\widetilde {\bf M}{\bf p}^{ss} = {\bf p}_0$. We manage to reduce it as follows,
 \begin{eqnarray}
\label{ArbKss}
 \begin{pmatrix}
 1          & 0 &\tikzmark{k}         &                           &       &             & \\
\tikzmark{i}& 1 &-\frac{\kl{1}}{q(1)} &                           & \mbox{\Huge  0}& & \\
            &   & 1                   & -\frac{\kl{2} q(1)}{q(2)} &       &             &  \\
            &   &                     & 1                         & \ddots&             & \tikzmark{l}\\
            &   &\mbox{\Huge  0}   &                           & \ddots&1            & -\frac{\kl{n-1}q(n-2)}{q(n-1)} \\
            &   &                     &                           &       &\tikzmark{j} & 1 \\
 \end{pmatrix}
 {\bf p}^{ss} =
 \begin{pmatrix}
 1 \\
 \left( \frac{1}{\kr{1}} + \frac{\kl{0}}{\kr{1}\kr{2}} \right)^{-1} \kr{2}^{-1} \\
 \left( \frac{1}{\kr{1}} + \frac{\kl{0}}{\kr{1}\kr{2}} + \frac{\kl{0}\kl{1}}{\kr{1}\kr{2}\kr{3}} \right)^{-1} \kr{3}^{-1}  \\
 \vdots \\
 \left( \frac{1}{\kr{1}} + \dots + \frac{\prod_{i=0}^{n-1}\kl{i}}{\prod_{i=1}^{n+1}\kr{i}} \right)^{-1} \kr{n+1}^{-1}  \\
 \left( \frac{1}{\kr{1}} + \dots + \frac{\prod_{i=0}^{n}\kl{i}}{\Gamma_A \prod_{i=1}^{n+1}\kr{i}} \right)^{-1} \Gamma_A^{-1}
 \label{eq:ssTTc}
 \end{pmatrix}
\end{eqnarray}
\MyLine[thick]{i}{j}
\MyLine[thick]{k}{l}
Here, $q(1) = \kr{2} + \kl{0}$, $q(2) = \kr{2}\kr{3} + \kr{3} \kl{0} + \kl{0} \kl{1}$,
$q(3)=\kl{2}\kl{0}\kl{1} + \kr{4} \kr{2}\kr{3} + \kr{4} \kr{3}\kl{0}+\kr{4}\kl{0}\kl{1}$, and so on.
To generate these combinations,
we define the sequence $s(x)$=$\left\{\kr{2}, \dots, \kr{x+1}, \kl{0}, \kl{1}, \dots, \kl{x-1} \right\}$;
 $q(x)$ is a sum of $x+1$ products of nearest neighbors elements in the sequence $s(x)$.
For convenience, we use the notation  $\kr{n+2} = \Gamma_A$.

The steady state flux is given by $\Gamma_A p_{n+1}$, which can
be found directly from the last row of Eq.\@ (\ref{ArbKss}).
The other elements of the column vector on
the right-hand side of Eq.\@ (\ref{ArbKss}) are the steady state populations
for chains that end with the site corresponding to that row.

In what follows, we exemplify our results for the MFPT and SSTT on several models,
see Figs.\@ \ref{Figs1}-\ref{Figs2}:
(i) Donor-bridge-acceptor configuration,
where the bridge's energy are set constant and higher than the D and A levels.
(ii) Biased chain, representing a system under a constant electric field.
(iii) Co-polymer motifs, either alternating or stacked,
representing charge transport in quasi 1D chains such
as motion along the pi stacking in a double stranded DNA.
Our results in these three cases assume that transitions are induced by a heat bath at
thermal equilibrium,
therefore we enforce the detailed balance relation. We work with
dimensionless energy parameters, scaled by temperature.

\begin{figure}[htbp]
\includegraphics[scale=0.3]{1.pdf}
\caption{One-dimensional models:
(a) Donor-bridge-acceptor chain with an energy gap (scaled by the temperature) $\Delta_B$.
(b) Biased system, exemplifying here a potential bias $\Delta_F$ towards the acceptor.
}
\label{Figs1}
\end{figure}

\begin{figure}[htbp]
\includegraphics[scale=0.28]{2.pdf}
\caption{Examples of co-polymer motifs.
(A1)-(A2) Alternating chains with D connected either to the high or low energy monomers.
(S1)-(S2) Stacked chains with D connected either to the high or low energy monomers.
The D (A) sites are placed in resonance with the first (last) sites.
There are even number of sites in the system in the presented examples. In the odd case,
an additional site is added right before the acceptor site.
}
\label{Figs2}
\end{figure}

\subsection{Example I: Donor-Bridge-Acceptor system}

We consider here the ubiquitous donor-bridge-acceptor setup.
We assume that bridge energies are uniform, and that they are
higher than both the donor and the acceptor levels, see Fig. \ref{Figs1}(a).
The MFPT is calculated from Eq.\@  (\ref{eq:arbres}). Enforcing detailed balance we set
 $\kr{1} = e^{-\Delta_B}$, $\kl{n} = e^{-\Delta_B}$, and all other $\kr{i}, \kl{i} = 1$;
 $n$ is the number of bridge sites and $\Delta_B$ is the dimensionless energy gap between the donor/acceptor and the bridge, (scaled by the temperature).
The residence times are given by
\begin{eqnarray} \bf{r} =
\begin{pmatrix}
		\frac{1}{\Gamma_A} + (n+1)e^{\Delta_B} \\
		\frac{e^{-\Delta_B}}{\Gamma_A} + n \\
		\frac{e^{-\Delta_B}}{\Gamma_A} + (n - 1) \\
		\frac{e^{-\Delta_B}}{\Gamma_A} + (n - 2) \\ \vdots \\
		\frac{e^{-\Delta_B}}{\Gamma_A} + 1 \\
		\frac{1}{\Gamma_A}
\end{pmatrix}
\end{eqnarray}
Summing them up while setting for simplicity $\Gamma_A = 1$ we get
\begin{eqnarray}
\tau_m = ne^{-\Delta_B} + (n+1)e^{\Delta_B} + \frac 12 n^2 + \frac 12 n + 2.
\label{eq:bridgeMFTT}
\end{eqnarray}
When $\Delta_B =0$, this formula reduces to  known results
$\tau_m =(n+2)(n+3)/2 $,  see  \cite{BeratanIs}.

In the opposite limit, when $\Delta_B$ is large, the linear term dominates Eq.\@ (\ref{eq:bridgeMFTT}),
and it approaches the scaling $\tau_m \propto (n+1)e^{\Delta_B}$.

The steady state transfer time is derived from the last row of Eq.\@ (\ref{ArbKss}),
\begin{eqnarray}
k^{-1}_{ss} &=&
\frac{1}{\kr{1}} + \dots + \frac{\prod_{i=0}^{n}\kl{i}}{\prod_{i=1}^{n+1}\kr{i}}
\nonumber\\
&=&
e^{\Delta_B} + \dots + e^{\Delta_B} + 1,
\end{eqnarray}
resulting in
\begin{eqnarray}
\tau_{ss} = (n+1)e^{\Delta_B} + 1.
\label{KSS}
\end{eqnarray}
This measure is linear in $n$ for all $\Delta_B$.
We therefore conclude that the MFPT and the SSTT agree so long as
$\Delta_B \gg 1$. In this case, the bridge population is small,
which agrees with our analysis of Sec. \ref{Sec-relation}.

In Fig.\@ \ref{fig-B}, we display the MFPT and the SSTT while
increasing the system size and the bridge height $\Delta_B$.
The SSTT scales linearly with $n$,
while the MFPT shows a transition from  quadratic to linear scalings upon the  increase in $\Delta_B$.

\begin{figure}[h]
\includegraphics[scale=0.34]{3.pdf}
\caption{Dependence of (a) steady state transfer time
and (b) mean first-passage time on $n$ for  donor-bridge-acceptor systems.}
\label{fig-B}
\end{figure}

\subsection{Example II: Biased chain}
We consider now a biased chain, as illustrated in Fig. \ref{Figs1}(b).
The system can be biased in opposite polarities, such that energies decrease or increase towards the acceptor.
In what follows, we consider both cases, which we refer to as negative (decreasing) and positive (increasing) potential bias.

{\it Negative potential bias, $E_D>E_A$.}
The donor energy is placed above the acceptor, and the total gap is denoted by $\Delta_F\equiv |E_D-E_A|$.
Recall that the chain comprises the donor, acceptor, and $n$ intermediate states.
For a non-increasing potential bias,
we get from Eq. (\ref{eq:arbmftt}) the MFPT
\begin{eqnarray}
\tau_m &=& (n+2) + \sum_{i=0}^n \kl{i}
+ \sum_{i=1}^n \left( \kl{i} \kl{i-1}\right)
\nonumber\\
&+& \sum_{i=2}^n \left( \kl{i} \kl{i-1} \kl{i-2}\right) + \dots + \left(\kl{0}\kl{1}\dots\kl{n} \right).
\label{eq:descendmftt}
\end{eqnarray}
We now specifically assume a linear, constant gradient profile such that
 $\kl{no} = e^{-\frac{\Delta_F}{n+1}}$  and $\kr{no} = 1$.
Using the analytical results of Secs. \ref{3MFPT} and \ref{3SSTT}, we find  the MFPT
\begin{eqnarray}
\tau_m &=& (n+2) + (n+1)e^{-\frac{\Delta_F}{n+1}}
	+ ne^{-2\frac{\Delta_F}{n+1}}
\nonumber\\
	&+& (n-1)e^{-3\frac{\Delta_F}{n+1}}
	+ \dots + e^{-\Delta_F},
\end{eqnarray}
and the SSTT
\begin{eqnarray}
	\tau_{ss} = 1+
	 e^{-\frac{\Delta_F}{n+1}}
	+ e^{-2\frac{\Delta_F}{n+1}}
	+ \dots + e^{-\Delta_F}.
\end{eqnarray}
The MFPT is dominated by the $(n+2)$ term, while the SSTT is dominated by a constant (explicitly, by the $\Gamma_A^{-1}$ rate constant) and
the $e^{-\frac{\Delta_F}{n+1}}$ term.
Therefore, by performing both transient and steady state measurements
one can experimentally determine both the number of sites $n$ and the potential drop $\Delta_F$.

{\it Positive potential bias, $E_D<E_A$.}
When the potential energy linearly increases in the direction of the trap,
the rate constants satisfy $\kr{no} = e^{-\frac{\Delta_F}{n+1}}\text{, }
\kl{no} = 1$. Note that we define the gap by its magnitude, $\Delta_F=|E_A-E_D|$.
The MFPT and the SSTT are
\begin{eqnarray}
\tau_m = 1 + (n+2)e^{\frac{\Delta_F}{n+1}} + (n+1) e^{2\frac{\Delta_F}{n+1}} + \dots + 2 e^{\Delta_F}
\nonumber\\
\end{eqnarray}
and
\begin{eqnarray}
\tau_{ss}
= e^{\frac{\Delta_F}{n+1}} + e^{2\frac{\Delta_F}{n+1}} + \dots + 2e^{\Delta_F}.
\end{eqnarray}
As $\Delta_F$ grows, both mean first-passage time and steady state transfer time approach
$2e^{\Delta_F}$. This agreement is expected: when the forward energy gap is large, the
steady state populations on  intermediate sites become small, and the timescales agree.

\subsection{Example III: Co-polymer motifs}

We consider polymers with two monomers, $B$ and $b$ of high energy ($E_B$) and low energy ($E_b$),
with the scaled energy difference ${\Delta_P}=E_B-E_b$.
The polymer may be alternating or stacked, e.g. with sequences BbBbBb or BBBbbb, respectively.
For simplicity, the donor (acceptor) is placed in resonance with the first (last) site.
Examples of examined co-polymers are presented in Fig.\@ \ref{Figs2}.
In what follows, we study the dependence of the MFPT and the SSTT on $n$,
the number of intermediate sites, the parity of $n$,
and ${\Delta_P}$, the energy difference between monomers.
Trends also depend on whether the donor state is coupled to the high or low energy monomer.
Our results are organized in Tables I and II.

\subsubsection{Alternating Polymers}
Focusing first on the MFPT, we observe the following, see Table I:
(i) Polymers that end on the high energy monomer (sequences A2 and A3, such as DbBbBbBA or  DBbBbBA, respectively)
show the functional form $e^{{\Delta_P}}$, but not $e^{-{\Delta_P}}$.
This is because the residence times for low energy monomers
increase with $e^{{\Delta_P}}$ while the residence time of the B monomer here does not depend on ${\Delta_P}$.
In contrast, polymers that end on the lower strand (DBbBbBbA or DbBbBbA) have both $e^{{\Delta_P}}$ and
$e^{-{\Delta_P}}$ terms. In this case, high energy monomer have residence times that
depend on $e^{-{\Delta_P}}$ while low-energy monomer contribute the residence time $e^{{\Delta_P}}$.
(ii) For all four types of alternating chains, the MFPT scales as $n^2 e^{{\Delta_P}}$. In contrast, the SSTT
shows two different scaling laws, depending on whether the donor is in resonance with B or b.
In the former case, $\tau_{ss}\propto n$, while in the latter case, the barrier energy shows up
and  $\tau_{ss}\propto ne^{{\Delta_P}}$.

\label{table1}
\begin{center}
\vspace{4mm}
Table I: Alternating polymer motifs and characteristic transfer times (leading terms in bold)\\
\begin{tabular}{|l|l|l|l|}
\hline
\,\,\,	&	{sequence} &{ \textbf{MFPT} } & {\textbf{SSTT}} \\
\hline
A1. (even)\,\,     &  { D(Bb)$_m$A} \,\,    &  {${\bf e^{{\Delta_P}}}\left( {\bf \frac 14 n^2} - \frac 12 n \right) + \left(n + 2\right) e^{-{\Delta_P}}   + \frac 14 n^2 + 2n + 1$}
& {${\bf n}+ 2e^{-{\Delta_P}} $} \\
 & $m=n/2$& & \\
 \hline
A2. (even)\,\, &  {D(bB)$_{m}$A} \,\,\, 		&
{${\bf e^{{\Delta_P}}}\left( {\bf \frac 14 n^2} + 2n + 1 \right)  + \frac 14 n^2 + \frac 12 n + 2$} &
{${\bf e^{\Delta_P}}({\bf n} + 1) + 1$ }\\
 & $m=n/2$& & \\
 \hline
A3. (odd)\,\,  	&		{D(Bb)$_{m}$BA } \,\,\, &
			{${\bf e^{{\Delta_P}}}\left( {\bf \frac 14 n^2} + \frac 12 n - \frac 34 \right)  + \frac 14 n^2 + 2 n + \frac{15}{4}$}   & {${\bf n}+2$} \\
& $m=(n-1)/2$& & \\
\hline
A4. (odd)\,\,	&
{D(bB)$_{m}$bA} \,\,\,  	&
{${\bf e^{{\Delta_P}}} \left({\bf \frac 14 n^2} + n - \frac 54 \right) + (n-1)e^{-{\Delta_P}}   + \frac 14 n^2 + \frac 12 n + \frac{21}{4} $}  & {${\bf e^{{\Delta_P}}}({\bf n} - 1)  + 3 $} \\
 &$m=(n-1)/2$ & & \\
	\hline

\end{tabular}
\end{center}

\begin{comment}
		\begin{tabular}{l|l|l}
		sequence & \textbf{SSTT} \\ \hline
			$n$ odd,  D(Bb)$_k$BA & ${\bf n}+2$ \\
			$n$ odd,  D(bB)$_k$bA & ${\bf e^{\Delta_P}}({\bf n} - 1)  + 3 $\\
			$n$ even, D(Bb)$_k$ A & ${\bf n}+ 2e^{-{\Delta_P}} $  \\
			$n$ even, D(bB)$_k$ A& ${\bf e^{\Delta_P}}({\bf n} + 1) + 1$\\
			\hline
		\end{tabular}\\

\noindent \begin{tabular}{l|l|l}
\textbf{MFTT} & DB...A & Db....A \\ \hline
			$n$ odd &
				${\bf e^{\Delta_P}}\left( {\bf \frac 14 n^2} + \frac 12 n - \frac 34 \right)  + \frac 14 n^2 + 2 n + \frac{15}{4}$
				& ${ \bf e^{\Delta_P} }\left({\bf \frac 14 n^2} + n - \frac 54 \right) + (n-1)e^{-{\Delta_P}}   + \frac 14 n^2 + \frac 12 n + \frac{21}{4} $\\
			$n$ even &
				${\bf e^{{\Delta_P}}}\left( {\bf \frac 14 n^2} - \frac 12 n \right) + \left(n + 2\right) e^{-{\Delta_P}}   + \frac 14 n^2 + 2n + 1$
				& ${\bf e^{{\Delta_P}}}\left( {\bf \frac 14 n^2} + 2n + 1 \right)  + \frac 14 n^2 + \frac 12 n + 2$
		\end{tabular} \\
		\begin{tabular}{l|l|l}
		\textbf{SSTT} & DB... A & Db....A \\ \hline
			$n$ odd & ${\bf n}+2$ & ${\bf e^{\Delta_P}}({\bf n} - 1)  + 3 $\\
			$n$ even & ${\bf n}+ 2e^{-{\Delta_P}} $ & ${\bf e^{\Delta_P}}({\bf n} + 1) + 1$
		\end{tabular}\\
		(dominating terms bolded)
\end{comment}
\subsubsection{Stacked polymers}
In this model, the co-polymer is made of two halves of different content, which could
correspond for example to different base-pairings in DNA molecules, see Ref. \cite{DNAS1,DNAS2}.
In this case, all rate constants $\kl{no}, \kr{no}$ are set equal to 1,
besides a single rate where the monomers switch.
For the mean first-passage time, this splits the sum of residence times into two groups:
residence times corresponding to sites
after the change in energy, which do not depend on ${\Delta_P}$,
and times corresponding to sites before the change in energy, which do manifest a ${\Delta_P}$-dependence.
The results are quite natural, see Table II:
If we switch from high energy sites towards low energy sites as we move towards the acceptor, we do not pay an energetic price
and $\tau_m \propto n^2$, $\tau_{ss}\propto n$.
In the opposite case, these times are increased by the $e^{{\Delta_P}}$ factor.

\begin{center}
\vspace{4mm}
Table II: Stacked Polymer motifs and characteristic transfer times (leading terms in bold)\\
\label{table1}
\begin{tabular}{|l|l|l|l|}
\hline
\,\,\,\,	&	{sequence} &{ \textbf{MFPT} } & {\textbf{SSTT}} \\
\hline
S1. (even)\,  	&		{DB$_{m}$b$_m$A } \,\,\,  & $ { \bf \frac 14 n^2 }+ \frac 32 n + 2 + \left( \frac 14 n^2 + n + 1 \right) e^{-{\Delta_P}} $  &
$ {\bf \frac 12 n} + 1 + \left( \frac 12 n + 1 \right) e^{-{\Delta_P}}  $\\
& $m=n/2$ & & \\
\hline
S2. (even)\,  	&		{Db$_{m}$B$_m$A } \,\,\, &  ${\bf e^{\Delta_P} } \left( {\bf \frac 14 n^2} + \frac 32 n + 2 \right) + \frac 14 n^2 + n + 1$ & ${\bf e^{\Delta_P}}\left( {\bf \frac 12 n} + 2 \right) + \frac 12 n $ \\
& $m=n/2$& & \\
\hline
S3. (odd)\,  	& {DB$_{m}$b$_{m+1}$A	} \,\,\, 	&   ${ \bf \frac 14 n^2} + \frac 32 n + \frac 94 + \left( \frac 14 n^2 + n + \frac 34 \right) e^{-{\Delta_P}} $ &
$ { \bf \frac 12 n }+ \frac 12 + \left( \frac 12 n + \frac 32 \right)e^{-{\Delta_P}} $\\
&$m=(n-1)/2$ && \\
\hline
S4. (odd)\,  	&		{Db$_{m}$B$_{m+1}$A } \,\,\, &  ${\bf e^{{\Delta_P}}}\left( {\bf \frac 14 n^2} + \frac 32 n + \frac 54 \right)  + \frac 14 n^2 + n + \frac 74 $ &
${\bf e^{{\Delta_P}}}\left({\bf \frac 12 n} + \frac 52 \right)  + \frac 12 n - \frac 12 $\\
& $m=(n-1)/2$ && \\
	\hline
\end{tabular} \\
\end{center}

\begin{comment}

		\begin{tabular}{l|l|l}
		\textbf{MFTT} & Donates to higher-energy strand & Donates to lower-energy strand \\ \hline
			$n$ odd &
				${ \bf \frac 14 n^2} + \frac 32 n + \frac 94 + \left( \frac 14 n^2 + n + \frac 34 \right) e^{-{\Delta_P}} $
				& ${\bf e^{{\Delta_P}}}\left( {\bf \frac 14 n^2} + \frac 32 n + \frac 54 \right)  + \frac 14 n^2 + n + \frac 74 $\\
			$n$ even &
				$ { \bf \frac 14 n^2 }+ \frac 32 n + 2 + \left( \frac 14 n^2 + n + 1 \right) e^{-{\Delta_P}} $
				& ${\bf e^{\Delta_P} } \left( {\bf \frac 14 n^2} + \frac 32 n + 2 \right) + \frac 14 n^2 + n + 1$
		\end{tabular} \\
		\begin{tabular}{l|l|l}
			\textbf{SSTT} & High & Low \\ \hline
				$n$ odd & $ { \bf \frac 12 n }+ \frac 12 + \left( \frac 12 n + \frac 32 \right)e^{-{\Delta_P}} $&
					${\bf e^{{\Delta_P}}}\left({\bf \frac 12 n} + \frac 52 \right)  + \frac 12 n - \frac 12 $\\
				$n$ even & $ {\bf \frac 12 n} + 1 + \left( \frac 12 n + 1 \right) e^{-{\Delta_P}}  $&
					${\bf e^{\Delta_P}}\left( {\bf \frac 12 n} + 2 \right) + \frac 12 n $
		\end{tabular}\\

		\end{comment}

Overall, we find that:
(i) The MFTT and SSTT of alternating configurations
are typically longer than for a stacked one.
It is interesting to note that higher
resistances were indeed measured for alternating DNA configurations,
relative to stacked sequences \cite{DNAS1,DNAS2},
in line with our calculations.
Nevertheless, since coherent quantum effects are not included in this work,
this agreement may be incidental.
(ii) If $n$ and ${\Delta_P}$ are large enough such that only the leading term matters,
it is generally possible to determine both ${\Delta_P}$ and $n$ by jointly studying  the
steady state lifetime and the mean first-passage time,  assuming we know the nature of
the polymer (from the configurations S1-S4 and A1-A4), but not necessarily its length or energetics.
First, we note that $\tau_m / \tau_{ss} \propto n$ in all cases, besides A1 and A3.
In these special cases, the dominating term of steady state lifetime is itself proportional to $n$.
As well, all co-polymers besides S1 and S3,
show  $\frac 14 n^2e^{{\Delta_P}}$ as the leading term in the mean first-passage time.
In all these configurations, ${\Delta_P}$ can be readily determined once we find $n$.

\vspace{10mm}
\begin{comment}
As an example, we detail the derivation of the MFPT for S1.
In this case, there is only a single rate that is not equal to one, that
is $\kl{n/2} = e^{-{\Delta_P}}$.
From (\ref{eq:descendmftt}),
The mean first-passage time is the sum of products of $\kl{no}$,
all of which are either 1 or $e^{-{\Delta_P}}$.
The constant term (without the energy prices) is
$(n+2) + \frac{n}{2}(\frac{n}{2}+1)=n^2/4+3/2n+2$.
We now collect terms with the factor $e^{-{\Delta_P}}$.
For products with  $s \leq \frac n2$ factors, there are $s$ terms with $e^{-{\Delta_P}}$.
For $s \geq \frac n2 + 1$, there are $n+2 - s$.
The total $e^{-{\Delta_P}}$ term is
 $e^{-{\Delta_P}} \left(\frac{n^2}{4} + n + 1 \right)$.
\end{comment}

\begin{figure}[htbp]
\includegraphics[scale=0.3]{4.pdf}
\caption{Illustrations of energy profiles that minimize the (a)-(b) MFPT  and (c)-(d) SSTT
under forward or reversed biases.
The energy gap between the D and A sites is fixed.
The position of the intermediate levels is determined by minimizing the transfer time.
Under low bias (not shown), the energy profile minimizing the MFPT is nonmonotonic,
and it generates an internal strong field  as we show in Figs.
\ref{fig-negative-energy-profile} and \ref{fig-positive-energy-profile}.
}
\label{fig-opt}
\end{figure}

\section{Optimization of the energy profile}

Achieving fast rate of charge transport through molecules is desirable for many applications. Experiments
manifest an enhancement of hole transport in DNA by modifying the injection site, the sequence,
and the chemical environment. These modifications alter the molecular energy profile
thus the driving force for charge migration \cite{speed, lipid,Grozema}.

In this Section, we ask the following question: Assuming a fixed donor and acceptor sites energies,
what is the optimal energy profile for the intermediate $n$ sites,
so as to minimize the transfer time---thus enhance the speed of
transfer? We explore this question analytically and numerically for positive and negative biases.
Central observations are that the MFPT and the SSTT are minimized under different principles, and
that the profile generally deviates from linearity and monotonicity.
In Fig. \ref{fig-opt} we exemplify optimized setups under large bias, derived in the following discussion.

\subsection{Negative bias, $E_D>E_A$}
\subsubsection{MFPT}
Let us first consider the negative bias case, with a total energy gap of $\Delta_F=|E_D-E_A|$
between the donor and acceptor.
As a constraint, we demand that the energies of the sequence of $n$ internal sites are non-increasing.
We assign a (temperature scaled) energy $E_i$ to site $i$. The transition rate constant from site $i$
to $i-1$ is set at $\kl{i-1}=e^{E_{i} - E_{i-1}}$ if $E_i < E_{i-1}$ and $1$ otherwise.
For simplicity, the trapping rate constant is taken as $\Gamma_A = 1$.
Since the energies do not increase going towards the trap, all $\kr{i} = 1$, and the MFPT follows Eq.\@
(\ref{eq:descendmftt}). The overall energy gap satisfies the constraint
\begin{eqnarray}
\prod_{i=0}^{n} \kl{i} = e^{-\Delta_F}.
\end{eqnarray}
The optimal rates in Eq.\@ (\ref{eq:descendmftt}) fulfill the constraint and $n+1$ equations of the form
\begin{align}
\label{LMTmfpt}
	\frac{\partial \tau_m}{\partial \kl{i}} =&
	1 + \kl{i-1} +  \kl{i+1}+
	\kl{i-2}\kl{i-1} + \kl{i-1}\kl{i+1} +  \kl{i+1}\kl{i+2}  \dots
	\nonumber\\
	+& \kl{0} \kl{1} \dots \kl{i-1}\kl{i+1} \dots \kl{n}
	\nonumber\\
	=& \lambda \kl{0} \kl{1} \dots \kl{i-1}\kl{i+1} \dots \kl{n}
\end{align}
where the constant $\lambda$ is the Lagrange multiplier;   $\kl{j} = 0$ if $j<0$.
As an example, let us consider the case with two intermediate sites, $n = 2$.
The MFPT is
\begin{eqnarray}
\tau_m= 4+ \kl{0}+\kl{1}+\kl{2} + \kl{0}\kl{1} + \kl{1}\kl{2} + \kl{0}\kl{1}\kl{2},
\end{eqnarray}
and the following four equations should be jointly solved
\begin{subequations}
 \begin{align}
                1 + \kl{1} + \kl{1}\kl{2} &= \lambda \kl{1}\kl{2} \\
                1 + \kl{0} + \kl{2} + \kl{0}\kl{2} &= \lambda \kl{0}\kl{2}
\label{eq:b}
\\
                1 + \kl{1} + \kl{0}\kl{1} &= \lambda \kl{0}\kl{1} \\
                \kl{0}\kl{1}\kl{2} &= e^{-\Delta_F}
\end{align}
\end{subequations}
From symmetry, note that the first and last transition rates are identical, $\kl{2}=\kl{0}$.
In general,  since Eq.\@  (\ref{LMTmfpt}) is symmetric with respect to replacing each $\kl{i}$ with $\kl{n-i}$,
pairs of rate constants of equal distance from the first and last sites will be identical.
For $n=2$, the minimization problem involves solving
a third-degree polynomial in $\sqrt{\lambda}$,
\begin{eqnarray}
0 = \lambda \sqrt{\lambda} - 2 \lambda + \sqrt{\lambda} - e^{\Delta_F}.
\label{eq:LMTpolynomial}
\end{eqnarray}
where
\begin{eqnarray} \kl{0} = \kl{2} = \frac{1}{\sqrt{\lambda} - 1}, \, \kl{1} = \frac{1}{\sqrt{\lambda}}.
\end{eqnarray}
We can also organize Eq. (\ref{eq:LMTpolynomial})  as $x(x-1)^2 = e^{\Delta_F }$ with  $x=\sqrt{\lambda}$.
As expected, when $\Delta_F$ becomes large, $x$ grows and approaches $x\sim e^{\Delta_F/3}$,
which corresponds to constant spacings (linear profile).
Optimization of the model with $n$ intermediate sites
requires solving a similar $n+1$-degree polynomial.

\begin{figure} [htbp]
\includegraphics[scale=.4]{5a.pdf}
\includegraphics[scale=.4]{5b.pdf}
\includegraphics[scale=.4]{5c.pdf
\caption{Simulations of the mean first-passage
time for configurations with two internal sites (D, 1, 2, A) for negative bias, $E_D>E_A$.
As $\Delta_F$ increases, the optimal configuration (minimum MFPT)
approaches the profile  $(\frac{2}{3}\Delta_F,  \frac{1}{3}\Delta_F)$.}
\label{fig-2d-negative-bias}
\end{figure}

The analytic calculation above constrains the energies to be non-increasing.
This constraint is not enforced in simulations that in fact reveal situations in which
the transfer times are minimized with nonmonotonic profiles.
In Figures \@ \ref{fig-2d-negative-bias}-\ref{fig-negative-energy-profile}
we demonstrate such calculations using gradient descent.
We find that when $n$ is small ($n=2$) and $\Delta_F$ is large,
starting from the D, the energies descend slowly (small local bias), then more quickly towards the
center (large local bias), and again slowly at the end of the chain.
In contrast, when $n$ is large and $\Delta_F$ small, the first internal site is placed {\it above} the donor state.
The rest of the sites descend linearly, reaching below the acceptor state, see Fig. \ref{fig-negative-energy-profile}(a)-(b).
This result can be rationalized as follows: The system pays the cost of a
large energy jump in the first transition (slow-down of transfer), yet this early barrier
prevents the carrier from going backwards towards the donor from the internal sites.
Nevertheless, if we force the internal sites to take values only in between the donor and acceptor levels,
we obtain profiles that are similar to the small-$n$ large-$\Delta_F$ case.

Non-monotonic internal profiles allow fast transfer when the external bias $\Delta_F$ is small relative to the
temperature.
In such low-field cases, the levels organize to create a large internal field,  opposing thermal
fluctuations and facilitating fast transport.
This principle could be used in the design of biomolecules that  support  fast charge migration \cite{Grozema}.
\begin{figure} [htbp]
\includegraphics[scale=.7]{6.pdf}
\caption{Negative bias, $E_D>E_A$.
Energy profile obtained numerically using gradient descent
by minimizing the MFPT for systems with eight internal sites (D, 1, \dots 8, A). }
\label{fig-negative-energy-profile}
\end{figure}

\subsubsection{SSTT}
Continuing with the same model, with a non-increasing profile, we turn to the steady state transfer time,
\begin{eqnarray}
\label{descendkss}
 \tau_{ss} = \kl{0} + \kl{0}\kl{1} + \kl{0} \kl{1} \kl{2} + \dots + \prod_0^{n} \kl{i}.
\end{eqnarray}
This expression is minimized when $\kl{0} = e^{-\Delta_F}$, $\kl{i} = 1$ for
 $i \neq 0$, resulting in the  minimum steady state transfer time of $(n+1)e^{-\Delta_F}$.
In other words, the fastest process occurs when all the internal sites are aligned with the acceptor,
i.e., the energy jump occurs immediately, from the donor to the first site, $n=1$.

\subsection{Positive bias, $E_D<E_A$}
\subsubsection{MFPT}
We now consider a low-energy donor,
$n$ non-decreasing energy levels, and a high energy acceptor site.
We define the gap as $\Delta_F=|E_D-E_A|$.
This arrangement implies that $\kl{no} = 1$, while the forward rates are to be optimized.
The MFPT is
\begin{align}
\tau_m &=
\left(\frac 1 \Gamma_A + \frac{1}{\kr{n+1}} + \dots + \frac{1}{\kr{1}} \right)
\nonumber\\
&+ \left(\frac{1}{\Gamma_A \kr{n+1}} + \frac{1}{\kr{n+1}\kr{n}} + \dots +
\frac{1}{\kr{2}\kr{1}} \right)
 \nonumber\\
&+ \dots +
\frac{1}{\Gamma_A \prod_{i=1}^{n+1} \kr{i}
}.
\end{align}
For simplicity, taking $\Gamma_A = 1$, we get
\begin{align}
\tau_m &=
\left(\frac{2}{\kr{n+1}} + \frac{1}{\kr{n}} \dots + \frac{1}{\kr{1}} \right)
\nonumber\\
&+
\left(\frac{2}{\kr{n+1}\kr{n}} + \frac{2}{\kr{n}{\kr{n-1}}} +  \dots +
\frac{1}{\kr{2}\kr{1}} \right)
\nonumber\\
&
+ \dots +
\frac{2}{\prod_{i=1}^{n+1} \kr{i}}.
\end{align}
Minimizing the MFPT subjected to the constraint $\prod_{i=1}^{n+1} \kr{i} = e^{-\Delta_F}$,
the equations have no convenient symmetries.
For example, $n = 1$ yields
\begin{eqnarray}
\tau_m=1 + \frac{1}{\kr{1}} + \frac{2}{\kr{2}} +\frac{2}{\kr{1}\kr{2}}.
\end{eqnarray}
While this equation can be readily minimized with the constraint, we
proceed with the general formalism of Lagrange multipliers
and solve the system,
\begin{subequations}
\begin{align}
-\frac{1}{\kr{1}^2} - \frac{2}{\kr{1}^2\kr{2}} &= \lambda \kr{2}\\
-\frac{2}{\kr{2}^2} - \frac{2}{\kr{1}\kr{2}^2} &= \lambda \kr{1}\\
\kr{1}\kr{2} &= e^{-\Delta_F}.
\label{eq:c}
\end{align}
\end{subequations}
This problem can be solved by  noting that
\begin{eqnarray}
-\frac{1}{\kr{1}^2\kr{2}} - \frac{2}{\kr{1}^2\kr{2}^2}
= \lambda =
-\frac{2}{\kr{1}\kr{2}^2} - \frac{2}{\kr{1}^2\kr{2}^2},
\end{eqnarray}
which implies that
$\kr{2} = 2\kr{1}$, or explicitly
\begin{eqnarray}
\kr{1} = e^{\frac 12 \left( -\ln 2 - \Delta_F \right)}, \quad
\kr{2} = e^{\frac 12 \left( \ln 2 - \Delta_F \right)}.
\end{eqnarray}
Figures \@  \ref{fig-2d-positive-bias}-\ref{fig-positive-energy-profile} present simulation results, where
we allowed the internal potential profile to become nonmonotonic (unlike the analytic derivation).
We find that for large $n$ and large $\Delta_F$,
the internal sites cluster at a level slightly above the midgap
between the donor and acceptor.
In contrast,  for very small $\Delta_F$ the energy profile first increases,
thus artificially creating a large internal field, which facilities fast transfer.
In fact, when $\Delta_F$ is small the results for positive and negative biases are
very similar, compare Fig. \ref{fig-negative-energy-profile}(a) to Fig. \ref{fig-positive-energy-profile}(a).

\begin{figure} [htbp]
\includegraphics[scale=.4]{7a.pdf}
\includegraphics[scale=.4]{7b.pdf}
\includegraphics[scale=.4]{7c.pdf}
\caption{Simulations of the
mean first-passage time for a two-state system with $E_D<E_A$.
\label{fig-2d-positive-bias}
As the gap $\Delta_F$ increases, the optimal configuration is of
two states placed together at around $\Delta_F/2$.}
\end{figure}

\begin{figure} [htbp]
\includegraphics[scale=.7]{8.pdf
\caption{Energy profile obtained numerically by minimizing the
MFPT for a system with eight intermediate sites under a positive bias $E_D<E_A$.
At small $\Delta_F$, the profiles under positive and negative biases look identical,
compare panel (a) to Fig. \ref{fig-positive-energy-profile}(a).
}
\label{fig-positive-energy-profile}
\end{figure}

\subsubsection{SSTT}
We calculate the SSTT using Eq. (\ref{eq:ssTTc}), assuming now a non-decreasing profile,
\begin{eqnarray}
\tau_{ss} = \left[\kr{1}^{-1}  + \left(\kr{1} \kr{2}\right)^{-1} + \dots +
\prod_{i=1}^{n+1} \kr{i}^{-1}. \right]
\label{eq:ssfincrease}
\end{eqnarray}
The minimum MFPT, subject to the constraint $\prod_{i=1}^{n+1} =
e^{-\Delta_F}$, can be intuitively obtained:
each term in Eq.\@ (\ref{eq:ssfincrease})
is greater  or equal of the preceding term. Therefore, $\tau_{ss}$ is minimized
once we take $\kr{n+1} =
e^{-\Delta_F}$ and $\kr{i} = 1$ for all $i \neq n+1$.
This setup is illustrated in Fig. \ref{fig-opt}(d): besides the last site, all sites align with the donor state.

\section{Summary}

In summary, using kinetic equations we studied particle transfer in one-dimensional systems with different motifs:
linear, branched, uniform, biased, homogeneous and multi-component systems.
Central results of our work are:
(i) We derived an intuitive relationship between the MFPT and the SSTT.
(ii) We obtained closed-form expressions for the MFPT and the SSTT in nearest-neighbor 1D chains.
(iii) We exemplified our results on experimentally relevant setups, such as donor-bridge-acceptor systems, biased
chains, and stacked and alternating co-polymers, and discussed the physical information
that can be gained from the MFPT and the SSTT.
(iv) We minimized the transit time through chains
by optimizing the energy profile.
Here, we found that the MFPT and the SSTT were minimized under fundamentally different design rules.
Most strikingly, under shallow external potentials of the order of the thermal energy, $\Delta_F\leq 1$,
the system achieved fast transit time if the levels were set so as to create a strong internal
field.

This work clarifies on the relationship between transient and steady state measures in classical kinetic networks.
It is worthwhile to mention studies of the MFPT in complex scale-invariant media \cite{klafter}
and in the context of enzymatic chemical reactions in biochemistry \cite{Udo07,Cao11}.
Nevertheless, in these works the networks included loops, a motif that was not investigated in the present study.
It is also interesting to consider extensions of this work, particularly the enhancement of the transfer speed,
to systems under thermal gradients \cite{NitzanH}.
The analysis of transfer processes in networks beyond 1D, and the role of quantum coherent
effects in corresponding quantum systems, will be the focus of future works.

\begin{acknowledgments}
DS acknowledges the Natural Sciences and Engineering Research Council (NSERC) of Canada Discovery Grant
and the Canada Chair Program.
NK was supported by the NSERC Undergraduate Student Research Award (USRA)  program.
\end{acknowledgments}

\renewcommand{\theequation}{A\arabic{equation}}
\setcounter{equation}{0}
\section*{Appendix: Graphs with multiple acceptors}

We consider here a system with a single donor and multiple acceptors.
Each acceptor is connected to the trap site,
therefore we define the trapping mean first-passage time as in Eq. (\ref{eq:MFPTdef}).
The definition of the MFPT in terms of the
residence times, Eq. (\ref{eq:MFPT1}), holds even with multiple acceptors.

For simplicity, let us consider
a chain that is branched at point $c$ into two chains, $\alpha$ and $\beta$, with
two acceptors, which are identified by $\alpha_A$ and $\beta_A$,  see Fig. \ref{FigA}.
We explore next the MFPT to be trapped leaving from either acceptors.
The normalization of the trap population  leads to
$\int_0^{\infty}dt \left(\Gamma_{\alpha}p_{\alpha_A}(t) +  \Gamma_{\beta}p_{\beta_A}(t)\right)  =1$.
In the language of residence time this condition translates to
$\Gamma_{\alpha}r_{\alpha_A}+\Gamma_{\beta}r_{\beta_A}=1$.
Furthermore, it is useful to define the yield to each acceptor as
\begin{eqnarray} \Phi_{l} \equiv r_{l_A} \Gamma_{l}, \,\,\,\ l=\alpha,\beta \end{eqnarray}
with the total yield $\sum_l{\Phi_l}=1$.

Based on the discussion of Sec. \ref{3MFPT}, the MFPT
for a graph with no loops is determined by a flux condition,
\begin{eqnarray} \kr{no} r_i - \kl{no} r_{i+1} = \Phi,
\end{eqnarray}
with $\Phi$ the yield on the branch considered.
To find the trapping MFPT, we need to resolve the residence times at every site.
Particularly, if we find
the yields (proportional to residence times) at the acceptors,
we can use the flux condition and moving backwards generate the residence times at every site.
We now describe how to find the yields at each acceptor for a system with a single branching point, Fig. \ref{FigA}.
The method can be furthermore generalized to describe a more complex graph, with additional branches.

\begin{figure}[h]
\includegraphics{9.pdf}
\caption{A network with a single branching point $c$.
The two branches are identified by $\alpha$ and $\beta$.}
\label{FigA}
\end{figure}

In the model sketched in Fig. \ref{FigA},  site $c$ is connected to three sites,
$c-1$, $\alpha_1$, and $\beta_1$, where
$\alpha_1$ and $\beta_1$ are the first sites in chains that end in acceptors
$\alpha_A$ and $\beta_A$, respectively. These sites have respective rates to the trap, $\Gamma_\alpha$, $\Gamma_\beta$.
Once we determine the yields
 $\Phi_{\alpha}$ and  $\Phi_{\beta}$, we can proceed and find the residence times at all sites
by the flux condition.

To find the yield $\Phi_{\alpha}$, we consider a modified system composed of
the sites $\alpha_1 \dots \alpha_A$,
with $\alpha_1$ as an additional acceptor,
 with an irreversible rate $\kl{\alpha_1 \to c}$. For this modified system, we define
the yield through the `true' acceptor as $\Phi^*_{\alpha}$,
which is the probability that, given a particle arriving at $\alpha_1$, it will exit at
$\alpha_A$ and not at $\alpha_1$. In the original model, this yield represents the
probability that a particle
coming to $\alpha_1$ will leave by $\alpha_A$ without returning to $c$.

The yield $\Phi^*_{\alpha}$ can be readily obtained analytically, since
the matrix corresponding to the new
system is tridiagonal. The solution is found using the Thomas algorithm;
the rate matrix has the same form as in Eq. (\ref{eq:arbratematrix}) except the top left element
combines transitions from $\alpha_1$ to both $\alpha_2$ and the `trap' $c$,
 $-\kr{\alpha_1\to\alpha_2}-\kl{\alpha_1\to c}$.
The yield obtained is
\begin{eqnarray}
\Phi^*_{\alpha}
= \left( 1
+ \kl{\alpha_1 \to c} \left( \sum_{i=1}^{n-1} \frac{\prod_{j=1}^i\kl{\alpha_{j+1}\to \alpha_j}}{\prod_{j=1}^{i+1} \kr{\alpha_{j-1} \to \alpha_j}}
+ \frac{\prod_{j=1}^{n}\kl{\alpha_{j+1}\to \alpha_j}}{\Gamma_\alpha\prod_{j=1}^{n+1}\kr{\alpha_{j-1} \to \alpha_j}} \right)\right)^{-1},
\end{eqnarray}
where $n$ is the number of non-acceptor states, and site $\alpha_{n+1} \equiv \alpha_{A}$.
As expected, if $\kl{\alpha_1 \to c} = 0$ then $\Phi^*_{\alpha} = 1$;  increasing $\kl{no}$ relative to $\kr{no}$ reduces the yield. The yield $\Phi^*_{\beta}$ can be calculated in a similar way.

We now relate the true yields $\Phi_{\alpha,\beta}$ to the auxiliary ones $\Phi_{\alpha,\beta}^*$.
The probability that the particle exits at $\alpha_A$ after visiting $c$ only once
is
\begin{eqnarray}
\frac{\kr{c \to \alpha_1}}{\kr{c \to \alpha_1} + \kr{c \to \beta_1}} \Phi^*_{\alpha}
\end{eqnarray}
Here, the probability to enter $\alpha_1$ is multiplied by the chance
to leave from $\alpha_A$ without returning to $c$, given that the particle reaches $\alpha_1$.

Likewise, the probability that the particle exits from $\alpha_A$, leaving $c$ to the
right exactly twice is
\begin{eqnarray}
\left( \frac {\kr{c\to\alpha_1}\left(1-\Phi^*_{\alpha}\right) +
\kr{c\to\beta_1}\left(1-\Phi^*_{\beta}\right)}{\kr{c\to\alpha_1} + \kr{c\to\beta_1} }\right)
\frac{\kr{c \to \alpha_1}}{\kr{c \to \alpha_1} + \kr{c \to \beta_1}}  \Phi^*_{\alpha},
\end{eqnarray}
where the new term in the product is the probability to return to $c$ after leaving to the right once.
Overall, the probability of leaving site $c$ $n$ times to the right is
\begin{eqnarray}
\left( \frac {\kr{c\to\alpha_1}\left(1-\Phi^*_{\alpha}\right) +
\kr{c\to\beta_1}\left(1-\Phi^*_{\beta}\right)}{\kr{c\to\alpha_1} + \kr{c\to\beta_1} }\right)^{n-1}
\frac{\kr{c \to \alpha_1}}{\kr{c \to \alpha_1} + \kr{c \to \beta_1}}  \Phi^*_{\alpha}.
\end{eqnarray}
The total yield is a geometric series, which collapses to the simple result,
\begin{eqnarray}
\label{eq:fluxfromlinflux}
\Phi_{\alpha} &=& \sum_{n=1}^\infty
\left( \frac {\kr{c\to\alpha_1}\left(1-\Phi^*_{\alpha}\right) +
\kr{c\to\beta_1}\left(1-\Phi^*_{\beta}\right)}{\kr{c\to\alpha_1} + \kr{c\to\beta_1} }\right)^{n-1}
\frac{\kr{c \to \alpha_1}}{\kr{c \to \alpha_1} + \kr{c \to \beta_1}}  \Phi^*_{\alpha}
\nonumber\\
&=& \frac{ \kr{c \to \alpha_1}\Phi^*_{\alpha} }
{ \kr{c\to\alpha_1}\Phi^*_{\alpha} + \kr{c\to\beta_1}\Phi^*_{\beta} }.
\end{eqnarray}
This yield provides the residence time on the acceptor,
$r_{\alpha A} = \Gamma_{\alpha}^{-1} \Phi_{\alpha}$. Based
on the flux condition, we can now achieve the residence times of all other sites, recursively.

This technique can be generalized to treat roots with more than two branches by the modification of
the factor in the geometric series of (\ref{eq:fluxfromlinflux}) to include the probability
 of returning from additional branches.

Note that this approach allows us to  find the yield of
systems suffering from losses, which could physically correspond to particle trapping in the
context of charge transfer of exciton recombination.
Back to Fig. \ref{FigA}, we include a lossy process  of rate $\Gamma_s$ at site $c$, by
 modifying the `branch' at that site,
 $\kr{c \to \alpha_1} = \Gamma_s$, $\kl{\alpha_1\to c} = 0$.

\begin{comment}

\subsection{Other Graphs}
From above it is possible to find yields and residence times
\subsubsection{Reducing graphs}
Take a graph with a group of connected states $\alpha = \left\{\alpha_1, \alpha_2, \dots \alpha_f \right\} $, which has at least one acceptor and
call the rest of the states $R = \left\{R_1, \dots , R_f \right\}$. The only connection between states of $\alpha$ and states of $R$ is between $A_1$ and $R_f$, with $\kr{R_f \to \alpha_1}$,
$\kl{\alpha_1 \to R_f}$ rates to and from $\alpha_1$ respectively. All the states of $\alpha$ can be replaced by a single
acceptor state
$A^*$ which has as rate to trap
\begin{eqnarray} \Gamma^* =  \left( \tau_m^\alpha \right)^{-1} \end{eqnarray} and rate back to $R$
\begin{eqnarray} \kl{A \to R}^* = \frac{r_{\alpha_1}}{\tau^\alpha_m } * \kl{\alpha_1 \to R_f} \end{eqnarray}

where $\tau_m^\alpha$  is the mean first-passage time
of the system when $\alpha_1$ as the initial state and
flow into $R$ is forbidden:  $\kl{\alpha_1 \to R_f} = 0$.
This process does not change mean residence time for any part
of R, and so it leaves all yields the same.

Proof: note that the rate matrix has the block form
\begin{eqnarray}
{\bf M} = \left(
\begin{array}{ccc|ccc}
&&&&& \\
&\mbox{\Huge $\bf R$}&&&\mbox{\Huge 0}& \\
&&&\kl{\alpha_1 \to R_f}&& \\ \hline
&&\kr{R_f \to \alpha_1}&&& \\
&\mbox{\Huge 0}&&&\mbox{\Huge \boldmath$\alpha$}& \\
&&&&& \\
\end{array}
\right)
\end{eqnarray}
where {\boldmath$\alpha$} is a rate matrix describing transitions between the states of $\alpha$, and ${\bf R}$ is the rate matrix for transitions in R.
The formula for the inversion of such $2\times 2$ block matrices gives that
\begin{eqnarray}
\label{eq:blockinv}
\left[ M^{-1} p_0 \right]_i =
\begin{cases}
\left[{\bf \widetilde R}^{-1} p_{R}\right]_i = -r_{\widetilde R_i}& i \in R \\
-\left[{\bf \widetilde R}^{-1} p_{R}\right]_{R_f} *
\left[{\bf \alpha}^{-1} p_{\alpha}\right]_i
= -r_{\widetilde R_f} * r_{\alpha_i} & i \in A
\nonumber\\
\end{cases}
\end{eqnarray}
where ${\bf \widetilde R}$ is a rate matrix similar to ${\bf R}$ except the $R_f$ state is an acceptor with rate to trap
\begin{eqnarray} \Gamma_{{\widetilde R}_f} = \frac{\kr{R_f\to \alpha_1}}{1 + \kl{\alpha_1\to R_f}r_{\alpha_1}} \end{eqnarray}
and $p_R$, $p_\alpha$ are initial condition vectors corresponding to the state $R_1, \alpha_1$ respectively.

Replacing $\alpha$ with $A^*$ leaves $\widetilde R$ unchanged:
\begin{eqnarray}
\Gamma^*_{\widetilde R_f^*} & =& (1 - \overleftarrow{k_{\alpha_1 \to R_f}^*} r_{A^*}) \kr{R_f \to \alpha_1}
= (1 - \frac{r_{\alpha_1}}{\langle \tau^\alpha_m \rangle} * \kl{\alpha_1 \to R_f}
* \langle \tau^\alpha_m \rangle ) \kr{R_f \to \alpha_1}
\nonumber\\
&=& (1 - \kl{\alpha_1 \to R_f}r_{\alpha_1}) \kr{R_f \to \alpha_1}  = \Gamma_{\widetilde R_f}
\end{eqnarray}
So no part of \ref{eq:blockinv} changes, and so the residence times at every
site in $R$ are the same. The yields to any part of $R$ and to $\alpha$ do not change under this reduction.

Example: for a chain of equal energy with $n$ intermediate sites. Take $\alpha$ to be all sites after and including site $i$.
Replacing these $n- i$ with $A^*$ which has $\Gamma^* = \left(\tau_m^A\right)^{-1} = \left((n - i)(n-i+1)\right)^{-1}$ and
$\kl{A^* \to R_f} = \frac{r_{\alpha_1}}{\tau_m^\alpha} = \frac{n-i+1}{(n-i)(n-i+1)} = (n-i)^{-1}$
does not change the residence times on $R_f$. This can be seen by examining the flux condition $\kr{R_f \to \alpha_1} r_{R_f} - \kl{\alpha_1 \to R_f} r_{\alpha_1} = 1$
and noticing that $\kl{\alpha_1 \to R_f} r_{\alpha_1} = n-i+1 = \kl{A^* \to R_f} * r_{A^*}$. The residence time on $R_f$ does not change. Recursively applying the flux condition means that residence times on the other sites in $R$ also do not change.

\subsubsection{Mean first-passage time}
Finding the total mean first-passage time complicated system can be done by reducing all branch points into a single state depth-first, until the system has only one branch point. Then the formula of the first subsection applies to find the total passage time. The reduction must be done depth first, because reducing a branch into a single state requires finding mean-first passage time for the region, which can only be done if it contains only one branch point.
\begin{figure}
\includegraphics{multiple-branch}
\caption{}
\label{treefigure}
\end{figure}

Example: The scheme shown in Figure \ref{treefigure} has three acceptors, shown by the arrows.
We take all the nearest-neighbors rates to be 1. The first step is to reduce the sites $\left\{\alpha_1, \alpha_2, \beta_1, \beta_2, c_5\right\}$
into a single acceptor state $c_5^A$, which requires finding $\tau_m^{c_5^A}$.
The relative yield to $\alpha_2$ and $\beta_2$ are both $\frac 12$. Therefore, following the formula for a system with a single branch point, $\tau_m^{c_5^A} = 5$. The rate to trap is $\frac 15$ and the rate to $c_4$ is $\frac 25$. The entire system is reduced to one with a single branch, and the total mean first-passage time can be computed.
\end{comment}

redpajama_set_name: RedPajamaArXiv


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Discussion in 'Chit Chat' started by jonford123, Oct 29, 2009.
Naked juice, Might Mango flavor.
For those uneducated on it, it's 100% juice, rather overpriced, and very tasty.
Coffee. It is morning here.
I m drinking a cup of coffee right now.
Last night at the beer bar. Sprecher Black Bavarian, Pauwel Kwak, Great Lakes Dortmunder Gold and Bell's Two Hearted Ale.
Chicken soup bouillon, its a mystery drink makes one wonder how the fuq they can produce this.


text:
Posted on August 19, 2009 in Uncategorized
Previously, Mick Upbraids Apathy…
Reprinted from:
http://cryptome.org/0001/nz-tax-charade.htm
I'm a long time Inland Revenue NZ employee of 20 years I have many frustrations with incompetent managers, absolute waste of taxpayer money, illiterate team leaders ( 'waist bin' in team minutes is not a typo, it's a spelling mistake), a workplace bully, Carol Buckenham, who has had extraordinary protection, even by my union delegates, failed payroll systems, failing tools, and last but not least a tax system which is designed to overdeduct tax, but which the IRD adverts sold as 'you never have to file a tax return again' allowing a company to set up a very succesful business giving people back the money the government hoped they wouldn't claim …
There's also penalising of low income taxpayers but almost no scrutiny of huge refunds going out the door — my job was to check refunds of over $100,000 and I was told on Friday to do this at the rate of 9 per hour, if that was the work given to me.
In this case it was me the payroll system failed to pay — twice — and they had no idea why and asked me to get a bank statement to prove I wasn't paid as they thought I had been. Complaints got me nowhere, requests for information got me nowhere and finally four official information requests (our FOIA equivalent) resulted in either no information, or on the last day an extension could be requested in a well-defined form, simply a statement that I'd be charged NZ$38 per half hour, without specifying what would be provided, when, or even whether they'd search all night for non-existent info.
Basically the New Zealand Inland Revenue Department believes it's above the law.
Our code of conduct states in the preamble that the actions of the department should be able to withstand the closest public scrutiny — however Inland Revenue doesn't open itself to any scrutiny. In an effort to draw attention to this and a major abuse of process …
… I drove my car through the Christchurch NZ Inland Revenue's closed front entrance, through two sets of glass doors, through the staff only door, then backed out into the foyer to wait for police.
You sure as hell wouldn't know it from Google or the papers. It was on Saturday morning and it's now Monday night.
I filmed an interview, across from the patched up hole, done by our local television station CTV. I popped in there today because I'm out on bail of NZ$500. In fact today they forgot to serve a tresspass order so I walked inside, showed my tag, and almost got access through to the staff only area. This is symptomatic of the place — I've dealt with stupidity by pointing it out when and where it occurs and can give you megabytes of examples because I just BCC'd myself in on the responses. Walking out with taxpayer information (which I haven't done — I have absolute integrity with regard to taxpayers) would be as simple as photographing the terminal the information is on.
Some of this is comic, some serious, but all worthy of public discussion. However the NZ Inland Revenue while driving staff harder for less, doesn't permit any discussion. Discussion at team meetings is cut off if the team leader can even understand it.
This all sounds surreal, but I've played music, my job supported that, and part of their action screwed with my music. So one group have added a webpage where we intend to put everything up relating to the crash.
A sample page is at:
https://theaxemen.wordpress.com/2009/08/17/mick-elborado-the-car-crash-set/
We're taking the piss and they know it. But Inland revenue have some pretty heavy lawyers — they've just upped the charge to Criminal damage which is 7 years maximum.
Can you add this to the Cryptome Collection? I assume then it would be archived on the DVD and any gag orders redundant. I did everything I could through the system. The checks and balances that should catch incompetence or bias failed. Also the checks that should stop Carol, the bully mentioned above from calling staff native to the country 'my little black jungle bunny' and then claiming it was a joke. In fromt of maybe 20 staff who all look the other way. She also tried to get me committed but my sanity hopefully is reasonably evidenced by the fact I'm on bail and not in jail.
It's late … I'm tired … far more of this if you want it, my stack of paer is about 6 feet high. I've been with the place 20 years and believe it's time the public scrutiny test was applied. I'll name names, give detailed analyses of abuses, failures, incompetence. If our media is too scared to deal with it I will pass it to you until I'm gagged. I've played this for laughs to keep sane and am proud of some of there responses and the fact they couldn't fire me, and it was I that "was the decider" as our HRSouls put it.
If my kinetic statement about 'moving forward' won't do it, I invite you to. I'll give you the information until I'm stopped, principally so the low income NZ taxpayers see that a tax system called fair by IRD certainly isn't t fair to them
Dave Theobald. (aka Mick Elborrado)
Posted on August 19, 2009 August 19, 2009 in 2009 happenings, cultural studies, government
Elborado Showdown – The Shit Hits The Fan pt 2
Show Your Support… join MICK ELBORADO IS INNOCENT now!
http://www.stuff.co.nz/dominion-post/news/2764914/IRD-drove-me-to-this
IRD drove me to this
By STACEY WOOD – The Dominion Post
TAXBREAK: David Theobald said he was prepared to accept the consequences after driving his car into an IRD building.
A taxman driven around the bend by continuing work woes drove his car through the building where he had worked for 25 years.
David Jerrold Theobald, 47, drove his Mazda 626 through the foyer of the Christchurch Inland Revenue building on Cashel St at 6.30am on Saturday after a three-year employment dispute.
He crashed through two sets of glass doors and smashed a third on the other side of the foyer before coming to a stop.
Mr Theobald said he had gone to great lengths to avoid risk to any staff.
"I checked with the cleaners the night before to make sure no-one would be in the building. I drove right up to the doors, looked both ways, then slowly pushed forward till the doors broke."
Mr Theobald said he was fed up with what he saw as concealment of workplace bullying and incompetent management at Inland Revenue.
"This has been going on for three years, and now I've got four official information requests in with them and they're making that as onerous as possible … This was just a way to make a gesture."
He also wanted to highlight how easy it would be for someone with terrorist intentions to ramraid the building with a truck full of explosives. "It's just another 9/11 waiting to happen."
Mr Theobald is a fixture of the Christchurch music scene. He is known as Mick Elborado when he plays with his band The Axemen and in other groups.
He appeared in Christchurch District Court yesterday charged with intentional damage and reckless driving and was remanded on bail till August 31.
He told The Dominion Post he had not decided how he would plead, but was prepared to accept the consequences of his actions.
Inland Revenue would not comment on the incident as it was before the court, but Mr Theobald has received a letter from human resources head Patrick Crawford.
In it, Mr Crawford said: "Information has come to my attention which indicates that you may have intentionally driven a car through Inland Revenue's Christchurch building … I am concerned that your conduct may be inconsistent with the Code of Conduct."
Mr Theobald could face up to seven years in prison if found guilty of criminal damage.
http://news.msn.co.nz/article.aspx?id=851637
'Bullied' taxman drives car through workplace
By MSN NZ staff
A Christchurch tax worker smashed his car through his own Inland Revenue office because he was fed up with "incompetent management and workplace bullying".
David Jerrold Theobald, 47, drove through two sets of glass doors and smashed a third at the Christchurch Inland Revenue building before coming to a stop.
Mr Theobald, a well-known musician in Christchurch who has worked for the IRD for 25 years, said he had been careful not to put lives in danger.
"I checked with the cleaners the night before to make sure no-one would be in the building," he told The Dominion Post.
"I drove right up to the doors, looked both ways, then slowly pushed forward till the doors broke."
Mr Theobald posted pictures of the damage on his blog , along with copies of legal documents he received from police and his employer following the incident.
One of the documents was a letter from Inland Revenue human resources head Patrick Crawford.
"Information has come to my attention which indicates that you may have intentionally driven a car through Inland Revenue's Christchurch building," Mr Crawford wrote.
"I am concerned that your conduct may be inconsistent with the Code of Conduct."
Mr Theobald said he was upset with what he saw as workplace bullying and incompetent management at Inland Revenue.
"This has been going on for three years and now I've got four official information requests in with them and they're making that as onerous as possible," he said.
"This was just a way to make a gesture."
He also said he wanted to highlight the potential for a terrorist attack if someone were to drive a car full of explosives into the building.
"It's just another 9/11 waiting to happen," he said.
Mr Theobald appeared in court yesterday. He has been charged with intentional damage and reckless driving and could face up to seven years in prison if found guilty of criminal damage.
http://keepingstock.blogspot.com/2009/08/taxman-cometh.html
The Taxman Cometh…
We had a bit of a chuckle when we read this story on Stuff this morning:
Doubtless there will be many of us who will at some point in our lives have been tempted, sorely tempted to lash out at the IRD. But the overwhelming majority manage to resist said temptation. And there's a certain delicious irony in an employee of the IRD accusing the department of workplace bullying when we taxpayers regularly feel bullied by the likes of Mr Theobald and his colleagues!
And mindful of Mr Theobald apparently being a serial litigant, we also chuckled when we read this:
The letter is extremely careful in its wording is it not? We would have thought that this was a pretty clear case of serious misconduct by way of the wilful destruction of departmental property; something that would normally warrant summary dismissal. Mr Theobald's concerns about workplace bullying could very soon be addressed – but not in the way he may have expected!!
ird mick Mick Elborado
Posted on August 19, 2009 August 19, 2009 in Uncategorized
Reprinted From The Christchurch Press, 19 August 2009
http://www.stuff.co.nz/the-press/news/christchurch/2764971/Its-OK-officer-I-work-here/
"It's OK Officer, I work here"
By IAN STEWARD – The Christchurch Press
DRIVE-IN: David Theobald poses for a self-portrait of himself with his car "parked'' inside IRD's Christchurch office.
A Christchurch tax worker fed up with his bosses has made a spectacular protest by driving his car through three plate-glass windows in the Inland Revenue building.
David Theobald, 47, a musician and 25-year Inland Revenue employee, posted photographs on the internet of his protest, which he made at 6.30am on Saturday. The pictures showed his car "parked" in the Cashel St building.
A blog accompanying the photos said Theobald, a long-time member of rock and punk band The Axemen, "decided he had had enough and wasn't going to take it any more after years of abuse" from Inland Revenue.
"Driving into the place that had been his nemesis for the past 25 years, he felt an eerie calm as he took the final turnoff, revved the vehicle and drove through three plate-glass windows and into the reception area," it said.
Theobald said he was careful not to trample broken glass into the department's new carpet because "I didn't want to get in trouble".
He waited "calmly" for police and quipped when they arrived: "It's OK officer, I work here."
Theobald appeared in the Christchurch District Court yesterday charged with intentional damage and reckless driving.
He was re-arrested and his bail conditions were altered, ordering him to stay away from the Inland Revenue building, after he stood outside and sent text messages to his workmates "to show them I'm not in the loony bin".
Theobald is known in music circles as Mick Elborado. He has played in several bands since the early 1980s.
A picture on the blogsite of Theobald and his Mazda 626 was captioned: "Mick at grand opening of Inland Revenue's new 24hr drive-thru in Chch."
An Inland Revenue spokeswoman yesterday confirmed the incident and said no-one had been hurt. The damage "did not compromise activities", she said.
She declined to comment further.
Theobald posted the letter suspending him from work, written by a human resources manager, on the internet.
"Information has come to my attention which indicates that you may have intentionally driven a vehicle through Inland Revenue's Christchurch building," the manager wrote.
Theobald said his actions were the culmination of a three-year employment dispute centred on "workplace bullying".
Theobald said he believed destruction of property was a last resort after all other checks and balances had failed. "I take all responsibility for these actions. You've got an evangelist on your hands here," he said.
A founding member of The Axemen, Steve McCabe, began a Facebook support group for Theobald titled "Mick Elborado is innocent". McCabe said Theobald was innocent because he had been provoked into the act by years of bureaucratic and institutional bullying.
Reprinted from NZPA, 19 August 2009
http://nz.news.yahoo.com/a/-/mp/5836958/man-in-court-after-driving-into-ird-building/
Man in court after driving into IRD building
NZPA, NZPA August 19, 2009, 7:26 am
A disgruntled Inland Revenue employee who intentionally drove through the organisation's Christchurch building has appeared in court.
David Jerrold Theobald, 47, appeared in Christchurch District Court yesterday charged with intentional damage and reckless driving, after driving into the building at 6.30am on Saturday.
The car crashed through two sets of glass doors and smashed a third on the other side of the foyer before coming to a stop.
He was remanded on bail to reappear on August 31.
Theobald, who had worked at IRD for 25 years, told the Dominion Post the incident followed a three-year employment dispute.
He said he was fed up with what he saw as concealment of workplace bullying and incompetent management at Inland Revenue.
Inland Revenue would not comment on the incident, but Theobald said he had received a letter from IRD's human resources saying: "Information has come to my attention which indicates that you may have intentionally driven a car through Inland Revenue's Christchurch building … I am concerned that your conduct may be inconsistent with the Code of Conduct."
David Theobold ird mick Mick Elborado
Posted on August 17, 2009 August 19, 2009 in 2009 happenings, government, law, News, NZ bands
Mick Elborado & the Car Crash Set
Click Pics for enlarged image…
Posted on August 17, 2009 August 19, 2009 in 2009 happenings, government, law, Uncategorized
A statement from Mick – Monday 17 August 2009
ask the same of anyone awake or woken up.
1: appear in court and behave
2: walk to the crime scene gathering a motley crowd of the slightly curious hoping for more action.
3: make a very short speech calling upon tony mccone senior human resources officer to resign within 24 hours if he believes the actions of ir human resources staff will not withstand the main code of conduct test of 'the closest public scrutiny'
4: advise that because of the results of his incompetence i require payment in front of ir at this time tomorrow.
5: advise that if he does not resign and he's found to have sanctioned any actions which arouse public disgust it's 10 times his salary or everything he owns except his home.
6: give the commisioner of inland revenue one week to hear whether all actions can withstand public scrutiny – if not the same thing.
7: the next steps escalate to ir shifting all its banking to Kiwibankor another bank with nz ownership no matter what the cost and if they work theres leverage to change the world. You all helped!
I bet this is the first plan for world undomination transmitted by txt, facebook and hopefully youtube, if anyone films it
– let's see anyone shut down fcbk youtube and txting in one week.
If this crazy strategy works we all work 3 day weeks and live better, all meaning NZ and the world – if it gets shut down then it was already too late, but then any opposition will be visible.
All the planets are in line and they won't be again – forget scrabble and soaps – just for a day or so.
Huge thanks to all:
special mention to Amy, Ange, Alex, Ros, Steve, Joanne, Peter Hall-Jones, my union, and Emma Goldman.
Posted on August 16, 2009 August 19, 2009 in bile, cultural studies, government, News, Photograsps
Elborados Fury – Drive In Saturday
Mick at Grand Opening of Inland Revenue's new 24 hr drive thru in Chch
On Saturday August 16, 2009, Axemen Taxman, grunty bass player and disgruntled employee Mick Elborado decided he had had enough and wasn't going to take it any more after years of abuse from the IRD.
Driving into the place that had been his nemesis for the past 25 years he felt an eery calm as he took the final turnoff, revved the vehicle and drove through 3 plate glass windows and into the reception area, then stepped from the car taking care not to trample the glass into the new carpet – "I didn't want to get in trouble" – and calmly waited for the police to arrive.
Te Tari Taake or Te Tari Turkey?
On their arrival he quipped "Its OK officer, I work here!" as they cuffed him and hauled him away.
Its OK Officer I work here!
When asked for a quote about the split he cited musical differences and wanting to spend more time on his private projects – "My dentistry business is really picking up!" he chirpily quoted to one reporter.
It was fairly obvious robbery was not the motive, in fact as with many people that come through these doors he later noted he later checked his wallet and realised he had left with $150 less than what he came in with. "Even after all these years I still don't know how they do that" he noted, shaking his head.
Mick doing the hard yards awaiting bail
After 6 hours in the infamous Sydenham Prison he was put out on the street on bail with his brother Des.
We will be following his case with interest and keeping you posted.
Post: Amble Testicle Evict, with Micks Blessings
Posted on August 11, 2009 August 11, 2009 in 2009 happenings, Audio, Buses, cultural studies, Gigs, NZ bands, Photograsps, poetry, video
What about these guys??? the Puddle Live-to-air at Radio Volcano, sweet as, direct from another universe
the puddle, live to air, radio volcano, lyttelton, one romantic gesture
and this one with different scenery – needs more mick though if you ask me….
The Puddle - better scenery in this vid... wish george the best after his stroke - oops sorry that was that other guy

redpajama_set_name: RedPajamaCommonCrawl


text:
Ripley Snell's 6 Natural Flavors 4.23.14
In the five months since his November release of Fall Denim, Ripley Snell has been busy working on collaborations and clothing lines, and is already putting out his sophomore album, Natural Flavors, which is a six-course menu of imaginative lyricism. He draws inspiration from be-bop, hiphop and R&B, funk, metaphysics, and his Portland surroundings.
The titles of each track on 6 Natural Flavors are named after the often unnatural and additive ingredients listed on the back of a Coca-Cola. Tied with Snell's lyrical content, the album serves as a metaphor for the sugarcoating of political and socioeconomic problems which have grown over time into the everyday norm. The album is said to be inspired by and dedicated to the individuals that live on W. 3rd Ave in downtown Portland.
Speaking his mind and keeping it real, Ripley Snell is a storyteller giving insight on Portland nights, homelessness, and racism with candor. Jazz infused "Caramel Colors" speaks on "gentri-fried chicken" and feeling isolated from the Portland Spirit because of his style. "Phosphoric Acid" has whirling string loops, primal urges, and some hilarious personal hygiene disclosures. "Caffeine" begins with a sub octave vocal sample of the words "easy beats" then quickly gets to tackling the topics of STD's and dying on the street. "High Fructose Corn Syrup" begins with some good advice: "Let's do things well... let's do 'em well," then jumps into a flow of colorful metaphors about Monsanto's lucid dreams and hanging out with fat polar bears on frozen lakes "drinking blue chip shares." While Snell is getting lyrically deep inside issues, Neill Von Tally's beats throughout the album feature jazzy and playful instrumentation and vintage audio sampling including everything from the triangle bell to string trios and brassy melodies.
Ripley Snell will celebrate the release of Natural Flavors by performing the album in it's entirety with a public all-ages show at the NW Portland art gallery and clothing store Upper Playground at 7pm sharp. There will be access to a free download of the album at Ripleysnell.com for a limited time when the album drops April 23rd. In the meantime, check out his album Fall Denim.
- Brandy Crowe
Ripley Snell
This Emerging Artist is based in Portland,
for our Portland Artist of the Month poll below!
Portland New Bands With Buzz
Explore The Deli Portland Music Charts!


text:
Q: Create generic function - Official MongoDB Driver when using the mgo driver from globalsign, i could save some time reusing a function to return all the elements from a collection, no matter what models i was using.
But now, with the Official Driver from MongoDB, i need to specific which interface I want to decode, so in this way I can't reuse this method for other interfaces.
Has anyone got to this point?
Function using the mgo driver:
func ReturnAll(collection string, model interface{}, skip int, limit int) error {
 session := GetSession()
 defer session.Close()
 return session.DB(DBName).C(collection).Find(nil).Skip(skip).Limit(limit).All(modelo)
}

A: Use Cursor.All in version >= 1.1.0 of the driver:
var result []Example
err := cursor.All(&result)
if err != nil {
    // handle error
}

For earlier versions,
use the reflect package to decode all values to a slice:
// decodeAll decodes all values to the slice pointed to by result.
func decodeAll(cur *mongo.Cursor, result interface{}) error {
    rv := reflect.ValueOf(result).Elem()

    // reset to beginning of the slice.
    sv := rv.Slice(0, rv.Cap())

    for cur.Next(context.Background()) {

        // Allocate new element value and decode to it.
        pev := reflect.New(sv.Type().Elem())
        if err := cur.Decode(pev.Interface()); err != nil {
            return err
        }

        // Append the element value.
        sv = reflect.Append(sv, pev.Elem())
    }

    rv.Set(sv)
    return cur.Err()
}

Call it like this:
var result []Example
err := decodeAll(cursor, &result)
if err != nil {
    // handle error
}

redpajama_set_name: RedPajamaStackExchange


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text:
Venture Class Launch Services
Virgin's satellite launcher reaches orbit for first time
An air-launched rocket built by Richard Branson's Virgin Orbit reached orbit Sunday for the first time, delivering 10 experimental CubeSats for NASA and positioning the company for the start of commercial operations.
NASA takes a chance on Virgin Orbit with company's second test launch
On its second launch attempt after a propulsion problem prevented its light-class air-dropped rocket from reaching space last year, Virgin Orbit will try to deliver 10 experimental CubeSats into orbit as soon as Sunday on a mission booked by NASA five years ago for $4.7 million.
Smallsat launch providers readying for first missions of 2021
Virgin Orbit and Rocket Lab teams are gearing up for their first missions of the year in the coming days, with Virgin's air-launched rocket set for its second demonstration flight and Rocket Lab's Electron booster poised to launch a small German-owned communications satellite.
NASA, Rocket Lab partner on successful satellite launch from New Zealand
Rocket Lab's commercial Electron booster fired into orbit from New Zealand on Sunday, carrying a flock of 13 CubeSats on the company's first mission chartered by NASA, and closing out a landmark year for the new smallsat launch provider as Rocket Lab aims to grow its flight rate to at least one per month in 2019.
Mission timeline for Rocket Lab's launch of NASA's ELaNa-19 CubeSats
Rocket Lab's light-class Electron launcher is set to take off on its fourth flight from New Zealand, aiming to deliver 13 CubeSats to orbit on the company's first mission for NASA.
Live coverage: Rocket Lab's Electron rocket launches on NASA mission
Rocket Lab's Electron rocket, a commercially-developed light-class booster, lifted off at 1:33 a.m. EST (0633 GMT) Sunday, carrying 13 CubeSats into orbit under contract to NASA.
Rocket Lab preps to launch flock of NASA-funded CubeSats
Rocket Lab's Electron rocket is set to launch 13 CubeSats Saturday night (U.S. time) under contract to NASA from the company's privately-owned spaceport in New Zealand, a pathfinder mission that could pave the way for more NASA satellites to hitch a ride on future Electron flights.
Rocket Lab sets new target date for first operational launch
Rocket Lab announced this week it has added two more nanosatellites to the first operational launch of the company's Electron rocket, which is now scheduled to take off from New Zealand no earlier than Nov. 10, U.S. time, after a multi-month delay to resolve a technical problem.
NASA to fly CubeSats on three new commercial launchers
NASA announced Wednesday it is joining a chorus of companies pushing for easier access to space, awarding $17.1 million in contracts to three firms developing new launchers tailored for lifting lightweight satellites into orbit.