\section{Introduction}
Recent work suggests that Loop Quantum Gravity (LQG) may be capable of resolving the singularities that are inevitable in classical general relativity. Because of the inherent difficulty in solving the complete system, the focus has been
on dimensionally reduced, mini-superspace models of quantum cosmology \cite{lqc} and  spherically symmetric black hole spacetimes \cite{Ashtekar05, modesto06, boehmer07, pullin08}. As observed in \cite{Ashtekar05} and \cite{modesto06}, for example, singularity avoidance in these two cases is deeply connected because the interior Schwarzschild spacetime coincides with the homogeneous anistropic Kantowski-Sachs cosmology.  While significant progress has been made, there is as yet no clear evidence as to which theory of quantum gravity will ultimately be proven correct, LQG, string theory, or perhaps something else.  Nor is there, to the best of our knowledge, a rigorous and unambiguous path from the full loop quantum gravity theory to the quantization techniques used in the mini-superspace models. One particularly fruitful technique that has been used recently to great effect \cite{boehmer07, pullin08} is a semi-classical polymerization that preserves aspects of the underlying discreteness of spacetime suggested by LQG but considers the limit in which quantum effects are vanishingly small. Different polymerizations can give qualitatively different regularized spacetimes, so that it is of great interest to examine more fully  a wider class of models and methods.

In the following, we describe quantum corrections that arise from the semi-classical polymerization of the interior of generic black holes in a family of theories known collectively as generic 2-D dilaton gravity. Of prime importance for the present work is that this family includes spherically symmetric black holes in spacetime dimension three or higher. We investigate two different polymerization schemes, and show that the results differ qualitatively: in one case the resulting non-singular spacetime generically has only a single horizon while in the second there are multiple horizons.
In our considerations, the polymerization scale is taken to be a constant but we note that in the context of loop quantum cosmology, as well as in some black hole scenarios \cite{nelson}, a dynamical polymerization scale has been considered.

Our key result is the analytic solution of the semi-classical equations that is obtained in 4-D when only area is polymerized. This solution can be extended analytically to a complete non-singular spacetime with only a single horizon. This has the advantage over other candidates for loop quantum corrected black holes \cite{boehmer07, pullin08} of avoiding the problem of mass inflation \cite{mass_inflation} normally associated with inner horizons.  The exterior has non-zero quantum stress energy but closely approximates the classical spacetime for macroscopic black holes. There are two interior regions, one in the past and one in the future. Both exhibit a bounce at a microscopic scale and then asymptote (one in the infinite past and the other in the infinite future) to a non-singular product Kantowski-Sachs \cite{kantowski-sachs} type cosmological spacetime containing an anisotropic fluid, with product topology of a spacelike R and an expanding 2+1 positive curvature FRW cosmology. The polymer dynamics thus drives the system into an asymptotic interior end-state that is not a small correction to the classical spacetime. In the limit that the polymerization scale goes to zero, the interior cosmological regions ``pinch off'' leaving behind the standard singular Schwarzschild interior. The complete, non-singular semi-classical spacetime is suggestive of past proposals for ``universe creation'' inside black holes \cite{frolov90}.

\section{Classical Theory}
Our formalism  begins with the most general (up to point reparametrizations) $1+1$-dimensional, second order, diffeomorphism
invariant action that can be built from a 2-metric $g_{\mu \nu}$ and a scalar $\phi$ \cite{gru,dil1}:
\begin{equation} \label{eq:act} S[g,\phi ] = \frac{1}{2G}\int d^2x \sqrt{-g}\, \bigg( \phi R(g)
	+ \frac{V(\phi)}{l^2}\bigg),
\end{equation}
where $l$ is a positive constant with
a dimension of length and $G$ is the dimensionless two-dimensional Newton's constant. This action provides a convenient representation of spherically symmetric black hole spacetimes in $D\equiv n+2$ dimensions. It obeys a generalized Birkhoff theorem \cite{dil1} with general solution:
\begin{equation} \label{eq:ds} ds^2 = -\big[ 2lGM-j(\phi )\big]^{-1}l^2 d\phi^2+\big[ 2lGM-j(\phi )\big]dx^2.
\end{equation}
where $j(\phi )$ satisfies $dj/d\phi=V(\phi )$. For our purposes, it is convenient to assume that $j(\phi)\rightarrow 0$ when \mbox{$\phi\rightarrow 0$}. The integration constant $M$ is the Arnowitt-Deser-Misner (ADM) mass, and we take $M>0$.
For monotonic functions $j(\phi )$ the solution contains precisely one Killing horizon \cite{dil1} at $\phi_\text{H}$, such that $j(\phi_\text{H}) = 2lGM$. The metric (\ref{eq:ds}) and dilaton can be related to a spherically symmetric metric in $D$ dimensions as follows:
\begin{equation}
ds^2_\text{phys}= \frac{ds^2}{j(\phi )} + r^2(\phi) d\Omega^2_{(D-2)},
\label{eq:physical_metric}
\end{equation}
where $d\Omega^2_{(D-2)}$ is the line element of the unit $(D-2)$-sphere. Eq. (\ref{eq:physical_metric}) corresponds in particular to the interior of the 4-D Schwarzschild solution with the identifications $2Gl^2=G^{\text{(4)}}$, $V=1/(2\sqrt{\phi})$ and $\phi=r^2/(4l^2)$.

In order to address the question of singularity resolution in the semi-classical polymerized theory, we restrict to homogeneous slices with metric parametrization
\begin{equation} \label{eq:adm}
ds^2 = e^{2\rho(t)}\big( -\sigma^2(t)dt^2+dx^2\big).
\end{equation}
After suppressing an irrelevant integration over the spatial coordinate, the resulting action is that of a parametrized system:
\begin{equation} I=\frac{1}{2G}\int dt\Big(\Pi_\rho\dot{\rho}+\Pi_\phi\dot{\phi} +\sigma \mathcal{G}\Big),
\label{eq:action}
\end{equation}
where a dot denotes a derivative with respect to the time coordinate $t$ and the (Hamiltonian) constraint is
\begin{equation}
\mathcal{G}=G\Pi_\rho \Pi_\phi +e^{2\rho} \frac{V}{2l^2G}\approx 0.
\label{eq:constraint}
\end{equation}
In spherically symmetric 4-D spacetime, this Hamiltonian can be converted to the LQG Hamiltonian of \cite{pullin08} by a simple point canonical transformation.

The simplicity of the Hamiltonian in (\ref{eq:constraint}) makes it rather straightforward to find analytic solutions for the components of the physical metric. These solutions depend on two parameters, the ADM mass $M$, and its canonical conjugate $P_M$. In the full (inhomogeneous) spherically symmetric theory the latter corresponds to the Schwarzschild time separation of spatial slices \cite{kuchar94}. In the present case, the arbitrariness of $P_M$ represents the residual invariance of the theory under rescalings of the Schwarzschild ``time'' coordinate $x$.

\section{Semi-Classical Polymer Approach}
In the polymer representation of quantum mechanics \cite{ash,halvorson} one effectively studies
the Hamiltonian dynamics on a discrete spatial lattice.
In many cases the full polymer theory is rather challenging to analyze
but fortunately one can get interesting results  by investigating the
semi-classical limit of the theory, which corresponds formally to the limit in which
quantum effects are small ($\hbar\to 0$), but the polymerization scale $\mu$ stays finite.
This approximation is the basis for recent analyses of black hole interiors \cite{boehmer07,pullin08}. It can be derived  by studying the action of the fully quantized operators on coherent states and expanding in the width of the states \cite{husain:semiclass,ding08}. The end result is to simply replace the classical momentum variable $p$ in the classical Hamiltonian function by $\sin (\mu p)/\mu$. After the replacement, one studies the \hbox{(semi-)classical} dynamics of the resulting polymer Hamiltonian by means of standard techniques. We assume in the following that this effective description is valid without further state-dependent corrections, at least for states that approach semi-classical behaviour at scales large compared to $\mu$.

We first polymerize only the generalized area variable $\phi$. While this may seem {\it ad hoc}, it is perhaps not unreasonable to introduce a fundamental discreteness for the geometrical variable that corresponds to area in the spherically symmetric theory while leaving the coordinate dependent conformal mode of the metric continuous.  This approach is motivated by (but not derived from) full LQG, where it is the area operator that is naturally discretised. For completeness we will subsequently illustrate the result of polymerizing both variables. Details of both polymerizations will be presented elsewhere \cite{AK:prep}. Note that we choose to work with a state-independent (constant) polymerization scale, despite the fact that in the context of loop quantum cosmology consistency with predictions requires a state-dependent discreteness scale\cite{lqc}. Here we choose the simplest approach that produces reasonable semi-classical behaviour and leave the study of state-dependent polymerization scales for future research.

The partially polymerized Hamiltonian constraint is:
\begin{equation} \label{eq:polyH}\mathcal{G} =  G \Pi_\rho\frac{\sin (\mu\Pi_\phi)}{\mu}
  + e^{2\rho}\frac{V(\phi)}{2l^2G}\approx 0.
\end{equation}
In this equation $\mu$ has a dimension of length, which means that $\phi$ has a discrete polymer structure with edge length of $\mu/l$. The essence of the  singularity resolution mechanism is evident from the equation of motion for $\phi$:
\begin{eqnarray}
  \frac{\dot{\phi}}{G\sigma}&=& -\Pi_\rho \cos(\mu\Pi_\phi).
\label{eq:phidot}
\end{eqnarray}
$\dot{\phi}$ now vanishes at two turning points: the ``classical'' turning point  $\Pi_\rho=0$ and semi-classical turning point: $\cos(\mu\Pi_\phi)=0$. The former condition turns out to be satisfied at the horizon as expected while the latter occurs first at a microscopic scale proportional to $\mu$.

The solution to the corresponding H-J equation is:
\begin{equation} \label{eq:HJF} S=-\frac{\alpha}{4G}\, e^{2\rho}+\frac{l}{\mu}\int \arcsin
	\bigg( \frac{\mu V}{\alpha lG}\bigg)\, d\phi +C,
\end{equation}
where $\alpha$ and $C$ are constants.  For convenience, we shall take $\alpha >0$.
The expression for $\Pi_\phi$ now reads:
\begin{equation}
\label{eq:pi1}\Pi_\phi =\frac{1}{l}\frac{\partial S}{\partial \phi } =
	\frac{1}{\mu}\arcsin \bigg( \frac{\mu V}{\alpha lG}\bigg).
\end{equation}
We concentrate henceforth on spherically symmetric gravity for which $V\propto\phi^{-1/n}$. In this case $\phi$ has an minimum given by $\mu V(\phi_{\text{min}})=\alpha lG$. Note that $\phi_\text{min}$ is located at the roots of the cosine function as expected from (\ref{eq:phidot}).

To find the solutions, we  require:
\begin{equation} \label{eq:beta2} \frac{\partial S}{\partial\alpha}=-\beta,
\end{equation}
where $\beta$ is again a constant of motion that is conjugate to $\alpha$. The constants $\beta$ and $\alpha$ are related to the usual canonical pair $(M,P_M)$ by a simple canonical transformation. Eq. (\ref{eq:beta2}) yields a solution for $e^{2\rho}$ in terms of $\phi$ and the two constants $\alpha$ and $\beta$ that are determined by initial conditions. The solution is:
\begin{equation} \label{eq:rho soln} \frac{1}{4G}\, e^{2\rho}+\frac{l}{\mu} I^{(n)}(\phi ) = \beta ,
\end{equation}
where
\begin{eqnarray}
I^{(n)}&:=& \frac{\epsilon}{\alpha}\int\frac{V}{\sqrt{\left(\alpha lG/\mu\right)^2-V^2}}\,d\phi \, .
\label{eq:I}
\end{eqnarray}
In (\ref{eq:I}), the value of $\epsilon=\pm 1$ depends on a branch of $\mu\Pi_\phi$. The upper sign is valid in the branches where the cosine function is positive, which include the principal branch $(-\pi/2,\pi/2)$, whereas the lower sign is used elsewhere.

One now has a complete solution in terms of a single arbitrary function that must be fixed by specifying a time variable. It is illustrative to write the physical metric using $\phi$ as a coordinate, which corresponds to the area of the throat in the interior of the extended Schwarzschild solution of the unpolymerized theory. Suppressing the angular part, one obtains
\begin{equation} ds^2_\text{phys}=\frac{1}{j(\phi)}\Big(\frac{-4l^2d\phi^2}{\alpha^2
e^{2\rho}[1-(\mu V/(\alpha lG))^2]}+e^{2\rho}dx^2\Big),
\end{equation}
which shows that the solution has a horizon at $\phi=\phi_{\text{H}}$ for which $e^{2\rho}=0$.

Taking $\Pi_\phi$ as the time variable, one can deduce, generically, the following qualitative time evolution. One starts at the horizon $\phi=\phi_\text{H}$ , with corresponding initial $\Pi_\phi^\text{H}$ given by (\ref{eq:pi1}). Without loss of generality, we can take $\mu\Pi_\phi^\text{H}$ to be in the principle branch of the arcsin function, and because $\phi$ is positive, it takes values between $(0,\pi/2)$. As $\mu\Pi_\phi$ increases, $\phi$ decreases until it reaches its minimum value at $\mu\Pi_\phi =\pi/2$. After that $\phi$ increases as $\Pi_\phi$ increases. When $\mu\Pi_\phi$ is in the range $(\pi/2, \pi)$, $\epsilon$ in (\ref{eq:I}) necessarily changes sign, and one can verify that after the bounce $e^{2\rho}$ does not vanish again, but the throat area expands to $\phi\to\infty$ in finite coordinate time $\Pi_\phi$. The expansion takes an infinite amount of proper time so that our semi-classical polymerization has produced a solution that avoids the singularity but does not oscillate. The time evolution of the physical conformal mode, $e^{2\rho}/j(\phi)$, is illustrated in Fig. \ref{fig:bounce}.

\begin{figure}[htb!]
\begin{center}
\includegraphics{4Dbounce2.eps}
\caption{The physical conformal mode in 4-D spacetime. The calculations use $\alpha=G^\text{(4)}=M=1$ and $\mu=0.1$.}
\label{fig:bounce}
\end{center}
\end{figure}

We now contrast the above behavior with that of the fully polymerized theory. The Hamiltonian constraint is:
\begin{equation} \label{eq:fullyH}\mathcal{G} =  G \frac{\sin(\mu\Pi_\rho)}{\mu}\frac{\sin (\mu\Pi_\phi)}{\mu}
  + e^{2\rho}\frac{V}{2l^2G}\approx 0.
\end{equation}
Going through precisely the same Hamilton-Jacobi analysis as before, we are lead to the following expression for the
conformal mode of the metric:
\begin{equation}
e^{2\rho}= \frac{2lG}{\alpha\mu}\sin\big( 2\mu\alpha\beta/l-2\alpha I^{(n)}(\phi)\big),
\label{eq:fullrho}
\end{equation}
where $I^{(n)}$ is given in Eq. (\ref{eq:I}). All other $\phi$-dependence is unchanged so that the net change from the partially polymerized case is that the conformal mode is now an oscillating function of $\phi$. There will now be inner horizons whenever the sin function vanishes, giving rise to a qualitatively different quantum corrected spacetime. In fact the number of horizons  will vary depending on the relative magnitude of $M$ and the quantity $\mu /\alpha$ \cite{AK:prep}.

\section{Four-Dimensional Schwarzschild Black Hole}
We now write the 4-D partially polymerized metric in terms of the radius $r$ and the Schwarzschild ``time'' $x$:
\begin{eqnarray}
\!\!\!\!\!\!\!\!\!\!\!ds^2_\text{phys}\!\!&=&\! -\frac{dr^2}{\Big(\frac{2MG^{(4)}}{r} - \epsilon\sqrt{1-\frac{k^2}{r^2}}\Big)\Big(1-\frac{k^2}{r^2}\Big)} \nonumber\\
& &+  \bigg(\!\frac{2MG^{(4)}}{r} - \epsilon\sqrt{1-\frac{k^2}{r^2}}\,\bigg)\!\bigg(\!\frac{2dx}{\alpha}\!\bigg)^{\!\!\!2} \!+\!r^2d\Omega^2.
\label{eq:four_metric}
\end{eqnarray}
In the above, $M \equiv\alpha^2 \beta/(2l)$ and $k\equiv \mu/(\alpha G)$.  As per our earlier claim, $P_M=2l/\alpha$ completes the canonical transformation from the pair $(\alpha,\beta)$  to $(M,P_M)$, and hence the conjugate $P_M$ does indeed rescale the $x$ coordinate. These rescalings do affect the bounce radius $k$ which is a consequence of the fact that the introduction of the discrete scale has broken the scale invariance of the theory.

The metric (\ref{eq:four_metric}) has remarkable properties. There is a single bifurcative horizon at  $r_\text{H} := \sqrt{(2MG^{(4)})^2+k^2}$ which exhibits a quantum correction due to the polymerization. The solution evolves from the horizon at $r_\text{H}$ to the minimum radius $k$ in finite proper time, and then expands to $r=\infty$ in infinite proper time. As the throat expands in the interior, the metric approaches:
\begin{equation}
ds^2_\text{phys} = -\frac{dr^2}{1+2MG^{(4)}/r}+\Big(1+\frac{2MG^{(4)}}{r}\Big)dx^2,
\label{eq:asympt_interior}
\end{equation}
where we have absorbed $2/\alpha$ into $x$ and suppressed the angles. This asymptotic interior solution does not obey the vacuum Einstein equations, but has non-vanishing stress tensor with $T^{\,r}_r = T^{\,x}_x \propto -1/r^2$ which does not depend on $k$. This corresponds to an anisotropic perfect fluid that has been recently considered in a model of the Schwarzschild interior \cite{culetu}.

It is possible to continue the metric (\ref{eq:four_metric}) analytically across the horizon to the exterior region. The validity of this extension is an open question given that our chosen foliation does not extend to the exterior, but the procedure seems natural in the present context and has been used before to construct a complete semi-classical black hole spacetime.  In the present case, the resulting black hole exterior has interesting and physically reasonable properties: in particular, it is asymptotically flat and closely approximates a Schwarzschild black hole of mass $M$.

The fact that $r=k$ in (\ref{eq:four_metric}) is a coordinate singularity can be explicitly illustrated by the coordinate transformation $r=k\cosh(y)$ \cite{ftnote}. The resulting metric is regular at the bounce $y=0$ and has a horizon at $\sinh(y_\text{H})=2MG^{(4)}/k$. For large $y$, the metric describes the exterior of the black hole, while the asymptotic interior region corresponds to $y\to -\infty$. The conformal diagram of the complete spacetime is shown in Fig. \ref{fig:conformal}. It includes two exterior regions (I and I'), the black hole and the white hole interior regions (II and II'), and two ``quantum corrected'' interior regions (III and III'). The classical singularity is replaced by a bounce at $r=r_\text{min}$ and subsequent expansion to $r=\infty$.

The Ricci and Kretschmann scalars are everywhere non-singular and vanish rapidly in the exterior (large, positive $y$).
A calculation of the Einstein tensor reveals that while the solutions violate the classical energy conditions, the violations are of order $k^2/r^4$ and hence vanish far from the bounce radius $r=k$. This means in particular that the exterior is endowed with non-zero quantum stress energy that is vanishingly small for macroscopic black holes ($r_\text{H}\gg k$) so that the Schwarzschild solution is well approximated everywhere outside the horizon. This also leads naturally to a quantum corrected Hawking temperature of order $\mathcal{O}(k^2/M^3)$ as well as a logarithmic correction to the Bekenstein-Hawking entropy \cite{AK:prep}.

The interior spacetime on the other hand describes an expanding anisotropic cosmology with stress energy that approximately satisfies the energy conditions but does not vanish far from $k$, nor does it vanish in the limit that $k \to 0$. Instead, the asymptotic region ``pinches off'' in this limit at the curvature singularity at $r=0$, leaving behind the standard, complete but singular Schwarzschild spacetime and two disconnected, time-reversed copies of the (singular) cosmological spacetime.

\begin{figure}[htb!]
\begin{center}
\includegraphics{ConfDiag2.eps}
\caption{Conformal diagram of the partially polymerized Schwarzschild spacetime. }
\label{fig:conformal}
\end{center}
\end{figure}

\section{Conclusion}
We have presented an intriguing candidate for a complete, non-singular quantum corrected black hole spacetime. This spacetime was derived by the semi-classical polymerization of only the area in the interior of spherically symmetric black hole spacetimes. Our treatment neglects polymeric corrections to the conformal mode of the metric, but this somewhat speculative procedure is perhaps justified by the interesting results that emerge. The singularity is resolved at a bounce radius determined by the polymerization scale and the exterior black hole spacetime has small, but non-zero quantum-energy. Remarkably, the solution in the interior does not oscillate, but instead re-expands indefinitely to a Kantowski-Sachs spacetime with anisotropic fluid stress-energy that is non-vanishing in the limit that the polymerization scale vanishes. The generation via polymerization of an interior cosmology is reminiscent of earlier work that explored universe creation in black hole interiors \cite{frolov90}.

One may also note that the solution does not reduce to Minkowski space in the limit that the Schwarzschild mass $M$ goes to zero. In fact, Eq. (\ref{eq:four_metric}) shows that there is still a horizon in this limit, located at the bounce radius $r=r_\text{min}$. While it is tempting to speculate about quantum remnants, it must be remembered that the semi-classical approximation employed here will likely break down for microscopic black holes. This is certainly worthy of further investigation.

{\bf Note Added:} After this paper was completed, a paper by Modesto \cite{modesto08} appeared which presents an interesting and thorough analysis of the complete LQG corrected, multiple horizon 4-D black hole spacetimes that emerge from a generalization of the procedure in \cite{pullin08}.

\section*{Acknowledgements}

We thank J. Ziprick, R. Daghigh, J. Gegenberg, J. Louko, the theory group at CECS and H. Maeda in particular for useful discussions.  GK acknowledges the kind hospitality of the University of Nottingham, the University of New Brunswick and CECS where parts of this work were carried out. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada.


\section{Introduction}

A Brownian particle moving in a periodic potential and subected
to a spatially non-uniform temperature profile gives rise
to a net current, acting like a Brownian motor. This device
is autonomous since it is entirely driven by thermal fluctations.
Several properties of such a Brownian motor, such as current,
heat and thermodynamic eficiency have been studied.

Another important performance characteristic of such a Brownian
device is the transport coherence as measured by the Peclet
number, which is the ratio of the thermal velocity times the
period of the substrate potential and the effective diffusion
coefficient of the motor. It is desirable to design motors
which produce the maximum velocity with the minimum dispersion
i.e. small diffusion coefficient.

While the net current of such a Brownian ratchet has been derived
for an overdamped system in the works of Landauer, Van Kampen and
B{\"u}ttiker the determination of the effective diffusion coefficient had
remained a challenging task until the early twenty-first century.
Reimann {\it{et al.}} first determined the effective diffusion coefficient
for

In the past decade several studies have appeared regarding the
coherent transport of Brownian motors. Most studies have been carried out based on
uniform temperature. In this work we obtain analytical expressions for current, effective diffusion
coefficient analytically and numerically. In the low temperature regime, we determine
the transition rates and from that the current and effective diffusion coefficient.

\section{System}

The potential and temperature profile are respectively:-

 $$
  U(x) =
  \begin{cases}
    \frac{U_{0}x}{\alpha L}, & \text{for } 0 \leq x < \alpha L \\
    \newline\\
    \frac{U_{0}(L-x)}{(1-\alpha)L} & \text{for } \alpha L \leq x < L \\

  \end{cases}
$$

$$
  T(x) =
  \begin{cases}
    T_H, & \text{for } 0 \leq x < \alpha L \\
    \newline\\
    T_C & \text{for } \alpha L \leq x < L \\

  \end{cases}
$$
Both Potential and Temperatue profiles are periodic i.e. $U(x+L)=U(x)$ and $T(x+L)=T(x)$.

\begin{figure}[tbh]
\includegraphics[width=3.0in]{fig1drawing.eps}
\caption{\label{fig:profile}(Color online)Schematic of piecewise linear potential alternately sujected to hot and cold baths. }
\end{figure}

$\alpha$ is the potential asymmetry parameter, $U_{0}$ is the barrier height and $L$ is the spatial period of the potential. Without loss of generality we consider $U_{0}=1$ and $L=1$.

The Langvin equation used to study the motion of the Brownian particle is
given by,
\begin{equation}
 m\ddot{x}=-\gamma\dot{x}-U^\prime(x)+\sqrt{2k_{B}T(x)\gamma}\xi(t)
\end{equation}
We set $k_{B}=1$ and $\gamma=1$.

\begin{figure}
 \includegraphics[width=3.0in]{figpot.eps}
 \caption{\label{fig:potential}(Color online) $U(x)$, $\psi(x)$, and $\phi(x)$.}
 \end{figure}

In the overdamped limit, we ignore the inertial term. However, for temperature dependent on position, an additional term needs to be added as pointd out in earlier works. As per the Stratonovich interpretation, the overdamped Langevin equation is given by,

\begin{equation}
  \dot{x}=-U^\prime(x) -\frac{1}{2}\frac{dT(x)}{dx} + \sqrt{2T(x)\gamma}\xi(t)
\end{equation}
Here, $<\xi(t)>=0$ and $<\xi(t)\xi(t^\prime)>=\delta(t-t^\prime)$.
We set $\gamma=1$. This equation will be understood according to the Stratonovivh
interpretation.

The Fokker-Planck equation corresponding to this overdamped Langevin equation is given by,

\begin{equation}
 \frac{\partial P(x,t)}{\partial t}=\frac{\partial }{\partial x}\left[U^\prime(x)P(x,t)\right]+
 \frac{\partial^{2}}{\partial x^{2}}\left[T(x) P(x,t)\right]
\end{equation}

The current  is given by,
\begin{equation}
 <\dot{x}>=\frac{x(t_f)-x(t_s)}{t_f - t_s}
\end{equation}
where, $t_s \gg 1$ is the time taken to reach the steady state and $t_f$
is the final time upto which the simulations are carried out.

The effective diffusion coefficient is computed as per the following relation:
\begin{equation}
 D_{eff}=\frac{<(x(t_f)-x(t_s))^{2}>-<x(t_f)-x(t_s)>^2}{2(t_f-t_s)}
\end{equation}

Our numerical calculations are carrier out as per the Stochastic
Euler-Maruyama algorithm.

The analytical calculation were carried out as per the following formulas
provided in Ref.~\cite{lindner}.

\begin{equation}
 <\dot{x}>=L\frac{1-\exp(\psi(L))}{\int_{0}^{L} dx I_+(x)/g(x)}
\end{equation}
where, $g(x)=\sqrt{T(x)}$ and
\begin{equation}
 I_+(x)=\exp(-\psi(x))\int_{x}^{x+L} dy \exp(\psi(y))/g(y)
\end{equation}

The analytical expression for the effective diffusion coefficient is,
\begin{equation}
 D_{eff}=(L^{2})\frac{\int_{0}^{L} dx I_+(x)^{2}I_-(x)/g(x)}{[\int_{0}^{L} dx I_+(x)/g(x)]^{3}}
\end{equation}
where,
\begin{equation}
 I_-(x)=\exp(\psi(x))\int_{x-L}^{x} dy \exp(-\psi(y))/g(y)
\end{equation}
We can also determine the Peclet number which determines the coherence of transport. It's
given by,
\begin{equation}
 Pe=L <\dot{x}>/D_{eff}
\end{equation}

We also carried out numerical simulations to test the validity of our analytical
results using the Euler-Mayuram algorithm. The time-step chosen was $h=0.001$ and
he number of relizations was $5000$.

\section{Exact Expressions}
The first step to solve this model is to recognize that the
non-uniform temperature breaks the symmetry and simultaneously results in
a non-equilibriun condition, the minimal ingredient to
produce directed motion. In order  break the left right symmetry there should
be a phase difference between them. Due to the spatial dependence of the temperature
profile the noise term is multiplicative and Brownian particles subject
to such a temperature profile move under the influence of the generalized
potential given by,
\begin{equation}
 \psi(x)=\int_{0}^{x} dx^\prime [U^{\prime}(x^{\prime})+(1/2)T^{\prime}(x^{\prime})]/T(x^{\prime})
\end{equation}
The condition to achieve  directed transport ($ <\dot{x}>\neq 0$) is for this potential to have an
effective bias such that $\psi(L=1)-\psi(0) \neq 0$.

For the piecewise linear potential  and piecewise constant temperature profile,
it is simple to calculate $\psi(x)$, which is given by,
\begin{widetext}
 $$
  \psi(x) =
  \begin{cases}
    \frac{U_0}{T_C}+\frac{U_{0}(x+1-\alpha)}{\alpha T_H}, & \text{for } -1 \leq x < -1+\alpha  \\
    \newline\\
    \frac{-U_{0}x}{(1-\alpha)T_C} +\frac{1}{2}\log(\frac{T_C}{T_H}) & \text{for } -1+\alpha \leq x < 0 \\
    \newline \\
     \frac{U_{0}x}{\alpha T_H}    & \text{for } 0 \leq x <\alpha \\
     \newline \\
     \frac{U_0}{T_H}-\frac{U_{0}(x-\alpha)}{(1-\alpha)T_{C}}+\frac{1}{2}\log(\frac{T_C}{T_H})  & \text{for }  \alpha \leq x < 1 \\
     \newline \\
     \frac{U_0}{T_H}-\frac{U_0}{T_C}+\frac{U_0(x-1)}{\alpha T_H} & \text{for }  1 \leq x < 1+\alpha \\
     \newline \\
     \frac{2U_0}{T_H}-\frac{U_0}{T_C}-\frac{U_{0}(x-1-\alpha)}{(1-\alpha)T_C}+\frac{1}{2}\log(\frac{T_C}{T_H})  &    \text{for }  1+\alpha \leq x < 2 \\

  \end{cases}
$$

\end{widetext}

Using a suitable transformation of variables one can convert the overdamped LAngevin equation with multiplicatie noise to one with additive noise.
The required transformation is given by,
\begin{equation}
 y(x)=\int_{0}^{x} \,dz/\sqrt{T(z)}
\end{equation}
and the corresponding Langevin equation is given by,

\begin{equation}
 \dot{y}=\dot{x}/\sqrt{T(x)}=-\frac{d\phi(y)}{dy}+\sqrt{2}\xi(t)
\end{equation}
where,
\begin{equation}
 \phi(y)=\int_{0}^{y} \,dy^{*} \frac{U^{\prime}[x(y^{*})]+(1/2)T^{\prime}[x(y^{*})]}{\sqrt{T[x(y^{*})]}}
\end{equation}
and  $\phi(y)=\psi[x(y)]\,.$

For our potential and temperature profiles, the relation between the original and transformed coordinates is given by,

 $$
  y =
  \begin{cases}
    \frac{\alpha-1}{\sqrt{T_C}}-\frac{\alpha-x-1}{\sqrt{T_H}}, & \text{for } -1 \leq x < -1+\alpha  \\
    \newline\\
    \frac{x}{\sqrt{T_C}}& \text{for } -1+\alpha \leq x < 0 \\
    \newline \\
     \frac{x}{\sqrt{T_H}}    & \text{for } 0 \leq x <\alpha \\
     \newline \\
     \frac{1}{\sqrt{T_H}}+\frac{x-\alpha}{\sqrt{T_{C}}}  & \text{for }  \alpha \leq x < 1 \\
     \newline \\
     \frac{1}{\sqrt{T_H}}+\frac{1-\alpha}{\sqrt{T_C}}+\frac{(x-1)}{\sqrt{T_H}} & \text{for }  1 \leq x < 1+\alpha \\
     \newline \\
     \frac{2\alpha}{\sqrt{T_H}}+\frac{1-\alpha}{\sqrt{T_C}}+\frac{(x-1-\alpha)}{\sqrt{T_C}}  &    \text{for }  1+\alpha \leq x < 2 \\

  \end{cases}
$$

Finally, the effective potential in the transformed coordinates is given by,

 \begin{widetext}
 $$
  \phi(y) =
  \begin{cases}
    \frac{U_0}{T_C}+\frac{U_{0}(y+[(1-\alpha)/\sqrt{T_C}])}{\alpha \sqrt{T_H}}, & \text{for } \frac{\alpha-1}{\sqrt{T_C}}-\frac{\alpha}{\sqrt{T_H}} \leq y < \frac{\alpha-1}{\sqrt{T_C}}  \\
    \newline\\
    \frac{-U_{0}y}{(1-\alpha)\sqrt{T_C}} +\frac{1}{2}\log(\frac{T_C}{T_H}) & \text{for } \frac{-1+\alpha}{\sqrt{T_C}} \leq y < 0 \\
    \newline \\
     \frac{U_{0}y}{\alpha \sqrt{T_H}}    & \text{for } 0 \leq y < \frac{\alpha}{\sqrt{T_H}} \\
     \newline \\
     \frac{U_0}{T_H}-\frac{U_{0}(y-\alpha/\sqrt{T_H})}{(1-\alpha)\sqrt{T_{C}}}+\frac{1}{2}\log(\frac{T_C}{T_H})  & \text{for }  \frac{\alpha}{\sqrt{T_H}} \leq y < \frac{\alpha}{\sqrt{T_H}} + \frac{1-\alpha}{\sqrt{T_C}} \\
     \newline \\
     -\frac{U_0}{T_C}+\frac{U_0(y+(\alpha-1)/\sqrt{T_C})}{\alpha \sqrt{T_H}} & \text{for }  \frac{\alpha}{\sqrt{T_H}} + \frac{1-\alpha}{\sqrt{T_C}} \leq y < \frac{2\alpha}{\sqrt{T_H}} + \frac{1-\alpha}{\sqrt{T_C}} \\
     \newline \\
     \frac{2U_0}{T_H}-\frac{U_0}{T_C}-\frac{U_{0}(y-2\alpha/\sqrt{T_H})}{(1-\alpha)\sqrt{T_C}}+\frac{1}{2}\log(\frac{T_C}{T_H})  &    \text{for }  \frac{2\alpha}{\sqrt{T_H}} + \frac{1-\alpha}{\sqrt{T_C}}  \leq y < \frac{2\alpha}{\sqrt{T_H}} + \frac{2(1-\alpha)}{\sqrt{T_C}}  \\

  \end{cases}
$$

\end{widetext}
The period of $\phi(y)$ is $L_{y}=\frac{\alpha}{\sqrt{T_H}}+\frac{1-\alpha}{\sqrt{T_C}}$.

The effective diffusion coefficient computed as per the transformed dynamics with additive noise
is given by,
\begin{equation}
 D_{eff,y}=\frac{\int_{0}^{L_y} \,dx [I_-(x)]^{2}I_+(x)/L_y}{[\int_{0}^{L_y} \,dx I_-(x)/L_y]^{3}}
\end{equation}
and
\begin{equation}
 I_\pm (x)=\mp e^{\pm \phi(x)}\int_{x}^{x \mp L_y} \,dy \, e^{\mp \phi(y)}
\end{equation}

The velocity in the transformed coordinates is given by,
\begin{equation}
 v_{y}=\frac{1-e^{\phi(L_y)}}{[\int_{0}^{L_y} \,dx \, I_-(x)/L_y]}
\end{equation}

The relation between the effective diffusion coefficient and the particle current
in the original and transformed coordinates is given by,
\begin{equation}
 D_{eff}=\frac{D_{eff,y}}{{L_{y}}^{2}} , \,\,\, <\dot{x}>=\frac{v_{y}}{L_{y}}
 \end{equation}

 We will calculate the effective diffusion coefficient using Eq. 8.
 The Integral in the denominatior is given by,
 \begin{widetext}
 \begin{equation}
  I_d=\int_{0}^{\alpha} \,dx \frac{e^{-\psi(x)}}{\sqrt{T_H}} \int_{x}^{x+1} \,dy \frac{e^{\psi(y)}}{\sqrt{T(y)}} +   \int_{\alpha}^{1} \,dx \frac{e^{-\psi(x)}}{\sqrt{T_C}} \int_{x}^{x+1} \,dy \frac{e^{\psi(y)}}{\sqrt{T(y)}}= A+B
 \end{equation}
\end{widetext}
such that
\begin{equation}
 A=\int_{0}^{\alpha} \,dx \frac{e^{-\psi(x)}}{\sqrt{T_H}} \int_{x}^{x+1} \,dy e^{\psi(y)}/\sqrt{T(y)}
\end{equation}
and,
\begin{equation}
 B=\int_{\alpha}^{1} \,dx \frac{e^{-\psi(x)}}{\sqrt{T_C}} \int_{x}^{x+1} \,dy e^{\psi(y)}/\sqrt{T(y)}
\end{equation}
we find that,
\begin{widetext}
\begin{equation}
 A=\int_{0}^{\alpha} \,dx \frac{e^{-\frac{U_0{x}}{\alpha T_H}}}{\sqrt{T_H}}\left[ \int_{x}^{\alpha} \,dy \frac{e^{\frac{U_0 {y}}{\alpha T_H}}}{\sqrt{T_H}} + \int_{\alpha}^{1} \,dy \frac{e^{\frac{U_0}{T_H}-\frac{U_{0}(y-\alpha)}{(1-\alpha)T_{C}}+\frac{1}{2}\log\left(\frac{T_C}{T_H}\right) }}{\sqrt{T_C}} + \int_{1}^{x+1} \,dy \frac{e^{\frac{U_0}{T_H}-\frac{U_0}{T_C}+\frac{U_0(y-1)}{\alpha T_H}}}{\sqrt{T_H}} \right]
\end{equation}
and,
\begin{equation}
\begin{split}
 B   = \int_{\alpha}^{1} \,dx \frac{e^{-\left[{\frac{U_0}{T_H}-\frac{U_{0}(x-\alpha)}{(1-\alpha)T_{C}}+\frac{1}{2}\log(\frac{T_C}{T_H})  }\right]}}{\sqrt{T_C}} \left[ \int_{x}^{1}  \,dy \frac{e^{\frac{U_0}{T_H}-\frac{U_{0}(y-\alpha)}{(1-\alpha)T_{C}}+\frac{1}{2}\log\left(\frac{T_C}{T_H}\right) }}{\sqrt{T_C}} +\int_{1}^{1+\alpha} \,dy \frac{e^{\frac{U_0}{T_H}-\frac{U_0}{T_C}+\frac{U_0(y-1)}{\alpha T_H}}}{\sqrt{T_H}} \right.  \\
 \left.   +\int_{1+\alpha}^{x+1}\,dy \frac{e^{\frac{2U_0}{T_H}-\frac{U_0}{T_C}-\frac{U_{0}(y-1-\alpha)}{(1-\alpha)T_C}+\frac{1}{2}\log(\frac{T_C}{T_H})}} {\sqrt{T_C}}  \right]
\end{split}
 \end{equation}

\end{widetext}
Here,
\begin{equation}
 a_{11}=\int_{x}^{\alpha} \,dy  \frac{e^{\frac{U_0 {y}}{\alpha T_H}}}{\sqrt{T_H}}=-{\frac {\alpha\,\sqrt {T_{H}}}{U_{0}} \left( {{\rm e}^{{\frac {U_{0}
\,x}{\alpha\,T_{H}}}}}-{{\rm e}^{{\frac {U_{0}}{T_{H}}}}} \right) }
\end{equation}
\begin{widetext}
\begin{equation}
 a_{12}=\int_{\alpha}^{1} \,dy  \frac{e^{\frac{U_0}{T_H}-\frac{U_{0}(y-\alpha)}{(1-\alpha)T_{C}}+\frac{1}{2}\log\left(\frac{T_C}{T_H}\right) }}{\sqrt{T_C}} =
 -{\frac { \left( \alpha-1 \right) T_{C}}{U_{0}\,\sqrt {T_{H}}}{{\rm e}
^{{\frac {U_{0}\, \left( T_{C}-T_{H} \right) }{T_{C}\,T_{H}}}}}
 \left( {{\rm e}^{{\frac {U_{0}}{T_{C}}}}}-1 \right) }

\end{equation}
\end{widetext}
and,
\begin{widetext}
\begin{equation}
 a_{13}=\int_{1}^{x+1} \frac{e^{\frac{U_0}{T_H} -\frac{U_0}{T_C} + \frac{U_{0}(y-1)}{\alpha T_H} }   } {\sqrt{T_H}} = {\frac {\alpha\,\sqrt {T_{H}}}{U_{0}} \left( {{\rm e}^{{\frac {U_{0}\,
 \left( \alpha+x \right) }{\alpha\,T_{H}}}}}-{{\rm e}^{{\frac {U_{0}}{
T_{H}}}}} \right) {{\rm e}^{-{\frac {U_{0}}{T_{C}}}}}}

\end{equation}
\end{widetext}
\begin{widetext}
Similarly,
\begin{equation}
 b_{11}={\frac { \left( \alpha-1 \right) T_{C}}{U_{0}\,\sqrt {T_{H}}} \left( -
{{\rm e}^{{\frac {U_{0}\, \left( T_{C}\,\alpha+T_{H}\,x-T_{C}-T_{H}
 \right) }{T_{H}\, \left( \alpha-1 \right) T_{C}}}}}+{{\rm e}^{{\frac
{U_{0}}{T_{H}}}}} \right) {{\rm e}^{-{\frac {U_{0}}{T_{C}}}}}}

\end{equation}
\begin{equation}
 b_{12}={\frac {\alpha\,\sqrt {T_{H}}}{U_{0}}{{\rm e}^{{\frac {U_{0}\, \left(
T_{C}-T_{H} \right) }{T_{C}\,T_{H}}}}} \left( {{\rm e}^{{\frac {U_{0}
}{T_{H}}}}}-1 \right) }

\end{equation}
\begin{equation}
 b_{13}={\frac { \left( \alpha-1 \right) T_{C}}{U_{0}\,\sqrt {T_{H}}} \left( {
{\rm e}^{{\frac {U_{0}\, \left( 2\,T_{C}\,\alpha-\alpha\,T_{H}+T_{H}\,
x-2\,T_{C} \right) }{T_{H}\, \left( \alpha-1 \right) T_{C}}}}}-{
{\rm e}^{2\,{\frac {U_{0}}{T_{H}}}}} \right) {{\rm e}^{-{\frac {U_{0}
}{T_{C}}}}}}

\end{equation}

\begin{equation}
\begin{split}
 A=-\frac {\alpha}{{U_{0}}^{2}} \left(  \left(  \left( -T_{C}+T_{H}-U_{0
} \right) \alpha+T_{C} \right) {{\rm e}^{{\frac {U_{0}\, \left( T_{C}-
T_{H} \right) }{T_{C}\,T_{H}}}}}+ \right.  \\
\left. \left(  \left( T_{C}-T_{H} \right)
\alpha-T_{C} \right) {{\rm e}^{-{\frac {U_{0}}{T_{C}}}}}+
\left(
 \left( T_{C}-T_{H} \right) \alpha-T_{C} \right) {{\rm e}^{{\frac {U_{
0}}{T_{H}}}}}+ \left( -T_{C}+T_{H}+U_{0} \right) \alpha+T_{C} \right)

\end{split}
\end{equation}

\begin{equation}
 \begin{split}
 B=\frac {\alpha-1}{{U_{0}}^{2}} \left(  \left(  \left( -T_{C}+T_{H}-U_{
0} \right) \alpha+T_{C}+U_{0} \right) {{\rm e}^{{\frac {U_{0}\,
 \left( T_{C}-T_{H} \right) }{T_{C}\,T_{H}}}}}+ \right. \\
 \left. \left(  \left( T_{C}-T
_{H} \right) \alpha-T_{C} \right) {{\rm e}^{-{\frac {U_{0}}{T_{C}}}}}+
 \left(  \left( T_{C}-T_{H} \right) \alpha-T_{C} \right) {{\rm e}^{{
\frac {U_{0}}{T_{H}}}}}+ \left( -T_{C}+T_{H}+U_{0} \right) \alpha+T_{C
}-U_{0} \right)
 \end{split}
\end{equation}
Then $I_d$ is given by,
\begin{equation}
\begin{split}
 I_d=\frac {1}{{U_{0}}^{2}} \left(  \left(  \left( T_{C}-T_{H}+2\,U_{0}
 \right) \alpha-T_{C}-U_{0} \right) {{\rm e}^{{\frac {U_{0}\, \left( T
_{C}-T_{H} \right) }{T_{C}\,T_{H}}}}}+ \right. \\
\left. \left(  \left( -T_{C}+T_{H}
 \right) \alpha+T_{C} \right) {{\rm e}^{-{\frac {U_{0}}{T_{C}}}}}+
 \left(  \left( -T_{C}+T_{H} \right) \alpha+T_{C} \right) {{\rm e}^{{
\frac {U_{0}}{T_{H}}}}}+ \left( T_{C}-T_{H}-2\,U_{0} \right) \alpha-T_
{C}+U_{0} \right)
\end{split}
\end{equation}

\end{widetext}

The numerator in the expression for effective diffusion coefficient can be written as,
\begin{widetext}
\begin{equation}
\begin{split}
 num=\int_{0}^{\alpha} \,dx \frac{e^{-\frac{U_0{x}}{\alpha T_H}}}{\sqrt{T_H}} \left[\int_{x}^{x+L}\,dy \frac{\exp(\psi(y))}{\sqrt{T(y)}}\right]^{2}\left(\int_{x-L}^{x}\,dy \frac{\exp[-\psi(y)]}{\sqrt{T(y)}}\right) + \\
 \int_{\alpha}^{1} \,dx  \frac{e^{-\left[{\frac{U_0}{T_H}-\frac{U_{0}(x-\alpha)}{(1-\alpha)T_{C}}+\frac{1}{2}\log(\frac{T_C}{T_H})  }\right]}}{\sqrt{T_C}}\left[\int_{x}^{x+L}\,dy \frac{\exp(\psi(y))}{\sqrt{T(y)}}\right]^{2}\left(\int_{x-L}^{x}\,dy \frac{\exp[-\psi(y)]}{\sqrt{T(y)}}\right)
 \end{split}
\end{equation}
such that,
\begin{equation}
\begin{split}
  num=\int_{0}^{\alpha} \,dx \frac{e^{-\frac{U_0{x}}{\alpha T_H}}}{\sqrt{T_H}} \left[ a_{11}+a_{12}+a_{13} \right]^{2}\left(\int_{x-L}^{x}\,dy \frac{\exp[-\psi(y)]}{\sqrt{T(y)}}\right) + \\
 \int_{\alpha}^{1} \,dx  \frac{e^{-\left[{\frac{U_0}{T_H}-\frac{U_{0}(x-\alpha)}{(1-\alpha)T_{C}}+\frac{1}{2}\log(\frac{T_C}{T_H})  }\right]}}{\sqrt{T_C}}\left[ b_{11}+b_{12}+b_{13}\right]^{2}\left(\int_{x-L}^{x}\,dy \frac{\exp[-\psi(y)]}{\sqrt{T(y)}}\right)
 \end{split}
 \end{equation}
\end{widetext}
For $0 \leq x < \alpha$,
\begin{equation}
 P=\int_{x-1}^{x} \,dy \frac{\exp[-\psi(y)]}{\sqrt{T(y)}}=p_{11}+p_{12}+p_{13}
\end{equation}
where,
\begin{equation}
 p_{11}=
 {\frac {\alpha\,\sqrt {T_{H}}}{U_{0}} \left( {{\rm e}^{{\frac {U_{0}\,
 \left( \alpha-x \right) }{\alpha\,T_{H}}}}}-1 \right) {{\rm e}^{-{
\frac {U_{0}}{T_{C}}}}}}
\end{equation}
\begin{equation}
 p_{12}=-{\frac {\sqrt {T_{H}} \left( \alpha-1 \right) }{U_{0}} \left( {
{\rm e}^{{\frac {U_{0}}{T_{C}}}}}-1 \right) {{\rm e}^{-{\frac {U_{0}}{
T_{C}}}}}}

\end{equation}
\begin{equation}
 p_{13}=-{\frac {\alpha\,\sqrt {T_{H}}}{U_{0}} \left( {{\rm e}^{-{\frac {U_{0}
\,x}{\alpha\,T_{H}}}}}-1 \right) }

\end{equation}

For $\alpha \leq x < 1$,
\begin{equation}
 Q=\int_{x-1}^{x} \,dy \frac{\exp[-\psi(y)]}{\sqrt{T(y)}}=q_{11}+q_{12}+q_{13}
\end{equation}
where,
\begin{widetext}
\begin{equation}
 q_{11}=\int_{x-1}^{0} \,dy \frac{e^{\frac{U_{0}y}{(1-\alpha)T_C}-\frac{1}{2}\log(T_{C}/T_{H})}}{\sqrt{T_C}}=
 {\frac { \left( \alpha-1 \right) \sqrt {T_{H}}}{U_{0}} \left( {{\rm e}
^{-{\frac {U_{0}\, \left( x-1 \right) }{ \left( \alpha-1 \right) T_{C}
}}}}-1 \right) }
\end{equation}
\end{widetext}
\begin{equation}
 q12=\int_{0}^{\alpha} \,dy \frac{e^{\frac{-U_{0}y}{\alpha T_H}}}{\sqrt{T_H}}=
 {\frac {\alpha\,\sqrt {T_{H}}}{U_{0}} \left( {{\rm e}^{{\frac {U_{0}}{
T_{H}}}}}-1 \right) {{\rm e}^{-{\frac {U_{0}}{T_{H}}}}}}
\end{equation}

\begin{equation}
\begin{split}
 q13=\int_{\alpha}^{x}\,dy \frac {e^{\frac{-U_{0}}{T_H}+\frac{U_{0}(y-\alpha)}{(1-\alpha)T_{C}}-\frac{1}{2}\log(\frac{T_C}{T_H})} } {\sqrt{T_C}}
 =\\
 -{\frac { \left( \alpha-1 \right) \sqrt {T_{H}}}{U_{0}} \left( {
{\rm e}^{{\frac {U_{0}\, \left( \alpha-x \right) }{ \left( \alpha-1
 \right) T_{C}}}}}-1 \right) {{\rm e}^{-{\frac {U_{0}}{T_{H}}}}}}
\end{split}
\end{equation}

Finally, we have
\begin{widetext}
\begin{equation}
\begin{split}
num=\int_{0}^{\alpha} \,dx \frac{e^{-\frac{U_0{x}}{\alpha T_H}}}{\sqrt{T_H}} \left[ a_{11}+a_{12}+a_{13} \right]^{2}( p_{11}+p_{12}+p_{13} ) + \\
\int_{\alpha}^{1} \,dx  \frac{e^{-\left[{\frac{U_0}{T_H}-\frac{U_{0}(x-\alpha)}{(1-\alpha)T_{C}}+\frac{1}{2}\log(\frac{T_C}{T_H})  }\right]}}{\sqrt{T_C}}\left[ b_{11}+b_{12}+b_{13}\right]^{2} ( q_{11}+q_{12}+q_{13})
\end{split}
\end{equation}
\end{widetext}
The final expression is obtained as,
\begin{equation}
 num=num1+num2
\end{equation}
where,
\begin{widetext}
\begin{equation}
\begin{split}
 num1=\frac{1}{\sqrt{T_H}}\left({\varphi_{0}}^{3}\alpha +\frac{\alpha T_H}{U_0}\left(\exp\left(\frac{U_{0}}{T_H}\right)-1\right)\left[ {\varphi_{0}}^{2}{\tilde{\varphi_{1}}}{\varphi_{c}}+\varphi_{1}\varphi_{c}(2{\varphi_{0}}^{2}   +\varphi_{1} {\tilde{\varphi_{1}}} {{\varphi_{c}}^{2}}   \exp\left(\frac{U_0}{T_H}\right) \right] \right. \\
 \left. +2\alpha\varphi_{0}\varphi_{1}\tilde{\varphi_{1}}{\varphi_{c}}^{2}\exp(U_{0}/T_{H}) +\varphi_{0}{\varphi_{1}}^{2}{\varphi_{c}}^{2}\frac{\alpha T_H}{2 U_0} \left(exp(2U_{0}/T_{H})-1 \right) \right)
 \end{split}
\end{equation}
\begin{equation}
\begin{split}
num2=\frac{{{\mu_{0}}^{2}}\lambda_{0}(1-\alpha)T_{C}}  {2U_0} \left[ \exp \left( \frac{2U_0}{T_C} \right) -1\right] +\frac{\xi_{1}(1-\alpha)T_C}{U_0} \left[ \exp \left(\frac{U_0}{T_C} \right) -1\right] +\xi_{0}(1-\alpha) \\
+\mu_{2}\lambda_{1}\frac{\alpha-1}{U_0}T_{C}\left[\exp\left( -\frac{U_0}{T_C} \right)-1 \right]
\end{split}
\end{equation}
\end{widetext}

The numerator is then given by $num=num1+num2$. The effective diffusion coefficient then obtained as
\begin{equation}
 D_{eff}=\frac{num}{{I_d}^3}
\end{equation}
The various terms are provided in the Appendix.
The current is  calculated as,
\begin{equation}
 <\dot{x}>=L\frac{1-\exp(\psi(L))}{I_d}
\end{equation}

In the low temperature limit, the particle current and effective diffusion coefficient
can be obtained in terms of the transition rates in forward and reverse directions given
by,
\begin{equation}
 r_{f}=\frac{1}{\alpha}\frac{1}{\alpha T_{H} Y^2 +(\alpha-1)T_{C}Z}
\end{equation}
where,
\begin{equation}
 Y=\exp(\frac{U_{0}}{2T_{H}})-\exp(\frac{U_{0}}{2T_{C}})
\end{equation}
and,
\begin{equation}
 Z=(\exp(\frac{-U_{0}}{T_{C}})-1)(\exp(\frac{U_{0}}{T_{H}})-1)
\end{equation}
and,
\begin{equation}
 r_{b}=\frac{1}{1-\alpha}\frac{1}{\alpha T_{H} X +(1-\alpha) T_{C} R^{2}}
\end{equation}
where,
\begin{equation}
 X=(1-\exp(-\frac{1}{T_{H}}))(\exp(\frac{1}{T_{C}})-1)
\end{equation}
and,
\begin{equation}
 R=\exp(\frac{U_{0}}{2T_{C}})-\exp(-\frac{U_{0}}{2T_{C}})
\end{equation}
So, the current and effective diffusion coefficient are given by,
\begin{equation}
 <\dot{x}>=r_{f}\alpha-r_{b}(1-\alpha)
\end{equation}
and,
\begin{equation}
 D_{eff}=\frac{r_{f}\alpha+r_{b}(1-\alpha)}{2}
\end{equation}

{\it{\underline{Results.-}}} In Fig.~2, we plot the current as a function of the asymmetry parameter.
In FIg.~3 we show the effective diffusion coefficient as a function of the temperature
of the hot bath. Fig.~4 shows the Peclet number. Good agreement is obtained between theory
and simulation result.

\begin{figure}[htb]
\includegraphics[width=3.0in]{deff1.eps}
\caption{\label{fig:Q-F_fridge}(Color online) Effective diffusion coefficient as a function
of the asymmetry parameter $\alpha$.}
\end{figure}

\begin{figure}[htb]
\includegraphics[width=3.0in]{deff_vs_Thot.eps}
\caption{\label{fig:Q-F_fridge}(Color online) Effective diffusion coefficient as a function
of the temperature of the hot bath.}
\end{figure}

\begin{figure}[htb]
\includegraphics[width=3.0in]{vel1.eps}
\caption{\label{fig:Q-F_fridge}(Color online) Current as a function
of the asymmetry parameter $\alpha$.}
\end{figure}

\begin{figure}[htb]
\includegraphics[width=3.0in]{pec1.eps}
\caption{\label{fig:Q-F_fridge}(Color online) Peclet number as a function
of the asymmetry parameter $\alpha$.}
\end{figure}

{\it{\underline{Conclusion.-}}} In this work, we have analytically and numerically obtained
the current and effective diffusion coefficient of a Brownian particle in a piecewise
linear potential subject alternately to hot and cold baths. Good agreement is obtained between
our numerical and analytical results. In some parameter regimes transport is enhanced.

\begin{acknowledgments}
The author acknowledges
\end{acknowledgments}


\section{Introduction}

Outer automorphims of a free group divide into two categories, polynomially
growing and exponentially growing, according to the behaviour of word lengths
under iteration. Subgroups of $\mathrm{Out}(F_r)$ that consist of only polynomially
growing outer automorphisms are known as \emph{Kolchin} subgroups and are
the $\mathrm{Out}(F_r)$ analog of unipotent subgroups of a linear group.

Clay and Pettet~\cite{clay-pettet} and Gultepe~\cite{gultepe} give sufficient
conditions for when a subgroup of $\mathrm{Out}(F_r)$ generated by two Dehn twists
contains an exponentially growing outer automorphism. This article gives an
algorithmic criterion that characterizes when a subgroup of $\mathrm{Out}(F_r)$
generated by two Dehn twists is Kolchin, in terms of a combinatorial
invariant of the gnerators known as the \emph{edge-twist} directed graph (see
Definition~\ref{def:et}).

\begin{theorem}[Main Theorem]
	Suppose $\sigma, \tau \in \mathrm{Out}(F_r)$ are Dehn twists. The subgroup
	$\langle \sigma,\tau\rangle$ is Kolchin if and only if the edge
	twist digraph of the defining graphs of groups is directed acyclic.
\end{theorem}

\section{Background}

\subsection{Graphs of groups and Dehn twists}

A \emph{graph} $\Gamma$ is a collection of vertices $V(\Gamma)$, edges
$E(\Gamma)$, initial and terminal vertex maps $o, t: E\to V$ and an involution
$\bar{\cdot}: E\to E$ satisfying $\bar{e}\neq e$ and $o(\bar{e}) = t(e)$. A
\emph{directed graph (digraph)} omits the involution.

\begin{definition}
	A \emph{graph of groups} is a pair $(G,\Gamma)$ where
	$\Gamma$ is a connected graph and $G$ is an assignment of groups to the
	vertices and edges of $\Gamma$ satisfying $G_e = G_{\bar{e}}$ and
	injections $\iota_e: G_e\to G_{t(e)}$. The assignment will often be
	suppressed and $\Gamma_v, \Gamma_e$ used instead.
\end{definition}

\begin{definition}
	The \emph{fundamental groupoid} $\pi_1(\Gamma)$ of a graph
	of groups $\Gamma$ is the groupoid with vertex set $V(\Gamma)$
	generated by the path groupoid of $\Gamma$ and the groups $G_v$ subject
	to the following relations. We require that for each $v\in V(\Gamma)$
	the group $G_v$ is a subgroupoid based at $v$ and that the group and
	groupoid structures agree. Further, for all $e\in E(\Gamma)$ and $g\in
	G_e$ we have $\bar{e}\iota_{\bar{e}}(g)e = \iota_e(g)$.
\end{definition}

The \emph{fundamental group} of $\Gamma$ based at $v$, $\pi_1(\Gamma, v)$ is
the vertex subgroup of $\pi_1(\Gamma)$ based at $v$. It is standard that
changing the basepoint gives an isomorphic group~\cites{trees, higgins}.

Let $(e_1,\ldots,e_n)$ be a possibly empty edge path in $\Gamma$ starting at
$v$ and $(g_0,\ldots, g_n)$ be a sequence of elements $g_i\in G_{t(e_i)}$ with
$g_0 \in G_v$. These data represent an arrow of $\pi_1(\Gamma)$ by the groupoid
product \[ g_0e_1g_1\cdots e_ng_n.\] A non-identity element of $\pi_1(\Gamma)$
expressed this way is \emph{reduced} if either $n = 0$ and $g_0\neq \mathrm{id}$, or $n
> 0$ ans for all $i$ such that $e_i = \bar{e}_{i+1}$, $g_i \not\in
\iota_{e_i}(G_{e_i})$. The edges appearing in a reduced arrow are uniquely
determined.
Further, if $t(e_n) = o(e_1)$ the arrow is cyclically reduced if either $e_n
\neq \bar{e}_0$ or $e_n = \bar{e}_0$ and $g_ng_0\not\in \iota_{e_n}(G_{e_n})$.
For an element $g\in \pi_1(\Gamma, v)$, the edges appearing in a cyclically
reduced arrow conjugate to $g$ in $\pi_1(\Gamma)$ is a conjugacy class
invariant.\footnote{These edges are covered by the axis of $g$ in the
Bass-Serre tree of $\Gamma$.}

\begin{definition}
	Given a graph of groups $\Gamma$ a subset of edges
	$E'\subseteq E(\Gamma)$ and edge-group elements $\{z_e\}_{e\in E'}$
	satisfying $z_e \in Z(\Gamma_e)$ and $z_{\bar{e}} = z_e^{-1}$, the
	\emph{Dehn twist} of $\Gamma$ about $E'$ by $\{z_e\}$ is the
	fundamental groupoid
	automorphism $D_z$ given on the generators by
	\begin{align*}
		D_z(e) &= ez_e & e\in E' \\
		D_z(e) &= e & e\not\in E' \\
		D_z(g) &=g & g\in \Gamma_v \\
	\end{align*}
	As this groupoid automorphism preserves vertex subgroups it induces a
	well-defined outer automorphism class $D_z\in\mathrm{Out}(\pi_1(\Gamma, v))$,
	which we will also refer to as a Dehn twist.
\end{definition}

For a group $G$ we say $\sigma\in\mathrm{Out}(G)$ is a \emph{Dehn twist} if it can be realized
as a Dehn twist about some graph of groups $\Gamma$ with $\pi_1(\Gamma, v)
\cong G$.

Specializing to $\mathrm{Out}(F_r)$, when $\sigma\in\mathrm{Out}(F_r)$ is a Dehn twist there
are many graphs of groups $\Gamma$ with $\pi_1(\Gamma, v) \cong F_r$ that can
be used to realize $\sigma$. However, Cohen and Lustig~\cite{cohen-lustig}
define the notion of an \emph{efficient graph of groups} representative of a
Dehn twist and show that each Dehn twist in $\mathrm{Out}(F_r)$ has a unique efficient
representative. For a fixed $\sigma$ let $\mathcal{G}(\sigma)$ denote the graph
of groups of its efficient representative; edge groups of
$\mathcal{G}(\sigma)$ are infinite cyclic~\cite{cohen-lustig}.

\begin{remark}\label{rem:twist-power}
	If $\sigma,\tau\in\mathrm{Out}(F_r)$ are Dehn twists with a common power, then
	$\mathcal{G}(\sigma) = \mathcal{G}(\tau)$.
\end{remark}

\subsection{Topological representatives and the Kolchin theorem}

Given a graph $\Gamma$ the \emph{topological realization} of $\Gamma$ is a
simplicial complex with zero-skeleton $V(\Gamma)$ and one-cells joining $o(e)$
and $t(e)$ for each edge in a set of $\bar{\cdot}$ orbit representatives. It
will not cause confusion to use $\Gamma$ for both a graph and its topological
representative. If $\gamma \subset \Gamma$ is a based loop, denote the
associated element of $\pi_1(\Gamma)$ by $\gamma^\ast$. Given $\sigma\in
\mathrm{Out}(F_r)$, a \emph{topological realization} is a homotopy equivalence
$\hat{\sigma}:\Gamma\to\Gamma$ so that $\hat{\sigma}_\ast : \pi_1(\Gamma,v)\to
\pi_1(\Gamma,\hat{\sigma}(v))$ is a representative of $\sigma$. A homotopy
equivalence $\hat{\sigma}:\Gamma\to\Gamma$ is \emph{filtered} if there is a
filtration $\emptyset = \Gamma_0\subsetneq \Gamma_1\subsetneq \cdots\subsetneq
\Gamma_k = \Gamma$ preserved by $\hat{\sigma}$.

\begin{definition}
	A filtered homotopy equivalence $\hat{\sigma}:\Gamma\to\Gamma$ is
	\emph{upper triangular} if
	\begin{enumerate}
		\item $\hat{\sigma}$ fixes the vertices of $\Gamma$,
		\item Each stratum of the filtration $\Gamma_i\setminus
			\Gamma_{i-1} = e_i$ is a single topological edge,
		\item Each edge $e_i$ has a preferred orientation and with this
			orientation there is a closed path $u_i\subseteq
			\Gamma_{i-1}$ based at $t(e_i)$ so that
			$\hat{\sigma}(e_i) = e_iu_i$.
	\end{enumerate}
\end{definition}

The path $u_i$ is called the suffix associated to $u_i$. A filtration assigns
each edge a height, the $i$ such that $e\in \Gamma_i\setminus\Gamma_{i-1}$, and
taking a maximum this definition extends to edge paths.

Every Dehn twist in $\mathrm{Out}(F_r)$ has an upper-triangular
representative~\cites{bfh-ii, cohen-lustig}. In a previous
paper~\cite{bering-lg} I describe how to construct $\mathcal{G}(\sigma)$ from
an upper-triangular representative, following a similar construction of
Bestvina, Feighn, and Handel~\cite{bfh-ii}. The following is an immediate
consequence of my construction.

\begin{lemma}\label{lem:filter}
	Suppose $\Gamma$ is a filtered graph and $\sigma\in\mathrm{Out}(F_r)$ is a Dehn
	twist that is upper triangular with respect to $\Gamma$. Then
	\begin{enumerate}
		\item there is a height function $ht: E(\mathcal{G}(\sigma))\to
			\mathbb{N}$ so that for any loop
			$\gamma\subseteq\Gamma_i$ the height of the edges in a
			cyclically reduced representative of the conjugacy
			class of $\gamma^\ast$ in
			$\pi_1(\mathcal{G}(\sigma))$ is at most $i$,
		\item For each edge $e\in E(\mathcal{G}(\sigma))$, $ht(e) =
			ht(\bar{e})$, and the edge
			group $\mathcal{G}(\sigma)_e$ is a conjugate of a
			maximal cyclic subgroup of $F_r$ that contains
			$u_i^\ast$ for some suffix $u_i$, and $ht(e) >
			\min_{\gamma \sim u_i} \{ht_\Gamma(\gamma)\}$, where
			$\gamma$ ranges over loops freely homotopic to $u_i$.
	\end{enumerate}
\end{lemma}

Bestvina, Feighn, and Handel proved an $\mathrm{Out}(F_r)$ analog of the classical
Kolchin theorem for $\mathrm{Out}(F_r)$, which provides simultaneous upper-triangular
representatives for Kolchin-type subgroups of $\mathrm{Out}(F_r)$.

\begin{theorem}[\cite{bfh-ii}]\label{thm:kolchin}
	Suppose $H\leq\mathrm{Out}(F_r)$ is a Kolchin subgroup. Then there is a finite
	index subgroup $H' \leq H$ and a filtered graph $\Gamma$ so that each
	$\sigma \in H'$ is upper triangular with respect to $\Gamma$.
\end{theorem}

\subsection{Twists and polynomial growth}

In a previous paper~\cite{bering-lg} I introduced the edge-twist digraph of two
Dehn twists and used it to provide a sufficient condition for a subgroup of
$\mathrm{Out}(F_r)$ generated by two Dehn twists to be Kolchin.

\begin{definition}[\cite{bering-lg}]\label{def:et}
	The edge-twist digraph $\mathcal{ET}(A,B)$ of two graphs of groups $A$,
	$B$ with isomorphic fundamental groups and infinite cyclic edge
	stabilizers is the digraph with vertex set
	\[ V(\mathcal{ET}(A,B)) = \{(e, \bar{e}) | e\in E(A)\}\cup
	\{(f,\bar{f})|f\in E(B)\} \]
	directed edges $((e,\bar{e}), (f,\bar{f}))$ $e\in E(A), f\in E(B)$ when a generator of $A_e$
	uses $f$ or $\bar{f}$ in its cyclically reduced representation in
	$\pi_1(B)$, and directed edges $((f,\bar{f}),(e,\bar{e}))$, $f\in E(B),
	e\in E(A)$ when a generator of $B_f$ uses $e$ or $\bar{e}$ in a
	cyclically reduced representation in $\pi_1(A)$.
\end{definition}

\begin{remark}
	This is well-defined, using an edge is a conjugacy invariant, and using
	an edge or its reverse is preserved under taking inverses.
\end{remark}

\begin{lemma}[\cite{bering-lg}]\label{lem:pg}
	If $\sigma, \tau\in\mathrm{Out}(F_r)$ are Dehn twists and
	$\mathcal{ET}(\mathcal{G}(\sigma),\mathcal{G}(\tau))$ is directed acyclic, then
	$\langle \sigma,\tau\rangle$ is Kolchin.
\end{lemma}

\section{Proof of the main theorem}

\begin{proof}
	It suffices to prove the converse to Lemma \ref{lem:pg}. Suppose
	$\langle \sigma, \tau\rangle$ is Kolchin.
	By Theorem \ref{thm:kolchin}, there is a finite
	index subgroup $H\leq \langle \sigma, \tau\rangle$
	where every element of $H$ is upper triangular with respect
	respect to a fixed filtered graph $\Gamma$. Since $H$ is finite index, there are
	powers $m, n$ so that $\sigma^m, \tau^n \in H$, so that $\sigma^m$
	and $\tau^n$ are upper triangular with respect to $\Gamma$.

	By Lemma \ref{lem:filter}, the filtration of $\Gamma$ induces height
	functions on $E(\mathcal{G}(\sigma^m))$ and $E(\mathcal{G}(\tau^n))$,
	combining these gives a height function on the vertices of
	$\mathcal{ET}(\mathcal{G}(\sigma^m),\mathcal{G}(\tau^n))$. Every
	directed edge $((e,\bar{e}), (f,\bar{f}))$ satisfies $ht(e) > ht(f)$.
	Indeed, suppose $(e,\bar{e}) \in E(\mathcal{G}(\sigma^m))$. Let
	$[g] \subset F_r$ be the conjugacy class of a generator of
	$\mathcal{G}(\sigma^m)_e$. By
	Lemma \ref{lem:filter} (ii), there is a representative $g\in [g]$ such that
	$g^k = u_i^\ast$ for some $\sigma$-suffix $u_i$. Take a minimum
	height loop $\gamma$ representing $[g]$, so that $\gamma \subseteq
	\Gamma_{ht(\gamma)}$. Again by Lemma \ref{lem:filter} (ii), $ht(e) >
	ht(\gamma)$. Finally, by Lemma \ref{lem:filter} (i), each edge $f$ in a
	cyclically reduced $\pi_1(\mathcal{G}(\tau^n))$ representative of
	$[g]$ satisfies $ht(f) \leq ht(\gamma)$. Thus $ht(e) > ht(f)$ for each
	directed edge with origin $(e,\bar{e})$, as required. The argument for
	generators of the edge groups of
	$\mathcal{G}(\tau^n)$ is symmetric. Therefore any directed path
	in $\mathcal{ET}(\mathcal{G}(\sigma^m),\mathcal{G}(\tau^n))$ has
	monotone decreasing vertex height, which implies that $\mathcal{ET}$
	is directed acyclic. To conclude, observe that by
	Remark~\ref{rem:twist-power}
	$\mathcal{ET}(\mathcal{G}(\sigma^m),\mathcal{G}(\tau^n)) =
	\mathcal{ET}(\mathcal{G}(\sigma), \mathcal{G}(\tau))$.
\end{proof}

Cohen and Lustig~\cite{cohen-lustig-ii} give an algorithm to find efficient
representatives. Computing the edge-twist digraph and testing if it is acyclic
are straightforward computations, so the criterion in the main theorem is
algorithmic.

\section*{Acknowledgements}

I thank Mladen Bestvina for a helpful conversation that led to this note.

\begin{bibdiv}
	\begin{biblist}
		\bibselect{bibliography}
	\end{biblist}
\end{bibdiv}

\end{document}