# Virial-like Thermodynamic Uncertainty Relation in the Tight-Binding Regime

N. J. López-Alamilla${}^{\rm\,a}$ R. U. L. Cachi${}^{\rm\,a,b}$
${}^{\rm\,a\,}$Department of Physics, University of Otago, P. O. Box 56,
Dunedin 9054, New Zealand  ${}^{\rm\,b\,}$Department of Chemistry, KU Leuven,
Leuven, Belgium

(August 28, 2024)

###### Abstract

We presented a methodology to approximate the entropy production for Brownian
motion in a tilted periodic potential. The approximation stems from the well
known thermodynamic uncertainty relation. By applying a virial-like expansion,
we provided a tighter lower limit solely in terms of the drift velocity and
diffusion. The approach presented is systematically analysed in the tight-
binding regime. We also provide a relative simple rule to validate using the
tight-binding approach based on drift and diffusion relations rather than
energy barriers and forces. We also discuss the implications of our results
outside the tight-binding regime.

##  I Introduction

Recently, fluctuation theorems have attracted a great deal of research in non-
equilibrium physics Barato and Seifert (2015, 2016); Pietzonka  _et al._
(2016); Pietzonka and Seifert (2018); Gingrich  _et al._ (2016); Rosas  _et
al._ (2017); Hwang and Hyeon (2018); Brandner  _et al._ (2018); Fischer  _et
al._ (2018). In particular, the _thermodynamic uncertainty relations_ (TURs)
that provide a trade-off between the cost (entropy generation) and the
precision of the system Barato and Seifert (2015, 2016); Pietzonka  _et al._
(2016); Pietzonka and Seifert (2018). In the original derivation of the TUR
the system considered was a motor protein, Barato and Seifert (2015); Gingrich
_et al._ (2016) and it has been shown that the TUR provides a lower bound for
the entropy generation of this system. Generalizations of the TUR have been
proposed recently, for systems with multiple degrees of freedom, Dechant
(2018) quantum systems, Macieszczak  _et al._ (2018); Falasco  _et al._
(2020); Lee  _et al._ (2021) and two bodies interacting systems Saryal  _et
al._ (2021). However, experiments and analytic studies show that systems
usually operate far above the lower bound predicted by the TUR Song and Hyeon
(2020); Jack  _et al._ (2020), additionally, violations to this bound are
possible in quantum systems Paneru  _et al._ (2020); Cangemi  _et al._ (2020);
Kalaee  _et al._ (2021).

For the case of motor proteins, we can consider them as overdamped Brownian
particles converting chemical energy into work Svoboda and Block (1994);
Magnasco (1994); Astumian and Derényi (1998); Itoh  _et al._ (2004); Toyabe
_et al._ (2011); Kolomeisky (2013). Usually, the motor protein will transform
the released energy from the hydrolysis of fuel molecules (ATP or GTP) into
the mechanical motion of the motor by a specific distance ‘$\delta$’ commonly
refereed as step size ($\delta\sim 8\,{\text{nm}}$ for kinesin/dynein and
$50\sim 90\,{\text{nm}}$ for myosin, $\delta\sim 2\pi/3\,{\text{rad}}$ for
F0F1ATPase). When this chemical energy conversion is tightly-coupled, the
motion of the motor can be effectively described as one-dimensional Magnasco
(1994). In the long time limit, the system will reach a steady-state, which in
turn can be described by its rate of entropy production, drift velocity and
diffusion Challis and Jack (2013); Nguyen  _et al._ (2016); Challis (2018).

Several descriptions of the directed Brownian motion observed in motor
proteins have been proposed Astumian and Derényi (1998); Astumian (2007);
Astumian  _et al._ (2016). One widely spread approach is to consider over-
damped Brownian motion over a time-independent tilted periodic free-energy
potential. From this free-energy approach one can numerically evaluate or in
some cases derive closed solutions for steady-state dynamics Reimann  _et al._
(2001). In this paper, we will use this approach in a system for which exact
solutions exist. Next, we will briefly review the main issues surrounding the
usage of the TUR entropy production lower bound. Then, we will introduce a
formalism to derive a tighter bound based on a virial-like expansion of the
original TUR and its application.

##  II background

We are interested in the steady-state rate of entropy generation ‘$\sigma^{\rm
ss}$’ for over-damped Brownian motion over a tilted periodic potential

$V(\hat{x})=V_{0}(\hat{x})-k_{\rm B}T\hat{f}\hat{x},$ (1)

where $\hat{x}=x/\delta$ is the dimensionless position coordinate, and
$\hat{f}=f\delta/k_{\rm B}T$ the dimensionless force, driving the system out
of equilibrium, $V_{0}(\hat{x}+\delta)=V_{0}(\hat{x})$ is the periodic part,
$k_{\rm B}$ is the Boltzmann constant and $T$ the temperature. For a free-
energy potential of the form of Eq. (1), it has been shown that the rate of
entropy generation has the form Gardiner (2009); Barato and Seifert (2015)

$\sigma^{\rm ss}=k_{\rm B}\hat{v}\hat{f},$ (2)

with $\hat{v}=v/\delta$ the steady-state drift velocity. For the same system,
the TUR cost-precision trade-off ‘$\mho$’ is stated as follows Barato and
Seifert (2015, 2016)

$\mho=\sigma^{\rm ss}\times\frac{\hat{D}}{\hat{v}^{2}}\geq k_{\rm B},$ (3)

with $\hat{D}=D/\delta^{2}$ the steady-state diffusion.

From the above inequality, we obtain a lower bound ‘$\sigma^{\rm
ss}_{\textsc{tur}}$’ for the entropy generation of the system Barato and
Seifert (2015, 2016)

$\sigma^{\rm ss}\geq\sigma^{\rm ss}_{\textsc{tur}},$ (4)

with

$\sigma^{\rm ss}_{\textsc{tur}}=k_{\rm B}\frac{\hat{v}^{2}}{\hat{D}}.$ (5)

The usefulness of Eq.(4) and its tightness for several regimes and systems has
been explored Jack  _et al._ (2020). Here, we will briefly review some
important aspects of this bound.

Let us inspect the steady-state features of a model system. For simplicity, we
will assume that Eq. (1) periodic part $V_{0}(\hat{x})$ is:

$V_{0}(\hat{x})=k_{\rm B}TA_{0}\cos(2\pi(\hat{x}-\alpha)).$ (6)

With such periodic part, the potential minima points are at $\hat{x}_{\rm
c}=1/2+\arcsin{(\hat{f}/8\pi)}/2\pi\pm n$ and the maxima points are at
$\hat{x}_{\rm a}=1-\arcsin{(\hat{f}/8\pi)}/2\pi\pm n$, with $n\in\mathbb{Z}$
(we have assumed $\delta=1$). Let us denote the energy barriers in the
direction of the driving force of this potential as $E_{\rm
a}(\hat{f}\,)=V(\hat{x}_{\rm a})-V(\hat{x}_{\rm c})$, in particular $E_{\rm
a}(0)=2A_{0}$. In Figure 1 we show this free-energy potential. It is
convenient to define the mean time for the system to cross barrier $E_{\rm a}$
at equilibrium $\tau_{0}$ defined by

$\tau_{0}=\frac{\gamma}{k_{\rm B}T}\int_{\hat{x}_{\rm c}}^{\hat{x}_{\rm
a}}d\hat{y}\,e^{V_{0}(\hat{y})/k_{\rm B}T}\int_{\hat{y}}^{\hat{x}_{\rm
a}}d\hat{z}\,e^{-V_{0}(\hat{z})/k_{\rm B}T}\,,$ (7)

with $\gamma$ the viscous drag coefficient, and the associated transition rate
is $\kappa_{0}=\tau_{0}^{-1}$.

Figure 1: Scheme of energy potential Eq. (1), with periodic part Eq. (6), for
reference we show critic points of the potential and energy barrier in the
direction of the applied tilting force.

By applying a variable force on potential Eq. (1) and periodic part Eq. (6),
we can observe how the energy barrier $E_{\rm a}$, drift velocity $\hat{v}$
and diffusion $\hat{D}$ evolve. This is shown in Figure 2.

(c)(b)(a)

Figure 2: Characteristic properties of Brownian motion over an energy
potential Eq. (1) with periodic part Eq. (6) as function of the tilting force.
(a) Energy barrier. (b) Steady-state drift velocity, showing (patched)
transient regime, linear regime and (inset) tight-binding regime, for
reference (dashed) identity function for the tilting force. (c) Steady-state
diffusion. Parameters used $A_{0}=4$.

The values of $\hat{v}$, $\hat{D}$ where evaluated from the closed solutions
by Stratonovich Stratonovich (1958) and Reimann, et al Reimann  _et al._
(2001) respectively. We can identify three different regimes for the drift
velocity. These regimes correspond to the particular features observed in each
of them. In the tight-binding regime both drift and diffusion grow
exponentially with the force Gardiner (2009); Challis (2016). In the transient
regime the diffusion display an stochastic resonance Reimann  _et al._ (2001).
Finally, in the linear regime $\hat{v}\propto\hat{f}$, and $\hat{D}\rightarrow
k_{\rm B}T/\gamma$ Reimann  _et al._ (2001).

###  II.1 Entropy generation and the TUR

Now, let us compare Eqs. (2) and (5) for variable $\hat{f}$, see Figure 3.

Figure 3: Steady-state entropy production of potential Eq. (1) with periodic
part Eq. (6) as function of the tilting force, with (solid) Eq. (2) and
(dashed) Eq. (5). Inset shows the tight-binding regime. Parameters used
$A_{0}=4$.

As we can see, even in the tight-binding regime, the lower bound provided by
the TUR is a relative loose bound. One can expect that for very small force
$\hat{f}\rightarrow 0$, $\sigma^{\rm ss}\gtrsim\sigma^{\rm
ss}_{\textsc{tur}}$, implying that $\hat{f}\gtrsim\hat{v}/\hat{D}$.

Let us look at potential Eq. (1) with periodic part Eq. (6). By setting
$A_{0}\gg 1$ and $\hat{f}\approx 0$, this corresponds to the tight-binding
regime. Here, $\hat{v}$ and $\hat{D}$ have the following functional dependence
on the driving force

$\hat{v}=2\kappa_{0}\sinh\left(\frac{\hat{f}}{2}\right),$ (8)
$\hat{D}=\kappa_{0}\cosh\left(\frac{\hat{f}}{2}\right).$ (9)

In this regime the ratio $\hat{v}/\hat{D}$ becomes

$\frac{\hat{v}}{\hat{D}}=2\tanh\left(\frac{\hat{f}}{2}\right).$ (10)

If we consider the Taylor series of $\tanh(\hat{f}/2)$ for small $\hat{f}$
values we have

$2\tanh\left(\frac{\hat{f}}{2}\right)\approx\hat{f}-\frac{\hat{f}^{3}}{12}+\frac{\hat{f}^{5}}{120}-\ldots$
(11)

if $\sigma^{\rm ss}\gtrsim\sigma^{\rm ss}_{\textsc{tur}}$ Eq. (11) shows that
agrees with $\hat{f}$ as a 1st order approximation. However, the other terms
of the Taylor series are not insignificant, explaining the increasing
disagreement between Eq. (2) and Eq. (5) as $\hat{f}$ increases observed in
Figure 3. This opens the question, is there a comprehensive way to approximate
$\hat{f}$ in terms solely of $\hat{v}$ and $\hat{D}$?

##  III A tighter entropy bound using a Virial-like TUR approach

Let us start from the assumption that we can express $f$ as a series of $g$
functions that depend on both $v$ and $D$.

$\hat{f}\approx\sum_{n=1}^{N}g(a_{n}v^{n},b_{n}D^{n})+\mathcal{O}^{N+1}$ (12)

Looking at the Taylor series for $\tanh(f/2)$, it is clear that by raising to
the right power $\tanh(f/2)$ and multiplying by a proper coefficient, one can
reproduce the higher order terms in expansion Eq. (11). Thus, after some
cumbersome algebra, we find that the following sum satisfies

$\frac{\hat{f}}{2}\gtrsim\tanh\left(\frac{\hat{f}}{2}\right)+\frac{\tanh\left(\frac{\hat{f}}{2}\right)^{3}}{3}+\frac{\tanh\left(\frac{\hat{f}}{2}\right)^{5}}{5}+\ldots$
(13)

in other words,

$\lim\limits_{\hat{f}\rightarrow
0}\hat{f}\gtrsim\sum_{n=1}^{N}\frac{\hat{v}^{2n-1}}{2^{2n-2}(2n-1)\hat{D}^{2n-1}}+\mathcal{O}^{N+1}$
(14)

Then, in the regime where Eqs. (8)-(9) are valid, we can provide a better
approximation to the entropy generation of the system by implementing a higher
and higher order sum of Eq. (14), resulting in a corrected lower bound of
$N$-th order

$k_{\rm B}\hat{v}\hat{f}\geq\sigma^{N\text{-th}}_{\textsc{tur}}=k_{\rm
B}\sum_{n=1}^{N}\frac{\hat{v}^{2n}}{2^{2n-2}(2n-1)\hat{D}^{2n-1}}$ (15)

In Figure 4 we compare the bounds provided by Eq. (15) with the original TUR
bound Eq. (5) and the actual entropy generation for the system Eq. (2) in the
Tight-Binding regime. Notice that for $N=1$, Eq. (15)= Eq. (5).

Figure 4: Steady-state entropy production of potential Eq. (1) with periodic
part Eq. (6) as function of the tilting force, via: (solid) Eq. (2), (dashed)
TUR Eq. (5), (dash-dotted) Eq. (15) 2-nd order, (dotted) Eq. (15) 7-th order,
(dash-dot-dot-dashed) Eq. (15) 29-th order. Parameters used $A_{0}=4$.

One of the applications of the TUR lower bound is to be able to estimate the
entropy production in the system without having knowledge of applied force
value. Since, Eq. (15) validity depends on being in the tight-binding regime.
We need to understand how to evaluate if the system is in this regime without
prior knowledge of the force value. The relative height of the energy barrier
$E_{\rm a}$ (see Figure 1) compared to the thermal energy has previously been
used to validate the usage the tight-binding approach Challis (2016, 2018).
Based on this metric, many authors set $E_{\rm a}\geq 3k_{\rm B}T$ as a valid
regime Astumian (2007); Astumian  _et al._ (2016); Challis (2018), this
requires knowledge of the energy landscape which in many cases is unknown too.
Here, we aim to systematically determine if we are in the tight-binding regime
using only the observables of the system. Equation (15) will only converge
with increasing $N$-terms if $2\hat{D}$ and $\hat{v}$ grow at a similar rate.
Thus, we will focus on the grow rate of $2\hat{D}$ as $\hat{v}$ changes, in
other words $2\partial\hat{D}(\hat{v})/\partial\hat{v}$. Regardless of how the
drift changes, $\partial\hat{v}/\partial\hat{v}\equiv 1$, then $2\hat{D}$ and
$\hat{v}$ growing at a similar rate implies

$\frac{2\partial\hat{D}(\hat{v})}{\partial\hat{v}}\approx 1\,,$ (16)

ensuring we are in the tight-binding regime. In Figure 5 we examine
$d\hat{D}(\hat{v})/d\hat{v}$ for the model periodic potential Eq. (6), we
display this plot in a log scale for readability purposes.

Figure 5: Evaluation of tight-binding regime validity for Eq. (1) with
periodic part Eq. (6), showing: (a) Log-log plot of (solid)
$2\hat{D}(\hat{v})$ and (dashed) $\hat{v}$. (b) Semi-log plot fo (solid)
$2\partial\hat{D}(\hat{v})/\partial\hat{v}$ and (dashed)
$\partial\hat{v}/\partial\hat{v}$ for reference. Shaded region shows
$2\partial\hat{D}(\hat{v})/\partial\hat{v}\approx 1\pm 0.1$. Parameters used
$A_{0}=4$.

###  III.1 Beyond the tight-binding regime

Outside the tight-binding regime the approach presented in this paper starts
to break down. In the linear regime $\hat{v}$ grows faster than $\hat{D}$
which additionally starts to decrease, see Figure 2. Thus, it is evident that
Eq. (14) and therefore Eq. (15) will diverge as we consider larger $N$ values
in the sum. In the transient regime if we evaluate Eq. (15) up to very large
$N$ values we encounter the same issue, however by cutting down the sum at
2-nd or 3-rd order improves the approximation compared to the TUR but
ultimately diverges if the force is larger than the critic force $\hat{f}>2\pi
A_{0}$. This is shown in Figure 6.

Figure 6: Steady-state entropy production potential Eq. (1) with periodic part
Eq. (6) in the transient regime, via: (solid) Eq. (2) ,(dashed) TUR Eq. (5),
(dash-dotted) Eq. (15) 2-nd order. Parameters used $A_{0}=4$.

###  III.2 Bi chromatic potential

We now analyse the validity of Eq. (15) for a new model potential. We consider
Eq. (1) periodic part is now

$V_{0}(\hat{x})=k_{\rm B}TA_{0}\cos{(2\pi\nu_{A}\hat{x})}+k_{\rm
B}TB_{0}\cos{(2\pi\nu_{B}\hat{x})}\,.$ (17)

The different regimes for Eq. (1) and periodic part Eq. (17) as we vary the
tilting force can be identify via its diffusion López-Alamilla  _et al._
(2020), see Figure 7. Again, we compare the TUR lower bound Eq. (5) with the
bounds provided by Eq. (15) and the actual entropy generation Eq. (2) in the
tight-binding regime. This is shown in Figure 8, and in Figure 9 beyond the
tight binding regime.

Figure 7: Diffusion as function of tilting force with for Eq. (1) with
periodic part Eq. (17), showing: (patched) transient regime, (inset) the
tight-binding regime and linear regime. The horizontal axis is in log10 scale
for readability purposes. Parameters used $A_{0}=4$, $\nu_{A}=1$, $B_{0}=1.2$,
$\nu_{B}=8$. Figure 8: Steady-state entropy production for Eq. (1) with
periodic part Eq. (17) as function of the tilting force, via: (solid) Eq. (2),
(dashed) TUR Eq. (5), (dash-dotted) Eq. (15) 2-nd order, (dotted) Eq. (15)
11-th order, (dash-dot-dot-dashed) Eq. (15) 29-th order. Parameters used
$A_{0}=4$, $\nu_{A}=1$, $B_{0}=1.2$, $\nu_{B}=8$. Figure 9: Steady-state
entropy production for Eq. (1) with periodic part Eq. (17) in the transient
regime, via: (solid) Eq. (2), (dashed) TUR Eq. (5) and (dash-dotted) Eq. (15)
2-nd order. Parameters used $A_{0}=4$, $\nu_{A}=1$, $B_{0}=1.2$, $\nu_{B}=8$.

We can use the criteria Eq. (16) to quantify the range of the tight-binding
regime, for the model periodic potential Eq. (17) and the parameters values of
used in Figs. 8, 9. This is shown in Figure 10.

Figure 10: Evaluation of tight-binding regime validity for Eq. (1) with
periodic part Eq. (17), showing: (a) Log-log plot of (solid)
$2\hat{D}(\hat{v})$ and (dashed) $\hat{v}$. (b) Semi-log plot fo (solid)
$2\partial\hat{D}(\hat{v})/\partial\hat{v}$ and (dashed)
$\partial\hat{v}/\partial\hat{v}$ for reference. Shaded region shows
$2\partial\hat{D}(\hat{v})/\partial\hat{v}\approx 1\pm 0.1$. Parameters used
$A_{0}=4$, $\nu_{A}=1$, $B_{0}=1.2$, $\nu_{B}=8$.

##  IV Results

We presented a methodology to approximate the entropy production for Brownian
motion in a tilted periodic potential. The approximation stems from the well
known thermodynamic uncertainty relation. By implementing a virial-like
expansion of the original TUR we improve the approximation . Equation (15) is
particularly useful in the tight-binding regime were both drift and diffusion.
In fact, in this regime by increasing the order of the approximation it will
converge to the actual value of the entropy generation in the system, see
Figures 4, 8. It is worth mentioning that the convergence in the tight-binding
regime of Eq. (15) to Eq. (2) is quite slow, this is because in this regime
$\hat{v}/\hat{D}$ is proportional to the hyperbolic tangent of the applied
force. In the transient regime were the stochastic resonance of the diffusion
takes place, Eq. (15) improves the approximation only for small $N$ values in
the expansion, see Figures 6, 9. The results presented here for the two model
periodic parts considered can be generalized to potentials with more
periodicity components. However, the transient regime is larger for more
complicated potentials. Thus, it is expected that the force range where those
potentials are in the tight-binding regime will be smaller. Ultimately, for
any type of periodic potential in the linear regime, Eq. (15) will rapidly
diverge, except when $N=1$ which in turn returns the original TUR bound of
entropy generation.

## References

  * Barato and Seifert (2015) A. C. Barato and U. Seifert, Phys. Rev. Lett. 114, 158101 (2015).
  * Barato and Seifert (2016) A. C. Barato and U. Seifert, Phys. Rev. X 6, 041053 (2016).
  * Pietzonka  _et al._ (2016) P. Pietzonka, A. C. Barato, and U. Seifert, J. Stat. Mech: Theory Exp. 2016, 124004 (2016).
  * Pietzonka and Seifert (2018) P. Pietzonka and U. Seifert, Phys. Rev. Lett. 120, 190602 (2018).
  * Gingrich  _et al._ (2016) T. R. Gingrich, J. M. Horowitz, N. Perunov, and J. L. England, Phys. Rev. Lett. 116, 120601 (2016).
  * Rosas  _et al._ (2017) A. Rosas, C. Van den Broeck, and K. Lindenberg, Phys. Rev. E 96, 052135 (2017).
  * Hwang and Hyeon (2018) W. Hwang and C. Hyeon, J. Phys. Chem. Lett. 9, 513 (2018).
  * Brandner  _et al._ (2018) K. Brandner, T. Hanazato, and K. Saito, Phys. Rev. Lett. 120, 090601 (2018).
  * Fischer  _et al._ (2018) L. P. Fischer, P. Pietzonka, and U. Seifert, Phys. Rev. E 97, 022143 (2018).
  * Dechant (2018) A. Dechant, J. Phys. A: Math. Theor. 52, 035001 (2018).
  * Macieszczak  _et al._ (2018) K. Macieszczak, K. Brandner, and J. P. Garrahan, Phys. Rev. Lett. 121, 130601 (2018).
  * Falasco  _et al._ (2020) G. Falasco, M. Esposito, and J.-C. Delvenne, New J. Phys. 22, 053046 (2020).
  * Lee  _et al._ (2021) S. Lee, M. Ha, and H. Jeong, Phys. Rev. E 103, 022136 (2021).
  * Saryal  _et al._ (2021) S. Saryal, O. Sadekar, and B. K. Agarwalla, Phys. Rev. E 103, 022141 (2021).
  * Song and Hyeon (2020) Y. Song and C. Hyeon, J. Phys. Chem. Lett. 11, 3136 (2020).
  * Jack  _et al._ (2020) M. W. Jack, N. J. López-Alamilla, and K. J. Challis, Phys. Rev. E 101, 062123 (2020).
  * Paneru  _et al._ (2020) G. Paneru, S. Dutta, T. Tlusty, and H. K. Pak, Phys. Rev. E 102, 032126 (2020).
  * Cangemi  _et al._ (2020) L. M. Cangemi, V. Cataudella, G. Benenti, M. Sassetti, and G. De Filippis, Phys. Rev. B 102, 165418 (2020).
  * Kalaee  _et al._ (2021) A. A. S. Kalaee, A. Wacker, and P. P. Potts, “Violating the thermodynamic uncertainty relation in the three-level maser,” (2021), arXiv:2103.07791 [quant-ph] .
  * Svoboda and Block (1994) K. Svoboda and S. M. Block, Cell 77, 773 (1994).
  * Magnasco (1994) M. O. Magnasco, Phys. Rev. Lett. 72, 2656 (1994).
  * Astumian and Derényi (1998) R. D. Astumian and I. Derényi, Eur. Biophys. J. 27, 474 (1998).
  * Itoh  _et al._ (2004) H. Itoh, A. Takahashi, K. Adachi, H. Noji, R. Yasuda, M. Yoshida, and K. Kinosita, Nature 427, 465 (2004).
  * Toyabe  _et al._ (2011) S. Toyabe, T. Watanabe-Nakayama, T. Okamoto, S. Kudo, and E. Muneyukia, Proc. Natl. Acad. Sci. 108, 17951 (2011).
  * Kolomeisky (2013) A. B. Kolomeisky, J. Phys.: Condens. Matter 25, 463101 (2013).
  * Challis and Jack (2013) K. J. Challis and M. W. Jack, Phys. Rev. E 88, 062136 (2013).
  * Nguyen  _et al._ (2016) P. T. T. Nguyen, K. J. Challis, and M. W. Jack, Phys. Rev. E 94, 052127 (2016).
  * Challis (2018) K. J. Challis, Phys. Rev. E 97, 062158 (2018).
  * Astumian (2007) R. D. Astumian, Phys. Chem. Chem. Phys. 9, 5067 (2007).
  * Astumian  _et al._ (2016) R. D. Astumian, S. Mukherjee, and A. Warshel, ChemPhysChem 17, 1719 (2016), https://chemistry-europe.onlinelibrary.wiley.com/doi/pdf/10.1002/cphc.201600184 .
  * Reimann  _et al._ (2001) P. Reimann, C. Van den Broeck, H. Linke, P. Hänggi, J. M. Rubi, and A. Pérez-Madrid, Phys. Rev. Lett. 87, 010602 (2001).
  * Gardiner (2009) C. W. Gardiner,  _Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences_ , 4th ed. (Springer, 2009).
  * Stratonovich (1958) R. L. Stratonovich, Radiotekh Elektron (Moscow) 3, 497 (1958).
  * Challis (2016) K. J. Challis, Phys. Rev. E 94, 062123 (2016).
  * López-Alamilla  _et al._ (2020) N. J. López-Alamilla, M. W. Jack, and K. J. Challis, Phys. Rev. E 102, 042405 (2020).