Gauge fixing and field redefinitions of the
Hartle–Hawking wavefunction path integral***Based on a talk given at “Beyond
Standard Model: From Theory to Experiments” (BSM-2021), 29 March – 2 April
2021, online.

Hervé<EMAIL_ADDRESS>Nicolaos
<EMAIL_ADDRESS>and Balthazar de
<EMAIL_ADDRESS>
1 CPHT, CNRS, Ecole polytechnique, IP Paris,
F-91128 Palaiseau, France

2 Department of Physics, University of Cyprus,
Nicosia 1678, Cyprus

Abstract

> We review some recent results concerning the Hartle–Hawking wavefunction of
> the universe. We focus on pure Einstein theory of gravity in the presence of
> a positive cosmological constant. We carefully implement the gauge-fixing
> procedure for the minisuperspace path integral, by identifying the single
> modulus and by using diffeomorphism-invariant measures for the ghosts and
> the scale factor. Field redefinitions of the scale factor yield different
> prescriptions for computing the no-boundary ground-state wavefunction. They
> give rise to an infinite set of ground-state wavefunctions, each satisfying
> a different Wheeler–DeWitt equation, at the semi-classical level. The
> differences in the form of the Wheeler–DeWitt equations can be traced to
> ordering ambiguities in constructing the Hamiltonian upon canonical
> quantization. However, the inner products of the corresponding Hilbert
> spaces turn out to be equivalent, at least semi-classically. Thus, the model
> yields universal quantum predictions.

##  1 Introduction

The inflationary universe scenario [1, 2, 3] has been very successful in
accounting for key cosmological puzzles of the hot Big Bang model, such as the
flatness, the horizon and monopole problems, and in generating the primordial
fluctuations that eventually led to the large scale structure observed today
[4]. Furthermore, inflation has gained support by observational data
concerning the anisotropies in the cosmic microwave background radiation.
Despite the many successes however, there are still open questions regarding
the UV completion of inflationary models and a lack of understanding of the
initial conditions from first principles. To date there is no complete
embedding of phenomenologically viable inflationary models in string
theory.111See e.g. [5] for seminal work towards this end. We also lack of a
definite understanding of how the Universe could have entered naturally into
such an inflationary phase in the early past. It is likely that a quantum,
probabilistic explanation exists, in terms of a wavefunction that favors
suitable conditions to initiate inflation. See e.g. [6, 7, 8, 9, 10, 11, 12,
13, 14, 15] for different perspectives and discussions.

A very appealing possibility to explore is to apply the no-boundary proposal
of Hartle and Hawking [7]. In this context, the wavefunction of the universe
is computed via a Euclidean path integral over all compact four-geometries
that end on a particular spatial slice. The induced metric on this slice and
the value of the inflaton field are fixed to be $h_{ij}$ and $\phi_{0}$,
respectively. The four-geometries summed over should have no boundaries other
than that of metric $h_{ij}$. As a result the wavefunction is expressed as a
functional of $h_{ij}$ and $\phi_{0}$. We refer to this wavefunction as the
“ground state” wavefunction, even though such a denomination may not be
appropriate since in quantum gravity all physical quantum states associated
with a closed universe are annihilated by the Hamiltonian. In fact, the
Hartle–Hawking wavefunction can be interpreted as a probability amplitude to
create from nothing a three-dimensional universe with metric $h_{ij}$ and
inflaton field $\phi_{0}$ [8, 9, 10, 11, 13]. As argued by Vilenkin [8, 9, 10,
11, 13] and also by Linde [12, 16] some time ago, a suitable continuation to
Euclidean time yields probability amplitudes favoring inflation.

In this work we revisit the Hartle–Hawking no-boundary proposal in the context
of pure Einstein’s theory of gravity with a positive cosmological constant
$\Lambda>0$. Our goal is to discuss a number of issues pertaining to this path
integral approach to quantum cosmology in a rather simpler setting, before
delving into analyzing more complex cosmological models in the presence of
matter (including inflationary ones). Indeed, in the minisuperspace
approximation, where the universe is taken to be homogeneous and isotropic,
the degrees of freedom reduce to a single scale factor depending only on time.
The issues we would like to discuss were recently raised in [17], in the
context of the minisuperspace approximation, and concern i) the proper gauge
fixing of the local symmetry group associated with time-reparametrization
invariance; ii) the construction of an infinite set of “ground state”
wavefunctions based on field redefinitions of the scale factor degree of
freedom; iii) the derivation of the corresponding Wheeler–DeWitt equations
[6]; and finally, iv) the equivalence of these prescriptions at the
semiclassical level and observable predictions. To our knowledge, these points
have not been adequately addressed in the literature before. We believe they
will prove to be important in properly applying the no-boundary proposal to
obtain probabilities in cosmological, inflationary settings.222Previous work
on the Hartle–Hawking wavefunction, related to our discussions but with some
different results, includes [18, 19, 20]. Further work and applications can be
found in [21, 22, 23, 24, 25, 26, 27].

To illustrate these issues, it is convenient to interpret the minisuperspace
model as a non-linear sigma model, where the Euclidean time parameterizes the
base manifold, which is a line segment. The scale factor parameterizes a one-
dimensional target space, which is a half line.

The theory is invariant under time-reparametrizations of the base manifold. In
section 2, we implement the gauge fixing procedure of Euclidean-time
reparameterizations. The path integral over the lapse function reduces to an
integral over the modulus of the base manifold, which can be identified to be
the proper length of the line segment. We express the Faddeev–Popov
determinant as a path integral over anticommuting ghost fields, and compute it
to be a constant, independent of the modulus of the line segment. It is
important to use gauge invariant measures in both the ghost and scale factor
path integrals to implement the gauge-fixing properly.

Field redefinitions of the scale factor, $a=A(q)$, amount to
reparameterizations of the target space and leave the classical sigma model
action invariant. At the quantum level, the path integral measures ${\cal D}a$
and ${\cal D}q$ are not equivalent in general, since they are related by a
non-trivial Jacobian. Since there is no preferred choice, an infinite number
of ground-state wavefunctions can be constructed, upon implementing the no-
boundary proposal (based on the different measures ${\cal D}q$). In section 3,
we compute the ground-state wavefunction for each choice of ${\cal D}q$, using
the steepest-descent method, expanding around instanton solutions to quadratic
order. The path integral over the fluctuations are obtained by applying the
methods of Ref. [28] – see also [29].

We proceed in section 4 to determine the Wheeler–DeWitt equation each ground
state wavefunction satisfies. Recall that there is an ambiguity in the exact
form of the Wheeler–DeWitt equation, due to an ordering ambiguity of $q$ and
its conjugate momentum $\pi_{q}$ in the quantum Hamiltonian. For each ${\cal
D}q$, we resolve this ambiguity in the Wheeler–DeWitt equation by comparing
with solutions via the WKB approximation. The inner product in each case is
determined by imposing hermiticity of the corresponding Hamiltonian. Despite
the fact that the precise form of the inner product depends on the choice
${\cal D}q$, the norms of the wavefunctions at the semiclassical level turn
out to be the same, leading to universal predictions, independent of the
${\cal D}q$ prescription. For the particular model at hand, the norm of the
wavefunctions turns out to be logarithmically divergent. At best, these
wavefunctions can be used to discuss relative probabilities. We conclude in
section 5 with further discussion and perspectives. Throughout we work in
Planck units, setting $M_{p}=\sqrt{8\pi G}=1$.

##  2 The ground-state wavefunction as a gauge fixed path integral

The Lorentzian theory is formulated on four-manifolds with space-like
boundaries at initial and final times. The slices at constant time $x^{0}$ are
taken to be compact and closed. In the minisuperspace approximation, these
slices are restricted to be homogeneous and isotropic $3$-spheres. As a
result, the physical degrees of freedom reduce to to a single scale factor
depending on time, $a(x^{0})$. The

metric is given by

${\rm d}s^{2}=-N(x^{0})^{2}({\rm d}x^{0})^{2}+a(x^{0})^{2}\,{\rm
d}\Omega_{3}^{2}\,,$ (2.1)

where $N(x^{0})\equiv\sqrt{g_{00}(x^{0})}$ is the lapse function and ${\rm
d}\Omega_{3}$ is the volume element of the unit 3-sphere of volume
$v_{3}=2\pi^{2}$. Einstein’s action, in the presence of a non-zero positive
cosmological constant $\Lambda$, takes the form

$S=3v_{3}\int_{x^{0}_{\rm i}}^{x^{0}_{\rm f}}{\rm
d}x^{0}\,N\bigg{[}\\!-\\!{a\over N^{2}}\Big{(}{{\rm d}a\over{\rm
d}x^{0}}\Big{)}^{2}+a-\lambda^{2}a^{3}\bigg{]}\,,\quad\mbox{where}\quad\lambda=\sqrt{\Lambda\over
3}\,.$ (2.2)

The classical equations of motion can be obtained by varying the action,
keeping the scale factor at initial and final times, $x^{0}_{\rm i}$ and
$x^{0}_{\rm f}$, fixed.333The boundary action cancels upon integrating by
parts a bulk term that involves the second derivative of the scale factor, see
e.g. [17] for details.

Notice that the kinetic energy term of the scale factor has a negative sign
compared to that of a conventional matter scalar field. This fact motivates us
to consider two alternative prescriptions for the continuation to Euclidean
time,

$x^{0}=s\,i\,x^{0}_{\rm E}\,,\leavevmode\nobreak\ \leavevmode\nobreak\
\quad\mbox{where}\leavevmode\nobreak\ \leavevmode\nobreak\ \quad
s\in\\{1,-1\\}\,,$ (2.3)

both of which have been advocated in the literature. Hartle and Hawking [7]
adopt the conventional prescription $s=-1$. In this case, the no-boundary
wavefunctions become large as $\lambda\to 0$, and so they seem to favor a
vanishing cosmological constant [14]. On the other hand, Vilenkin [8, 9, 10,
11, 13] and Linde [12, 16] have argued for $s=+1$, which favors conditions
amenable for inflation. The Euclidean action $S_{\rm E}=-iS$ in each case is
given by

$S_{\rm E}[g_{00},a]=3sv_{3}\int_{x^{0}_{\rm Ei}}^{x^{0}_{\rm Ef}}{\rm
d}x^{0}_{\rm E}\,\sqrt{g_{00}}\,\bigg{[}a\,g^{00}\Big{(}{{\rm d}a\over{\rm
d}x^{0}_{\rm E}}\Big{)}^{2}+V(a)\bigg{]}\,,$ (2.4)

where

$V(a)=a-\lambda^{2}a^{3}\,.$ (2.5)

This potential becomes negative when $\lambda a>1$. When $s=-1$, the action
can become arbitrarily large and negative due to rapidly oscillating
configurations of the scale factor. On the other hand, for $s=1$, there are
time-independent configurations, satisfying $\lambda a\gg 1$, that yield
arbitrarily large negative values for the action. We see that both choices
yield Euclidean actions, which are not bounded from below, and thus a suitable
continuation will be needed to obtain convergent path integrals.

Based on the form of the action $S_{\rm E}$, we interpret the theory as a non-
linear $\sigma$-model. The base manifold is a line segment of metric $g_{00}$,
parameterized by the Euclidean time $x^{0}$. The one-dimensional target space
is parameterized by the scale factor $a$. The metric is given by

$G_{aa}=6v_{3}a\,.$ (2.6)

The local symmetry group consists of Euclidean-time diffeomorphisms of the
base manifold. Under such a coordinate change, the metric $g_{00}$ transforms
as a tensor and the scale factor as a scalar:

$\xi(x^{0}_{\rm E})=x^{\xi 0}_{\rm E}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \quad g^{\xi}_{00}(x^{\xi 0}_{\rm E})=\bigg{(}{{\rm
d}x^{0}_{\rm E}\over{\rm d}x^{\xi 0}_{\rm E}}\bigg{)}^{2}\,g_{00}(x^{0}_{\rm
E})\,,\leavevmode\nobreak\ \leavevmode\nobreak\ \quad a^{\xi}(x^{\xi 0}_{\rm
E})=a(x^{0}_{\rm E})\,.$ (2.7)

In addition, the action is invariant under field redefinitions of the scale
factor, $a=A(q)$, which can be interpreted as reparameterizations of the
target space.

In order to implement the no-boundary proposal, we take the initial boundary
$3$-sphere to have vanishing radius, $a(x^{0}_{\rm Ei})\equiv a_{\rm i}=0$,
and fix the radius of the final sphere to an arbitrary value: $a(x^{0}_{\rm
Ef})\equiv a_{\rm f}=a_{0}$. We then define the ground-state wavefunction to
be given by the following Euclidean path integral [7]

$\Psi(a_{0})=\int{{\cal D}g_{00}\over{\rm Vol}({\rm
Diff}[g_{00}])}\int_{\textstyle{\,a_{\rm i}=0,\,\,a_{\rm f}=a_{0}}}\,\,{\cal
D}a\,e^{-{1\over\hbar}S_{\rm E}[g_{00},a]}\,,$ (2.8)

where we kept explicit the reduced Planck constant $\hbar$. According to
Vilenkin, the wavefunction thus defined can be interpreted as the probability
amplitude for creating a $3$-dimensional spherical universe of radius $a_{0}$
from nothing [8, 9, 10, 11]. Notice that we have divided the measure ${\cal
D}g_{00}$ in the path integral by the volume of the local symmetry group,
${\rm Vol}({\rm Diff}[g_{00}])$, in order to take care of the overcounting of
physical configurations, yielded by diffeomorphism-equivalent metrics
$g_{00}$. The measure ${\cal D}a$ must be invariant under Euclidean-time
diffeomorphisms. Such a gauge-invariant measure, however, is far from being
unique. As we will see later on, field redefinitions of the scale factor
provide us with an infinite set of inequivalent diffeomorphism-invariant
measures, ${\cal D}q$, leading to an infinite set of alternative definitions
for the ground state wavefunction.

We proceed now to discuss the gauge-fixing procedure, which allows us to
express the wavefunction Eq. (2.8) as an integral over physically distinct
configurations. First notice that not all metrics $g_{00}$ are diffeomorphism-
equivalent, since the proper length $\ell$ of the line segment remains
invariant under such transformations

$\ell=\int_{x_{\rm Ei}}^{x_{\rm Ef}}{\rm d}x^{0}_{\rm
E}\,\sqrt{g_{00}}=\int_{\xi(x_{\rm Ei})}^{\xi(x_{\rm Ef})}{\rm d}x^{\xi
0}_{\rm E}\,\sqrt{g^{\xi}_{00}}\,.$ (2.9)

Thus, the proper length $\ell$ behaves as a modulus, and its value can be used
to distinguish the classes of diffeomorphism-equivalent metrics. In [17] we
show that the line segment has no other moduli than the proper length $\ell$.
Let $\hat{g}_{00}[1]$, defined on a domain $[\hat{x}^{0}_{\rm
Ei},\hat{x}^{0}_{\rm Ef}]$, be a fiducial metric representing the class
$\ell=1$. Then all the other equivalence classes can be represented by
fiducial metrics of the form $\hat{g}_{00}[\ell]=\ell^{2}\hat{g}_{00}[1]$,
defined on the same interval $[\hat{x}^{0}_{\rm Ei},\hat{x}^{0}_{\rm
Ef}]$.444The Killing group of metric isometries reduces to a discrete
$\mathbb{Z}_{2}$ group, generated by the transformation that reverses the
orientation of the line segment.

Choosing such a metric $\hat{g}_{00}[\ell]$ for each equivalence class, we
insert in Eq. (2.8) a gauge fixing condition

$1=\Delta_{\rm FP}[g_{00}]\int_{0}^{+\infty}\\!\\!{\rm d}\ell\int_{{\rm
Diff}[\hat{g}_{00}[\ell]]}\\!\\!{\cal
D}\xi\;\delta\big{[}g_{00}-\hat{g}^{\xi}_{00}[\ell]\big{]}\,,$ (2.10)

where $\Delta_{\rm FP}[g_{00}]$ is the Faddeev–Popov determinant, which is
gauge invariant. Then integrating over $g_{00}$ fixes the metric to be
$\hat{g}^{\xi}_{00}[\ell]$ (defined on $[\xi(\hat{x}^{0}_{\rm
Ei}),\xi(\hat{x}^{0}_{\rm Ef})]$), as implied by the Dirac
$\delta$-functional, while integrating over the gauge orbits, together with
gauge invariance, lead to the cancellation of the volume of the local symmetry
group ${\rm Vol}({\rm Diff}[g_{00}])$. The wavefunction simplifies as follows

$\Psi(a_{0})=\int_{0}^{+\infty}\\!\\!{\rm d}\ell\;\Delta_{\rm
FP}[\hat{g}_{00}[\ell]]\int_{a(\hat{x}^{0}_{\rm
Ei})=0,\,\,\,a(\hat{x}^{0}_{\rm Ef})=a_{0}}\,{\cal
D}a\,e^{-{1\over\hbar}S_{\rm E}[\hat{g}_{00}[\ell],a]}\,,$ (2.11)

where the integral over the modulus $\ell$ is an ordinary integral.

The Faddeev–Popov determinant appearing in the expression above can be related
to a path integral over the diffeomorphisms that are connected to the identity
as follows555Due to the fact that the orientation reversal is the only Killing
isometry, the path integral over all diffeomorphisms is twice the contribution
of the diffeomorphisms connected to the identity.

${1\over\Delta_{\rm FP}[\hat{g}_{00}[\ell]]}=2\int_{0}^{+\infty}\,\,{\rm
d}\ell^{\prime}\int_{{\rm Diff}[\hat{g}_{00}[\ell^{\prime}]]_{{\rm
Id}}}\,\,{\cal
D}\xi\,\delta\big{[}\hat{g}_{00}[\ell]-\hat{g}^{\xi}_{00}[\ell^{\prime}]\big{]}\,.$
(2.12)

To compute it, we first examine the total variation of the metric
$\hat{g}_{00}[\ell]$ under infinitesimal diffeomorphisms in the vicinity of
the identity and small changes of the modulus field,

$\delta\hat{g}_{00}[\ell]\equiv\hat{g}^{{\rm
Id}+\delta\xi}_{00}[\ell+\delta\ell]-\hat{g}_{00}[\ell]=-2\hat{\nabla}_{0}\delta
x_{\rm E0}+2\hat{g}_{00}[\ell]\,{\delta\ell\over\ell}+\cdots\,,$ (2.13)

where $\hat{\nabla}$ is the covariant derivative with respect to
$\hat{g}_{00}[\ell]$. Then we introduce anticommuting ghost fields. Two such
fields are needed, $c_{0}$ corresponding to $\delta x_{\rm E0}$ and $b^{00}$
corresponding to the tensor field $\beta^{00}$ needed to express the
$\delta$-functional as a Fourier integral [17]. Moreover, one introduces an
anticommuting variable $\lambda$ corresponding to $\delta\ell$. Berezin
integration over $\lambda$ yields the following path integral expression [17]

$\Delta_{\rm FP}[\hat{g}_{00}[\ell]]=2i\pi\alpha\int_{c^{0}(\hat{x}^{0}_{\rm
Ei})=0,\,\,c^{0}(\hat{x}^{0}_{\rm Ef})=0}{\cal D}c\int{\cal
D}b\,\Big{(}b,{\hat{g}[\ell]\over\ell}\Big{)}_{\ell}\,\exp\Big{\\{}4i\pi\,(b,\hat{\nabla}c)_{\ell}\Big{\\}}\,,$
(2.14)

where $\alpha$ is an irrelevant constant and the tensor inner product is given
by

$(f,h)_{\ell}\equiv\displaystyle\int_{\hat{x}^{0}_{\rm Ei}}^{\hat{x}^{0}_{\rm
Ef}}{\rm d}\hat{x}^{0}_{\rm E}\,\sqrt{\hat{g}_{00}[\ell]}\;f^{00}\,h_{00}\,.$
(2.15)

The ghost path integrals can be readily computed by expanding the ghost fields
in Fourier modes on the interval $[\hat{x}^{0}_{\rm Ei},\hat{x}^{0}_{\rm
Ef}]$. To achieve this, we must take into account the boundary conditions and
use gauge invariant measures – we refer the reader to [17] for the detailed
computations. The Faddeev–Popov determinant turns out to be a constant,
independent of the modulus $\ell$. This is to be contrasted with the case of a
base manifold with the topology of a circle, where the Faddeev–Popov
determinant is non-trivial, being proportional to $1/\ell$, where $\ell$ is
the proper length of the circle. As a result the wavefunction further
simplifies to the following gauge-fixed path-integral expression

$\Psi(a_{0})=\Delta_{\rm FP}\int_{0}^{+\infty}\\!\\!{\rm
d}\ell\;\int_{a(\hat{x}^{0}_{\rm Ei})=0,\,\,\,a(\hat{x}^{0}_{\rm
Ef})=a_{0}}\,{\cal D}a\,e^{-{1\over\hbar}S_{\rm E}[\hat{g}_{00}[\ell],a]}\,,$
(2.16)

where $\Delta_{\rm FP}$ is an irrelevant constant.

##  3 Scale factor path integral and field redefinitions

Next we compute the path integral over the scale factor and the integral over
the modulus $\ell$. Since the path-integral expression (2.16) for the
wavefunction is gauge invariant, we choose to work in a convenient gauge,
setting the lapse function to be a constant,

$\hat{g}_{00}[\ell](\tau)=\ell^{2}\quad\mbox{defined on}\quad[\hat{x}^{0}_{\rm
Ei},\hat{x}^{0}_{\rm Ef}]=[0,1]\,.$ (3.17)

The Euclidean-time coordinate $\hat{x}^{0}_{\rm E}$ is denoted by $\tau$. This
time variable is proportional to the “cosmological Euclidean time $t_{\rm
E}$,” which satisfies ${\rm d}t_{\rm E}=\ell\,{\rm d}\tau$. The wavefunction
becomes

$\Psi(a_{0})=\Delta_{\rm FP}\int_{0}^{+\infty}\,{\rm
d}\ell\int_{a(0)=0,\,\,a(1)=a_{0}}\,{\cal D}a\,e^{-{1\over\hbar}S_{\rm
E}[\ell^{2},a]}\,,$ (3.18)

with the action (2.4) written as

$S_{\rm E}[\ell^{2},a]=3sv_{3}\int_{0}^{1}{\rm
d}\tau\,\bigg{[}{a\over\ell}\Big{(}{{\rm d}a\over{\rm
d}\tau}\Big{)}^{2}+\ell\,V(a)\bigg{]}\,.$ (3.19)

This action is not quadratic, and so we will approximate the path integral via
the method of steepest-descent. To this end, we first expand the action around
its extrema to quadratic order, and then carry out the resulting Gaussian
integrals over the fluctuations. This steepest-descent approximation becomes
accurate in the semiclassical limit, where $\hbar\to 0$.

Let us denote an extremum of the action by $(\bar{\ell}^{2},\bar{a})$, where
we require the solution $\bar{a}$ to satisfy the boundary conditions
$\bar{a}(0)=0$ and $\bar{a}(1)=a_{0}$. Varying with respect to the modulus
$\ell$ gives

$0=\\!\left.{{\rm d}S_{\rm E}\over{\rm
d}\ell}\right|_{(\bar{\ell}^{2},\bar{a})}=3sv_{3}\int_{0}^{1}{\rm
d}\tau\,\bigg{[}\\!-\\!{\bar{a}\over\bar{\ell}^{2}}\Big{(}{{\rm
d}\bar{a}\over{\rm d}\tau}\Big{)}^{2}+V(\bar{a})\bigg{]}\,,$ (3.20)

while the equation of motion of the scale factor can be integrated to be

$-{\bar{a}\over\bar{\ell}^{2}}\Big{(}{{\rm d}\bar{a}\over{\rm
d}\tau}\Big{)}^{2}+V(\bar{a})={{\cal E}\over 3v_{3}}\,,$ (3.21)

where ${\cal E}$ is an arbitrary integration constant. Eq. (3.20) implies
$0=s{\cal E}$, and so it suffices to solve the Friedmann equation in order to
determine the extrema of the action.

It is useful to write the Friedmann equation in the form

$\left({{\rm d}(\lambda\bar{a})\over{\rm
d}(\lambda\bar{\ell}\tau)}\right)^{2}+(\lambda\bar{a})^{2}=1\,,$ (3.22)

with solution $\lambda\bar{a}(\tau)=\pm\sin(\lambda\bar{\ell}\tau+{\rm
cst.})$. The boundary conditions $\bar{a}(0)=0$, $\bar{a}(1)=a_{0}$ set the
constant to be zero and fix the modulus $\bar{\ell}$. In this work and in
[17], we consider the case

$0<\lambda a_{0}<1\,,$ (3.23)

leaving the case $\lambda a_{0}>1$ for future work. Then, there are two real
instanton solutions

$\displaystyle\lambda\bar{a}_{\epsilon}(\tau)\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\sin(\lambda\bar{\ell}_{\epsilon}\tau)\,,\quad\epsilon\in\\{+1,-1\\}\,,$
$\displaystyle\mbox{where}\quad\leavevmode\nobreak\ \leavevmode\nobreak\
\lambda\bar{\ell}_{+}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\arcsin(\lambda
a_{0})\,,\quad\lambda\bar{\ell}_{-}=\pi-\arcsin(\lambda a_{0})\,,$ (3.24)

corresponding to parts of a $4$-sphere of radius $1/\lambda$. The
$\epsilon=+1$ solution describes a cap smaller than a hemisphere, while the
$\epsilon=-1$ solution describes a cap bigger than a hemisphere. The instanton
actions are given by

$\bar{S}_{\rm
E}^{\epsilon}=s\,{2v_{3}\over\lambda^{2}}\Big{[}1-\epsilon\big{(}1-(\lambda
a_{0})^{2}\big{)}^{3\over 2}\Big{]}\,.$ (3.25)

We now proceed to expand the action around the extremal solutions. We set

$\ell=\bar{\ell}_{\epsilon}+\delta\ell\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \quad a(\tau)=\bar{a}_{\epsilon}(\tau)+\delta a(\tau)\,,$
(3.26)

where the fluctuation $\delta a(\tau)$ satisfies the boundary conditions
$\delta a(0)=0$ and $\delta a(1)=0$. Thus, $\delta a(\tau)$ is an element of
the Hilbert space of square integrable real functions on $[0,1]$, vanishing at
the boundary points. This space of functions is equipped with the inner
product

$(\delta a_{1},\delta a_{2})_{\bar{\ell}_{\epsilon}}=\int_{0}^{1}{\rm
d}\tau\bar{\ell}_{\epsilon}\,\delta a_{1}\,\delta a_{2}\,.$ (3.27)

As a result, $\delta a(\tau)$ can be expanded in terms of the orthonormal sine
Fourier series

$\big{\\{}\sqrt{2/\bar{\ell}_{\epsilon}}\sin(k\pi\tau),k\in\mathbb{N}^{*}\big{\\}}\,.$
(3.28)

To quadratic order in the fluctuations, the action (3.19) can be written as
follows

$S_{\rm E}[\ell^{2},a]=\bar{S}_{\rm E}^{\epsilon}+3sv_{3}\int_{0}^{1}{\rm
d}\tau\bar{\ell}_{\epsilon}\Big{[}\delta a\,{\cal S}_{\epsilon}\delta
a+2\,\delta
a\,V_{a}(\bar{a}_{\epsilon})\,{\delta\ell\over\bar{\ell}_{\epsilon}}+{\delta\ell\over\bar{\ell}_{\epsilon}}\,V(\bar{a}_{\epsilon})\,{\delta\ell\over\bar{\ell}_{\epsilon}}\Big{]}\\!+{\cal
O}(\delta^{3})\,.$ (3.29)

where the linear operator ${\cal S}_{\epsilon}$, given by

${\cal
S}_{\epsilon}=-{\bar{a}_{\epsilon}\over\bar{\ell}_{\epsilon}^{2}}\,{{\rm
d}^{2}\over{\rm d}\tau^{2}}-{1\over\bar{\ell}_{\epsilon}^{2}}\,{{\rm
d}\bar{a}_{\epsilon}\over{\rm d}\tau}\,{{\rm d}\over{\rm
d}\tau}-2\lambda^{2}\bar{a}_{\epsilon}\,,$ (3.30)

is self-adjoint with respect to the inner product (3.27): $(\delta a_{1},{\cal
S}_{\epsilon}\delta a_{2})_{\bar{\ell}_{\epsilon}}=({\cal S}_{\epsilon}\delta
a_{1},\delta a_{2})_{\bar{\ell}_{\epsilon}}$. Here also, $V_{a}\equiv{\rm
d}V/{\rm d}a$. Moreover, as will be seen later on, this operator is invertible
when $0<\lambda a_{0}<1$, a fact that allows us to diagonalize the integrand
in Eq. (3.29). For this purpose we set666We consider the odd periodic
extension of the function $V_{a}(\bar{a}_{\epsilon})$ on the real line, so
that both $\delta a$ and $\delta\check{a}$ can be expanded in terms of the
same sine Fourier series.

$\displaystyle\delta a\,{\cal S}_{\epsilon}\delta a+2\,\delta
a\,V_{a}(\bar{a}_{\epsilon})\,{\delta\ell\over\bar{\ell}_{\epsilon}}+{\delta\ell\over\bar{\ell}_{\epsilon}}\,V(\bar{a}_{\epsilon})\,{\delta\ell\over\bar{\ell}_{\epsilon}}\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\delta\check{a}\,{\cal
S}_{\epsilon}\delta\check{a}+{\delta\ell\over\bar{\ell}_{\epsilon}}\big{[}V(\bar{a}_{\epsilon})-V_{a}(\bar{a}_{\epsilon}){\cal
S}_{\epsilon}^{-1}V_{a}(\bar{a}_{\epsilon})\big{]}{\delta\ell\over\bar{\ell}_{\epsilon}}\,,$
$\displaystyle\mbox{where}\quad\delta\check{a}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\delta a+{\delta\ell\over\bar{\ell}_{\epsilon}}\,{\cal
S}_{\epsilon}^{-1}V_{a}(\bar{a}_{\epsilon})\,.$ (3.31)

Using Eq. (3.31) and defining

${\cal K}_{\epsilon}=\int_{0}^{1}{\rm
d}\tau\bar{\ell}_{\epsilon}\,\big{[}V(\bar{a}_{\epsilon})-V_{a}(\bar{a}_{\epsilon}){\cal
S}_{\epsilon}^{-1}V_{a}(\bar{a}_{\epsilon})\big{]}\,,$ (3.32)

we obtain the following expression for the wavefunction (3.18) in the
steepest-descent approximation,

$\displaystyle\Psi(a_{0})\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\Delta_{\rm FP}\sum_{\epsilon=\pm
1}e^{-{1\over\hbar}\bar{S}_{\rm E}^{\epsilon}}\,Z_{\epsilon}(a_{0})\int{\rm
d}\delta\ell\,\exp\Big{\\{}\\!-\\!{3sv_{3}\over\hbar}\,{\cal
K}_{\epsilon}\Big{(}{\delta\ell\over\bar{\ell}_{\epsilon}}\Big{)}^{2}\Big{\\}}\,(1+{\cal
O}(\hbar))$ $\displaystyle\mbox{where}\quad Z_{\epsilon}(a_{0})\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\int_{\delta\check{a}(0)=0,\,\,\delta\check{a}(1)=0}{\cal
D}\delta\check{a}\,\exp\Big{\\{}\\!-\\!{3sv_{3}\over\hbar}\,(\delta\check{a},{\cal
S}_{\epsilon}\delta\check{a})_{\bar{\ell}_{\epsilon}}\Big{\\}}\,.$ (3.33)

The operator ${\cal S}_{\epsilon}$ is self-adjoint and so it can be
diagonalized in an orthonormal basis. Let us denote its eigenvectors by
$\phi^{\epsilon}_{k}$ and the corresponding eigenvalues by
$\nu_{k}^{\epsilon}$. These satisfy

${\cal
S}_{\epsilon}\phi^{\epsilon}_{k}=\nu^{\epsilon}_{k}\phi^{\epsilon}_{k}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ k\in\mathbb{N}^{*}\,,\quad\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{where}\leavevmode\nobreak\ \leavevmode\nobreak\
\quad(\phi^{\epsilon}_{k},\phi^{\epsilon}_{k^{\prime}})_{\bar{\ell}_{\epsilon}}=\delta_{kk^{\prime}}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \nu^{\epsilon}_{k}\in\mathbb{R}\,.$ (3.34)

Then we may expand the scale factor fluctuation as

$\delta{\check{a}}(\tau)=\sum_{k\geq
1}{\boldmath\delta{\check{a}}_{k}}\,\phi^{\epsilon}_{k}(\tau)\,,$ (3.35)

and use zeta regularization to get

$\displaystyle Z_{\epsilon}(a_{0})\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\prod_{k\geq 1}\int
d{\boldmath\delta{\check{a}}_{k}}\,e^{-{3sv_{3}\over\hbar}\nu^{\epsilon}_{k}({\boldmath\delta{\check{a}}_{k}})^{2}}=\prod_{k\geq
1}\sqrt{\hbar\,\pi\over 3sv_{3}\,\nu^{\epsilon}_{k}}$ (3.36) $\displaystyle=$
$\displaystyle\\!\\!\\!\left({3sv_{3}\over\hbar\,\pi}\right)^{1\over
4}{1\over\sqrt{\det{\cal S}_{\epsilon}}}\,.$

In order to define the Gaussian integrals, we have used the following
prescription: The Fourier mode ${\boldmath{\delta{\check{a}}_{k}}}$ is
integrated from $-\infty$ to $+\infty$ when $s\nu_{k}^{\epsilon}>0$, and from
$-i\infty$ to $+i\infty$ when $s\nu_{k}^{\epsilon}<0$. There is no vanishing
eigenvalue, $\nu_{k}^{\epsilon}=0$, since ${\cal S}_{\epsilon}$ is invertible
(see below). In fact, $\det{\cal S}_{+}$ and $\det{\cal S}_{-}$ turn out to
have opposite signs, independently of the sign of $s$. Hence, rotating some
contours of integration along the imaginary axis is necessary for both
$Z_{+}(a_{0})$ and $Z_{-}(a_{0})$ to exist, irrespectively of the choice of
continuation to Euclidean time.

The determinant of ${\cal S}_{\epsilon}$ can be computed via the method of
Ref. [28]. It is given by

$\det{\cal S}_{\epsilon}={\cal N}_{\epsilon}\,\varphi^{\epsilon}_{0}(1)\,,$
(3.37)

where ${\cal N}_{\epsilon}$ is a universal constant and the function
$\varphi^{\epsilon}_{0}(\tau)$ (to be evaluated at $\tau=1$) solves the system

$\left\\{\begin{array}[]{l}{\cal
S}_{\epsilon}\varphi^{\epsilon}_{0}(\tau)=0\,,\\\
\varphi^{\epsilon}_{0}(\tau_{\epsilon})=0\,,\quad\displaystyle{{\rm
d}\varphi^{\epsilon}_{0}\over{\rm
d}\tau}(\tau_{\epsilon})=1\,.\end{array}\right.$ (3.38)

Here, $\tau_{\epsilon}\in(0,1)$ is a regulator to be sent to $0$ at the end of
the calculations. The universal constant ${\cal N}_{\epsilon}$ can be obtained
by finding the determinant of an operator that is identical to ${\cal
S}_{\epsilon}$ up to terms involving no derivatives. The computations of
${\cal N}_{\epsilon}$ and $\varphi^{\epsilon}_{0}(1)$ have been carried out in
great detail in [17], giving the net result

$\det{\cal S}_{\epsilon}=2\,\Big{(}{\theta_{*}\over\lambda}\Big{)}^{1\over
4}\ln{1\over\theta_{*}}\times\epsilon\,a_{0}^{1\over 4}\sqrt{1-(\lambda
a_{0})^{2}}\,,$ (3.39)

where $\theta_{*}=\lambda{\bar{\ell}}_{\epsilon}\tau_{\epsilon}$ is to be sent
to zero. Note that

$\det{\cal S}_{+}>0\,,\quad\det{\cal S}_{-}<0\,,\quad\mbox{when}\quad
0<\lambda a_{0}<1\,,$ (3.40)

demonstrating that both ${\cal S}_{+}$ and ${\cal S}_{-}$ are invertible.

The integral over the fluctuation $\delta\ell$ is Gaussian. To evaluate it we
need to determine ${\cal K}_{\epsilon}$ given in Eq. (3.32), which requires to
find the function ${\cal S}_{\epsilon}^{-1}V_{a}(\bar{a}_{\epsilon})$, or
equivalently the function $f_{\epsilon}$ satisfying ${\cal
S}_{\epsilon}f_{\epsilon}=V_{a}(\bar{a}_{\epsilon})$ along with the boundary
conditions $f_{\epsilon}(\tau_{\epsilon})=f_{\epsilon}(1)=0$. These yield the
following net result [17]

$\int{\rm d}\delta\ell\,\exp\Big{\\{}\\!-\\!{3sv_{3}\over\hbar}\,{\cal
K}_{\epsilon}\Big{(}{\delta\ell\over\bar{\ell}_{\epsilon}}\Big{)}^{2}\Big{\\}}=\sqrt{{\pi\hbar\over
3sv_{3}}\,\ln{1\over\theta_{*}}}\,,$ (3.41)

where the domain of integration is from $-\infty$ to $+\infty$ for $s=+1$, and
from $-i\infty$ to $+i\infty$ for $s=-1$.

Collecting all results in the expression for the wavefunction, Eq. (3.33), we
obtain the result

$\Psi(a_{0})={\cal C}_{s}(\theta_{*})\sum_{\epsilon=\pm
1}{1\over\sqrt{\epsilon}}\,{\exp\\!\Big{[}\epsilon
s\,\displaystyle{2v_{3}\over\hbar\lambda^{2}}\,\big{(}1-(\lambda
a_{0})^{2}\big{)}^{3\over 2}\Big{]}\over a_{0}^{1\over 8}\,\big{(}1-(\lambda
a_{0})^{2}\big{)}^{1\over 4}}\,(1+{\cal O}(\hbar))\,,\quad 0<\lambda
a_{0}<1\,,$ (3.42)

where

${\cal C}_{s}(\theta_{*})=\alpha\,\sqrt{i\pi}\,\Big{(}{\pi\hbar\over
3sv_{3}}\Big{)}^{1\over
4}\exp\\!\Big{[}\\!-\\!s\,{2v_{3}\over\hbar\lambda^{2}}\Big{]}\Big{(}{\lambda\over\theta_{*}}\Big{)}^{1\over
8}$ (3.43)

is a regulator-dependent coefficient, which is irrelevant once $\Psi(a_{0})$
is normalized or when we discuss relative probabilities.

### Field Redefinitions:

Let us now discuss the issue of field redefinitions. As we have already
remarked, they leave the classical action invariant. They can be thought of as
reparameterizations of the target space. Let us consider such a field
redefinition

$a=A(q)\leavevmode\nobreak\ \leavevmode\nobreak\
\Longleftrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ q=Q(a)\,,$ (3.44)

where $Q=A^{-1}$ is an invertible function defined for $a>0$. The field
$q(\tau)$ satisfies the following fixed boundary conditions

$q(1)\equiv q_{0}=Q(a_{0})\,,\quad\leavevmode\nobreak\ \leavevmode\nobreak\
q(0)=Q(0)\,.$ (3.45)

The fluctuations around the instanton solutions satisfy

$\delta a=A^{\prime}(\bar{q}_{\epsilon})\delta q+{\cal O}((\delta
q)^{2})\,,\leavevmode\nobreak\ \leavevmode\nobreak\
\quad\mbox{where}\quad\leavevmode\nobreak\ \leavevmode\nobreak\
\bar{q}_{\epsilon}=Q(\bar{a}_{\epsilon})\,,$ (3.46)

where a prime denotes a derivative.

At the quantum level, the path integral measures ${\cal D}a$ and ${\cal D}q$
will not be equivalent in general, since they will be related by a non-trivial
Jacobian. As a result, we can define a quantum wavefunction, as in Eq. (3.18),

$\widetilde{\Psi}(q_{0})=\Delta_{\rm FP}\int_{0}^{+\infty}\\!\\!{\rm
d}\ell\int_{q(0)=Q(0),\,\,q(1)=q_{0}}{\cal
D}q\,e^{-{1\over\hbar}\widetilde{S}_{\rm E}[\ell^{2},q]}\,,$ (3.47)

based on the gauge invariant path-integral measure ${\cal D}q$. The tilde
action satisfies

$S_{\rm E}[\ell^{2},a]\equiv\widetilde{S}_{\rm E}[\ell^{2},q]\,.$ (3.48)

Following similar steps as before, we may calculate $\widetilde{\Psi}(q_{0})$
in the semi-classical limit to get [17]

$\widetilde{\Psi}(q_{0})={\widetilde{\cal
C}}_{s}(\theta_{*})\sum_{\epsilon=\pm 1}{1\over\sqrt{\epsilon\,{\rm
sign}(Q^{\prime})}}\,{\exp\\!\Big{[}\epsilon
s\,\displaystyle{2v_{3}\over\hbar\lambda^{2}}\,\big{(}1-(\lambda
a_{0})^{2}\big{)}^{3\over 2}\Big{]}\over a_{0}^{1\over
8}\;|Q^{\prime}(a_{0})|^{1\over 4}\,\big{(}1-(\lambda
a_{0})^{2}\big{)}^{1\over 4}}\,(1+{\cal O}(\hbar))\,,\quad 0<\lambda
a_{0}<1\,,$ (3.49)

where $\widetilde{\cal C}_{s}(\theta_{*})$ is a regulator-dependant
coefficient

$\widetilde{\cal C}_{s}(\theta_{*})=\alpha\,\sqrt{i\pi}\,\Big{(}{\pi\hbar\over
3sv_{3}}\Big{)}^{1\over
4}\exp\\!\Big{[}\\!-\\!s\,{2v_{3}\over\hbar\lambda^{2}}\Big{]}\Big{(}{\lambda\over\theta_{*}}\Big{)}^{1\over
8}\,\Big{|}Q^{\prime}\Big{(}{\sin\theta_{*}\over\lambda}\Big{)}\Big{|}^{-{1\over
4}}\,.$ (3.50)

The wavefunction can also be expressed in terms of $q_{0}$. The expression is

$\widetilde{\Psi}(q_{0})={\widetilde{\cal
C}}_{s}(\theta_{*})\sum_{\epsilon=\pm 1}{1\over\sqrt{\epsilon\,{\rm
sign}(A^{\prime})}}\,{\exp\\!\Big{[}\epsilon
s\,\displaystyle{2v_{3}\over\hbar\lambda^{2}}\,\big{(}1-(\lambda
A(q_{0}))^{2}\big{)}^{3\over 2}\Big{]}\over A(q_{0})^{1\over
8}\;|A^{\prime}(q_{0})|^{-{1\over 4}}\,\big{(}1-(\lambda
A(q_{0}))^{2}\big{)}^{1\over 4}}\,(1+{\cal O}(\hbar))\,.$ (3.51)

We conclude that there are infinitely many prescriptions to define the “ground
state” wavefuction. In the next section we will show that these yield
identical observable predictions.

##  4  Wheeler–DeWitt equation and universality

For each choice ${\cal D}q$, the corresponding ground-state wavefunction
satisfies a Wheeler–DeWitt equation. To see this let us first note that the
path integral of a total functional derivative must vanish

$0=\int{{\cal D}N\over{\rm Vol}({\rm Diff}[N^{2}])}\,{\delta\over\delta
N(x^{0})}\,e^{i\widetilde{S}[N^{2},q]}\,,\quad\mbox{for all $x^{0}$}\,.$
(4.52)

In this formula, $\widetilde{S}$ is the Lorentzian action expressed in terms
of the field $q$ and corresponding to the classical Lagrangian

$\widetilde{L}(N,q,\dot{q})=3v_{3}\Big{(}\\!-\\!{A(q)A^{\prime}(q)^{2}\over
N}\,\dot{q}^{2}+N\widetilde{V}(q)\Big{)}\,.$ (4.53)

Using this expression, it is easy to see that Eq. (4.52) further yields the
constraint identity

$0=-i\int_{\cal C}{{\cal D}N\,{\cal D}q\over{\rm Vol}({\rm
Diff}[N^{2}])}\left.{\widetilde{H}\over
N}\right|_{x^{0}}\,e^{i\widetilde{S}[N^{2},q]}\,,$ (4.54)

where

$\widetilde{H}=N\Big{(}\\!-\\!{1\over 12v_{3}}\,{\pi_{q}^{2}\over AA^{\prime
2}}-3v_{3}\widetilde{V}\Big{)}\,$ (4.55)

is the classical Hamiltonian. Here, $\pi_{q}$ is the momentum conjugate to $q$
given by

$\pi_{q}={\partial\widetilde{L}\over\partial\dot{q}}=-6v_{3}{AA^{\prime
2}\over N}\,\dot{q}\,.$ (4.56)

The implication of the constraint identity is the vanishing of all matrix
elements of the quantum Hamiltonian divided by the lapse function.
Equivalently, the quantum Hamiltonian (divided by the lapse function) must
annihilate all physical states. The corresponding wavefunctions must satisfy
the Wheeler–DeWitt equation.

As usual, the canonical quantization of the classical expression for
$\widetilde{H}/N$ can be obtained by replacing

$q\longrightarrow q_{0}\,,\leavevmode\nobreak\ \leavevmode\nobreak\
\quad\pi_{q}\longrightarrow-i\hbar\,{{\rm d}\over{\rm d}q_{0}}\,,$ (4.57)

which satisfy the canonical commutation relation $[q,\pi_{q}]=i\hbar$.
However, because the first term in the classical expression of
$\widetilde{H}/N$ involves a product of functions $q$ and $\pi_{q}$, there are
ordering ambiguities in constructing the quantum operator. These ambiguities
are induced in the precise form of the Wheeler–DeWitt equation. They can be
parameterized in terms of two functions of $q$, $\widetilde{\rho}$ and
$\widetilde{\omega}$, as follows [17]

${\widetilde{H}\over N}\,\widetilde{\Psi}_{\cal C}\equiv{\hbar^{2}\over
12v_{3}}\,{1\over AA^{\prime 2}}\bigg{[}{1\over\widetilde{\rho}}\,{{\rm
d}\over{\rm d}q_{0}}\Big{(}\widetilde{\rho}\,{{\rm d}\widetilde{\Psi}_{\cal
C}\over{\rm d}q_{0}}\Big{)}+\widetilde{\omega}\widetilde{\Psi}_{\cal
C}\bigg{]}-3v_{3}\widetilde{V}\widetilde{\Psi}_{\cal C}=0\,,$ (4.58)

where $\Psi_{\cal C}$ denotes a generic solution. Setting

$\Psi_{A{\cal C}}(a_{0})\equiv\widetilde{\Psi}_{\cal C}(Q(a_{0}))\,,$ (4.59)

we may alternatively write the above equation in terms of the scale factor as

$\displaystyle{\widetilde{H}\over N}\,\widetilde{\Psi}_{\cal
C}\equiv{H_{A}\over N}\,\Psi_{A{\cal C}}\\!\\!\\!$ $\displaystyle\equiv$
$\displaystyle\\!\\!\\!{\hbar^{2}\over 12v_{3}}\,{1\over
a_{0}}\bigg{[}{1\over\rho_{A}}\,{{\rm d}\over{\rm
d}a_{0}}\Big{(}\rho_{A}\,{{\rm d}\Psi_{A{\cal C}}\over{\rm
d}a_{0}}\Big{)}+\omega_{A}\Psi_{A{\cal C}}\bigg{]}-3v_{3}V\Psi_{A{\cal
C}}=0\,,$ $\displaystyle\mbox{where}\quad\leavevmode\nobreak\
\leavevmode\nobreak\ \rho_{A}(a_{0})\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!{\widetilde{\rho}(Q(a_{0}))\over|Q^{\prime}(a_{0})|}\,,\leavevmode\nobreak\
\leavevmode\nobreak\
\quad\omega_{A}(a_{0})=\widetilde{\omega}(Q(a_{0}))\,Q^{\prime}(a_{0})^{2}\,.$
(4.60)

We can lift the ambiguity in the form of the Wheeler–DeWitt equation by
imposing that the ground-state wavefunctions have to satisfy it. Indeed, the
generic solutions at the semi-classical can be obtained by applying the WKB
method [30], which leads to [17]

$\widetilde{\Psi}_{\cal C}(q_{0})=\sum_{\epsilon=\pm 1}N_{{\cal
C}\epsilon}\,{\exp\\!\Big{[}\epsilon
s\,\displaystyle{2v_{3}\over\hbar\lambda^{2}}\,\big{(}1-(\lambda
A(q_{0}))^{2}\big{)}^{3\over 2}\Big{]}\over|\widetilde{\rho}(q_{0})|^{1\over
2}\,A(q_{0})^{1\over 2}\,|A^{\prime}(q_{0})|^{1\over 2}\,\big{(}1-(\lambda
A(q_{0}))^{2}\big{)}^{1\over 4}}\,(1+{\cal O}(\hbar))\,,\quad 0<\lambda
A(q_{0})<1\,,$ (4.61)

where $N_{{\cal C}\epsilon}$ are two integration constants. Comparing with Eq.
(3.51) we find $\widetilde{\rho}$,

$\tilde{\rho}(q_{0})=A(q_{0})^{-{3\over 4}}\,|A^{\prime}(q_{0})|^{-{3\over
2}}\,.$ (4.62)

Notice that the unknown function $\widetilde{\omega}$ is absorbed in ${\cal
O}(\hbar)$ terms, and so it cannot be determined at the semi-classical level.
The expression for $\rho_{A}$ is

$\rho_{A}(a_{0})=a_{0}^{-{3\over 4}}\,|Q^{\prime}(a_{0})|^{1\over 2}.$ (4.63)

Both $\widetilde{\rho}(q_{0})$ and $\rho_{A}(a_{0})$ are positive for
$0<\lambda a_{0}<1$. The values $N_{\epsilon}$ of the mode coefficients
$N_{{\cal C}\epsilon}$ that select the corresponding ground-state wavefunction
are given by

$N_{\epsilon}={1\over\sqrt{\epsilon\,{\rm sign}(Q^{\prime})}}\,.$ (4.64)

### Quantum equivalence at the semi-classical level:

A natural question that arises is whether different wavefunction prescriptions
based on the path integral measures ${\cal D}q$, and the corresponding
Wheeler–DeWitt equations, define different quantum gravity models with same
classical limits. The answer to this question is negative. The reason is that
all these prescriptions yield the same observable predictions at the semi-
classical level.

Indeed to obtain probability amplitudes, we need to define a suitable inner
product in each Hilbert space. This takes the form

$\langle\Psi_{A1},\Psi_{A2}\rangle_{A}=\int_{0}^{+\infty}{\rm
d}a_{0}\,\mu_{A}(a_{0})\,\Psi_{A1}(a_{0})^{*}\,\Psi_{A2}(a_{0})\,,$ (4.65)

for some real positive measure $\mu_{A}$.

Based on the form of the inner product, we obtain the following identity

$\big{\langle}\Psi_{A1},{H_{A}\over
N}\,\Psi_{A2}\big{\rangle}=\big{\langle}{H_{A}^{\dagger}\over
N}\,\Psi_{A1},\Psi_{A2}\big{\rangle}+{\hbar^{2}\over
12v_{3}}\left[\rho_{A}\\!\left({\mu_{A}\over
a_{0}\rho_{A}}\,\Psi^{*}_{A1}\,{{\rm d}\Psi_{A2}\over{\rm d}a_{0}}-{{\rm
d}\over{\rm d}a_{0}}\Big{(}{\mu_{A}\over
a_{0}\rho_{A}}\,\Psi^{*}_{A1}\Big{)}\Psi_{A2}\right)\right]_{0}^{+\infty}\,,$
(4.66)

where integration by parts gives

${H^{\dagger}_{A}\over N}\,\Psi_{A{\cal C}}\equiv{\hbar^{2}\over
12v_{3}}\,{1\over a_{0}}\bigg{[}{a_{0}\over\mu_{A}}\,{{\rm d}\over{\rm
d}a_{0}}\Big{(}\rho_{A}\,{{\rm d}\over{\rm d}a_{0}}\Big{(}{\mu_{A}\over
a_{0}\rho_{A}}\,\Psi_{A{\cal C}}\Big{)}+\omega_{A}\Psi_{A{\cal
C}}\bigg{]}-3v_{3}V\Psi_{A{\cal C}}\,.$ (4.67)

Imposing hermiticity of the Hamiltonian gives rise to a differential equation,
which determines the measure $\mu_{A}$ in terms of $\rho_{A}$ [17]:

$\mu_{A}(a_{0})=a_{0}\,\rho_{A}(a_{0})\,.$ (4.68)

Furthermore the Wheeler–DeWitt equation ensures the vanishing of the boundary
term in Eq. (4.66).

It follows that at the semi-classical level, the probability amplitudes
$\sqrt{\mu_{A}}\,\Psi_{A{\cal C}}$ are universal, since

$\sqrt{\mu_{A}(a_{0})}\Psi_{A{\cal C}}(a_{0})=\sum_{\epsilon=\pm 1}N_{{\cal
C}\epsilon}\,{\exp\\!\Big{[}\displaystyle\epsilon
s\,{2v_{3}\over\hbar\lambda^{2}}\,\big{(}1-(\lambda a_{0})^{2}\big{)}^{3\over
2}\Big{]}\over\displaystyle\big{(}1-(\lambda a_{0})^{2}\big{)}^{1\over
4}}\,(1+{\cal O}(\hbar))\,,\quad 0<\lambda a_{0}<1\,.$ (4.69)

This universality relation can be extended also for $\lambda a_{0}>1$ [17]. So
all probabilities and relative probabilities are independent of the choice of
the path integral measure ${\cal D}q$, at least at the semi-classical level.

An important consequence however is that none of the solutions of the
Wheeler–DeWitt equation is normalizable. Indeed by examining the large $a_{0}$
behavior of these functions, we can infer that
$|\sqrt{\mu_{A}(a_{0})}\Psi_{A{\cal C}}(a_{0})|^{2}$ scales as $1/a_{0}$ in
this limit, giving rise to a logarithmically divergent norm. So at best we can
use these wavefunctions to define relative probabilities, in terms of ratios
of the probability densities evaluated at different points of minisuperspace
in this model. It would be interesting to extend the analysis to more
realistic cases, in the presence of matter, in order to see if normalizable
wavefunctions, based on the no-boundary proposal, can be constructed.
Interesting attempts to extract observables in quantum cosmology includes
Refs. [13, 31, 32, 33, 34].

##  5 Conclusions

In this work we have considered the Hartle–Hawking wavefunction for spatially
closed universes, with positive cosmological constant $\Lambda>0$. We focused
on the simpler minisuperspace version, considering homogeneous and isotropic
universes. The system can be seen as a non-linear $\sigma$-model with a line
segment for the base and a one-dimensional target space parameterized by the
scale factor. The gauge fixing of time reparameterizations is achieved by
integrating over the proper length of the line-segment base, introducing the
necessary Faddeev–Popov determinant, which turns out to be trivial, and using
gauge invariant measures for the scale factor path integral. The
reparametrizations of the scale factor, that is the coordinate of the target
space, yield different gauge invariant measures and path integrals, but the
corresponding Hilbert spaces are equivalent, at least semi-classically.

## References

  * [1] A. H. Guth, “The inflationary universe: A possible solution to the horizon and flatness problems,” Phys. Rev. D 23, 347-356 (1981).
  * [2] A. D. Linde, “A new inflationary universe scenario: A possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems,” Phys. Lett. B 108, 389-393 (1982).
  * [3] A. Albrecht and P. J. Steinhardt, “Cosmology for grand unified theories with radiatively induced symmetry breaking,” Phys. Rev. Lett. 48, 1220-1223 (1982).
  * [4] V. F. Mukhanov and G. V. Chibisov, “Quantum fluctuations and a nonsingular universe,” JETP Lett. 33, 532-535 (1981); V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger, “Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions,” Phys. Rept. 215, 203-333 (1992).
  * [5] S. Kachru, R. Kallosh, A. D. Linde, J. M. Maldacena, L. P. McAllister and S. P. Trivedi, “Towards inflation in string theory,” JCAP 10, 013 (2003) [arXiv:hep-th/0308055 [hep-th]].
  * [6] B. S. DeWitt, “Quantum theory of gravity. I. The canonical theory,” Phys. Rev. 160 (1967), 1113-1148.
  * [7] J. B. Hartle and S. W. Hawking, “Wave function of the universe,” Phys. Rev. D 28 (1983) 2960 [Adv. Ser. Astrophys. Cosmol. 3 (1987) 174].
  * [8] A. Vilenkin, “Creation of universes from nothing,” Phys. Lett. B 117 (1982) 25.
  * [9] A. Vilenkin, “The birth of inflationary universes,” Phys. Rev. D 27 (1983) 2848.
  * [10] A. Vilenkin, “Quantum creation of universes,” Phys. Rev. D 30 (1984), 509.
  * [11] A. Vilenkin, “Boundary conditions in quantum cosmology,” Phys. Rev. D 33 (1986), 3560.
  * [12] A. D. Linde, “Quantum creation of an inflationary universe,” Sov. Phys. JETP 60, 211 (1984) [Zh. Eksp. Teor. Fiz. 87, 369 (1984)]; A. D. Linde, “Quantum creation of the inflationary universe,” Lett. Nuovo Cim. 39, 401 (1984).
  * [13] A. Vilenkin, “Predictions from quantum cosmology,” NATO Sci. Ser. C 476 (1996), 345-367 [arXiv:gr-qc/9507018 [gr-qc]].
  * [14] S. W. Hawking, “The cosmological constant is probably zero,” Phys. Lett. B 134 (1984), 403.
  * [15] S. W. Hawking and D. N. Page, “How probable is inflation?,” Nucl. Phys. B 298, 789-809 (1988).
  * [16] A. D. Linde, “Particle physics and inflationary cosmology,” Contemp. Concepts Phys. 5 (1990), 1-362 [arXiv:hep-th/0503203 [hep-th]].
  * [17] H. Partouche, N. Toumbas and B. de Vaulchier, “Wavefunction of the universe: Reparametrization invariance and field redefinitions of the minisuperspace path integral,” [arXiv:2103.15168 [hep-th]].
  * [18] J. J. Halliwell, “Derivation of the Wheeler–DeWitt equation from a path integral for minisuperspace models,” Phys. Rev. D 38 (1988), 2468.
  * [19] J. J. Halliwell and J. Louko, “Steepest descent contours in the path integral approach to quantum cosmology. 1. The de Sitter minisuperspace model,” Phys. Rev. D 39 (1989), 2206.
  * [20] J. Feldbrugge, J. L. Lehners and N. Turok, “Lorentzian quantum cosmology,” Phys. Rev. D 95 (2017) no.10, 103508 [arXiv:1703.02076 [hep-th]].
  * [21] J. J. Halliwell and S. W. Hawking, “The origin of structure in the universe,” Phys. Rev. D 31 (1985), 1777.
  * [22] K. Schleich, “Semiclassical wave function of the universe at small three geometries,” Phys. Rev. D 32 (1985), 1889-1898.
  * [23] J. Diaz Dorronsoro, J. J. Halliwell, J. B. Hartle, T. Hertog and O. Janssen, “The real no-boundary wave function in Lorentzian quantum cosmology,” Phys. Rev. D 96 (2017) no.4, 043505 [arXiv:1705.05340 [gr-qc]].
  * [24] S. Cespedes, S. P. de Alwis, F. Muia and F. Quevedo, “Lorentzian vacuum transitions: Open or closed universes?,” [arXiv:2011.13936 [hep-th]].
  * [25] A. Davidson, D. Karasik and Y. Lederer, “Wavefunction of a brane-like universe,” Class. Quant. Grav. 16 (1999), 1349-1356 [arXiv:gr-qc/9901003 [gr-qc]].
  * [26] A. Davidson and B. Yellin, “Quantum black hole wave packet: Average area entropy and temperature dependent width,” Phys. Lett. B 736 (2014), 267-271 [arXiv:1404.5729 [gr-qc]].
  * [27] J. J. Halliwell and R. C. Myers, “Multiple sphere configurations in the path integral representation of the wave function of the universe,” Phys. Rev. D 40 (1989), 4011.
  * [28] S. Coleman, “Aspects of symmetry: Selected Erice lectures,” Cambridge University Press (2010).
  * [29] C. G. Callan, Jr. and S. R. Coleman, “The fate of the false vacuum. II. First quantum corrections,” Phys. Rev. D 16 (1977), 1762-1768.
  * [30] See e.g. D. J. Griffiths, “Introduction to quantum mechanics,” Cambridge University Press (2016).
  * [31] G. W. Gibbons, S. W. Hawking and J. M. Stewart, “A natural measure on the set of all universes,” Nucl. Phys. B 281 (1987), 736.
  * [32] J. B. Hartle, S. W. Hawking and T. Hertog, “Quantum probabilities for inflation from holography,” JCAP 01 (2014), 015 [arXiv:1207.6653 [hep-th]].
  * [33] J. Hartle, S. W. Hawking and T. Hertog, “Local observation in eternal inflation,” Phys. Rev. Lett. 106 (2011), 141302 [arXiv:1009.2525 [hep-th]].
  * [34] J. B. Hartle, S. W. Hawking and T. Hertog, “No-boundary measure of the universe,” Phys. Rev. Lett. 100 (2008), 201301 [arXiv:0711.4630 [hep-th]].


# Transport of neutral and charged nanorods across varying-section channels

Paolo Malgaretti<EMAIL_ADDRESS>Max-Planck-Institut für
Intelligente Systeme, Heisenbergstr. 3, D-70569 Stuttgart, Germany  IV.
Institut für Theoretische Physik, Universität Stuttgart, Pfaffenwaldring 57,
D-70569 Stuttgart, Germany  Helmholtz Institute Erlangen-Nürnberg for
Renewable Energy (IEK-11), Forschungszentrum Jülich, Fürther Straße 248, 90429
Nürnberg, Germany  Jens Harting  Helmholtz Institute Erlangen-Nürnberg for
Renewable Energy (IEK-11), Forschungszentrum Jülich, Fürther Straße 248, 90429
Nürnberg, Germany  Department of Chemical and Biological Engineering and
Department of Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg,
Fürther Straße 248, 90429 Nürnberg, Germany

###### Abstract

We study the dynamics of neutral and charged rods embedded in varying-section
channels. By means of systematic approximations, we derive the dependence of
the local diffusion coefficient on both the geometry and charge of the rods.
This microscopic insight allows us to provide predictions for the permeability
of varying-section channels to rods with diverse lengths, aspect ratios and
charge. Our analysis shows that the dynamics of charged rods is sensitive to
the geometry of the channel and that their transport can be controlled by
tuning both the shape of the confining walls and the charge of the rod.
Interestingly, we find that the channel permeability does not depend
monotonically on the charge of the rod. This opens the possibility of a novel
mechanism to separate charged rods.

##  I Introduction

The transport of molecules, proteins and small particles across pores,
channels and, in general, porous materials is of paramount relevance in
several biological, environmental and technological scenarios Malgaretti  _et
al._ (2019a). In biology and technological applications, ions and molecules
are transported across membrane pores Calero  _et al._ (2011); Peyser  _et
al._ (2014); Lee  _et al._ (2017), RNA is transported across the nuclear
membrane Melnikov  _et al._ (2017); Bacchin (2018); Berezhkovskii  _et al._
(2019) and the crowded environment of cell cytoplasm provides dynamical
obstacles to the motion of organelles and vesicles Alberts  _et al._ (2007).
Similarly, in environmental sciences, pollutants and plant nutrients spread
across the porous matrix of rocks and soil. In this perspective, substantial
effort has been dedicated to the synthesis Yu  _et al._ (2017); Chen  _et al._
(2019) and characterization Yang  _et al._ (2019a) of anisotropic nano-
colloids and stiff filaments Fakhri  _et al._ (2010); Domínguez and Abreu
(2016). Finally, in recent years the development of experimental techniques
such as DNA Bonthuis  _et al._ (2008); Sakaue (2018) and protein Chinappi and
Cecconi (2018) sequencing, resistive-pulse sensing techniques Saleh and Sohn
(2003); Ito  _et al._ (2004); Heins  _et al._ (2005); Arjmandi  _et al._
(2012) and chromatography Martinez Cristancho and Seidel-Morgenstern (2016);
Ghosh (2002); Swernath  _et al._ (2014) have been developed by exploiting
transport across pores.

On the theoretical side, tackling these problems is typically complicated
since it requires detailed numerical simulations that take into account the
interactions between the suspended object and the confining walls. Recently,
several groups have contributed developing the so-called Fick-Jacobs
approximationZwanzig (1992); Reguera and Rubi (2001); Kalinay and Percus
(2008); Martens  _et al._ (2011); Chacón-Acosta  _et al._ (2013); Malgaretti
_et al._ (2013). This scheme relies on the assumption that when the net
longitudinal velocity is slow enough, the transported object will explore the
directions transverse to the motion with a probability distribution that is
very close to the equilibrium one. Thanks to this approximation, the geometry
of the pore as well as the interaction with the walls can be accounted for by
the local equilibrium free energy. This approach has been shown to be
applicable to the transport of colloids Reguera  _et al._ (2006, 2012); Marini
Bettolo Marconi  _et al._ (2015); Malgaretti  _et al._ (2016a); Puertas  _et
al._ (2018), ions Malgaretti  _et al._ (2014, 2015, 2016b); Chinappi and
Malgaretti (2018); Malgaretti  _et al._ (2019b), and polymers Bianco and
Malgaretti (2016); Malgaretti and Oshanin (2019); Bodrenko  _et al._ (2019)
just to mention a few among others (see recent review articles Burada  _et
al._ (2009); Malgaretti  _et al._ (2013) for a more comprehensive list).

Figure 1: Top: scheme of a rod (orange) with major axis of length $2L$ tilted
by an angle $\theta$ and embedded in a plane channel with half section
$h_{0}$. The minimum, $\theta_{\text{min}}(z)$ and maximum,
$\theta_{\text{max}}(z)$ angles for a given position of the center of mass are
reported. Bottom: a rod in a varying-section channel with half-section $h(x)$.

In this article we study the dynamics of charged rods confined in varying
section channels. In particular, we propose an extension of the Fick-Jacobs
approximation that allows us to account for both the geometry of the rod as
well as for its net charge. From this perspective, our work goes beyond a
recent contribution (see Ref. Karimi  _et al._ (2020)) in which an
approximated equation governing the transport of uncharged rods across varying
section channels has been proposed. First, we test our approach against known
experimental results Yang  _et al._ (2019b) for neutrally charged rods. Our
model retrieves both the local dependence of the diffusion coefficient as well
as the dependence of the Mean First Passage Time (MFPT) profile. Second, we
use our model to predict the dependence of the diffusion coefficient on the
geometry of both the channel and the rod as well as on the charge of the rod.
For neutrally charged rods, in agreement with Ref. Yang  _et al._ (2019b), we
find that the local diffusion coefficient maximizes at the channel bottleneck,
i.e. where the rod is mainly oriented along the channel axes. Interestingly,
the modulation of the local diffusion coefficient is maximized when the length
of the rod major axis is comparable to the channel average section and it
becomes vanishing small when these two length scales differ. Finally, our data
show that such a dependence can be even amplified by the electrostatic
interaction between the rod and the channel walls.

##  II Model

We characterize the transport of rigid rods across a $2D$ varying-section
channel whose half-section is given by

$h(x)=h_{0}\left(1-h_{1}\cos\left(\frac{2\pi}{L_{0}}x\right)\right).$ (1)

where $h_{0}$ is the average section, $h_{1}$ is the amplitude of the section
modulation and $L_{0}$ is the period of the channel. Accordingly, the
$2D$-Smoluchowski equation reads

$\partial_{t}\rho(x,z,\theta,t)=\partial_{x}J_{x}+\partial_{z}J_{z}+\partial_{\theta}J_{\theta}\,,$
(2)

with

$\displaystyle J_{x}$ $\displaystyle=$
$\displaystyle\mathbf{e}_{x}\cdot\mathbf{D}(\theta)\cdot\left[\bar{\nabla}\rho+\rho\beta\bar{\nabla}W\right],$
(3) $\displaystyle J_{z}$ $\displaystyle=$
$\displaystyle\mathbf{e}_{z}\cdot\mathbf{D}(\theta)\cdot\left[\bar{\nabla}\rho+\rho\beta\bar{\nabla}W\right],$
(4) $\displaystyle J_{\theta}$ $\displaystyle=$ $\displaystyle
D_{\theta}\partial_{\theta}\rho\,,$ (5)

where $\rho$ is the probability distribution function, $\beta^{-1}=k_{B}T$ is
the inverse thermal energy, $k_{B}$ is the Boltzmann constant, $T$ is the
absolute temperature. Moreover, $\mathbf{D}(\theta)$ is the translational
diffusion matrix that accounts for the fact that the diffusion along the major
axis of the rod differs from that perpendicular to it, $D_{\theta}$ is the
rotational diffusion coefficient and $W$ accounts for the both, the
conservative forces acting on the rod and the confinement. In the following,
we focus on the case in which the local radius of curvature is much larger
than the length of the rod. In such a regime the channel walls can be
approximated as locally parallel to the longitudinal axis. Accordingly we have
that the potential $W$ reads

$W(x,z,\theta)\\!=\\!\begin{cases}\phi(x,z,\theta)-fx&\\!\\!|z|<h(x)\,\,\&\,\,\,\theta_{m}\\!<\\!\theta\\!<\\!\theta_{M}\\\
\infty&\text{else}\end{cases}$ (6)

with

$\displaystyle\theta_{m}=$
$\displaystyle\begin{cases}\frac{\pi}{2}-\arccos\left(\frac{h(x)-z}{L}\right)\\!\\!&\\!h(x)-L\leq|z|\leq
h(x)\\\ 0\\!\\!&|z|<h(x)-L\end{cases}$ (7) $\displaystyle\theta_{M}=$
$\displaystyle\begin{cases}\frac{\pi}{2}+\arccos\left(\frac{h(x)-z}{L}\right)\\!\\!&\\!h(x)-L\leq|z|\leq
h(x)\\\ \pi\\!\\!&|z|<h(x)-L\end{cases}$ (8)

where $\phi(x,z,\theta)$ accounts for the equilibrium conservative forces and
$f$ for the longitudinal force responsible for the transport along the
channel. In Eqs. (7), (8) we assume that the aspect ratio between the long
axis, $2L$, and the minor axis, $2l$, of the rod is such that $L\gg l$ and
hence we can disregard the geometric corrections to Eqs. (7), (8) due to the
finiteness of $l$.

We further assume that translation along the channel axis is slow enough such
that $\rho$ retains its equilibrium profiles along $z$ and $\theta$, which
implies the absence of fluxes in $z$ and $\theta$, i.e.

$\displaystyle J_{z}$ $\displaystyle=$ $\displaystyle 0\,,$ (9) $\displaystyle
J_{\theta}$ $\displaystyle=$ $\displaystyle 0\,.$ (10)

Accordingly, we perform the standard Fick-Jacobs approximation Zwanzig (1992);
Reguera and Rubi (2001); Kalinay and Percus (2008); Martens  _et al._ (2011);
Chacón-Acosta  _et al._ (2013)

$\displaystyle\rho(x,z,\theta,t)$ $\displaystyle=p(x,t)\frac{e^{-\beta
W(x,z,\theta)}}{e^{-\beta A(x)}}\,,$ (11)

where

$\displaystyle\beta A(x)$
$\displaystyle=-\ln\left[\frac{1}{2h_{0}\pi}\int\limits_{0}^{\pi}\int\limits_{-\infty}^{\infty}e^{-\beta
W(x,z,\theta)}dzd\theta\right],$ (12)

is the local equilibrium free energy Zwanzig (1992); Reguera and Rubi (2001).

Figure 2: Validation of the model (solid lines) against experimental data
taken from Ref. Yang  _et al._ (2019b) (dots). (a) Dependence of the diffusion
coefficient $\mathcal{D}(x)$ normalized by the value at the bottleneck. In
order to check the validity of the model the values of the parameters are
chosen to be close to those used in Fig. 3a of Ref. Yang  _et al._ (2019b).
Accordingly, we have: $L_{0}=12\mu m$, $h_{0}=3\mu m$, $l=0.15\mu m$ and
$L=1,1.2,1.6\mu m$. Lighter colors stand for larger values of $L$. We remark
the agreement between the model and the experimental data improves upon
increasing the length of the rods. (b) Mean First Passage Time (MFPT) with
reflecting boundary conditions at the origin and absorbing at $x$ for
$L=1.04\mu m,h_{0}=\mu m,h_{1}=\mu m$ (black), $L=1.14\mu m,h_{0}=\mu
m,h_{1}=\mu m$ (dark green), $L=1.18\mu m,h_{0}=\mu m,h_{1}=\mu m$ (light
grey). As shown in the panel, the agreement improves upon increasing the
corrugation of the channel.

We remark that $p(x)$ (see Eq. (11)) is proportional to the probability of
finding the center of mass at a position $x$. In the frame of reference of the
rod we have

$\bar{\mathbf{D}}=D_{0}\left[\begin{array}[]{cc}\frac{l}{L}&0\\\
0&1\end{array}\right],$ (13)

where $D_{0}\frac{l}{L}$ is the diffusion coefficient along the minor axis of
the rod (of size $l$), $\mathbf{e}_{\perp}$, and $D_{0}$ is the diffusion
coefficient along the major axis of the rod (of size $L$),
$\mathbf{e}_{\parallel}$, and the off-diagonal terms are zero due to the axial
symmetry of the rod Happel and Brenner (1983). Integrating Eq. (3) along $z$
and $\theta$ and using Eqs. (9)-(11) we obtain

$\displaystyle\int\limits_{0}^{\pi}\\!\\!\int\limits_{-\infty}^{\infty}\\!\\!J_{x}\frac{dzd\theta}{2h_{0}\pi}=$
$\displaystyle\,\partial_{x}\int\limits_{0}^{\pi}\\!\\!\int\limits_{-\infty}^{\infty}\\!\\!\mathbf{e}_{x}\cdot\mathbf{D}(\theta)\cdot\mathbf{e}_{x}\left[\partial_{x}\rho+\rho\beta\partial_{x}W\right]\frac{dzd\theta}{2h_{0}\pi}$
$\displaystyle=$
$\displaystyle\partial_{x}\left[\mathcal{D}(x)\left(\partial_{x}p+p\beta\partial_{x}A\right)\right]\,.$
(14)

Here,

$\mathcal{D}(x)=\int\limits_{0}^{\pi}\int\limits_{-\infty}^{\infty}\mathbf{e}_{x}\cdot\mathbf{D}(\theta)\cdot\mathbf{e}_{x}\frac{e^{-\beta
W(x,z,\theta)}}{e^{-\beta A(x)}}\frac{dzd\theta}{2h_{0}\pi}$ (15)

is the effective local diffusion coefficient. According to Fig. 1 we have
$\mathbf{e}_{x}=\cos\theta\mathbf{e}_{\parallel}-\sin\theta\mathbf{e}_{\perp}$
which leads to the following expression for the local diffusion coefficient,

$\mathbf{e}_{x}\cdot\mathbf{D}(\theta)\cdot\mathbf{e}_{x}=D_{0}\left(\cos^{2}\theta+\frac{l}{L}\sin^{2}\theta\right)$
(16)

which, once substituted into Eq. (15), leads to

$\dfrac{\mathcal{D}(x)}{D_{0}}=\int\limits_{0}^{\pi}\int\limits_{-\infty}^{\infty}\left(\cos^{2}\theta+\frac{l}{L}\sin^{2}\theta\right)\frac{e^{-\beta
W(x,z,\theta)}}{e^{-\beta A(x)}}\frac{dzd\theta}{2h_{0}\pi}\,.$ (17)

We remark that for $l=L$ Eq. (17) reduces to $\mathcal{D}(x)=D_{0}$, which is
the diffusion coefficient of spherical particles of size $l$ Zwanzig (1992);
Reguera and Rubi (2001). Finally, integrating Eq. (2) along $z$ and $\theta$
and using Eqs. (14) leads to

$\partial_{t}p(x,t)=-\partial_{x}\left\\{\mathcal{D}(x)\left[\partial_{x}p(x,t)+\beta
p(x,t)\partial_{x}A(x)\right]\right\\}.$ (18)

This is the extension of the Fick-Jacobs approximation to the case of rigid
rods embedded in varying-section channels. The steady-state solution of Eq.
(18) reads

$p(x)=e^{-\beta A(x)}\left[J\int_{0}^{x}\frac{e^{\beta
A(x^{\prime})}}{\mathcal{D}(x^{\prime})}dx^{\prime}+\Pi\right]$ (19)

where $J$ and $\Pi$ are integration constants. For periodic boundary
conditions and imposing the normalization of the probability we obtain

$\displaystyle\Pi$ $\displaystyle=$
$\displaystyle-J\left[\frac{\int_{0}^{L}\frac{e^{\beta
A(x)}}{\mathcal{D}(x)}dx}{e^{-\beta(A(0)-A(L))}-1}\right]=-J\Pi_{0}$ (20)
$\displaystyle J$ $\displaystyle=$
$\displaystyle-\left[\int_{0}^{L}\\!\\!\\!\\!\\!e^{-\beta
A(x)}\left[\int_{0}^{x}\frac{e^{\beta
A(x^{\prime})}}{\mathcal{D}(x^{\prime})}dx^{\prime}+\Pi_{0}\right]dx\right]^{-1}\\!\\!\\!\\!\\!\\!.$
(21)

Finally we define the dimensionless permeability as

$\displaystyle\mu=\frac{J}{J_{0}}=\frac{J}{\beta D_{0}f/L},$ (22)

where $f$ is the magnitude of the external force and $J_{0}=\beta D_{0}f/L$ is
the flux of a spherical colloid of radius $l$ in a flat channel of half-width
$h_{0}$.

##  III Results

###  III.1 Neutral rods

At first we focus on neutral rods for which the potential $W$ reads

$W(x,z)=\begin{cases}-fx&|z|<h(x)\,\&\,\theta_{m}<\theta<\theta_{M}\\\
\infty&\mbox{else}\end{cases}$ (23)

and the local free energy reads

$\displaystyle A(x)$ $\displaystyle=-fx-
k_{B}T\ln\left[\int\limits_{0}^{\pi}\\!\int\limits_{-h(x)}^{h(x)}\\!\\!\Gamma(x,z,\theta)\frac{dzd\theta}{2h_{0}\pi}\right],$
(24)

with

$\displaystyle\Gamma(x,z,\theta)=\Theta\left(\theta-\theta_{m}(x,z)\right)\Theta\left(\theta_{M}(x,z)-\theta\right),$
(25)

where $\Theta(\theta)$ is the Heaviside step function. Accordingly, the
effective diffusion coefficient reads

$\displaystyle\dfrac{\mathcal{D}(x)}{D_{0}}=\frac{\int\limits_{0}^{\pi}\int\limits_{-h(x)}^{h(x)}\\!\\!\left[\cos^{2}\theta+\frac{l}{L}\sin^{2}\theta\right]\Gamma(x,z,\theta)dzd\theta}{\int\limits_{0}^{2\pi}\int\limits_{-h(x)}^{h(x)}\Gamma(x,z,\theta)dzd\theta},$
(26)

where $L$ is the half-length of the major axis of the rod (see Fig. 1).

####  III.1.1 Diffusion

Figure 3: Ratio of the effective diffusion coefficient at the maximum channel
amplitude ($D(h_{max})$) and at the bottleneck ($D(h_{min})$) as a function of
$L$ and for different values of $\beta\Delta A_{gas}=0,0.2,0.6,1.1,1.7,2.9$
(from blue to cyan). Rods have a fixed magnitude of the small axis
$l/h_{0}=0.01$.

Figure 4: Equilibrium free energy barrier $\Delta A$. (a) and (b) depict the
dependence of the free energy barrier, $\Delta A$, for rods of different
length $L/h0=0.05,0.1,0.5,0.7,1,2$ (from blue to cyan) as a function of the
free energy barrier of the ideal gas. (c) Dependence of the free energy
barrier on the length, $L$, of the rod for different channel corrugations
$\Delta A_{gas}\simeq 0.4,0.6,1.1,1.7,3$.

At first we compare our analytical predictions for the diffusion coefficient
against experimental results. As already shown in Ref. Yang  _et al._ (2019b),
the diffusion coefficient is maximum at the channel bottleneck where the rod
is mainly parallel to the axis of the channel. Fig. 2a shows that predictions
of Eq. (26) qualitatively agree with the experimental data. In particular, the
model properly captures the enhancement of the sensitivity of the local
diffusion coefficient upon increasing the length of the rod. Indeed, we remark
that for longer rods, for which the dependence of the diffusion coefficient is
more significant, the predictions of the model are quantitatively reliable,
whereas the agreement with the experimental data becomes weaker when the
dependence of $D$ on $x$ becomes milder. Once the agreement for the diffusion
coefficient has been assessed we focus on the validity of the Fick-Jacobs
approximation for what concerns the MFPT between the origin and an arbitrary
position $x$:

$T(x)=\int\limits_{0}^{x}dxe^{\beta
A(x^{\prime})}\int\limits_{0}^{x^{\prime}}\frac{e^{-\beta
A(z)}}{\mathcal{D}(z)}dz$ (27)

Interestingly, the predictions of Eq. (27) match well with the data reported
in Ref. Yang  _et al._ (2019b) in the case of more corrugated channels whereas
it is just qualitative for more shallow shapes. Interestingly, this trend is
similar to that of the model (based on finite-element numerical simulations)
reported in Ref. Yang  _et al._ (2019b).

Next, we characterize the dependence of the ratio between the maximum and the
minimum local diffusion coefficients on the length of the rod. As expected,
Fig. 3 shows that in the asymptotic limits, $L/h_{0}\ll 1$ and $L/h_{0}\gg 1$,
the rods either reduce to a spherical particle of radius $l=L$ or to a very
long stiff filament that, due to the confinement is almost always aligned with
the channel axis. In both cases the effective diffusion coefficient reduces to
$\mathcal{D}\simeq D_{0}$. In contrast, for $L\simeq h_{0}$, Fig. 3 shows that
increasing the corrugation of the channel increases the range of values of the
rod length for which the local diffusion coefficient is sensitive to the local
channel section. Clearly, if the aspect ratio $L/l$ approaches unity the
diffusion coefficient becomes homogeneous along the channel.

####  III.1.2 Free energy barrier

Having an explicit formula for the free energy, it allows us to discuss the
dependence of the equilibrium ($f=0$) free energy barrier defined as the
difference between the free energy at the bottleneck and the one at the
channel’s widest section:

$\Delta A_{eq}=A(h_{min},f=0)-A(h_{max},f=0)$ (28)

For comparison we recall that the free energy difference for an ideal gas
depends solely on the geometry of the channel:

$\Delta A_{gas}=k_{B}T\ln\left[\frac{h_{max}}{h_{min}}\right]$ (29)

Our model shows that the dependence of the free energy barrier of rods
strongly depends on their length. Indeed, while for shorter rods,
$L/h_{0}\lesssim 0.5$, the free energy barrier of the rod increases upon
increasing channel corrugation (i.e. increasing $\Delta A_{gas}$) for longer
rods, $L/h_{0}\gtrsim 1$, the free energy barrier of the rod decreases upon
increasing $\Delta A_{gas}$, as shown in Fig. 4. Additionally, for every
geometry of the channel the maximum departure of $\Delta A$ from $\Delta
A_{gas}$ is attained when the length of the rod is comparable to the channel
average section, $L\simeq h_{0}$. Interestingly, for rods much longer than the
channel section we have that $\Delta A\rightarrow 2\Delta A_{gas}$. In
particular, very long rods $L\gg h_{0}$ imply that $z\simeq 0$, i.e. that the
center of mass is confined close to the channel axis. Hence, this asymptotic
behavior can be understood by expanding Eq. (24) about $z=0$. Indeed, for
$L/h_{0}\gg 1$ the integrand of Eq. (24) can be approximated by $h(x)/L$ and
hence we obtain that $A(x)\propto 2\ln\left[h(x)\right]$ which eventually
leads to $\Delta A\rightarrow 2\Delta A_{gas}$.

####  III.1.3 Transport

Next, we analyze the transport of rods under the action of a constant force.
In order to simplify the analysis we assume that the fluid rods are suspended
and keep at rest, i.e. there is no advection of rods due to fluid motion. In
such a regime, Fig. 5 shows that the dependence of the dimensionless
permeability $\mu$ on the length of the major axis of the rod, $L$, is quite
complex. Indeed, for flat channels ($\beta\Delta A_{gas}=0$) the transport is
determined solely by the effective diffusion coefficient. In particular, for
$L\gg h_{0}$ the rod is almost aligned with the axis of the channel and the
effective diffusion coefficient approaches the one of the minor axis (see
Fig.3) and, according to Eq.(22) $\mu\rightarrow 1$ for $L\gg h_{0}$. In
contrast, for $\beta\Delta A_{gas}=0$ and $L\ll h_{0}$ the rod can freely
rotate and it experiences a reduction in the effective diffusion coefficient
(see Fig.3). For $\beta\Delta A_{gas}\neq 0$, the channel is not flat and the
dependence of the dimensionless permeability $\mu$ on $L$ becomes more
involved. Indeed, Fig. 5 shows that for intermediate values of $\beta\Delta
A_{gas}$, $\mu$ displays a non-monotonous dependence on $L/h_{0}$. This
behavior is similar to that observed for polymers confined between corrugated
plates (see Ref. Bianco and Malgaretti (2016)). Finally, for larger values of
$\beta\Delta A_{gas}$, $\mu$ monotonically decreases upon increasing
$L/h_{0}$.

###  III.2 Charged rods

Figure 5: Dimensionless channel permeability, $\mu$, to non-charged rods as
function of the normalized length of the major axis of the rods, $L/h_{0}$ for
rods with $l=10^{-2}h_{0}$ and for different values of $\beta\Delta
A_{gas}=0.2,0.6,1.1,1.7,2.9$ (from blue to cyan), whereas the grey dots are
for $\beta\Delta A_{gas}=0$.

Next, we analyze the case of charged rods. For simplicity we assume that the
charge, $q$, is localized at the center of the rod and that the channel walls
are characterized by a constant charge density $\sigma$. In addition, we
assume that a dilute monovalent electrolyte is suspended in the fluid phase so
that the system is electrically neutral. In this regard, the rod brings an
extra charge, whose magnitude is assumed to be much smaller than that of the
charge of the double layer such that it does not affect significantly the
local electrostatic field111Such an approximation becomes exact in the limit
of vanishing charge of the rod.. Accordingly, the (unperturbed) electrostatic
potential inside the channel reads

$\phi=\frac{\sigma}{\varepsilon\kappa}\frac{\cosh\left(\kappa
z\right)}{\sinh\left(\kappa
h\left(x\right)\right)}=\phi_{0}\frac{\cosh\left(\kappa
z\right)}{\sinh\left(\kappa h\left(x\right)\right)}\,,$ (30)

with $\phi_{0}=\frac{\sigma}{\varepsilon\kappa}$. The local free energy
becomes

$\displaystyle A(x)$ $\displaystyle=-fqx-
k_{B}T\ln\left[\int\limits_{0}^{\pi}\\!\int\limits_{-h(x)}^{h(x)}\\!\\!e^{-\beta
q\phi(x,z)}\Gamma(x,z,\theta)\frac{dzd\theta}{2h_{0}\pi}\right],$ (31)

where the diffusion coefficient reads

$\dfrac{\mathcal{D}(x)}{D_{0}}=\dfrac{\int\limits_{0}^{\pi}\int\limits_{-h(x)}^{h(x)}\\!\\!\left[\cos^{2}\theta+\frac{l}{L}\sin^{2}\theta\right]e^{-\beta
q\phi(x,z)}\Gamma(x,z,\theta)dzd\theta}{\int\limits_{0}^{\pi}\int\limits_{-h(x)}^{h(x)}e^{-\beta
q\phi(x,z)}\Gamma(x,z,\theta)dzd\theta}.$ (32)

Figure 6: Dimensionless channel permeability, $\mu$, as function of the charge
of the rods. (a): $\mu$ as a function of $\beta q\phi_{0}$ for different
values of the channel corrugation, $\beta\Delta A_{gas}=1.1,1.7,2.2,2.9$, with
$\kappa h_{0}=1$. Lighter colors stand for larger values of $\beta\Delta
A_{gas}$, whereas the black solid line stands for $\beta\Delta A_{gas}=0$. The
force is proportional to the charge and, for a unit charge, it amounts to
$\beta feL=0.1$. (b): $\beta\Delta A_{eq}$ (see Eq. (28)) as function of the
interaction potential with the walls $\beta q\phi_{0}$. Note that the maxima
in panel (a) occur for values of $q$ for which $\beta\Delta A_{eq}\simeq 0$.

The dependence of the channel permeability on the electrostatic interaction of
the rods with the walls is shown in Fig. 6a. In particular, Fig. 6a shows that
the dependence of $\mu$ on the interaction potential of the rods with the
walls is non-monotonous. For a given charge density on the walls, when the
magnitude of the charge is very large (for both signs of the charge) the
permeability becomes vanishingly small. Indeed, for such cases the effective
free energy barrier to be overcome is very large and hence $\mu$ becomes very
small. In contrast, upon reducing the magnitude of the charge, $\mu$ increases
and it reaches its maximum for weakly negatively charged rods, i.e. when rods
are attracted by the channel walls. As shown in Fig. 6a the optimal value of
the charge depends on the channel corrugation and it decreases upon increasing
the channel corrugation. This non-monotonous dependence, and in particular the
location of the maximum, can be understood by looking at the equilibrium free
energy difference $\beta\Delta A_{eq}$. Indeed, Fig. 6b shows that the maxima
observed in Fig. 6a occur for values of $q$ for which $\beta\Delta A_{eq}=0$
(see Fig. 6b) i.e., when the equilibrium overall free energy barrier drops and
the rods have only to overcome smaller local free energy barriers.

Fig. 6a shows that the geometry of the channel can be used to tune the sorting
of rods depending on their charge. In order to exploit the geometry of the
channel and to separate charged rods depending on their length, it is
mandatory to find the set of parameters for which the sensitivity of the
dimensionless channel permeability on the channel geometry is maximized. As
discussed, our model highlights a direct correlation between the equilibrium
free energy barrier $\Delta A_{eq}$ and the dimensionless channel permeability
$\mu$. This implies that the region of the parameter space in which $\mu$ is
likely to be sensitive to the geometry, $\Delta A_{gas}$, can be guessed by
inspecting the equilibrium free energy difference $\Delta A_{eq}$.

Figure 7: Contour lines of $\beta\Delta A_{eq}=0$ for $kh_{0}=0.5$ (dashed
lines) and $kh_{0}=1$ (solid lines). Different colors are for diverse values
of $\beta q\phi_{0}=-0.6,-0.8,-1$ where lighter colors stand for larger values
of the absolute value of $\beta q\phi_{0}$.

Figure 8: Absolute value of the dimensionless channel permeability, $\mu$, to
charged rods as function of the normalized length of the major axis of the
rods, $L/h_{0}$, for $\kappa h_{0}=0.5$, $\beta\Delta A_{gas}=1.7$ (panel
(a)), $\beta\Delta A_{gas}=0.6$ (panel (b)), $L/l=100$ and for different
values of the wall potential, $\beta q\phi_{0}=-0.5,-0.3,-0.1,0,0.1,0.3,0.5$,
that is color coded: blue (red) lines stand for negative (positive) charges
and lighter colors stands for smaller magnitudes of the wall potential. The
black solid line stands for $z=0$. All results are for $\beta qfL=0.1$.
Figure 9: Absolute value of the dimensionless channel permeability, $\mu$, to
charged rods upon varying the inverse dimensionless Debye length, $\kappa
h_{0}$, for $\beta\Delta A_{gas}=1.7$, $L/l=100$ and for $\beta
q\phi_{0}=-0.5$ (blue lines) and $\beta q\phi_{0}=0.5$ (red lines). In blue
colors lighter colors stands for larger values of $L/h_{0}=0.3,0.5,1$ and
$\beta q\phi_{0}=0.5$ in red colors lighter colors stands for larger values of
$L/h_{0}=0.3,0.5,1$. All results are for $\beta qfL=0.1$.

Fig. 7 shows that for $kh_{0}\gtrsim 1$, the isolines $\beta\Delta A_{eq}=0$
are parallel to the ordinate axis for $L\gtrsim h_{0}$. Hence, all rods with
length $L\gtrsim h_{0}$ are expected to have relatively similar values of
$\mu$. Accordingly, for this set of parameters only rods with length
$L\lesssim h_{0}$ can be separated. In contrast, for $kh_{0}\simeq 0.5$ and
for $\beta\Delta A_{gas}\simeq 0.75$ the contour line $\beta\Delta A_{eq}=0$
bends. This can be the signal of a strong dependence of the velocity on the
rod size. Fig. 8a shows that for larger values of $\beta\Delta A_{gas}$ of and
$\beta q\phi_{0}\gtrsim-0.1$, $\mu$ is sensitive to $L/h_{0}$ only for
$L\lesssim h_{0}$ whereas $\mu$ shows a plateau for $L\gtrsim h_{0}$, in
agreement with our argument based on $\beta\Delta A_{eq}$. In contrast, for
smaller values of $\beta\Delta A_{gas}$, Fig. 8b shows that for $\beta
q\phi_{0}\lesssim-0.1$, $\mu$ is more sensitive to $L$ for the full range of
values explored in Fig. 8b, again in agreement with our argument.

Finally we have studied the dependence of the flux of charged rods as a
function of the Debye length. Interestingly, Fig. 9 shows that the flux of
negatively charged rods (hence attracted by the channel walls) has a non-
monotonous dependence on the dimensionless inverse Debye length $\kappa h_{0}$
and it displays a maximum around $\kappa h_{0}\lesssim 1$ for all the
geometries of the channel that we have explored. In contrast, the net flow of
positively charged rods increases monotonously with the inverse Debye length
$\kappa h_{0}$.

###  III.3 Conclusions

We have studied the dynamics of charged rods embedded in varying-section
channels. Under the assumption of slowly varying channel sections,
$\partial_{x}h(x)\ll 1$ we have extended the Fick-Jacobs approximation to the
case of charged rods. Our approximation allows us to derive an expression for
the local diffusion coefficient, $\mathcal{D}(x)$ (see Eq. (17)). We have
tested our prediction for the case of neutrally charged rods against
experimental and numerical results (see Ref. Yang  _et al._ (2019b)).
Interestingly, our predictions (see Fig. 2) match well with the
experimental/numerical data for both the local diffusion coefficient as well
as for the Mean First Passage Time. We remark that, while in experiments
hydrodynamic coupling between the rods and the channel walls is naturally
accounted for, it is not so in our model. Hence, a first result of our
analysis is that, in the regime under study, hydrodynamic coupling plays a
minor role in the diffusion of rods within corrugated channels.

In order to grasp the relevance of the rod geometry on the free energy
barrier, $\Delta A$, we have compared it to that of point particles, $\Delta
A_{gas}$. Interestingly, we found that the enhancement in the free energy
difference, $\Delta A$, is at most twofold (see Fig. 4c). Next, we have
characterized the channel permeability, $\mu$, to neutral rods as a function
of their length and found that for intermediate values of $\beta\Delta
A_{eq}$, $\mu$ displays a non-monotonous dependence (see Fig. 5) on the major
axis of the rod, $L$, similarly to what has been observed for confined
polymers Bianco and Malgaretti (2016).

Finally, we have characterized the channel permeability to charged rods. In
particular, we have focused on the dependence of $\mu$ on the rod size and
charge. Interestingly, we found that $\mu$ displays a non-monotonous
dependence on the rod charge and that the value of the charge that maximizes
$\mu$ is the one for which the overall equilibrium free energy barrier,
$\Delta A_{eq}$ vanishes. Hence, our model allows us to rationalize the non-
monotonous dependence of $\mu$ with respect to the charge of the rod. We have
exploited this feature to identify those set of parameters for which $\mu$ is
likely to be more sensitive to the length, $L$, of the major axis of the rods.
Interestingly, this approach has revealed to be reliable in identifying those
regimes for which the sensitivity of $\mu$ on $L$ is maximized. All in all our
results show that $\mu$ can be tuned by properly combining the channel
geometry and the length and charge of the rod. Moreover, our model allows to
rationalize the appearance of non-monotonous dependence of $\mu$ on both, the
charge of the rod and its length. Our results can be useful for the design of
novel micro- and nano-fluidic devices aiming at sorting stiff filaments.

## Acknowledgements

This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) – Project-ID 416229255 – SFB 1411.

## References

  * Malgaretti  _et al._ (2019a) P. Malgaretti, G. Oshanin, and J. Talbot, J. Phys.: Condens. Matt. 31, 270201 (2019a).
  * Calero  _et al._ (2011) C. Calero, J. Faraudo, and M. Aguilella-Arzo, Phys. Rev. E 83, 021908 (2011).
  * Peyser  _et al._ (2014) A. Peyser, D. Gillespie, R. Roth, and W. Nonner, Biophysical Journal 107, 1841 (2014).
  * Lee  _et al._ (2017) H. Lee, D. Segets, S. Süß, W. Peukert, S.-C. Chen, and D. Y. Pui, Journal of Membrane Science 524, 682 (2017).
  * Melnikov  _et al._ (2017) D. V. Melnikov, Z. K. Hulings, and M. E. Gracheva, Physical Review E 95, 063105 (2017).
  * Bacchin (2018) P. Bacchin, Membranes 8, 10 (2018).
  * Berezhkovskii  _et al._ (2019) A. M. Berezhkovskii, L. Dagdug, and S. M. Bezrukov, J. Chem. Phys. 151, 054113 (2019).
  * Alberts  _et al._ (2007) B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter,  _Molecular Biology of the Cell_ (Garland Science, Oxford, 2007).
  * Yu  _et al._ (2017) Q. Yu, K. Wang, J. Zhang, M. Liu, Y. Liu, and C. Cheng, RSC Adv. 7, 37542 (2017).
  * Chen  _et al._ (2019) C. Chen, L. Xie, and Y. Wang, Nano Research 12, 1267 (2019).
  * Yang  _et al._ (2019a) L. Yang, Z. Zhou, J. Song, and X. Chen, Chem. Soc. Rev. 48, 5140 (2019a).
  * Fakhri  _et al._ (2010) N. Fakhri, F. C. MacKintosh, B. Lounis, L. Cognet, and M. Pasquali, Science 330, 1804 (2010).
  * Domínguez and Abreu (2016) M. S. Domínguez and C. R. Abreu, eds.,  _Nanocolloids_ (Elsevier, Amsterdam, 2016).
  * Bonthuis  _et al._ (2008) D. J. Bonthuis, C. Meyer, D. Stein, and C. Dekker, Physical review letters 101, 108303 (2008).
  * Sakaue (2018) T. Sakaue, J. Phys.: Condens. Matt. 30, 244004 (2018).
  * Chinappi and Cecconi (2018) M. Chinappi and F. Cecconi, J. Phys.: Condens. Matt. 30, 204002 (2018).
  * Saleh and Sohn (2003) O. A. Saleh and L. L. Sohn, Proc. Natl. Acad. Sci. U. S. A. 100, 820 (2003).
  * Ito  _et al._ (2004) T. Ito, L. Sun, M. A. Bevan, and R. M. Crooks, Langmuir 20, 6940 (2004).
  * Heins  _et al._ (2005) E. A. Heins, Z. S. Siwy, L. A. Baker, and R. C. Martin, Nano Lett. 5, 1824 (2005).
  * Arjmandi  _et al._ (2012) N. Arjmandi, W. Van Roy, L. L., and G. Borghs, Anal. Chem. 84, 8490 (2012).
  * Martinez Cristancho and Seidel-Morgenstern (2016) C. A. Martinez Cristancho and A. Seidel-Morgenstern, Journal of Chromatography A 1434, 29 (2016).
  * Ghosh (2002) R. Ghosh, Journal of Chromatography A 952, 13 (2002).
  * Swernath  _et al._ (2014) S. Swernath, M. Kaspereit, and A. Kienle, Chemical Engineering and Technology 37, 643 (2014).
  * Zwanzig (1992) R. Zwanzig, J. Phys. Chem. 96, 3926 (1992).
  * Reguera and Rubi (2001) D. Reguera and J. M. Rubi, Phys. Rev. E 64, 061106 (2001).
  * Kalinay and Percus (2008) P. Kalinay and J. K. Percus, Phys. Rev. E 78, 021103 (2008).
  * Martens  _et al._ (2011) S. Martens, G. Schmidt, L. Schimansky-Geier, and P. Hänggi, Phys. Rev. E 94, 2492 (2011).
  * Chacón-Acosta  _et al._ (2013) G. Chacón-Acosta, I. Pineda, and L. Dagdug, J. Chem. Phys. 139, 214115 (2013).
  * Malgaretti  _et al._ (2013) P. Malgaretti, I. Pagonabarraga, and J. Rubi, Frontiers in Physics 1, 21 (2013).
  * Reguera  _et al._ (2006) D. Reguera, G. Schmid, P. S. Burada, J. M. Rubi, P. Reimann, and P. Hänggi, Phys. Rev. Lett. 96, 130603 (2006).
  * Reguera  _et al._ (2012) D. Reguera, A. Luque, P. S. Burada, G. Schmid, J. M. Rubi, and P. Hänggi, Phys. Rev. Lett. 108, 020604 (2012).
  * Marini Bettolo Marconi  _et al._ (2015) U. Marini Bettolo Marconi, P. Malgaretti, and I. Pagonabarraga, J. Chem. Phys. 143, 184501 (2015).
  * Malgaretti  _et al._ (2016a) P. Malgaretti, I. Pagonabarraga, and J. Rubi, Entropy 18, 394 (2016a).
  * Puertas  _et al._ (2018) A. Puertas, P. Malgaretti, and I. Pagnabarraga, J. Chem. Phys. 149, 174908 (2018).
  * Malgaretti  _et al._ (2014) P. Malgaretti, I. Pagonabarraga, and J. M. Rubi, Phys. Rev. Lett 113, 128301 (2014).
  * Malgaretti  _et al._ (2015) P. Malgaretti, I. Pagonabarraga, and J. M. Rubi, Macromol. Symposia 357, 178 (2015).
  * Malgaretti  _et al._ (2016b) P. Malgaretti, I. Pagonabarraga, and J. Miguel Rubi, J. Chem. Phys. 144, 034901 (2016b).
  * Chinappi and Malgaretti (2018) M. Chinappi and P. Malgaretti, Soft Matter 14, 9083 (2018).
  * Malgaretti  _et al._ (2019b) P. Malgaretti, M. Janssen, I. Pagonabarraga, and J. M. Rubi, J. Chem. Phys. 151, 084902 (2019b).
  * Bianco and Malgaretti (2016) V. Bianco and P. Malgaretti, J. Chem. Phys. 145, 114904 (2016).
  * Malgaretti and Oshanin (2019) P. Malgaretti and G. Oshanin, Polymers 11, 251 (2019).
  * Bodrenko  _et al._ (2019) I. V. Bodrenko, S. Salis, S. Acosta-Gutierrez, and M. Ceccarelli, J. Chem. Phys. 150, 211102 (2019).
  * Burada  _et al._ (2009) P. S. Burada, P. Hänggi, F. Marchesoni, G. Schmid, and P. Talkner, ChemPhysChem 10, 45 (2009).
  * Karimi  _et al._ (2020) H. Karimi, M. R. Setare, and A. Moradian, Phys. Rev. E 102, 012610 (2020).
  * Yang  _et al._ (2019b) X. Yang, Q. Zhu, C. Liu, W. Wang, Y. Li, F. Marchesoni, P. Hänggi, and H. P. Zhang, Phys. Rev. E 99, 020601 (2019b).
  * Happel and Brenner (1983) J. Happel and H. Brenner,  _Low Reynolds number hydrodynamics_ (Kluwer, The Hague, 1983).