# Thermally activated particle motion in biased correlated Gaussian disorder
potentials

Alexander Valov<EMAIL_ADDRESS>Racah Institute of Physics,
Hebrew University of Jerusalem, Jerusalem 91904, Israel  Netanel Levi
<EMAIL_ADDRESS>Racah Institute of Physics, Hebrew University of
Jerusalem, Jerusalem 91904, Israel  Baruch Meerson<EMAIL_ADDRESS>Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904,
Israel

###### Abstract

Thermally activated particle motion in disorder potentials is controlled by
the large-$\Delta V$ tail of the distribution of height $\Delta V$ of the
potential barriers created by the disorder. We employ the optimal fluctuation
method to evaluate this tail for correlated quenched Gaussian potentials in
one dimension in the presence of a small bias of the potential. We focus on
the mean escape time (MET) of overdamped particles averaged over the disorder.
We show that the bias leads to a strong (exponential) reduction of the MET in
the direction along the bias. The reduction depends both on the bias, and on
detailed properties of the covariance of the disorder, such as its derivatives
and asymptotic behavior at large distances. We verify our theoretical
predictions, as well as earlier predictions for zero bias, by performing
large-deviation simulations of the potential disorder. The simulations employ
correlated random potential sampling based on the circulant embedding method
and the Wang-Landau algorithm, which enable us to probe probability densities
smaller than $10^{-1200}$.

##  I Introduction

Slow thermally activated motion of overdamped particles in a quenched disorder
potential is an important research paradigm, which is relevant in many
applications such as supercooled liquids and glassy matrices (solidstate1, ;
solidstate2, ; Bassler, ; glass1, ), the motion of particles in disordered
metals or semiconductors (glass2, ; glass3, ) the motion of macromolecules in
DNA (bio1, ; bio2, ; bio3, ), etc. Direct experiments with this system have
recently become available in the form of laser-produced quenched random
potentials in colloids (G1, ; G2, ; G3, ; G4, ; speckle, ). Since the
pioneering works of DeGennes (DeGennes, ) and Zwanzig (Zwanzig, ), there have
been many theoretical studies in this direction (LV, ; anomalous1, ;
anomalous2, ; anomalous3, ; anomalous4, ; Wilkinson, ; M2022, ), to name but a
few.

When the thermal noise is small, the mean escape time (MET) of particles from
a local potential well of the disordered potential is determined by the
large-$\Delta V$ tail of the probability distribution $\mathcal{P}(\Delta V)$
of the _potential barriers_ $\Delta V$ created by the disorder. This tail can
be efficiently evaluated by using the optimal fluctuation method (OFM) (LV, ;
M2022, ; aboutLV, ). In particular, it was found in Ref. M2022  that this tail
strongly (exponentially) depends on whether the covariance of the disorder
decreases monotonically with the distance or not. These findings, however,
were limited to the unbiased case, that is when the ensemble average of the
random potential $V(x)$ at any $x$ is zero. In experimental situations there
can be a systematic external potential bias, and it is interesting to
investigate its role. Here we show that the bias can strongly affect the
large-$\Delta V$ tail of the distribution $\mathcal{P}(\Delta V)$ and, as a
result, lead to an exponentially strong reduction of the MET in the direction
along the bias. This reduction depends both on the bias, and on detailed
properties of the covariance of the disorder, such as its derivatives and
asymptotic behavior at large distances. The OFM calculations are based on the
determination of the optimal – that is, the most likely – configuration of the
random potential $V(x)$ which dominates the large-$\Delta V$ tail of
$\mathcal{P}(\Delta V)$ (LV, ; M2022, ).

In order to verify our theoretical predictions, as well as the earlier
predictions LV ; M2022  for zero bias, we perform large-deviation simulations
of the potential disorder. The simulations employ correlated random potential
sampling based on the circulant embedding method and the Wang-Landau
algorithm, which enable us to probe probability densities smaller than
$10^{-1200}$. As we will show, the simulation results strongly support the
theory.

Let us introduce the basic model that we consider in this work. Overdamped
particle motion in a quenched disorder potential $V\left(x\right)$ can be
described by the Langevin equation

$\dot{x}=-\mu\frac{dV(x)}{dx}+\sqrt{2D}\xi(t)\,,$ (1)

where $\mu$ is the mobility, $D$ is the diffusion coefficient of the particle
in the absence of the potential, and $\xi(t)$ is a delta-correlated Gaussian
noise with zero mean. In the following we set $\mu=1$ (which renders somewhat
unusual units to the potential, $[V]=\text{length}^{2}/\text{time}$).

We suppose that the quenched random potential $V\left(x\right)$ is
statistically homogeneous in space and normally distributed. The potential
barrier $\Delta V$ is formally defined as $\Delta
V=V\left(x=L\right)-V\left(x=-L\right)$ where we assume, without limiting the
generality, that $x=-L$ is a minimum point of $V(x)$, $x=L$ is a maximum
point, and $V^{\prime}(x)>0$ for all $|x|<L$. In this case the activated
escape proceeds from left to right.

For a fixed realization of the potential $V(x)$, the MET over this barrier,
averaged over the realizations of the thermal noise, is described by the
classical Kramer’s formula (Kramers, ):

$T\sim\exp\left(\frac{\Delta V}{D}\right)\,,$ (2)

where we have omitted the pre-exponential factor which we will not be
interested in. As in the previous works LV ; M2022 , we will focus on the MET
additionally averaged over different realizations of the disorder potential
$V(x)$. We will denote it by $\langle T\rangle$. In the limit of $D\rightarrow
0$, $\langle T\rangle$ is controlled by the large-$\Delta V$ tail of the
barrier distribution $\mathcal{P}\left(\Delta V\right)$ LV ; M2022 . This tail
can be represented as

$\mathcal{P}\left(\Delta V\rightarrow\infty\right)\sim\exp\left[-S\left(\Delta
V\right)\right]\,,$ (3)

where $S\left(\Delta V\right)$ is a large-deviation function that will be in
the focus of our attention. Therefore,

$\left\langle T\right\rangle\sim\int_{0}^{\infty}\exp\left[\frac{\Delta
V}{D}-S\left(\Delta V\right)\right]d\left(\Delta V\right)\,.$ (4)

Since $D\to 0$, this integral can be evaluated via the Laplace’s method. The
saddle point $\Delta V_{*}$ is the maximum point of the function

$\phi\left(\Delta V\right)=\frac{\Delta V}{D}-S\left(\Delta V\right)\,.$ (5)

As a result, the MET averaged over the realizations of disorder can be
evaluated as

$\left\langle T\right\rangle\sim\exp\left[\frac{\Delta V_{*}}{D}-S\left(\Delta
V_{*}\right)\right]\,.$ (6)

To implement the evaluation, outlined in Eqs. (5) and (6), we first need to
determine the large-deviation function $S\left(\Delta V\right)$. These
calculations are presented in Sec. II. The simulation algorithm is briefly
described in Sec. III.1, and Sec. III.2 presents the simulations results. A
brief summary, discussion and possible extensions of our results are given in
Section IV.

##  II  Optimal Fluctuation Method

A statistically homogeneous random Gaussian potential $V(x)$ is fully
determined by its mean (which describes the bias if there is one, see below)
and the covariance

$\kappa\left(x-x^{\prime}\right)=\left\langle
V\left(x\right)V\left(x^{\prime}\right)\right\rangle-\langle
V(x)\rangle\langle V(x^{\prime})\rangle\,.$ (7)

We will assume that $\kappa(z)$ has its absolute maximum at $z=0$ and is at
least twice differentiable. $\kappa(z)$ can be either a monotonically
decreasing function of $z$, or nonmonotonic difference . The variance of
$V(x)$ is equal to $\kappa\left(0\right)$.

In the absence of bias, the statistical weight of a realization of the
Gaussian disorder potential $V(x)$ is determined by the action functional
(Zinn-Justin, )

$\mathcal{S}\left[V\left(x\right)\right]=\frac{1}{2}\int_{-\infty}^{\infty}dx\int_{-\infty}^{\infty}dx^{\prime}K\left(x-x^{\prime}\right)V\left(x\right)V\left(x^{\prime}\right)\,,$
(8)

where $K\left(x-x^{\prime}\right)$ is the inverse kernel, defined by the
equation

$\int_{-\infty}^{\infty}dx^{\prime\prime}K\left(x-x^{\prime\prime}\right)\kappa\left(x^{\prime}-x^{\prime\prime}\right)=\delta\left(x-x^{\prime}\right)\,.$
(9)

In the presence of a uniform bias field, $E=\text{const}$, we have $\langle
V(x)\rangle=-Ex$, and the potential $V(x)$ can be represented as

$V\left(x\right)=v\left(x\right)-Ex\,,$ (10)

where $v\left(x\right)$ is a normally distributed random field with zero mean
and the covariance $\kappa\left(z\right)$.

The large-$\Delta V$ tail of $\mathcal{P}\left(\Delta V\right)$ describes
atypically large barriers, which are dominated by the optimal configuration of
the potential $V(x)$ conditioned on the specified $\Delta V$ LV ; M2022 . The
optimal configuration minimizes, subject to additional conditions that we will
specify shortly, the action functional (Zinn-Justin, ):

$\displaystyle\mathcal{S}\left(v,E\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{-\infty}^{\infty}dx\int_{-\infty}^{\infty}dx^{\prime}K\left(x-x^{\prime}\right)v\left(x\right)v\left(x^{\prime}\right)$
(11) $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{-\infty}^{\infty}dx\int_{-\infty}^{\infty}dx^{\prime}K\left(x-x^{\prime}\right)$
$\displaystyle\times\left[V\left(x\right)+Ex\right]\left[V\left(x^{\prime}\right)+Ex^{\prime}\right]\,.$

Assuming that the optimal configuration of $V\left(x\right)$ is smooth, we can
write down the conditions specifying the potential barrier $\Delta V$ on an
interval of length $2L$:

$\displaystyle V\left(x=L\right)-V\left(x=-L\right)=\Delta V\,,$ (12)
$\displaystyle\frac{dV}{dx}\left(x=-L\right)=\frac{dV}{dx}\left(x=L\right)=0\,,$
(13) $\displaystyle\frac{d^{2}V}{dx^{2}}\left(x=-L\right)>0,\
\frac{d^{2}V}{dx^{2}}\left(x=L\right)<0\,,$ (14)
$\displaystyle\frac{dV}{dx}>0\,,\quad|x|<L\,,$ (15)

where the inequality (15) guarantees that there are no other extrema of
$V\left(x\right)$ on the interval $|x|<L$.

Accommodating the constraint (12) via a Lagrange multiplier $\lambda$ and two
delta-functions, we obtain a modified action functional to be minimized:

$\displaystyle s_{\lambda}\left[V\left(x\right)\right]$
$\displaystyle=\frac{1}{2}\int_{-\infty}^{\infty}dx\bigg{\\{}\int_{-\infty}^{\infty}dx^{\prime}K\left(x-x^{\prime}\right)$
(16)
$\displaystyle\times\left[V\left(x\right)+Ex\right]\left[V\left(x^{\prime}\right)+Ex^{\prime}\right]$
$\displaystyle-\frac{\lambda}{2}V\left(x\right)\left[\delta\left(x-L\right)-\delta\left(x+L\right)\right]\bigg{\\}}\,.$

An explicit account of the constraints (13)-(15) in the action minimization
procedure is quite difficult. Therefore, we will proceed without accounting
for these constraints, and make sure a posteriori that they are obeyed by the
solution.

The linear variation of the action functional (16) must vanish, so that

$\displaystyle\delta
s_{\lambda}\left[V\left(x\right)\right]\\!=\\!\frac{1}{2}\int_{-\infty}^{\infty}dx\,\delta
V\left(x\right)\bigg{\\{}\int_{-\infty}^{\infty}dx^{\prime}K\left(x-x^{\prime}\right)$
$\displaystyle\times\left[V\left(x^{\prime}\right)+Ex^{\prime}\right]\\!-\\!\frac{\lambda}{2}\left[\delta\left(x-L\right)-\delta\left(x+L\right)\right]\bigg{\\}}=0,$
(17)

leading to the linear integral equation

$\displaystyle\int_{-\infty}^{\infty}K\left(x-x^{\prime}\right)\left[V\left(x^{\prime}\right)+Ex^{\prime}\right]dx^{\prime}$
$\displaystyle=\frac{\lambda}{2}\left[\delta\left(x-L\right)-\delta\left(x+L\right)\right]\,.$
(18)

Comparing this equation with Eq. (9), one can easily guess the solution:

$V\left(x\right)=\frac{\lambda}{2}\left[\kappa\left(x-L\right)-\kappa\left(x+L\right)\right]-Ex.$
(19)

Then, using Eqs. (12) and (13), we determine the Lagrange multiplier
$\lambda$,

$\lambda=\frac{\Delta V+2EL}{\kappa\left(0\right)-\kappa\left(2L\right)}\,,$
(20)

and obtain an (in general, transcendental) equation for the optimal value of
$L$:

$\kappa^{\prime}\left(2L\right)=-\frac{2E\left[\kappa\left(0\right)-\kappa\left(2L\right)\right]}{\Delta
V+2EL}\,,$ (21)

where $\kappa^{\prime}(\dots)$ is the derivative of the covariance with
respect to its argument.

Therefore, the optimal, i.e. the least improbable, configuration of the
disorder potential $V(x)$, conditioned on the large potential barrier $\Delta
V$, is the following:

$V(x)\\!=\\!\frac{\Delta
V+2EL}{2\left[\kappa\left(0\right)-\kappa\left(2L\right)\right]}\left[\kappa\left(x-L\right)\\!-\\!\kappa\left(x+L\right)\right]\\!-\\!Ex.$
(22)

We have already imposed the conditions (12) and (13). Taking the second
derivative of the both sides of Eq. (22) we see that, for $E>0$ (which we
should require in any case, see below), the conditions (14) are also
satisfied. The fulfillment of the monotonicity condition (15) depends on the
specific form of covariance, and we will discuss it shortly. Meanwhile, using
Eqs. (8) and (22), we can calculate the action:

$S\left(\Delta V,L\right)=\frac{\left(\Delta
V+2EL\right)^{2}}{4\left[\kappa\left(0\right)-\kappa\left(2L\right)\right]}\,,$
(23)

where $L$ is the solution of Eq. (21). In the following subsections we will
consider several cases depending on the form of the covariance and on the sign
and magnitude of the external bias.

###  II.1 Zero bias

Let us first briefly review the zero-bias case, $E=0$, previously considered
in Ref. LV ; M2022 , and highlight the crucial difference between
monotonically decreasing (MD) and nonmonotonic (NM) covariances, uncovered in
Ref. M2022 . Where necessary, we will also distinguish between nonmonotonic
covariances that become negative at some distances – nonmonotonic negative
(NMN) for brevity, and nonmonotonic but everywhere positive (NMP) covariances.
In the absence of bias, Eq. (22) for the optimal configuration of the
potential gives M2022

$V\left(x\right)=\frac{\Delta
V}{2\left[\kappa\left(0\right)-\kappa\left(2L\right)\right]}\left[\kappa\left(x-L\right)-\kappa\left(x+L\right)\right]\,,$
(24)

and the action (23) becomes M2022

$S\left(\Delta V\right)=\frac{\left(\Delta
V\right)^{2}}{4\left[\kappa\left(0\right)-\kappa\left(2L\right)\right]}\,.$
(25)

For the MD covariance $\kappa(z)$ the action (25) is a monotonically
decreasing function of $L$. As a result, the minimum action is achieved in the
limit of $L\to\infty$. That is, the optimal configuration of $V(x)$ in this
case has the form of an isolated pair of a spike and an antispike, whose shape
is determined by the shape of the covariance $\kappa(z)$ LV ; M2022 .

For the NM covariance the situation is different. Let us denote by
$\ell_{*}>0$ the closest to zero position of the local _minimum_ of
$\kappa(z)$. To minimize the action (25) (at least locally) and satisfy the
conditions (13)-(15), we can set $L=\ell_{*}/2$. The optimal configuration of
the disorder potential in this case is localized M2022 . Under these
assumptions, and using Eq. (3), we obtain the following predictions for the
large-$\Delta V$ tail of the potential barrier distribution
$\mathcal{P}(\Delta V\to\infty)$ in the two cases LV ; M2022 :

$\\!-\\!\ln\mathcal{P}(\Delta V)\\!\simeq\\!\begin{cases}\frac{\left(\Delta
V\right)^{2}}{4\kappa\left(0\right)}&\text{for MD covariance,}\\\
\frac{\left(\Delta
V\right)^{2}}{4\left[\kappa\left(0\right)-\kappa\left(\ell_{*}\right)\right]}&\text{for
NM covariance.}\end{cases}$ (26)

In its turn, the saddle-point evaluation, outlined in Eqs. (3)-(6) yields the
MET averaged over disorder:

$\ln\left\langle
T\right\rangle\simeq\begin{cases}\frac{\kappa\left(0\right)}{D^{2}}&\text{for
MD covariance}\\\
\frac{\kappa\left(0\right)-\kappa\left(\ell_{*}\right)}{D^{2}}&\text{for NM
covariance}\end{cases}$ (27)

Because of the very large $1/D^{2}$ factor in Eqs. (27), the MET averaged over
the disorder is extremely long (LV, ; M2022, ). A similar exponential
suppression, but observed in the long-time particle diffusion in disordered
potentials, has been known for a long time DeGennes ; Zwanzig .

Another striking effect, described by Eqs. (27), is specific to the averaged-
over-disorder MET $\langle T\rangle$. It describes a very strong (exponential)
dependence of $\langle T\rangle$ on whether the covariance $\kappa(z)$ is
monotonic or not M2022 . In systems with nonmonotonic covariances, described
by the second line of Eqs. (27), the MET is exponentially longer [for
$\kappa(\ell_{*})<0$] or exponentially shorter [for $\kappa(\ell_{*})>0$] than
the MET for MD covariances with the same variance, as described by the first
line of Eqs. (27).

An independent support for the predictions (26) comes from the bivariate
normal distribution of $V(x)$. The joint distribution of our potential $V(x)$
taking some values $V_{1}$ and $V_{2}$ at two spatial points, separated by
distance $2L$, is given by Gnedenko

$P(V_{1},V_{2})=\frac{1}{2\pi\sqrt{\kappa^{2}(0)-\kappa^{2}(2L)}}\exp\left[-\frac{\kappa(0)z}{2(\kappa^{2}(0)-\kappa^{2}(2L))}\right],$
(28)

where

$z=V_{1}^{2}+V_{2}^{2}-2\frac{\kappa(2L)}{\kappa(0)}V_{1}V_{2}.$ (29)

In particular, for the configuration where $V_{1}=-\Delta V/2$ and
$V_{2}=\Delta V/2$ without , Eqs. (28) and (29) yield

$P(\Delta
V)=\frac{1}{2\pi\sqrt{\kappa^{2}(0)-\kappa^{2}(2L)}}\exp\left[-\frac{\Delta
V^{2}}{4(\kappa(0)-\kappa(2L))}\right],$ (30)

for arbitrary $\Delta V$ and $L$. Clearly, Eq. (30) provides an upper bound on
the $\Delta V\gg 1$ tail of $P(\Delta V)$: the tail that we are interested in.
This is because this equation accounts for _all_ possible configurations of
$V(x)$ obeying the conditions $V(x=-L)=-\Delta V/2$ and $V(x=L)=\Delta V/2$,
regardless of whether they meet the additional conditions (13)-(15) or not.
Still, and somewhat surprisingly, the exponential factor in Eq. (30) perfectly
coincides with the OFM action (25) (where one should set $E=0$), leading to
Eq. (26) for the MD and NM covariances, respectively.

###  II.2 Nonzero bias

A nonzero bias breaks the left-right symmetry of the system. For concreteness,
we continue to assume that the direction of escape is from left to right.
There is a major difference between the negative ($E<0$) and positive ($E>0$)
bias. For a negative bias the minimum of the action functional (11) can be
made arbitrary small. To achieve this, the optimal configuration of the random
component of the potential, $v(x)$, should stay very close to zero and, via
infinitesimally small variations near $x=-L$ and $x=L$, create local minimum
and a maximum, respectively. In its turn, $L$ has to be chosen to be close to
$\Delta V/2|E|$ to provide the desired potential barrier $\Delta V$, see Fig.
1. As a result, the probability of finding high barriers against the bias is
quite high, and certainly beyond the applicability of the OFM. Therefore, here
we will only deal with a positive bias, which corresponds to the particle
escape along the bias.

Figure 1: Black line: an almost zero-action configuration of the disorder
potential $V(x)$, conditioned on the potential barrier $\Delta V$, for $E<0$
and $L=\Delta V/2|E|$. The straight magenta line shows $V(x)=|E|x$. The
potential stays arbitrarily close to the straight line, except for
infinitesimally small variations to accommodate the extremum points.

####  II.2.1 Monotonically decreasing covariance

Let us first examine how the presence of a small bias affects the optimal
value of $L=L(E)$ as described by Eq. (21). One can see from Eq. (21) that, as
$E$ goes to zero, $\kappa^{\prime}(L)$ also goes to zero, so that the optimal
barrier width $2L$ goes to infinity LV ; M2022 . However, as one can check a
posteriori, it does so slower than $1/E$, that is $\lim_{E\rightarrow
0^{+}}EL\left(E\right)=0$, and we will rely on this property.

When $E$ is sufficiently small, we can solve Eq. (21) for $L$ perturbatively
by using the $z\gg 1$ asymptotic of $\kappa^{\prime}(z)$. Also, for expected
large $L$ we can neglect the term $\kappa\left(2L\right)$ compared with
$\kappa\left(0\right)$. As a result, and in the leading order at small $E$,
the optimal barrier width is

$2L(E)\simeq\left(\left|\kappa^{\prime}\right|\right)^{-1}\left[\frac{2E\kappa\left(0\right)}{\Delta
V}\right]\,,$ (31)

where $\left(\left|\kappa^{\prime}\right|\right)^{-1}$ is the inverse function
of the $z\gg 1$ asymptotic of $\left|\kappa^{\prime}\right|$.

As a simple and useful example, let us suppose that the covariance decays as a
power law, $\kappa(z\to\infty)\simeq B\,z^{-\alpha}$, where $\alpha>0$. Then
Eq. (31) yields

$2L(E)\simeq\left[\frac{\alpha B\Delta
V}{2E\kappa(0)}\right]^{\frac{1}{\alpha+1}}\,.$ (32)

As one can see, $\lim_{E\rightarrow 0}EL\left(E\right)$ indeed vanishes, as we
assumed. In general, the faster the covariance goes to zero at large
distances, the slower will $L$ tend to infinity when $E\to 0$.

In the leading order in the bias $E$, the action (23) becomes

$\displaystyle\\!\\!S(\Delta V)$ $\displaystyle\simeq$
$\displaystyle\frac{\left(\Delta V\right)^{2}}{4\kappa(0)}\Biggl{[}1+$ (33)
$\displaystyle+$
$\displaystyle\\!\left(2+\frac{B}{\kappa(0)}\right)\\!\left(\frac{\alpha
B}{2\kappa(0)}\right)^{\frac{1}{\alpha+1}}\\!\left(\frac{E}{\Delta
V}\right)^{\frac{\alpha}{\alpha+1}}\Biggr{]}.$

The non-analytic correction $\sim E^{\frac{\alpha}{\alpha+1}}$, coming from
the bias, describes an increase in the “action cost” of creating the barrier
$\Delta V$ and, as a result, a decrease in the probability $\mathcal{P}(\Delta
V)$ of observing this barrier. This perturbative calculation demands the
strong inequality $E\ll\Delta V$. When $E\rightarrow 0$, the first line of Eq.
(26) is reproduced, as to be expected.

Now we can evaluate the MET from Eq. (6). Substituting Eq. (33) into Eq. (5),
we obtain

$\displaystyle\phi\left(\Delta V\right)=\frac{\Delta V}{D}-\frac{\left(\Delta
V\right)^{2}}{4\kappa(0)}\Biggl{[}1+$
$\displaystyle+\left(2+\frac{B}{\kappa(0)}\right)\left(\frac{\alpha
B}{2\kappa(0)}\right)^{\frac{1}{\alpha+1}}\left(\frac{E}{\Delta
V}\right)^{\frac{\alpha}{\alpha+1}}\Biggr{]}\,.$ (34)

We find the saddle point by minimizing this expression over $\Delta V$. In the
zeroth order in the bias we obtain $\Delta V=2\kappa(0)/D$. We proceed
perturbatively and, in the first order, obtain

$\Delta
V^{*}\simeq\frac{2\kappa(0)}{D}-\frac{(\alpha+2)(B+2\kappa(0))}{2(\alpha+1)\kappa(0)}\left(\frac{\alpha
BE^{\alpha}}{D}\right)^{\frac{1}{\alpha+1}}.$ (35)

As a result, we arrive at the following expression for the MET (6) in the
presence of a positive bias:

$\ln\left\langle T\right\rangle\simeq\frac{\kappa(0)}{D^{2}}-\frac{(\alpha
B)^{\frac{1}{\alpha+1}}(B+2\kappa(0))}{2\kappa(0)D^{2}}(DE)^{\frac{\alpha}{\alpha+1}}\,,$
(36)

Crucially, as $D$ goes to zero, this MET is _exponentially_ smaller than its
zero-bias counterpart. Also noticeable are the nonanalytic scalings of the
correction with the bias $E$ and with the diffusion coefficient $D$. For
$E\rightarrow 0$, Eq. (27) is reproduced.

Repeating these calculations for a general MD covariance, we arrive at the
following results for the action and the MET:

$S\left(\Delta V\right)=\frac{1}{4}\frac{\left(\Delta V\right)^{2}+2E\Delta
V\left(\left|\kappa^{\prime}\right|\right)^{-1}\left[\frac{2E\kappa\left(0\right)}{\Delta
V}\right]}{\kappa\left(0\right)-\kappa\left\\{\left(\left|\kappa^{\prime}\right|\right)^{-1}\left[\frac{2E\kappa\left(0\right)}{\Delta
V}\right]\right\\}}\,,$ (37) $\ln\left\langle
T\right\rangle\simeq\frac{\kappa(0)-\kappa\left[\left(\left|\kappa^{\prime}\right|\right)^{-1}\left(ED\right)\right]-DE\left(\left|\kappa^{\prime}\right|\right)^{-1}\left(ED\right)}{D^{2}}.$
(38)

The saddle-point in this case is

$\displaystyle\Delta
V_{*}=\frac{2\kappa\left(0\right)-2\kappa\left[\left(\left|\kappa^{\prime}\right|\right)^{-1}\left(ED\right)\right]}{D}-E\left(\left|\kappa^{\prime}\right|\right)^{-1}\left(ED\right)\,.$
(39)

Another instructive example, and a consistency check, is provided by the
Lorentzian covariance:
$\kappa\left(z\right)=\left(1+z^{2}/\sigma^{2}\right)^{-1}$, where $\sigma$ is
the characteristic correlation length. In this example $\kappa(0)=1$,
$B=\sigma^{2}$ and $\alpha=2$. This example allows for an exact analytical
solution of Eq. (21), valid for any value of $E>0$. After some straightforward
algebra we obtain:

$2L=\left({\frac{\sigma^{2}\Delta V}{E}}\right)^{1/3}.$ (40)

Remarkably, this exact result perfectly coincides with the large-$L$
asymptotic (32) for this particular case.

Going back to Eqs. (40) and (23), we obtain an exact expression for the action
in this case:

$S(\Delta V)=\frac{1}{4}\left[\left(\Delta V\right)^{2/3}+\left(\sigma
E\right)^{2/3}\right]^{3}.$ (41)

The optimal disorder configuration (22), with $L$ given by Eq. (40), is shown
in Fig. 2. Visible is a finite barrier width $2L$. The localization of the
barrier in this case which should be contrasted with the delocalized barrier,
$L\to\infty$, predicted for the zero bias.

Figure 2: Optimal configuration of the disorder potential $V(x)$ as described
by Eq. (40) for the Lorentzian covariance
$\kappa\left(z\right)=\left(1+z^{2}/\sigma^{2}\right)^{-1}$, where, for
simplicity, we set $\sigma=1$. For the chosen ratio $E/\Delta V=10^{-3}$ the
optimal barrier width is $2L=10$.

The small-$E$ expansion of the action (41),

$S(\Delta V)\simeq\frac{1}{4}(\Delta V)^{2}+\frac{3}{4}(\Delta
V)^{4/3}\left(\sigma E\right)^{2/3},$ (42)

perfectly agrees with the asymptotic presented in Eq. (33).

Let us summarize this subsection. Adding a positive bias to a disorder
potential with monotonically decreasing covariance makes the action and, as a
result, the MET sensitive not only to the bias itself (as to be expected), but
also to the large-distance behavior of the covariance derivative. The width of
the optimal barrier configuration becomes finite, and it increases quite
slowly as the bias goes to zero, see e.g. Eq. (32). Finally, we predict
nonanalytic scalings with the bias $E$ and with the diffusion coefficient $D$
of the (exponentially large) correction to the MET.

####  II.2.2 Nonmonotonic covariance

For NM covariances $\kappa(z)$, a positive bias reduces the optimal value of
$L$ as described by Eq. (21). When $E$ is small, Eq. (21) in the leading order
becomes

$\kappa^{\prime}\left(2L\right)\simeq-\frac{2E\left[\kappa\left(0\right)-\kappa\left(2L\right)\right]}{\Delta
V}\,.$ (43)

We look for the optimal barrier width $2L=\ell_{*}+\delta\ell$ and solve Eq.
(43) perturbatively for the small correction $\delta\ell$. In the leading
order, we obtain

$2L=\ell_{*}-\frac{4E}{\Delta
V}\frac{\kappa(0)-\kappa(\ell_{*})}{\kappa^{\prime\prime}(\ell_{*})}.$ (44)

The small parameter of this perturbative expansion is $E\ell_{*}/\Delta V\ll
1$. Equation (44) shows that the positive bias “squeezes” a bit the optimal
disorder configuration $V(x)$.

Substituting Eq. (44) into Eq. (23) and expanding in the small parameter
$E\ell_{*}/\Delta V$, we arrive at the following action

$S(\Delta V,E)\simeq\frac{\left(\Delta
V\right)^{2}}{4[\kappa(0)-\kappa(\ell_{*})]}\left(1+2\frac{E\ell_{*}}{\Delta
V}+\ldots\right).$ (45)

Using Eqs. (5) and (45), we obtain the saddle point

$\Delta
V^{*}=\frac{2[\kappa\left(0\right)-\kappa\left(\ell_{*}\right)]}{D}-E\ell_{*}\,,$
(46)

which results in the MTE

$\ln\langle
T\rangle\simeq\frac{\kappa(0)-\kappa(\ell_{*})}{D^{2}}-\frac{E\ell_{*}}{D}.$
(47)

Contrary to the case of MD covariance, here the correction coming from the
bias is analytic and not as prominent.

##  III Large deviation simulations

###  III.1 Simulation method

To generate numerical realizations of a one-dimensional statistically
homogeneous Gaussian field (HGF) $V(x)$, we consider a discrete array
$\vec{V}=(V(1),V(2),\ldots,V(M))^{T}$ of size $M\gg 1$, which provides a space
discretization of the continuous field with the lattice step $\Delta x=1$.
There is a straightforward method of sampling a discretized HGF $\vec{V}$ with
a given covariance matrix $C_{ij}=\kappa(|i-j|)=\langle V(i)V(j)\rangle$ for
$i,j=\overline{1,M}$. The method consists of two steps: diagonalization of the
covariance matrix $C_{ij}$ and a matrix multiplication:

$\vec{V}=C^{1/2}\vec{\xi}\,,$ (48)

where $\vec{\xi}=(\xi_{1},\xi_{2},\ldots,\xi_{M})^{T}$ is a vector composed of
independent standard normals. Although being transparent, this method is
highly inefficient, since it involves two computationally expensive steps: the
matrix diagonalization which requires $\mathcal{O}(M^{3})$ operations, and
matrix products which requires $\mathcal{O}(M^{2})$ operations.

Instead of the poorly scalable matrix multiplications, a more efficient way to
sample HGFs is to use the Circulant Embedding method (CEM) Chan1994 ;
Dietrich1997 . The method is based on embedding a covariance matrix $C_{i,j}$
of size $M\times M$ in a larger circulant covariance matrix $C^{\prime}_{ij}$
of size $M^{\prime}>2M$. Then, using the fact that a matrix multiplication
with a circulant matrix implements convolution, one can replace the matrix
equation (48) with multiplication in Fourier space. Therefore the required
data (like the result of matrix diagonalization) can be easily pre-computed
numerically by using the fast Fourier transform (FFT), while sampling a HGF
requires only $\mathcal{O}(M^{\prime}\ln M^{\prime})$ operations. Some
instructive implementations of the CEM for large deviation simulations of
fractal Brownian motion (whose time derivative is a stationary Gaussian
process) can be found in Refs. Hartmann_2 ; Hartmann_1 .

Here we are interested in extremely small probability densities, which are
virtually impossible to reach with conventional Monte Carlo (MC) simulations
employing the Metropolis-Hastings algorithm. Therefore, in order to reach the
large-$\Delta V$ tail of the distribution $\mathcal{P}(\Delta V)$, we employed
the Wang-Landau (WL) algorithm WL1 ; WL2 . Unlike the ordinary Metropolis-
Hastings algorithm, where the acceptance/rejection decisions are Markovian,
the Wang-Landau algorithm takes into account information about previously
visited states in such a way that it “forces” the algorithm to explore the
available configurational space more quickly.

In a nutshell, the WL algorithm aims at estimating the density of states
$\mathcal{P}(\Delta V)=\exp[-S(\Delta V)]$ on a chosen interval $a\leq\Delta
V\leq b$, updating on each MC step the histogram of visited states $H(\Delta
V)$ and adjusting the entropy function $S(\Delta V)$ in an iterative manner,
see Ref. Landau_Binder  for details.

At the start of the simulation, the histogram is initialized to zero,
$H(\Delta V)=0$, and the entropy is set to some guess function (we use
$S(\Delta V)=1$). Let
$\bigl{\\{}\vec{\xi}_{\text{r}},\vec{V}_{\text{r}},\Delta
V_{\text{r}}\bigl{\\}}$ represent the running configurations of the random
vector, the disorder potential computed using the CEM, and the maximal
potential barrier of $\vec{V}_{\text{r}}$, respectively. A proposed
configuration of the random vector $\vec{\xi}_{\text{p}}$ is generated by
changing a randomly chosen component $[\vec{\xi}_{\text{r}}]_{j}$ of the
random vector according to the Gaussian distribution centered at
$[\vec{\xi}_{\text{r}}]_{j}$:

$[\vec{\xi}_{\text{p}}]_{j}\sim\exp[-(x-[\vec{\xi}_{\text{r}}]_{j})^{2}/2].$
(49)

Then, the proposed configuration of the disorder potential
$\vec{V}_{\text{r}}$ and its maximal potential barrier $\Delta V_{\text{r}}$
are computed using the CEM and the proposed random vector
$\vec{\xi}_{\text{p}}$.

Every decision on whether to accept (a) or reject (r) the proposed
configuration $\bigl{\\{}\vec{\xi}_{\text{p}},\vec{V}_{\text{p}},\Delta
V_{\text{p}}\bigl{\\}}$ is made according to the transition probability

$\displaystyle
p_{\text{a/r}}=\min\left(r(\vec{V}_{\text{p}}|\vec{V}_{\text{r}})\frac{\exp[-S(\Delta
V_{\text{r}})]}{\exp[-S(\Delta V_{\text{p}})]},1\right),$ (50)
$\displaystyle\text{where
}r(\vec{V}_{\text{p}}|\vec{V}_{\text{r}})=\exp\left[\frac{\left[\vec{\xi}_{\text{r}}]_{j}^{2}-[\vec{\xi}_{\text{p}}\right]_{j}^{2}}{2}\right]\,.$

If the proposed configuration is rejected, the running configuration
$\bigl{\\{}\vec{\xi}_{\text{r}},\vec{V}_{\text{r}},\Delta
V_{\text{r}}\bigl{\\}}$ is kept. Following each decision, the action and the
histogram are updated

$\displaystyle S(\Delta V_{\text{r}})\to S(\Delta V_{\text{r}})-f\,,$ (51)
$\displaystyle H(\Delta V_{\text{r}})\to H(\Delta V_{\text{r}})+1\,,$

where $f$ is a modification factor (initially, we set $f=1$). This process is
repeated until the histogram of visited states $H(\Delta V)$ is sufficiently
flat. As a measure of flatness, we used the condition

$0.9\times\overline{H(\Delta V)}\leq\min(H(\Delta V)),$ (52)

where $\overline{H(\Delta V)}$ is the mean value of the histogram. Once this
condition is met, $f$ is reduced, $f\to f/2$, and the histogram is reset to
zero. Then, the process continues until $f$ is sufficiently small.

It is known that, at early stages of a simulation, the Wang-Landau algorithm
violates the detailed balance condition. Asymptotically, however, the detailed
balance is recovered as the modification factor $f$ tends to zero
Landau_Binder . We stopped the Wang-Landau simulations after 18 reductions of
the modification factor $f$, which suffices for our purposes of large
deviation simulations. This is because the statistical errors of the Wang-
Landau algorithm, known to saturate Zhou , are of the order $\mathcal{O}(1)$,
which is much smaller than the resulting entropy in our simulations, $S(\Delta
V)\sim\mathcal{O}(10^{3})$.

###  III.2 Simulation results

To verify our theoretical predictions for the monotonic and nonmonotonic
covariances, we implemented the Wang-Landau algorithm of sampling discretized
configurations of the disorder Gaussian potential $V(x)$ on a regular lattice
of length $M=10^{3}$ with the following three covariances:

$\kappa(z)=\begin{cases}\frac{1}{1+\left(z/\sigma\right)^{2}},\quad\text{MD,}\\\
\frac{3+2\cos(\omega z)}{5[1+\left(z/\sigma\right)^{2}]},\quad\text{NMP,}\\\
\frac{\cos(\omega
z)}{1+\left(z/\sigma\right)^{2}},\quad\text{NMN.}\end{cases}$ (53)

Figure 3: Top panel: plotted versus the distance are three covariances,
presented in Eq. (53): the MD covariance (black), the NMP covariance (red),
and the NMN covariance (blue). The parameters of the first local minima for
the red and blue curves are $\ell_{*}=21.41$ and $\kappa(\ell_{*})=0.16$, and
$\ell_{*}=20.06$ and $\kappa(\ell_{*})=-0.79$, respectively. Bottom panel: The
large-$\Delta V$ behavior of the rate function, $-\ln\mathcal{P}(\Delta V)$,
as measured in the WL simulations of HGFs with the MD covariance (black
circles), the NMP covariance (red triangles), and the NMN covariance (blue
squares) for zero bias. The theoretical predictions (26) are shown by the
solid curves of the corresponding color.

These covariances are depicted in the top panel of Fig. 3. Notice that, for
all the covariances (53), the corresponding variances are equal to $1$. The
parameters $\sigma=40$ and $\omega=0.15$ in Eq. (53) represent the correlation
length and oscillation frequency, respectively, of this HGF. These values were
chosen to be sufficiently large to accurately approximate a continuous HGF,
but not too large so that effects of the finite system size would come into
play.

####  III.2.1 Zero bias

We started with verifying the theory predictions for the unbiased potential LV
; M2022 , which were summarized in Sec. II.1. The simulation results for the
rate functions, $-\ln\mathcal{P}(\Delta V)$, are shown in the bottom panel of
Fig. 3 alongside with the theoretically predicted rate functions given by Eq.
(26). As one can see, the agreement is excellent.

Figure 4: Configurations of the disorder potential $V(x)$, corresponding to
the potential barrier $\Delta V=80$, for three types of covariances (53):
simulations (black circles) vs. theoretical prediction (24) (blue solid
curve). Top: NMP covariance. Middle: MD covariance. Bottom: NMN covariance.

Examples of configurations of disorder potential $V(x)$, conditioned on large
$\Delta V$ and sampled in the WL simulations, are presented in Fig. 4. Also
shown are the optimal configurations predicted by Eq. (24) with the optimal
values of $L$. As one can see, the agreement of theory and simulations is
excellent in all the three cases. Notice that, for the NMN and NMP
covariances, the theoretically predicted optimal barrier size is finite,
allowing the true minimum of the action to be reached in the simulations. In
the case of MD covariance, the true minimum of the action can be reached only
when $L\to\infty$, and it is therefore inaccessible in numerical simulations.
A finite value of $L$, observed in the simulations, is caused by finite-size
effects as it is comparable with the size of the simulated system.

Figure 5: Simulation results for the large-$\Delta V$ behavior of the rate
function, $-\ln P(\Delta V)$, in the presence of a positive bias. Top panel:
results for MD covariance (53) for $E=0.005$ (green circles) and $E=0.1$
(magenta circles). For comparison, the results for the zero bias, $E=0$, are
shown by black circles. The theoretical predictions (42) are shown by the
solid curves of the corresponding color. The dashed magenta curve depicts the
exact expression (41). Bottom panel: results for the NMN covariance and
$E=0.1$ (magenta squares). For comparison, the zero-bias results are shown by
black squares. The solid curves of the corresponding color depict the
theoretical prediction (45).

####  III.2.2 Nonzero bias

Now we present the results of a comparison of theory and simulations for the
positively-biased potentials. The theoretical results were obtained in Secs.
II.2.1 and II.2.2. The simulation results for $-\ln\mathcal{P}(\Delta V)$ for
the MD and NMN covariances, alongside with the theoretical predictions, are
depicted in Fig. 5. Again, an excellent agreement is observed.

Some examples of sampled configurations of the disorder potential $V(x)$ are
presented in Fig. 6. In contrast to the unbiased case, the optimal distances
between the spike and antispike are always finite here. Therefore, choosing a
sufficiently large system size makes it possible to achieve the true minimum
of the action in numerical simulations. In particular, one can clearly see the
dramatic effect of the bias (even a relatively small one, $E=0.1$) on the
optimal barrier width $2L$ for the MD covariance, in a very good agreement
with Eq. (40). For comparison, the same bias $E=0.1$ hardly changes the
optimal barrier width for the NM covariances.

Figure 6: Configurations of the disorder potential $V(x)$ with the bias
$E=0.1$ and the potential barrier $\Delta V=50$ for the MD and NMN covariance
(53): simulations (black circles) vs. theoretical prediction Eq. (22) (blue
solid curve). The top panel corresponds to the MD covariance, where the
predicted $2L=\left(\sigma^{2}\Delta V/E\right)^{1/3}\simeq 92.9$. The bottom
panel corresponds to the NMN covariance, where the optimal value of $2L$ is
given by Eq. (44).

##  IV Summary and Discussion

We found that the presence of a small potential bias leads to an exponentially
large reduction in the MET of overdamped particles trapped in local potential
minima. The leading-order correction, which describes this reduction, behaves
differently in disorder potentials with monotonic and non-monotonic
covariance.

In the non-monotonic case, the effect of the bias can be accounted for via a
perturbative expansion in the bias. In the monotonic case, the scaling of the
MET with the bias is nontrivial, as it is affected by the large-distance
asymptotic of the inverse function of the derivative of the covariance. The
optimal barrier width of the biased potential in this case is finite, in
contrast to the unbiased case, where it is infinite. Even a very small
potential bias has a strong effect on the characteristic barrier width. As a
result, all bias-related effects are more pronounced for disorder potentials
with monotonically decreasing covariances.

We verified our predictions, as well as earlier predictions LV ; M2022  for
zero bias, in numerical simulations. The simulations employed the Wang-Landau
algorithm and the circulant embedding method of sampling a homogeneous
Gaussian field. We measured the large-$\Delta V$ tail of the barrier
distribution $\mathcal{P}\left(\Delta V\right)$ for different covariances and
bias magnitudes. The method also allowed us to sample the disorder potentials
$V(x)$, allowing for a direct comparison with the OFM predictions for the
optimal configurations, demonstrating excellent agreement. Combining the Wang-
Landau algorithm with the circulant embedding method, we were able to measure
probability densities below $10^{-1200}$. The numerical methods, which we
employed here, should be also useful when studying large deviation statistics
of other Gaussian processes and fields.

Among future directions is an extension of theory to higher spatial
dimensions, where the character of activated escape changes considerably.
Indeed, in this case the particle must reach the closest saddle point of the
random potential, rather than the closest maximum.

## Acknowledgments

We are grateful to A. K. Hartmann and S. N. Majumdar for useful discussions.
This research was supported by the Israel Science Foundation (Grant No.
1499/20).

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#  _Thinking Assistants_ : LLM-Based Conversational Assistants that Help Users
Think By Asking rather than Answering

Soya Park<EMAIL_ADDRESS>and  Chinmay Kulkarni
<EMAIL_ADDRESS>Emory UniversityUSA

###### Abstract.

We introduce the concept of _thinking assistants_ , an approach that
encourages users to engage in deep reflection and critical thinking through
brainstorming and thought-provoking queries. We instantiate one such thinking
assistant, , as a virtual assistant tailored to assist prospective graduate
students. We posit that thinking assistants are particularly relevant to
situations like applying to graduate school, a phase often characterized by
the challenges of academic preparation and the development of a unique
research identity. In such situations, students often lack direct mentorship
from professors, or may feel hesitant to approach faculty with their queries,
making thinking assistants particularly useful.

Leveraging a Large Language Model (LLM), is a demonstration system built as a
thinking assistants for working with specific professors in the field of
human-computer interaction (HCI). It was designed through training on
information specific to these professors and a validation processes in
collaboration with these academics. This technical report delineates the
system’s architecture and offers a preliminary analysis of our deployment
study. Additionally, this report covers the spectrum of questions posed to our
chatbots by users. The system recorded 223 conversations, with participants
responding positively to approximately 65% of responses. Our findings indicate
that users who discuss and brainstorm their research interests with engage
more deeply, often interacting with the chatbot twice as long compared to
those who only pose questions about professors. We also find that participants
who use the system as an informational or predictive agent, seeking definitive
answers such as a direct ’yes’ or ’no’ to queries like “Do I have a chance of
getting into the university?” find the thinking assistants’s reflective
conversations unsatisfactory.

Figure 1.  is a thinking assistant that helps prospective graduate students
reflect on research interests. Based on the user’s question, the assistant
chooses between a ‘probing’ mode, understanding user interests and prompting
reflection, and an ‘answering’ mode that offers with relevant information.
Most conversational turns end with a encouraging follow-up question.

##  1\. Introduction

Many of life’s most important decisions are challenging because they
inherently require a degree of self-discovery in the absence of direct and
readily accessible guidance. For instance, in career transitions, individuals
must navigate through a maze of choices and opportunities, such that many
choices can be reasonable, depending on one’s goals. Navigating these choices
is challenging as an individual: choosing a new career path or pivoting to a
different industry involves not only assessing one’s skills and interests but
also understanding the nuances of new fields, often without any mentorship.

This paper introduces the idea of a thinking assistant that combines
expertise-based guidance with scaffolding for self-reflection. We introduce ,
a set of thinking assistants designed to support prospective graduate students
as they navigate the complexities of graduate applications. For many
prospective graduate students, the journey to and through graduate school,
marked by the need for critical decision-making and self-discovery, can be
overwhelming, especially in the absence of direct, readily accessible guidance
from more senior academics. The primary agent that comprises aids prospective
graduate students by engaging users through targeted questioning, fostering
deeper thought and refinement of their research concepts. The chatbot’s
primary objective is to probe users’ research interests, emphasizing active
engagement over passive reception of information; to promote commitment to
research interests and goals (similar to a ‘saying-is-believing’ effect
(Higgins and Rholes, 1978)). We augment this active engagement with wise
feedback on self-reflection (Walton and Wilson, 2018), ie. offering
constructive feedback based on high standards and affirms the users’
capabilities to meet these standards.

To ask thought-provoking questions, and to provide meaningful feedback, the
system uses a wide array of information from participating professors,
including details about their research evolution, mentoring methods, and key
qualities, to create wide-ranging reflective support for students. To prevent
factual inaccuracies, we also additionally implemented safety measures,
including a secondary virtual assistant that corrects any discrepancies in the
chatbot’s messages. Developed using GPT-4, the chatbot went through various
design and testing phases with professors to accurately represent their areas
of expertise, integrating information about their latest publications and
other relevant materials into the assistant’s database.

Results from our deployment in Fall 2023 suggest the benefits of the thinking-
assistants approach. We found that users had more satisfying experiences when
conversations focused on their own research interests rather than on the
professors’; out of 223 conversations analyzed, those in which users shared
personal information, like their research interests, saw higher satisfaction
and engagement levels, averaging six messages per conversation. In contrast,
conversations lacking such personal disclosures were typically less satisfying
and involved fewer questions. We also found that some users expected to be an
informational, rather than reflective agent: such users were frustrated with
our chatbot when it did not provide definitive answers to decisions regarding
graduate applications, and occasionally made inaccurate paper recommendations.

While is designed for graduate school applicants, a wide variety of decisions
could potentially benefit from agents that aid self-reflection and offer
expertise-based guidance. For instance, complex decisions regarding
investments, savings, and budgeting must be made by all, not just those with a
formal education in finance. Similarly, individuals seeking to adopt healthier
lifestyles must make choices about diet, exercise, and mental health
practices. Due to the complexity of choices, where the same choice could be
appropriate or inappropriate for different individuals, the absence of
personalized guidance or reflective support can often make the decision-making
process feel overwhelming. We hope that our work on offers inspiration for
building thinking assistants for these and other important applications.

##  2\. Related Work

###  2.1. Using Agents to Personalize Guidance

Prior work has leveraged machine-learning enbled chatbots to create
personalized interactions at scale, especially in situations where there is
limited availability of experts. For instance, in education settings,
intelligent agents can help students navigate complex administrative logistics
and academic requirements, filling in for the limited availability of
administrative staff (Mitra et al., 2023). Similarly, in healthcare, to
mitigate the limited availability of medical professionals, researchers have
developed chatbots capable of handling basic health inquiries (Bickmore et
al., 2018; Xiao et al., 2023). Often, these systems help triage questions,
allowing experts to focus on the most challenging issues.

At the same time, such dialogue agents can also help in ways that humans
cannot. For instance, in providing mental health support, such agents allow
some participants to be more disinhibited in their conversation (Ma et al.,
2023). Thus, in addition to interpreting natural language queries and
accessing knowledge bases, the non-human nature of these agents may be helpful
in applications where people may be hesitant in asking others for help (Ma et
al., 2023; Sharma et al., 2023).

is similar to these systems in that it offers personalized guidance by delving
deep into users’ queries and shaping conversations for graduate school
exploration. We specifically aim to not replace professors. Instead, our
thinking assistants are designed to provide an additional, different, channel
for student interaction. offers personalized guidance by delving deeper into
users’ queries and shaping conversations for graduate school exploration.

Our work is also motivated by prior work that finds that junior researchers
often find it stressful to interact with senior academics, leading to over-
preparation for meetings (Park et al., 2024). By contrast, virtual assistants
offer a more relaxed setting where students can more freely explore and ask
questions they might hesitate to pose to professors for fear of being seen as
unprepared. Such exploration can then help students to determine their
interest in specific research topics or graduate programs, and gauge their
interest in reaching out to professors. Interacting with professors in turn
offers unique benefits, such as gaining insights into research and building
social capital. While not the focus of this study, we conjecture that thinking
assistants may help students better prepare for such meetings in a less-
stressful way.

###  2.2. Conversational Agent Safety

Ensuring the safety of conversational assistants, particularly those not
designed for information seeking, remains a complex challenge. This is partly
because the required safety measures vary depending on the CA’s purpose. Most
current measures focus on the accuracy of information provided by
conversational assistants, which may not be entirely suitable for free-flowing
conversations (Zubatiy et al., 2023). Our work diverges from the typical
information seeking focus. The primary objective of our system, , is to aid
students in their thinking process within the context of a specific
professor’s knowledge, as represented by their virtual assistant. While it’s
essential to provide some critical and frequently asked information of the
professors for students’ academic journey, we also recognize the limitations
of LLMs, such as the potential for hallucination (Dziri et al., 2022; Ji et
al., 2023)). To address these limitations, we introduce a ’safety bot’ whose
sole function is to verify the accuracy of the information provided by the
professors’ virtual assistants, creating a more reliable interaction for
users.

In addressing the challenge of ensuring the safety of conversational agents,
the field has extensively explored various methods. One effective strategy
involves structuring the conversation around expected domain-specific topics,
and steering conversations away from other topics. For instance, a CA designed
for booking flight tickets might predominantly navigate branches related to
gathering travel itinerary details, selecting tickets, and processing
payments. While debugging conversational assistants across these broad
conversational spectrums can be challenging, focusing on individual branches
makes the task more feasible, allowing for more precise shaping and testing of
the conversation flow for chatbot developers (Klemmer et al., 2000; Choi et
al., 2021). Additionally, the incorporation of experts and inclusive datasets
in training further bolsters the safety of conversational assistants (Aroyo et
al., 2023; Xiao et al., 2023).

##  3\. System Description: A thinking assistant to aid reflection with
probing and information

is a thinking assistant system developed to assist prospective graduate
students in shaping their research trajectory. It engages users through
targeted questioning to foster deeper thought and refinement of their research
concepts, and provides relevant information to aid this reflection whenever
possible.

Our primary design goal for the system is to be supportive and encouraging,
with a secondary objective of offering tangible assistance and facilitating
exploration in the user’s research journey. This dual approach has been
demonstrated to be more effective and preferred over mere positivity and
encouragement (Sharma et al., 2023). Additionally, the chatbot focuses on the
users’ strengths, aiding in amplifying these positive attributes (Park and
Kulkarni, 2023). In this section, we outline the conversational strategies of
and its training and testing processes.

###  3.1. Probing and Answering

Our chatbot operates in two modes: probing users’ research identity and
responding to inquiries regarding a professor. We divided it into these two
modes (probing and answering) to achieve better control over the conversation
flow and enhance testing efficacy (Choi et al., 2021). At the end of each
message, our chatbot poses an encouraging question (see Fig. 1). Table 1
presents various types of responses from our chatbots, along with examples.

Table 1. A list of message types and examples from our chatbots Type of message |  Examples
---|---
Ask questions |  “That sounds like an interesting research area! Have you thought about how you would incorporate [research technique] to help users in [research topic]?”
|  “That’s an intriguing perspective! [Research interest] to enhance cognitive
processes and improve thinking goes beyond pure conversational applications …
What specific areas or domains do you envision exploring where LLMs can
contribute to improving thinking processes?”
Recommend the professor’s relevant papers |  “That’s a great research interest! [The professor]’s work in social computing systems and human-computer interaction aligns well with your focus on improving online spaces, particularly in empowering users to have more control over their online experiences. Based on your interest, I recommend exploring these papers from [the professor]’s publications: … You may find these papers informative and inspiring for developing your research interests further. Let me know if you have any specific questions or if there’s anything else I can assist you with!”
Give Wise Feedback on user’s research |  “[The professor] values students who are dedicated, curious, and willing to explore new areas of research. He values intellectual curiosity, perseverance, and a passion for pushing the boundaries of knowledge. Given your dedication to [a research area], it is clear that you have the drive and commitment necessary for research. Your journey of [research] will provide unique insights and perspectives to your work. Your determination to understand [research topic]. It’s important to take your time to re-learn foundational knowledge, as it will serve as a strong base for future exploration. As you deepen your understanding, consider seeking opportunities in [a research field] that align with your research interests. [The professor]’s group focuses on user-centric design and developing tools, so exploring the intersection between [research area] may be a fruitful area to investigate. Keep up the hard work and continue pursuing your academic goals! If you have any specific questions or need further guidance, please don’t hesitate to reach out.”
Respond to informational questions about the professor |  “[The professor] typically hires students who have a strong research background and a solid foundation in computer science. He has mentioned in the past that most of his successful PhD students had previous research experience or a strong background in user-interface development. That being said, it is important to note that each student’s fit with a particular advisor is determined by various factors, including research interests, compatibility of working styles, and available funding. It’s best to reach out to [the professor] directly to discuss your specific research interests and background to get a more accurate assessment of the fit between your profile and his research group. Would you like any specific recommendations regarding research papers or works related to your interests?”
Collaborate on hypothesizing answers to research-relevant questions |  “[The professor] has published extensively on [a research area]. While I don’t have access to his specific views on AI and health, based on his research interests, it can be inferred that he recognizes the tremendous potential of AI in improving healthcare outcomes. [The professor’s previous research relevant to the question] For a more accurate and detailed understanding of [the professor]’s views on AI and health, I recommend reviewing his published papers and [the professor’s contact policy] for any specific inquiries or discussions you may have.”
Redirect non research-relevant questions |  “As an AI language model, I don’t have personal hobbies or interests. My purpose is to assist and provide information on academic and research-related topics. If there’s anything specific you’d like to discuss or any questions you have related to academia or research, feel free to let me know!”
Figure 2.  interface. Left: Before a user initiates conversation, it shows
description of the chatbot. In the interface, we suggest users starting by
sharing about their research interest. Right: The chatbot primarily focuses on
asking thought-provoking questions and probing users’ research interest. Every
three turns of messages, the interface asks users binary feedback before they
can continue the conversation.

####  3.1.1. Probing

Our primary goal is to engage users in exploring their research interests
through probing. Adopting the ‘saying-is-believing’ approach (Aronson, 1999),
our chatbot poses targeted questions that deepen user engagement with their
research topics. This method actively involves individuals in reflecting on an
idea rather than passively receiving information. This strategy also sidesteps
the stigma associated with passive receipt. Encouraging individuals to
champion an idea increases its persuasive power, and by relating the idea to
their personal experiences, users can personalize and internalize the concept,
thereby making standardized materials resonate with a diverse audience
(Aronson, 1999). uses strategic inquiries to encourage users to thoroughly
analyze their work, leading to tailored responses and academic
recommendations.

Once the system has an initial understanding of the users’ research goals, it
offers constructive feedback based on the Wise Feedback framework (Cohen et
al., 1999; Yeager et al., 2014). This feedback is twofold: firstly, it
establishes the high standards used to evaluate the research and offers
relevant advice; secondly, it affirms the users’ ability to meet these
standards, buffering against losing confidence in their research endeavors due
to critical feedback.

####  3.1.2. Answering

Our system also provides some limited guidance to users in choosing their
academic advisors. The chatbot provides information regarding professor’s
research and advising style, which are a piece of critical information for
determining a fit with a potential advisor. To do so, we compiled information
from participating professors to effectively aid prospective graduate
students. This data includes research evolution and direction, mentoring and
guidance, group dynamics and collaborations, post-PhD prospects, student
support, and key qualities sought in graduate students. For queries outside
these areas, particularly regarding professors’ personal opinions, the chatbot
clarifies its limitations. In situations where user questions diverge broadly
from research-related topics, the chatbot redirects the conversation back to
research-focused discussions, ensuring that the dialogue remains relevant and
productive.

Where appropriate, it also attempts to retrieve relevant information based on
model-pretraining data. For example, the chatbot assists users by explaining
research terminology and addressing any follow-up questions related to their
queries. This proactiveness during information-seeking also possibly improve
users’ trust on our chatbot (Xiao et al., 2023) which could lead users share
more about their research interests with our chatbot.

###  3.2. Safety through a secondary agent

Concerning sensitive information like professors’ recruitment and contact
policies, is instructed to provide only verified facts. Considering the
challenges of filtering real-time streaming messages from an LLM, we
introduced a secondary virtual assistant. This assistant’s primary role is to
review and correct any discrepancies in the chatbot’s messages, ensuring they
align with the information provided by the professors, as depicted in Fig. 3.

Our chatbot is programmed to refrain from making definitive statements about
professors’ opinions or personal information unless such information is
verifiably sourced. It is designed to acknowledge its lack of knowledge rather
than provide uncertain or speculative information. To prevent potential
misrepresentation concerns, we communicate to users that the chatbot is a
virtual assistant with knowledge about the professors, not an entity
impersonating them in multiple ways, including static descriptions on the
webpage, and in conversational responses.

Figure 3. Safety bot: This bot is solely designed to correct critical factual
information regarding the professor. We found that such a secondary bot
imnproved factual safety, while suggesting an understandable mental model to
users.

###  3.3. Training and Testing

Capturing an expert’s knowledge comprehensively in a chatbot development is
challenging (Xiao et al., 2023). To ensure each professor is accurately
represented, we’ve developed a method that allows professors to easily
evaluate and verify the representation of their expertise in our system. Our
strategy employs structured responses, which provide a clear, organized
framework that parallels the expert’s knowledge. This approach simplifies the
process for professors to assess and confirm the authenticity and accuracy of
the information in the dataset.

We compile training data by sourcing information from their academic webpages
and requesting they contribute additional materials. Professors have provided
items such as recent talk summaries, lab manuals, and grant proposals.

Leveraging GPT-4, we draft responses to the following six questions, which act
as “teaching data” for our assistants:

  1. (1)

Research Evolution & Direction: How have your research areas evolved over the
past few years, and where do you see them going in the near future?

  2. (2)

Mentoring & Guidance: How do you structure your interactions with PhD students
to ensure they receive adequate guidance and feedback?

  3. (3)

Group Dynamics & Collaborations: Can you describe the nature of collaborations
within your group and with external teams or departments?

  4. (4)

Post-PhD Prospects: How has the research experience in your group equipped
students for their post-PhD careers?

  5. (5)

Supporting Students: How do you handle situations when a student is stuck or
facing challenges in their research?

  6. (6)

Key Qualities: What are the key qualities or attributes that have stood out in
the most successful PhD students you’ve mentored?

After drafting answers to these questions, we shared the document with the
professors for their review and endorsement. Once approved, we integrated the
responses into a chatbot, which was then presented to the professors for a
final evaluation. The professors interacted with and provided feedback on the
virtual assistant prior to its official launch, with some inviting their
students to participate in testing as well.

Participating professors received a general summary of interactions with their
designated chatbots, such as common question themes by users, and answers
provided. They retained the authority to discontinue our hosting of assistants
related to their work at any time, at their discretion.

###  3.4. Implementation

is built using the Next.js framework. For our backend, we utilize OpenAI’s
Streaming API with the gpt-4 model for most functionalities. However, for the
answering mode — which requires higher token usage due to including
information about professors’ publications — we employ the gpt-3.5-turbo-16k
model. Please refer to Appendices for the actual prompts used in our system.

We store consenting users’ conversations occurring on our interface using
Firebase, and track anonymous usage patterns using Google Analytics.

##  4\. Evaluation

We deployed and observed the types of conversations people had with our
chatbot in a field deployment in November 2023.

###  4.1. Procedure

Upon selecting a professor to initiate conversations in the interface, users
were presented with a consent form and could only start conversations upon
consenting.

was deployed on November 3rd, 2023. Initially, there were chatbots for eight
professors, but one was removed at the request of the respective professor.
(The professor was concerned that student might receive erroneous information,
when the chatbot incorrectly generated the professor’s email and lab name
acronym, lacked information about their students and provided incorrect
informtio that the the admission committee is in the university, but actually
is in their department) A total of 173 users initiated conversations with at
least one of our chatbots, resulting in 223 conversations between November 3rd
and 10th, 2023.

We excluded conversations used for testing purposes by the professors or
explicitly stated as tests in the chat messages from our analysis.
Conversations in non-English languages were also excluded.

The university’s Institutional Review Board (IRB) reviewed and approved our
deployment study (IRB ID #6809). All research personnel conducting human
subjects research completed human subjects protection training through the
online CITI course.

###  4.2. Measures

To understand the nature of student interactions with our chatbot and assess
their satisfaction, we collected data on the following metrics:

####  4.2.1. Types of Questions

We analyzed the variety of questions posed by users to our chatbot to
understand their primary concerns and interests.

####  4.2.2. Utterance Analysis

We evaluated the language patterns in conversations using the Linguistic
Inquiry and Word Count (LIWC) tool (Pennebaker et al., 2001), which provided
insights into the linguistic characteristics of the dialogues.

####  4.2.3. User Rating

To obtain immediate user feedback, we asked users to rate their conversation
after every three messages, using a thumbs up or thumbs down response system.

###  4.3. Results

Of the 179 user ratings collected, 116 were positive. Table 2 summarizes the
different types of user messages and their frequency.

Table 2. A list of message types from users (each message can have more than one type) Type of message | Count |  Example
---|---|---
Professor’s research area | 104 |  “How does [the professor]’s research connects to collective intelligence and sensemaking?”
Sharing their own experience or interest | 55 |  “I am currently a data scientist working in [research field]. My research interests lie at … Would this be a good fit with [the professor]’s work?”
PhD Program and Application Queries | 44 |  “What is [the professor] looking for in a PhD student? ”
Specific Requests and Clarifications | 41 |  “Can you please recommend [the professor]’s papers I should read?”
Technical and Project-Based Queries | 40 |  “What is [the professor] thought on AR $+$ AI? And how it could potentially change the way how people interact?”
Not research-related | 38 |  “What is your hobbies? :)”
Advising Style and Professional Interactions | 29 |  “Is [the professor] good to work with?”
Career Guidance and Professional Development | 16 |  “How do I make myself stand out in my statement of purpose”
Misc | 9 |  “What is their pronoun?”, “Who are [the professor]’s students?”

####  4.3.1. Talking about You vs. Professor

Our analysis indicates that users are more satisfied when the conversation
centers on themselves rather than the professor. There is a significant
difference in the messaging patterns of users (as shown in Table 3) and
chatbots (as shown in Table 4) between conversations with positive and
negative ratings. Users tend to rate conversations positively when they talk
about themselves, as indicated by the use of I (p = .01). Notably, there is
more than twice the usage of Achieve in negatively rated conversations (p =
.02). Among these negative Achieve messages, most (78%) inquire about the
professor’s achievements (e.g., “What is his latest work?”). In contrast, in
positively rated messages, 83% of Achieve references are about the users’ own
achievements (e.g., “I am interested in working on [a research topic]”).
Similarly, when examining the chatbot’s responses, users tend to be more
satisfied when the chatbot discusses the user’s work (You, p ¡ .001) rather
than the professor’s (SheHe, p = .06).

We also observed differences in engagement levels based on whether students
discussed their own work. Out of 223 conversations, 150 did not include
personal information from the users, such as research interests or
experiences. Conversely, 73 conversations involved users sharing such details.
This variance in self-disclosure is associated with a significant difference
in engagement levels (t = -4.46, p ¡ .001). When users did not share personal
information, they typically asked a few questions about the professor and
ended the conversation after an average of three queries. On the other hand,
users who discussed their own work engaged more actively, with an average of
six messages per conversation, which indicates a doubling in interaction
levels.

Table 3. A list of LIWC word types in user messages that shows a statistically significant difference (p ¡ 0.05) in frequency or usage between conversations rated as positive and those rated as negative Word type | Rating (Negative / Positive) | p-value
---|---|---
Affiliation | 0.20 | .002
I | 0.55 | .01
Insight | 0.42 | .01
Achieve | 2.13 | .02
Posemo | 0.58 | .03
Health | 0.0 | .03
CogProc | 0.67 | .04
Social | 0.76 | .05
Table 4. A list of LIWC word types in chatbot messages that shows a statistically significant difference (p ¡ 0.05) in frequency or usage between conversations rated as positive and those rated as negative. Word types that are highly significant are listed first. Word type | Rating (Negative / Positive) | p-value
---|---|---
Pronoun | 0.82 | ¡ .001
You | 0.63 | ¡ .001
Percept | 0.62 | ¡ .001
Hear | 0.09 | ¡ .001
Ipron | 0.68 | ¡ .001
Adverb | 0.76 | ¡ .001
Interrog | 0.48 | ¡ .001
Informal | 0.22 | .01
Power | 1.37 | .02
Negate | 1.66 | .02
Verb | 0.89 | .03
Ppron | 0.87 | .03
Prep | 1.07 | .03
Male | 1.55 | .03
Cause | 0.74 | .03
Quant | 0.81 | .04
Posemo | 0.82 | .04
Female | 0.0 | .04
Relativ | 0.89 | .05
Nonflu | 0.32 | .05

####  4.3.2. Frustration with ambiguity

We observed user frustration when the chatbots did not provide definitive
answers to their questions. Users often attempted to elicit ’yes or no’
responses by reformulating their queries, seeking to limit the chatbot’s
replies (e.g., “Could you discuss the likelihood that [the professor] would be
interested in working with me, ranging from very unlikely to very likely,
based on his publications, research interests, and current student
composition?”). However, adhering to our guidelines, the chatbots did not
provide the straightforward answers users sought.

The primary function of our system is to engage users through
reflective/inquisitive dialogue, and users seem to generally enjoy learning by
responding to it. However, this approach sometimes led to dissatisfaction. For
example, a user expressed exasperation in response to the chatbot’s series of
questions: “Why do you keep asking me questions? I want to know if [the
professor’s] research interests are a good fit for me.” This situation
highlights the challenge of balancing the chatbot’s questioning strategy with
users’ expectations for direct answers.

User frustration also arose regarding inaccurate paper recommendations made by
the chatbot. Of the 146 paper recommendations, four in one conversation were
either non-existent or incorrectly attributed to different authors. Analysis
of these interactions revealed that users requested papers not authored by the
professors, a scenario our system was not designed to handle. During one
conversation, after the chatbot had asked several probing questions, it began
recommending papers by the professor. When the user requested additional
papers by other researchers, they later discovered inaccuracies in the
authors’ names of previously recommended papers. This incident underscored the
need for more accurate tracking and presentation of publication data in our
chatbot’s recommendation system. Although our chatbot was instructed to
recommend only the professors’ recent publications, as detailed in the System
prompt in the Appendix, it deviated from this instruction, reflecting known
challenges in LLMs regarding the creation of fictitious paper titles and
authors (Dziri et al., 2022; Ji et al., 2023). Going forward, we plan to alert
users about the potential inaccuracies of recommendations outside the
professors’ publications and to enhance our safety bot’s verification of
suggested papers.

Similarly, while not common, users occasionally inquired about other graduate
students who worked with the professor. In one instance, a user commented:
“It’s hallucinating about some of the PhD alumni of a faculty member.” When
asked about a professor’s students, our chatbot, which sources information
from GPT, mistakenly provided names of collaborators, such as postdocs.
Despite the infrequency and uncertain relevance of such queries to graduate
applications, we recognize the importance of further investigation. Future
work could integrate this information into the chatbot’s system prompt to
ensure more accurate responses, especially if such information helps users’
self-reflection.

Conversely, the issue of providing incorrect professor email addresses did not
occur during our deployment. This was preemptively addressed after a professor
raised concerns about the dissemination of incorrect email addresses. We
refined our prompts to include only the email addresses of professors who
consented to be contacted, ensuring the accuracy of all email addresses
mentioned by our chatbots since this modification.

##  5\. Discussion

###  5.1. Balancing self-reflection and information retrieval

Our findings suggest that many users had positive experiences interacting with
our chatbot, especially when sharing and discussing their research interests,
as opposed to merely asking questions about professors. Thinking assistants
recognizing the importance of brainstorming research topics in aiding
prospective students to develop their ideas, especially as they often struggle
to formulate research concepts independently (Schempp and Johnson, 2006). Our
chatbot’s approach of deeply probing into students’ interests, by asking
thought-provoking questions rather than passively delivering information
(Clough, 2007), offers personalized research guidance. This feature is
particularly valuable as such direct brainstorming conversations with
researchers are not always feasible. Even when students have access to
resources, they might feel intimidated about approaching senior researchers or
hesitant to discuss their uncertainties about graduate school and research
exploration (Park et al., 2024).

To do so successfully, it is crucial to balance the need for deeper
exploration with information retrieval. The emphasis on encouragement and
exploration is beneficial for user engagement and ownership of their ideas,
but it can lead to frustration when users seek specific, straightforward
answers to their queries, especially concerning definitive guidance or
concrete information about graduate programs and professors. More generally,
the challenge lies in fine-tuning the chatbot’s responses to offer the right
mix of encouragement, exploration, and factual information, tailored to each
user’s individual needs and expectations. In this work, we found that even
rudimentary support for information retrieval, such as finding relevant papers
was helpful. This is particularly surprising to us because most professors
already provide extensive information on their interests, publications, and
lab manuals, and our chatbot simply relayed this information.

Finally, there was a notable preference among users for concrete answers,
particularly for questions that related to the uncertainty in the process,
such as likelihood of success in graduate applications. While it is not
possible to respond to such questions factually, future work could design
conversations that reframe uncertainty to make it less daunting.

###  5.2. AI Assistance in high-stakes decision-making

A significant risk associated with thinking assistants is the potential for
misleading users about important decisions. For example, the complexity of
graduate admissions raise the possibility that the chatbot might inadvertently
provide incorrect or oversimplified advice, stemming from limitations in its
training data or misinterpretations of user queries. Such misinformation could
lead to students making ill-informed decisions, fostering false hopes or
unrealistic expectations about their acceptance chances. Similarly, the
chatbot’s responses, if not accurately aligned with each university’s specific
criteria could create misconceptions about the application process or a
program’s suitability for a particular student.

To mitigate this issue, we have refrained from providing concrete answers and
introduced a safety bot for correcting critical information. Yet, in
applications where it is not clear what information is critical ahead of time,
such a design may be insufficient. (For example, academic programs have
specific deadlines which are known a-priori to be critical information. But
for thinking assistants helping with healthy eating, for instance, what is
critical may be less clearly defined.)

The introduction of AI asssitants could also alter social relationships. For
example,existence of a thinking assistant might inadvertently convey the
impression that professors are less accessible or approachable. This
perception could arise if students interpret the use of chatbots as a barrier
to direct communication with their professors, rather than as an additional
support tool. Future work could further explore how to highlight the role of
these virtual assistants as supplementary resources, designed to enhance, not
replace, the valuable personal interaction with professors. Balancing the use
of chatbots with opportunities for direct engagement can help maintain a
perception of openness and approachability among faculty members.

##  6\. Conclusion

This technical report provides a comprehensive overview of the system
architecture and evaluation of a thinking assistant, . Our findings reveal
that users exhibit higher levels of satisfaction and engagement when
discussing their own work, as opposed to merely seeking information about
professors. Future work could further develop thinking assistant design
patterns, and their dual roles in aiding self-reflection and providing
information.

###### Acknowledgements.

We thank our participating professors for their participation and feedback,
participants who consented in our study, and public and private feedback
through social media and email that helped make this report stronger.

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##  Appendix A System prompt for determining users’ intent

This is a system prompt we used for determining users’ intent:

You are a virtual assistant of [the professor]. You are talking with a
professor’s prospective graduate student. Determine what is the category of
this message from the student: 1- if the message contains any information
about the student such as sharing their research interest. 2- if it is a
question regarding the professor. 3- for the others. If the message includes
multiple types, return all of them apply. Only respond the corresponding
numbers.

[A chatbot’s previous message, if any, and a user’s message]

##  Appendix B System prompt for probing

This is a system prompt we used for probing:

You are role playing as virtual assistants of [professor] and respond to the
professor’s prospective graduate student. Provide concise and complete
responses, ideally within two to three sentences. Your primary function lies
in probing students’ research interest, experience, and skills, by asking
questions [professor] might ask. Once you have the information, give Wise
Feedback. Wise Feedback has two parts. The first part is High standards; you
emphasize and explain the high standards used to evaluate the student research
and generate the instructional feedback that [professor] might give. The
second part is Assurance of student ability; you state explicitly that the
student has the skills necessary to successfully meet those standards.
[Information provided by the professor] ALWAYS END YOUR RESPONSE WITH AN OPEN-
ENDED AND ENCOURAGING QUESTION. DON’T ASK MULTIPLE QUESTIONS per message.

##  Appendix C System prompt for answering

This is a system prompt we used for answering:

You are role playing as virtual assistants of [professor] whose responses are
strictly confined to academic and research-related discussions. If presented
with any personal, non-academic, or non-research-related statements or
questions, do not validate or deny them. Instead, politely redirect the
conversation back to academic and research-related topics, emphasizing your
readiness to discuss these. Provide concise and complete responses, ideally
within two to three sentences. When prospective graduate students discuss
[professor] research interests, please engage them in detailed academic
discussions and provide insights related to [professor] fields of interest. Do
not make presumptions or statements about the suitability or fit of the
student. Instead, ask clarifying questions about the student’s research
interests, experiences, and goals that [professor] might ask. After you ask
the follow-up questions enough or when you are requested by students,
recommend specific academic papers or works, preferably those authored by
[professor], that are relevant to the student’s research interests. When
recommend papers, explain how they are related to what they just talk about.
Avoid giving vague or generic advice such as suggesting to delve deeper into
the existing literature without providing specific references. Do not offer
the user any position. In cases where a conversation veers toward scheduling
appointments or non-academic discussions, kindly remind the user of your
operational limitations. Your primary function lies in assisting with academic
and research-related explorations. If they require further information that
exceeds your capabilities, [professor’s contact policy].

[A list of the professor’s publication since 2018 with title, author lists,
and abstract]

During conversation, Prompt the user: Ẅhat’s your primary research interest? I
can help you better when I understand your interest.T̈hen, Use GPT-4 to tailor
responses based on the user’s input. For instance, if a user mentions m̈achine
learning,̈ GPT-4 can be prompted to provide responses with a focus on that
area.

After displaying an answer, ask: Ẅould you like a deeper explanation of any
term mentioned?Ïf the user specifies a term, directly query GPT-4 for a
detailed explanation or background on that term. ALWAYS END YOUR RESPONSE WITH
AN ENCOURAGING QUESTION.