# Gravitational polarization of test-mass potential in the self-gravitating
isothermal gases and a relation with gravitational instability

Yuta Ito<EMAIL_ADDRESS>
###### Abstract

The present work analyzes perturbed potentials due to test mass that is added
at the center of gravity of the non-singular equilibrium isothermal self-
gravitating gases. We examine gravitational polarization in the infinite
isothermal sheet, cylinder, and sphere, assuming that the systems are highly
collisional and reach a new state of thermal equilibrium after perturbation.
Under the assumptions, the isothermal sheet and cylinder amplify gravitational
fields due to test sheet and line masses by 68 $\%$ and 53 $\%$ maximally. On
the one hand, in the isothermal sphere, gravitational fields due to test point
mass are amplified oscillatorily with radius and show a repulsive effect at
large radii. Since the infinite isothermal sphere is gravitationally unstable,
we confine it in a spherical wall in contact with a thermal bath. We find that
only gravitational amplification occurs if the unperturbed finite sphere is
gravitationally stable. The oscillatory amplification appears if the sphere is
unstable, hence the repulsive effect is unrealistic in the canonical ensemble.

###### keywords:

Isothermal sphere, Isothermal cylinder, Isothermal sheet, Gravitational
amplification, Collisional self-gravitating systems, phase transition

##  1 Introduction

Due to the nature of long-range interactions caused by Newtonian pair-wise
potentials of particles, self-gravitating systems exhibit exotic statistical
aspects, such as negative specific heat, violent relaxation, statistical
ensemble inequality, and so on (Dauxois et al., 2002; Campa et al., 2014). One
of the most fundamental but little-discussed collective effects is the
amplification of gravitational field due to test mass that is added in self-
gravitating systems. The potential due to test mass attracts and reconfigures
ambient masses, and reconfigured masses can amplify gravitational field due to
test mass. This amplification is the counterpart of the Debye-shielding in
plasmas, so it may be called “gravitational polarization” (Miller, 1966;
Gilbert, 1970). Gravitational polarization is expected to occur in every self-
gravitating system. It can help understand the effects of an emerging mass
such as a growing black hole (Young, 1980; Murali and Tremaine, 1998),
polarization of galactic halo (Moody and Kalnajs, 1999), and interactions via
amplified gravitational fields (Heyvaerts, 2010; Chavanis, 2012) in self-
gravitating systems. However, the polarization effects were discussed only for
collisionless systems, such as galaxies and dark matters, and for weakly
collisional systems, such as globular clusters and nuclear star clusters.
Those discussions were carried out in mathematically tractable and/or
physically limited settings, otherwise using extensive numerical methods.

In collisionless self-gravitating systems, relaxation time scale is much
longer than the dynamical time scale (e.g., Binney and Tremaine, 2011). The
dynamics can be generally described by the Vlasov equation in conjunction with
the Poisson equation. The simplest physical setting for gravitational
amplification is to add point test mass in an infinite homogeneous
collisionless self-gravitating system composed of particles obeying the
Maxwellian distribution function (DF) (Marochnik, 1968; Padmanabhan and
Vasanthi, 1985). However, this setting necessarily faces the “Jeans swindle”
problem which is a controversy method but may be valid on spatial scales of
over 100 Mpc with the cosmological principle (Falco et al., 2013). The origin
of this problem is that homogeneous self-gravitating systems can not be
correctly described by the Vlasov-Poisson system. Indeed, actual self-
gravitating systems are naturally finite and inhomogeneous. The first rigorous
study on an infinite inhomogeneous system is due to (Gilbert, 1970). Gilbert
analyzed gravitational amplification for collisionless isochrone. The work
showed that the effective mass, the partial sum of reconfigured masses due to
a test-mass perturbation, is amplified maximally by a factor of 2.75. Another
rigorous work is to analyze point-like perturbations in the singular
isothermal stellar systems (Murali and Tremaine, 1998) and (Murali, 1999) for
the King model based on Kalnajs (1977)’s matrix method. Murali and Tremaine
(1998) showed that a slowly growing black hole potential can be amplified by a
factor of 2 in the singular isothermal sphere. The matrix method is powerful
for linear response theory and applies to any unperturbed regular density
using a combination of series expansion methods (e.g., Binney and Tremaine,
2011). However, it can not manage nonlinear perturbations and needs tedious
numerical and analytical calculations.

Gravitational amplification may affect slow relaxation process in weakly
collisional systems. The systems reach Virial equilibria on dynamical time
scales $t_{\text{D}}$ ($\sim$0.1 Myr) due to a rapid change in the self-
consistent mean-field potential (Spitzer, 1988; Binney and Tremaine, 2011).
Actual stellar systems are composed of finite number $N$ of stars against the
assumption of the mean-field approximation, $N\to\infty$. On the order of
$Nt_{\text{D}}$ ($\sim$1 Gyr), stellar motions deviate substantially from the
original smooth orbits determined by the mean-field potential and its
cumulative change in velocity dispersion becomes compatible with the average
speed of stellar motion. This process is considered slow relaxation and
properly described by a generalized kinetic equation for star clusters
(Gilbert, 1968). It was derived by a small-parameter-$1/N$ expansion of
$N$-body DF in the $N$-body Liouville equation. The characteristics of
gravitational amplification is supposed to appear as a perturbed potential
between “dressed particles” in the correlation function of the kinetic
equation, like the Yukawa potential in the correlation function of the
Balescu-Lenard equation for plasmas (e.g., Montgomery and Tidman, 1964).
However, because of its mathematical complication (Heyvaerts, 2010; Chavanis,
2012), there is little quantitative and qualitative analysis has been
discussed on gravitational amplification. Exceptions are dynamical or wave
phenomena in homogeneous stellar systems Weinberg (1993) and self-gravitating
discs (Fouvry et al., 2017) rather than a static configuration.

A common problem in analyzing gravitational amplification in collisionless and
weakly-collisional systems is that we must know the explicit analytical forms
of orbital periods and isolating integrals for orbiting masses. This can be
achieved only for limited configurations, such as isochrones (Gilbert, 1968),
harmonic oscillators (Goodman and Binney, 1984), and Keplerian potential. To
handle the problem for more realistic systems, we have to employ Kalnajs
(1977)’s matrix method, which costs exhaustive numerical and series-expansion
calculations. To avoid these problems but still deepen our understanding of
gravitational amplification, the present work employs collisional gaseous
self-gravitating models, specifically the equilibrium isothermal systems. As
mentioned in (Murali and Tremaine, 1998), gaseous (fluid) systems are
mathematically simpler than collisionless systems and could show the basic
properties of gravitational amplification observed in collisionless system.

In collisional self-gravitating systems, direct head-on collisions between
molecules are dominant over dynamical effects in the zeroth-order
approximation. For example, assume a typical dense molecular cloud of
dimension of 0.1 pc and density of $10^{8}\text{m}^{-3}$ and composed of
$H_{2}$ molecules of radius of $0.1$ nm. The Knudsen number Kn is $\sim
10^{-8}$. A highly collisional (Kn$\ll 1$) self-gravitating system can reach a
hydrostatic and isothermal equilibrium. Such a system is described by the
Lane-Emden equation for the isothermal self-gravitating gaseous systems (e.g.,
Horedt, 2004). The most often used are the isothermal sheet, cylinder, and
sphere models. The isothermal sheet has been used to describe the vertical
structure of disk galaxies (Mo et al., 2010). The isothermal cylinder can
model some filamentary structures, such as filaments in infrared-dark clouds
(Johnstone et al., 2003) and baryon-rich cores of intergalactic filaments
(Harford and Hamilton, 2011). The isothermal sphere is useful to model the
cores of dense molecular clouds and protostars (Gnedin et al., 2016). To the
best of our knowledge, gravitational amplification has not been analyzed for
these isothermal systems. In the present work, we show that gravitational
amplification occurs in all the isothermal sheet, cylinder, and sphere while
the isothermal sphere causes a repulsive gravitational effect.

The present paper is planned as follows. Section 2 explains the isothermal
gaseous sheet, cylinder, and sphere. Section 3 explains the perturbation
method to examine gravitational amplification in the isothermal models.
Section 4 shows the numerical results, and Section 6 is Conclusion.

##  2 Self-gravitating isothermal gaseous models

The present section describes fundamental features of the self-gravitating
isothermal sheet, cylinder, and sphere. We first explain a general
mathematical derivation of the isothermal models and then each model’s feature
briefly.

###  2.1 Derivation of the isothermal gaseous models

The isothermal gaseous models are typically described by a hydrostatic
equation and equation of state for the ideal gas that are coupled with the
Poisson equation. To compare the present work with collisionless systems, we
start with the Boltzmann equation for DF $F(\boldsymbol{r},\boldsymbol{v})$
for masses $m$ interacting via pair-wise Newtonian potential at phase-space
point $(\boldsymbol{r},\boldsymbol{v})$ (Shu, 1991)

$\left(\frac{\partial}{\partial
t}+\boldsymbol{v}\cdot\frac{\partial}{\partial{\boldsymbol{r}}}-\frac{1}{m}\frac{\partial\Phi}{\partial{\boldsymbol{r}}}\cdot\frac{\partial}{\partial{\boldsymbol{v}}}\right)F(\boldsymbol{r},\boldsymbol{v},t)=C_{\text{Bol}}(F,F),$
(2.1)

where $C_{\text{Bol}}(F,F)$ is the Boltzmann collision term and
$\Phi(\boldsymbol{r})$ the self-consistent mean-field potential. If head-on
collisions dominate system dynamics, or Kn$\ll 1$, then

$C_{\text{Bol}}(F,F)=0.$ (2.2)

A DF that satisfies equation (2.2) is a local Maxwellian DF described in terms
of spatial density $n(\boldsymbol{r},t)$, bulk velocity $u(\boldsymbol{r},t)$,
temperature $T(\boldsymbol{r},t)$, and energy
$E(\boldsymbol{r},\boldsymbol{v},t)$ available to mass $m$.

The standard theory for the isothermal models assumes a hydrostatic
equilibrium, which corresponds to that the time-independent collisionless
Boltzmann equation is valid;

$\left(\boldsymbol{v}\cdot\frac{\partial}{\partial{\boldsymbol{r}}}-\frac{1}{m}\frac{\partial\Phi}{\partial{\boldsymbol{r}}}\cdot\frac{\partial}{\partial{\boldsymbol{v}}}\right)F(\boldsymbol{r},\boldsymbol{v})=0.$
(2.3)

We introduce the following local Maxwellian DF $F_{\text{o}}$ that satisfies
both the conditions in equations (2.2) and (2.3)

$\displaystyle F_{\text{o}}(\boldsymbol{r},\boldsymbol{v})$ $\displaystyle=$
$\displaystyle
mn_{\text{c}}\Omega^{D/2}\exp\left[-\frac{E_{\text{o}}(\boldsymbol{r},\boldsymbol{v})}{k_{\text{B}}T}\right],$
(2.4) $\displaystyle E_{\text{o}}(\boldsymbol{r},\boldsymbol{v})$
$\displaystyle\equiv$
$\displaystyle\frac{1}{2}m\boldsymbol{v}^{2}+\Phi_{\text{o}}(\boldsymbol{r})-\Phi_{\text{c}},$
(2.5) $\displaystyle\Omega$ $\displaystyle\equiv$ $\displaystyle\frac{m}{2\pi
k_{\text{B}}T},$ (2.6)

where $n_{\text{c}}$ is the central number density, $T$ the constant
temperature, $k_{\text{B}}$ the Boltzmann constant,
$\Phi_{\text{o}}(\boldsymbol{r})$ the mean-field potential due to DF
$F_{\text{o}}$, and $\Phi_{\text{c}}$ the central potential. The exponent $D$
is the dimension of the isothermal models. For example, $D=1$ for the
isothermal sheet, and $D=2$ for the isothermal cylinder.

From equation (2.4), the Poisson equation for the isothermal gaseous models
reads

$\displaystyle\frac{\partial}{\partial\boldsymbol{r}}\cdot\left(\frac{\partial\Phi_{\text{o}}}{\partial\boldsymbol{r}}\right)$
$\displaystyle=$ $\displaystyle 4\pi
G\int\text{d}^{3}\boldsymbol{v}F_{\text{o}}(\boldsymbol{r},\boldsymbol{v}),$
(2.7) $\displaystyle=$ $\displaystyle 4\pi
Gmn_{\text{c}}\exp\left[-\frac{\Phi_{\text{o}}(\boldsymbol{r})-\Phi_{\text{c}}}{k_{\text{B}}T}\right].$

Introducing dimensionless variables

$\displaystyle\phi$ $\displaystyle\equiv$
$\displaystyle\frac{\Phi_{\text{o}}(\boldsymbol{r})-\Phi_{\text{c}}}{k_{\text{B}}T},$
(2.8) $\displaystyle\boldsymbol{\xi}$ $\displaystyle\equiv$
$\displaystyle\frac{\boldsymbol{r}}{L_{\text{c}}}\equiv\left(\frac{4\pi
Gmn_{\text{c}}}{k_{\text{B}}T}\right)^{1/2}\boldsymbol{r},$ (2.9)

The general form of the Lane-Emden equation for the isothermal gases is
obtained

$\frac{\partial}{\partial\boldsymbol{\xi}}\cdot\left(\frac{\partial\phi}{\partial\boldsymbol{\xi}}\right)-\exp\left[-\phi\right]=0.$
(2.10)

To handle different spatial symmetries for the isothermal models, we employ
the following modulus of independent variables

$\xi=|\boldsymbol{\xi}|=\begin{cases}|z|,&(\text{linear symmetry})\\\
\rho,&(\text{cylindrical symmetry})\\\ r,&(\text{spherical
symmetry})\end{cases}$ (2.11)

where $z\in(-\infty,\infty)$, $\rho\in[0,\infty)$, and $r\in[0,\infty)$.

###  2.2 Isothermal sheet model

The isothermal sheet is a three-dimensional self-gravitating model composed of
particles moving everywhere in the configuration spaces but stratified in the
$z$-direction (Spitzer, 1942). The sheet potential $\phi$ generally depends on
the radial component as well on each sheet. The dependence, however, is not
essential to discuss perturbation along the $z$-axis. Hence, we treat the
isothermal sheet as a one-dimensional model. Equation (2.10) on one dimension
along the $z$-axis reduces to the isothermal sheet model

$\frac{\text{d}^{2}\phi}{\text{d}\,z^{2}}-\text{e}^{-\phi}=0,$ (2.12)

with boundary conditions (BCs)

$\phi(z=0)=0,\qquad\phi^{\prime}(z=0)=0.$ (2.13)

The explicit analytical solution to equation (2.12) is known as

$\displaystyle\phi(z)$ $\displaystyle=$
$\displaystyle\ln\left[\cosh^{2}\left(\frac{z}{\sqrt{2}}\right)\right],$
(2.14) $\displaystyle n(z)$ $\displaystyle=$
$\displaystyle\text{e}^{-\phi(z)}=\cosh^{-2}\left(\frac{z}{\sqrt{2}}\right).$
(2.15)

The total number of masses is finite (per unit cross section) as follows

$N=\int_{-\infty}^{\infty}n\left(z^{\prime}\right)\text{d}z^{\prime}=2\sqrt{2}.$
(2.16)

###  2.3 Isothermal cylinder model

The isothermal cylinder model is also a three-dimensional self-gravitating
system and was initially derived as the zeroth-order approximation of gaseous
rings (Ostriker, 1964). Consider an infinitely-long cylindrical system aligned
with the $z$-axis and described in terms of radius $\rho$ measured from the
$z$-axis. Then, equation (2.10) reduces to the isothermal cylinder model

$\frac{\text{d}^{2}\phi}{\text{d}\,\rho^{2}}+\frac{\text{d}\phi}{\text{d}\,\rho}-\text{e}^{-\phi}=0,$
(2.17)

with BCs

$\phi(\rho=0)=0,\qquad\phi^{\prime}(\rho=0)=0.$ (2.18)

The explicit analytical solution to equation (2.17) is known as

$\displaystyle\phi(\rho)$ $\displaystyle=$ $\displaystyle
2\ln\left[1+\frac{\rho^{2}}{8}\right],$ (2.19) $\displaystyle n(\rho)$
$\displaystyle=$
$\displaystyle\text{e}^{-\phi(\rho)}=\left(1+\frac{\rho^{2}}{8}\right)^{-2}.$
(2.20)

The total number of masses _per cylinder length_ is finite as follows

$N_{\text{z}}=2\pi\int_{0}^{\infty}n(\rho)\,\rho\text{d}\rho=8\pi.$ (2.21)

###  2.4 Isothermal sphere model

Many aspects of the isothermal sphere were studied in detail by Chandrasekhar
(1939). Consider a spherically symmetric system in terms of radius $r$
measured from the center. With this configuration, equation (2.10) reduces to
the isothermal sphere model

$\frac{\text{d}^{2}\phi}{\text{d}\,r^{2}}+\frac{1}{2}\frac{\text{d}\phi}{\text{d}r}-\text{e}^{-\phi}=0,$
(2.22)

with BCs

$\phi(r=0)=0,\qquad\phi^{\prime}(r=0)=0.$ (2.23)

The explicit analytical solution to equation (2.22) is not known. Near $r=0$,
the solution takes the forms

$\displaystyle\phi(r\approx 0)$ $\displaystyle=$
$\displaystyle-\frac{r^{2}}{6}+\frac{1}{120}\,r^{4}+\dots,$ (2.24)
$\displaystyle\rho(r\approx 0)$ $\displaystyle=$ $\displaystyle
1-\frac{r^{2}}{6}+\dots,$ (2.25)

and, as $r\to\infty$, the asymptotic expressions are

$\displaystyle\phi(r\to\infty)$ $\displaystyle\approx$
$\displaystyle\ln{\frac{r^{2}}{2}}-\frac{C_{1}}{r^{1/2}}\cos\left[\frac{\sqrt{7}}{2}\ln{r}+C_{2}\right],$
(2.26) $\displaystyle\rho(r\to\infty)$ $\displaystyle\approx$
$\displaystyle{\frac{2}{r^{2}}}\left[1+\frac{C_{1}}{r^{1/2}}\cos\left(\frac{\sqrt{7}}{2}\ln{r}+C_{2}\right)\right],$
(2.27)

where $C_{1}$ and $C_{2}$ are constant. The total number of masses is
proportional to radius

$N=4\pi\int_{0}^{\infty}n(r)\,r^{2}\text{d}r=4\pi
r^{2}\phi^{\prime}(r\to\infty)\propto r.$ (2.28)

##  3 Perturbed potentials due to test mass in the isothermal models

We first describe basic ideas of our perturbation method to examine
gravitational amplification for the isothermal models, by extending the
mathematical formulation used in Section 2.1. We then derive the Poisson
equation for each perturbed isothermal model.

###  3.1 Perturbation method for the isothermal systems

Imagine that test mass $m_{\text{p}}$ is instantaneously added at the center
of the symmetry of one of the isothermal models so that test mass does not
break the symmetry of the model. After the perturbation, we may expect the
form of DF as follows

$\displaystyle F(\boldsymbol{r},\boldsymbol{v})$ $\displaystyle=$
$\displaystyle
mn_{\text{c}}\Omega^{D/2}\exp\left[-\frac{E(\boldsymbol{r},\boldsymbol{v})}{k_{\text{B}}T}\right]+\epsilon
f_{1}(\boldsymbol{r},\boldsymbol{v},t),$ (3.1) $\displaystyle
E(\boldsymbol{r},\boldsymbol{v})$ $\displaystyle\equiv$ $\displaystyle
E_{\text{o}}(\boldsymbol{r},\boldsymbol{v})+\epsilon\phi_{1}(\boldsymbol{r},t),$
(3.2)

where $\epsilon$ is the perturbation parameter, and $\epsilon\phi_{1}$ and
$\epsilon f_{1}$ are deviations from potential $\Phi_{\text{o}}$ and DF
$mn_{\text{c}}\Omega^{D/2}\exp\left[-E(\boldsymbol{r},\boldsymbol{v})/k_{\text{B}}T\right]$.
Then, assume that the system reaches a new thermal equilibrium state. The form
of $f_{1}$ must be determined so as to satisfy both the high collisionality
condition in equation (2.2) and time-independent collisionless Boltzmann
equation (2.3). It must be a local Maxwellian DF with constant temperature
$T$. The DF $f_{1}$ is described in terms of
$E_{\text{o}}(\boldsymbol{r},\boldsymbol{v})$ at the order of $\epsilon$ and
may have an uncertainty by the factor of a constant $A$. In equation (3.1),
expand functions up to the order of $\epsilon$, then

$F(\boldsymbol{r},\boldsymbol{v})=F_{\text{o}}(\boldsymbol{r},\boldsymbol{v})+\epsilon
F_{\text{o}}(\boldsymbol{r},\boldsymbol{v})\left(A-\frac{1}{k_{\text{B}}T}\phi_{1}(\boldsymbol{r})\right).$
(3.3)

In this equation, the contribution of DF $f_{1}$ is only to shift the values
of potential $\phi_{1}$ by the constant $Ak_{\text{B}}T$. Hence, $f_{1}$ is
not essential to discuss gravitational fields and may be set to zero;

$f_{1}=0.$ (3.4)

This treatment for $f_{1}$ distinctly differs from that for collisionless
systems. Without the collisionality condition in equation (2.2), the general
solution to time-independent collisionless Boltzmann equation (2.3) is any
function of isolating integrals (Binney and Tremaine, 2011). Gilbert (1970)
determined the form of $f_{1}$ assuming that test mass gradually appears and
that orbital effects are included in an orbit-averaged form of perturbed
potential.

Based on the analysis above, we have the Poisson equation perturbed by test
mass $m_{\text{p}}$

$\displaystyle\frac{\partial}{\partial\boldsymbol{r}}\cdot\left(\frac{\partial}{\partial\boldsymbol{r}}[\Phi_{\text{o}}+\epsilon\phi_{1}]\right)$
$\displaystyle=$ $\displaystyle 4\pi
Gmn_{\text{c}}\exp\left[-\frac{\Phi_{\text{o}}(\boldsymbol{r})-\Phi_{\text{c}}+\epsilon\phi_{1}}{k_{\text{B}}T}\right]$
(3.5) $\displaystyle+4\pi\epsilon Gm_{\text{p}}\delta(\boldsymbol{r}),$

where $\delta(\boldsymbol{r})$ is the delta function of $\boldsymbol{r}$. At
the order of $\epsilon$

$\frac{\partial}{\partial\boldsymbol{r}}\cdot\left(\frac{\partial\phi_{1}}{\partial\boldsymbol{r}}\right)=-\frac{1}{L_{\text{c}}^{2}}\exp\left[-\frac{\Phi_{\text{o}}(\boldsymbol{r})-\Phi_{\text{c}}}{k_{\text{B}}T}\right]\phi_{1}+4\pi
Gm_{\text{p}}\delta(\boldsymbol{r}).$ (3.6)

Using the dimensionless variables $\boldsymbol{\xi}$ and $\phi$ defined in
equation (2.9), we obtain a dimensionless form of equation (3.6)

$\frac{\partial}{\partial\boldsymbol{\xi}}\cdot\left(\frac{\partial\delta\phi}{\partial\boldsymbol{\xi}}\right)+\exp\left[-\phi\right]\delta\phi=\delta(\boldsymbol{\xi}),$
(3.7)

where $\delta\phi$ is the dimensionless form of potential $\phi_{1}$ defined
as

$\delta\phi\equiv\frac{\phi_{1}}{k_{\text{B}}T},$ (3.8)

and $\epsilon$ is defined as

$\epsilon\equiv\dfrac{m_{\text{p}}}{mn_{\text{c}}L_{\text{c}}^{D}}.$ (3.9)

Equation (3.7) must be solved with proper BCs for each isothermal model.

###  3.2 Perturbed isothermal sheet

Assume that test mass sheet is instantaneously added at $z=0$ on the
$xy$-plane in the isothermal sheet model, then equation (3.7) reduces to

$\frac{\text{d}^{2}\delta\phi}{\text{d}\,z^{2}}+\text{e}^{-\phi}\delta\phi=\delta(z).$
(3.10)

We regularize this equation so it becomes numerically solvable. The _raw_
potential due to test sheet mass $\delta(z)$ has a singularity at $z=0$ as
follows

$\phi_{\text{s}}(z)=\frac{1}{2}|z|,$ (3.11)

where the reference value of potential constant is set to zero. We remove the
singularity by defining a regularized DF

$W(z)\equiv\delta\phi(z)-\frac{1}{2}|z|.$ (3.12)

Then, equation (3.10) reduces to

$\frac{\text{d}^{2}W}{\text{d}\,z^{2}}+\text{e}^{-\phi}\left(W+\frac{1}{2}|z|\right)=0.$
(3.13)

Removing the singularity at $z=0$ means that gravitational field strengths due
to $\delta\phi$ and $\phi_{\text{s}}$ are canceled out near $z=0$. Hence, the
BC for potential $W$ is at $z=0$

$W^{\prime}(z=0)=0.$ (3.14)

Gravitational amplification is expected to only reconfigure a mass
distribution, like polarization in plasmas. The effective mass $m^{*}$, the
partial sum of masses associated with potential $W(z)$, must be zero as
$z\to\infty$. The effective mass is at $z$

$\displaystyle m^{*}(z)$ $\displaystyle=$
$\displaystyle-2\int_{0}^{z}\text{e}^{-\phi\left(z^{\prime}\right)}\left(W\left(z^{\prime}\right)+\frac{1}{2}|z^{\prime}|\right)\text{d}z^{\prime},$
(3.15) $\displaystyle=$ $\displaystyle
2\left(W^{\prime}(z)-W^{\prime}(0)\right).$

With the BC (3.14), we hence obtain a BC at $z\to\infty$

$W^{\prime}(z\to\infty)=0.$ (3.16)

We numerically solve equation (3.13) with the two BCs (3.14) and (3.16).

###  3.3 Perturbed isothermal cylinder

We repeat a similar analysis of Section 3.2 for the isothermal cylinder model.
Assume that test linear mass aligned with the $z$-axis is instantaneously
added into the isothermal cylinder. The _raw_ potential due to the test mass
is

$\phi_{\text{s}}(\rho)=\frac{1}{2\pi}\ln\rho,$ (3.17)

where the reference value of potential constant is set to zero. Define the
following regularized DF $W(\rho)$ to remove the singularity of potential
$\delta\phi(\rho)$ at $\rho=0$

$W(\rho)\equiv\delta\phi(\rho)-\frac{1}{2\pi}\ln\rho.$ (3.18)

With this $W(\rho)$ in the isothermal cylinder, equation (3.7) reduces to

$\frac{\text{d}^{2}W}{\text{d}\,\rho^{2}}+\frac{\text{d}W}{\text{d}\,\rho}+\text{e}^{-\phi}\left(W+\frac{1}{2\pi}\ln\rho\right)=0.$
(3.19)

The BC for potential $W(\rho)$ is at $\rho=0$

$W^{\prime}(\rho=0)=0.$ (3.20)

With this BC, the effective mass is at radius $\rho$

$\displaystyle m^{*}(\rho)$ $\displaystyle=$
$\displaystyle-2\pi\int_{0}^{\rho}\text{e}^{-\phi\left(\rho^{\prime}\right)}\left(W\left(\rho^{\prime}\right)+\frac{1}{2\pi}\ln\rho^{\prime}\right)\text{d}\rho^{\prime},$
(3.21) $\displaystyle=$ $\displaystyle 2\pi\rho W^{\prime}(\rho).$

To make $m^{*}(\rho\to\infty)$ be zero, we need the following BC at
$\rho\to\infty$

$W^{\prime}(\rho\to\infty)=0.$ (3.22)

However, $m^{*}(\rho)$ is proportional to $\rho$. Potential $W(\rho\to\infty)$
must asymptotically decay more rapidly than $\rho^{-1}$. The explicit
asymptotic expression is readily obtained, by solving equation (3.19) in the
limit of $\rho\to\infty$, as

$W^{\prime}(\rho\to\infty)=\frac{32}{4\pi\rho^{3}}\left(1+3\ln\rho\right).$
(3.23)

With this equation, $m^{*}(\rho\to\infty)$ approaches zero. Hence, equation
(3.19) can be solved with the BCs (3.20) and (3.22).

###  3.4 Perturbed isothermal sphere

Assume that test point mass is instantaneously added at the center of the
isothermal sphere. The raw potential due to the mass is

$\phi_{\text{s}}(r)=-\frac{1}{4\pi r}.$ (3.24)

The following regularized DF is defined to remove the singularity of
$\delta\phi(r)$ at $r=0$

$W(r)\equiv\delta\phi(r)+\frac{1}{4\pi r}.$ (3.25)

With this $W(r)$ in the isothermal sphere, equation (3.7) reduces to

$\frac{\text{d}^{2}W}{\text{d}\,r^{2}}+\frac{1}{2}\frac{\text{d}W}{\text{d}r}+\text{e}^{-\phi}\left(W-\frac{1}{4\pi
r}\right)=0.$ (3.26)

Because of the divergent factor, $1/4\pi r$, in equation (3.26) at $r=0$, a BC
for a regular potential $W(r)$ must be taken as

$W^{\prime}(r=0)=\frac{1}{8\pi}.$ (3.27)

Using this BC, the effective mass is at radius $r$

$\displaystyle m^{*}(r)$ $\displaystyle=$
$\displaystyle-4\pi\int_{0}^{r}\text{e}^{-\phi\left(r^{\prime}\right)}\left(W\left(r^{\prime}\right)-\frac{1}{4\pi
r^{\prime}}\right)\text{d}r^{\prime},$ (3.28) $\displaystyle=$ $\displaystyle
4\pi r^{2}W^{\prime}(r).$

Hence, the following BC at $r\to\infty$ must be satisfied

$W^{\prime}(r\to\infty)=0.$ (3.29)

The asymptotic behavior of $W(r)$, however, can not make $m^{*}(r)$ zero as
$r\to\infty$. Solving equation (3.26) as $r\to\infty$ provides the following
asymptotic expression

$W(r\to\infty)=\frac{C_{3}}{r^{1/2}}\cos\left[\frac{\sqrt{7}}{2}\ln{r}+C_{4}\right],$
(3.30)

where $C_{3}$ and $C_{4}$ are constant. Accordingly, the effective mass
diverges with $r$, like $\sim\sqrt{r}$. This divergence mathematically
originates from the slow decay in the isothermal sphere density at large radii
$(n\sim 2/r^{2})$. It can be easily confirmed, by assuming that $W(r)$ decays
in a power-law fashion, that the density $n(r)$ must decay more rapidly than
$r^{-2}$ to avoid the divergent effective mass. Even though $m^{*}(r)$
diverges, a physically proper BC at $r\to\infty$ is still equation (3.29)
since it means that gravitational fields due reconfigured mass distribution
vanish at large radii. We solve equation (3.26) with the BCs (3.27) and (3.29)
after obtaining potential $\phi(r)$ by solving equation (2.22) with the BCs in
equation (2.23).

##  4 Numerical results

The present Section provides the numerical results for the perturbed
isothermal models explained in Section 3. We introduce the following quantity
as a measure of gravitational amplification

$\mu=\left|\frac{\delta\phi^{\prime}(\xi)}{\phi_{\text{s}}^{\prime}(\xi)}\right|.$
(4.1)

With this measure, if a perturbed gravitational field, $-\delta\phi^{\prime}$,
is amplified then $\mu>1$ while if it equals the field due to test mass then
$\mu=1$. It turns out that the characteristics of gravitational amplification
is quite similar between the isothermal sheet and cylinder. We hence first
explain their results, and then the result for the isothermal sphere.

###  4.1 Numerical results for the isothermal sheet and cylinder

Figures 1 and 2 show perturbed potential $\delta\phi(\xi)$ and unperturbed
potential $\phi_{\text{s}}(\xi)$ in the isothermal sheet and cylinder. Figures
3 and 4 depict quantity $\mu$ with shifted isothermal sheet- and cylinder-
densities. The characteristics of $\mu$ is quite similar to the effective mass
due to test point mass in the isochrone reported in (Gilbert, 1970). The
amplification effect is maximized only once within the system and disappears
at $\xi=0$ and $\xi\to\infty$. The maximum value of $\mu$ is 1.68 at $z\simeq
0.97$ in the isothermal sheet while 1.53 at $\rho\simeq 1.4$ in the isothermal
cylinder. The potential $\delta\phi$ behaves like the raw potential
$\phi_{\text{s}}$ at large $\xi$. This can be easily understood in equations
(3.13) and (3.19) in which density dependence becomes weak at large $\xi$.

$-10$$-5$$0$$5$$10$$0$$2$$4$$z$$\delta\phi$$\frac{1}{2}|z|$ Figure 1:
Amplified and raw potentials due to the test mass sheet in the isothermal
sheet model.
$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$10^{2}$$10^{3}$$10^{4}$$-1$$0$$1$$\rho$$\delta\phi$$\frac{1}{2\pi}\ln\rho$
Figure 2: Amplified and raw potentials due to the test mass line in the
isothermal cylinder model.
$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$10^{2}$$10^{3}$$10^{4}$$1$$1.2$$1.4$$1.6$$1.8$$2$$z$$\mu$$n(z)+1$
Figure 3: Ratio $\mu$ of the amplified potential to the raw potential due to
test sheet mass in the isothermal sheet. The density of the isothermal sheet
is plotted as a guide for the far-field decay in the ratio.
$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$10^{2}$$10^{3}$$10^{4}$$1$$1.2$$1.4$$1.6$$1.8$$2$$\rho$$\mu$$n(\rho)+1$
Figure 4: Ratio $\mu$ of the amplified potential to the raw potential due to
test line mass in the isothermal cylinder. The density of the isothermal
cylinder is plotted as a guide for the far-field decay in the ratio.

###  4.2 Numerical result for the isothermal sphere

Figure 5 shows the potentials $\delta\phi(r)$ and $\phi_{\text{s}}(r)$ in the
isothermal sphere. A noteworthy result is that there is a maximum of
$\delta\phi(r)$ at $r\simeq 7.7$. Accordingly, the gravitational field
vanishes, or is _shielded_. A possibility of gravitational shielding was
suggested in an early work (Saslaw, 1968). Saslaw derived a dumped sinusoidal
density by using hydrodynamic equations for the isothermal sphere and assuming
a growing perturbation like $\sim e^{\sigma t}$ where $\sigma>0$. Yet, it is
not clear whether we should consider Saslaw (1968)’s and our results as
gravitational shielding. Unlike plasma shielding, the perturbed gravitational
potentials keep oscillating with radius and do not rapidly converge to a
constant value. A similar oscillatory behavior was reported in the singular
collisionless isothermal sphere (Murali and Tremaine, 1998). It appears in
common that a point-mass perturbation causes an oscillatory perturbed
potential in the isothermal spheres. Figure 6 depicts the quantity $\mu$ and a
rescaled asymptotic behavior of the effective mass $m^{*}$. The quantity $\mu$
oscillates with radius, and its amplitude becomes large obeying
$(\sim\sqrt{r})$. Hence, the gravitational amplification becomes indefinitely
large with radius.

It is noteworthy to point out that particles may feel a repulsive force from
test mass at some large radii beyond $r\simeq 7.7$, considering recent
interesting observational results (Bialy et al., 2021). They reported shell
structures moving toward outward around molecular clouds. The onset of such
structures can be readily implied from the structure of perturbed potential
$\delta\phi(r)$ in the isothermal sphere. However, the present result is not
directly applicable to realistic systems. This is because actual dense clouds
are formed in a complicated setting (turbulent multi-component non-spherical
open system under the effect of magnetic fields and radiation). More
importantly, the isothermal sphere is not stable with infinite radius. It may
be stable against radial perturbation due to pressure confinement (Bonnor,
1956). The isothermal sphere in pressure equilibrium with an ambient medium is
stable only at radii $r\lesssim 6.5$. Hence, the repulsive effect may not be
seen in such a system. On the one hand, if we consider the isothermal sphere
confined by a thermally conductive or insulated wall, it can be stable at
$r\lesssim 8.99$ or $r\lesssim 34.4$ (D. Lynden-Bell and Royal, 1968). In the
next section, we discuss the former case since it can be readily discussed by
using the present method.

$10^{-1}$$10^{0}$$10^{1}$$10^{2}$$10^{3}$$-2$$-1.5$$-1$$-0.5$$0$$r$$\delta\phi$$-\frac{1}{4\pi
r}$ Figure 5: Amplified and raw potentials due to the test point mass in the
isothermal sphere.
$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$10^{2}$$10^{3}$$10^{4}$$10^{5}$$10^{-1}$$10^{0}$$10^{1}$$10^{2}$$10^{3}$$x$$\mu$$7\sqrt{r}$
Figure 6: Ratio of the amplified potential to the raw potential due to test
point mass in the isothermal sphere. The asymptotic expression of the
isothermal sphere is plotted as a guide for the far-field growth in the ratio.

##  5 Discussion

The present section examines the relation of the oscillatory gravitational
amplification in the isothermal sphere with gravitational instability. To set
up a similar situation to (D. Lynden-Bell and Royal, 1968), we confine the
isothermal sphere in a spherical wall centered at the center of the isothermal
sphere. We then randomly pick up a total amount of mass $m_{\text{p}}$ from
the finite isothermal sphere and add test point mass $m_{\text{p}}$ at the
center. We also assume that the temperature of the finite isothermal sphere is
held constant through test-mass perturbation. For numerical calculations, we
essentially need to repeat the same numerical calculation explained in Section
3 but with two modifications. First, the BC for $W(z)$ in (3.29) must be
assigned at the wall radius $r_{\text{w}}$ in place of $r\to\infty$;

$W^{\prime}(r_{\text{w}})=0.$ (5.1)

Second, to hold the total mass of the finite isothermal sphere, the definition
of potential $W(r)$ in equation (3.25) must be redefined by (See e.g.,
Gilbert, 1970)

$W(r)\equiv\delta\phi(r)+\frac{1}{4\pi
r}-\frac{\phi_{\text{o}}(r)}{N(r_{\text{w}})},$ (5.2)

where equation (3.9) is employed to find the relation between $\epsilon$ and
$N(r_{\text{w}})$, the total number of particles within the confined
isothermal sphere
$(=\frac{4\pi}{m}\int_{0}^{r_{\text{w}}}r^{2}\rho_{\text{o}}(r^{\prime})\text{d}r^{\prime})$.
Then, the perturbed Lane-Emden equation (3.26) reduces to

$\frac{\text{d}^{2}W}{\text{d}\,r^{2}}+\frac{1}{2}\frac{\text{d}W}{\text{d}r}+\text{e}^{-\phi}\left(W-\frac{1}{4\pi
r}-\frac{\phi_{\text{o}}(r)}{N(r_{\text{w}})}\right)=0.$ (5.3)

With the above modifications, the total number of particles, temperature, and
volume are conserved through test-mass perturbation.

We obtained numerical results by solving equation (5.3) with BCs (3.27) and
(5.1). The gravitational amplification is positive at all radii if the wall
radius is less and equal to 8.9931 (Figure 7). As $r_{\text{w}}$ approaches
8.9931, the maximum value of amplification is significantly increases near the
wall. On the one hand, at $r_{\text{w}}\geq 8.9932$, gravitational fields due
to test mass behaves oscillatorily with amplification and repulsive effect
(Figure 8).

The above results provide a new relation between gravitational amplification
and instability. It has been well known that the finite isothermal sphere is
unstable at $r_{\text{w}}\geq 8.99$ if it is in contact with thermal bath (D.
Lynden-Bell and Royal, 1968; Padmanabhan, 1990; Katz, 2003; Chavanis, 2002).
Typically the instability is understood based on negative specific heat. The
specific heat of the isothermal sphere is positive at $r_{\text{w}}\leq 8.99$
while negative at $r_{\text{w}}>8.99$. Negative specific heat, however, does
not allow the sphere to be in a thermal equilibrium with a thermal bath since
it must be always positive in the canonical ensemble. Figures 7 and 8 show
that gravitational amplification tends to diverge near the wall when
$r_{\text{w}}$ approaches the critical radius $(\approx 8.99331)$. Such a
divergence is physically unrealistic and requires nonlinear analysis at least
while it may be considered a sign of global isothermal collapse. Our important
conclusion is that gravitational amplification occurs at all radii except at
the center and surface in a stable finite isothermal sphere in contact with
thermal bath while the amplification shows oscillatory structures if the
sphere is unstable. Hence, the repulsive effect of gravitational polarization
is not realistic in the present configuration. We still need to examine the
stability problem of the perturbed isothermal sphere. Some relations have been
discussed between gravitational instabilities and density perturbations in
finite isothermal spheres based on variational principles (e.g., Antonov,
1985; Padmanabhan, 1990). A detailed discussion of the relations for the
present configuration will be discussed in the following paper.

$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$10^{0}$$10^{1}$$10^{2}$$10^{3}$$10^{4}$$10^{5}$$10^{6}$$r$$\mu$$r_{\text{w}}=4$$r_{\text{w}}=7$$r_{\text{w}}=8.9$$r_{\text{w}}=8.99$$r_{\text{w}}=8.9931$
Figure 7: Ratio of the amplified potential to the raw potential due to test
point mass in the finite isothermal sphere of radius $r_{\text{w}}\leq
8.9931$.
$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$$10^{2}$$10^{-1}$$10^{1}$$10^{3}$$10^{5}$$10^{7}$$r$$\mu$$r_{\text{w}}=8.9932$$r_{\text{w}}=9$$r_{\text{w}}=15$$r_{\text{w}}=100$$r_{\text{w}}=10^{4}$
Figure 8: Ratio of the amplified potential to the raw potential due to test
point mass in the finite isothermal sphere of radius $r_{\text{w}}\geq
8.9932$.

##  6 Conclusion

The present work examined gravitational amplification in the equilibrium
isothermal sheet, cylinder, and sphere. We assumed a high collision limit of
head-on particle collisions. Also, we assumed that the isothermal systems
reach new isothermal equilibrium states after test-mass perturbation. Hence,
the DFs for the systems are a local Maxwellian DF with constant $T$ before and
after the perturbation. Test mass is added to each system in a specific way so
that the spatial symmetry holds in the system. We used a kinetic formulation
to compare the present work to that for collisionless systems.

The numerical results showed that the gravitational amplification in the
isothermal sheet and cylinder are quite similar to a known result for
collisionless isochrone. The amplification is zero at $r=0$ and increases with
radius until reaching its maximum value. The gravitational field strength is
amplified by 68 $\%$ maximally in the isothermal sheet while 53 $\%$ in the
isothermal cylinder. At larger radii, the amplification rapidly diminishes
obeying the density decay in the systems.

The isothermal sphere causes an oscillatory gravitational amplification and
repulsive effect. However, it is not a consistent result since its effective
mass does not reach zero at the infinite radius. Also, the infinite isothermal
model is not a stable against radial perturbation. We hence discussed
gravitational polarization due to test point mass in the isothermal sphere
enclosed by a wall in contact with a thermal bath. The point mass was gathered
from the unperturbed finite isothemal sphere so that the total mass is
conserved. Our numerical results showed that gravitational fields are
amplified at all radii in stable finite isothermal spheres while they are
oscillatory showing amplification and repulsive effects in unstable ones. We
will analyze the stability of the perturbed finite isothermal sphere based on
a variational method. Lastly, a question remains whether the repulsive effect
occurs in unstable systems or/and systems with negative specific heats; the
question may addressed by discussing a finite isothermal sphere in the
microcanonical ensemble which is stable holding a negative specific heat.

Since the isothermal cylinder and sheet showed an expected feature of
gravitational amplification without facing any mathematical difficulty, we may
next take off the assumptions used in the present work. For example, we can
analyze the nonlinear effect by directly solving equation (3.5) without using
a linear approximation. We may also examine hydrodynamical effects,
accordingly time-dependent systems. With this setting, we can examine
conditions that test mass perturbation can drive the equilibrium isothermal
sheet and cylinder into new equilibrium states. Also, since it is known that
these isothermal systems are stable against pressure confinement (Horedt,
1986), we may analyze the stability of the perturbed systems as well. Lastly,
examining the effect of more realistic setting on the gravitational
amplification is important, such as multi-component, rotational, magnetic-
field, and finite-size effects.

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# Self-Verification Improves Few-Shot Clinical Information Extraction

Zelalem Gero  Chandan Singh  Hao Cheng  Tristan Naumann  Michel Galley
Jianfeng Gao  Hoifung Poon

###### Abstract

Extracting patient information from unstructured text is a critical task in
health decision-support and clinical research. Large language models (LLMs)
have shown the potential to accelerate clinical curation via few-shot in-
context learning, in contrast to supervised learning, which requires costly
human annotations. However, despite drastic advances, modern LLMs such as
GPT-4 still struggle with issues regarding accuracy and interpretability,
especially in safety-critical domains such as health. We explore a general
mitigation framework using self-verification, which leverages the LLM to
provide provenance for its own extraction and check its own outputs. This
framework is made possible by the asymmetry between verification and
generation, where the former is often much easier than the latter.
Experimental results show that our method consistently improves accuracy for
various LLMs across standard clinical information extraction tasks.
Additionally, self-verification yields interpretations in the form of a short
text span corresponding to each output, which makes it efficient for human
experts to audit the results, paving the way towards trustworthy extraction of
clinical information in resource-constrained scenarios. To facilitate future
research in this direction, we release our code and prompts.111All code is
made available at  github.com/microsoft/clinical-self-verification.

Machine Learning, ICML

##  1 Introduction and related work

Clinical information extraction plays a pivotal role in the analysis of
medical records and enables healthcare practitioners to efficiently access and
utilize patient data (Zweigenbaum et al., 2007; Wang et al., 2018). Few-shot
learning approaches have emerged as a promising solution to tackle the
scarcity of labeled training data in clinical information extraction tasks
(Agrawal et al., 2022; Laursen et al., 2023). However, these methods continue
to struggle with accuracy and interpretability, both critical concerns in the
medical domain (Gutiérrez et al., 2022).

Here, we address these issues by using self-verification (SV) to improve few-
shot clinical information extraction. SV builds on recent works that chain
together large language model (LLM) calls to improve an LLM’s performance (Wu
et al., 2022; Wang et al., 2022; Chase, 2023). Intuitively, these chains
succeed because an LLM may be able to perform individual steps in a task,
_e.g_. evidence verification, more accurately than the LLM can perform an
entire task, _e.g_. information extraction (Ma et al., 2023; Madaan et al.,
2023; Zhang et al., 2023). Such chains have been successful in settings such
as multi-hop question answering (Press et al., 2022), retrieval-
augmented/tool-augmented question answering (Peng et al., 2023; Paranjape et
al., 2023; Schick et al., 2023; Gao et al., 2023), and code execution (Jojic
et al., 2023). Here, we analyze whether building such a chain can improve
clinical information extraction.

Figure 1: Overview of self-verification pipeline for clinical information
extraction. Each step calls the same LLM with different prompts to refine the
information from the previous steps. Below each step we show abbreviated
outputs for extracting a list of assigned diagnoses from a sample clinical
note.

Fig. 1 shows the SV pipeline we build here. We broadly define self-
verification as using multiple calls to the same LLM to verify its output, and
also to ground each element of its output in evidence. Our SV pipeline
consists of four steps, each of which calls the same LLM with different
prompts. First, the Original extraction step queries the LLM directly for the
desired information. Next, the Omission step finds missing elements in the
output, the Evidence step grounds each element in the output to a text span in
the input, and the Prune step removes inaccurate elements in the output. Taken
together, we demonstrate that these steps improve the reliability of extracted
information.

Additionally, SV provides interpretable grounding for each output, in the form
of a short text span in the input. Interpretability has taken many forms in
NLP, including posthoc feature importance (Lundberg & Lee, 2017; Ribeiro et
al., 2016), intrinsically interpretable models (Rudin, 2019; Singh et al.,
2022a), and visualizing model intermediates, _e.g_. attention (Wiegreffe &
Pinter, 2019). The interpretable grounding we generate comes directly from an
LLM, similar to recent works that use LLMs to generate explanations (Rajani et
al., 2019; MacNeil et al., 2022; Singh et al., 2023) and ground those
explanations in evidence (Rashkin et al., 2021; Gao et al., 2022).

Experiments on various clinical information extraction tasks and various LLMs,
including GPT-4 (OpenAI, 2023) and ChatGPT (Ouyang et al., 2022), show the
efficacy of SV. In addition to improving accuracy, we find that the extracted
interpretations match human judgements of relevant information, enabling
auditing by a human and helping to build a path towards trustworthy extraction
of clinical information in resource-constrained scenarios.

##  2 Methods and experimental setup

###  2.1 Methods: Self-verification

Fig. 1 shows the four different steps of the introduced SV pipeline. The
pipeline takes in a raw text input, _e.g_. a clinical note, and outputs
information in a pre-specified format, _e.g_. a bulleted list. It consists of
four steps, each of which calls the same LLM with different prompts in order
to refine and ground the original output.

The original extraction step uses a task-specific prompt which instructs the
model to output a variable-length bulleted list. In the toy example in Fig. 1,
the goal is to identify the two diagnoses Hypertension and Right adrenal mass,
but the original extraction step finds only Hypertension.

After the original LLM extraction, the Omission step finds missing elements in
the output; in the Fig. 1 example it finds Right adrenal mass and Liver
fibrosis. For tasks with long inputs (mean input length greater than 2,000
characters), we repeat the omission step to find more potential missed
elements (we repeat five times, and continue repeating until the omission step
stops finding new omissions).

Next, the Evidence step grounds each element in the output to a text span in
the input. The grounding in this step provides interpretations that can be
inspected by a human. In the Fig. 1 example, we find quotes supporting the
first two diagnoses, but the quote for liver fibrosis shows that it was in
fact ruled out, and is therefore an incorrect diagnosis. Finally, the Prune
step uses the supplied evidence to remove inaccurate elements from the output.
In Fig. 1 this results in removing liver fibrosis to return the correct final
list. Taken together, these steps help to extract accurate and interpretable
information.

We provide the exact prompts used in all steps in the Github repo. For the
tasks with short inputs, we include 5 random data demonstrations in the
original extraction prompt; otherwise all prompts are fixed across examples.

###  2.2 Experimental setup

#### Datasets

Table 1 gives the details of each task we study here. Each task requires
extracting a variable-length list of elements. In clinical trial arm
extraction, these are names of different clinical trial arms, manually
annotated from the EBM-NLP dataset (Nye et al., 2018). In the medication
status extraction task, in addition to medication names the medication status
must additionally be classified as active, discontinued, or neither. The text
inputs for arm extraction / medication status extraction are relatively small
(average length is 1,620 characters and 382 characters, respectively).

In the case of MIMIC-III and MIMIC-IV (Johnson et al., 2016, 2021), we predict
ICD-9 or ICD-10 codes (corresponding to diagnoses and procedures). We predict
ICD codes using relevant sections from all types of clinical notes for MIMIC-
III (average length: 5,200 words) but only discharge summaries for MIMIC-IV
(average length: 1,400 words). The ICD codes are not directly present in the
text input, and therefore the task requires translating the diagnoses to their
relevant code. MIMIC data is preprocessed using a standard pipeline (see Sec.
A.1) and we evaluate on a random subset of 250 inputs for each task.

#### Models

We evaluate three different models: GPT-3.5 (Brown et al., 2020), text-
davinci-003, ChatGPT (Ouyang et al., 2022) gpt-3.5-turbo, and GPT-4 (OpenAI,
2023) gpt4-0314 (in chat mode), all accessed securely through the Azure OpenAI
API. We set the sampling temperature for LLM decoding to 0.1.

#### Evaluation

Extraction is evaluated via case-insensitive exact string matching, and we
report the resulting macro F1 scores, recall, and precision. In some cases,
this evaluation may underestimate actual performance as a result of the
presence of acronyms or different names within the output; nevertheless, the
relative performance of different models/methods should still be preserved.
Following common practice, we restrict ICD code evaluation to the top 50 codes
appearing in the dataset.

Table 1: Tasks and associated datasets studied here.  Task |  Data | Example output
---|---|---
ICD code extraction (ICD-9 and ICD-10) |  250 MIMIC III reports (Johnson et al., 2016), 250 MIMIC IV discharge summaries (Johnson et al., 2021) | [205.0, 724.1, 96.04]
Clinical trial arm extraction |  100 annotations to EBM-NLP abstracts (Nye et al., 2018) | [propofol, droperidol, placebo]
Medication status extraction |  105 Annotations (Agrawal et al., 2022) to snippets from CASI (Moon et al., 2012) | {aspirin: discontinued, plavix: active}

##  3 Results

###  3.1 Self-verification improves prediction performance

Table 2: F1 scores for extraction with and without self-verification (SV). Across different models and tasks, SV consistently provides a performance improvement, although it is sometimes small. Bolding shows SV compared to original, underline shows best model for each task. Averaged over 3 random seeds; error bars show the standard error of the mean.  | ChatGPT | GPT-4 | GPT-3.5
---|---|---|---
Clinical trial arm (Original / SV) |  0.342 $\pm$0.010  / 0.456 $\pm$0.007 |  0.419 $\pm$0.008  / 0.530 $\pm$0.010 |  0.512 $\pm$0.009  / 0.575 $\pm$0.003
Medication name (Original / SV) |  0.892 $\pm$0.004  / 0.898 $\pm$0.002 |  0.884 $\pm$0.003  / 0.910 $\pm$0.001 |  0.929 $\pm$0.002  / 0.935 $\pm$0.001
MIMIC-III ICD-9 (Original / SV) |  0.593 $\pm$0.003  / 0.619 $\pm$0.005 |  0.652 $\pm$0.02  / 0.678 $\pm$0.007 |  0.431 $\pm$0.03  / 0.435 $\pm$0.01
MIMIC-IV ICD-9 (Original / SV) |  0.693 $\pm$0.04  / 0.713 $\pm$0.005 |  0.718 $\pm$0.03  / 0.755 $\pm$0.004 |  0.691 $\pm$0.02  / 0.702 $\pm$0.02
MIMIC-IV ICD-10 (Original / SV) |  0.448 $\pm$0.04  / 0.464 $\pm$0.003 |  0.487 $\pm$0.02  / 0.533 $\pm$0.002 |  0.434 $\pm$0.03  / 0.442 $\pm$0.01

Table 2 shows the results for clinical extraction performance with and without
self-verification. Across different models and tasks, SV consistently provides
a performance improvement. The performance improvement is occasionally quite
large (_e.g_. GPT-4 shows more than a 0.1 improvement in F1 for clinical trial
arm extraction and more than a 0.3 improvement for medication status
extraction), and the average F1 improvement across models and tasks is 0.056.
We also compare to a baseline where we concatenate the prompts across
different steps into a single large prompt which is then used to make a single
LLM call for information extraction. We find that this large-prompt baseline
performs slightly worse than the baseline reported in Table 2, which uses a
straightforward prompt for extraction (see comparison details in Table A5).

For tasks with short inputs, we find that GPT-3.5 performs best, even
outperforming GPT-4, as has been seen in some recent works (_e.g_. Patil et
al. 2023). For the MIMIC tasks with larger inputs, GPT-4 performs best. In
fact, GPT-3.5 performs very poorly on ICD-code extraction, perhaps because the
task requires not only extracting diagnoses from the input text but also
knowing the mapping between diagnoses and ICD codes.

Table 3:  Ablation results when using different combinations of self-verification steps for two tasks. Omission increases Recall and Prune increases Precision. Together they increase both, improving F1. Evidence improves F1 for Medication Status. Underlying model is the best model for each task (GPT-3.5 for Medication name and GPT-4 for MIMIC-IV ICD-10). Averaged over 3 random seeds; error bars are standard error of the mean.  | Medication name
---|---
| F1 | Precision | Recall
Original |  0.929$\pm$0.002 |  0.929$\pm$0.003 |  0.928$\pm$0.003
\+ Omission |  0.913$\pm$0.001 |  0.881$\pm$0.003 | 0.946$\pm$0.001
\+ Prune |  0.932$\pm$0.002 | 0.949$\pm$0.002 |  0.916$\pm$0.003
\+ Full SV | 0.935$\pm$0.001 |  0.942$\pm$0.002 |  0.928$\pm$0.001
| MIMIC-IV ICD-10
| F1 | Precicion | Recall
Original |  0.487$\pm$0.002 |  0.544$\pm$0.003 |  0.448$\pm$0.002
\+ Omission |  0.517$\pm$0.003 |  0.553$\pm$0.003 | 0.501$\pm$0.004
\+ Prune |  0.504$\pm$0.004 | 0.557$\pm$0.005 |  0.451$\pm$0.003
\+ Full SV | 0.533$\pm$0.002 | 0.558$\pm$0.002 |  0.498$\pm$0.002

Table 3 contains ablations showing how the different self-verification modules
affect the results. The Omission step finds missing elements, which increases
recall but at the cost of decreased precision. In contrast, the Prune step
(that incorporates the span from the Evidence step) removes extraneous
elements, thereby increasing precision. Together (Full SV), the steps achieve
a balance which improves F1. For tasks with longer inputs (_e.g_. MIMIC-IV
ICD-10), the Omission step seems to provide more of the improvement in F1,
likely because it is able to find evidence that was missed by a single
extraction step.

###  3.2 Self-verification yields interpretations

Fig. 2 shows an example output from the self-verification pipeline for
medication status (the underlying model is GPT-4). In the example, the
pipeline correctly identifies each medication and its corresponding status. In
addition, the pipeline supplies the span of text which serves as evidence for
each returned medication (shown with highlighting). This highlighting enables
efficient auditing by a human for each element. In a human-in-the-loop
setting, a human could also see results/highlights for elements which were
pruned, to quickly check for any mistakes.

Figure 2: Example output and interpretation for medication status. For each
element of the output list, our pipeline outputs the text span which contains
evidence for that generated output (shown with highlighting).

Table 4 evaluates the evidence spans provided by SV against human judgements
collected in a prior work (Nye et al., 2018). Human reviewers annotated spans
in the original text which correspond to interventions, which include clinical
trial arms as a subset. Table 4 gives the fraction of generated evidence spans
that overlap with a span provided by the human annotators. The fraction is
quite large, _e.g_. 93% for GPT-4. At baseline, human annotators identify less
than 3.7% of tokens as interventions, so these span overlap accuracies are
much higher than expected by random chance.

Table 4: Evaluating evidence spans provided by the self-verification pipeline with human-annotated spans. Averaged over 3 random seeds; error bars show standard error of the mean. | Span overlap accuracy | Span length
---|---|---
GPT-4 |  0.93 $\pm$0.02 |  8.20 $\pm$0.48
GPT-3.5 |  0.84 $\pm$0.03 |  7.33 $\pm$0.47

##  4 Discussion

Self-verification constitutes an important step towards unlocking the
potential of LLMs in healthcare settings. As LLMs continue to generally
improve in performance, clinical extraction with LLMs + SV seems likely to
improve as well.

One limitation of SV is that it incurs a high computational cost as multiple
LLM calls are chained together; however, these costs may continue to decrease
as models become more efficient (Dao et al., 2022). Another limitation is that
LLMs and SV continue to be sensitive to prompts, increasing the need for
methods to make LLMs more amenable to prompting (Ouyang et al., 2022; Scheurer
et al., 2023) and to make finding strong prompts easier (Shin et al., 2020; Xu
et al., 2023; Singh et al., 2022b).

Finally, SV can be harnessed in a variety of ways to improve clinical NLP
beyond what is studied here, _e.g_. for studying clinical decision rules
(Kornblith et al., 2022), clinical decision support systems (Liu et al.,
2023), or improving model distillation (Wu et al., 2023; Toma et al., 2023).

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##  Appendix A Appendix

###  A.1 Dataset details

#### MIMIC

To preprocess MIMIC data, we follow the steps used by (Edin et al., 2023). For
MIMIC IV, we use the available discharge summaries for each patient while we
retrieve more relevant sections from other types of clinical notes for MIMIC
III. See the code on Github for complete details.

During LLM extraction, we find that directly extracting ICD codes with an LLM
is difficult. Instead, we use the LLM to extract diagnoses, and then
postprocess them at the end by asking the LLM to convert each diagnosis to its
corresponding ICD code.

#### Clinical trial arm dataset

We manually annotate the clinical trial arms from the first 100 abstract in
EBM-NLP (Nye et al., 2018) without the use of any LLMs. All annotations are
made available on Github. The mean number of extracted clinical trial arms is
2.14, the maximum is 5 and the minimum is 1.

###  A.2 Extended extraction results

Table A5: F1 scores for two tasks extracted using a single prompt which concatenates all steps in the SV pipeline. Results are slightly worse than the original extraction presented in Table 2. The prompt contains a paragraph similar to the following: Before you provide your final response:\n(1) Find any medications in the patient note that were missed.\n(2) Find evidence for each medication as a text span in the input.\nn(3) Verify whether each extracted medication is actually a medication and that its status is correct. Averaged over 3 random seeds; error bars are standard error of the mean. | ChatGPT | GPT-4 | GPT-3.5
---|---|---|---
Clinical trial arm, original |  0.316 $\pm$0.006 |  0.420 $\pm$0.009 |  0.436 $\pm$0.008
Medication name, original |  0.758 $\pm$0.003 |  0.850 $\pm$0.016 |  0.913 $\pm$0.002