# General relativistic stochastic thermodynamics

Tao Wang, Yifan Cai, Long Cui, and Liu Zhao
School of Physics, Nankai University, Tianjin 300071, China email:
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###### Abstract

Based on the recent work [1, 2], we formulate the first law and the second law
of stochastic thermodynamics in the framework of general relativity. These
laws are established for a charged Brownian particle moving in a heat
reservoir and subjecting to an external electromagnetic field in generic
stationary spacetime background, and in order to maintain general covariance,
they are presented respectively in terms of the divergences of the energy
current and the entropy density current. The stability of the equilibrium
state is also analyzed.

Keywords: Stochastic thermodynamics; general covariance; first law; second
law; stability

##  1 Introduction

Since the publication of Sekimoto’s article[3] in the 1990s, stochastic
energetics has increasingly emerged as a prominent field of interest.
Especially when biological experiments served as evidence to validate the
rationality of stochastic energetics, this field has become one of the most
active research direction in non-equilibrium statistical physics. This
theoretical framework is based on the Langevin equation and the corresponding
Fokker-Planck equation (FPE), and the elegance of this theory lies in its
seamless establishment of the relationship between these equations and
thermodynamics [4]. Although this field has been developing for nearly thirty
years, there are still many interesting issues which are worth of further
exploration.

Apart from the current hot topics such as discussing specific bounds based on
this theory [5, 6], a more fundamental question is whether we can discuss this
theory within the framework of relativity. It is well known that all physics
laws should inherently abide by the principles of relativity. However, the
original Langevin equation was built on top of Newtonian mechanics. This
situation calls for a relativistic covariant treatment for the stochastic
motion of the Brownian particle. Recently, we established a reasonable
framework for relativistic stochastic mechanics in [1, 2], in which both the
Langevin and Fokker-Planck equations (FPEs) are fully general relativistic
covariant and are independent on the concrete choice of spacetime background
except for a requirement of stationarity. As a natural continuation of these
preceding works, it seems now the right time to revisit the existing
stochastic thermodynamics within the framework of general relativity.

The related problem in the context of special relativity has been discussed by
Pal and Deffner [7], our work extends the relevant discussions to curved
spacetime. Some unnatural assumptions are discarded in our analysis. The
clarification for the connection and difference between the coordinate system
and the observer in [1] provides us a solid working ground to maintain
covariance throughout the formalism. Thanks to such generality, our discussion
is irrelevant to any specific choice spacetime background.

This paper is organized as follows. We review the covariant formalism of
relativistic stochastic mechanics in Section 2, and the corresponding content
has been extended to the case of charged Brownian particle under the influence
of external Maxwell field. In Section 3 and Section 4, we discuss the first
and second laws of stochastic thermodynamics. To maintain fully relativistic
covariance, these laws are presented in the form of divergences of the energy
and entropy density currents, and the powers of gravity and of the
electromagnetic field as well as the heat transfer rate play essential roles
in the first law. Section 5 is devoted to the analysis on the stability of the
equilibrium state, which helps to elucidate the authentic underpinnings of the
long time limitation. In Section 6, we present a brief summary and outline
some open questions that still need to be answered along the line of the
current research.

The convention of notations follows our previous work [1, 2], and it is
important to distinguish random and deterministic variables, as well as
objects in different spaces. The variables with tilde are random variables,
and the corresponding un-tilded symbols represent their concrete realizations.
The objects in different spaces (including the $(d+1)$-dimensional spacetime
$\mathcal{M}$, future mass shell bundle $\Gamma_{m}^{+}$ and the future mass
shell $(\Gamma_{m}^{+})_{x}$ at the event $x$) are distinguished by the font
of the objects and their indices. We will provide the specific meaning of new
symbols when they appear.

##  2 Elements of relativistic stochastic mechanics

Starting with the geodesic equation on the spacetime manifold ${\mathcal{M}}$,
we built the covariant Langevin equation LEτ with evolution parameter $\tau$
(i.e. the proper time of the Brownian particle) [1] by supplementing the
former with contributions from the damping, the Gaussian stochastic and the
additional stochastic forces. The resulting system describes the motion of a
Brownian particle in the future mass shell bundle $\Gamma_{m}^{+}$,

$\displaystyle\left\\{\begin{aligned} {\rm
d}\tilde{x}_{\tau}^{\mu}&=\frac{\tilde{p}_{\tau}^{\mu}}{m}{\rm d}\tau,\\\ {\rm
d}\tilde{p}_{\tau}^{\mu}&=\left[\mathcal{R}^{\mu}{}_{\mathfrak{a}}\circ_{S}{\rm
d}\tilde{w}_{\tau}^{\mathfrak{a}}+\mathcal{F}^{\mu}_{\text{add}}{\rm
d}\tau\right]+\mathcal{K}^{\mu\nu}U_{\nu}{\rm
d}\tau-\frac{1}{m}\varGamma^{\mu}{}_{\alpha\beta}\tilde{p}_{\tau}^{\alpha}\tilde{p}_{\tau}^{\beta}{\rm
d}\tau.\end{aligned}\right.$ (1)

Here, $(\tilde{x}^{\mu}_{\tau},\tilde{p}^{\mu}_{\tau})$ represents the path of
the Brownian particle in the tangent bundle $T\mathcal{M}$, or more precisely
in the future mass shell bundle
$\Gamma_{m}^{+}=\bigcup_{x\in{\mathcal{M}}}(\Gamma_{m}^{+})_{x}$, with
$(\Gamma_{m}^{+})_{x}$ denoting the momentum space at the spacetime event $x$
(i.e. the mass shell). Thanks to the mass shell condition, the components of
$\tilde{p}^{\mu}_{\tau}$ are not all independent.
$\mathcal{R}^{\mu}{}_{\mathfrak{a}}$, $\mathcal{F}^{\mu}_{\text{add}}$,
$\mathcal{K}^{\mu\nu}$ are respectively the amplitudes of the Gaussian
stochastic force, the additional stochastic force and the friction tensor, and
they are all tensors on $(\Gamma_{m}^{+})_{x}$. ${\rm d}\tilde{w}_{\tau}$ is
the Gaussian increment taking the proper time of Brownian particle $\tau$ as
the evolution parameter. The assumption that the stationary solution of the
corresponding FPE is identical to the one particle distribution function
(1PDF) of the reservoir particles in the long time limit helps in determining
the form of the additional stochastic force [2]

$\displaystyle\mathcal{F}^{\mu}_{\text{add}}=\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}\mathcal{R}^{\mu}{}_{\mathfrak{a}}\nabla^{(h)}_{j}\mathcal{R}^{j}{}_{\mathfrak{b}},$
(2)

where $\nabla^{(h)}_{j}$ is the covariant derivative on
$(\Gamma_{m}^{+})_{x}$. $U_{\nu}$ is the proper velocity of the heat reservoir
and also of the reservoir comoving observer Bob.
$\varGamma^{\mu}{}_{\alpha\beta}$ appearing in the last term is the
Christoffel connection on $\mathcal{M}$. Notice that, in eq.(2),
$\mathcal{R}^{\mu}{}_{\mathfrak{a}}$ and $\mathcal{R}^{i}{}_{\mathfrak{a}}$
are the same thing but with different coordinate bases
$\displaystyle\left\\{\frac{\partial}{\partial p^{\mu}}\right\\}$ and
$\displaystyle\left\\{\frac{\partial}{\partial\breve{p}^{i}}:=\frac{\partial}{\partial{p}^{i}}-\dfrac{p_{i}}{p_{0}}\frac{\partial}{\partial{p}^{0}}\right\\}$,
and all tensors on the mass shell $(\Gamma_{m}^{+})_{x}$ have such different
representations.

In general, the proper time $\tau$ of the Brownian particle should become a
random variable with respect to an arbitrarily chosen, non-comoving observer
Alice. Therefore, a better description for the stochastic motion of the
Brownian particle should be parametrized by the proper time $t$ of Alice,
rather than that of the Brownian particle. This is achieved in [1] through a
reparametrization scheme and the corresponding Langevin equation is denoted
LEt. The concrete form of LEt will not be used in this work, so we omit it.

In order to establish a statistical description for the Brownian particle, the
first thing to be clarified is the space of states. It is crucial to notice
that the configuration space must be taken as an “equal time slice” of the
spacetime manifold, and as such it is inherently observer dependent. To keep
things as general as possible, we adopt Alice as the observer whose proper
velocity is denoted as $Z^{\mu}$. The proper time $t$ of Alice is made use of
in making the time slicing,

$\displaystyle\mathcal{S}_{t}=\\{x\in{\mathcal{M}}|t(x)=t={\rm const}.\\},$
(3)

where $t(x)$ is an extension of the proper time $t$ over $\mathcal{M}$ as a
scalar field. The space of states is a hypersurface $\Sigma_{t}$ in the mass
shell bundle [2],

$\displaystyle\Sigma_{t}:=\bigcup_{x\in\mathcal{S}_{t}}(\Gamma^{+}_{m})_{x}=\\{(x,p)\in\Gamma^{+}_{m}|x\in\mathcal{S}_{t}\\},$
(4)

which is accompanied by the timelike hypersurface orthogonal vector field
$\mathscr{Z}=Z^{\mu}e_{\mu}$, where $e_{\mu}$ is a part of the orthogonal
basis of the mass shell bundle $\\{e_{\mu},\partial/\partial p^{\mu}\\}$. As
mentioned earlier, LEτ is not the proper version of the equation that
describes the stochastic motion in $\Sigma_{t}$ and we should replace it by
LEt. From the point of view of stochastic thermodynamics, the more important
object is the corresponding FPE. After conducting both numerical and
analytical validations [1, 2], it has been proved that the physical
distribution $\varphi$ of the Brownian particle as observed by Alice is
encoded in the solution of the reduced FPE,

$\displaystyle\frac{1}{m}\mathscr{L}(\varphi)=\nabla^{(h)}_{i}\mathcal{I}^{i}[\varphi],$
(5)

where

$\mathscr{L}=p^{\mu}\dfrac{\partial}{\partial
x^{\mu}}-\varGamma^{\mu}{}_{\alpha\beta}p^{\alpha}p^{\beta}\dfrac{\partial}{\partial
p^{\mu}}$

is the Liouville vector field which is also the Hamiltonian vector field
associated with the Hamiltonian $\displaystyle
H=\frac{1}{2m}g_{\mu\nu}p^{\mu}p^{\nu}$ of a free relativistic particle,

$\displaystyle\mathcal{I}^{i}[\varphi]=\frac{1}{2}\mathcal{D}^{ij}\nabla^{(h)}_{j}\varphi-\mathcal{K}^{i\nu}U_{\nu}\varphi,$
(6)

and
$\mathcal{D}^{ij}=\delta^{\mathfrak{ab}}\mathcal{R}^{i}{}_{\mathfrak{a}}\mathcal{R}^{j}{}_{\mathfrak{b}}$
is the diffusion tensor in the momentum space. The vector field
$\mathcal{I}^{i}[\varphi]$ is tangent to the mass shell, thus it can carry
either a Latin index or a Greek one [2], which correspond to components under
different bases. Moreover, this vector field is closely connected to the heat
transfer rate from the reservoir to the Brownian particle,

$\displaystyle
Q[\varphi]:=\int_{(\Gamma^{+}_{m})_{x}}\eta_{(\Gamma^{+}_{m})_{x}}Z_{\nu}\mathcal{I}^{\nu}[\varphi].$
(7)

When $\mathcal{I}^{i}[\varphi]$ vanishes, the heat transfer rate also
vanishes.

Some characteristic quantities of the system including particle number current
and energy-momentum-stress tensor can be constructed using this distribution,

$\displaystyle
N^{\mu}[\varphi]=\int\eta_{(\Gamma_{m}^{+})_{x}}\dfrac{p^{\mu}}{m}\varphi
N:=Nn^{\mu}[\varphi],\qquad
T^{\mu\nu}[\varphi]=\int\eta_{(\Gamma_{m}^{+})_{x}}\dfrac{p^{\mu}p^{\nu}}{m}\varphi,$
(8)

where $N$ is a constant denoting the total number of the Brownian particle and
$n^{\mu}$ is the one-particle current. Here and below we use the notation
$\eta_{(\Gamma_{m}^{+})_{x}}$ to denote the invariant volume element on
$(\Gamma_{m}^{+})_{x}$. Similar notations such as $\eta_{\mathcal{M}}$,
$\eta_{{\mathcal{S}}_{t}}$, $\eta_{\Sigma_{t}}$ etc are all invariant volume
elements on the relevant manifolds given in the suffices. The divergences of
the above tensors are given as follows [2],

$\displaystyle\nabla_{\mu}N^{\mu}[\varphi]=N\nabla_{\mu}n^{\mu}[\varphi]=0,\qquad\nabla_{\mu}T^{\mu\nu}[\varphi]=-\int\eta_{(\Gamma_{m}^{+})_{x}}\mathcal{I}^{\nu}[\varphi].$
(9)

In the rest of this section, we will extend the basic equations of stochastic
mechanics in curved spacetime to the case of charged Brownian particle
subjecting to external electromagnetic field $A_{\mu}$. Let
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ be the corresponding
field strength, then the extension of LEτ should read

$\displaystyle\left\\{\begin{aligned} {\rm
d}\tilde{x}_{\tau}^{\mu}&=\frac{\tilde{p}_{\tau}^{\mu}}{m}{\rm d}\tau,\\\ {\rm
d}\tilde{p}_{\tau}^{\mu}&=\left[\mathcal{R}^{\mu}{}_{\mathfrak{a}}\circ_{S}{\rm
d}\tilde{w}_{\tau}^{\mathfrak{a}}+\mathcal{F}^{\mu}_{\text{add}}{\rm
d}\tau\right]+\mathcal{K}^{\mu\nu}U_{\nu}{\rm
d}\tau-\frac{1}{m}\varGamma^{\mu}{}_{\alpha\beta}\tilde{p}_{\tau}^{\alpha}\tilde{p}_{\tau}^{\beta}{\rm
d}\tau+\frac{q}{m}F^{\mu}{}_{\nu}\tilde{p}^{\nu}_{\tau}{\rm
d}\tau.\end{aligned}\right.$ (10)

The modification to the reduced FPE is fully encoded in the modified Liouville
vector field [9]

$\displaystyle\mathscr{L}_{F}=p^{\mu}\dfrac{\partial}{\partial
x^{\mu}}+\left(qF^{\mu}{}_{\nu}p^{\nu}-\varGamma^{\mu}{}_{\alpha\beta}p^{\alpha}p^{\beta}\right)\dfrac{\partial}{\partial
p^{\mu}}.$ (11)

Apart from the different form of the Liouville vector field, the reduced FPE
takes the same form as eq.(5), which can be verified by use of the diffusion
operator approach (see the appendix in [2]),

$\displaystyle\dfrac{1}{m}\mathscr{L}_{F}(\varphi)=\nabla_{i}^{(h)}\mathcal{I}^{i}[\varphi].$
(12)

In particular, the vector field $\mathcal{I}^{i}[\varphi]$ maintains its
original form as given in eq.(6). Using the probability current of the
Brownian particle defined as

$\displaystyle\mathscr{J}[\varphi]:=\frac{\varphi}{m}\mathscr{L}_{F}-\mathcal{I}[\varphi],$
(13)

the reduced FPE (12) can also be written as a current conservation equation

$\displaystyle\hat{\nabla}^{(\hat{h})}_{A}\mathscr{J}^{A}[\varphi]=0,$ (14)

where $\hat{\nabla}^{(\hat{h})}$ is covariant derivative operator on the mass
shell bundle.

We assume that the heat reservoir is in its intrinsic equilibrium state with
the 1PDF [10, 9, 11, 12]

$\varphi_{R}=\mathrm{e}^{-\alpha_{R}+U_{\mu}p^{\mu}/T_{B}},$

where $T_{B}$ is the temperature as measured by the comoving observer Bob. In
the long time limit, the 1PDF for the Brownian particle should approach the
same form as the equilibrium distribution of the reservoir, i.e.

$\displaystyle\varphi_{\text{eq}}=\operatorname{e}^{-\alpha+\beta_{\mu}p^{\mu}},\qquad\beta_{\mu}=U_{\mu}/T_{B},$
(15)

and it should stop receiving heat from the reservoir, i.e.
$\mathcal{I}^{i}[\varphi]=0$. On such occasions, the reduced FPE reduces into
the Liouville equation $\mathscr{L}_{F}\varphi=0$, which leads to the
following constraints over $\alpha$ and $\beta_{\mu}$ (the equilibrium
conditions)

$\displaystyle\nabla_{\mu}\alpha+q\beta^{\nu}F_{\mu\nu}=0,\qquad\nabla_{(\mu}\beta_{\nu)}=0.$
(16)

The condition over $\beta_{\mu}$ remains the same as in the case of neutral
Brownian particle, which implies that $\beta_{\mu}$ should be future directed
and timelike Killing. The condition over $\alpha$ is dependent on the
electromagnetic field strength, thus making a difference from the neutral
case.

It is easy to check that the condition $\mathcal{I}^{i}[\varphi]=0$ calls for
an un-altered Einstein relation as in the case of neutral Brownian particle,

$\displaystyle\mathcal{D}^{ij}=2T_{B}\mathcal{K}^{ij},$ (17)

and the particle number current remains divergence-free. However, the
divergence of the energy-momentum tensor needs to be modified which reflects
the consequence of the action of the electromagnetic field,

$\displaystyle\nabla_{\mu}T^{\mu\nu}[\varphi]=qF^{\nu}_{\
\mu}n^{\mu}[\varphi]-\int\eta_{(\Gamma_{m}^{+})_{x}}\mathcal{I}^{\nu}[\varphi].$
(18)

##  3 First law of relativistic stochastic thermodynamics

The first law of relativistic Brownian particle in the absence of an external
field has been established in our previous work [2], which is presented in the
form of the divergence of the energy current rather than in the change in the
energy itself as in the ordinary thermodynamics,

$\displaystyle\nabla_{\mu}E^{\mu}[\varphi]=P_{\text{grav}}[\varphi]+Q[\varphi],$
(19)

where $Q[\varphi]$ is the heat transfer rate as given in eq.(7), and

$\displaystyle P_{\text{grav}}[\varphi]$
$\displaystyle=-\int\eta_{(\Gamma_{m}^{+})_{x}}\dfrac{\varphi}{m}p^{\mu}p^{\nu}\nabla_{\mu}Z_{\nu}=-T^{\mu\nu}[\varphi]\nabla_{\mu}Z_{\nu}$
(20)

is the gravitational power as measured by Alice [14]. In this section, we will
build the complete first law of relativistic Brownian particle in the presence
of electromagnetic field.

It should be mentioned that, in the presence of electromagnetic field, the
definition of energy of the charged Brownian particle is non-unique. Such non-
uniqueness arises from different choices of the proper momentum of the
particle, i.e. the kinematic momentum $p^{\mu}$ (which is proportional to the
proper velocity of the particle) and the physical momentum
$P^{\mu}=p^{\mu}+qA^{\mu}$. Each choice gives rise to a separate definition of
the energy, i.e.

$\displaystyle E_{p}=-Z_{\mu}p^{\mu}$ (21)

and [15]

$\displaystyle E_{P}=-Z_{\mu}P^{\mu}.$ (22)

$E_{p}$ corresponds to the kinematic energy and $E_{P}$ corresponds to the
Hamiltonian. It will be clear that the different choices of energy leads to
different forms of the first law of relativistic stochastic thermodynamics,
both forms are physically correct.

Following the discussions made in Appendix A, we now introduce the energy
currents associated with $E_{p}$ and $E_{P}$ as follows,

$\displaystyle
E_{p}^{\mu}[\varphi]=\int\eta_{(\Gamma_{m}^{+})_{x}}\dfrac{p^{\mu}}{m}\varphi
E_{p},\qquad
E_{P}^{\mu}[\varphi]=\int\eta_{(\Gamma_{m}^{+})_{x}}\dfrac{p^{\mu}}{m}\varphi
E_{P}.$ (23)

These currents are related to the energy-momentum tensor via the following
equations,

$\displaystyle E_{p}^{\mu}[\varphi]=-Z_{\nu}T^{\mu\nu}[\varphi],\qquad
E_{P}^{\mu}[\varphi]=-Z_{\nu}T^{\mu\nu}[\varphi]-qA^{\mu}n^{\nu}[\varphi],$
(24)

where $T^{\mu\nu}[\varphi]$ and $n^{\mu}[\varphi]$ are given in eq.(8). Using
the trick discussed in Appendix A, the divergences of the above currents are
calculated to be

$\displaystyle\nabla_{\mu}E_{p}^{\mu}[\varphi]$
$\displaystyle=-\int\eta_{(\Gamma_{m}^{+})_{x}}\left(\dfrac{\varphi}{m}p^{\mu}p^{\nu}\nabla_{\mu}Z_{\nu}+\dfrac{q\,\varphi}{m}F_{\mu\nu}p^{\nu}Z^{\mu}-\mathcal{I}^{\mu}[\varphi]Z_{\mu}\right),$
(25)

and

$\displaystyle\nabla_{\mu}E_{P}^{\mu}[\varphi]$
$\displaystyle=-\int\eta_{(\Gamma_{m}^{+})_{x}}\left(\dfrac{\varphi}{m}p^{\mu}p^{\nu}\nabla_{\mu}Z_{\nu}+\dfrac{q\,\varphi}{m}p^{\mu}\pounds_{Z}A_{\mu}-\mathcal{I}^{\mu}[\varphi]Z_{\mu}\right),$
(26)

in which $\pounds_{Z}A_{\mu}$ is the Lie derivative of $A_{\mu}$ along the
direction of $Z^{\mu}$. The first and the third terms on the right hand side
of eqs.(25) and (26) can be easily recognized to be the gravitational power
and the heat transfer rate respectively as presented in eqs.(20) and (6). The
middle terms on the right hand side of eqs.(25) and (26) can be respectively
denoted as

$\displaystyle P_{\text{em}}[\varphi]$
$\displaystyle=-\int\eta_{(\Gamma_{m}^{+})_{x}}\dfrac{q\,\varphi}{m}F_{\mu\nu}p^{\nu}Z^{\mu}=-qF_{\mu\nu}n^{\mu}[\varphi]Z^{\nu},$
$\displaystyle P_{\text{nc}}[\varphi]$
$\displaystyle=-\int\eta_{(\Gamma_{m}^{+})_{x}}\dfrac{q\,\varphi}{m}p^{\mu}\pounds_{Z}A_{\mu}=-qn^{\mu}[\varphi]\pounds_{Z}A_{\mu}.$
(27)

$P_{\text{em}}[\varphi]$ is obviously the power of the electromagnetic field,
while $P_{\text{nc}}[\varphi]$ should be understood as the power of the non-
conservative part of the electromagnetic field observed by Alice. It might be
helpful to point out that $P_{\text{nc}}[\varphi]\neq 0$ only when
$\pounds_{Z}A_{\mu}\neq 0$, i.e. when the proper velocity of Alice is not a
symmetry generator of the electromagnetic field. The difference between
eq.(25) and eq.(26) lies in the fact that $E_{P}$ absorbs the potential energy
of the particle into the total energy.

Finally, both versions of the first law can be expressed in a concise form,

$\displaystyle\nabla_{\mu}E_{p}^{\mu}=P_{\text{grav}}[\varphi]+P_{\text{em}}[\varphi]+Q[\varphi],$
(28)
$\displaystyle\nabla_{\mu}E_{P}^{\mu}=P_{\text{grav}}[\varphi]+P_{\text{nc}}[\varphi]+Q[\varphi].$
(29)

##  4 Second law of relativistic stochastic thermodynamics

The research on macroscopic irreversibility has taken central stage in the
history of thermodynamics and statistical physics. After about 150 years of
development, this field has grown considerably. For example, it was found that
the second law of thermodynamics can also be established in stochastic
thermodynamics [3], and we will give a relativistic version of the law in this
section. It is worth mentioning that the fluctuation theorem gives a fairly
good explanation for the origin of macroscopic irreversibility [16, 17, 18,
19], but the relevant analysis is relatively complex, so we leave it to a
separate upcoming work.

Let us now precede by separating the probability current (13) of the Brownian
particle into the time-reversible and dissipative parts as did in [20, 21,
22],

$\displaystyle\mathscr{J}_{\text{r}}[\varphi]=\dfrac{\varphi}{m}\mathscr{L}_{F},\qquad\mathscr{J}_{\text{d}}[\varphi]=-\mathcal{I}[\varphi].$
(30)

The reversibility of $\mathscr{J}_{\text{r}}[\varphi]$ can be easily
understood because $\mathscr{L}_{F}$ is the Hamiltonian vector field which is
clearly time-reversible. The dissipative part is caused by the diffusion and
friction forces which breaks the time-reversal symmetry. It will be clear
shortly that the entropy production is closely related to the breaking of this
symmetry.

The definition of entropy in non-equilibrium states is not as clear as the
entropy in the equilibrium [23], especially in the relativistic context. We
could of course imitate the form of Gibbs entropy but since we have at least
three different FPEs for different distribution functions [2], and each
distributions have reasonable meanings in different spaces, it seems that the
definition for the entropy is not unique. Fortunately, to fulfill our aim to
formulate the second law of relativistic stochastic thermodynamics, it
suffices to have one definition for the entropy which is non-decreasing in the
course of time. Following the discussions made in [2], it is reasonable to
start with the 1PDF $\varphi$ which obeys the reduced FPE. This object is the
physical distribution on the future mass shell bundle. Therefore, we introduce
the entropy density on the mass shell bundle of the Brownian particle as

$\displaystyle s(x,p)=-\log\varphi(x,p).$ (31)

This entropy density characterizes the properties of the micro states of the
Brownian particle. In stochastic thermodynamics, the entropy production comes
not only from the Brownian particle but also from the heat reservoir.
Therefore, the relative entropy is more suitable to portray the
irreversibility of the whole system. The relative entropy density reads

$\displaystyle
h(x,p)=-\log\dfrac{\varphi(x,p)}{\varphi_{\text{eq}}(x,p)}=-\log\varphi-\alpha+\beta_{\mu}p^{\mu},$
(32)

where $\varphi_{\text{eq}}$ is the equilibrium distribution of the Brownian
particle, hence $\alpha$ and $\beta_{\nu}$ satisfy the equilibrium condition
(16).

What we actually need to make use of while formulating the second law of
relativistic stochastic thermodynamics is not the the total entropy but rather
the entropy density current on the space time. The entropy density current of
the Brownian particle on the spacetime is given as

$\displaystyle
S^{\mu}[\varphi]=-\int\eta_{\left(\Gamma_{m}^{+}\right)_{x}}\dfrac{p^{\mu}}{m}\varphi\log\varphi,$
(33)

and the relative entropy density current can be expressed as the linear
combination of entropy density current, the particle number current, and the
energy-momentum tensor

$\displaystyle H^{\mu}[\varphi]=S^{\mu}[\varphi]-\alpha
n^{\mu}[\varphi]+\beta_{\nu}T^{\mu\nu}[\varphi].$ (34)

We will shortly prove that the relative entropy production rate
$\nabla_{\mu}H^{\mu}[\varphi]$ is the sum of the entropy production rate of
the Brownian particle and that of the heat reservoir.

Since the heat reservoir is assumed to be in local thermodynamic equilibrium,
its entropy current can be expressed as [12, 13]

$\displaystyle
S^{\mu}_{R}=(1+\alpha_{R})N^{\mu}_{R}-\beta_{\nu}T^{\mu\nu}_{R}.$ (35)

This expression is not affected by the presence of the electromagnetic field,
but the equilibrium condition of the reservoir will take a form similar to
that of the Brownian particle,

$\displaystyle\nabla_{\mu}\alpha_{R}+q_{R}\beta^{\nu}F_{\mu\nu}=0,\qquad\nabla_{(\mu}\beta_{\nu)}=0,$
(36)

where $q_{R}$ characterizes the charge of the reservoir particle which could
either be vanishing or not. We assume that the reservoir is consisted purely
of classical particles with no chemical reactions, its particle number current
should be conserved, $\nabla_{\mu}N^{\mu}_{R}=0$. However, since the reservoir
exchanges energy with the Brownian particle and also with the electromagnetic
field, the divergence of the total energy-momentum tensor needs to obey

$\displaystyle\nabla_{\mu}T^{\mu\nu}_{R}+\nabla_{\mu}T^{\mu\nu}[\varphi]=F^{\nu}{}_{\mu}(qn^{\mu}[\varphi]+q_{R}N_{R}^{\mu}),$
(37)

which means that the non-conservation of the total energy-momentum tensor is
originated solely from the external field. With the help of eqs.(35–37) and
eq.(18), we finally find the covariant divergence of entropy density current
of the reservoir,

$\displaystyle\nabla_{\mu}S_{R}^{\mu}$
$\displaystyle=N_{R}^{\mu}\nabla_{\mu}\alpha_{R}-\beta_{\nu}\nabla_{\mu}T_{R}^{\mu\nu}$
$\displaystyle=-q_{R}N_{R}^{\mu}\beta^{\nu}F_{\mu\nu}-\beta_{\nu}\left[F^{\nu}{}_{\mu}(qn^{\mu}+q_{R}N_{R}^{\mu})-\nabla_{\mu}T^{\mu\nu}[\varphi]\right]$
$\displaystyle=-n^{\mu}[\varphi]\nabla_{\mu}\alpha+\beta_{\nu}\nabla_{\mu}T^{\mu\nu}[\varphi],$
(38)

where the equilibrium condition of reservoir particles (36) is used in the
second line and the equilibrium condition of Brownian particle (16) is used in
the third line. The total entropy production rate is then

$\displaystyle\nabla_{\mu}S^{\mu}_{\text{tot}}$
$\displaystyle=\nabla_{\mu}S^{\mu}[\varphi]+\nabla_{\mu}S_{\text{R}}^{\mu}$
$\displaystyle=\nabla_{\mu}S^{\mu}[\varphi]-n^{\mu}[\varphi]\nabla_{\mu}\alpha+\beta_{\nu}\nabla_{\mu}T^{\mu\nu}[\varphi]=\nabla_{\mu}H^{\mu}[\varphi],$
(39)

which is the desired result advocated earlier.

Using eq.(54) in Appendix A we can evaluate the total entropy production rate
to be

$\displaystyle\nabla_{\mu}S^{\mu}_{\text{tot}}=\nabla_{\mu}H^{\mu}[\varphi]$
$\displaystyle=\int\eta_{(\Gamma^{+}_{m})_{x}}\mathscr{J}[\varphi](h)$
$\displaystyle=\int\eta_{(\Gamma_{m}^{+})_{x}}\dfrac{\mathcal{D}^{\mu\nu}}{2\varphi}\left(\frac{\partial}{\partial
p^{\mu}}\varphi-\beta_{\mu}\varphi\right)\left(\frac{\partial}{\partial
p^{\nu}}\varphi-\beta_{\nu}\varphi\right),$ (40)

where the Einstein relation and the vanishing boundary condition for $\varphi$
at the infinity of the mass shell have been used. Since the diffusion tensor
$\mathcal{D}^{ij}=\delta^{\mathfrak{ab}}\mathcal{R}^{i}{}_{\mathfrak{a}}\mathcal{R}^{j}{}_{\mathfrak{b}}$
must be a semi-positive definite quadratic form, the total entropy production
rate is always nonnegative. Therefore, we arrive at the second law of
relativistic stochastic thermodynamics

$\displaystyle\nabla_{\mu}S^{\mu}_{\text{tot}}\geq 0.$ (41)

The entropy production is always closely related to the breaking of the time-
reversal symmetry. We can divide the total entropy production rate into that
of the Brownian particles and of the heat reservoir, both will be shown to be
proportional to the dissipative part of the probability current.

The entropy production rate of the Brownian particle reads

$\displaystyle\nabla_{\mu}S^{\mu}[\varphi]$
$\displaystyle=\int\eta_{(\Gamma_{m}^{+})_{x}}\mathscr{J}[\varphi](-\log\varphi)$
$\displaystyle=\int\eta_{(\Gamma_{m}^{+})_{x}}\varphi^{-1}\left(\frac{\partial\varphi}{\partial
p^{\mu}}\right)\mathscr{J}^{\mu}_{\mathrm{d}}[\varphi],$ (42)

where the trick mentioned in Appendix A is used again, and this derivation is
similar to that leads to eq.(4). Substituting eq.(18) into eq.(4), the entropy
production rate of the reservoir caused by the heat exchange can be expressed
as

$\displaystyle\nabla_{\mu}S_{R}^{\mu}$
$\displaystyle=-\beta_{\nu}\int\eta_{(\Gamma_{m}^{+})_{x}}\mathcal{I}^{\nu}[\varphi]=\int\eta_{(\Gamma_{m}^{+})_{x}}\beta_{\mu}\mathscr{J}^{\mu}_{\text{d}}[\varphi].$
(43)

Similar result was also mentioned in Pal and Deffner’s paper [7], but now it
is made fully covariant. It is obvious that the time-reversible part
$\mathscr{J}_{\mathrm{r}}^{\mu}[\varphi]$ of the probability current does not
contribute to the entropy production rate. All entropy productions arise as
consequences of the dissipative part $\mathscr{J}_{{\rm d}}^{\mu}[\varphi]$,
which is the same as in the non-relatrivistic case [22, 7]. It is worth
noticing that eq.(43) is actually the relativistic version of Clausius’
identity for the heat reservoir,

$\displaystyle\nabla_{\mu}S^{\mu}_{R}=-\frac{Q_{B}[\varphi]}{T_{B}}=\frac{Q_{R}}{T_{B}},$
(44)

where $Q_{B}[\varphi]$ is the heat transfer rate from the heat reservoir to
the Brownian particle from the perspective of Bob, and hence
$Q_{R}=-Q_{B}[\varphi]$ is the heat transfer rate from the Brownian particle
to the heat reservoir.

There are some interesting consequences from the equation of entropy
production rate. Assuming that the diffusion tensor $\mathcal{D}^{ij}$ is a
full-rank matrix and hence strictly positive definite, the condition for zero
entropy production rate becomes

$\displaystyle\left(\frac{\partial}{\partial
p^{\mu}}-\beta_{\mu}\right)\varphi=0.$ (45)

This is equivalent to the thermal equilibrium condition
$\mathcal{I}[\varphi]=0$, thanks to the covariant Einstein relation (17).
Therefore, the equilibrium state is the unique state which saturates the
second law. However, if the diffusion tensor is not full-rank, implying that
there exists some nonzero vector field $\mathcal{C}^{i}[\varphi]$ on the
momentum space obeying
$\mathcal{D}_{ij}\mathcal{C}^{i}[\varphi]\mathcal{C}^{j}[\varphi]=0$, the
condition for zero entropy production rate will become

$\displaystyle\left(\frac{\partial}{\partial
p^{\mu}}-\beta_{\mu}\right)\varphi=\mathcal{C}_{\mu}[\varphi].$ (46)

This leads to the possibility for the existence of certain state which is
different from the prescribed equilibrium state but still with zero entropy
production rate. This situation is not unexpected. The presence of a
degenerate diffusion tensor implies the existence of certain degrees of
freedom that are decoupled from the heat reservoir. The 1PDF of these states
obeys the Liouville equation and the entropy remains conserved. In the next
section, we will discuss how the rank of the diffusion tensor affects the
ability of the Brownian particle to reach the equilibrium state.

##  5 The stability of equilibrium state

In Ref. [2], we assumed that the Brownian particle can always reach thermal
equilibrium with the heat reservoir in the long time limit, without delving
into the dynamical process for the Brownian particle to evolve from a non-
equilibrium state to the equilibrium. To tackle this problem, the Lyapunov
theorem offers a powerful tool.

For the sake of convenience, we make an equivalent statement of the Lyapunov
theorem: if there exists a functional for a given system that satisfies the
condition of possessing at least one maximum point, with its time derivative
strictly positive except at these maximum points, then the system is deemed
asymptotically stable [24]. This functional is commonly referred to as the
Lyapunov functional. The maximum points of the Lyapunov functional represent
the stable states of the system. In the event that a system in a stable state
experiences minor disturbances, as long as these disturbances do not surpass
the local minimum of the Lyapunov functional, the system will return to its
original stable state. Notably, if the maximum point of the Lyapunov
functional is unique, no matter which initial state the system starts from, it
will always reach the stable state. Such a property is called global
asymptotically stable.

In our case, the expectation value of the relative entropy

$\displaystyle\bar{H}_{t}[\varphi]$
$\displaystyle:=\int_{\Sigma_{t}}\eta_{\Sigma_{t}}\dfrac{Z_{\mu}p^{\mu}}{m}\varphi\log\dfrac{\varphi}{\varphi_{\text{eq}}}$
(47)

plays the role of Lyapunov functional of the Brownian particle. We now try to
impose a small perturbation to the distribution function and calculate the
variation of the above functional. The effect of the perturbation is to drive
the system from one distribution to another, so the variation only acts on the
distribution function. The result of the variation reads

$\displaystyle\delta\bar{H}_{t}[\varphi]$
$\displaystyle=\int\eta_{\Sigma_{t}}\dfrac{Z_{\mu}p^{\mu}}{m}\delta\left[\varphi\left(\log\varphi+\alpha-\beta_{\nu}p^{\nu}\right)\right]$
$\displaystyle=\int\eta_{\Sigma_{t}}\dfrac{Z_{\mu}p^{\mu}}{m}\left[\left(1+\log\varphi+\alpha-\beta_{\nu}p^{\nu}\right)\delta\varphi+\varphi^{-1}(\delta\varphi)^{2}+O((\delta\varphi)^{2})\right]$
$\displaystyle=\int\eta_{\Sigma_{t}}\dfrac{Z_{\mu}p^{\mu}}{m}\left[\left(\log\varphi+\alpha-\beta_{\nu}p^{\nu}\right)\delta\varphi+\varphi^{-1}(\delta\varphi)^{2}+O((\delta\varphi)^{2})\right],$
(48)

where, in the third line, the particle conservation has been used. It is
obvious that the first order variation is zero and the second order variation
is negative at the equilibrium state, so the equilibrium state is a maximum
point of $\bar{H}_{t}[\varphi]$. Furthermore, the second order variation is
not only negative at the equilibrium state but also at any states. This
implies that no critical points like the one illustrated in Fig.1 could appear
in such systems, and hence the relative entropy of the Brownian particle is
globally convex with only one maximum.

Using eq.(56) and the second law, the time derivative of
$\bar{H}_{t}[\varphi]$ can be confirmed to be non-negative

$\displaystyle\frac{{\rm d}}{{\rm
d}t}\bar{H}_{t}[\varphi]=\int_{\mathcal{S}_{t}}\eta_{\mathcal{S}_{t}}\nabla_{\mu}H^{\mu}[\varphi]|\nabla
t|^{-1}\geq 0.$ (49)

However, the Lyapunov theorem requires it to be strictly positive except at
the equilibrium state. According to the discussion made in the end of the last
section, if the diffusion tensor is full-rank, the above inequality can only
be saturated in the equilibrium state. Therefore, $\bar{H}_{t}[\varphi]$ is a
Lyapunov functional of the Brownian particle. Fig.1 illustrates the general
behavior of a Lyapunov functional with the inclusion of an equilibrium state
and an artificial metastable state. In such cases, it is possible to drive the
system away from the equilibrium state into the metastable state or vice versa
by appropriate perturbations. Now, assuming a full-rank diffusion tensor, the
above possibility is excluded because of the global asymptotic stability.
Therefore, we can draw the conclusion that the relativistic charged Brownian
particle system exhibits a tendency to evolve towards an increase in relative
entropy, and the eventual state of evolution is characterized by the unique
equilibrium distribution.

Figure 1: The curve for an artificial Lyapunov functional. In the case with a
full-rank diffusion tensor, our discussion rules out the possibility for the
Lyapunov functional of the charged relativistic Brownian particle to exhibit
metastable and critical (i.e. inflection) points.

However, if the diffusion tensor is not full-rank, there will be some states
other than the equilibrium state with zero entropy production rate. The
Lyapunov functional may stop growing when the Brownian particle enters such
states, and the distribution function no longer evolves toward the equilibrium
one. Therefore, in the situation that the diffusion tensor is not full-rank,
the Brownian particle may not be able to reach the equilibrium state.

##  6 Concluding remarks

With the aid of covariant formalism of relativistic stochastic mechanics, the
first law and the second law of general relativistic stochastic thermodynamics
have been established for a system of charged Brownian particle subjecting to
an external electromagnetic field. The fully covariant formulation has
assisted in elucidating numerous conceptual issues. The final discussion about
the stability of the equilibrium distribution helps us to clarify that the
equilibrium state of Brownian particle is the state with maximum entropy.

As a potential application scenario of the generic construction, we expect
that a charged Brownian particle moving in Reissner-Nordström spacetime may be
an excellent example for which the theory is applicable. In doing so, the
relationship between non-equilibrium statistical physics in curved spacetime
and black hole thermodynamics might become an interesting subject of future
study.

On the other hand, we have discussed only the first and second laws in this
work, but a deeper understanding about the origin of the irreversibility in
curved spacetime still calls for further investigation. At present, the best
approach for the origin of irreversibility is based on various fluctuation
theorems [16, 17, 18, 19]. However, fluctuation theorems have not yet been
realized in the framework of general relativity. We hope to come back to this
subject in some upcoming works.

##  Appendix A Currents on spacetime and their divergences

Let $\mathscr{O}(x,p)$ be a scalar field on the mass shell bundle, then the
expectation value of $\mathscr{O}$ observed by Alice reads

$\displaystyle\bar{\mathscr{O}}_{t}[\varphi]$
$\displaystyle:=-\int\eta_{\Sigma_{t}}\mathscr{Z}_{A}\mathscr{J}^{A}[\varphi]\mathscr{O}=-\int\eta_{\mathcal{S}_{t}}Z_{\mu}\int\eta_{(\Gamma^{+}_{m})_{x}}\frac{p^{\mu}}{m}\varphi\mathscr{O}.$
(50)

Since $\mathscr{Z}=Z^{\mu}e_{\mu}$ is the unit normal vector of $\Sigma_{t}$
and $\mathscr{J}[\varphi]$ is the probability current,
$-\mathscr{Z}_{A}\mathscr{J}^{A}$ is clearly the probability density on
$\Sigma_{t}$. The momentum space integral in eq.(50) is simply a current on
spacetime manifold,

$\displaystyle\mathscr{O}^{\mu}[\varphi]:=\int\eta_{(\Gamma^{+}_{m})_{x}}\frac{p^{\mu}}{m}\varphi\mathscr{O}.$
(51)

Therefore, the expectation value can also be written as a surface integral of
$\mathscr{O}^{\mu}[\varphi]$ on the configuration space,

$\displaystyle\bar{\mathscr{O}}_{t}[\varphi]=-\int_{\mathcal{S}_{t}}\eta_{\mathcal{S}_{t}}Z_{\mu}\mathscr{O}^{\mu}[\varphi],$
(52)

which implies that $\mathscr{O}^{\mu}[\varphi]$ is the density current of
$\mathscr{O}$.

Let $V$ be an arbitrary region in the spacetime manifold $\mathcal{M}$, and
$\Gamma=\\{(x,p)\in\Gamma^{+}_{m}|x\in V\\}$ is the corresponding region in
the mass shell bundle. Let $y^{\mu}$ be the unit normal vector of the boundary
$\partial V$, the corresponding unit normal vector of $\partial\Gamma$ is then
$\mathscr{Y}=y^{\mu}e_{\mu}$. By the Gauss theorem, we have

$\displaystyle\int_{V}\eta_{\mathcal{M}}\nabla_{\mu}\mathscr{O}^{\mu}[\varphi]$
$\displaystyle=\int_{\partial V}\eta_{\partial
V}y_{\mu}\mathscr{O}^{\mu}[\varphi]=\int_{\partial\Gamma}\eta_{\partial\Gamma}\frac{y_{\mu}p^{\mu}}{m}\varphi\mathscr{O}=\int_{\partial\Gamma}\eta_{\partial\Gamma}\mathscr{Y}_{A}\mathscr{J}^{A}[\varphi]\mathscr{O}$
$\displaystyle=\int_{\Gamma}\eta_{\Gamma^{+}_{m}}\mathscr{J}^{A}[\varphi]\hat{\nabla}^{(\hat{h})}_{A}\mathscr{O}=\int_{V}\eta_{\mathcal{M}}\int_{(\Gamma^{+}_{m})_{x}}\eta_{(\Gamma^{+}_{m})_{x}}\mathscr{J}[\varphi](\mathscr{O}).$
(53)

Since the region $V$ is arbitrary, the integrand on the right hand side and
the left hand side should be equal to each other. Hence, the divergence of
$\mathscr{O}^{\mu}[\varphi]$ is

$\displaystyle\nabla_{\mu}\mathscr{O}^{\mu}[\varphi]=\int_{(\Gamma^{+}_{m})_{x}}\eta_{(\Gamma^{+}_{m})_{x}}\mathscr{J}[\varphi](\mathscr{O}).$
(54)

In relativistic thermodynamic relation, the time derivative of a physical
quantity is usually replaced by the divergence of the associated current. This
can be achieved as follows. Let $\mathcal{S}_{t_{1}}$ and
$\mathcal{S}_{t_{2}}$ be two Cauchy surfaces and $V$ be the region enclosed by
a cylindrical surface with $\mathcal{S}_{t_{1}}$ and $\mathcal{S}_{t_{2}}$
playing as the upper and bottom. Then, integrating eq.(54) over $V$, we get,
by Gauss theorem and co-area formula[25, 26], the following equation,

$\displaystyle\bar{\mathscr{O}}_{t_{2}}[\varphi]-\bar{\mathscr{O}}_{t_{1}}[\varphi]=\int_{V}\eta_{\mathcal{M}}\nabla_{\mu}\mathscr{O}^{\mu}[\varphi]=\int_{t_{1}}^{t_{2}}{\rm
d}t\int_{\mathcal{S}_{t}}\eta_{\mathcal{S}_{t}}\nabla_{\mu}\mathscr{O}^{\mu}[\varphi]|\nabla
t|^{-1}.$ (55)

Therefore, the time derivative of the expectation value is the integral of the
divergence of the current times $|\nabla t|^{-1}$ on the configuration space,

$\displaystyle\frac{{\rm d}}{{\rm
d}t}\bar{\mathscr{O}}_{t}[\varphi]=\int_{\mathcal{S}_{t}}\eta_{\mathcal{S}_{t}}\nabla_{\mu}\mathscr{O}^{\mu}[\varphi]|\nabla
t|^{-1}.$ (56)

## Acknowledgement

This work is supported by the National Natural Science Foundation of China
under the grant No. 12275138.

## Data Availability Statement

This research has no associated data.

## Declaration of competing interest

The authors declare no competing interest.

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Corresponding author: D. Shy, e-mail<EMAIL_ADDRESS>
DISTRIBUTION A: Approved for public release, distribution is unlimited.

# Development of the ComPair gamma-ray telescope prototype

Daniel Shy  National Research Council Research Associate resident at the U.S.
Naval Research Laboratory, 4555 Overlook Ave., SW, Washington, DC, 20375, USA
Carolyn Kierans  NASA Goddard Space Flight Center, Greenbelt, MD, USA
Nicholas Cannady  NASA Goddard Space Flight Center, Greenbelt, MD, USA  Center
for Space Sciences and Technology, University of Maryland, Baltimore County,
1000 Hilltop Circle, Baltimore, MD 21250, USA  Center for Research and
Exploration in Space Science and Technology, NASA/GSFC, Green- belt, MD 20771,
USA  Regina Caputo  NASA Goddard Space Flight Center, Greenbelt, MD, USA  Sean
Griffin  Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin-Madison, 222 W Washington Ave Unit 500, Madison, WI 53703, USA  J.
Eric Grove  Space Science Division, Naval Research Laboratory, 4555 Overlook
Ave SW, Washington, DC 20375  Elizabeth Hays  NASA Goddard Space Flight
Center, Greenbelt, MD, USA  Emily Kong  Technology Service Corporation,
Arlington, VA, 22202, USA  Nicholas Kirschner  The Department of Physics, The
George Washington University, 725 21st NW, Washington, DC 20052, USA  Iker
Liceaga-Indart  NASA Goddard Space Flight Center, Greenbelt, MD, USA  Catholic
University of America, 620 Michigan Ave NE, Washington, DC 20064, USA  Julie
McEnery  NASA Goddard Space Flight Center, Greenbelt, MD, USA  John Mitchell
NASA Goddard Space Flight Center, Greenbelt, MD, USA  A. A. Moiseev  NASA
Goddard Space Flight Center, Greenbelt, MD, USA  University of Maryland at
College Park, College Park, MD 20742, USA  Center for Research and Exploration
in Space Science and Technology, NASA/GSFC, Green- belt, MD 20771, USA  Lucas
Parker  Los Alamos National Laboratory, Los Alamos, NM 87544, USA  Jeremy S.
Perkins  NASA Goddard Space Flight Center, Greenbelt, MD, USA  Bernard Phlips
Space Science Division, Naval Research Laboratory, 4555 Overlook Ave SW,
Washington, DC 20375  Makoto Sasaki  NASA Goddard Space Flight Center,
Greenbelt, MD, USA  University of Maryland at College Park, College Park, MD
20742, USA  Center for Research and Exploration in Space Science and
Technology, NASA/GSFC, Green- belt, MD 20771, USA  Adam J. Schoenwald  NASA
Goddard Space Flight Center, Greenbelt, MD, USA  University of Maryland,
Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, USA  Clio Sleator
Space Science Division, Naval Research Laboratory, 4555 Overlook Ave SW,
Washington, DC 20375  Jacob Smith  George Mason University, resident at the
Naval Research Laboratory, 4555 Overlook Ave SW, Washington, DC 20375  Lucas
D. Smith  University of Maryland at College Park, College Park, MD 20742, USA
Sambid Wasti  Catholic University of America, 620 Michigan Ave NE, Washington,
DC 20064, USA  NASA Goddard Space Flight Center, Greenbelt, MD, USA  Center
for Research and Exploration in Space Science and Technology, NASA/GSFC,
Green- belt, MD 20771, USA  Richard Woolf  Space Science Division, Naval
Research Laboratory, 4555 Overlook Ave SW, Washington, DC 20375  Eric Wulf
Space Science Division, Naval Research Laboratory, 4555 Overlook Ave SW,
Washington, DC 20375  Anna Zajczyk  NASA Goddard Space Flight Center,
Greenbelt, MD, USA  Center for Space Sciences and Technology, University of
Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, USA
Center for Research and Exploration in Space Science and Technology,
NASA/GSFC, Green- belt, MD 20771, USA

###### Abstract

There is a growing interest in the science uniquely enabled by observations in
the MeV range, particularly in light of multi-messenger astrophysics. The
Compton Pair (ComPair) telescope, a prototype of the AMEGO Probe-class
concept, consists of four subsystems that together detect and characterize
gamma rays in the MeV regime. A double-sided strip silicon Tracker gives a
precise measure of the first Compton scatter interaction and tracks pair-
conversion products. A novel cadmium zinc telluride (CZT) detector with
excellent position and energy resolution beneath the Tracker detects the
Compton-scattered photons. A thick cesium iodide (CsI) calorimeter contains
the high-energy Compton and pair events. The instrument is surrounded by a
plastic anti-coincidence (ACD) detector to veto the cosmic-ray background. In
this work, we will give an overview of the science motivation and a
description of the prototype development and performance.

###### keywords:

Gamma-ray astrophysics, gamma-ray instrumentation, Compton imaging, pair-
conversion telescope

##  1 INTRODUCTION

Gamma rays have an integral part in multi-messenger discoveries such as the
combined detection with neutrinos from supernova SN1987A [1, 2] and
gravitational waves (GWs) from GW 170817 [3]. However, the MeV domain remains
largely under explored; COMPTEL on the Compton Gamma Ray Observatory was the
last major imaging instrument sensitive to that energy range and was de-
orbited in 2000 [4].

The All-sky Medium Energy Gamma-ray Observatory (AMEGO) is a probe-class
mission concept designed to address the MeV gap [5]. AMEGO is sensitive to
photons from $0.2\ \mathrm{MeV}$ to $10\ \mathrm{GeV}$, which covers both the
Compton and pair-conversion regime. To accomplish the detection and imaging of
both these processes, we employ a unique combination of a ‘Tracker’ with both
a low- and high-energy calorimeter that are all surrounded by an anti-
coincidence detector. These four major subsystems are briefly summarized here:

  * •

Double Sided Silicon Strip Detector (DSSD) Tracker acts as the Compton-
scattering element for low-energy gamma rays and the pair-conversion material
for high-energy gamma rays by measuring the position and energy of the
particles moving through the Tracker layers.

  * •

Cadmium Zinc Telluride (CZT) calorimeter, or low-energy calorimeter, is
optimized for detecting the scattered photon and enhances the line
sensitivity, as well as providing improved imaging and polarization
capabilities.

  * •

Cesium Iodide (CsI) calorimeter, or high-energy calorimeter, is designed to
extend the energy range by detecting the ensuing electromagnetic showers.

  * •

Anti-coincidence detector (ACD) consisting of a plastic scintillator to reject
charged particle background.

The simulated performance of the AMEGO concept is available in Ref. 6 which is
optimized for continuum sensitivity across a broad energy range. There are
other astrophysical instruments with a slightly different mission set and
technological solution in various stages of conception/development. They
include the Compton Spectrometer and Imager (COSI) [7], ASTROGAM [4], Galactic
Explorer with a Coded Aperture Mask Compton Telescope (GECCO), Advanced
Particle-astrophysics Telescope (APT) [8], SMILE-2+ [9], and AMEGO-X [10, 11].

To demonstrate the instrument’s potential, the AMEGO team has developed a
prototype, ComPair, to advance the hardware and software used in the probe
concept. Like the AMEGO probe, it consists of all four subsystems but with an
energy range of $0.5-100\ \mathrm{MeV}$. After in-lab calibration and
qualification, the prototype is slated to perform a balloon test flight.

This work presents an update on the development and initial performance of the
ComPair instrument. Sec. 2 presents an overview of the ComPair prototype while
Sec. 3 presents the preliminary performance of each subsystem as well as the
integrated instrument. The manuscript also describes the planned future work.

(a)

(b)

Figure 1: (a) Cutaway view of the CAD showing ComPair’s subsystems and their
dimensions. (b) Image of the ComPair prototype, without the ACD, integrated
with 6 Tracker layers, CZT, and the CsI calorimeters. Two peltier coolers are
attached to the Tracker layers for heat removal.

##  2 ComPair: An AMEGO Prototype

Fig. 1a presents a computer-aided design (CAD) model of the ComPair balloon
prototype, while Fig. 1b presents the actual experimental setup without the
ACD. The three core subsystems (Tracker, CZT and CsI) all fit within an
envelope of roughly $43\times 30\times 41\ \mathrm{cm}^{3}$. The DSSD Tracker
layers are at the top of the instrument. Below the Tracker are arrays of the
CZT detectors, and the CsI calorimeter is at the bottom of the instrument.

For the actual balloon flight, the CZT housing will be re-designed to be
hermetically sealed. The entire balloon instrument with the associated
electronics weighs less than $100\ \mathrm{kg}$. Additional information on the
balloon instrument is available in Sec. 4. The rest of this section overviews
the different subsystems, trigger module, and flight computer.

###  2.1 Tracker: Double Sided Silicon Strip Detector

The ComPair Tracker has been designed, built, and tested by NASA’s Goddard
Space Flight Center (GSFC), with firmware support from Los Alamos National
Laboratory (LANL). Its goal is to achieve position sensitivity within each
Tracker layer to measure the Compton-scattered electron or the
$e^{-}\scalebox{0.75}[1.0]{$-$}e^{+}$ tracks from the pair production. During
a Compton scatter, should the recoiled electron proliferate into other layers,
the Tracker also has the potential to track the electron and produce Compton
arcs rather than rings [12].

Fig. 2a presents a single layer of the Tracker. The DSSD is positioned on the
lower-right side of the image and has an active area of $10\times 10\
\mathrm{cm}^{2}$. Fabricated by Micron Semiconductor Ltd, the silicon wafers
are $500\ \mu\mathrm{m}$ thick with strips on each side that are orthogonal to
each other to reconstruct the $X$ and $Y$ positions of interaction [13]. The
strips are $60\ \mu\mathrm{m}$ wide with a pitch of $\sim 510\ \mu\mathrm{m}$.
This results in 192 AC-coupled strips per side.

The wafers are mounted in a custom carrier that routes the signals to front-
end electronics that use the IDEAS VATA460.3 application-specific integrated
circuits (ASIC) [14]. Each side of the DSSD requires 6 ASICs with 32 channels
on each ASIC. A complete description of the readout is available in Ref. 15.
The ComPair prototype aims to use ten layers. Fig. 2b presents 6 fully
integrated layers.

(a)

(b)

Figure 2: The ComPair DSSD Tracker subsystem with the custom analog and
digital readouts.

###  2.2 Low-Energy Calorimeter: CZT

The low-energy calorimeter uses 3D position-sensitive virtual Frisch-grid
Cadmium Zinc Telluride (CZT) developed at Brookhaven National Laboratory [16,
17]. Its performance is optimized to act as the absorber plane for gamma rays
scattered in the Tracker. Its major advantage over other semiconductors is
that it can operate at room temperature due to its wide bandgap. Each CZT
crystal, manufactured by Redlen Technologies, is $6\times 6\times 20\
\mathrm{mm}^{3}$ in size. Fig. 3a illustrates a single CZT crystal. Each end
has a gold electrode representing the cathode and anode. Due to the severe
trapping of the holes in CZT, single polarity charge sensing techniques need
to be applied [18]. In this work, we utilize a virtual Frisch-grid technique
whereby the shielding electrode pads are placed on the sides of the crystal.
This allows for the reconstruction of the 3D position of interaction.

First, the cathode-to-anode ratio (CAR) calculates the depth of interaction
($Z$ coordinate). Due to the single-polar nature of CZT, the signal on the
cathode is depth-dependent, while the virtual Frisch grid allows for an
induced charge on the anode that is depth-independent. Therefore, the ratio
results in a relative depth of interaction. Using the technique, the depth
resolution is estimated to be less than 0.5 mm. Next, the side pads and the
relative amplitudes induced on each pad could be used to calculate the
$X\scalebox{0.75}[1.0]{$-$}Y$ coordinates of interaction [19]. Therefore,
assuming a linear approximation for the response function, the
$X\scalebox{0.75}[1.0]{$-$}Y$ coordinate could be calculated as

$\begin{split}X=\frac{A^{x}_{1}}{A^{x}_{1}+A^{x}_{2}},\\\
Y=\frac{A^{Y}_{1}}{A^{Y}_{1}+A^{Y}_{2}},\end{split}$ (1)

where $A^{x}_{1}$ and $A^{x}_{2}$ represent the recorded amplitude of pad 1
that is opposite of pad 2 in the $X$ direction. $A^{y}_{1}$ and $A^{y}_{2}$
similarly represent the pads in the $Y$ direction. This method can produce a
position resolution of better than $1\ \mathrm{mm}$ in $X-Y$. Since the 3D
position of each interaction can be reconstruction, a 3D voxel correction can
be applied. This technique can compensate for non-uniformity in the detector
material and results in a resolution of around 1% FWHM at 662 keV.

As a subsystem for ComPair, each crystal is packaged in a crate that encloses
a $4\times 4$ array of crystals, shown in Fig. 3b. The crate is read out by a
custom low-noise ASIC called the advanced virtual Frisch-Grid 2 (AVG2) [20].
An Altera Arria GX FPGA then handles the data recording and ASIC controls. In
ComPair, there are a total of 9 crates arranged in a $3\times 3$ array, shown
in Fig. 3c. The bars are biased to 2500 volts which requires the need for it
to be placed in a pressure vessel for the balloon flight.

(a)

(b)

(c)

Figure 3: Images of the CZT Subsystem. (a) Shows a single CZT bar with the
pads that create the virtual Frisch grid clear visible at the edge of the
crystal. (b) A semi-populated CZT crate. (c) The assembled CZT subsystem with
the FPGA board is visible on the right side of the image.

###  2.3 High-Energy Calorimeter: CsI

Developed by the U.S. Naval Research Laboratory, the high-energy calorimeter
is composed of CsI:Tl bars and takes heritage from Fermi’s Large Area
Telescope (LAT) CsI:Tl Calorimeter [21]. Each bar, grown by either Amcrys in
Kharkov, Ukraine or Saint-Gobain, has dimension $1.67\times 1.67\times 10.0\
\mathrm{cm}^{3}$. Each face of the scintillator is roughened with 180-grit
sandpaper to increase light asymmetry along the length of the bar. The
crystals are then wrapped with Tetratex to increase light output [22]. Each
end of the bar is read out by a $2\times 2$ array of $6\times 6\
\mathrm{mm}^{2}$ J-series SiPMs manufactured by ON Semiconductors. Fig. 4a
shows the SiPMs being bonded with epoxy to the crystal bars.

The CsI calorimeter makes use of 30 bars arranged in a hodoscopic fashion. In
other words, the bars are arranged in rows of 6 that alternate direction
between each layer. The bars are placed in a 3D printed plastic housing such
that the center-to-center pitch between each bar is $1.9\ \mathrm{cm}$. The 60
SiPMs are read out by an IDEAS ROSSPAD [23], which is a module with 64
channels that is distributed to four SiPHRAs ASICS [24]. If $E_{1}$ and
$E_{2}$ represent the energy recorded on each end of the bar, the depth of
interaction in the bar is calculated as
$\mathrm{DOI}=(E_{1}-E_{2})/(E_{1}+E_{2})$ and the energy deposited is
calculated as $E_{\mathrm{dep}}=\sqrt{E_{1}\times E_{2}}$. The gain and SiPM
voltage is tuned such that each bar has a dynamic energy range of roughly 250
keV to 30 MeV. Fig. 4b shows the CsI calorimeter in its final configuration.

(a)

(b)

Figure 4: (a) The CsI bars at various stages of SiPM bonding. (b) The CsI
calorimeter subsystem within its housing. The black assembly in the upper-left
corner of the picture contains the arrays of CsI. Immediately to the right is
the ROSSPAD front-end readout. Below are electronics associated with the
readout and synchronization with the trigger module.

###  2.4 ACD

Similarly to the Fermi-LAT, the ComPair ACD uses plastic scintillator panels,
specifically Eljen EJ-208. Unlike the Fermi LAT, the panels are not tiled as
there is less concern of self-vetoing at the lower AMEGO energies. Two
wavelength shifters are appended to two adjacent edges, shown in Fig. 5a. At
the edge where the two shifters meet, the edges are chamfered out such that
two $2\times 2$ arrays of $6\times 6\ \mathrm{mm^{2}}$ C-Series SiPMs by On
Semiconductors can be placed and read out the scintillation signals. The
panels are then placed in aluminum frames for structural support.

Five panels compose the entire subsystem and cover the instrument’s four sides
and its top. The ACD’s dimensions span $72.1\times 56.1\times 45.2\
\mathrm{cm^{2}}$, as seen in Fig. 5b. The ACD is then mounted onto a base
plate with a bracket that elevates the panel slightly such that harnessing can
pass from the core instrument to outside the ACD to connect to their relevant
electronics. The front-end readout is duplicated from the CsI calorimeter.

(a)

(b)

(c)

Figure 5: (a) The plastic scintillator panel is in the center with the two
wavelength shifters shown in yellow. The readout edge and the SiPMs are shown
at the bottom of the image. The panel is then enclosed by the aluminum frame
(darker green). The inset image shows the joint between two frame enclosures.
(b) Shows the assembly of the wavelength shifters onto the plastic panel. (c)
Exploded CAD view of the 5 ACD panels as they would assemble in the final
configuration.

###  2.5 Trigger Module

The Trigger Module (TM) is designed and built at GSFC to check for
coincidences between the subsystems within a programmable window [25]. It uses
a Xilinx Zynq-7000 FPGA evaluation kit with a custom breakout board for the 14
separate front-end boards: 10 for the Tracker layers, and one each for the
CZT, CsI, and ACD. Fig. 6 shows the trigger module in its housing. Each
subsystem produces a fast trigger signal with the Tracker’s layers acting
independently from each other. If an interaction occurs in a subsystem, it
will send a trigger to the TM, which will check for coincidence between other
subsystems and flag it appropriately. If the triggers pass the coincidence
logic, a trigger acknowledgment along with an Event ID is distributed to all
the subsystems. The TM is programmable to enable 16 coincidence logic modes
with various permutations of the different subsystems.

The ROSSPAD, which is the front-end readout for the CsI calorimeter and the
ACD, does not have the general-purpose input/output capability to accept the
Event ID from the TM. Therefore, we utilize an Arduino Due to accept EventIDs
and synchronize them with the ROSSPAD triggers. The system also accepts a
pulse-per-second (PPS) signal that is converted with a TTL-LVDS board to
maintain synchronization between the different components. The Arduino and the
TTL-LVDS converter boards are shown in Fig. 4b.

Figure 6: Picture of the ComPair Trigger Module. This module utilizes the
ZC706 Xilinx Zynq-7000 kit, which will be replaced with the ZC702 for the
balloon flight.

##  3 Preliminary Performance

This section presents the performance of individual subsystems as well as the
performance of the integrated system.

###  3.1 Individual System Performance

Before full system integration, each subsystem has individually undergone
testing and calibrations to validate performance. The Tracker underwent a
series of inspections to study the uniformity, depletion voltage, and inter-
strip capacitance of the DSSDs.

At the time of writing, six of the ten layers have been assembled and
characterized. Fig. 7 presents an Am-241 spectrum from all the live strips in
layer 2 with the $59.5\ \mathrm{keV}$ line measured with a full-width-at-half-
maximum (FWHM) of $22.1\ \mathrm{keV}$, or $37.1\%$. The energy range of each
Tracker strip is estimated to be around $40-700\ \mathrm{keV}$.

Figure 7: Americium 241 spectrum as recorded by the live strips on a single
Tracker layer. Note the second peak left of the americium peak is an artifact
of the ASIC’s pedestal.

The CZT subsystem has undergone rigorous testing with its performance reported
numerous times [19, 20, 16]. Fig. 8 presents the CZT’s response to an Na-22
source showing $1.8\%$ FWHM resolution at the $1.274\ \mathrm{MeV}$. The
energy range of the CZT subsystem is estimated to be 0.2-10 MeV per bar.

Figure 8: CZT spectrum of a Na-22 check source. Note that the $0.511\
\mathrm{MeV}$ line shows Doppler broadening.

We tested the CsI subsystem extensively in the lab with various check sources
at energies such as $0.511\ \mathrm{MeV}$ (Na-22) and $2.6\ \mathrm{MeV}$ from
Th-228. The energy resolution across all individual bars in the system
averages around $8.0\%$ FWHM at $662\ \mathrm{keV}$. The energy range of a
single bar is set to be $0.25\scalebox{0.75}[1.0]{$-$}30\ \mathrm{MeV}$. In
addition, the CsI calorimeter has undergone various tests from monoenergetic
beams. Fig. 9a plots the energy response of a single bar to a monoenergetic
$5\ \mathrm{MeV}$ gamma-ray beam. Fig. 9b plots the calorimeter’s response to
muons showing a clear Landau distribution.

(a)

(b)

Figure 9: (a) 5 MeV beam spectral response of single log from the CsI
calorimeter spectral response. The $5\ \mathrm{MeV}$ full energy peak is
clearly seen along with its escape peaks. At the lower end, the $0.511\
\mathrm{MeV}$ annihilation peak is visible. (b) The CsI calorimeter’s response
to muons at around sea level using all channels. Note that (b) treats each
interaction in the calorimeter as an independent event.

###  3.2 Fully Integrated System

We conducted several tests and experiments to validate the integrated system
during the instrument commissioning phase. To test the timing and position
calibration of the CZT and CsI subsystem, a Na-22 check source was placed in
between the active area of the two systems. As the Na-22 source undergoes
$\beta^{+}$ decay, it emits two characteristic $0.511\ \mathrm{MeV}$
annihilation photons in anti-parallel directions. Simultaneous detection by
the CZT and CsI subsystems of the $0.511\ \mathrm{MeV}$ photons indicates that
the triggering mechanism is actively working and assures that the geometry is
appropriately modeled. Fig. 10 plots the lines of response between the
locations of the detected $0.511\ \mathrm{MeV}$ photons. Since a clear hot
spot is visible, this implies a simultaneous detection of the $0.511\
\mathrm{MeV}$ photons and a verification of coincidence. The hot spot also
indicates that position reconstruction is functioning.

Figure 10: A tomographic image plotting the lines of responses of Na-22
annihilation lines when the source is placed between the CZT and CsI. This is
essentially a positron emission tomography image and only considers the top
layer of the CsI instrument.

When the Tracker was incorporated, cosmic rays presented a convenient tool to
validate its integration [26]. Muons are an attractive source in which to
verify the coincident triggering mechanism and the geometry of the system due
to their greater mass and electric charge. They are therefore very penetrating
and travel in straight lines. Fig. 11 plots two different muon tracks
traversing ComPair. In the plot, the positive Z represents the top of the
instrument, where the Tracker is located. The red box represents the error
associated with the position reconstructed by the CZT. No error considerations
or visual aids are added to the CsI calorimeter tracks.

The linear tracks indicate the coincidence logic that requires a trigger among
all subsystems. Next, since the muons travel in a straight line, recording
such a behavior is another indication that integration is successful.

(a)

(b)

Figure 11: Muons tracks in the core ComPair instrument. Note that the position
resolution of the CsI calorimeter is $1\ \mathrm{cm}$ FWHM. The top set of
dots represent hits seen by the Tracker while the middle red cube represents
the error boundaries of the hit in CZT. The bottom set of points represent
hits in the CsI calorimeter.

###  3.3 Preliminary results from HIGS Beam Test 2022

In April 2022, a beam test of the prototype without the ACD was performed at
Triangle Universities Nuclear Laboratory’s High Intensity Gamma-ray Source
(HIGS) facility, which delivers monoenergetic gamma-ray beams. The energies
studied include 2, 5, 7, 15, and 25 MeV. We present the preliminary results
here.

Fig. 12 shows the experimental setup at the HIGS facility. During the runs,
the implemented coincidence allows for full-system readout after any two
subsystems have triggered. Fig. 13 presents the spectral responses to a
monoenergetic beam for each subsystem and the summed energies across all
subsystems, showing a full-energy peak at 2 MeV. Note that the results are
preliminary as more advanced calibrations are currently under investigation.

Figure 12: Experimental setup of the ComPair system at the HIGS facility. In
this configuration, the system is offset 15 degrees with respect to the beam.
The beam’s vector is into the page aimed at the leftmost side of the DSSD
active area. The non-offset configuration has the beam aimed towards the
center of the CZT array.

To study the energy responses of the subsystem, Fig. 14 plots the bivariate
nature of the energies deposited in the CsI and CZT calorimeters for (a)
experimental and (b) simulated data. Several features are visible in both
experiment and simulation, mainly the double escape peak in the CZT and an
escaped annihilating photon detected in the CsI. In addition, small angle
scatters in the CZT with a photo-absorption in the CsI region are clearly
visible in both the experiment and simulation.

Figure 13: Spectral responses of each subsystem as well as the full instrument
response to monoenergetic 2 MeV photons.

(a)

(b)

Figure 14: Experimental and simulated bivariate histograms of energy responses
in the CZT and CsI calorimeters. The 1 MeV band in the CZT represents the
double-escape peak with one of the annihilation photons detected in the CsI.
The 1.5 MeV band in CZT represents the single escape peak with a single
annihilation photon in the CsI. The diagonal of the lower triangle represents
the fully contained gamma ray in the two subsystem. The island in the diagonal
in the CZT energies between $0.25-1\ \mathrm{MeV}$ range represent scatter in
the CZT and a full energy deposition in the CsI.

##  4 Future Work

Analysis of HIGS beam test is underway with more results expected in the
future. Next, in fall of 2023, the system is slated to fly as a short-duration
balloon experiment out of Fort Sumner, NM, hosted by NASA’s Columbia
Scientific Balloon Facility (CSBP). The system is therefore undergoing design
upgrades to a balloon flight-ready prototype. This includes upgrading the CZT
housing to a hermetically sealed container to maintain appropriate high-
voltage safety and ensure high performance. The upgrade also includes the
integration of the ACD. Finally, each subsystem will undergo thermal-vacuum
testing to characterize and study its performance. Fig. 15a presents the
instrument’s block diagram with Fig. 15b presenting the notional balloon
instrument.

(a)

(b)

Figure 15: (a) Block diagram of the instrument detailing data flow, power and
tertiary systems. (b) CAD of the balloon instrument.

###  4.1 Flight Computer

The ComPair DAQ computer is under development by the Los Alamos National
Laboratory (LANL) and acts as the data acquisition CPU for the different
subsystems. As the main controller, it will send appropriate commands to each
subsystem and store data during the flight. Its architecture is based on a
Versalogic BayCat single board computer.

During the balloon flight, the computer will supervise subsystem-specific
long-running processes, manage data collection, control the power distribution
unit and readout thermometry. The instrument will also connect to the CSBP
miniature science instrument package to telemeter housekeeping and a subset of
data during the flight.

###  4.2 Further Technology Development

Several technologies are currently in development for future observatories.
The Tracker group is looking into implementing Si Pixel technology [27]. The
CZT group is developing a digital system that can digitize the waveforms that
yield better energy and position resolution. The CsI group is developing Dual-
Gain SiPMs [28] to extend the dynamic range while maintaining energy linearity
with high position and energy resolution. Also in development is a ‘finger’
array rather than a hodoscopic geometry which has vertical short chunks placed
in an array and readout from the top and bottom, rather than the sides. This
has the potential of delivering higher position resolution. Improvements to
the image processing code, MEGALib, is being done by advancing pair
reconstruction techniques and optimization for ComPair and AMEGO. This
includes using single-site events to improve the detection and localization
efficiency of gamma sources [29].

##  5 Conclusion

The ComPair instrument has been developed as a prototype for AMEGO to
demonstrate its technological potential for a MeV gamma ray observatory. Each
subsystem and the integrated system as a whole have undergone testing and
verification. An augmentation to AMEGO, named AMEGO-X [11], which uses Silicon
pixels and a CsI calorimeter, is currently proposed as a medium-explorer class
instrument. The technology developed in ComPair directly supports AMEGO-X and
the training of early career scientists in MeV gamma-ray instrument
development.

###### Acknowledgements.

This work is supported under NASA Astrophysics Research and Analysis (APRA)
grants NNH14ZDA001N-APRA, NNH15ZDA001N-APRA, NNH18ZDA001N-APRA,
NNH21ZDA001N-APRA.

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