# Many-body effective thermal conductivity in phase-change nanoparticle chains
due to near-field radiative heat transfer

Minggang Luo  Junming Zhao<EMAIL_ADDRESS>Linhua Liu  Brahim Guizal  Mauro
Antezza<EMAIL_ADDRESS>School of Energy Science and
Engineering, Harbin Institute of Technology, 92 West Street, Harbin 150001,
China  Laboratoire Charles Coulomb (L2C) UMR 5221 CNRS-Université de
Montpellier, F- 34095 Montpellier, France  Key Laboratory of Aerospace
Thermophysics, Ministry of Industry and Information Technology, Harbin 150001,
China  School of Energy and Power Engineering, Shandong University, Qingdao
266237, China  Institut Universitaire de France, 1 rue Descartes, F-75231
Paris Cedex 05, France

###### Abstract

In dense systems composed of numerous nanoparticles, direct simulations of
near-field radiative heat transfer (NFRHT) require considerable computational
resources. NFRHT for the simple one-dimensional nanoparticle chains embedded
in a non-absorbing host medium is investigated from the point of view of the
continuum by means of an approach combining the many-body radiative heat
transfer theory and the Fourier law. Effects of the phase change of the
insulator-metal transition material (VO2), the complex many-body interaction
(MBI) and the host medium relative permittivity on the characteristic
effective thermal conductivity (ETC) are analyzed. The ETC for VO2
nanoparticle chains below the transition temperature can be as high as 50
times of that above the transition temperature due to the phase change effect.
The strong coupling in the insulator-phase VO2 nanoparticle chain accounts for
its high ETC as compared to the low ETC for the chain at the metallic phase,
where there is a mismatch between the characteristic thermal frequency and
resonance frequency. The strong MBI is in favor of the ETC. For SiC
nanoparticle chains, the MBI even can double the ETC as compared to those
without considering the MBI effect. For the dense chains, a strong MBI
enhances the ETC due to the strong inter-particles couplings. When the chains
go more and more dilute, the MBI can be neglected safely due to negligible
couplings. The host medium relative permittivity significantly affects the
inter-particles couplings, which accounts for the permittivity-dependent ETC
for the VO2 nanoparticle chains.

###### keywords:

effective thermal conductivity, near-field radiative heat transfer, many-body
interaction , insulator-metal phase-change material, nanoparticles

††journal: Int. J Heat Mass Transf.

##  1 Introduction

Near-field radiative heat transfer (NFRHT) is currently attracting a lot of
interests for its fundamental and applicative facets [1, 2, 3, 4, 5]. In dense
particulate systems, the separation distance between two nanoparticles is
often comparable to or less than the characteristic thermal wavelength [6].
Due to the near-field effect (e.g., evanescent wave tunneling), the heat flux
will exceed the Planck's blackbody limit by several orders of magnitude, which
has been predicted thanks to the fluctuational electrodynamics theory [7, 8,
9, 10, 11, 12] and proved by recent experimental observations [13, 14, 15, 16,
17, 18].

Many important progresses have been reported on direct simulations of NFRHT
for systems composed of nanoparticles. On the one hand, the inter-ensemble
NFRHT between two nanoparticle ensembles (e.g., three-dimensional (3D)
clusters of hundreds of nanoparticles [6, 19, 20] and two-dimensional (2D)
nanoparticle ensembles [21, 22, 23], as well as the simple ensembles composed
of only a few nanoparticles [24, 25]) has been analyzed. On the other hand,
the intra-ensemble NFRHT inside the nanoparticle ensemble itself has also been
reported. The fractional diffusion theory was applied to describe NFRHT along
3D plasmonic nanostructure networks, as well as one-dimensional (1D) ones, and
the heat superdiffusion behavior was found [26]. For 2D fractal structures,
the effects of the structure morphology on the collective properties were
analyzed and the heat flux has no large-range character, in contrast to non-
fractal structures [27]. Thermal radiation behavior along a 1D nanoparticle
chain has been shown to be significantly affected by another nanoparticle
chain in proximity due to strong couplings [28]. Radiative heat flux along a
linear chain considering an external magnetic field was also analyzed [29]. In
addtion, thermal transport behaviors along the atomic chains due to quantum
effects have been reported recently [30, 31, 32]. In general, the direct
simulation of NFRHT for dense particulate systems composed of hundreds of
thousands of nanoparticles will introduce considerable unknowns, which will be
very time-consuming and will require considerable computational resources.

From the point of view of continuum, the method applying the effective thermal
conductivity (ETC) to characterize the NFRHT in dense particulate systems
based on the diffusion assumption is really time-saving, as compared to direct
simulation methods (e.g., many-body radiative heat transfer theory [24, 25],
scattering matrix method [33, 34], trace formulas method [35, 36], thermal
discrete dipole approximation method (T-DDA) [37, 38, 39], fluctuating surface
currents approach (FSC)[40], boundary element method (BEM) [41], finite
difference time domain method (FDTD) [42, 43, 44] and the quasi-analytic
solution [45], to name a few). The kinetic theory (KT) framework was applied
to obtain the ETC for 1D nanoparticle chains [46, 47]. Recently, the
limitations of KT framework to describe NFRHT was analyzed systematically: 1)
the KT framework is not suitable for materials with resonant modes outside the
Planck's window (e.g., metal Ag) and 2) the KT framework cannot be applied
directly to 2D and 3D systems due to the lack of dispersion relations for
these systems [48, 49]. Most recently, a new method based on the many-body
radiative heat transfer theory and the Fourier law (MF method) was proposed by
Tervo et al. [50] to obtain the ETC for arbitrary nanoparticle collections
allowing the comparison between different materials and different heat
transfer modes.

Due to the limitation of the KT framework describing ETC for nanoparticle
ensembles NFRHT, the investigation on ETC for materials supporting resonances
outside the Planck's window is still missing. Furthermore, VO2 attracts lots
of interests because of its special insulator-metal transition behavior around
its phase transition temperature. Besides, there is still lack of the
investigation on the ETC for phase-change VO2 nanoparticle ensembles,
especially for its metallic phase. Based on the promising phase-changing
characteristics for VO2, many potential applications were proposed recently:
1) near-field applications (e.g., the radiative thermal rectifier [51, 52],
thermal transistor [53], conductive thermal diode [54], dynamic radiative
cooling [55] and the scalable radiative thermal logic gates [56]); 2) far-
field applications (e.g., radiative thermal memristor [57] and radiative
thermal rectifier [58]). Hence, it's worth analyzing the effects of the phase
change on the ETC for VO2 nanoparticle ensembles, in addition to the phase-
change effect on the thermal conductance reported very recently [21].

Nanoparticles in a dense particulate system often lie in the near field of
each other, which results in the many-body interaction (MBI) making the NFRHT
mechanism more complex [24, 50, 59]. Though the complex MBI effects on the
radiative heat flux and thermal conductance for various nanoparticle systems
(three-nanoparticle system [24, 60, 61], 1D nanoparticle chains [28], 2D
nanoparticle ensembles [21, 23, 27], 3D nanoparticle ensembles [6, 19, 20])
have been analyzed, the MBI effect on the ETC is still missing.

We extract simple 1D nanoparticle chains from realistic 3D nanoparticle
ensembles embedded in a non-absorbing host medium and focus on the thermal
property (i.e., effective thermal conductivity) describing and characterizing
the NFRHT from the point of view of continuum. The relative permittivity of
the host medium significantly affects the inter-particle couplings [49, 59].
Effects of the host medium relative permittivity on the ETC of closely spaced
1D metallic nanoparticle chains have already been analyzed [46]. However,
effects of the host medium relative permittivity on the ETC for the specific
case of phase-change VO2 nanoparticles have not yet been investigated.

We address the aforementioned missing points in this paper, where the ETC for
the 1D nanoparticle chains of interest is obtained by means of the MF method.
In Sec. 2, we give a brief description of the theoretical models for the MF
method, as well as the formulas concerning the ETC for 1D nanoparticle chains.
In Sec. 3, effects of the phase change of the insulator-metal transition
material (i.e., VO2), complex many-body interaction and host medium relative
permittivity ($\epsilon_{\rm m}$) on the effective thermal conductivity due to
NFRHT are analyzed. The optical properties for the materials used in this work
are also given in this section.

##  2 Models

In this section, we describe in brief the physical system (the schematic is
shown in Fig. 1) and the theoretical aspects of the MF method for the ETC due
to NFRHT in 1D nanoparticle chains. The nanoparticle chain is divided into two
parts $L$ and $R$ by an imaginary plane.

Figure 1: Schematic of the ordered nanoparticle chain embedded in a non-
absorbing host medium with permittivity $\epsilon_{\rm m}$. The radiative heat
flux in the chain is the sum of the net heat exchange between nanoparticles
from part $L$ and nanoparticles from part $R$. Parts $L$ and $R$ are separated
by the imaginary surface (dash line). The lattice spacing is $h$. Nanoparticle
radius is $a$. A small linear temperature gradient d$T$/d$x$ along the chain
is assumed. All particles are assumed near thermal equilibrium.

The ETC due to near-field radiative heat transfer is defined as follows:

$k_{eff}=\frac{Q}{A\cdot|\textrm{d}T/\textrm{d}x|},$ (1)

where $A$ is the cross section, $Q$ is the net radiative heat flux in the
chain near thermal equilibrium with a small linear temperature gradient
d$T$/d$x$, which is given as follows [50].

$Q=\sum_{\textrm{i}\in L}\sum_{\textrm{j}\in
R}G_{\textrm{ij}}(T)d_{\textrm{ij}}\left|\frac{\textrm{d}T}{\textrm{d}x}\right|,$
(2)

where $d_{\textrm{ij}}$ is the separation distance center to center between
the nanoparticle i from the part $L$ of the chain and nanoparticle j from the
part $R$ of the chain, $G_{\textrm{ij}}(T)$ is the radiative thermal
conductance between nanoparticles i and j, which yields [23, 26]

$G_{\textrm{ij}}(T)=3\int_{0}^{+\infty}\frac{\textrm{d}\omega}{2\pi}\frac{\partial\Theta(\omega,T)}{\partial
T}\mathcal{T}_{\rm i,j}(\omega),$ (3)

where $\Theta(\omega,T)$ is the mean energy of a harmonic Planck's oscillator,
$\omega$ is the angular frequency, the parameter $\mathcal{T}_{\rm
i,j}(\omega)=\frac{4}{3}\frac{k^{4}}{\epsilon_{\rm m}}{\rm Im}(\chi_{E}^{\rm
i}){\rm Im}(\chi_{E}^{\rm j})\textrm{Tr}(G_{\rm ij}^{EE}G_{\rm
ij}^{EE\dagger})$, $\chi_{E}=\alpha_{E}-\frac{ik^{3}}{6\pi}|\alpha_{E}|^{2}$ ,
$\alpha_{E}$ is the nanoparticle polarizability, $k=\sqrt{\epsilon_{\rm
m}}~{}\omega/c$ is the wave vector in the host medium, $\epsilon_{\rm m}$ is
the host medium relative permittivity, $c$ is the speed of light in vacuum,
the Green 's function $G_{\rm ij}^{EE}$ in the many-particle system naturally
includes the many-body interaction and is the element of the following left
matrix.

$\begin{pmatrix}0&G_{12}^{EE}&\cdots&G_{1N}^{EE}\\\
G_{21}^{EE}&0&\ddots&\vdots\\\ \vdots&\vdots&\ddots&G_{(N-1)N}^{EE}\\\
G_{N1}^{EE}&G_{N2}^{EE}&\cdots&0\end{pmatrix}=\begin{pmatrix}0&G_{0,12}^{EE}&\cdots&G_{0,1N}^{EE}\\\
G_{0,21}^{EE}&0&\ddots&\vdots\\\ \vdots&\vdots&\ddots&G_{0,(N-1)N}^{EE}\\\
G_{0,N1}^{EE}&G_{0,N2}^{EE}&\cdots&0\end{pmatrix}\mathbb{A}^{-1},$ (4)

where $G_{\rm 0,ij}^{EE}=\frac{e^{ikr}}{4\pi
r}\left[\left(1+\frac{ikr-1}{k^{2}r^{2}}\right)\mathbb{I}_{3}+\frac{3-3ikr-k^{2}r^{2}}{k^{2}r^{2}}\hat{\textbf{r}}\otimes\hat{\textbf{r}}\right]$
is the free space Green's function connecting two nanoparticles at
$\textbf{r}_{\rm i}$ and $\textbf{r}_{\rm j}$, $r$ is the magnitude of the
separation vector $\textbf{r}=\textbf{r}_{\rm i}-\textbf{r}_{\rm j}$,
$\hat{\textbf{r}}$ is the unit vector $\textbf{r}/r$, $\mathbb{I}_{3}$ is the
$3\times 3$ identity matrix and the matrix $\mathbb{A}$ including many-body
interactions is defined as

$\mathbb{A}=\mathbb{I}_{3N}-k^{2}\begin{pmatrix}0&\alpha_{E}^{1}G_{0,12}^{EE}&\cdots&\alpha_{E}^{1}G_{0,1N}^{EE}\\\
\alpha_{E}^{2}G_{0,21}^{EE}&0&\ddots&\vdots\\\
\vdots&\vdots&\ddots&\alpha_{E}^{N-1}G_{0,(N-1)N}^{EE}\\\
\alpha_{E}^{N}G_{0,N1}^{EE}&\cdots&\alpha_{E}^{N}G_{0,N(N-1)}^{EE}&0\end{pmatrix},$
(5)

where $\mathbb{I}_{3N}$ is the $3N\times 3N$ identity matrix. Hence, the
effective thermal conductivity will be rearranged as

$k_{eff}=\frac{1}{A}\sum_{\textrm{i}\in L}\sum_{\textrm{j}\in R}G_{\rm
ij}(T)d_{\rm ij}.$ (6)

The effective thermal conductivity $k_{eff}$ can also be expressed as the
frequency integral of the spectral effective thermal conductivity
$k_{\omega}$: $k_{eff}=\int_{0}^{+\infty}k_{\omega}~{}\text{d}\omega$. For
materials (e.g., metal Ag) where the magnetic-magnetic polarized eddy-current
Joule dissipation dominates the radiative heat transfer, rather than the
electric-electric polarized displacement current dissipation, the magnetic
dipole contribution to the radiative heat transfer can be taken into
consideration in the parameter $\mathcal{T}_{\rm i,j}(\omega)$ by the coupled
electric and magnetic dipole approach [6, 21].

##  3 Results and discussion

In this section, the optical properties of phase-change VO2 and polar SiC
nanoparticles are introduced. Effects of the phase change, complex many-body
interaction and host medium relative permittivity on the ETC of the 1D
nanoparticle chains due to the NFRHT are analyzed by means of the MF method.
We consider particles with radius $a=25$ nm forming a chain with 250 elements
in part $L$ and as many in part $R$. This is large enough to reach convergent
results for all the calculations of interest considered here [49].

###  3.1 Dielectric function and polarizability of nanoparticles

VO2 is a kind of phase-change materials, which undergoes an insulator-metal
transition around 341 K (phase transition temperature). Below 341 K, VO2 is an
uniaxial anisotropic insulator, of which the dielectric function can be
described by the following tensor [51, 52, 62]:

$\begin{pmatrix}\epsilon_{\rVert}&0&0\\\ 0&\epsilon_{\bot}&0\\\
0&0&\epsilon_{\bot}\end{pmatrix},$ (7)

where $\epsilon_{\bot}$ and $\epsilon_{\rVert}$ are the ordinary and
extraordinary dielectric function component relative to the optical axis,
respectively. Both $\epsilon_{\bot}$ and $\epsilon_{\rVert}$ can be described
by the Lorentz model as follows:

$\epsilon(\omega)=\epsilon_{\infty}+\sum_{n=1}^{N_{L}}\frac{S_{n}\omega_{n}^{2}}{\omega_{n}^{2}-i\gamma_{n}\omega-\omega^{2}},$
(8)

where $S_{n}$, $\omega_{n}$ and $\gamma_{n}$ are the phonon strength, phonon
frequency and damping coefficient of the $n^{th}$ phonon mode. $N_{L}$ is the
number of phonon modes ($N_{L}=8$ for $\epsilon_{\bot}$ and $N_{L}=9$ for
$\epsilon_{\rVert}$). All the necessary parameters for both $\epsilon_{\bot}$
and $\epsilon_{\rVert}$ can be found in Ref.[62]. Above 341K, the dielectric
function of the metallic-phase VO2 is described by the Drude model as follows
[51, 52, 62]:

$\epsilon(\omega)=-\epsilon_{\infty}\frac{\omega_{p}^{2}}{\omega^{2}+i\omega\gamma},$
(9)

where $\epsilon_{\infty}=9$, $\omega_{p}$ = $1.51\times 10^{15}$ rad$\cdot$s-1
and $\gamma$ =$1.88\times 10^{15}$ rad$\cdot$s-1. In addition to the phase-
change VO2, the polar SiC is also used. The dielectric functions of SiC is
described by the Drude-Lorentz model
$\epsilon(\omega)=\epsilon_{\infty}(\omega^{2}-\omega_{l}^{2}+i\gamma\omega)/(\omega^{2}-\omega_{t}^{2}+i\gamma\omega)$
with parameters $\epsilon_{\infty}$ = 6.7, $\omega_{l}$ = 1.827 $\times$ 1014
rad$\cdot$s-1, $\omega_{t}$ = 1.495 $\times$ 1014 rad$\cdot$s-1, and $\gamma$
= 0.9 $\times$ 1012 rad$\cdot$s-1 [63].

For an isotropic material embedded in the host medium with $\epsilon_{\rm m}$,
the polarizability can be obtained from the first order Lorenz-Mie scattering
coefficient [64, 65].

$\alpha_{E}=\frac{i6\pi}{k^{3}}a_{1},$ (10)

where $a_{1}$ is the first order Lorenz-Mie scattering coefficient defined as

$a_{1}=\frac{\epsilon/\epsilon_{\rm m}j_{1}(y)[xj_{1}(x)]\textquotesingle-
j_{1}(x)[yj_{1}(y)]\textquotesingle}{\epsilon/\epsilon_{\rm
m}j_{1}(y)[xh_{1}^{(1)}(x)]\textquotesingle-
h_{1}^{(1)}(x)[yj_{1}(y)]\textquotesingle},$ (11)

where $x=ka$, $y=\sqrt{\epsilon/\epsilon_{\rm m}}ka$, $a$ is the nanoparticle
radius, $\epsilon$ is the relative permittivity,
$j_{1}(x)=\sin(x)/x^{2}-\cos(x)/x$ and
$h_{1}^{(1)}(x)=\textrm{e}^{ix}(1/ix^{2}-1/x)$ are the first order Bessel
function and spherical Hankel function. For 1D nanoparticle chains composed of
many anistropic insulator-phase VO2 nanoparticles, we assume that
nanoparticles’ anisotropic axes are randomly oriented. For this reason, we
decide to use the well-known $1/3-2/3$ description given in Ref. [66] and
consisting of two steps: first calculate polarizability for nanoparticle using
$\epsilon_{\bot}$ and $\epsilon_{\rVert}$ separately, and then add up the
results according to the $1/3-2/3$ rule [21]:

$\alpha_{E}=\frac{1}{3}\alpha_{E}(\epsilon_{\rVert})+\frac{2}{3}\alpha_{E}(\epsilon_{\bot}).$
(12)

The polarizabilities for both insulator-phase and metallic-phase VO2
nanoparticles are shown in Fig. 2. In order to compare the resonance frequency
to the characteristic thermal frequency, the spectral radiance of the
blackbody at 400 K is also added in Fig. 2(b) for reference. The
characteristic thermal frequency mismatches with the polarizability resonance
frequency of metallic VO2 nanoparticle.

(a) Insulator-phase VO2

(b) metallic-phase VO2

Figure 2: Polarizability of VO2 nanoparticle: (a) insulator phase and (b)
metallic phase. Nanoparticle radius $a$ is 25 nm. $\epsilon_{\rm m}=1$. For
insulator VO2 particle, the “$1/3-2/3$” rule is applied to calculate the
polarizability with the help of $\epsilon_{\rVert}$ and $\epsilon_{\bot}$
[66]. The spectral radiance of the blackbody at 400 K is also added for
reference.

For the metallic-phase VO2, the spectral thermal conductance $G_{\omega}$
between two metallic-phase VO2 nanoparticles with a separation 500 nm center
to center is shown in Fig. 3. Nanoparticle radius $a$ is 20 nm. Temperature
$T$ is 300 K. All the four contributions, i.e., electric-electric polarized
displacement current dissipation (EE), magnetic-electric polarized eddy-
current Joule dissipation contribution (ME contribution), electric-magnetic
polarized displacement current dissipation (EM contribution) and magnetic-
magnetic polarized eddy-current Joule dissipation (MM) are calculated by the
coupled electric and magnetic dipole approach [21, 25]. Unlike the simple
metal Ag, it is still the EE contribution that dominates the total thermal
conductance, rather than the MM contribution, as shown in the Fig. 3. Hence,
when calculating radiative thermal conductance between phase-change VO2
nanoparticles even in metallic phase, we can consider only the electric dipole
and neglect the magnetic dipole safely.

Figure 3: Spectral thermal conductance between two metallic-phase VO2
nanoparticles, $G_{\omega}$, due to EE, EM, ME and MM contribution,
respectively. Nanoparticle radius $a$ is 20 nm, temperature $T$ is 300 K.
Separation between the two nanoparticle is 500 nm center to center.

###  3.2 Effect of the phase change on ETC

The ETC of the ordered nanoparticle chains (as shown in Fig. 1) as a function
of temperature $T$ is shown in Fig. 4. Here $a$ = 25 nm and $h$ = 75 nm. SiC
and insulator-metal phase-change VO2 nanoparticles chains are embedded in a
host medium with $\epsilon_{\rm m}=5$. The temperature $T$ ranges from 300 K
to 500 K, including the transition temperature of the VO2.

Figure 4: Effective thermal conductivity $k_{eff}$ of nanoparticle chains as a
function of temperature. Phase-change VO2 and polar SiC are considered.
Nanoparticle radius $a$ = 25 nm. Lattice spacing $h$ =75 nm. $\epsilon_{\rm
m}=5$.

For the non-phase-change SiC nanoparticle chain, the ETC increases
monotonically with temperature. While for the insulator-metal phase-change VO2
nanoparticle chain, an obvious transition of the ETC can be observed around
the transition temperature of the VO2. In the temperature range of interest,
the ETC for the metallic-phase VO2 nanoparticle chain at high temperature is
even much lower than that of the insulator-phase VO2 nanoparticle chain at low
temperature, which is due to the insulator-metal phase change of VO2. As shown
in Fig. 2(b), an obvious mismatch between the resonance frequency of the
metallic-phase VO2 nanoparticle and the characteristic thermal frequency
(Planck's window), which accounts for the low ETC. However, for the insulator-
phase VO2 nanoparticles, the resonance frequency matches well with the
characteristic thermal frequency as shown in Fig. 2(a), which accounts for the
high ETC.

To give a quantitative description on the phase change effect, the dependence
of the ETC on $h/a$ is shown in Fig. 5. SiC (300K), insulator-phase VO2 (300K)
and metall-phase VO2 (400K) nanoparticles chains in vacuum are considered. The
dependence of the ratio of $k_{eff}^{i}$ (the ETC for insulator-phase VO2) to
$k_{eff}^{m}$ (the ETC for metallic-phase VO2) on $h/a$ is also shown in Fig.
5. In general, the ratio $k_{eff}^{i}/k_{eff}^{m}$ is much larger than unity.
The ETC for the insulator-phase VO2 nanoparticle chains is much larger than
that of the metallic-phase VO2 nanoparticle chains. The ratio
$k_{eff}^{i}/k_{eff}^{m}$ increases to its maximum (around 50) and then
decreases with increasing $h/a$. The phase change effect is significant when
the chain is compact and decreases when the chain goes dilute. In addition,
the ETC decreases with increasing $h/a$. The inter-particle coupling decreases
when the lattice spacing of the nanoparticle chain $h$ increases. The
decreasing coupling accounts for the decreasing ETC when increasing $h/a$.

Figure 5: Dependence of the ETC on $h/a$. SiC (300K), insulator-phase VO2
(300K) and metallic-phase VO2 (400K) nanoparticles chains in vacuum are
considered. The dependence of the ratio of $k_{eff}^{i}$ (the ETC for
insulator-phase VO2) to $k_{eff}^{m}$ (the ETC for metallic-phase VO2) on
$h/a$ is also added. Nanoparticle radius $a=25$ nm.

###  3.3 Effect of the many-body interaction on ETC

To evaluate the effects of the many-body interaction on the ETC, we define the
following parameter [6]:

$\varphi=\frac{k_{eff}}{k_{eff}^{0}},$ (13)

where $k_{eff}$ is the ETC evaluated with the help of the radiative thermal
conductance in Eq.(3) and the Green's function including the MBI in Eq.(4),
$k_{eff}^{0}$ is the ETC without the MBI evaluated with the help of the Eq.(3)
and the free space Green's function. Generally speaking, the MBI inhibits the
ETC when $\varphi<1$, enhances it when $\varphi>1$ and can be neglected safely
when $\varphi\approx 1$.

We quantitatively evaluate the MBI effect on the ETC for polar SiC, metallic-
phase VO2 and insulator-phase VO2 nanoparticle chains. The dependence of the
parameter $\varphi$ defined by Eq.(13) on the geometrical dimensionless
parameter $h/a$ is shown in Fig. 6. Nanoparticles of three different sizes
have been considered $a$ = 5 nm, 25 nm and 50 nm. As shown in Fig. 6,
$\varphi$ is never less than unity, which indicates that the MBI does not
inhibit the ETC for the chains composed of the considered materials. When
$h/a>8$, $\varphi$ starts to approach unity. The MBI decreases with the
increasing lattice spacing $h$. When $h/a<8$, $\varphi>1$. The MBI is
favorable to the ETC. Small lattice spacing (i.e., $h/a<8$) results in strong
inter-particles couplings, which resulting in a significant MBI.

Figure 6: The dependence of the ratio $\varphi$ of the effective thermal
conductivity $k_{eff}$ of nanoparticle chains with the MBI to that without MBI
on $h/a$. Insulator-phase VO2, metallic-phase VO2 and polar SiC are
considered. Nanoparticles of three different sizes have been considered $a$ =
5 nm, 25 nm and 50 nm. $\epsilon_{\rm m}=1$.

The maximal $\varphi$ for SiC nanoparticle chains is around 2. The MBI can
double the ETC for SiC chains. For SiC nanoparticle chains, an extremum value
for $\varphi$ was observed at $h/a\approx 3.5$, of which the insight is still
unclear and remains to be explored in the future. It's worth mentioning that a
similar extremum value for ratio of radiative heat flux between two
nanoparticles with insertion of a third nanoparticle to that of the two
isolated nanoparticles when increasing the distance between the two
nanoparticles has already been reported [24]. It's also observed in Fig. 6
that $\varphi$ does not change with varying the nanoparticle size. That is to
say that the MBI is independent of the nanoparticle size. $\varphi$ for the
metallic-phase VO2 and insulator-phase VO2 nanoparticle chains is similar to
each other, though $\varphi$ for the metallic-phase VO2 chains is a little bit
larger than that of the insulator-phase VO2 chains.

###  3.4 Effect of the host medium relative permittivity on ETC

The dependence of total ETC on the host medium permittivity $\epsilon_{\rm m}$
for insulator-metal phase-change VO2 and SiC is show in Fig. 7. Nanoparticles
with radius $a$ = 25 nm and $h=75$ nm are used. ETC increases with the host
medium relative permittivity $\epsilon_{\rm m}$ for both insulator-metal
phase-change VO2 and SiC. The host medium relative permittivity significantly
affects the inter-particle coupling, which finally significantly affects the
ETC. High relative permittivity is favorable to enhancing the ETC and
radiative heat transfer in the nanoparticle chain, which is consistent with
the reported results for closely spaced metallic nanoparticle chains [46]. In
addition, at low relative permittivity $\epsilon_{\rm m}$, the difference
between the ETC for the VO2 nanoparticle chains at different phases (i.e.,
metallic phase and insulator phase) is much smaller than that at high relative
permittivity, of which the insight will be analyzed in the following from the
thermal conductivity spectrum standpoint.

Figure 7: The dependence of ETC on the host medium permittivity $\epsilon_{\rm
m}$ for insulator-metal phase-change VO2 and SiC. Insulator-phase VO2 (300 K),
metallic-phase VO2 (400 K) and SiC (300 K) are considered. Nanoparticle radius
$a$ = 25 nm. $h=75$ nm.

The dependence of the spectral effective thermal conductivity $k_{\omega}$ on
the angular frequency $\omega$ and the relative permittivity of the host
medium $\epsilon_{\rm m}$ is shown in Fig. 8: (a) insulator-phase VO2 (300 K)
and (b) metallic-phase VO2 (400 K) ($a=25$ nm and $h=75$ nm). For a fixed
angular frequency, the value of the spectral effective thermal conductivity
increases significantly with increasing $\epsilon_{\rm m}$, which is
consistent with the dependence of total ETC on $\epsilon_{\rm m}$, as shown in
Fig. 7. Increasing the relative permittivity is in favor of enhancing the
radiative effective thermal conductivity. For metallic-phase VO2 nanoparticle
chains, the frequency peak of the spectral effective thermal conductivity
corresponds to the characteristic thermal frequency, as shown in Fig. 8(b).
However, for insulator-phase VO2 nanoparticle chains, besides the peak of the
spectral effective thermal conductivity corresponding to the characteristic
thermal frequency, there are several secondary peaks, as shown in Fig. 8(a).
It's worthwhile to mention that the peak of the spectral effective thermal
conductivity shows a red-shift behavior with the increase of relative
permittivity $\epsilon_{\rm m}$ for both insulator-phase and metallic-phase
VO2, as can be seen from Fig. 8.

(a) Insulator-phase VO2

(b) metallic-phase VO2

Figure 8: Dependence of the spectral effective thermal conductivity
$k_{\omega}$ on the $\omega$ and relative permittivity of the host medium
$\epsilon_{m}$: (a) insulator phase (300 K) and (b) metallic phase (400 K).
Nanoparticle radius $a$ is 25 nm. $h=75$ nm.

From the formulas for the ETC (i.e., Eq. (6) combined with the Eq. (3)), the
polarizability for the nanoparticle plays a significant role in determining
the ETC for the nanoparticle chains. To understand the insight of the red-
shift behavior of the peaks with increasing the relative permittivity
$\epsilon_{\rm m}$ and different spectral behaviors of the effective thermal
conductivity for the insulator-phase VO2 and metallic-phase VO2 nanoparticle
chains, the polarizability of the VO2 nanoparticles embedded in the host
medium with several different relative permittivities $\epsilon_{\rm m}$ is
given in Fig. 9 (a) for insulator-phase VO2 nanoparticles and (b) for
metallic-phase VO2 nanoparticles with the following parameters. Nanoparticle
radius $a$ = 25 nm. Relative permittivity $\epsilon_{\rm m}$= 1, 3, 5 and 7,
respectively. The increasing directions of the $\epsilon_{\rm m}$, as well as
the angular frequency corresponding to the main peaks, are also added for
reference.

(a) Insulator-phase VO2

(b) metallic-phase VO2

Figure 9: Polarizability of VO2 nanoparticle at several different relative
permittivities $\epsilon_{\rm m}$ : (a) insulator phase and (b) metallic
phase. Nanoparticle radius $a$ is 25 nm. Relative permittivity $\epsilon_{\rm
m}$= 1, 3, 5 and 7, respectively. The increasing directions of the
$\epsilon_{\rm m}$, as well as the angular frequency corresponding to the main
peaks, are added for reference.

As shown in Fig. 9, the peak of the polarizability shows an obvious red-shift
behavior for both the insulator-phase VO2 and metallic-phase VO2
nanoparticles, which accounts for the red-shift behaviors of the peaks for the
spectral effective thermal conductivity as shown in Fig. 8. This red-shift
behavior of the peaks for the polarizability with increasing $\epsilon_{\rm
m}$ results in the increasing match between the peak frequency of the
polarizability and the characteristic thermal frequency (Planck's window, as
shown in Fig. 2(b)), which accounts for the increasing total ETC with
increasing $\epsilon_{\rm m}$ as shown in Fig. 7.

The main reason for the observed increasing ETC of the VO2 chain in the
metallic phase when changing $\epsilon_{\rm m}$ is that the resonance is red-
shifted so that the heat flux is enhanced in the infra-red regime by the low
frequency side of the resonance. This red-shift for the resonances is also
known for simple metallic (Drude model
$\epsilon=\epsilon_{\infty}-\omega_{p}^{2}/{\omega}^{2}$) nanoparticles where
the localized resonances are given by
$\omega_{p}/\sqrt{\epsilon_{\infty}+2\epsilon_{\rm m}}$. For the metallic-
phase VO2 nanoparticle chain, we also can observe a competition between the
following two processes: (1) the decreasing peak value of the metallic-phase
VO2 nanoparticle polarizability with increasing $\epsilon_{\rm m}$ and (2) the
increasing match degree between the peak frequency of the polarizability and
the characteristic thermal frequency (Planck's window) with increasing
$\epsilon_{\rm m}$. As shown in Fig. 7, the total ETC increases with
increasing the $\epsilon_{\rm m}$, therefore, the match degree between the
peak frequency of the polarizability and the characteristic thermal frequency
(Planck's window) is the influencing factor prior to the exact value of the
polarizability peak.

As shown in Fig. 7, for the materials supporting coupled resonant modes in the
infra-red regime (i.e., SiC and metallic-phase VO2), increasing the host
medium permittivity will significantly enhance the ETC as compared to the
materials supporting no coupled resonant modes in the infra-red regime. The
main reason is that the resonances modes in the infra-red regime are
significantly enhanced when increasing the host medium permittivity, which can
be characterized by the increased propagation length defined as $\frac{{\rm
d}{\rm Re}(\omega)}{{\rm d}k}\frac{1}{2{\rm Im}(\omega)}$ [48]. We take the
simple dielectric SiC as an example to show the effect of the permittivity on
the propagation length of coupled modes in infra-red regime. The dependence of
the propagation length on the angular frequency for the SiC nanoparticle chain
is shown in Fig. 10. Nanoparticle radius $a=$ 25 nm. Lattice spacing $h=$ 75
nm. Two different host medium permittivities are considered, $\epsilon_{\rm
m}=1$ and 4, respectively. When increasing the permittivity of the host
medium, the propagation length increases significantly. Hence more energy can
be transported and the ETC is significantly enhanced by the increased coupled
resonant modes in infra-red regime. In addition, the red-shift of the
corresponding resonant frequency of the coupled modes also can be observed in
the Fig. 10, when increasing the host medium permittivity $\epsilon_{\rm m}$.

Figure 10: The propagation length of the SiC nanoparticle chain. Two host
medium permittivities are considered, $\epsilon_{\rm m}=1$ and 4,
respectively. Nanoparticle radius $a$ = 25 nm. Lattice spacing $h=75$ nm. The
symbols $\rVert$ and $\bot$ are corresponding to the parallel and
perpendicular polarizations, respectively.

##  4 Conclusion

Near-field radiative heat transfer for 1D nanoparticle chains embedded in a
non-absorbing host medium is investigated from the point view of the continuum
by means of the MF method combining the many-body radiative heat transfer
theory and the Fourier law together. Effects of the phase change of materials,
complex many-body interaction and host medium relative permittivity on the
effective thermal conductivity ETC are analyzed.

The value of the ETC for VO2 nanoparticle chains below the transition
temperature can reach around 50 times that above the transition temperature
due to the phase change effect. The strong coupling in the insulator-phase VO2
nanoparticle chain accounts for its high ETC as compared to the low ETC for
the metallic-phase VO2 nanoparticle chain, where there is a mismatch between
the characteristic thermal frequency and the polarizability resonance
frequency.

Strong MBI is in favor of the ETC. For dense chains (the ratio of the lattice
spacing to nanoparticle radius $h/a<8$), the MBI enhances the ETC, which is
due to the strong coupling in the dense chains. When the chains go more and
more dilute ($h/a>8$), the MBI can be neglected safely, which is due to
negligible coupling. It's worthwhile to mention that for the SiC nanoparticle
chain the MBI can even double the ETC of the chain as compared to that without
considering MBI. It is still remaining unknown whether there are some other
materials in nature supporting an even larger MBI effect on ETC than SiC.

The host medium relative permittivity significantly affects the inter-particle
coupling, which accounts for the permitivity-dependent ETC for the VO2
nanoparticle chains. For the materials supporting no resonant modes in the
infra-red regime (e.g., simple metal or metallic-phase VO2), the red-shift
behavior of the peaks for the polarizability with increasing $\epsilon_{\rm
m}$ results in the increasing degree of match between the peak frequency of
the polarizability and the characteristic thermal frequency (Planck's window),
which accounts for increasing the total ETC with increasing $\epsilon_{\rm
m}$. For the materials supporting resonant modes in the infra-red regime
(e.g., simple dielectric SiC and insulator-phase VO2), more energy can be
transported and then ETC can be enhanced due to the increased coupled
resonance when increasing $\epsilon_{\rm m}$.

It is noted that the topological phase transition has a significant effect on
the radiative heat flux through the 1D chain of nanoparticles [67]. However,
the effect of such topological phase transition on the thermal transport
characteristics (e.g., ETC) still remains unclear and needs to be investigated
in the future.

## Acknowledgements

The support of this work by the National Natural Science Foundation of China
(No. 51976045) is gratefully acknowledged. M.A. acknowledges support from the
Institute Universitaire de France, Paris, France (UE). M.G.L. also thanks for
the support from China Scholarship Council (No.201906120208).

## References

  * [1] S.-A. Biehs, R. Messina, P. S. Venkataram, A. W. Rodriguez, J. C. Cuevas, P. Ben-Abdallah, Near-field radiative heat transfer in many-body systems (2020).  arXiv:2007.05604.
  * [2] P. Ben-Abdallah, Multitip near-field scanning thermal microscopy, Phys. Rev. Lett. 123 (2019) 264301.  doi:10.1103/PhysRevLett.123.264301.
  * [3] J. DeSutter, L. Tang, M. Francoeur, A near-field radiative heat transfer device, Nat. Nanotechnol. 14 (8) (2019) 751–755.  doi:10.1038/s41565-019-0483-1.
  * [4] W. Shen, J. Y. Xiao, Y. Wang, S. H. Su, J. C. Guo, J. C. Chen, Intermediate-band-assisted near-field thermophotovoltaic devices with InAs, GaSb, and Si based absorbers, J Appl. Phys. 128 (3) (2020) 035105\.  doi:10.1063/5.0010965.
  * [5] R. St-Gelais, B. Guha, L. X. Zhu, S. H. Fan, M. Lipson, Demonstration of strong near-field radiative heat transfer between integrated nanostructures, Nano Lett. 14 (12) (2014) 6971–6975.  doi:10.1021/nl503236k.
  * [6] M. G. Luo, J. Dong, J. M. Zhao, L. H. Liu, M. Antezza, Radiative heat transfer between metallic nanoparticle clusters in both near field and far field, Phys. Rev. B 99 (2019) 134207.  doi:10.1103/PhysRevB.99.134207.
  * [7] S. M. Rytov, Y. A. Kravtsov, V. I. Tatarskii, Priniciples of statistical radiophysics, Vol. 3, Springer-Verlag, 1989.
  * [8] D. Polder, M. Van Hove, Theory of radiative heat transfer between closely spaced bodies, Phys. Rev. B 4 (10) (1971) 3303–3314.  doi:10.1103/PhysRevB.4.3303.
  * [9] A. I. Volokitin, B. N. J. Persson, Resonant photon tunneling enhancement of the radiative heat transfer, Phys. Rev. B 69 (2004) 045417.  doi:10.1103/PhysRevB.69.045417.
  * [10] A. Narayanaswamy, G. Chen, Thermal near-field radiative transfer between two spheres, Phys. Rev. B 77 (7) (2008) 075125.
  * [11] Z. H. Zheng, Y. M. Xuan, Theory of near-field radiative heat transfer for stratified magnetic media, Int. J Heat Mass Transf. 54 (5) (2011) 1101 – 1110\.  doi:10.1016/j.ijheatmasstransfer.2010.11.012.
  * [12] X. L. Liu, R. Zhang, Z. M. Zhang, Near-field radiative heat transfer with doped-silicon nanostructured metamaterials, Int. J Heat Mass Transf. 73 (2014) 389 – 398.  doi:https://doi.org/10.1016/j.ijheatmasstransfer.2014.02.021.
  * [13] M. Lim, J. Song, S. S. Lee, J. Lee, B. J. Lee, Surface-plasmon-enhanced near-field radiative heat transfer between planar surfaces with a thin-film plasmonic coupler, Phys. Rev. Applied 14 (2020) 014070.  doi:10.1103/PhysRevApplied.14.014070.
  * [14] P. Sabbaghi, L. S. Long, X. Y. Ying, L. Lambert, S. Taylor, C. Messner, L. P. Wang, Super-planckian radiative heat transfer between macroscale metallic surfaces due to near-field and thin-film effects, J Appl. Phys. 128 (2) (2020) 025305.  doi:10.1063/5.0008259.
  * [15] M. Ghashami, H. Geng, T. Kim, N. Iacopino, S. K. Cho, K. Park, Precision measurement of phonon-polaritonic near-field energy transfer between macroscale planar structures under large thermal gradients, Phys. Rev. Lett. 120 (2018) 175901.  doi:10.1103/PhysRevLett.120.175901.
  * [16] B. Song, Y. Ganjeh, S. Sadat, D. Thompson, A. Fiorino, V. Fernández-Hurtado, J. Feist, F. J. Garcia-Vidal, J. C. Cuevas, P. Reddy, E. Meyhofer, Enhancement of near-field radiative heat transfer using polar dielectric thin films, Nat. Nanotechnol. 10 (2015) 253.  doi:10.1038/nnano.2015.6.
  * [17] S. Shen, A. Narayanaswamy, G. Chen, Surface phonon polaritons mediated energy transfer between nanoscale gaps, Nano Lett. 9 (8) (2009) 2909–2913.  doi:10.1021/nl901208v.
  * [18] E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, J.-J. Greffet, Radiative heat transfer at the nanoscale, Nat. Photonics 3 (9) (2009) 514–517.  doi:10.1038/nphoton.2009.144.
  * [19] J. Dong, J. M. Zhao, L. H. Liu, Near-field radiative heat transfer between clusters of dielectric nanoparticles, J Quant. Spectrosc. Radiat. Transf. 197 (2017) 114 – 122.  doi:10.1016/j.jqsrt.2016.10.015.
  * [20] J. Chen, C. Y. Zhao, B. X. Wang, Near-field thermal radiative transfer in assembled spherical systems composed of core-shell nanoparticles, J Quant. Spectrosc. Radiat. Transf. 219 (2018) 304 – 312.  doi:10.1016/j.jqsrt.2018.08.024.
  * [21] M. G. Luo, J. M. Zhao, L. H. Liu, M. Antezza, Radiative heat transfer and radiative thermal energy for two-dimensional nanoparticle ensembles, Phys. Rev. B 102 (2020) 024203.  doi:10.1103/PhysRevB.102.024203.
  * [22] A. D. Phan, T.-L. Phan, L. M. Woods, Near-field heat transfer between gold nanoparticle arrays, J. Appl. Phys. 114 (21) (2013) 214306.  doi:10.1063/1.4838875.
  * [23] M. G. Luo, J. M. Zhao, M. Antezza, Near-field radiative heat transfer between twisted nanoparticle gratings, Appl. Phys. Lett. 117 (5) (2020) 053901.  doi:10.1063/5.0018329.
  * [24] P. Ben-Abdallah, S.-A. Biehs, K. Joulain, Many-body radiative heat transfer theory, Phys. Rev. Lett. 107 (2011) 114301.  doi:10.1103/PhysRevLett.107.114301.
  * [25] J. Dong, J. M. Zhao, L. H. Liu, Radiative heat transfer in many-body systems: Coupled electric and magnetic dipole approach, Phys. Rev. B 95 (2017) 125411.  doi:10.1103/PhysRevB.95.125411.
  * [26] P. Ben-Abdallah, R. Messina, S.-A. Biehs, M. Tschikin, K. Joulain, C. Henkel, Heat superdiffusion in plasmonic nanostructure networks, Phys. Rev. Lett. 111 (2013) 174301.  doi:10.1103/PhysRevLett.111.174301.
  * [27] M. Nikbakht, Radiative heat transfer in fractal structures, Phys. Rev. B 96 (2017) 125436.  doi:10.1103/PhysRevB.96.125436.
  * [28] M. G. Luo, J. M. Zhao, L. H. Liu, Near-field radiative heat transfer in a chain of nanoparticles with another chain in proximity, J Quant. Spectrosc. Radiat. Transf. 243 (2020) 106801.  doi:10.1016/j.jqsrt.2019.106801.
  * [29] I. Latella, P. Ben-Abdallah, Giant thermal magnetoresistance in plasmonic structures, Phys. Rev. Lett. 118 (2017) 173902.  doi:10.1103/PhysRevLett.118.173902.
  * [30] P. Doyeux, R. Messina, B. Leggio, M. Antezza, Excitation injector in an atomic chain: Long-range transport and efficiency amplification, Phys. Rev. A 95 (2017) 012138.  doi:10.1103/PhysRevA.95.012138.
  * [31] P. Doyeux, S. A. H. Gangaraj, G. W. Hanson, M. Antezza, Giant interatomic energy-transport amplification with nonreciprocal photonic topological insulators, Phys. Rev. Lett. 119 (2017) 173901.  doi:10.1103/PhysRevLett.119.173901.
  * [32] B. Leggio, R. Messina, M. Antezza, Thermally activated nonlocal amplification in quantum energy transport, Europhys. Lett. 110 (4) (2015) 40002.  doi:10.1209/0295-5075/110/40002.
  * [33] R. Messina, M. Antezza, Casimir-lifshitz force out of thermal equilibrium and heat transfer between arbitrary bodies, Europhys. Lett. 95 (6) (2011) 61002.  doi:10.1209/0295-5075/95/61002.
  * [34] R. Messina, M. Antezza, Three-body radiative heat transfer and Casimir-Lifshitz force out of thermal equilibrium for arbitrary bodies, Phys. Rev. A 89 (5) (2014) 052104.  doi:10.1103/PhysRevA.89.052104.
  * [35] M. Krüger, G. Bimonte, T. Emig, M. Kardar, Trace formulas for nonequilibrium casimir interactions, heat radiation, and heat transfer for arbitrary objects, Phys. Rev. B 86 (2012) 115423.  doi:10.1103/PhysRevB.86.115423.
  * [36] B. Müller, R. Incardone, M. Antezza, T. Emig, M. Krüger, Many-body heat radiation and heat transfer in the presence of a nonabsorbing background medium, Phys. Rev. B 95 (8) (2017) 085413.  doi:10.1103/PhysRevB.95.085413.
  * [37] S. Edalatpour, M. Francoeur, The Thermal Discrete Dipole Approximation (T-DDA) for near-field radiative heat transfer simulations in three-dimensional arbitrary geometries, J Quant. Spectrosc. Radiat. Transf. 133 (2014) 364 – 373.  doi:10.1016/j.jqsrt.2013.08.021.
  * [38] S. Edalatpour, M. Čuma, T. Trueax, R. Backman, M. Francoeur, Convergence analysis of the thermal discrete dipole approximation, Phys. Rev. E 91 (2015) 063307.  doi:10.1103/PhysRevE.91.063307.
  * [39] S. Edalatpour, J. DeSutter, M. Francoeur, Near-field thermal electromagnetic transport: An overview, J Quant. Spectrosc. Radiat. Transf. 178 (2016) 14 – 21\.  doi:10.1016/j.jqsrt.2015.12.027.
  * [40] A. W. Rodriguez, M. T. H. Reid, S. G. Johnson, Fluctuating-surface-current formulation of radiative heat transfer for arbitrary geometries, Phys. Rev. B 86 (2012) 220302.  doi:10.1103/PhysRevB.86.220302.
  * [41] M. T. H. Reid, J. White, S. G. Johnson, Computation of casimir interactions between arbitrary three-dimensional objects with arbitrary material properties, Phys. Rev. A 84 (2011) 010503.  doi:10.1103/PhysRevA.84.010503.
  * [42] A. W. Rodriguez, O. Ilic, P. Bermel, I. Celanovic, J. D. Joannopoulos, M. Soljačić, S. G. Johnson, Frequency-selective near-field radiative heat transfer between photonic crystal slabs: A computational approach for arbitrary geometries and materials, Phys. Rev. Lett. 107 (2011) 114302.  doi:10.1103/PhysRevLett.107.114302.
  * [43] A. Datas, D. Hirashima, K. Hanamura, Fdtd simulation of near-field radiative heat transfer between thin films supporting surface phonon polaritons: Lessons learned, J Therm. Sci. Technol. 8 (1) (2013) 91–105.  doi:10.1299/jtst.8.91.
  * [44] A. Didari, M. P. Mengüç, Analysis of near-field radiation transfer within nano-gaps using fdtd method, J Quant. Spectrosc. Radiat. Transf. 146 (2014) 214 – 226.  doi:10.1016/j.jqsrt.2014.04.002.
  * [45] B. Czapla, A. Narayanaswamy, Thermal radiative energy exchange between a closely-spaced linear chain of spheres and its environment, J Quant. Spectrosc. Radiat. Transf. 227 (2019) 4 – 11.  doi:10.1016/j.jqsrt.2019.01.020.
  * [46] P. Ben-Abdallah, K. Joulain, J. Drevillon, C. Le Goff, Heat transport through plasmonic interactions in closely spaced metallic nanoparticle chains, Phys. Rev. B 77 (2008) 075417.  doi:10.1103/PhysRevB.77.075417.
  * [47] E. Tervo, O. S. Adewuyi, J. S. Hammonds, B. Cola, High thermal conductivity in polaritonic SiO2 nanoparticle beds, Mater. Horiz. 3 (2016) 434–441.  doi:10.1039/C6MH00098C.
  * [48] C. Kathmann, R. Messina, P. Ben-Abdallah, S.-A. Biehs, Limitations of kinetic theory to describe near-field heat exchanges in many-body systems, Phys. Rev. B 98 (2018) 115434.  doi:10.1103/PhysRevB.98.115434.
  * [49] E. J. Tervo, B. A. Cola, Z. M. Zhang, Comparison of kinetic theory and fluctuational electrodynamics for radiative heat transfer in nanoparticle chains, J Quant. Spectrosc. Radiat. Transf. 246 (2020) 106947.  doi:10.1016/j.jqsrt.2020.106947.
  * [50] E. Tervo, M. Francoeur, B. Cola, Z. M. Zhang, Thermal radiation in systems of many dipoles, Phys. Rev. B 100 (2019) 205422.  doi:10.1103/PhysRevB.100.205422.
  * [51] P. Zhang, P. P. Yang, Z. H. Zheng, W. Yu, Effect of monolayer graphene on the performance of near-field radiative thermal rectifier between doped silicon and vanadium dioxide, Int. J Heat Mass Transf. 155 (2020) 119707.  doi:10.1016/j.ijheatmasstransfer.2020.119707.
  * [52] Z. H. Zheng, X. L. Liu, A. Wang, Y. M. Xuan, Graphene-assisted near-field radiative thermal rectifier based on phase transition of vanadium dioxide (VO2), Int. J Heat Mass Transf. 109 (2017) 63 – 72.  doi:10.1016/j.ijheatmasstransfer.2017.01.107.
  * [53] P. Ben-Abdallah, S.-A. Biehs, Near-field thermal transistor, Phys. Rev. Lett. 112 (2014) 044301.  doi:10.1103/PhysRevLett.112.044301.
  * [54] J. Ordonez-Miranda, J. M. Hill, K. Joulain, Y. Ezzahri, J. Drevillon, Conductive thermal diode based on the thermal hysteresis of VO2 and nitinol, J Appl. Phys. 123 (8) (2018) 085102.  doi:10.1063/1.5019854.
  * [55] S. Taylor, Y. Yang, L. P. Wang, Vanadium dioxide based fabry-perot emitter for dynamic radiative cooling applications, J Quant. Spectrosc. Radiat. Transf. 197 (2017) 76 – 83.  doi:10.1016/j.jqsrt.2017.01.014.
  * [56] C. Kathmann, M. Reina, R. Messina, P. Ben-Abdallah, S.-A. Biehs, Scalable radiative thermal logic gates based on nanoparticle networks, Sci. Rep. 10 (1) (2020) 3596.  doi:10.1038/s41598-020-60603-4.
  * [57] J. Ordonez-Miranda, Y. Ezzahri, J. A. Tiburcio-Moreno, K. Joulain, J. Drevillon, Radiative thermal memristor, Phys. Rev. Lett. 123 (2019) 025901\.  doi:10.1103/PhysRevLett.123.025901.
  * [58] C. L. Gomez-Heredia, J. A. Ramirez-Rincon, J. Ordonez-Miranda, O. Ares, J. J. Alvarado-Gil, C. Champeaux, F. Dumas-Bouchiat, Y. Ezzahri, K. Joulain, Thermal hysteresis measurement of the VO2 emissivity and its application in thermal rectification, Sci. Rep. 8 (2018).  doi:10.1038/s41598-018-26687-9.
  * [59] E. Tervo, Z. M. Zhang, B. Cola, Collective near-field thermal emission from polaritonic nanoparticle arrays, Phys. Rev. Materials 1 (2017) 015201.  doi:10.1103/PhysRevMaterials.1.015201.
  * [60] Y. H. Wang, J. Z. Wu, Radiative heat transfer between nanoparticles enhanced by intermediate particle, AIP Adv. 6 (2) (2016) 025104.  doi:10.1063/1.4941751.
  * [61] J. L. Song, Q. Cheng, Z. X. Luo, X. P. Zhou, Z. M. Zhang, Modulation and splitting of three-body radiative heat flux via graphene/SiC core-shell nanoparticles, Int. J Heat Mass Transf. 140 (2019) 80 – 87.  doi:10.1016/j.ijheatmasstransfer.2019.05.102.
  * [62] A. S. Barker, H. W. Verleur, H. J. Guggenheim, Infrared optical properties of vanadium dioxide above and below the transition temperature, Phys. Rev. Lett. 17 (1966) 1286–1289.  doi:10.1103/PhysRevLett.17.1286.
  * [63] E. Palik, Handbook of Optical Constants of Solids, Academic, New York, 1998.
  * [64] C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles, John Wiley and Sons, New York, 1983.
  * [65] G. W. Mulholland, C. F. Bohren, K. A. Fuller, Light scattering by agglomerates: Coupled electric and magnetic dipole method, Langmuir 10 (8) (1994) 2533–2546.  doi:10.1021/la00020a009.
  * [66] M. Quinten, Optical properties of nanoparticle systems: Mie and beyond, John Wiley & Sons, 2010.
  * [67] A. Ott, S.-A. Biehs, Radiative heat flux through a topological Su-Schrieffer-Heeger chain of plasmonic nanoparticles, Phys. Rev. B 102 (2020) 115417.  doi:10.1103/PhysRevB.102.115417.


# A Relativistic Gas of Inflatons as an Initial State for Inflation

Lasha Berezhiani  Max-Planck-Institut für Physik, Föhringer Ring 6, 80805
München, Germany  Mark Trodden  Center for Particle Cosmology, Department of
Physics and Astronomy, University of Pennsylvania,
209 South 33rd St, Philadelphia, PA 19104

###### Abstract

The possibility of the resilience of the beginning of inflation under
unfavorable conditions is examined by considering the initial state of the
inflaton field to be in the form of a relativistic gas with some of its
properties in close proximity to the black body spectrum. It is demonstrated
that the initial potential energy budget in such an environment is suppressed
beyond the minimal value required for inflation. This is the extension of our
earlier work, where we have shown that the rare regions which happen to host
favorable initial conditions for the beginning of inflation could come to
dominate the late-time Universe only if they started out from above the so-
called self-reproduction threshold.

## Introduction

Cosmic Inflation provides a mechanism for producing the entire observable
Universe from a single causal patch, many orders of magnitude smaller than the
current Hubble radius Guth:1980zm ; Linde:1981mu ; Albrecht:1982wi . By now it
is straightforward to construct models in which an initial ultra-cold
homogenous (potential energy dominated) and featureless Hubble patch evolves
into a macroscopic universe which is in spectacular agreement with
observations. An interesting question around the inflationary paradigm
concerns the origin of this initial patch. Much effort has been invested in
understanding how much (if any) fine-tuning of the initial conditions is
required for inflation to begin Hartle:1983ai ; Linde:1985ub ;
Brandenberger:1990wu ; Linde:1993xx ; Farhi:1986ty ; Farhi:1989yr ;
Vachaspati:1998dy ; Kaloper:2002cs ; Albrecht:2004ke ; Gibbons:2006pa ;
Hartle:2008ng ; Schiffrin:2012zf ; Ijjas:2013vea ; Guth:2013sya ;
Linde:2014nna ; Ijjas:2014nta ; Carroll:2014uoa ; Mukhanov:2014uwa ;
East:2015ggf ; Kleban:2016sqm ; Berezhiani:2015ola ; Aurrekoetxea:2019fhr ;
Clough:2017efm ; Clough:2016ymm . A tantalizing thought is that inflation is
highly resilient and that it will begin under arbitrary initial conditions.
Barring an additional special mechanism for producing preferable initial
state, there are two possible scenarios for starting inflation in a universe
with generic initial conditions:

  * i)

Even if regions with conditions favouring the beginning of inflation are
initially rare, they are rewarded with an exponential volume factor due to
accelerated expansion. Thus, they still may come to dominate the physical
volume of the universe.

  * ii)

In spite of a given region being highly inhomogeneous, it may evolve into one
capable of seeding inflation if, in the expanding universe, all forms of
energy redshift away quickly, except for the slowly-rolling potential energy.

There are a number of different techniques for investigating these questions,
including, most popularly, simulating given inflationary models with varying
initial conditions in order to understand the evolution towards inflation in
the spirit of option (ii). Recently a significant progress has been made in
this direction East:2015ggf ; Clough:2016ymm ; Clough:2017efm ;
Aurrekoetxea:2019fhr . There one usually considers highly inhomogeneous
initial inflaton profile of the form

$\displaystyle\varphi=\varphi_{0}+\delta\varphi\,.$ (1)

The constant $\varphi_{0}$ is chosen to be capable of giving sufficient
inflation if one were to remove inhomogeneities. The latter component
$\delta\varphi$ is considered to average to zero over the simulation box;
which is chosen to be of the size of the curvature radius sourced by
$\varphi_{0}$ only. The simulations are performed for quite a broad (with
$|\delta\varphi|\gg\phi_{0}$), yet limited, class of classical
inhomogeneities, demonstrating that for such initial conditions inflation
persists. However, if $\varphi_{0}$ is significantly smaller than the above-
mentioned critical value, the same simulations show that the outcome is not as
favourable.

In previous work Berezhiani:2015ola , we have taken a complementary analytic
approach, in which we describe the initial state in terms of constituent
quanta. In order to quantify the probability distribution for the pre-
inflationary universe, one then has to make assumptions about the production
mechanism. We considered the possibility that the inflaton quanta were
produced through high energy processes, and used this technique to explore the
option (i) for the onset of inflation. In this way we demonstrated that
regions smooth enough to seed inflation are quite rare, so that even the
exponential volume factor is insufficient to overcome it, unless one enters
the regime of slow-roll eternal inflation. This underlined the importance of
gaining an improved understanding of this stage of evolution. This approach
was motivated by the corpuscular description of inflation developed in
Dvali:2013eja , where classical field configurations are represented as a
coherent state of inflatons and gravitons. Semi-classically, the quantum
dynamics of a spectator scalar field on a fixed de Sitter background is
considered as a fair proxy for eternal inflation, in which case the question
of the beginning of inflation is rendered far less pressing. However, it has
also been argued Dvali:2013eja  that the fundamental treatment of classical
backgrounds such as quasi-de-Sitter space may well be in terms of the coherent
state built around the well-defined (Minkowski) vacuum. One of the
justifications for this picture is the S-matrix formulation of quantum gravity
Dvali:2020etd . Using this method one concludes that there is a significant
loss of coherence for regions that start out deep within the slow-roll eternal
inflationary regime.

The goal of this paper is to extend the arguments of Berezhiani:2015ola , in
the context of option (ii). Our work is intended to complement the results
obtained through numerical simulations, by assessing whether a sufficiently
homogeneous component $\varphi_{0}$, which is the input for those simulations,
is expected to be present under the assumption that the initial state is in
the form of the relativistic inflaton gas. It is clear from the very beginning
that our assumptions deliberately suppress the initial potential energy
budget, i.e. $\varphi_{0}$. The point we would like to understand is by how
much.

To reiterate the concept, we are probing the resilience of inflation by
pinpointing the types of initial conditions for which its onset is
inefficient. Moreover, we do so by considering the initial state to have
properties that one encounters in actual physical systems. For instance, as is
well known, a gas of relativistic particles (with vanishing chemical
potential) in a black body cavity at temperature $T$ obeys Planck’s
distribution per unit volume

$\displaystyle dn_{k}\propto\frac{k^{2}dk}{e^{k/T}-1}\,,$ (2)

with $\int dn_{k}\propto T^{3}$.

Our proposal is to consider initial states of the inflaton gas having
properties resembling this well known system, so that the expected value of
the initial potential energy budget, and correspondingly $\varphi_{0}$, can be
estimated as the energy density of nonrelativisitic particles. This paper is
dedicated to analyzing the consequences of such assumptions.

## The Basic Idea

For simplicity, let us focus on a simplest model, in which the inflaton is a
non-self-interacting massive scalar field, minimally coupled to
gravity111Although it is disfavoured by observations, this model represents a
convenient playground for exploring theoretical questions.

$\displaystyle\mathcal{L}_{\varphi}=\sqrt{-g}\left(-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi-\frac{1}{2}m^{2}\varphi^{2}\right)\,,$
(3)

with the mass $m$ significantly smaller than the energy and curvature scales
of interest.

The approach proposed in Berezhiani:2015ola , was to treat the initial state
of $\varphi$ as we would treat that of any other particle - namely to consider
its initial configuration as being comprised of $\varphi$-quanta defined
around the true vacuum of the theory, which we assume to be Minkowski space,
setting the Cosmological Constant to zero for simplicity. Obviously, the
collection of such quanta will not always behave as a quasi-ideal gas of on-
shell particles, especially in the setting of the early universe. In fact,
depending on the density and momentum distribution they could be significantly
off-shell in some situations (see e.g. Berezhiani:2016grw ). We will estimate
how many on-shell inflaton particles with certain kinematic properties should
be produced in a given region in order for them to go off-shell and
collectively behave as a desirable classical configuration.

In order to make any useful statement, it is necessary to have some idea about
the mechanism through which the field is produced. We choose to do this by
considering the production of $\varphi$ degrees of freedom through
perturbative high-energy scattering and decay processes of some relativistic
ultra-violet (UV) degrees of freedom. This implies that quanta would be
predominantly produced in high momentum states, and we account for this by
assuming that in a region with energy density $\rho$ it is unlikely for a
given inflaton quantum to be much softer than the characteristic inter-
particle separation. We quantify this suppressed probability as

$\displaystyle p(k<k_{0})=\left(\frac{k_{0}}{E}\right)^{\alpha}\,,$ (4)

with $\alpha\geq 1$, for $k_{0}\ll E\equiv\rho^{1/4}$. Moreover, we assume
that the corresponding probability distribution contains a single energy scale
$E$ and obeys $\langle k\rangle\simeq E$. This behavior is satisfied by the
black body spectrum outlined at the end of the previous section, but is also
true of a more general family of distributions.

The key point here is that we would like to consider an initial state in which
inflatons behave as radiation on average and thus are not predisposed to aid
the beginning of inflation. We then study the resilience of inflation in this
environment; one not designed ahead of time to ensure accelerated stage of
expansion. Notice that (4) leads to the suppression of the homogeneous
component of the field configuration, and correspondingly of the potential
energy budget in the initial state. Taking into account that a priori we have
no information about the pre-inflationary universe, it is vital to look into
such initial states to have a full understanding of the question of robustness
of the paradigm. Such challenges are important for understanding just how
strong of an attractor we are dealing with, and to inform us about what we may
need to further understand about the dynamics of the universe before the
observable stage222This is related to the idea Aurrekoetxea:2019fhr  that
inflation can be a test of fundamental theories not only through its
constraints on dynamical models, but also via constraints on the space of
initial conditions..

For example, it was demonstrated in Berezhiani:2015ola  that within the
corpuscular approach to the initial state (even requiring that the homogeneous
component is suppressed) the problem could be ameliorated if slow-roll could
be extended beyond the self-reproduction threshold. In that case it was not
even necessary to appeal to the potentially infinite inflated volume predicted
by the semi-classical analysis of eternal inflation; classical slow-rolling
was sufficient. See the next section for the overview of this.

## Option (i)

Having discussed the basic idea, we now revisit the steps taken in
Berezhiani:2015ola  to estimate the prior probability for a given region to be
smooth enough to initiate inflation. In order for a region of size $\ell$ to
be well described by a homogeneous configuration of the inflaton field, it
must be dominated by $\varphi$-quanta with wavelength longer than $\ell$.
Assuming $\langle k\rangle_{k\ell<1}\simeq\ell^{-1}$ and $\langle
k\rangle_{k\ell>1}\simeq E$, for the situation of physical interest
$\ell^{-1}\ll E$, one arrives at the following requirement

$\displaystyle N_{k\ell<1}\ell^{-1}\gg
N_{k\ell>1}E\,,\quad\Rightarrow\quad\frac{N_{k\ell<1}}{N_{k\ell>1}}\gg\sqrt{\ell
M_{\rm Pl}}\,.$ (5)

Here, in the last step, we have used $E=\rho^{1/4}$ and have required the
curvature radius to be shorter than the size of the patch. In other words,
most of the $\varphi$-quanta must have wavelengths longer than the size of the
patch in order to be able to treat their configuration as a homogeneous
classical background333Note that $\ell M_{\rm Pl}\gg 1$.. Furthermore, the
energy density should be high-enough that the curvature radius is shorter than
the size of the region. Considering (5), the probability of a given patch
being dominated by soft quanta is well-approximated by the probability for all
quanta within the patch to be soft. Therefore, the probability for a given
patch being capable of seeding inflation with curvature scale $H$ can be
estimated as444Strictly speaking, this is the probability for having a
homogeneous inflaton configuration. In order for most of its energy to be in
the form of slowly-rolling potential energy, the constituent quanta also need
to be non-relativistic. We will come to this point later on.

$\displaystyle\mathcal{P}_{inf}(H)=\mathcal{P}_{N}\cdot
p\left(k<\ell^{-1}\right)^{N}\,,$ (6)

where $N$ stands for the number of quanta required to drive inflation with
curvature scale $H$, within a patch of size $\ell$; $\mathcal{P}_{N}$ is the
probability of having $N$ quanta within the region in question; and
$p(k<\ell)$ is the probability that a given quantum has wavelength longer than
the size of this region (given by (4)). It is straightforward to derive a
lower bound on the particle number, defined as that value $N_{c}$ for which
inflation takes place with curvature radius precisely equal to $\ell$. Using
the Friedmann equation, this value is

$\displaystyle N\geq N_{c}\equiv 3\ell^{2}M_{\rm Pl}^{2}\,.$ (7)

In other words, $N_{c}$ is the number of quanta with average energy
$\ell^{-1}$ per particle, such that the cumulative energy density sources
inflation with curvature radius $\ell$.

Taking into account that $\mathcal{P}_{N}$ may not exceed unity for obvious
reasons, and using (4), we obtain

$\displaystyle\mathcal{P}_{inf}(N\geq N_{c})<\left(\ell E\right)^{-\alpha
N}\,.$ (8)

Since $E$ represents the expected energy of a given quantum, and is set by the
energy density, we have $NE\ell^{-3}=E^{4}$, which allows us to recast the
above expression into the simple form

$\displaystyle\mathcal{P}_{inf}(N\geq N_{c})<N^{-\alpha N/3}\,.$ (9)

Considering that $N\gg 1$, we can conclude that in the initial state the
regions capable of initiating inflation are exponentially rare. However, as we
have already commented, the exponential volume factor they are rewarded with
may be sufficiently large that the late time volume is dominated by inflated
patches. The necessary condition for this to happen is given by

$\displaystyle\mathcal{P}_{inf}e^{3\mathcal{N}}>1\,,$ (10)

with $\mathcal{N}$ denoting the number of e-folds inflation lasted for.
Assuming $\frac{\alpha}{3}{\rm ln}N\sim{\rm few}$, we arrive at

$\displaystyle\mathcal{N}>N\geq N_{c}\,.$ (11)

It is straightforward to show that $\mathcal{N}>N_{c}$ translates into the
following bound on the Hubble scale for the efficient beginning of inflation

$\displaystyle H>H_{*}\equiv\sqrt{mM_{\rm Pl}}\,.$ (12)

Interestingly, $H_{*}$ represents the curvature scale for the onset of the
self-reproduction regime (i.e., slow-roll eternal inflation, in which the
amplitude of quantum fluctuations on horizon scales dominates over the
classical variation of the field-value over the Hubble time.) It is worth
mentioning that, although this regime is reasonable if it is well-described by
the semi-classical picture, there exist arguments suggesting that regions that
begin with such a small slow-roll parameter may cease to be described by semi-
classical physics before reheating Dvali:2013eja ; Berezhiani:2016grw . The
essence of the argument is the following. Semi-classically the inflationary
background pays no price for producing particles such as, for instance,
gravitational waves. However, quantum mechanically it must lose at least one
out of the $N$ quantum constituents of the classical background during
particle production. Therefore, a configuration with $N$ quantum constituents
is expected to expire within $N$ e-folds555In Berezhiani:2015ola , it was
quoted from Dvali:2013eja  that the maximal number of e-folds for which one
could trust the semi-classical picture was $N^{2/3}$. Later, in
Berezhiani:2016grw , the estimate was improved to establish that for inflation
$\mathcal{N}_{\rm max}=N$, matching the same result for pure de Sitter space
obtained in Dvali:2014gua ; Dvali:2017eba .. These argument suggest that
option (i) may be an inefficient way of beginning inflation, or that at best
it may be sensitive to our understanding of the quantum aspects of eternal
inflation.

## Option (ii)

Now let us turn to the second option for the beginning of inflation. The idea
we would like to reconsider is whether it is plausible for an inhomogeneous
universe to become homogeneous as a result of a decelerating expansion. One
often-discussed possibility, which is supported by numerical simulations for
broad but limited sets of initial conditions, is that any inhomogeneities will
be eventually washed away, leaving behind the potential energy of the
inflaton, which is essentially frozen at curvature scales greater than the
inflaton mass. We would like to revisit some aspects of this scenario within
the corpuscular description of the initial state, in which different forms of
energy correspond to the distribution of energy among quanta of different
wavelengths.

In order to start gaining some intuition, let us consider a Hubble patch
filled with inflaton quanta with wavelengths significantly shorter than the
curvature radius. (Say, if the average energy density were around Grand
Unification scale, $\rho\sim M_{\rm GUT}^{4}$, then imagine that all $\varphi$
quanta have momenta around $M_{\rm GUT}\gg H_{\rm GUT}$). Let us further
assume that on sufficiently large scales, averaging over all components, we
have homogeneous decelerated expansion. We may then ask how much energy is in
the form of potential energy, which remains essentially unredshifted. Naively,
one might be inclined to estimate this as the initial rest energy, in the hope
that any kinetic energy would be washed away by the expansion. However, the
number density of particles having momenta shorter than the horizon scale is
certainly redshifted, so that in this case the potential energy will never
come to dominate. Thus, sub-horizon quanta do not contribute to the potential
energy budget. Thus, the only quanta that can contribute to the potential
energy are the super-horizon ones.

Now, among super-horizon modes, keeping in mind that $m\ll H$, we need not say
much about quanta with $k\ll m$, since these definitely contribute to the
potential energy because they are practically frozen. The energy density of
relativistic super-horizon modes, on the other hand, redshifts until they
become non-relativistic666This is an oversimplification in favour of
inflation, as the presence of certain super-horizon perturbations could in
fact hinder its onset Clough:2016ymm ..

We begin by estimating the potential energy density budget of a region, within
our corpuscular parametrization of the initial state. For this, let us
consider a region with relatively homogeneous coarse grained energy density
(over scales longer than the average inter-particle separation) $\rho$ and
with the curvature scale $H\gg m$. As we mentioned earlier, the modes that are
guaranteed to contribute to the potential energy are the non-relativistic
modes. To estimate their number, first note (see also the previous section),
that since the energy density is given by $\rho\equiv E^{4}$, and the typical
energy of a quantum is $E$, then the expected number of quanta within the
Hubble patch is given by

$\displaystyle N=\frac{E^{3}}{H^{3}}\ \,.$ (13)

From this, and the crucial property of the distribution function (4), we can
then estimate the number of non-relativistic quanta as

$\displaystyle N_{\rm
soft}=Np(k<m)=\left(\frac{E}{H}\right)^{3}\left(\frac{m}{E}\right)^{\alpha}\,.$
(14)

Using this, the corresponding energy density reduces to

$\displaystyle\rho_{\rm soft}=mH^{3}N_{\rm soft}\ .$ (15)

For $\alpha=1$, the smallest possible value consistent with our assumptions,
we finally arrive at

$\displaystyle\rho_{\rm soft}=m^{2}E^{2}\,.$ (16)

Note that higher values of $\alpha$ lead to a stronger suppression of this
potential energy density budget.

Assuming that this energy density does not redshift as a result of expansion,
it is not possible to show that it is adequate for eventually driving
inflation. The reason is simple, the absolute lower bound on the energy
density for inflation (within the model at hand) is given by $m^{2}M_{\rm
Pl}^{2}$ and consequently requiring that $\rho_{\rm soft}$ exceed this lower
bound implies

$\displaystyle E>M_{\rm Pl}\,,$ (17)

which is outside the validity of the effective field theory.

Having shown that the initial energy stored in non-relativistic quanta is
insufficient to provide the required potential energy for inflation, we switch
our attention to super-horizon modes that are initially relativistic, but
which could eventually contribute to the slowly-rolling potential energy,
because of redshift effects. We will refer to these modes as softening modes.
One might imagine that these modes would most likely soon reenter the horizon
and then be unable to contribute. However, if the expansion is dominated by
negative spatial curvature, then $H\sim a^{-1}$. Therefore, the super-horizon
modes will remain super-horizon throughout the expansion, while redshifting.
The question we would like to ask is whether these initially relativistic
super-horizon modes might contribute an appreciable amount to the potential
energy. To examine this, assume an initial Hubble patch with curvature scale
$H_{0}$, filled with $\varphi$ quanta with the properties we have adopted
above. As before, their total number is given by

$\displaystyle N=\frac{E^{3}}{H_{0}^{3}}\,.$ (18)

Suppose further that we are interested in inflation beginning at the curvature
scale $H_{\rm inf}>m$. Then, besides modes that were already non-relativistic
at early, pre-inflationary times, modes with

$\displaystyle k\lesssim\frac{H_{0}}{H_{\rm inf}}m\,$ (19)

will also contribute to the potential energy by the time that the curvature
has reduced to $H_{\rm inf}$. We may calculate the total number of such modes
using

$\displaystyle dN_{k}=Nf(k)dk\,,$ (20)

with $f(k)$ denoting the 1-particle probability distribution that corresponds
to (4).

To be conservative, we assume that the energy density of such modes redshifts
as $\rho\propto a^{-2}$ until they become non-relativistic777If it redshifts
more rapidly, our conclusions would be even stronger., and that it freezes
after that, we can estimate their cumulative energy density by the time the
curvature scale reaches $H_{\rm inf}$ as follows

$\displaystyle\rho_{\rm softening}$ $\displaystyle=$
$\displaystyle\int_{m}^{\frac{H_{0}}{H_{\rm
inf}}m}dN_{k}~{}\frac{kH_{0}^{3}}{(k/m)^{2}}$ (21) $\displaystyle=$
$\displaystyle\rho_{\rm soft}\cdot\frac{-1+\left(\frac{H_{0}}{H_{\rm
inf}}\right)^{\alpha-1}}{\alpha-1}\,,$

(for $\alpha\neq 1$) with $\rho_{\rm soft}$ denoting the energy density for
the initially non-relativistic quanta given by (15). For the case of least
suppressed homogeneous component, $\alpha=1$, the above integral yields

$\displaystyle\rho_{\rm softening}=\rho_{\rm soft}\cdot{\rm
ln}\left(\frac{H_{0}}{H_{\rm inf}}\right)\,.$ (22)

Interestingly, after all the optimistic assumptions, the softening modes in
question dominate somewhat over the modes that were non-relativistic from the
beginning; namely, the largest possible value of the logarithmic enhancement
factor is obtained for $H_{0}\simeq M_{\rm Pl}$ and $H_{\rm inf}\simeq
m=10^{-6}M_{\rm Pl}$, corresponding to ${\rm ln}(H_{0}/H_{\rm inf})\simeq 6$.
Nevertheless, the enhancement is not parametrically significant even for
$\alpha=1$. While it is true that for higher values of $\alpha$, the softening
modes are increasingly dominant over the initially soft modes, this is of
little help because all modes are increasingly suppressed as we increase
$\alpha$. For example, even for $\alpha=2$, $\rho_{\rm soft}$ itself is
suppressed so much that one requires $E>M_{\rm Pl}$ to ensure enough potential
energy for a minimal amount of inflation.

Having demonstrated that our assumptions about the initial state of Universe
do not provide sufficient potential energy for the eventual onset of inflation
when averaging over large scales, we now ask the question in a way that
combines the best of options (i) and (ii). In other words, our analysis of
option (ii) has shown the rareness of regions with sufficient potential energy
to drive inflation (once other forms are washed away), which begs the question
whether this rareness can be compensated for by the exponential volume factor,
in the spirit of option (i). For this, we begin with a Hubble patch of
curvature $H_{0}$ and total number of particles $N$ given by (18) and estimate
the probability of having $N_{\rm soft}$ modes with $k<m$, keeping in mind
that the minimal number of such modes888We are ignoring the softening modes
here for the sake of simplicity. should be such that their initial energy
density is greater than $m^{2}M_{\rm Pl}^{2}$. Assuming $N\gg 1$ and $Np$
being finite (with $p$ given by (4)), we can approximate the probability that
$N_{\rm soft}$ out of $N$ quanta are nonrelativistic using the Poisson
distribution

$\displaystyle P(N_{\rm soft})=\frac{(Np)^{N_{\rm soft}}e^{-{Np}}}{N_{\rm
soft}!}\,.$ (23)

This needs to be multiplied by the volume factor ${\rm exp}(3\mathcal{N})$,
with number of inflationary e-folds being determined in terms of $N_{\rm
soft}$ in a straightforward manner. In particular, having $N_{\rm soft}$
nonrelativistic modes corresponds to their energy density being $N_{\rm
soft}mH_{0}^{3}$. Assuming that this energy density is constant during the
course of the decelerated expansion (an assumption that makes inflation more
likely), the number of e-folds can be estimated from $N_{\rm
soft}mH_{0}^{3}\simeq H_{\rm inf}^{2}M_{\rm Pl}^{2}$ and $\mathcal{N}\simeq
H_{\rm inf}^{2}/m^{2}$ (valid for $\mathcal{N}$ at least a few). This reduces
to

$\displaystyle\mathcal{N}=\frac{N_{\rm soft}H_{0}^{3}}{mM_{\rm Pl}^{2}}\,.$
(24)

This makes the minimal required value of $N_{\rm soft}$ to be

$\displaystyle N_{\rm soft}^{min}=\frac{mM_{\rm Pl}^{2}}{H_{0}^{3}}\,.$ (25)

Taking into account that this number must be significantly greater than one
and that $Np/N^{min}_{\rm soft}<1$, it is straightforward to show that
$P(N_{\rm soft})\cdot{\rm exp}(3\mathcal{N})$ is exponentially small.

## Conclusions

We have examined the question of the prevalence of inflation under a concrete
set of assumptions about the prior state. Taking into account that the
inflaton is an ultralight particle in the early Universe, we have envisioned
the pre-inflationary state of these quanta to be in the form of a relativistic
gas. This is a particularly fair assumption if they were produced through
perturbative decays of high-energy quanta that could have populated the
Universe beforehand. Moreover, we have assumed there to be a small probability
for a given quantum to be produced soft rather than relativistic. As a result,
we expect the existence of rare regions within which inflation may begin,
either from the beginning, due to accidentally favorable conditions, or as a
result of further evolution and homogenization. Our estimations demonstrate
that if the initial state possesses the outlined properties, then the
efficient beginning of inflation seems to require access to the part of the
potential which is flat enough for hosting slow-roll eternal inflation (a.k.a
self-reproduction).

Our assumptions about the initial state include that the scalar field is
initially in the vacuum, corresponding to vanishing energy density, before the
perturbative population of the inflaton state began. As such, it is not
surprising that we obtained a highly suppressed density in ultra-soft modes
(and a correspondingly small contribution to the initial potential energy).

Alternatively, consider the scenario in which the inflaton potential has
degenerate minima. In this case, e.g. for the potential
$V(\varphi)=\lambda\left(\varphi^{2}-v^{2}\right)^{2}$, one usually expects
some regions of the universe to be found in one vacuum, and the others in the
opposite one, inevitably leaving us with regions in which the scalar field
assumes intermediate values. Such intermediate regions, representing
topological defects, would not be exponentially rare, and thus our assumptions
might require revision in this context999We would like thank Alex Vilenkin for
raising this point.. Within each region in the same vacuum, at the moment of
formation our assumptions regarding the state of $\varphi$ quanta should
apply. The only peculiar parts are the domain walls themselves, and their
implications are sensitive to their width. If the potential is such that the
domain wall is thinner than the curvature radius inside the core, then it will
not seed inflation. If its width exceeds the curvature scale, on the other
hand, then it would mean that we have sufficiently large regions of
homogeneous scalar field inside the topological defects, known as ’topological
inflation’ Vilenkin:1994pv ; Linde:1994hy ; deLaix:1998sf . Semi-classically
these regions then undergo slow-roll eternal inflation. Therefore, we once
again encounter the relevance of this quantum regime.

To summarize, an unequivocally tantalizing idea that makes these issues
irrelevant is slow-roll eternal inflation. It is therefore extremely important
to understand the true quantum dynamics of that process. Our findings, for the
initial conditions at hand, underscore its importance in an interesting way.
As we have shown, even if one does not invoke self-reproduction (which yields
a practically infinite homogeneous volume) but assumes a simple slow-roll of
the inflaton from field values deep inside this quantum regime, most of the
late-time volume can be still dominated by inflated patches. However, as we
already pointed out, this regime is far from being fully understood. For
series of recent papers discussing the issues with the semiclassical picture
of this quantum regime see Dvali:2013eja ; Berezhiani:2016grw ; Dvali:2017eba
.

## Acknowledgements

We would like to thank Gia Dvali, Aaron Levy and Eugene Lim for valuable
discussions. The work of MT is supported in part by US Department of Energy
(HEP) Award DE-SC0013528, by NASA ATP grant 80NSSC18K0694, and by the Simons
Foundation Origins of the Universe Initiative.

## References

  * (1) A. H. Guth, “The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems,” Phys. Rev. D 23, 347 (1981).
  * (2) A. D. Linde, “A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems,” Phys. Lett. B 108, 389 (1982).
  * (3) A. Albrecht and P. J. Steinhardt, “Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking,” Phys. Rev. Lett. 48, 1220 (1982).
  * (4) J. B. Hartle and S. W. Hawking, “Wave Function of the Universe,” Phys. Rev. D 28, 2960 (1983).
  * (5) A. D. Linde, “Initial Conditions For Inflation,” Phys. Lett. B 162, 281 (1985).
  * (6) R. H. Brandenberger and J. H. Kung, “Chaotic Inflation as an Attractor in Initial Condition Space,” Phys. Rev. D 42, 1008 (1990).
  * (7) A. D. Linde, D. A. Linde and A. Mezhlumian, “From the Big Bang theory to the theory of a stationary universe,” Phys. Rev. D 49, 1783 (1994) [gr-qc/9306035].
  * (8) E. Farhi and A. H. Guth, “An Obstacle to Creating a Universe in the Laboratory,” Phys. Lett. B 183, 149 (1987).
  * (9) E. Farhi, A. H. Guth and J. Guven, “Is It Possible to Create a Universe in the Laboratory by Quantum Tunneling?,” Nucl. Phys. B 339, 417 (1990).
  * (10) T. Vachaspati and M. Trodden, “Causality and cosmic inflation,” Phys. Rev. D 61, 023502 (1999) [gr-qc/9811037].
  * (11) N. Kaloper, M. Kleban, A. Lawrence, S. Shenker and L. Susskind, “Initial conditions for inflation,” JHEP 0211, 037 (2002) [hep-th/0209231].
  * (12) A. Albrecht and L. Sorbo, “Can the universe afford inflation?,” Phys. Rev. D 70, 063528 (2004) [hep-th/0405270].
  * (13) G. W. Gibbons and N. Turok, “The Measure Problem in Cosmology,” Phys. Rev. D 77, 063516 (2008) [hep-th/0609095].
  * (14) J. B. Hartle, S. W. Hawking and T. Hertog, “The Classical Universes of the No-Boundary Quantum State,” Phys. Rev. D 77, 123537 (2008) [arXiv:0803.1663 [hep-th]].
  * (15) J. S. Schiffrin and R. M. Wald, “Measure and Probability in Cosmology,” Phys. Rev. D 86, 023521 (2012) [arXiv:1202.1818 [gr-qc]].
  * (16) A. Ijjas, P. J. Steinhardt and A. Loeb, “Inflationary paradigm in trouble after Planck2013,” Phys. Lett. B 723, 261 (2013) [arXiv:1304.2785 [astro-ph.CO]].
  * (17) A. H. Guth, D. I. Kaiser and Y. Nomura, “Inflationary paradigm after Planck 2013,” Phys. Lett. B 733, 112 (2014) [arXiv:1312.7619 [astro-ph.CO]].
  * (18) A. Linde, “Inflationary Cosmology after Planck 2013,” arXiv:1402.0526 [hep-th].
  * (19) A. Ijjas, P. J. Steinhardt and A. Loeb, “Inflationary schism after Planck2013,” arXiv:1402.6980 [astro-ph.CO].
  * (20) S. M. Carroll, “In What Sense Is the Early Universe Fine-Tuned?,” arXiv:1406.3057 [astro-ph.CO].
  * (21) L. Berezhiani and M. Trodden, “How Likely are Constituent Quanta to Initiate Inflation?,” Phys. Lett. B 749, 425 (2015) [arXiv:1504.01730 [hep-th]].
  * (22) W. E. East, M. Kleban, A. Linde and L. Senatore, “Beginning inflation in an inhomogeneous universe,” JCAP 1609 (2016) 010 [arXiv:1511.05143 [hep-th]].
  * (23) K. Clough, E. A. Lim, B. S. DiNunno, W. Fischler, R. Flauger and S. Paban, “Robustness of Inflation to Inhomogeneous Initial Conditions,” JCAP 1709 (2017) 025 [arXiv:1608.04408 [hep-th]].
  * (24) K. Clough, R. Flauger and E. A. Lim, “Robustness of Inflation to Large Tensor Perturbations,” JCAP 1805 (2018) 065 [arXiv:1712.07352 [hep-th]].
  * (25) J. C. Aurrekoetxea, K. Clough, R. Flauger and E. A. Lim, “The Effects of Potential Shape on Inhomogeneous Inflation,” JCAP 05, 030 (2020) [arXiv:1910.12547 [astro-ph.CO]].
  * (26) V. Mukhanov, “Inflation without Selfreproduction,” Fortsch. Phys. 63, 36-41 (2015) doi:10.1002/prop.201400074 [arXiv:1409.2335 [astro-ph.CO]].
  * (27) M. Kleban and L. Senatore, “Inhomogeneous Anisotropic Cosmology,” JCAP 1610 (2016) 022 [arXiv:1602.03520 [hep-th]].
  * (28) G. Dvali and C. Gomez, “Quantum Compositeness of Gravity: Black Holes, AdS and Inflation,” JCAP 1401, 023 (2014) [arXiv:1312.4795 [hep-th]].
  * (29) G. Dvali, “$S$-Matrix and Anomaly of de Sitter,” Symmetry 13, no.1, 3 (2020) doi:10.3390/sym13010003 [arXiv:2012.02133 [hep-th]].
  * (30) L. Berezhiani, “On Corpuscular Theory of Inflation,” Eur. Phys. J. C 77, no. 2, 106 (2017) [arXiv:1610.08433 [hep-th]].
  * (31) G. Dvali and C. Gomez, “Quantum Exclusion of Positive Cosmological Constant?,” Annalen Phys. 528, 68 (2016) [arXiv:1412.8077 [hep-th]].
  * (32) G. Dvali, C. Gomez and S. Zell, “Quantum Break-Time of de Sitter,” JCAP 1706, 028 (2017) [arXiv:1701.08776 [hep-th]].
  * (33) A. Vilenkin, “Topological inflation,” Phys. Rev. Lett. 72, 3137-3140 (1994) [arXiv:hep-th/9402085 [hep-th]].
  * (34) A. D. Linde, “Monopoles as big as a universe,” Phys. Lett. B 327, 208-213 (1994) [arXiv:astro-ph/9402031 [astro-ph]].
  * (35) A. A. de Laix, M. Trodden and T. Vachaspati, “Topological inflation with multiple winding,” Phys. Rev. D 57, 7186-7191 (1998) [arXiv:gr-qc/9801016 [gr-qc]].