# Energetics of quantum vacuum friction. II: Dipole fluctuations and field
fluctuations

Xin Guo ID<EMAIL_ADDRESS>H. L. Dodge Department of Physics and Astronomy,
University of Oklahoma, Norman, Oklahoma 73019, USA  Kimball A. Milton ID
<EMAIL_ADDRESS>H. L. Dodge Department of Physics and Astronomy, University of
Oklahoma, Norman, Oklahoma 73019, USA  Gerard Kennedy ID<EMAIL_ADDRESS>School of Mathematical Sciences, University of Southampton, Southampton SO17
1BJ, United Kingdom  William P. McNulty ID<EMAIL_ADDRESS>H. L.
Dodge Department of Physics and Astronomy, University of Oklahoma, Norman,
Oklahoma 73019, USA  Nima Pourtolami ID<EMAIL_ADDRESS>National
Bank of Canada, Montreal, Quebec H3B 4S9, Canada  Yang Li ID<EMAIL_ADDRESS>Department of Physics, Nanchang University, Nanchang 330031, China

###### Abstract

Quantum vacuum friction experienced by an atom, where the only dissipative
mechanism is through its interaction with the radiation field, has been
studied in our recent paper [Phys. Rev. D 104, 116006 (2021)]. Quantum vacuum
friction on an intrinsically dissipative particle is different in that the
friction arises not only from the field fluctuations but also from the dipole
fluctuations intrinsic to the particle. As a result, the dissipative particle
can be out of the nonequilibrium steady state (NESS), and therefore loses or
gains internal energy (rest mass). Only if the temperature of the particle
equals a special NESS temperature will the particle be in NESS. In this paper,
general NESS conditions are derived which give the NESS temperature of the
particle as a function of the temperature of the radiation and the velocity of
the particle. Imposing the NESS conditions, we also obtain an expression for
the quantum vacuum friction in NESS. The NESS quantum vacuum friction is shown
to be negative definite (opposing the motion of the particle) and equivalent
to that found in our previous paper if the dissipation mechanism is restricted
to radiation reaction. The NESS temperature and quantum vacuum friction are
calculated numerically for various models. In particular, we show that, for a
gold nanosphere, the deviation of its NESS temperature from the temperature of
the radiation can be substantial and it is also possible to detect the NESS
quantum vacuum friction directly at sufficiently high temperatures. Out of
NESS, even though the quantum vacuum friction no longer has a definite sign in
the rest frame of the radiation, the friction in the rest frame of the
particle is still negative definite. Also, the external force needed to keep
the particle moving must be in the same direction as the motion of the
particle, therefore excluding the possibility of a perpetual motion machine,
which could convert the vacuum energy into useful mechanical work. In
addition, we find that the deviation of the temperature of the particle from
its NESS temperature causes the particle to lose or gain internal energy in
such a way that the particle would return to NESS after deviating from it.
This enables experimental measurements of the NESS temperature of the particle
to serve as a feasible signature for these quantum vacuum frictional effects.

##  I introduction

Quantum friction (also known as Casimir friction) [1] is usually associated
with two typical configurations: two sliding plates [2, 3] or a particle
moving parallel to a plate [4, 5, 6], even though the notion can be applied
more broadly to contexts like the expanding universe, rotating black holes,
moving mirrors and even activities of sub-cellular biosystems [7]. Although
much effort has been invested over several decades to calculate quantum
friction in both configurations, discrepancies in the theoretical results and
even doubts concerning the existence of such friction remain [8]. However,
relative motion between two macroscopic bodies is not a necessity for quantum
vacuum friction to occur. A neutral polarizable particle moving through free
space will experience a force in the opposite/same direction as the particle’s
motion due to its interaction with surrounding blackbody radiation. This is
what we refer to as quantum vacuum friction in Ref. [9] and in the present
paper.

The key ingredient of quantum vacuum friction is the presence of fluctuations.
Due to the dissipative nature of the fluctuations, the electromagnetic vacuum
behaves like a complex fluid, which modifies the motion of objects in it [10].
Fluctuations are related to the imaginary part of the corresponding
susceptibilities according to the fluctuation-dissipation theorem (FDT) [11].
Typical examples are the Green’s dyadic for electromagnetic field fluctuations
and the polarizability of the particle for dipole fluctuations. Both
fluctuations can be responsible for quantum vacuum friction. In Ref. [9],
where the intrinsic polarizability, $\bm{\alpha}$, of the particle is
considered to be purely real, it is the imaginary part of the electromagnetic
Green’s dyadic, $\bm{\Gamma}$, that allows for the existence of fluctuations
of the electromagnetic field and the corresponding induced dipole
fluctuations. In this paper, we continue to study quantum vacuum friction, but
for a neutral particle whose intrinsic electric polarizability is complex,
$\imaginary\bm{\alpha}(\omega)\neq 0$.111For simplicity of the discussion, we
assume the particle to possess no intrinsic magnetic polarizability,
$\bm{\beta}=0$, and therefore we do not consider magnetic dipole fluctuations
or those of higher multipoles. This setting exhibits some new features beyond
those already considered in Ref. [9]. First, the temperature of the particle,
$T^{\prime}$, comes into play, through the intrinsic dipole fluctuations.
Second, the particle now has the freedom to leave the nonequilibrium steady
state (NESS) because it can absorb or emit net energy.

We still focus on the NESS situation from Sec. II to Sec. IV. In Sec. II, we
find the NESS conditions by requiring the particle to absorb and emit net
energy at the same rate. The NESS conditions provide a relation between the
temperature of the particle, $T^{\prime}$, and the temperature of the
radiation, $T$, which defines the NESS temperature of the particle,
$\tilde{T}$. Interestingly, we are able to prove that the NESS temperature of
the particle is, quite generally, greater than the Planck-Einstein transformed
temperature of the blackbody radiation, that is, $\tilde{T}>T/\gamma$. Using
the NESS conditions, the NESS temperature ratio, $\tilde{T}/T$, can be
calculated as a function of the particle’s velocity $v$ and the radiation
temperature $T$ once $\imaginary\bm{\alpha}(\omega)$ has been specified. For
simplicity, we first work out the NESS conditions explicitly for the resonance
model, $\imaginary\bm{\alpha}(\omega)\propto\delta(\omega-\omega_{0})$, and
the monomial model, $\imaginary\bm{\alpha}(\omega)\propto\omega^{n}$. Even
though these models are simple, they provide insights into more realistic
situations because more realistic models often reduce to these simple models
in different temperature and velocity regimes. In Sec. III, we impose the NESS
conditions on the general formula for quantum vacuum frictional force
previously derived in Ref. [12] and find that the resultant expression for the
NESS quantum vacuum frictional force on the particle is negative definite and
equivalent to that in Ref. [9] if the dissipation mechanism is restricted to
radiation reaction. The nonrelativistic limit of the NESS friction reproduces
the famous Einstein-Hopf drag [13]. The classical high-temperature limit of
the NESS friction is found to be linear in $T$. The NESS quantum vacuum
friction for the resonance and monomial models are also explicitly calculated.
In Sec. IV, we estimate the NESS temperature ratio and NESS quantum vacuum
friction for a gold nanosphere moving in vacuum. The Lorenz-Lorentz relation
is used for the polarizability of the particle and the Drude model for the
permittivity of gold. We assume a constant damping parameter for the gold
nanosphere in IV.1. The model for the gold nanosphere turns out to reduce to
the $n=1$ monomial model in the low-temperature limit and the $n=-3$ monomial
model in the high-temperature limit. In the transition region between these
two extreme limits, there also exists a temperature regime where the model
reduces to the resonance model. In reality, the damping parameter is
temperature dependent. Therefore, in IV.2, we employ the Bloch-Grüneisen model
to describe the temperature dependence of the damping parameter. The effect on
the NESS temperature ratio and the NESS quantum vacuum friction of including
this temperature dependence in the damping parameter is also discussed.

We then extend our analysis to the out-of-NESS situation in Sec. V, and draw
several interesting conclusions. Even though the frictional force, $F$, in the
rest frame of radiation ($\mathcal{R}$), no longer has a definite sign, the
frictional force, $F^{\prime}$, in the rest frame of particle ($\mathcal{P}$),
is equal to the NESS quantum vacuum friction, $\tilde{F}$, and is therefore
negative definite. Also, the external force, $F_{\rm{ext}}$, required to
maintain the configuration is the negative of $F^{\prime}$, and always pushes
the particle forward. In addition, depending on whether the temperature of the
particle is higher or lower than the NESS temperature, the sign of the total
force, $F_{\rm{tot}}=F+F_{\rm{ext}}$, on the particle is negative or positive,
respectively. And the sign of the total force reflects the loss or gain of the
internal energy (rest mass) of the particle. Finally, we numerically obtain
the condition for the quantum vacuum friction, $F$, on a gold nanosphere (with
constant damping) to be zero in frame $\mathcal{R}$. The zeroes for $F$ occur
only in the high-temperature regime unless the velocities are
ultrarelativistic.

In Appendix A, we provide a proof of the equivalence between the Lorentz force
law and the principle of virtual work for friction on a moving dipole. In
Appendix B, we supply high-temperature asymptotic expressions for the
frictional power and force in the case of the Bloch-Grüneisen model. In
Appendix C, we extend the discussion about the zeroes of $F$ to the low-
temperature, high velocity regime.

In this paper we use Heaviside-Lorentz (rationalized) electromagnetic units
and set $k_{B}=c=\hbar=1$ in the formulas. SI units are used in the numerical
evaluations.

##  II The NESS conditions

In this paper, we explore again the energetics of the so-called quantum vacuum
friction, the electromagnetic force that a neutral but polarizable particle
experiences while it is moving through vacuum (free space) with constant
velocity and interacting with the surrounding blackbody radiation. Unlike our
previous investigation in Ref. [9], we study a particle that has intrinsic
dissipation. This imparts to the particle the freedom of absorbing/releasing
net energy and allows for the existence of independent dipole fluctuations.

There are three main players in the scene: the particle, the radiation, and an
external agent which compensates for the energy lost into the particle and the
radiation fields and maintains the uniform motion of the particle [14]. The
external agent and the radiation both directly interact with the particle.
Even though the particle itself is an open quantum system, the overall energy
of all three players should be conserved, which can be summarized into a power
balance relation,

$P_{\rm{par}}=P_{\rm{ext}}+P.$ (1)

Here, $P_{\rm{ext}}=F_{\rm{ext}}v$ is the time rate of the external agent
doing work on the particle. $P$ is the time rate of the radiation field doing
work on the particle. $P_{\rm{par}}$ is the net power absorbed by the
particle.222Of course, this results in a change in the internal dynamics of
the particle, which we will not discuss in detail in the current paper.

If the particle does not absorb net energy, $P_{\rm{par}}=0$, we say that it
is in the nonequilibrium steady state (NESS). A neutral particle (e.g., a gold
atom) that has no intrinsic dissipation is ensured to be in NESS. But for an
intrinsically dissipative particle (e.g., a gold nanosphere that is made of
millions of atoms), $P_{\rm{par}}=0$ defines a NESS condition that turns out
to be a relation between the temperature of the particle, $T^{\prime}$, and
the temperature of the radiation, $T$.

In frame $\mathcal{R}$, NESS requires the external force to balance the
electromagnetic force, $F_{\rm{ext}}+F=0$.333It will be clear in Sec. V that
this is not the case out of NESS due to the change in the particle’s mass. As
a result, the NESS condition also translates to a power-force relation for the
radiation

$P=-P_{\rm{ext}}=-F_{\rm{ext}}v=Fv.$ (2)

In frame $\mathcal{P}$, the external agent cannot do any work on the particle,
$P^{\prime}_{\rm{ext}}=0$. And the NESS condition simplifies to

$P^{\prime}_{\rm{par}}=P^{\prime}=0.$ (3)

Therefore, we could calculate either $P-Fv$ or $P^{\prime}$ in order to find
the NESS conditions explicitly.

###  II.1 The frictional power

We now calculate the electromagnetic power on the polarizable particle. Since
the electromagnetic force that corresponds to this power is precisely the
quantum vacuum friction, we may also call it the frictional power.

Before proceeding to the calculation, let us define the problem more
concretely. We will use $\bm{\alpha}(\omega)$ for the electric polarizability
of the neutral particle. To incorporate intrinsic dissipation,
$\bm{\alpha}(\omega)$ must be complex. Without loss of generality, we assume
that the particle is moving relative to the blackbody radiation in the $x$
direction with constant velocity $\mathbf{v}=v\hat{\mathbf{x}}$ and therefore
lies on the trajectory $\mathbf{r}(t)=\mathbf{v}t$. In the rest frame of the
particle $\mathcal{P}$, the particle sits at a fixed position, which we will
assume to be the origin, $\mathbf{r^{\prime}}=\mathbf{0}$. Throughout the
paper, we will use primes on quantities and coordinates in frame
$\mathcal{P}$, except for the polarizability $\bm{\alpha}$, which is always
defined in frame $\mathcal{P}$. We will now calculate the rate at which the
electromagnetic force does work (frictional power) on the particle.

In Ref. [9], the frictional power $P$ in frame $\mathcal{R}$ is derived using
the Joule heating law

$P(t)=\int
d\mathbf{r}\,\mathbf{j}(t,\mathbf{r})\cdot\mathbf{E}(t,\mathbf{r}),$ (4)

while the frictional power $P^{\prime}$ in frame $\mathcal{P}$ is computed by
differentiating the free energy $\mathcal{F^{\prime}}$ in $\mathcal{P}$

$P^{\prime}(t^{\prime})=\frac{\partial}{\partial
t^{\prime}}\mathcal{F^{\prime}}=-\frac{\partial}{\partial
t_{1}^{\prime}}\left[\mathbf{d}(t_{0}^{\prime})\cdot\mathbf{E}(t_{1}^{\prime},\mathbf{0})\right]|_{t_{0}^{\prime}=t_{1}^{\prime}\to
t^{\prime}}.$ (5)

Here, we use the same notation as in Ref. [9]: $t_{0}^{\prime}$ for the time
coordinate of the dipole operator and $t_{1}^{\prime}$ for that of the field
operator and they are set equal after the differentiation. Effectively, this
apparent separation of coordinates serves as a prescription that only the
field coordinates are to be differentiated.

One might be questioning the consistency between Eq. (4) and Eq. (5) and, in
particular, the validity of the differentiation prescription of Eq. (5). Here
we eliminate any such doubt by showing that the electromagnetic power on a
moving dipole according to Eq. (4) can always be written as a derivative of a
free energy. And in frame $\mathcal{P}$ where the velocity of the particle is
zero, it reduces to the special form Eq. (5).

For a uniformly moving dipole $\mathbf{d}(t)$ with velocity $\mathbf{v}$, the
corresponding classical current density is

$\mathbf{j}(t,\mathbf{r})=\left[\dot{\mathbf{d}}(t)-\mathbf{v}\gradient\cdot\mathbf{d}(t)\right]\delta(\mathbf{r}-\mathbf{v}t).$
(6)

When this current is inserted into Eq. (4), we obtain the power

$P(t)=\dot{\mathbf{d}}(t)\cdot\mathbf{E}(t,\mathbf{v}t)+\left[\mathbf{d}(t)\cdot\gradient\right]\left[\mathbf{v}\cdot\mathbf{E}(t,\mathbf{v}t)\right],$
(7)

where the first term can be rewritten as

$\dot{\mathbf{d}}(t)\cdot\mathbf{E}(t,\mathbf{v}t)=\frac{d}{dt}\left[\mathbf{d}(t)\cdot\mathbf{E}(t,\mathbf{v}t)\right]-\mathbf{d}(t)\cdot\frac{\partial}{\partial
t}\mathbf{E}(t,\mathbf{v}t)-\left[\mathbf{v}\cdot\gradient\right]\left[\mathbf{d}(t)\cdot\mathbf{E}(t,\mathbf{v}t)\right].$
(8)

Note the second term of Eq. (7) and the last term of Eq. (8) combine and give

$\left[\mathbf{d}(t)\cdot\gradient\right]\left[\mathbf{v}\cdot\mathbf{E}(t,\mathbf{v}t)\right]-\left[\mathbf{v}\cdot\gradient\right]\left[\mathbf{d}(t)\cdot\mathbf{E}(t,\mathbf{v}t)\right]=\left[\mathbf{d}(t)\crossproduct\mathbf{v}\right]\cdot\left[\gradient\crossproduct\mathbf{E}(t,\mathbf{v}t)\right]=-\left[\mathbf{d}(t)\crossproduct\mathbf{v}\right]\cdot\frac{\partial}{\partial
t}\mathbf{B}(t,\mathbf{v}t),$ (9)

where we have used the Faraday’s law in the last equality. As a result, the
power is written as

$P(t)=\frac{d}{dt}\left[\mathbf{d}(t)\cdot\mathbf{E}(t,\mathbf{v}t)\right]-\mathbf{d}(t)\cdot\frac{\partial}{\partial
t}\mathbf{E}(t,\mathbf{v}t)-\left[\mathbf{d}(t)\crossproduct\mathbf{v}\right]\cdot\frac{\partial}{\partial
t}\mathbf{B}(t,\mathbf{v}t).$ (10)

The total derivative term in Eq. (10) is generally present but it will not
contribute to the quantum vacuum frictional power considered in this paper.
The quantum vacuum frictional power is induced entirely by the quantum
fluctuations of the dipole operator $\mathbf{d}$ or the field operator
$\mathbf{E}$. These fluctuations are determined by the imaginary part of the
corresponding susceptibilities via the FDT. It follows from the linear
response assumption (embedded in the FDT) and the time-translational
invariance of the susceptibilities that the dipole interaction energy
$\mathbf{d}\cdot\mathbf{E}$ is time independent. The next two terms in Eq.
(10) each contains a partial time derivative of the field operator. Indeed, in
the above derivation, the prescription that only the time coordinates in the
field operators should be differentiated follows straightforwardly from the
Joule heating law. Similar to Eq. (5), we can now also write $P$ as a time
derivative of a free energy $\mathcal{F}$,

$P(t)=\frac{\partial}{\partial t}\mathcal{F}=-\frac{\partial}{\partial
t_{1}}\left[\mathbf{d}(t_{0})\cdot\mathbf{E}(t_{1},\mathbf{v}t)+\bm{\mu}_{\mathbf{v}}(t_{0})\cdot\mathbf{B}(t_{1},\mathbf{v}t)\right]|_{t_{0}=t_{1}\to
t},$ (11)

where we have identified
$\mathbf{d}(t)\crossproduct\mathbf{v}=\bm{\mu}_{v}(t)$ as the magnetic dipole
moment induced by the movement of the electric dipole. The induced magnetic
dipole term in $\mathcal{F}$ vanishes if we set $\mathbf{v}=\mathbf{0}$ in Eq.
(11) and the expression for $P^{\prime}$ in Eq. (5) is reproduced.

If we apply the same reasoning for the frictional force, we obtain a proof for
the equivalence of the Lorentz force law and the principle of virtual work.
This is detailed in Appendix A.

The NESS condition can be expressed as either $P-Fv=0$ in frame $\mathcal{R}$
or $P^{\prime}=0$ in frame $\mathcal{P}$. Let us work in frame $\mathcal{P}$,
where the calculation is simpler.

Recall the FDT for dipole fluctuations and field fluctuations in frame
$\mathcal{P}$:

$\langle\mathbf{d}^{\prime}(\omega^{\prime})\mathbf{d}^{\prime}(\nu^{\prime})\rangle=2\pi\delta(\omega^{\prime}+\nu^{\prime})\imaginary\bm{\alpha}(\omega^{\prime})\coth\left(\frac{\beta^{\prime}\omega^{\prime}}{2}\right),$
(12a)
$\langle\mathbf{E^{\prime}}(\omega^{\prime},\mathbf{k}^{\prime}_{\perp};z^{\prime})\mathbf{E^{\prime}}(\nu^{\prime},\bar{\mathbf{k}}^{\prime}_{\perp};\bar{z}^{\prime})\rangle=(2\pi)^{3}\delta(\omega^{\prime}+\nu^{\prime})\delta^{(2)}(\mathbf{k}^{\prime}_{\perp}+\bar{\mathbf{k}}^{\prime}_{\perp})\imaginary\mathbf{g}^{\prime}(\omega^{\prime},\mathbf{k}^{\prime}_{\perp};z^{\prime},\bar{z}^{\prime})\coth\left(\frac{\beta\gamma(\omega^{\prime}+k_{x}^{\prime}v)}{2}\right),$
(12b)

where $\beta^{\prime}=1/T^{\prime}$ is the inverse temperature of the particle
while $\beta=1/T$ is the inverse temperature of the radiation, each in their
respective frame and $\gamma$ is the relativistic dilation factor. Here,
$\mathbf{g}^{\prime}$ is the reduced Green’s dyadic, which is the Fourier
transform of the retarded Green’s dyadic,

$\mathbf{\Gamma}(\omega;\mathbf{r},\mathbf{r^{\prime}})=\int\frac{d^{2}\mathbf{k}_{\perp}}{(2\pi)^{2}}e^{i\mathbf{k}_{\perp}\cdot(\mathbf{r}_{\perp}-\mathbf{r^{\prime}}_{\perp})}\mathbf{g}(\omega,\mathbf{k}_{\perp};z,z^{\prime}),$
(13)

where $\mathbf{\Gamma}$ is only transformed in the transverse spatial
directions so that our analysis can be readily extended to a geometry with a
planar symmetry. The specific form of the Green’s dyadic is detailed in our
previous paper, Ref. [9].

In general, the symbol $\imaginary$ in Eq. (12) denotes the generalized
imaginary part of the corresponding matrices, i.e.,

$\imaginary\bm{\chi}=\frac{\bm{\chi}-\bm{\chi}^{\dagger}}{2i},$ (14)

so that $\imaginary\bm{\chi}$ is Hermitian, which is required because the
product of operators in Eq. (12) are symmetrized. In the calculation for the
quantum vacuum friction, however, $\imaginary\mathbf{g}^{\prime}$ reduces to
the ordinary imaginary part, since the vacuum Green’s dyadic is symmetric. And
as we will see, only the diagonal components of the polarization tensor
$\bm{\alpha}$ contributes to the quantum vacuum friction due to the additional
symmetry of the vacuum problem.

The interaction free energy in $\mathcal{P}$ is

$\mathcal{F}^{\prime}(t^{\prime})=-\mathbf{d}^{\prime}(t^{\prime})\cdot\mathbf{E}^{\prime}(t^{\prime},\mathbf{0})=-\int\frac{d\omega}{2\pi}e^{-i\omega
t^{\prime}}\mathbf{d}^{\prime}(\omega)\cdot\int\frac{d\nu}{2\pi}e^{-i\nu
t^{\prime}}\mathbf{E}^{\prime}(\nu;\mathbf{0}).$ (15)

To leading order in the intrinsic polarizability $\bm{\alpha}(\omega)$, the
free energy is split into a dipole-fluctuation-induced part

$\mathcal{F}^{\prime}_{dd}(t^{\prime})=-\int\frac{d\omega}{2\pi}e^{-i\omega
t^{\prime}}\left\langle\mathbf{d}^{\prime}(\omega)\cdot\int\frac{d\nu}{2\pi}e^{-i\nu
t^{\prime}}\int\frac{d^{2}\mathbf{k}_{\perp}}{(2\pi)^{2}}\mathbf{g}^{\prime}(\nu,\mathbf{k}_{\perp})\cdot\mathbf{d}^{\prime}(\nu)\right\rangle,$
(16a) and a field-fluctuation-induced part
$\mathcal{F}^{\prime}_{EE}(t^{\prime})=-\int\frac{d\omega}{2\pi}e^{-i\omega
t^{\prime}}\left\langle\left(\bm{\alpha}(\omega)\cdot\int\frac{d^{2}\mathbf{k}_{\perp}}{(2\pi)^{2}}\mathbf{E}^{\prime}(\omega,\mathbf{k}_{\perp})\right)\cdot\int\frac{d\nu}{2\pi}e^{-i\nu
t^{\prime}}\int\frac{d^{2}\mathbf{k}^{\prime}_{\perp}}{(2\pi)^{2}}\mathbf{E}^{\prime}(\nu,\mathbf{k}^{\prime}_{\perp})\right\rangle.$
(16b)

Now we proceed to differentiate $\mathcal{F^{\prime}}$ for the power
$P^{\prime}$ using Eq. (5). With the understanding that the derivative is only
taken with respect to the time dependence of the original field operator
$\mathbf{E}^{\prime}(t^{\prime},\mathbf{0})$ in Eq. (15), we find

$P^{\prime}_{dd}=\frac{\partial}{\partial
t^{\prime}}\mathcal{F^{\prime}}_{dd}=-\int\frac{d\omega}{2\pi}\frac{d^{2}\mathbf{k}_{\perp}}{(2\pi)^{2}}\,\omega\tr\imaginary\bm{\alpha}(\omega)\cdot\imaginary\mathbf{g}^{\prime}(\omega,\mathbf{k}_{\perp})\coth\frac{\beta^{\prime}\omega}{2},$
(17a) $P^{\prime}_{EE}=\frac{\partial}{\partial
t^{\prime}}\mathcal{F^{\prime}}_{EE}=\int\frac{d\omega}{2\pi}\frac{d^{2}\mathbf{k}_{\perp}}{(2\pi)^{2}}\,\omega\tr\imaginary\bm{\alpha}(\omega)\cdot\imaginary\mathbf{g}^{\prime}(\omega,\mathbf{k}_{\perp})\coth\left[\frac{\beta}{2}\gamma(\omega+k_{x}v)\right].$
(17b)

The $dd$ and $EE$ contributions are summed to yield $P^{\prime}$

$P^{\prime}=\int\frac{d\omega}{2\pi}\frac{d^{2}\mathbf{k}_{\perp}}{(2\pi)^{2}}\,\omega\tr\imaginary\bm{\alpha}(\omega)\cdot\imaginary\mathbf{g}^{\prime}(\omega,\mathbf{k}_{\perp})\left\\{\coth\left[\frac{\beta}{2}\gamma(\omega+k_{x}v)\right]-\coth(\frac{\beta^{\prime}\omega}{2})\right\\}.$
(18)

For the vacuum background, the Green’s dyadic is invariant in different
frames,

$\displaystyle\mathbf{g}^{\prime}(z,z^{\prime};\omega,\mathbf{k}_{\perp})=\mathbf{g}(z,z^{\prime};\omega,\mathbf{k}_{\perp})=\frac{1}{2\kappa}e^{-\kappa|z-z^{\prime}|}$
$\displaystyle\matrixquantity(\omega^{2}-k_{x}^{2}&-k_{x}k_{y}&-ik_{x}\kappa\operatorname{sgn}(z-z^{\prime})\\\
-k_{x}k_{y}&\omega^{2}-k_{y}^{2}&-ik_{y}\kappa\operatorname{sgn}(z-z^{\prime})\\\
ik_{x}\kappa\operatorname{sgn}(z^{\prime}-z)&\quad
ik_{y}\kappa\operatorname{sgn}(z^{\prime}-z)&k^{2}).$ (19)

When the Green’s functions in Eq. (19) are inserted into Eq. (18), it is
immediately seen that the off-diagonal polarizations ($i\neq j$) do not
contribute to $P^{\prime}$ because of the oddness of the integrand in $k_{y}$
or the vanishing of the signum function in the coincident spatial coordinate
limit. The diagonal contributions are found to be, respectively,

$P^{\prime\rm{X}}=\frac{1}{4\pi^{2}\gamma
v}\int_{0}^{\infty}d\omega\imaginary\alpha_{xx}(\omega)\,\omega^{4}\int_{y_{-}}^{y_{+}}dy\left[1-\frac{1}{\gamma^{2}v^{2}}(y-\gamma)^{2}\right]\left[\frac{1}{e^{\beta\omega
y}-1}-\frac{1}{e^{\beta^{\prime}\omega}-1}\right],$ (20a)
$P^{\prime\rm{Y}}=\frac{1}{8\pi^{2}\gamma
v}\int_{0}^{\infty}d\omega\imaginary\alpha_{yy}(\omega)\,\omega^{4}\int_{y_{-}}^{y_{+}}dy\left[1+\frac{1}{\gamma^{2}v^{2}}(y-\gamma)^{2}\right]\left[\frac{1}{e^{\beta\omega
y}-1}-\frac{1}{e^{\beta^{\prime}\omega}-1}\right],$ (20b)

where the integral limits on $y$ are $y_{-}=\gamma(1-v)$ and
$y_{+}=\gamma(1+v)$. Note we have taken advantage of the integrand’s evenness
in $\omega$ and the cancellation of the divergent piece in Eq. (20). If the
particle is isotropic, the contributions to $P^{\prime}$ from all diagonal
polarization states sum to

$P^{\prime\rm{ISO}}=\frac{1}{2\pi^{2}\gamma
v}\int_{0}^{\infty}d\omega\imaginary\alpha(\omega)\,\omega^{4}\int_{y_{-}}^{y_{+}}dy\left[\frac{1}{e^{\beta\omega
y}-1}-\frac{1}{e^{\beta^{\prime}\omega}-1}\right].$ (21)

Using the momentum distribution functions introduced in Ref. [9],

$f^{\rm{P}}(y)=\left\\{\begin{array}[]{lcrcl}\frac{3}{2\gamma
v},&\qquad\rm{P}=\rm{ISO}\\\ \\\ \frac{3}{4\gamma
v}\left[1-\frac{1}{\gamma^{2}v^{2}}(y-\gamma)^{2}\right],&\qquad\rm{P}=\rm{X}\\\
\\\ \frac{3}{8\gamma
v}\left[1+\frac{1}{\gamma^{2}v^{2}}(y-\gamma)^{2}\right],&\qquad\rm{P}=\rm{Y},\rm{Z}\\\
\end{array}\right.$ (22)

these formulas can be summarized as

$P^{\prime\rm{P}}=\frac{1}{3\pi^{2}}\int_{0}^{\infty}d\omega\imaginary\alpha_{\rm{P}}(\omega)\,\omega^{4}\int_{y_{-}}^{y_{+}}dyf^{\rm{P}}(y)\left(\frac{1}{e^{\beta\omega
y}-1}-\frac{1}{e^{\beta^{\prime}\omega}-1}\right).$ (23)

###  II.2 The NESS temperature of the neutral particle and its lower bound

Given a particular model for the particle’s polarizability, the NESS
condition, $P^{\prime P}=0$, defines a special NESS temperature of the
particle $\tilde{T}$ (the corresponding inverse temperature is denoted as
$\tilde{\beta}$) for a fixed radiation temperature $T$ and polarization state
$\rm{P}$.

Interestingly, we find that, independent of the model for
$\imaginary\alpha(\omega)$ and the polarization state of the particle, the
NESS temperature of the particle $\tilde{T}$ must be greater than the Planck-
Einstein transformed temperature of the blackbody radiation $T/\gamma$ [15].
We need the following assumptions to prove this theorem:

$\imaginary\alpha(\omega)\geq 0\quad\rm{and}\quad\lim_{\omega\to
0}\omega^{4}\imaginary\alpha(\omega)=0.$ (24)

The first assumption is usually satisfied by a realistic particle made of
ordinary lossy materials [16]. The second assumption is to avoid an infrared
divergence in the power formula Eq. (23).

Let us define a function

$I(\xi)=\int_{0}^{\infty}d\omega\,\imaginary\alpha(\omega)\,\omega^{4}\frac{1}{e^{\xi\omega}-1}.$
(25)

The general NESS condition $P^{\prime P}=0$ for different polarization states
can then be written as

$\int_{y_{-}}^{y_{+}}dy\,f^{\rm{P}}(y)I(\tilde{\beta})=\int_{y_{-}}^{y_{+}}dy\,f^{\rm{P}}(y)I(\beta
y).$ (26)

Noting $\gamma$ is the midpoint of the interval $[y_{-},y_{+}]$ and $I(\xi)$
is a decreasing and convex function, that is, $I^{\prime}(\xi)<0$ and
$I^{\prime\prime}(\xi)>0$, the following inequality follows,

$\int_{y_{-}}^{y_{+}}dy\,I(\beta y)>\int_{y_{-}}^{y_{+}}dy\,I(\beta\gamma).$
(27)

Because the momentum distribution functions $f^{\rm{P}}(y)$ are all even with
respect to $y=\gamma$, we also have

$\int_{y_{-}}^{y_{+}}dy\,f^{\rm{P}}(y)I(\beta
y)>\int_{y_{-}}^{y_{+}}dy\,f^{\rm{P}}(y)I(\beta\gamma).$ (28)

Combining Eq. (28) and Eq. (26), we find

$\int_{y_{-}}^{y_{+}}dy\,f^{\rm{P}}(y)I(\tilde{\beta})>\int_{y_{-}}^{y_{+}}dy\,f^{\rm{P}}(y)I(\beta\gamma).$
(29)

Since both $I(\tilde{\beta})$ and $I(\beta\gamma)$ are independent of $y$,
they can be taken out of the integral. In addition, the remaining integral
$\int_{y_{-}}^{y_{+}}dy\,f^{\rm{P}}(y)$ is positive definite. It then follows
that $I(\tilde{\beta})>I(\beta\gamma)$. Recalling $I(\xi)$ is a decreasing
function, we conclude

$\tilde{T}>\frac{T}{\gamma}.$ (30)

The theorem therefore predicts a lower bound for the NESS temperature of the
particle.

The nonrelativistic limit of the NESS temperature is also independent of the
model for $\imaginary\alpha(\omega)$ and the polarization state. Expanding the
NESS condition Eq. (26) in $v$ and keeping only the terms first order in $v$,
we find the nonrelativistic limit of the NESS temperature is precisely the
temperature of the surrounding blackbody radiation,

$\tilde{T}\sim T,\qquad v\to 0.$ (31)

Therefore, the deviation of the NESS temperature of the particle from the
temperature of the radiation is a relativistic effect.

###  II.3 The NESS temperature ratio for resonance and monomial models

Let us define the NESS temperature ratio to be

$\tilde{r}=\frac{\tilde{T}}{T}.$ (32)

In order to study the behavior of $\tilde{r}$ concretely, we will discuss two
ideal models here, namely the resonance model and the monomial model. The more
realistic situation that we consider later reduces to these ideal models in
various limits. We will also work with isotropic particles, the momentum
distribution functions for which are much simpler.

The resonance model is characterized by a sharp resonance at $\omega_{0}$,

$\imaginary\alpha(\omega)\propto\delta(\omega-\omega_{0}),$ (33)

where the numerical coefficient of the delta function is irrelevant for
studying the NESS temperature ratio $\tilde{r}$. Inserting Eq. (33) into Eq.
(26) for the isotropic polarization, $\tilde{r}$ can be determined in terms of
the dimensionless frequency $x_{0}=\beta\omega_{0}/2$,

$\tilde{r}=x_{0}\left\\{\coth^{-1}\left[\frac{1}{2\gamma
vx_{0}}\ln\frac{\sinh(x_{0}y_{+})}{\sinh(x_{0}y_{-})}\right]\right\\}^{-1}.$
(34)

The NESS temperature ratio $\tilde{r}$ determined by Eq. (34) is plotted as a
function of velocity $v$ for different $x_{0}$ in Fig. 1a. As seen in Fig. 1a,
the temperature ratio increases with $x_{0}$ for a given velocity $v$.
Therefore, the ratio is bounded below by its zero frequency/high-temperature
limit ($\omega_{0}\to 0$ or $\beta\to 0$),

$\tilde{r}\sim\frac{1}{\gamma v}\ln y_{+},\qquad x_{0}\to 0,$ (35)

which is illustrated by the dashed magenta curve in Fig. 1a. Note that this is
still well above the dashed grey curve, which is the lower bound of the NESS
temperature ratio given by Eq. (30), $1/\gamma$. In addition, independent of
the resonance frequency or radiation temperature, the NESS temperature ratio
$\tilde{r}$ drops to zero as the velocity of the particle approaches the speed
of light. In fact, the ultrarelativistic limit of $\tilde{r}$ can be shown to
be

$\tilde{r}\sim\frac{2x_{0}}{\ln(4\gamma
x_{0})-\ln\ln(\gamma/x_{0})},\qquad\gamma\to\infty,$ (36)

so the decay of NESS temperature of the particle is logarithmic in $\gamma$.

In Fig. 1b, the NESS temperature ratio $\tilde{r}$ is plotted as a function of
$x_{0}$ for various velocities $v$. It is confirmed that $\tilde{r}$ generally
grows with $x_{0}$. The figure also shows that the separatrix between
$\tilde{T}/T>1$ and $\tilde{T}/T<1$ is close to $x_{0}=1.3$ for most
velocities unless the particle becomes quite relativistic. To the order of
$v^{2}$, the position of the separatrix is found to be $x_{0}=1.288$ by
setting $\tilde{r}=1$ in Eq. (34).

(a)

(b)

Figure 1: For the resonance model, the NESS temperature ratio $\tilde{r}$ in
Eq. (34) is illustrated. (a) $\tilde{r}$ is plotted as a function of the
velocity $v$ for various values of $x_{0}$. All of the curves with different
$x_{0}$ drop to $0$ in the limit of $v\to 1$. (b) $\tilde{r}$ is plotted as a
function of $x_{0}$ for different values of $v$.

We now turn to the monomial model,

$\imaginary\alpha(\omega)\propto\omega^{n}.$ (37)

Of course, physically, $n$ must be an odd integer to make sure
$\alpha(t-t^{\prime})$ is real. We will ease this restriction in the following
discussion because the NESS temperature ratio $\tilde{r}$ is mathematically
well defined for monomial models with any real power $n$. However, we will see
in the next section that the frictional force would only be convergent for
monomial models with $n>-4$.

Inserting Eq. (37) into the NESS condition Eq. (26) for the isotropic
polarization, it’s easier to first take the integral on $\omega$ and then that
on $y$ with the result

$\tilde{r}=\left[\frac{1}{2\gamma
v(4+n)}\left(y_{+}^{4+n}-y_{-}^{4+n}\right)\right]^{\frac{1}{5+n}}.$ (38)

In particular, $n=3$ corresponds to the low-frequency radiation-reaction model
[9], for which the NESS temperature ratio is

$\tilde{r}=\left[\gamma^{6}\left(1+5v^{2}+3v^{4}+\frac{1}{7}v^{6}\right)\right]^{\frac{1}{8}},$
(39)

which agrees with the NESS temperature ratio obtained from $P^{\prime}=0$
given in Eq. (8.150) of Ref. [17].

The low velocity approximation of the NESS temperature ratio for a general
power $n$ is readily obtained by expanding Eq. (38) in $v$,

$v\ll 1:\qquad\tilde{r}\to 1+\frac{1}{6}(n+3)v^{2}+\cdots.$ (40)

Just as expected, the NESS temperature of the particle and the radiation
temperature coincide in the nonrelativistic limit. The high velocity behavior
of the NESS temperature ratio, however, is rather different for different
values of $n$,

$\displaystyle\gamma\gg 1$
$\displaystyle:\qquad\tilde{r}\to\left\\{\begin{array}[]{lcrcl}\left[\frac{(2\gamma)^{3+n}}{4+n}\right]^{\frac{1}{5+n}},&\qquad
n>-4\\\ \\\
\left[\frac{1}{-(4+n)}\right]^{\frac{1}{5+n}}\frac{1}{2\gamma},&\qquad n<-4\\\
\\\ \frac{\ln 2\gamma}{\gamma},&\qquad n=-4.\end{array}\right.$ (46)

We note $n=-5$ is included in the second situation, where the ratio evaluates
to $e/2\gamma$. In the limit of $\gamma\to\infty$, the NESS temperature ratio
diverges for $n>-3$ but vanishes for $n<-3$. Only for $n=-3$, as we can see
from the exact expression Eq. (38), the temperature ratio for the $n=-3$ model
equals $1$ at all velocities.

These limits are more clearly seen and confirmed in Fig. 2 where we plot the
NESS temperature ratio $\tilde{r}$ in Eq. (38) as a function of the velocity
$v$ for various powers $n$. At the same velocity, the temperature ratio is
generally greater for a bigger power $n$. Apart from $n=-3$, another special
power is $n=-6$, for which the NESS temperature ratio precisely equals
$1/\gamma$. As a result, the inequality Eq. (30) actually holds beyond the
assumption of the theorem Eq. (24) ($n>-4$ for these monomial models).

Figure 2: For the monomial model, the NESS temperature ratio $\tilde{r}$ in
Eq. (38) is plotted as a function of the velocity $v$ for various powers $n$.
The separatrix $n=-3$ is shown by the solid magenta curve. The curve for
$n=-4$ is $\ln y_{+}/\gamma v$ and the curve for $n=-5$ is $ey_{-}^{1/v}$,
obtained by taking the corresponding limits of Eq. (38). The red dotted curve
for $n=-6$ exactly coincides with $1/\gamma$.

##  III Quantum vacuum friction in NESS

###  III.1 General features of quantum vacuum friction

A formula for quantum friction in a general background, including both the
$\mathbf{dd}$ and $\mathbf{EE}$ contributions has been been worked out in Ref.
[12],

$F=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\frac{d^{2}\mathbf{k}_{\perp}}{(2\pi)^{2}}(k_{x}+\omega
v)\tr\imaginary\bm{\alpha}(\omega)\cdot\imaginary\mathbf{g}^{\prime}(\omega,\mathbf{k}_{\perp})\left[\coth\left(\frac{\beta\gamma(\omega+k_{x}v)}{2}\right)-\coth\left(\frac{\beta^{\prime}\omega}{2}\right)\right].$
(47)

Of course, one can derive the above formula by calculating the Lorentz force
directly. Or, one can find $F$ by applying the principle of virtual work in
frame $\mathcal{R}$, $F=-\partial_{x}\mathcal{F}$. Finally, one can also first
apply the principle of virtual work in frame $\mathcal{P}$,
$F^{\prime}=-\partial_{x^{\prime}}\mathcal{F^{\prime}}$ and then find $F$
using the relation $F=F^{\prime}+P^{\prime}v$ [9]. The equivalence of the
Lorentz force law and the principle of virtual work is established in Appendix
A for a moving electric dipole.

For the vacuum background, we insert the Green’s dyadic in Eq. (19) into Eq.
(47). Again, due to the symmetry of the Green’s dyadic, only the diagonal
polarizations contributes to the quantum vacuum friction. The vacuum
frictional force on a particle only polarizable in direction $\rm{P}$ is found
to be

$F^{\rm{P}}=\frac{1}{6\pi^{2}\gamma
v}\int_{0}^{\infty}d\omega\,\omega^{4}\imaginary\alpha_{\rm{P}}(\omega)\int_{y_{-}}^{y_{+}}dy\left(y-\frac{1}{\gamma}\right)f^{\rm{P}}(y)\left[\coth(\frac{\beta\omega
y}{2})-\coth(\frac{\beta^{\prime}\omega}{2})\right],$ (48)

where the momentum distribution functions $f^{\rm{P}}(y)$ are illustrated in
Eq. (22). In Eq. (48), the first term in the bracket originates from the field
fluctuations ($\mathbf{EE}$) and the second term from the dipole fluctuations
($\mathbf{dd}$). Both the $\mathbf{EE}$ and $\mathbf{dd}$ contributions have
ultraviolet divergent pieces unless $\imaginary\alpha(\omega)$ falls off
faster than $1/\omega^{5}$ for high frequencies. This, however, is not an
issue for the total frictional force because of the exact cancellation of the
divergent pieces between the two contributions. After the cancellation, the
frictional force becomes

$F^{\rm{P}}=\frac{1}{3\pi^{2}\gamma
v}\int_{0}^{\infty}d\omega\,\omega^{4}\imaginary\alpha_{\rm{P}}(\omega)\int_{y_{-}}^{y_{+}}dy\left(y-\frac{1}{\gamma}\right)f^{\rm{P}}(y)\left[\frac{1}{e^{\beta\omega
y}-1}-\frac{1}{e^{\beta^{\prime}\omega}-1}\right].$ (49)

Therefore, it is crucial to keep both $\mathbf{dd}$ and $\mathbf{EE}$
contributions. Otherwise, one would encounter an unphysical divergence in the
quantum vacuum friction. The quantum vacuum friction for the case of an
isotropic particle has been worked out by various authors. Volokitin and
Persson omitted the $\mathbf{dd}$ contribution for the blackbody friction in
one of their early papers on quantum friction [18].444Even though the
frictional force they got in Eq. (49) of Ref. [18] agrees with our NESS
quantum vacuum friction formula Eq. (50) for the isotropic case, it is clear
that they had not imposed the NESS condition and failed to consider an
independent temperature of the particle because of their omission of the
$\mathbf{dd}$ contribution. This error was pointed out by Dedkov and Kyasov in
Ref. [19] and it was corrected in later works of Volokitin and Persson, such
as Ref. [17]. In addition, Pieplow and Henkel [20] also consider both
contributions. The quantum vacuum frictional force we obtain in our Eq. (49)
is in agreement with those found in Refs. [19, 17, 20] where both
$\mathbf{dd}$ and $\mathbf{EE}$ contributions to the force are correctly
included.

In Ref. [12], we have also pointed out that the quantum vacuum frictional
force is not of a definite sign when both dipole and field fluctuations
coexist. Only when some particular model for polarizability is selected, for
example, the monomial model with power $n\geq-2$, is the frictional force on
an isotropic particle shown to be negative definite.555For these monomial
models with power $n$, a simple calculation shows that the field fluctuation
($\mathbf{EE}$) contribution to the quantum vacuum frictional force changes
sign when $n=-2$. To be more specific, the $\mathbf{EE}$ contribution is
positive for $n<-2$, zero for $n=-2$ and negative for $n>-2$. The dipole
fluctuation ($\mathbf{dd}$) contribution is always negative, independent of
the power $n$.

In order to find out the quantum vacuum friction that the particle feels in
NESS, we impose the NESS condition Eq. (26) on Eq. (49) so that the
independent temperature of the particle could be eliminated in the expression
for the force in favor of the the radiation temperature,

$\tilde{F}^{\rm{P}}=\frac{1}{3\pi^{2}\gamma
v}\int_{0}^{\infty}d\omega\,\omega^{4}\imaginary\alpha(\omega)\int_{y_{-}}^{y_{+}}dy\left(y-\gamma\right)f^{\rm{P}}(y)\frac{1}{e^{\beta\omega
y}-1},$ (50)

where the tilde on $F$ is to emphasize that it is the NESS quantum vacuum
friction. We note that the NESS frictional force is negative definite as long
as the neutral particle is made of ordinary lossy material,
$\imaginary\alpha(\omega)\geq 0$, independent of the particular model for the
polarizability.

The polarizability in Eq. (50) should be understood as the effective
polarizability, including all possible sources of dissipation in its imaginary
part. Let us temporarily resume our notation in Ref. [9] for the effective
polarizability, $\hat{\bm{\alpha}}$, for clarity. Suppose the particle is
dissipative only due to its interaction with the surrounding blackbody
radiation, then the imaginary part of the effective polarizability is related
to a real, intrinsic polarizability to the second order as

$\imaginary\hat{\bm{\alpha}}(\omega)=\frac{\omega^{3}}{6\pi}\bm{\alpha}^{2}(\omega).$
(51)

If we substitute $\omega^{3}\alpha_{\rm{P}}^{2}(\omega)/6\pi$ for
$\imaginary\alpha_{\rm{P}}$ in Eq. (50), the second order friction formulas
Eqs. (3.17), (3.20), (3.21), (3.22) in Ref. [9] are recovered precisely.

###  III.2 The limits of quantum vacuum friction in NESS

From Eq. (50), it is not hard to see that the NESS quantum vacuum friction
vanishes both in the zero temperature limit $\beta\to\infty$ and in the zero
velocity limit $v\to 0$.

Let us examine more closely, for the isotropic polarization, the velocity
dependence of the NESS quantum vacuum friction Eq. (50), which is only
contained in the following function

$J(x,v)=\frac{1}{\gamma^{2}v^{2}}\int_{y_{-}}^{y_{+}}dy\frac{y-\gamma}{e^{2xy}-1},\quad
x=\frac{\beta\omega}{2}.$ (52)

The function $J(x,v)$ can be explicitly expressed as

$J(x,v)=\frac{1}{2\gamma
vx}\left[\ln(1-e^{-2xy_{-}})+\ln(1-e^{-2xy_{+}})\right]+\frac{1}{4\gamma^{2}v^{2}x^{2}}\left[\operatorname{Li}_{2}(e^{-2xy_{-}})-\operatorname{Li}_{2}(e^{-2xy_{+}})\right].$
(53)

Though this form for $J(x,v)$ is exact, it is not particularly illuminating.
See Eq. (57) for a useful series representation of it.

In the low velocity limit, $J(x,v)$ can be expanded in series of $v$,

$J(x,v)\sim-\frac{v}{3}\frac{x}{\sinh^{2}(x)}-\frac{v^{3}}{60}\frac{x}{\sinh^{4}(x)}\left[-5+8x^{2}+(5+4x^{2})\cosh(2x)-10x\sinh(2x)\right],$
(54)

where we have kept the first two terms in the series. If we insert only the
linear term in Eq. (54) into Eq. (50), we obtain

$\tilde{F}^{\rm{ISO}}\sim-\frac{v}{6\pi^{2}}\int_{0}^{\infty}d\omega\,\omega^{4}\imaginary\alpha(\omega)\frac{\beta\omega/2}{\sinh^{2}(\beta\omega/2)},$
(55)

which is a generalization of the Einstein-Hopf drag [13, 21, 22]. In Ref.
[12], we obtained this formula by taking the nonrelativistic limit of Eq. (48)
and mandating the particle to have the same temperature as the blackbody
radiation. But now we understand the physics better: the radiation temperature
$T$ is indeed the nonrelativistic limit of the NESS temperature $\tilde{T}$
and the Einstein-Hopf drag is therefore simply the nonrelativistic NESS
quantum vacuum friction!

Let us now turn to the high velocity limit of $J(x,v)$. In Eq. (52), $J(x,v)$
can be recast into a sum,

$J(x,v)=\frac{1}{\gamma^{2}v^{2}}\sum_{k=1}^{\infty}\int_{y_{-}}^{y_{+}}dy\,(y-\gamma)e^{-2xyk}.$
(56)

The integral on $y$ for each term in the sum is easily done, yielding

$J(x,v)=-\frac{1}{\gamma^{2}v^{2}}\sum_{k=1}^{\infty}\left\\{\frac{\gamma
v}{2xk}\left[e^{-2xy_{+}k}+e^{-2xy_{-}k}\right]+\frac{1}{(2xk)^{2}}\left[e^{-2xy_{+}k}-e^{-2xy_{-}k}\right]\right\\}.$
(57)

This expression is so far exact and equivalent to Eq. (53). It can be used to
get an approximation for $J(x,v)$ to arbitrary precision by truncating the
series. In the high velocity limit with $\gamma\gg 1$ and $y_{+}\to 2\gamma$,
we can drop the exponentials involving $y_{+}$ in Eq. (57). If we further
assume $\gamma x\gg 1$ (no assumption on the relative magnitude $\gamma/x$
needs to be made), the expression in the braces can be approximated by only
keeping the second term of the first bracket, after which the sum is easily
written in closed form,

$J(x,v)\sim\frac{1}{2x\gamma v}\ln\left(1-e^{-2xy_{-}}\right).$ (58)

In Fig. 3, we illustrate the velocity dependence of the exact function $J$
together with its low velocity approximation [the linear term in Eq. (54)] and
high velocity approximation [Eq. (58)]. The exact function $J$ clearly
exhibits a minimum. That is to say, the magnitude of the contribution from a
particular frequency $x=\beta\omega/2$ to the NESS quantum vacuum friction is
maximized at an intermediate velocity. For smaller $x$ (e.g., $x=1$), the
minimum of $J$ lies in the transition region between the low velocity regime
and the high velocity regime. For larger $x$ (e.g., $x=10$), the minimum is
well captured by the high velocity approximation in Eq. (58). In the
approximation of $v\to 1$ and $y_{-}\to 1/2\gamma$, the location of the
minimum is found analytically from Eq. (58) to be $\gamma=x/\ln 2$.

(a)

(b)

Figure 3: The velocity dependence of $J$ in Eq. (52) is illustrated. The exact
expression in Eq. (53) is shown by the solid blue curves. The dashed orange
curves show the linear term in Eq. (54). The dashed purple curves show the
high velocity approximation of $J$ in Eq. (58). (a) $J$ is plotted as a
function of $v$ for $x=1$. (b) $J$ is plotted as a function of $\gamma$ for
$x=10$.

Finally, we also want to examine the classical limit of vacuum quantum
friction Eq. (50). In order to render the thermal contributions more manifest,
we temporarily reinsert $\hbar$ in Eq. (50) with change of variable $y=\gamma
z$ and obtain

$\tilde{F}^{\rm{ISO}}=\frac{\hbar}{2\pi^{2}v^{2}}\int_{0}^{\infty}d\omega\,\omega^{4}\imaginary\alpha(\omega)\int_{1-v}^{1+v}dz\frac{z-1}{e^{\hbar\beta\omega\gamma
z}-1}.$ (59)

In the limit of $\beta\gamma\to 0$, that is the high-temperature and modest-
velocity regime, the exponential factor can be expanded in $\beta\gamma$.
Keeping only the first term in the expansion, we find $\hbar$ disappears
explicitly and the integration on $z$ can be readily carried out, leading to

$\tilde{F}^{\rm{ISO}}_{\rm{CL}}=\frac{v-\ln y_{+}}{\pi^{2}\beta\gamma
v^{2}}\int_{0}^{\infty}d\omega\imaginary\alpha(\omega)\,\omega^{3},$ (60)

which can also be derived by using the classical FDT from the outset. In its
nonrelativistic limit, the friction in Eq. (60) further reduces to

$\tilde{F}^{\rm{ISO}}_{\rm{CL},\rm{NR}}=-\frac{v}{3\pi^{2}\beta}\int_{0}^{\infty}d\omega\imaginary\alpha(\omega)\,\omega^{3},$
(61)

linear in temperature and in velocity.666Obviously, the above arguments for
the high-temperature limit of quantum vacuum friction do not apply to the
situation when the imaginary part of the polarizability is temperature
dependent, e.g., in the Bloch-Grüneisen model discussed in Sec. IV.2. In
addition, $\imaginary\alpha(\omega)$ must be properly behaved in both infrared
and ultravilolet regimes so that the $\omega$ integral in Eq. (61) is
convergent.

###  III.3 The NESS quantum vacuum friction for resonance model and monomial
model

As simple examples, let us work out the NESS quantum vacuum friction for the
resonance and monomial models.

In order to evaluate the NESS quantum vacuum friction for the resonance model
in Eq. (33), the coefficient of the delta function is needed. So, let us be
specific and consider a simple model of an harmonically bound electron with
vanishing damping for the polarizability. See, for example, Ref. [23, 24] for
details. The imaginary part of the polarizability is therefore

$\imaginary\alpha(\omega)=\frac{\pi
e^{2}}{2m\omega_{0}}\delta(\omega-\omega_{0}),$ (62)

where $e$ and $m$ are the charge and mass of the electron and $\omega_{0}>0$
is the resonance frequency. We will assume the polarizability to be isotropic
for simplicity here. Inserting Eq. (62) into Eq. (50) with the dimensionless
variables $x_{0}=\beta\omega_{0}/2$, we obtain

$\tilde{F}^{\rm{ISO}}=\frac{2e^{2}}{\pi m\beta^{3}}x_{0}^{3}J(x_{0},v),$ (63)

where $J$ is defined in Eq. (52).

At the room temperature $T=300\,\rm{K}$, the dimensional coefficient in Eq.
(63) is evaluated as $1.60\crossproduct 10^{-24}\rm{N}$, 777The conversion
factors used in the estimate are $k_{B}=8.62\crossproduct 10^{-5}\rm{eV/K}$
and $\hbar c=1.97\crossproduct 10^{-5}\,\rm{eV}\cdot\rm{cm}$. after converting
to SI units. The velocity dependence of the friction is completely controlled
by the function $J$ illustrated in Fig. 3. Here, we illustrate the $x_{0}$
dependence of the magnitude of the friction in Eq. (63) for different
velocities in Fig. 4.

Figure 4: For the resonance model in Eq. (62), the magnitude of the NESS
quantum vacuum friciton $-\tilde{F}^{\rm{ISO}}$ in Eq. (63) at room
temperature $T=300\,\rm{K}$ is plotted as a function of $x_{0}$ for different
velocities.

As is seen in the figure, the magnitude of the friction exibits a peak at a
certain frequency ($x_{0}$) for each fixed velocity. And for greater
velocities, the peak is higher with the position of the peak blueshifted. The
resultant quantum friction for the typical velocities shown in the figure is
on the order of $10^{-25}\,\rm{N}$, which is still far from the reach of
current experiments.

Next, we turn to the monomial model in Eq. (37). To be specific, let us
consider the low-frequency radiation reaction model with $n=3$,

$\imaginary\alpha(\omega)=\frac{\omega^{3}}{6\pi}\alpha_{0}^{2}.$ (64)

After inserting this model into Eq. (50), again it’s easier to first work out
the integral on $\omega$ followed by the integral on $y$. The quantum vacuum
friction for various polarization states is found to be

$\tilde{F}^{\rm{P}}=-\frac{32\alpha_{0}^{2}\pi^{5}}{4725\beta^{8}}\gamma^{6}v\left\\{\begin{array}[]{lcrcl}15v^{4}+70v^{2}+35,&\qquad\rm{P}=\rm{ISO},\\\
\\\ -3v^{4}-4v^{2}+7,&\qquad\rm{P}=\rm{X},\\\ \\\
9v^{4}+37v^{2}+14,&\qquad\rm{P}=\rm{Y,Z}.\\\ \end{array}\right.$ (65)

The expected identity
$\tilde{F}^{\rm{ISO}}=\tilde{F}^{\rm{X}}+\tilde{F}^{\rm{Y}}+\tilde{F}^{\rm{Z}}$
can be immediately confirmed.

For a neutral gold atom, the recommmended value of its static scalar
polarizability found in Ref. [25] is $36\,\rm{a.u.}=5.33\crossproduct
10^{-24}\rm{cm^{3}}$. Using this value for $\alpha_{0}$, the dimensional
factor in Eq. (65) is estimated to be $-1.63\crossproduct 10^{-43}\,\rm{N}$ at
$T=300\,\rm{K}$. We plot the magnitude of the resultant NESS quantum vacuum
friction on a gold atom as a function of $v$ for different polarizations in
Fig. 5. From the figure, it is seen that the friction increases with velocity
and the friction for the parallel ($x$) polarization is always smaller than
that for the transverse ($y$ and $z$) polarizations.

Figure 5: For the low-frequency radiation reaction model in Eq. (64), the
magnitude of the NESS quantum vacuum friction $-\tilde{F}^{\rm{P}}$ for
different polarization states in Eq. (65) are plotted as functions of $v$ at
room temperature $T=300\,\rm{K}$. The static polarizability of a gold atom is
used for $\alpha_{0}$ in the numerical evaluation.

If the gold atom is replaced by a perfectly conducting sphere, the
polarizability becomes directly proportional to the volume of the sphere,
$\alpha_{0}=4\pi a^{3}$. The magnitude of the friction can be enhanced for a
sphere bigger than an atom. For example, the dimensional prefactor in Eq. (65)
becomes $9.09\crossproduct 10^{-25}\,\rm{N}$ for a perfectly conducting sphere
of radius $a=100\,\rm{nm}$, much greater than that for the gold atom.

Of course, the radiation reaction model in Eq. (64) is only a low-frequency
approximation of the more complete model already discussed in Sec. V of Ref.
[9]. Considering the exponential factor in Eq. (50), the low-frequency
contributions dominate so long as the temperature is low.888For a gold atom
moving at a velocity $v=0.5$, the low frequency approximation is accurate up
to a temperature $T=10^{6}\,\rm{K}$. See Fig. 2b of Ref. [9] Indeed, the
friction for isotropic polarization in Eq. (65) agrees with the low-
temperature friction obtained in Eq. (5.4a) of Ref. [9].

Contrary to the resonance model, quantum friction for the model in Eq. (64)
($n=3$ monomial model) diverges as the velocity approaches the speed of light.
This turns out to be true for all monomial models with a power $n>-3$. For a
monomial model with an arbitrary power $n$,
$\imaginary\alpha(\omega)=\alpha_{n}\omega^{n}$ ($\alpha_{n}$ is a constant in
the model which has the proper dimension according to the power $n$), the NESS
quantum friction for the isotropic polarization can be calculated explicitly
from Eq. (50),

$\tilde{F}^{\rm{ISO}}=\frac{\alpha_{n}\Gamma(5+n)\zeta(5+n)}{2\pi^{2}\beta^{5+n}}M_{n}(v),\quad
M_{n}(v)\equiv\frac{1}{\gamma^{2}v^{2}}\left[\frac{1}{3+n}\left(y_{+}^{3+n}-y_{-}^{3+n}\right)-\frac{\gamma}{4+n}\left(y_{+}^{4+n}-y_{-}^{4+n}\right)\right].$
(66)

The velocity dependence of the friction in Eq. (66) is all contained in the
function $M_{n}(v)$. The high velocity behavior ($\gamma\to\infty$) of $M_{n}$
depends on $n$,

$\lim_{\gamma\to\infty}M_{n}=\left\\{\begin{array}[]{lcrcl}-\frac{2^{4+n}}{4+n}\gamma^{(3+n)},&\qquad
n>-4\\\ \\\
\left(\frac{2^{-(4+n)}}{4+n}-\frac{2^{-(3+n)}}{3+n}\right)\gamma^{-(5+n)},&\qquad
n<-4\\\ \\\ -\frac{2}{\gamma}\ln\gamma,&\qquad n=-4.\\\ \end{array}\right.$
(67)

Even though $M_{n}$ can be defined for an arbitrary value of $n$,
$\tilde{F}^{\rm{ISO}}$ in Eq. (66) is not well defined for negative integers
$n\leq-4$ due to the singularities of the prefactor. As a result, for integer
powers $n$, the friction is almost always divergent in the high velocity
limit, except for $n=-3$,

$\tilde{F}^{\rm{ISO}}=\frac{\alpha_{-3}}{12\beta^{2}}\frac{1}{\gamma^{2}v^{2}}\left[\ln(\frac{1+v}{1-v})-2\gamma^{2}v\right].$
(68)

It is not surprising that $n=-3$ is special, recalling that the corresponding
NESS temperature ratio $\tilde{r}=1$ for all velocities. In addition, the full
radiation reaction model in Ref. [9] reduces to $n=-3$ monomial model in the
high-frequency limit. Indeed, the friction in Eq. (68) precisely equals the
high-temperature quantum vacuum friction recorded in Eq. (5.4b) of Ref. [9] if
we set $\alpha_{-3}=6\pi$ in accordance with Eq. (5.1) of Ref. [9].

##  IV NESS temperature ratio and NESS friction for a gold nanosphere

As a realistic example, we will in this section evaluate the NESS temperature
ratio and friction for a nanosphere made of gold. For simplicity, we assume
the nanosphere is isotropic and ignore any surface effect of the nanosphere.
The polarizability of the nanosphere $\alpha(\omega)$ can be expressed in
terms of its radius $a$ and the permittivity of gold $\varepsilon(\omega)$
through the Lorenz-Lorentz relation

$\alpha(\omega)=4\pi
a^{3}\,\frac{\varepsilon(\omega)-1}{\varepsilon(\omega)+2}.$ (69)

An introduction of the Lorenz-Lorentz relation can be found in Ref. [23]. The
permittivity of gold $\varepsilon(\omega)$ is often described by the Drude
model

$\varepsilon(\omega)=1-\frac{\omega_{p}^{2}}{\omega^{2}+i\omega\nu},$ (70)

where $\omega_{p}$ is the plasma frequency and $\nu$ the damping parameter of
gold. In IV.1, the damping parameter will be treated as a temperature-
independent constant. In IV.2, we will instead consider the damping parameter
to be temperature dependent and describe it using the Bloch-Grüneisen model.

###  IV.1 Constant damping model

Combining Eq. (69) and Eq. (70), we find

$\imaginary\alpha(\omega)=V\frac{\omega_{p}^{2}\,\omega\nu}{(\omega_{1}^{2}-\omega^{2})^{2}+\omega^{2}\nu^{2}}.$
(71)

where $V=\frac{4}{3}\pi a^{3}$ denotes the volume of the nanosphere and we
have introduced the rescaled plasma frequency
$\omega_{1}=\omega_{p}/\sqrt{3}$. The imaginary part of the polarizability is
temperature independent if all the parameters in Eq. (71) are constant. In the
following numerical evaluation, we use the nominal room temperature value for
the plasma frequency and damping parameter given in Ref. [26]:
$\omega_{p}=9.00\,\rm{eV}$ and $\nu=0.0350\,\rm{eV}$.

When the gold nanosphere moves through vacuum in a constant velocity $v$, the
NESS temperature $\tilde{T}$ deviates from the temperature of the blackbody
radiation $T$ determined by the NESS condition Eq. (26). And in NESS, the
nanosphere experiences a frictional drag given by Eq. (50).

In order to evaluate the NESS temperature ratio $\tilde{r}=\tilde{T}/T$, we
insert Eq. (71) into Eq. (26) and introduce a new set of dimensionless
variables

$u=\frac{\omega}{\omega_{1}},\qquad\epsilon=\frac{\nu}{\omega_{1}},\qquad
x_{1}=\frac{\beta\omega_{1}}{2},\qquad\tilde{r}=\frac{\beta}{\tilde{\beta}}.$
(72)

We therefore obtain

$\int_{0}^{\infty}du\frac{u^{5}}{(1-u^{2})^{2}+u^{2}\epsilon^{2}}\frac{1}{e^{2x_{1}u/\tilde{r}}-1}=\int_{0}^{\infty}du\frac{u^{5}}{(1-u^{2})^{2}+u^{2}\epsilon^{2}}\frac{1}{4\gamma
vx_{1}u}\ln\left(\frac{1-e^{-2x_{1}y_{+}u}}{1-e^{-2x_{1}y_{-}u}}\right),$ (73)

which can be solved to determine the NESS temperature ratio $\tilde{r}$
numerically. Due to the cancellation of the volume factor on both sides of the
equation, the temperature ratio is independent of the size of the nanosphere
$a$. The dimensionless damping for gold is $\epsilon=0.00673$; the temperature
ratio $\tilde{r}$ then depends only on the velocity of the nanosphere $v$ and
the temperature of the blackbody radiation $T$.

(a)

(b)

Figure 6: The velocity dependence and temperature dependence of the NESS
temperature ratio $\tilde{r}$ for a gold nanosphere with constant damping
parameter (CD) is illustrated. (a) At three different temperatures of the
blackbody radiation $T=300\,\rm{K}$ $(x_{1}=101)$, $T=3000\,\rm{K}$
$(x_{1}=10.1)$ and $T=30\,000\,\rm{K}$ $(x_{1}=1.01)$, $\tilde{r}$ computed
numerically from Eq. (73) is plotted as a function of velocity for $0.01\leq
v\leq 0.99$. The dashed gray curve shows the lower bound of the temperature
ratio $1/\gamma$ given by the theorem in Eq. (30). (b) At fixed velocity
$v=0.5$, $\tilde{r}$ is plotted as a function of the radiation temperature for
the constant damping model (solid, blue), the $n=1$ monomial model (dashed,
black), the $n=-3$ monomial model (dashed, magenta) and the resonance model
(dotted, purple).

In Fig. 6a, we plot the NESS temperature ratio $\tilde{r}$ as a function of
velocity at different temperatures of the blackbody radiation. For
$T=300\,\rm{K}$ and $T=3000\,\rm{K}$, $\tilde{r}$ is greater than $1$ and
increasing with velocity while it is less than $1$ and decreasing with
velocity for $T=30\,000\,\rm{K}$. Increasing the background temperature from
$T=300\,\rm{K}$ to $T=3000\,\rm{K}$ enhances the temperature ratio for modest
velocities but suppresses it for sufficiently high velocities.

Figure 6b shows the NESS temperature ratio $\tilde{r}$ as a function of
radiation temperature $T$ for a fixed velocity $v=0.5$ for various models. The
NESS temperature ratio of the gold nanosphere with constant damping parameter
is shown by the blue curve, which has a peak at $T=2650\,\rm{K}$. And at the
peak, the deviation of the NESS temperature of the nanosphere from the
temperature of the environment reaches $36\%$, being quite noticeable.

It is also seen from Fig. 6b that the constant damping model for the
nanosphere in various temperature regimes can be mimicked by the simple models
discussed earlier. In the low-temperature regime (lower than
$10^{3}\,\rm{K}$), $\tilde{r}$ for the nanosphere (CD model) almost remains
constant $\tilde{r}=1.18$, the same as the NESS temperature ratio for the
$n=1$ monomial model with $v=0.5$. In the intermediate temperature regime
(roughly from the peak to the valley of the blue curve), the behavior of
$\tilde{r}$ for the nanosphere is well captured by the resonance model. In the
high-temperature regime (higher than $10^{8}\,\rm{K}$), $\tilde{r}$ for the
nanosphere reaches and stablizes at $\tilde{r}=1$, which coincides with the
NESS temperature ratio for the very special $n=-3$ monomial model.

Apart from the numerical result, the above behavior of $\tilde{r}$ as a
solution to Eq. (73) can be understood qualitatively. Due to the exponential
factor on both sides of Eq. (73), at a modest velocity (i.e., $v=0.5$), the
small $u$ (frequency) contribution dominates the integral at low temperatures
while the large $u$ (frequency) contribution dominates the integral at high
temperatures. In Eq. (71), $\imaginary\alpha(\omega)$ reduces to the $n=1$
monomial model and the $n=-3$ monomial model in the low and high-frequency
limits, respectively. In addition, due to the smallness of the damping
$\epsilon$, there ought to be a region where the integral is dominated by the
resonance contributions around $u=1$, corresponding to a resonance model with
the resonance frequency $\omega_{1}$.

Let’s now calculate the NESS quantum vacuum friction on the moving gold
nanosphere with a constant damping parameter. After inserting Eq. (71) into
Eq. (50) and introducing the dimensionless variables in Eq. (72), we find the
quantum vacuum friction to be

$\tilde{F}^{\rm{ISO}}=\frac{V\omega_{p}^{2}\omega_{1}^{3}}{2\pi^{2}\gamma^{2}v^{2}}\int_{0}^{\infty}du\frac{u^{5}\epsilon}{(1-u^{2})^{2}+u^{2}\epsilon^{2}}\int_{y_{-}}^{y_{+}}dy\,(y-\gamma)\frac{1}{e^{2x_{1}uy}-1}.$
(74)

For a gold nanosphere of radius $a=100\,\rm{nm}$, the dimensional coefficient
independent of $v$ and $T$ is evaluated to be $2.57\crossproduct
10^{-10}\rm{N}$.

(a)

(b)

(c)

(d)

Figure 7: The magnitude of the NESS quantum vacuum friction in Eq. (74),
$-\tilde{F}^{\rm{ISO}}$, for a gold nanosphere with constant damping parameter
is illustrated. (a) At three different temperatures of the blackbody radiation
$T=300\,\rm{K}$ $(x_{1}=101)$, $T=3000\,\rm{K}$ $(x_{1}=10.1)$ and
$T=30\,000\,\rm{K}$ $(x_{1}=1.01)$, $-\tilde{F}^{\rm{ISO}}$ is plotted as a
function of velocity for $0.01\leq v\leq 0.99$. (b) For three different
velocities $v=0.1$, $v=0.5$ and $v=0.9$, $-\tilde{F}^{\rm{ISO}}$ is plotted as
a function of the temperature of the blackbody radiation $T$. (c) For
$T=30\,000\,\rm{K}$, $-\tilde{F}^{\rm{ISO}}$ in Eq. (74) is shown again as a
function of $\gamma$ by the solid red curve. The dashed blue curve shows its
high velocity approximation using Eq. (58). The dotted purple curve plots only
the contributions coming from the resonance at $u=1$. (d) For $T=300\,\rm{K}$,
the exact friction is shown again as a function of $\gamma$ by the solid blue
curve. The dashed red curve plots the high velocity approximation of the
friction using Eq. (58). The dotted purple curve plots the resonance
contributions.

In Fig. 7a, the magnitude of the NESS quantum vacuum friction
$-\tilde{F}^{\rm{ISO}}$ is plotted as a function of velocity $v$ for the three
different radiation temperatures. Even though $-\tilde{F}^{\rm{ISO}}$
increases with velocity for $T=300\,\rm{K}$ and $T=3000\,\rm{K}$ in the
velocity range shown ($0.01\leq v\leq 0.99$), it does not monotonically
increase with velocity for $T=30\,000\,\rm{K}$.

Figure 7b plots $-\tilde{F}^{\rm{ISO}}$ as a function of the radiation
temperature for fixed velocities. It is seen that the magnitude of the NESS
quantum vacuum friction generally increases with temperature and the
temperature effects are more prominent in the low-temperature regime than the
high-temperature regime. Again, this can be understood as a result of the
different limiting behavior of the model for the nanosphere in Eq. (71). It
reduces to the $n=1$ monomial model in the low-temperature regime and the
$n=-3$ monomial model in the high-temperature regime. In Eq. (66), the NESS
quantum vacuum friction for a monomial model with power $n$ is found to be
proportional to $T^{5+n}$. Therefore, the temperature dependence of NESS
friction on the nanosphere is $T^{6}$ for low temperatures and then weakens to
$T^{2}$ for high temperatures.

In order to illustrate the high velocity behavior of the friction more
clearly, we plot the magnitude of the NESS quantum vacuum friction as a
function of $\gamma$ in Figs. 7c and 7d. Figure 7c illustrates that, for
$T=30\,000\,\rm{K}$, the peak of the magnitude of the friction occurs at
$\gamma=2.25$ and the behavior of the friction around the peak can be captured
by the resonance contribution due to the smallness of the damping parameter.
This reminds us of the nonmonotonic behavior of the classical friction on a
charged particle passing above a conducting plate [27], also due to the small
damping of the plate. For even higher $\gamma$, the resonance approximation
starts to show discrepancy and the actual friction is better approximated
using the high velocity approximation Eq. (58). Figure 7d further reveals that
the nonmonotonic behavior of the friction also occurs for lower temperatures,
only with the peak shifted towards more relativistic velocities. For low
temperatures, contributions to the friction from frequencies lower than the
resonance frequency tend to dominate. Therefore, the resonance approximation
becomes less accurate unless the velocity gets really large.

###  IV.2 Bloch-Grüneisen model

Figure 8: The temperature-dependent damping parameter $\nu$ in Eq. (75) is
illustrated for gold with $\theta=175\,\rm{K}$ and $\nu_{0}=0.0832\,\rm{eV}$.

Even though the plasma frequency $\omega_{p}$ has very weak temperature
dependence, it is more realistic to adopt a temperature-dependent damping
parameter $\nu$ in Eq. (71). Damping of a simple metal is mainly due to the
scattering of electrons by phonons and can be well described by the Bloch-
Grüneisen (BG) model [28, 29],

$\nu(T)=\nu_{0}\left(\frac{T}{\theta}\right)^{5}\int_{0}^{\frac{\theta}{T}}dx\frac{x^{5}e^{x}}{(e^{x}-1)^{2}}.$
(75)

For gold, the Bloch-Grüneisen temperature $\theta$ is $175\,\rm{K}$. And the
constant $\nu_{0}$ in Eq. (75) is determined to be $0.0832\,\rm{eV}$ by the
room temperature ($300\,\rm{K}$) value of the damping parameter
$\nu=0.0350\,\rm{eV}$ [26, 30, 31].999The value for $\nu_{0}$ we use is
slightly different than that in Appendix D of [26] where the room temperature
is taken to be $295\,\rm{K}$. There is, of course, no definite consensus on
the meaning of the room temperature. Nonetheless, taking it to be
$300\,\rm{K}$ is more consistent with the source of the raw data in [31]. The
low and high temperature limits of the Bloch-Grüneisen damping can be easily
worked out,

$\nu(T)\to\left\\{\begin{array}[]{lcrcl}5\Gamma(5)\zeta(5)\nu_{0}\left(\frac{T}{\theta}\right)^{5},&\qquad
T\ll\theta,\\\ \\\ \frac{\nu_{0}}{4}\left(\frac{T}{\theta}\right),&\qquad
T\gg\theta.\\\ \end{array}\right.$ (76)

In Fig. 8, we plot the Bloch-Grüneisen damping parameter for gold as a
function of temperature. The transition between the low and high temperature
behavior of $\nu$ is seen to occur at a rather low temperature around
$T=40\,\rm{K}$.

(a)

(b)

Figure 9: The NESS temperature ratio $\tilde{r}$ for the BG model is compared
with that for the CD model. (a) At three different temperatures of the
blackbody radiation, $\tilde{r}$ is plotted as a function of $v$ for the BG
model and the CD model by the dotted curves and the solid curves,
respectively. The high-temperature prediction of the NESS temperature ratio
for the BG model is plotted by the dashed, gray curve. (b) At fixed velocity
$v=0.5$, $\tilde{r}$ is plotted as a function of the radiation temperature for
the CD model (solid, blue), the $n=1$ monomial model (dashed, black), the
$n=-3$ monomial model (dashed, magenta), the resonance model (dotted, purple)
and the BG model (dotted, red). The high-temperature prediction of $\tilde{r}$
for the BG model at $v=0.5$ is $1.03$, plotted by a dashed grey curve.

Incorporating the temperature dependence of the damping parameter,
$\imaginary\alpha(\omega)$ in Eq. (71) becomes dependent on the temperature of
the gold nanosphere $T^{\prime}$,

$\imaginary\alpha(\omega,T^{\prime})=V\frac{\omega_{p}^{2}\,\omega\nu(T^{\prime})}{(\omega_{1}^{2}-\omega^{2})^{2}+\omega^{2}\nu^{2}(T^{\prime})}.$
(77)

Equation (73) can still be used to find the NESS temperature ratio with the
only modification that the dimensionless damping $\epsilon$ now also depends
on the temperature of the nanosphere,
$\epsilon\to\epsilon(\tilde{T})=\epsilon(\tilde{r}T)$. The modified Eq. (73)
is numerically solved for the NESS temperature ratio of the gold nanosphere
with the temperature-dependent damping described by the BG model and the
results are compared with that for the CD model in Fig. 9.

Figure 9a shows that, at $T=300\,\rm{K}$, $\tilde{r}$ is almost the same for
the two models even though the temperature-dependent damping $\epsilon$ is
evaluated at $\tilde{r}T$ and will be different from the constant value used
for the CD model. This indicates that, at room temperatures, the NESS
temperature ratio $\tilde{r}$ is rather insensitive to the actual value of the
damping. The NESS temperature ratio for the BG model is seen to be generally
smaller than that for the CD model at $T=3000\,\rm{K}$ but larger at
$T=30\,000\,\rm{K}$.

The phenomenon is more obvious in Fig. 9b where we see both the peak ($\sim
3200\,\rm{K}$) and the valley ($\sim 42000\,\rm{K}$) of the NESS temperature
ratio are softened for the BG model in comparison to the CD model. In
addition, the behavior of the BG and CD models in the two extreme temperature
limits are also clearly shown in the figure. In the low-temperature limit, the
NESS temperature ratio shows no difference between the two models and both
models can be well approximated by the $n=1$ monomial model. In the high-
temperature limit, however, the NESS temperature ratio for the BG model is
raised to a constant above 1, while $\tilde{r}$ for the CD model approaches
$1$, which is the NESS temperature ratio for the $n=-3$ monomial model.

We can understand all these phenomena qualitatively as well. For temperatures
higher than room temperature, the damping $\epsilon$ in Eq. (73) must be
bigger for the BG model than the CD model, which will weaken the effect of the
resonance. As to why adding the temperature dependence in damping would alter
the high-temperature limit of $\tilde{r}$ but keep its low-temperature limit
unchanged, let us re-examine what happens to $\imaginary\alpha(\omega)$ in Eq.
(77) in both limits. Recall the damping $\nu(T^{\prime})$ grows as $T^{\prime
5}$ in the low-temperature limit and grows linearly in the high-temperature
limit. Therefore, the low-frequency (temperature) behavior for
$\imaginary\alpha(\omega)$ is still proportional to $\omega$. Yet, the high-
frequency behavior is no longer just proportional to $\omega^{-3}$, because in
the denominator, $\omega^{2}\nu^{2}$ can be comparable to $\omega^{4}$ in the
high-temperature limit. Indeed, keeping both terms enables us to provide an
analytical prediction of the correct high-temperature limit of $\tilde{r}$ for
the BG model, as is shown by the dashed, gray curves in both figures. The
analysis is detailed in Appendix B where $\tilde{r}$ is given as a solution of
an algebraic equation (106).

Finally, let us also calculate the NESS quantum vacuum friction for the BG
model. In this regard, we can still use Eq. (74), replacing the temperature-
independent damping $\epsilon$ with the temperature-dependent one evaluated at
the NESS temperature, $\epsilon(\tilde{T})$. The NESS temperature can be found
through $\tilde{T}=\tilde{r}T$, where $\tilde{r}$ has already been found
numerically as a function of velocity $v$ and radiation temperature $T$.

(a)

(b)

Figure 10: The magnitude of the NESS quantum vacuum friction,
$-\tilde{F}^{\rm{ISO}}$, for the BG model is compared with that for the CD
model. (a) At the three different temperatures, $-\tilde{F}^{\rm{ISO}}$ is
plotted as a function of velocity $v$ for the BG model and the CD model by the
dotted curves and the solid curves, respectively. (b) For a fixed velocity
$v=0.5$, $-\tilde{F}^{\rm{ISO}}$ is plotted as a function of the radiation
temperature $T$ for the BG model and the CD model by the dotted curves and the
solid curves, respectively. The high-temperature prediction of
$-\tilde{F}^{\rm{ISO}}$ for the BG model is plotted by the dashed, gray curve.

In Fig. 10a, the magnitude of the NESS quantum friction is plotted as a
function of velocity for both the BG model and the CD model. The behavior of
the discrepancy between the two models seems intricate. At $T=300\,\rm{K}$,
the discrepancy is larger for higher velocities. At $T=3000\,\rm{K}$, the
discrepancy is larger for smaller velocities instead. At $T=30\,000\,\rm{K}$,
the data for the two models pretty much agrees until the velocity becomes very
close to the speed of light. All of this can be understood as a result of
different limiting behaviors of the nanosphere model in different temperature
and velocity regimes. As the temperature increases, the behavior of the model
goes from the $n=1$ monomial model through the resonance model and finally to
the $n=3$ monomial model. Since the friction as seen in Eq. (74) is
independent of damping for the resonance model, the discrepancies between the
BG model and the CD model disappear when the resonant frequency dominates. And
whenever the model is better approximated by the monomial models, the friction
is proportional to the damping parameter. The damping for the BG model is
certainly bigger than that for the CD model at temperatures higher than
$300\,\rm{K}$. This is even the case when the radiation temperature is exactly
$300\,\rm{K}$ because, for the BG model, the damping is evaluated at the
temperature of the particle, which is still greater than $300\,\rm{K}$.

In Fig. 10b where the magnitude of quantum friction over a wide range of
temperature is plotted, it is seen that the temperature dependence in damping
lowers the friction for very low temperatures ($<100\,\rm{K}$) because of a
smaller damping parameter for the BG model. In the intermediate temperature
regime, the temperature dependence of the damping has a relatively smaller
effect on the magnitude of the friction. Whether we use the CD model or the BG
model for the gold nanosphere, the magnitude of the quantum vacuum friction on
the nanosphere reaches the order of picoNewtons101010It may be possible to
detect a fluctuation-induced force of such magnitude, since the precision
measurement for the static Casimir force can easily reach the order of
picoNewtons nowadays. See, for example, Ref. [32]. The main challenge for
experiments might then be keeping the particle moving at constant velocity in
a rather hot background. if it can be kept moving in a background with a
temperature as high as $10^{4}\,\rm{K}$, which is unambiguously much greater
than that for a gold atom calculated in Ref. [9]. Adding the temperature
dependence in the damping, the high-temperature behavior of the force is
enhanced from $T^{2}$ to $T^{3}$, which is clearly illustrated in the high-
temperature prediction of the NESS friction force given in Eq. (108).

##  V energetics of quantum vacuum friction out of NESS

The transformations of the frictional power and force between frame
$\mathcal{R}$ and frame $\mathcal{P}$ are [9]

$P=\frac{\partial}{\partial t}\mathcal{F}=\gamma\left(\frac{\partial}{\partial
t^{\prime}}-v\frac{\partial}{\partial
x^{\prime}}\right)\frac{1}{\gamma}\mathcal{F^{\prime}}=\left(\frac{\partial}{\partial
t^{\prime}}-v\frac{\partial}{\partial
x^{\prime}}\right)\mathcal{F^{\prime}}=P^{\prime}+vF^{\prime},$ (78a)
$F=-\frac{\partial}{\partial
x}\mathcal{F}=-\gamma\left(\frac{\partial}{\partial
x^{\prime}}-v\frac{\partial}{\partial
t^{\prime}}\right)\frac{1}{\gamma}\mathcal{F^{\prime}}=-\left(\frac{\partial}{\partial
x^{\prime}}-v\frac{\partial}{\partial
t^{\prime}}\right)\mathcal{F^{\prime}}=F^{\prime}+vP^{\prime}.$ (78b)

These relations are general and hold whether the particle is in or out of
NESS. They can be easily inverted as

$P^{\prime}=\gamma^{2}(P-vF),\qquad F^{\prime}=\gamma^{2}(F-vP).$ (79)

From Eq. (79), together with Eq. (23) for $P^{\prime\rm{P}}$ and Eq. (49) for
$F^{\rm{P}}$, we find the frictional power in frame $\mathcal{R}$ for a
particle in diagonal polarization state $\rm{P}$ is

$P^{\rm{P}}=\frac{1}{3\pi^{2}\gamma}\int_{0}^{\infty}d\omega\imaginary\alpha_{\rm{P}}(\omega)\,\omega^{4}\int_{y_{-}}^{y_{+}}dy\,yf^{\rm{P}}(y)\left(\frac{1}{e^{\beta\omega
y}-1}-\frac{1}{e^{\beta^{\prime}\omega}-1}\right).$ (80)

In the case of an isotropic particle, we confirm that this expression for
frictional power agrees with the time component of the four-force that Pieplow
and Henkel derive for blackbody friction in Ref. [20] using a fully covariant
formulation, and with the thermal radiation power (a different sign convention
is used there, though) obtained by Dedkov and Kyasov in Ref. [33].

When the particle is in NESS, the friction must be balanced by an external
force, in either frame $\mathcal{R}$ or frame $\mathcal{P}$,

$\tilde{F}_{\rm{tot}}=\tilde{F}+\tilde{F}_{\rm{ext}}=0,$ (81a)
$\tilde{F}_{\rm{tot}}^{\prime}=\tilde{F}^{\prime}+\tilde{F}^{\prime}_{\rm{ext}}=0.$
(81b)

Recall that in NESS, the friction is the same in frame $\mathcal{P}$ and frame
$\mathcal{R}$, $\tilde{F}^{\prime}=\tilde{F}$ [9]. As a result, the external
force needed is the same in frame $\mathcal{P}$ and frame $\mathcal{R}$,
$\tilde{F}_{\rm{ext}}^{\prime}=\tilde{F}_{\rm{ext}}$. Since we have
demonstrated that $\tilde{F}$ is negative definite, the external force is
positive definite in both frames,
$\tilde{F}_{\rm{ext}}^{\prime}=\tilde{F}_{\rm{ext}}>0$.

Now, let us discuss the different situation out of NESS. On the one hand, the
internal energy of the particle, or equivalently, the rest mass of the
particle, $m$, is allowed to change.111111The varying mass for moving dipoles
has also been discussed pedagogically in Ref. [34]. A quantum mechanical
illustration can be found in Ref. [35]. Therefore, the relativistic momentum
of the particle $\gamma mv$ in $\mathcal{R}$ can change, while the velocity
$v$ is kept constant by the external force. The varying mass results in a net
force on the particle in frame $\mathcal{R}$, so that Eq. (81a) must be
modified out of NESS,

$F_{\rm{tot}}=F+F_{\rm{ext}}=\gamma
v\frac{dm}{dt}=v\frac{dm}{dt^{\prime}}=vP^{\prime}\,\Rightarrow\,F_{\rm{ext}}=-F+vP^{\prime},$
(82)

where we have used the time dilation relation $dt=\gamma dt^{\prime}$ and that
the rate of rest mass change, $dm/dt^{\prime}$ is the same as the rate of
change in the particle’s internal energy, $P^{\prime}$. The rate $P^{\prime}$,
being precisely the frictional power in the rest frame of the particle, is
already derived in Eq. (23) for the different polarizations.

On the other hand, Eq. (81b) must still hold out of NESS because the
particle’s momentum in frame $\mathcal{P}$ remains zero,

$F_{\rm{tot}}^{\prime}=F^{\prime}+F_{\rm{ext}}^{\prime}=0\,\Rightarrow\,F_{\rm{ext}}^{\prime}=-F^{\prime}=-F+vP^{\prime},$
(83)

where we have used Eq. (78b) in the last equality. Comparing Eq. (82) with Eq.
(83), we find the external force needed in frame $\mathcal{R}$ and frame
$\mathcal{P}$ is still the same,
$F_{\rm{ext}}=F_{\rm{ext}}^{\prime}=-F^{\prime}$.

The quantum vacuum friction in frame $\mathcal{P}$, $F^{\prime}$, for a
particle in diagonal polarization state $\rm{P}$, can again be obtained from
Eq. (49) and Eq. (23),

$F^{\prime\rm{P}}=F^{\rm{P}}-vP^{\prime\rm{P}}=\frac{1}{3\pi^{2}\gamma
v}\int_{0}^{\infty}d\omega\,\omega^{4}\imaginary\alpha_{\rm{P}}(\omega)\int_{y_{-}}^{y_{+}}dy\,(y-\gamma)f^{\rm{P}}(y)\left(\frac{1}{e^{\beta\omega
y}-1}-\frac{1}{e^{\beta^{\prime}\omega}-1}\right).$ (84)

In Eq. (84), the term involving $\beta^{\prime}$ clearly does not contribute
to the integral on account of the oddness of the $y$ integrand around
$y=\gamma$. Physically, this reflects the fact that the emitted dipole
radiation does not have a momentum bias, and therefore does not contribute to
the frictional force in frame $\mathcal{P}$. See Ref. [9] for a detailed
explaination. As a result, $F^{\prime\rm{P}}$ is independent of the
temperature of the particle so long as $\imaginary\alpha$ does not depend on
temperature. In fact, $F^{\prime\rm{P}}$ in Eq. (84) for an out-of-NESS
particle precisely equals the NESS quantum vacuum friction
$\tilde{F}^{\rm{P}}$ shown in Eq. (50), $F^{\prime\rm{P}}=\tilde{F}^{\rm{P}}$.
Therefore, like $\tilde{F}^{\rm{P}}$, $F^{\prime\rm{P}}$ is also negative
definite.

The quantum vacuum frictional force in frame $\mathcal{R}$, $F$, consists of
two terms, i.e., $F=F^{\prime}+vP^{\prime}$. The first term, $F^{\prime}$, is
the steady state contribution $F^{\prime}=\tilde{F}$. The second term,
$vP^{\prime}$, is therefore the nonsteady part, which is due to the rest mass
change of the particle.121212This is purely an inertial effect. It even comes
into play in the electrodynamics of moving classical dipoles already discussed
in Ref. [12] where the friction $F$ simply equals $vP^{\prime}$. There, the
friction on the moving classical dipole in its rest frame, $F^{\prime}$, is
zero. This term does not have a definite sign because the particle could
either gain or lose rest mass.

As a result, $F$ is no longer negative definite out of NESS. That is to say,
the quantum vacuum “friction” in $\mathcal{R}$ could become a pushing force on
the particle by adjusting the temperature of the particle. Let us now attempt
to give another physical interpretation why such a positive friction should
occur. If we consider the combined system of particle and radiation, the only
net force on the system is $F_{\rm{ext}}$ provided by some external agent.
This external force will cause the total momentum of the combined system to
change. The rate of change of the momentum of the radiation equals $-F$, which
is the reaction force to the electromagnetic force (quantum frictional force)
exerted on the radiation fields by the particle. The rate of change of the
momentum of the particle due to its varying mass is given by $vP^{\prime}$.
Applying Newton’s second law to the combined system of particle and radiation
leads to $F_{\rm{ext}}=-F+vP^{\prime}$, just as given in Eq. (82). Here,
$F_{\rm{ext}}$ is positive definite while $vP^{\prime}$ could be positive or
negative depending on the radiation temperature, $T$, and the actual
temperature of the particle, $T^{\prime}$. As a result, $F$ could be of either
sign. In particular, if the particle gains mass in a rate greater than that
provided by the external force, $vP^{\prime}>F_{\rm{ext}}$, the
electromagnetic force exerted on the particle by the radiation fields will be
positive.

It is also interesting to study the temperature of the particle, $T_{0}$, at
which the quantum vacuum friction in frame $\mathcal{R}$ becomes zero. This
may be found by solving the equation, $F=0$, where $F$ is the quantum vacuum
friction in Eq. (49). For the temperature-independent model of the gold
nanosphere described in Sec. IV.1, the equation to be solved is

$\int_{0}^{\infty}du\frac{u^{5}}{(1-u^{2})^{2}+u^{2}\epsilon^{2}}\int_{y_{-}}^{y_{+}}dy\left(y-\frac{1}{\gamma}\right)\frac{1}{e^{2x_{1}uy}-1}=\int_{0}^{\infty}du\frac{u^{5}}{(1-u^{2})^{2}+u^{2}\epsilon^{2}}\int_{y_{-}}^{y_{+}}dy\left(y-\frac{1}{\gamma}\right)\frac{1}{e^{2x_{1}u/r_{0}}-1}.$
(85)

Here, the definitions for dimensionless variables in Eq. (72) still apply,
except that we use $r_{0}=T_{0}/T$ to denote the temperature ratio for $F=0$,
to be distinguished from the temperature ratio in NESS,
$\tilde{r}=\tilde{T}/T$. In the low-temperature regime ($x_{1}\gg 1$ or $T\ll
30\,000\,\rm{K}$), $\imaginary\alpha(\omega)$ reduces to the $n=1$ monomial
model, for which we already know that $F$ is negative definite [see footnote 5
for a detailed explanation] and therefore no solution could be found for Eq.
(85). In the high-temperature regime ($x_{1}\ll 1$ or $T\gg 30\,000\,\rm{K}$),
$\imaginary\alpha(\omega)$ reduces to the $n=-3$ monomial model and we
therefore obtain the high-temperature limit of the ratio,

$r_{0}=\sqrt{\frac{1}{\gamma^{2}v^{2}}\left(\frac{1}{2v}\ln\frac{1+v}{1-v}-1\right)}.$
(86)

For intermediate temperatures around the resonance ($x_{1}\sim 1$ or $T\sim
30\,000\rm{K}$), we may set $u=1$ and find the approximate ratio,

$r_{0}=2x_{1}\left(\ln\left\\{1+2\gamma^{2}v^{3}\left[\int_{y_{-}}^{y_{+}}dy\left(y-\frac{1}{\gamma}\right)\frac{1}{e^{2x_{1}y}-1}\right]^{-1}\right\\}\right)^{-1},$
(87)

where the $y$ integral in Eq. (87) can be expressed in terms of dilogarithms.

(a)

(b)

Figure 11: For the constant damping model of the gold nanosphere, the ratio
$r_{0}=T_{0}/T$ is solved numerically from Eq. (85) and illustrated. Here,
$T_{0}$ is the temperature of the particle at which the quantum vacuum
frictional force on it is zero in the rest frame of radiation. (a) The
velocity dependence of $r_{0}$ at $T=30\,000\,\rm{K}$ $(x_{1}=1.01)$ is
illustrated. The blue dots display the numerical solution obtained by solving
Eq. (85) directly. The dashed red curve shows $r_{0}$ calculated from the
resonance approximation Eq. (87). (b) The temperature dependence of $r_{0}$
for fixed velocity $v=0.5$ is illustrated. The exact numerical data are shown
by the blue dots. The dashed red curve plots the resonance approximation. The
dashed magenta curve shows the value of the high-temperature limit of the
ratio according to Eq. (86), $r_{0}=\sqrt{3(\ln 3-1)}=0.544$.

It is seen in Fig. 11a that, at $T=30\,000\,\rm{K}$, the prediction given by
the resonance formula Eq. (87) agrees very well with the exact numerical data
for $r_{0}$. In Fig. 11b, we see that the ratio $r_{0}$, for fixed velocity
$v=0.5$, is described well by the resonance approximation only up to about
$10^{5}\,\rm{K}$. Then the ratio goes through a transition region
($10^{5}-10^{9}\,\rm{K}$) and eventually decays to its high-temperature limit
given by Eq. (86) at around $10^{9}\,\rm{K}$. For the Bloch-Grüneisen model,
we also give a high-temperature prediction of $r_{0}$ in Eq. (110) of Appendix
B.

From Eq. (82), we see that the sign of the frictional power, $P^{\prime}$,
decides both the sign of the total force on the particle and the sign of its
mass change. Using the definition for $I(\xi)$ in Eq. (25), $P^{\prime\rm{P}}$
in Eq. (23) can be rewritten as

$P^{\prime\rm{P}}=\frac{1}{3\pi^{2}}\int_{y_{-}}^{y_{+}}f^{\rm{P}}(y)\left[I(\beta
y)-I(\beta^{\prime})\right],$ (88)

where the first term corresponds to the absorbed power due to the field
fluctuations and the second term corresponds to the emitted power due to the
dipole fluctuations. We can replace the first term using the NESS condition
Eq. (26)

$\int_{y_{-}}^{y_{+}}dy\,f^{\rm{P}}(y)I(\tilde{{\beta}})=\int_{y_{-}}^{y_{+}}dy\,f^{\rm{P}}(y)I(\beta
y).$ (89)

As a result, the frictional power now reads simply

$P^{\prime}=\frac{1}{3\pi^{2}}\left[I(\tilde{\beta})-I(\beta^{\prime})\right]\int_{y_{-}}^{y_{+}}dy\,f^{\rm{P}}(y),$
(90)

where the integral on $y$ becomes $3$ for the isotropic polarization and $1$
for other diagonal polarizations. Because $I(\xi)$ is a decreasing function,
the sign of $P^{\prime\rm{P}}$ is determined by the deviation of the
temperature of the particle from its NESS temperature as below,

$\left\\{\begin{array}[]{lcrcl}T^{\prime}<\tilde{T}\,\Rightarrow\,P^{\prime\rm{P}}>0,\\\
\\\ T^{\prime}=\tilde{T}\,\Rightarrow\,P^{\prime\rm{P}}=0,\\\ \\\
T^{\prime}>\tilde{T}\,\Rightarrow\,P^{\prime\rm{P}}<0.\\\ \end{array}\right.$
(91)

By now, we have come to an understanding of the issue of the stability of
NESS. Imagine the particle slightly deviates from NESS with a temperature
lower than the NESS temperature, $T^{\prime}<\tilde{T}$. According to Eq.
(91), the particle will absorb net energy thereafter, which will in turn raise
the temperature of the particle $T^{\prime}$ 131313Here, we assume the change
of the particle’s temperature is adiabatic so that it still has a well-defined
temperature while it is away from NESS, even though the particle is in neither
an equilibrium state nor a steady state.  so that it becomes closer to the
NESS temperature $\tilde{T}$. Therefore, the particle would tend to return to
NESS after deviating from it. This seems to provide theoretical support for
experimentally measuring the NESS temperature of the particle as a feasible
signature for the quantum vacuum frictional effects.

One might imagine that a positive quantum vacuum frictional force qualifies
the particle to function as a perpetual motion machine, which could extract
vacuum energy from the radiation fields and turn it into useful mechanical
work endlessly. We argue that this is not possible. First, the external agent
keeps providing energy to the particle-radiation system and the law of
conservation of energy is never violated. Therefore, the particle is not a
perpetual motion machine of the first kind. Second, the positive “friction”
only occurs if the temperature of the particle is lower than the NESS
temperature and the particle gains internal energy. Heat transfers to a
relatively colder reservoir.141414The particle is colder in the sense that its
actual temperature is lower than the temperature needed to reach the steady
state where there is no net heat transfer. In this process, the entropy should
increase and the second law of thermodynamics is respected. Therefore, the
particle is not a perpetual motion machine of the second kind either.

##  VI conclusions

In this paper, we studied the quantum vacuum frictional phenomenon associated
with an intrinsically dissipative particle. We not only derived conditions for
the particle to be in the nonequilibrium steady state (NESS) and found
expressions for quantum vacuum friction in NESS, but also extended the
analysis to the out-of-NESS scenario.

Independent of the particle’s polarizability, the NESS temperature,
$\tilde{T}$, has a general lower bound, $T/\gamma$, the Planck-Einstein
transformed temperature of the blackbody radiation. Furthermore, the
nonrelativistic limit of the particle’s NESS temperature always equals the
radiation temperature. Under the NESS conditions, the quantum vacuum friction
on the dissipative particle is shown to be negative definite and to be related
to the friction for a nondissipative particle, investigated in our previous
paper [9]. We also show that the NESS quantum vacuum friction reduces to the
well-known Einstein-Hopf drag in the nonrelativistic limit.

The NESS temperature ratio and the NESS quantum vacuum friction are first
calculated explicitly for the resonance model and the monomial models and then
for a gold nanosphere modelled either with a constant damping (constant
damping model) or a temperature-dependent damping (Bloch-Grüneisen model). The
NESS temperature ratio and friction for the resonance model and the monomial
models can be worked out analytically, providing insight to the numerical
results of the more realistic models used for the gold nanosphere. The
monomial models are good approximations for the gold nanosphere with constant
damping in the two extreme temperature limits. The resonance model
approximates the behavior in the intermediate temperature region. In addition,
the nonmonotonic behavior (in velocity) of the NESS friction on the gold
nanosphere in the ultra-relativistic region is mainly associated with the
resonance contributions. For both models (constant damping model and the
Bloch-Grüneisen model) of the gold nanosphere, the NESS temperature ratio is
found to be maximized around $T=3000\,\rm{K}$ and the deviation of the NESS
temperature from the radiation temperature is quite noticeable around this
maximum. The magnitude of the NESS quantum vacuum friction increases with
temperature and reaches the order of picoNewtons if the background temperature
could be raised to $10^{4}\,\rm{K}$. Comparing the Bloch-Grüneisen model with
the constant damping model of the gold nanosphere, the NESS friction is
enhanced for higher temperatures but suppressed for lower temperatures, and
the high-temperature limit of the NESS temperature ratio is raised to above
$1$.

The energetics out of NESS is different from that in NESS because the particle
can absorb or emit net energy, thereby changing its rest mass. The change of
the particle’s rest mass produces a change in its momentum in the rest frame
of radiation ($\mathcal{R}$). As a result, the quantum vacuum friction, $F$,
in frame $\mathcal{R}$, no longer balances the external force, $F_{\rm{ext}}$,
needed to keep the particle moving at constant velocity, and it is allowed to
take either sign or even zero. Nonetheless, the quantum vacuum friction,
$F^{\prime}$, in frame $\mathcal{P}$, is still negative definite. And we find
that $F_{\rm{ext}}$ must be opposite to $F^{\prime}$ and positive definite,
which excludes the possibility of converting the vacuum energy into useful
mechanical work using the moving particle as a perpetual motion machine. It is
precisely the difference between the particle’s actual temperature,
$T^{\prime}$, and its NESS temperature, $\tilde{T}$, that determines whether
the particle absorbs or emits energy. Only if $T^{\prime}=\tilde{T}$ is the
particle’s internal energy conserved. The particle will absorb energy if
$T^{\prime}<\tilde{T}$ and emit energy if $T^{\prime}>\tilde{T}$. To sum up,
the particle tends to return to NESS after deviating from it. If the
temperature of a nanosphere moving at constant velocity could be measured, it
is then expected to be the NESS temperature, which would show some deviation
from the temperature of the lab environment. Such an experiment could in
principle be used to identify the quantum vacuum frictional effects discussed
in this paper.

###### Acknowledgements.

We thank the US National Science Foundation, grants Nos. 1707511, 2008417, for
partial support of this work. We thank S. Fulling, P. Parashar and J. J.
Marchetta for insightful comments. We thank G. V. Dedkov for pointing us to
their papers so that we are able to confirm the agreement with their results.
This paper reflects solely the authors’ personal opinions and does not
represent the opinions of the authors’ employers, present and past, in any
way.

##  Appendix A THE EQUIVALENCE OF THE LORENTZ FORCE LAW AND THE PRINCIPLE OF
VIRTUAL WORK

In this appendix, we apply the Lorentz force law to a moving electric dipole
and illustrate that the Lorentz force can be equivalently calculated through
differentiating a free energy including both a electric dipole contribution
and a motion-induced magnetic dipole contribution.

The Lorentz force density reads

$\mathbf{f}(t,\mathbf{r})=\rho(t,\mathbf{r})\mathbf{E}(t,\mathbf{r})+\mathbf{j}(t,\mathbf{r})\crossproduct\mathbf{B}(t,\mathbf{r}).$
(92)

The charge and current densities corresponding to a moving electric dipole
with constant velocity $\mathbf{v}$ is151515If one only considers the
requirement of current continuity, $\partial_{t}\rho+\divergence\mathbf{j}=0$,
there might be another choice of the current,
$\mathbf{j}_{2}(t,\mathbf{r})=-\mathbf{d}(t)\divergence{\mathbf{v}}\delta(\mathbf{r}-\mathbf{v}t)+\dot{\mathbf{d}}(t)\delta(\mathbf{r}-\mathbf{v}t)$,
which differs from $\mathbf{j}(t,\mathbf{r})$ in Eq. (93b) by a total curl. In
fact, the current we use is associated with the convection current, which
includes both the polarization current $\mathbf{j}_{2}(t,\mathbf{r})$ and a
motion-induced magnetization current
$\mathbf{j}_{m}(t,\mathbf{r})=\curl[\mathbf{d}(t)\crossproduct\mathbf{v}\delta(\mathbf{r}-\mathbf{v}t)]$.
See Ref. [36] for a discussion. Therefore, if we use instead
$\mathbf{j}_{2}(t,\mathbf{r})$ in the derivation, we will not see the motion-
induced magnetic dipole term in Eq. (99). Such ambiguity of current for a
moving dipole can be avoided by deriving it from the electric and magnetic
polarization fields $\mathbf{P}$ and $\mathbf{M}$ in Ref. [12].

$\rho(t,\mathbf{r})=-\divergence\mathbf{d}(t)\delta(\mathbf{r}-\mathbf{v}t),$
(93a)
$\mathbf{j}(t,\mathbf{r})=-\mathbf{v}\divergence{\mathbf{d}}(t)\delta(\mathbf{r}-\mathbf{v}t)+\dot{\mathbf{d}}(t)\delta(\mathbf{r}-\mathbf{v}t).$
(93b)

After integrating the Lorentz force density $\mathbf{f}$ over all space, we
find the Lorentz force on the dipole to be

$\mathbf{F}(t)=\mathbf{d}(t)\cdot\gradient\mathbf{E}(t,\mathbf{v}t)+\mathbf{d}(t)\cdot\gradient\left[\mathbf{v}\crossproduct\mathbf{B}(t,\mathbf{v}t)\right]+\dot{\mathbf{d}}(t)\crossproduct\mathbf{B}(t,\mathbf{v}t).$
(94)

To prove the principle of virtual work, we are to write the right side of Eq.
(94) in the form of a total derivative on the spatial arguments.

The second term in Eq. (94) can be written as

$\displaystyle\mathbf{d}(t)\cdot\gradient\left[\mathbf{v}\crossproduct\mathbf{B}(t,\mathbf{v}t)\right]$
$\displaystyle=-\mathbf{d}(t)\crossproduct\left[\gradient\crossproduct\left(\mathbf{v}\crossproduct\mathbf{B}(t,\mathbf{v}t)\right)\right]+\gradient\left[\mathbf{d}(t)\cdot\left(\mathbf{v}\crossproduct\mathbf{B}(t,\mathbf{v}t)\right)\right]$
$\displaystyle=\mathbf{d}(t)\crossproduct\left[\mathbf{v}\cdot\gradient\mathbf{B}(t,\mathbf{v}t)\right]+\gradient\left[\mathbf{d}(t)\cdot\left(\mathbf{v}\crossproduct\mathbf{B}(t,\mathbf{v}t)\right)\right],$
(95)

where we have used $\divergence\mathbf{B}=0$ in the second equality. The third
term in Eq. (94) can be written as

$\dot{\mathbf{d}}(t)\crossproduct\mathbf{B}(t,\mathbf{v}t)=\frac{d}{dt}\left[\mathbf{d}(t)\crossproduct\mathbf{B}(t,\mathbf{v}t)\right]-\mathbf{d}(t)\crossproduct\frac{\partial}{\textbullet\partial
t}\mathbf{B}(t,\mathbf{v}t)-\mathbf{d}(t)\crossproduct\left[\mathbf{v}\cdot\gradient\mathbf{B}(t,\mathbf{v}t)\right],$
(96)

where the middle term can be broken into two pieces with the use of Faraday’s
law,

$-\mathbf{d}(t)\crossproduct\frac{\partial}{\textbullet\partial
t}\mathbf{B}(t,\mathbf{v}t)=\mathbf{d}(t)\crossproduct\left[\gradient\crossproduct\mathbf{E}(t,\mathbf{v}t)\right]=\gradient\left[\mathbf{d}(t)\cdot\mathbf{E}(t,\mathbf{v}t)\right]-\mathbf{d}(t)\cdot\gradient\mathbf{E}(t,\mathbf{v}t).$
(97)

When we put Eqs. (A)–(97) back into Eq. (94), several terms cancel and only
three terms remain in the expression for the Lorentz force:

$\mathbf{F}(t)=\gradient\left[\mathbf{d}(t)\cdot\mathbf{E}(t,\mathbf{v}t)\right]+\gradient\left[\left(\mathbf{d}(t)\crossproduct\mathbf{v}\right)\cdot\mathbf{B}(t,\mathbf{v}t)\right]+\frac{d}{dt}\left[\mathbf{d}(t)\crossproduct\mathbf{B}(t,\mathbf{v}t)\right].$
(98)

A more general derivation of the Lorentz force on neutral particles including
the contribution of an intrinsic magnetic dipole moment can be found in Ref.
[23]. Since the neutral particle considered in the paper does not possess a
magnetic dipole polarizability or magnetic dipole fluctuations, Eq. (98) is
already sufficient for our purpose. In addition, the last term in Eq. (98),
being a total derivative with respect to time, would not contribute to the
quantum frictional force, which is really the average time rate of change in
the frictional impulse over a large time period.

As a result, we are able to express the relevant Lorentz force $F$ as the
gradient on the same free energy $\mathcal{F}$ used in the power expression
(11) and therefore proves the principle of virtual work,

$\mathbf{F}=-\gradient\mathcal{F},\quad\mathcal{F}=-\mathbf{d}(t)\cdot\mathbf{E}(t,\mathbf{v}t)-\bm{\mu}_{v}(t)\cdot\mathbf{B}(t,\mathbf{v}t),$
(99)

where we have identified
$\mathbf{d}(t)\crossproduct\mathbf{v}=\bm{\mu}_{v}(t)$ as the magnetic dipole
moment induced by the movement of the electric dipole. The analysis above is
general and Eq. (99) is applicable in any frame of reference. In particular,
in the rest frame of the particle $\mathcal{P}$, the magnetic term in the free
energy vanishes and the principle of virtual work reads

$\mathbf{F}^{\prime}=-\gradient^{\prime}\mathcal{F^{\prime}},\quad\mathcal{F^{\prime}}=-\mathbf{d}^{\prime}(t^{\prime})\cdot\mathbf{E}^{\prime}(t^{\prime},\mathbf{0}).$
(100)

##  Appendix B THE BLOCH-GRÜNEISEN MODEL IN THE HIGH-TEMPERATURE LIMIT

In this appendix, we obtain high-temperature asymptotic expressions for the
power and the frictional force in the case that the damping parameter has the
linear high-temperature dependence of the Bloch-Grüneisen model [29, 28, 37]
as shown in Eq. (76),

$\nu=\frac{2\pi\eta}{\beta^{\prime}}=2\pi\eta T^{\prime},$ (101)

for some constant $\eta>0$.161616In terms of the parameters in Eq. (75),
$\eta=\nu_{0}/8\pi k_{B}\theta$ and it evaluates to be $\eta=0.219$ for gold.

The power in the rest frame of an isotropic particle is given by

$\displaystyle P^{\prime}$ $\displaystyle=\frac{1}{2\pi^{2}\gamma
v}\int_{0}^{\infty}d\omega\,\omega^{4}\operatorname{Im}\alpha(\omega)\int_{y_{-}}^{y_{+}}dy\left(\frac{1}{e^{\beta\omega
y}-1}-\frac{1}{e^{\beta^{\prime}\omega}-1}\right)$ (102)
$\displaystyle=\frac{V\omega_{p}^{2}\,\nu}{2\pi^{2}\gamma
v}\int_{0}^{\infty}d\omega\,\frac{\omega^{5}}{\left(\omega^{2}-\omega_{1}^{2}\right)^{2}+\omega^{2}\nu^{2}}\int_{y_{-}}^{y_{+}}dy\left(\frac{1}{e^{\beta\omega
y}-1}-\frac{1}{e^{\beta^{\prime}\omega}-1}\right)$
$\displaystyle=\frac{V\omega_{p}^{2}\,\eta}{\pi\gamma v\beta^{\prime
3}}\int_{y_{-}}^{y_{+}}dy\int_{0}^{\infty}dz\,\left(\frac{z^{5}}{r^{2}y^{2}\left[\left(z^{2}-\omega_{1}^{2}\beta^{\prime
2}r^{2}y^{2}\right)^{2}+4\pi^{2}\eta^{2}r^{2}y^{2}z^{2}\right]}-\frac{z^{5}}{\left(z^{2}-\omega_{1}^{2}\beta^{\prime
2}\right)^{2}+4\pi^{2}\eta^{2}z^{2}}\right)\frac{1}{e^{z}-1},$

where $r\equiv\frac{\beta}{\beta^{\prime}}=\frac{T^{\prime}}{T}$. In the high-
temperature limit, $\beta,\beta^{\prime}\to 0$, and $r=\beta/\beta^{\prime}$
tends to a finite value, the dependence on the shifted resonance frequency is
suppressed. Retaining only the leading term in this limit, we obtain

$\displaystyle P^{\prime}$
$\displaystyle=\frac{V\omega_{p}^{2}\,\eta}{\pi\gamma v\beta^{\prime
3}}\int_{y_{-}}^{y_{+}}dy\int_{0}^{\infty}dz\,\left(\frac{z^{3}}{r^{2}y^{2}\left(z^{2}+4\pi^{2}\eta^{2}r^{2}y^{2}\right)}-\frac{z^{3}}{z^{2}+4\pi^{2}\eta^{2}}\right)\frac{1}{e^{z}-1}$
(103) $\displaystyle=\frac{V\omega_{p}^{2}\,\pi\eta}{\gamma v\beta^{\prime
3}}\int_{y_{-}}^{y_{+}}dy\,\left\\{\frac{1}{r^{2}y^{2}}\left[\frac{1}{6}-2\eta^{2}r^{2}y^{2}\left(\ln(\eta
ry)-\psi(\eta ry)-\frac{1}{2\eta
ry}\right)\right]-\left[\frac{1}{6}-2\eta^{2}\left(\ln\eta-\psi(\eta)-\frac{1}{2\eta}\right)\right]\right\\}$
$\displaystyle=\frac{V\omega_{p}^{2}\,\pi\eta
r^{3}}{3\beta^{3}}\left[\frac{1}{r^{2}}+\frac{6\eta}{\gamma
vr}\ln\left(\frac{y_{+}\Gamma(\eta ry_{+})}{\Gamma(\eta
ry_{-})}\right)-1-6\eta-12\eta^{2}\left(\ln r+\psi(\eta)+\frac{\ln
y_{+}}{v}-1\right)\right],$

where we have employed the integral representation of the digamma function,

$\psi(s)\equiv\frac{d}{ds}\ln\Gamma(s)=\ln
s-\frac{1}{2s}-2\int_{0}^{\infty}dt\,\frac{t}{(t^{2}+s^{2})(e^{2\pi
t}-1)},\quad\operatorname{Re}s>0,$ (104)

which follows immediately on differentiation of Binet’s second integral
formula for the (log) gamma function [38, 39, 40], to deduce that

$\int_{0}^{\infty}dz\,\frac{z^{3}}{(z^{2}+u^{2})(e^{z}-1)}=\frac{\pi^{2}}{6}-\frac{u^{2}}{2}\left[\ln\left(\frac{u}{2\pi}\right)-\psi\left(\frac{u}{2\pi}\right)-\frac{\pi}{u}\right],\quad\operatorname{Re}u>0.$
(105)

When the NESS condition, $P^{\prime}=0$, holds, it is clear from Eq. (103)
that the NESS temperature ratio $\tilde{r}$ satisfies the implicit equation

$\tilde{r}^{2}\left[1+6\eta+12\eta^{2}\left(\ln\tilde{r}+\psi(\eta)+\frac{\ln
y_{+}}{v}-1\right)\right]-\frac{6\eta\tilde{r}}{\gamma
v}\ln\left(\frac{y_{+}\Gamma(\eta\tilde{r}y_{+})}{\Gamma(\eta\tilde{r}y_{-})}\right)=1$
(106)

in the high-temperature limit.

Similarly, the NESS frictional force in Eq. (50), which is equivalent to the
frictional force in frame $\mathcal{P}$ in Eq. (84), is given by

$\displaystyle\tilde{F}=F^{\prime}$
$\displaystyle=\frac{1}{2\pi^{2}\gamma^{2}v^{2}}\int_{0}^{\infty}d\omega\,\omega^{4}\operatorname{Im}\alpha(\omega)\int_{y_{-}}^{y_{+}}dy\,(y-\gamma)\,\frac{1}{e^{\beta\omega
y}-1}$ (107)
$\displaystyle=\frac{V\omega_{p}^{2}\,\nu}{2\pi^{2}\gamma^{2}v^{2}}\int_{0}^{\infty}d\omega\,\frac{\omega^{5}}{\left(\omega^{2}-\omega_{1}^{2}\right)^{2}+\omega^{2}\nu^{2}}\int_{y_{-}}^{y_{+}}dy\,(y-\gamma)\,\frac{1}{e^{\beta\omega
y}-1}$
$\displaystyle=\frac{V\omega_{p}^{2}\,\eta}{\pi\gamma^{2}v^{2}\beta^{\prime
3}}\int_{y_{-}}^{y_{+}}dy\,\frac{(y-\gamma)}{\tilde{r}^{2}y^{2}}\int_{0}^{\infty}dz\,\frac{z^{5}}{\left(z^{2}-\omega_{1}^{2}\beta^{\prime
2}\tilde{r}^{2}y^{2}\right)^{2}+4\pi^{2}\eta^{2}\tilde{r}^{2}y^{2}z^{2}}\,\frac{1}{e^{z}-1},$

which becomes, in the high-temperature limit,

$\displaystyle\tilde{F}=F^{\prime}$
$\displaystyle=\frac{V\omega_{p}^{2}\,\eta}{\pi\gamma^{2}v^{2}\beta^{\prime
3}}\int_{y_{-}}^{y_{+}}dy\,\frac{(y-\gamma)}{\tilde{r}^{2}y^{2}}\int_{0}^{\infty}dz\,\frac{z^{3}}{z^{2}+4\pi^{2}\eta^{2}\tilde{r}^{2}y^{2}}\frac{1}{e^{z}-1}$
(108)
$\displaystyle=\frac{V\omega_{p}^{2}\,\pi\eta}{\gamma^{2}v^{2}\beta^{\prime
3}}\int_{y_{-}}^{y_{+}}dy\,\frac{(y-\gamma)}{\tilde{r}^{2}y^{2}}\left[\frac{1}{6}-2\eta^{2}\tilde{r}^{2}y^{2}\left(\ln(\eta\tilde{r}y)-\psi(\eta\tilde{r}y)-\frac{1}{2\eta\tilde{r}y}\right)\right]$
$\displaystyle=\frac{2V\omega_{p}^{2}\,\pi\eta\tilde{r}^{3}}{\gamma^{2}v^{2}\beta^{3}}\left\\{\frac{1}{\tilde{r}^{2}}\left[\frac{\ln
y_{+}-\gamma^{2}v}{6}+\ln\left(\frac{G(1+\eta\tilde{r}y_{+})}{G(1+\eta\tilde{r}y_{-})}\right)\right]-\frac{\gamma\eta}{\tilde{r}}\left[v\ln(2\pi)+\ln\left(\frac{y_{+}\Gamma(\eta\tilde{r}y_{+})}{\Gamma(\eta\tilde{r}y_{-})}\right)\right]+\eta^{2}\left(\gamma^{2}v+\ln
y_{+}\right)\right\\},$

where $G$ is the Barnes (double gamma) $G$-function [41, 42, 43, 44, 45].

It follows from Eq. (103) and Eq. (108) that, in the high-temperature limit,
the force in the rest frame of the blackbody radiation,
$F=F^{\prime}+vP^{\prime}$, is given by

$\displaystyle F=\frac{2V\omega_{p}^{2}\,\pi\eta
r^{3}}{\gamma^{2}v^{2}\beta^{3}}$
$\displaystyle\left\\{\frac{1}{r^{2}}\left[\frac{\ln
y_{+}-v}{6}+\ln\left(\frac{G(1+\eta ry_{+})}{G(1+\eta
ry_{-})}\right)\right]-\frac{\gamma\eta}{r}\left[v\ln(2\pi)+\left(1-v^{2}\right)\ln\left(\frac{y_{+}\Gamma(\eta
ry_{+})}{\Gamma(\eta ry_{-})}\right)\right]\right.$ (109)
$\displaystyle-\left.\gamma^{2}v^{3}\left(\frac{1}{6}+\eta+2\eta^{2}\left(\ln
r+\psi(\eta)\right)\right)+\eta^{2}\left(\left(3\gamma^{2}-2\right)v+\left(3-2\gamma^{2}\right)\ln
y_{+}\right)\right\\}.$

In particular, when the frictional force becomes zero in frame $\mathcal{R}$,
$F=0$, the corresponding temperature ratio $r_{0}$ satisfies the implicit
equation in the high temperature limit

$\displaystyle
r_{0}^{2}\left[\gamma^{2}v^{3}\left(\frac{1}{6}+\eta+2\eta^{2}\left(\ln
r_{0}+\psi(\eta)\right)\right)+\eta^{2}\left(\left(2-3\gamma^{2}\right)v+\left(2\gamma^{2}-3\right)\ln
y_{+}\right)\right]$ (110) $\displaystyle\quad+\gamma\eta
r_{0}\left[v\ln(2\pi)+\left(1-v^{2}\right)\ln\left(\frac{y_{+}\Gamma(\eta
r_{0}y_{+})}{\Gamma(\eta r_{0}y_{-})}\right)\right]=\frac{\ln
y_{+}-v}{6}+\ln\left(\frac{G(1+\eta r_{0}y_{+})}{G(1+\eta
r_{0}y_{-})}\right).$

The exact analytical expressions in Eq. (106) and Eq. (110) enable exploration
of the variation of $r$ with $v$ across the entire $v$ domain, in the high-
temperature limit. They therefore provide a useful check of some of the
numerical results obtained by other means.

##  Appendix C SOLUTION FOR $r_{0}$ IN THE LOW-TEMPERATURE, HIGH-VELOCITY
REGIME

In Sec. V, we commented that quantum vacuum friction $F$ in frame
$\mathcal{R}$ can be made zero only for high temperatures, for example,
$30\,000\,\rm{K}$ for the gold nanosphere. In fact, this is only true in the
moderate velocity regime. It is possible to find solutions to Eq. (85) for
lower temperatures in the ultrarelativistic regime.171717Of course,
ultrarelativistic speed is hard to achieve experimentally for neutral
particles. This appendix explores that possibility.

Let us define the left hand side of Eq. (85) as $f_{L}(\gamma,x_{1},\epsilon)$
and the right hand side as $f_{R}(\gamma,x_{1},\epsilon,r_{0})$. In the low-
temperature regime, $x_{1}\gg 1$, for finite $r_{0}$, it is clear that $f_{R}$
is dominated by low frequency-contributions $u\ll 1$. The contribution around
the resonance $u\sim 1$ is suppressed because of the exponential factor so
that $f_{R}$ reduces to

$f_{R}(\gamma,x_{1},\epsilon,r_{0})\sim
2\Gamma(6)\zeta(6)\gamma^{2}\left(\frac{r_{0}}{2x_{1}}\right)^{6},$ (111)

where we have also applied the high velocity limit, $\gamma\gg 1$. For
$f_{L}$, we note the exponential factor contains $y$. In the ultrarelativistic
region, the integration on $y$ is essentially taken over the interval
$[1/2\gamma,2\gamma]$. The lower values of $y$ allow the resonance around
$u\sim 1$ to contribute significantly to the $u$ integration while the higher
values of $y$ are suppressed because $x_{1}$ is already large. As a result,
the $u$ integration in $f_{L}$ collapses, leaving

$f_{L}(\gamma,x_{1},\epsilon)\sim\frac{\pi}{2\epsilon}\int_{1/2\gamma}^{2\gamma}dy\left(y-\frac{1}{\gamma}\right)\frac{1}{e^{2x_{1}y}-1}.$
(112)

Equating $f_{R}$ in Eq. (111) and $f_{L}$ in Eq. (112) gives an approximation
for $r_{0}$ in the low-temperature, high-velocity regime,

$r_{0}=x_{1}\left[\frac{126}{\pi^{5}\gamma^{2}\epsilon}\int_{1/2\gamma}^{1}dy\left(y-\frac{1}{\gamma}\right)\frac{1}{e^{2x_{1}y}-1}\right]^{1/6},$
(113)

where we delibrately cut off the $y$ integration on the upper limit because
this will not hurt the accuracy in the large $x_{1}$ limit but improve the
efficiency of the numerical evaluation of the integral.

Let us note the zero of the integral inside the bracket of Eq. (113) gives
precisely the lower bound of the velocity for which Eq. (85) still has a
solution. At room temperature $T=300\,\rm{K}$ ($x_{1}=101$), this lower bound
is found to be $\gamma=115$.

Figure 12: The solution to Eq. (85) at $T=300\,\rm{K}$ is illustrated for
ultrarelativistic velocities. The exact numerical solution is shown by the
blue dots. The dashed red curve shows the approximate formula (113).

In Fig. 12, we show both the exact numerical solution and the approximation of
$r_{0}$ for ultrarelativistic velocites $\gamma\in[120,1000]$ at room
temperature. The approximate formula Eq. (113) works reasonably well in the
velocity region covered. It is seen that $r_{0}$ can become substantially
greater than $1$ in such a high velocity regime.

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