# Dynamic virial theorem at nonequilibrium and applications

Shi-Guo Peng<EMAIL_ADDRESS>State Key Laboratory of Magnetic Resonance
and Atomic and Molecular Physics, Innovation Academy for Precision Measurement
Science and Technology, Chinese Academy of Sciences, Wuhan 430071, China

(August 27, 2024)

###### Abstract

We show that a variety of nonequilibrium dynamics of interacting many-body
systems are universally characterized by an elegant relation, which we call
the dynamic virial theorem. The out-of-equilibrium dynamics of quantum
correlations is entirely governed by Tan’s contact. It gives rise to a series
of observable consequences and is closely related to experiments with
ultracold atoms. Especially, we show that the dynamic virial theorem provides
an experimentally accessible verification of maximum energy growth theorem [Qi
_et al._ , Phys. Rev. Lett. 126, 240401 (2021)], which is encoded in the
evolution of the atomic cloud size during expansion. In addition, the dynamic
virial theorem leads to a simple thermodynamic relation of strongly
interacting quantum gases in the framework of two-fluid hydrodynamic theory,
which holds in a wide range of temperature. This thermodynamic relation is a
kind of the out-of-equilibrium analog of Tan’s pressure relation at
equilibrium. Our results provide fundamental understanding of generic
behaviors of interacting many-body systems at nonequilibrium, and are readily
examined in experiments with ultracold atoms.

##  I Introduction

Investigating nonequilibrium dynamics of strongly-interacting many-body
systems is of fundamental importance and remains an open challenge in modern
physics, while it is broadly relevant to many phenomena in universe, ranging
from the evolution of neutron stars, procedure of chemical reactions to
complex living systems in biology. Discovering universal features in quantum
systems at nonequilibrium, irrelevant to the microscopic detail of objects
studied, is a long-standing challenge to date and attracts a great deal of
attention in both theory and experiment. Ultracold atoms, as a clean,
controllable and versatile quantum systems, provide an unprecedented platform
for exploration of a wide variety of phenomena in many-body physics, ranging
from thermodynamic equilibrium to out-of-equilibrium dynamics (Bloch2008M, ;
Giorgini2008T, ; Bloch2012Q, ; Polkovnikov2011C, ; Langen2015U, ; Eisert2015Q,
). Remarkably, ultracold atomic systems take unique advantages for studying
nonequilibrium dynamics within experimentally resolved intrinsic time scales
(typically milliseconds), holding promising opportunities to test universality
of many-body dynamics far from equilibrium (Makotyn2014U, ; Eigen2017U, ;
Eigen2018U, ; Prufer2018O, ; Erne2018U, ; Ko2019K, ; Dyke2021D, ).

A set of universal relations in cold atoms, connected by a simple contact
parameter, have been discovered (Tan2008E, ; Tan2008L, ; Tan2008G, ). These
relations simply follow from the short-range correlation of two-body physics
and provide remarkable understanding of profound properties of interacting
many-body systems. To date, an impressive amount of experimental and
theoretical efforts have been devoted to confirm universal relations and
explore their important consequences (Zwerger2011, ; Yu2015U, ; He2016C, ;
Luciuk2016E, ; Peng2016L, ; Fletcher2017Q, ; Peng2018C, ; Zhang2020U, ). While
most of them are focused on properties of many-body systems at equilibrium,
universal relations at nonequilibrium are rarely studied and still remain
elusive.

In this work, it is found that a variety of dynamic processes of many-body
systems driven away from equilibrium by either the time-dependent external
potential or time-dependent interactions are elegantly governed by the dynamic
virial theorem. These are favorite ways to study out-of-equilibrium dynamics
in experiments of cold atoms, such as the expansion dynamics (Elliott2014O, ;
Deng2016O, ) and the interaction quench (Makotyn2014U, ; Eigen2017U, ;
Eigen2018U, ; Prufer2018O, ; Dyke2021D, ). The dynamic virial theorem reveals
a deep insight into the precise energy conversion relation, i.e.,

$E\left(t\right)-2E_{ho}\left(t\right)=\frac{1}{4}\frac{d^{2}I\left(t\right)}{dt^{2}}-\frac{\hbar^{2}\mathcal{C}\left(t\right)}{8\pi
ma\left(t\right)},$ (1)

that imposes an intrinsic constrain on energy dynamics. Here, $m$ is the
atomic mass, $\hbar$ is Planck’s constant, and $a\left(t\right)$ is the
scattering length generally varying in time. $E_{ho}\left(t\right)$ is the
energy corresponding to the external harmonic potential,
$I\left(t\right)\equiv\left\langle mr^{2}\right\rangle$ is the moment of
inertia of systems, and $\mathcal{C}\left(t\right)$ is Tan’s contact
(Tan2008E, ; Tan2008L, ; Tan2008G, ). In the classical (or high-temperature)
limit, the kinetic energy dominates the internal energy. The dynamic virial
theorem immediately implies an interesting non-damping monopole oscillation
with twice the trapping frequency in a perfectly spherical trap, and systems
never reach thermal equilibrium. This long-predicted phenomenon by Boltzmann’s
equation has recently been observed by JILA’s group in a thermal Bose gas
(Lobser2015O, ). In the quantum (or low-temperature) limit, the interaction
between particles comes into play, and shifts the frequency of monopole
oscillation (Stringari1996C, ; Bruun1999H, ; Baranov2000L, ; Guey-Odelin2002M,
; SupplementalMaterial, ). We find that the dynamics of quantum correlations
is entirely governed by Tan’s contact. In the unitarity limit with divergent
scattering length, the systems display scale invariance. The dynamic virial
theorem requires a unitary gas to follow exactly the same dynamics of an ideal
gas (Elliott2014O, ; Deng2016O, ; Werner2006U, ). At equilibrium, the dynamic
virial theorem simply recovers that of (Tan2008G, ) at finite interaction
strength, and further reduces to the well-known equilibrium result of
$E=2E_{ho}$ for unitary and ideal gases (Thomas2005V, ).

The applications of the dynamic virial theorem are demonstrated in several
typical dynamic processes. The observable consequences in experiments of
ultracold atoms are discussed. There are two typical ways to study the
nonequilibirum dynamics in experiments with cold atoms, i.e., in a time-
dependent trap and with a time-dependent interaction. For the former, we show
that the release energy of cold atoms during free expansion is directly given
by the dynamic virial theorem. The frequencies of monopole oscillations are
simply derived according to the dynamic virial theorem. Besides, the dynamic
virial theorem simply characterizes the intriguing Efimovian expansion of
scale-invariant quantum gases in a time-dependent harmonic trap (Deng2016O, ).
For the latter, a fundamental issue has theoretically been addressed recently
as how fast the energy could be pumped into a noninteracting system by
increasing interactions (Qi2021M, ). Counterintuitively, it is not true that
the faster the interaction increases, the larger the rate of energy gain
becomes. Instead, there exists an upper limit of initial energy growth rate,
which could be reached only when the scattering length increases with time $t$
as $\sim\sqrt{t}$, known as the maximum energy growth theorem. However, the
direct measurement of the energy dynamics in experiments is difficult. We show
that the dynamic virial theorem provides an experimentally accessible way to
verify the maximum energy growth theorem according to the expansion behavior
of the atomic cloud size. Remarkably, a simple dynamic thermodynamic relation
is derived by using the dynamic virial theorem in the framework of the two-
fluid hydrodynamic theory for interacting quantum gases. This thermodynamic
relation is a kind of the out-of-equilibrium analog of Tan’s pressure relation
at equilibrium (Tan2008G, ), and holds in a wide range of temperature.

The rest of the paper is arranged as follows. In the next section, we present
the brief proof of the dynamic virial theorem for interacting many-body
systems. Afterwards, the dynamic virial theorem is applied to study the out-
of-equilibrium dynamics for cold atoms in time-dependent harmonic traps, i.e.,
free expansion (Sec. III), Efimovian expansion (Sec. IV), and monopole
oscillation (Sec. V). We show, in Sec. VI, that the dynamic virial theorem
provides an experimentally accessible verification of the maximum energy
growth theorem by increasing interatomic interactions, which is encoded in the
evolution of the atomic cloud size during expansion. An out-of-equilibrium
thermodynamic relations is obtained by using the dynamic virial theorem in the
framework of the two-fluid hydrodynamic theory in Sec. VII. Finally, the
remarks and conclusions are summarized in Sec. VIII.

##  II dynamic virial theorem

To prove the dynamic virial theorem (1), let us consider a system consisting
of $N$ atoms (either bosons or fermions) in a harmonic trap. The Hamiltonian
of the system takes the form of

$\hat{H}=\hat{H}_{0}+\hat{V}_{int},$ (2)

in which

$\hat{H}_{0}=\sum_{j=1}^{N}\left[{\bf p}_{j}^{2}/2m+\hat{V}_{ho}\left({\bf
r}_{j};t\right)\right]$ (3)

is the single-particle Hamiltonian, and

$\hat{V}_{int}=\sum_{ij}\hat{U}\left({\bf r}_{ij};a\left(t\right)\right)$ (4)

is the interatomic interactions with ${\bf r}_{ij}={\bf r}_{i}-{\bf r}_{j}$.
Here, we consider the most general case that the harmonic potential
$\hat{V}_{ho}$ and interatomic interactions both vary in time, i.e., with the
time-dependent trapping frequency $\omega\left(t\right)$ and $s$-wave
scattering length $a\left(t\right)$. The evolution of moment of inertia
$I\left(t\right)=\left\langle\sum_{j}mr_{j}^{2}\right\rangle\equiv\left\langle
mr^{2}\right\rangle$ is governed by Heisenberg equation, i.e.,

$\frac{dI}{dt}=\frac{1}{i\hbar}\left[I,\hat{H}\right],$ (5)

which simply yields $\dot{I}\equiv dI/dt=2\left\langle\hat{D}\right\rangle$,
and $\hat{D}=\sum_{j=1}^{N}\left({\bf r}_{j}\cdot{\bf p}_{j}+{\bf
p}_{j}\cdot{\bf r}_{j}\right)/2$ is the generator of scale transformation
named the dilatation operator (Pitaevskii1997B, ; Werner2006U, ; Maki2019Q, ).
By further taking the second-order derivative with respect to time, i.e.,
$\ddot{I}=2\left\langle\left[\hat{D},\hat{H}\right]\right\rangle/i\hbar$, we
obtain

$\frac{d^{2}I}{dt^{2}}=4\left\langle\hat{H}\right\rangle-8\left\langle\hat{V}_{ho}\right\rangle-4\left\langle\hat{V}_{int}\right\rangle+\frac{2}{i\hbar}\left\langle\left[\hat{D},\hat{V}_{int}\right]\right\rangle.$
(6)

To proceed, we notice that the two-body interaction $\hat{U}\left({\bf
r}_{ij};a\right)$, under the scale transformation, has the property of
$e^{-i\epsilon\hat{D}/\hbar}\hat{U}\left({\bf
r}_{ij};a\right)e^{i\epsilon\hat{D}/\hbar}=e^{2\epsilon}\hat{U}\left({\bf
r}_{ij};e^{\epsilon}a\right)$ for infinitesimal $\epsilon$. Expanding both
sides of this equation up to the first-order term of $\epsilon$, we obtain the
commutation relation of $\left[\hat{D},\hat{U}\right]$, which in turn gives

$\frac{1}{i\hbar}\left\langle\left[\hat{D},\hat{V}_{int}\right]\right\rangle=2\left\langle\hat{V}_{int}\right\rangle+a\left\langle\frac{\partial\hat{V}_{int}}{\partial
a}\right\rangle.$ (7)

Inserting Eq.(7) into Eq.(6) and combining with Hellmann-Feynman theorem and
Tan’s relation (Tan2008L, ), i.e.,

$\left\langle\frac{\partial\hat{V}_{int}}{\partial
a}\right\rangle=\frac{\partial E}{\partial a}=\frac{\hbar^{2}\mathcal{C}}{4\pi
ma^{2}},$ (8)

we finally arrive at Eq.(1) with
$E\left(t\right)=\left\langle\hat{H}\right\rangle$ and
$E_{ho}\left(t\right)=\left\langle\hat{V}_{ho}\right\rangle$. The similar
formulism has been obtained in discussing the breathing mode of two-
dimensional Fermi gases and corresponding scale-invariant dynamics, such as
the quantum anomaly (Hofmann2012Q, ; Gao2012B, ). Here, we emphasize that
Eq.(1), as a fundamental analogous relation of virial theorem at equilibrium
(Thomas2005V, ), characterizes a broad variety of nonequilibrium dynamics of
interacting quantum gases. It provides a deep insight into generic out-of-
equilibrium characters, which we are going to demonstrate as in the follows.

##  III Free expansion of quantum gases

Free expansion, simply releasing atoms from the trap, provides crucial
information on both equilibrium and dynamic properties of ultracold atomic
gases. It is widely used in experiments of cold atoms, such as the time of
flight. One of direct applications of dynamic virial theorem is to calculate
the release energy, i.e., $E_{rel}\equiv E\left(t\right)$. Since the release
energy is well understood in weakly-interacting and unitary gases, it is
inversely convenient to verify the validity of dynamic virial theorem. To this
end, the knowledge of evolution of Tan’s contact as well as that of moment of
inertia is required. As an example, let us consider free expansion of a two-
component Fermi gas, initially prepared at equilibrium in the trap. We
evaluate the evolution of Tan’s contact and moment of inertia of the system in
the framework of hydrodynamic theory (Menotti2002E, ; Hu2004C, ). To solve the
hydrodynamic equations, we adopt the scaling form of time-dependent density
profile, i.e., $n\left({\bf r},t\right)=n_{0}\left({\bf
r}/b\right)/b^{3}\left(t\right)$, where $n_{0}\left({\bf r}\right)$ is the
equilibrium density profile initially in the trap. The time dependence of
$n\left({\bf r},t\right)$ is entirely determined by the scaling factor
$b\left(t\right)$. The atomic cloud size $\left\langle
r^{2}\right\rangle\left(t\right)$ during expansion is related to the initial
size $\left\langle r^{2}\right\rangle_{0}$ by the scaling factor
$b\left(t\right)$ as $\left\langle
r^{2}\right\rangle\left(t\right)=b^{2}\left(t\right)\left\langle
r^{2}\right\rangle_{0}$, which in turn gives

$\ddot{I}\left(t\right)=\frac{4}{\omega_{0}}\left[\dot{b}^{2}\left(t\right)+b\left(t\right)\ddot{b}\left(t\right)\right]E_{ho}^{(0)},$
(9)

$E_{ho}^{(0)}$ is the initial potential energy and $\omega_{0}$ is the initial
trapping frequency.

In the weakly-interacting Bardeen-Cooper-Schrieffer (BCS) limit with small
negative scattering length, the scaling factor $b\left(t\right)$ satisfies
(Menotti2002E, ; Hu2004C, )

$\ddot{b}\left(t\right)-\frac{\omega_{0}^{2}}{b^{3}\left(t\right)}+\frac{3}{2}\chi\left(g\right)\omega_{0}^{2}\left[\frac{1}{b^{3}\left(t\right)}-\frac{1}{b^{4}\left(t\right)}\right]=0,$
(10)

where $\chi\left(g\right)=E_{int}^{(0)}/E_{ho}^{(0)}$ is the ratio of
interaction energy $E_{int}^{(0)}$ to potential energy $E_{ho}^{(0)}$
initially in the trap, and $g=4\pi\hbar^{2}a/m$ is the interaction strength
between atoms. The contact $\mathcal{C}\left(t\right)$ during expansion may be
evaluated according to the local density approximation (LDA) by making the
adiabatic ansatz (Qu2016E, ), i.e., $\mathcal{C}\left(t\right)=\int d{\bf
r}\mathcal{I}\left({\bf r},t\right)$ with local contact density
$\mathcal{I}\left({\bf r},t\right)\approx 4\pi^{2}n^{2}\left({\bf
r},t\right)a^{2}$ (Tan2008L, ), which in turn gives
$\mathcal{C}\left(t\right)=4\pi maE_{int}^{(0)}/\hbar^{2}b^{3}$. With all
these results in hands, we easily obtain the release energy according to the
dynamic virial theorem, i.e.,

$E_{rel}=E_{ho}^{(0)}\left\\{\frac{1}{\omega_{0}^{2}}\left[\dot{b}^{2}\left(t\right)+b\left(t\right)\ddot{b}\left(t\right)\right]-\frac{\chi\left(g\right)}{2b^{3}\left(t\right)}\right\\}$
(11)

From Eq.(10), we find (Derivation1, )

$\frac{1}{\omega_{0}^{2}}\left[\dot{b}^{2}\left(t\right)+b\left(t\right)\ddot{b}\left(t\right)\right]=1-\frac{1}{2}\chi\left(g\right)\left[1-\frac{1}{b^{3}\left(t\right)}\right].$
(12)

Then the release energy reads

$E_{rel}=\left[1-\frac{1}{2}\chi\left(g\right)\right]E_{ho}^{(0)}$ (13)

in the weakly-interacting limit, and further reduces to the well-known result
$E_{rel}=E_{ho}^{(0)}$ in the BCS limit (Giorgini2008T, ).

In the Bose-Einstein-Condensation (BEC) limit with small positive scattering
length, the scaling factor satisfies (Hu2004C, )

$\ddot{b}\left(t\right)-\frac{\omega_{0}^{2}}{b^{4}\left(t\right)}=0.$ (14)

It is nothing than that of weakly-interacting bosons (Castin1996B, ;
Kagan1996E, ). The evolution of contact is again determined by
$\mathcal{C}\left(t\right)=\int d{\bf r}\mathcal{I}\left({\bf r},t\right)$
with local contact density (Tan2008L, )

$\mathcal{I}\left({\bf r},t\right)\approx\frac{4\pi n\left({\bf
r},t\right)}{a}+\pi^{2}n^{2}\left({\bf r},t\right)a_{m}a,$ (15)

where $a_{m}\approx 0.6a$ is the scattering length between molecules
(Petrov2004, ). Then we have
$\mathcal{C}\left(t\right)=\mathcal{C}_{b}+\mathcal{C}_{m}b^{-3}$, in which
$\mathcal{C}_{b}=4\pi N/a$ is the contribution from the binding energy of
tightly bound molecules that is not released during expansion.
$\mathcal{C}_{m}=\int d{\bf r}\pi^{2}n_{0}^{2}\left({\bf r}\right)a_{m}a$ is
the initial contact corresponding to the interaction energy between molecules,
i.e., $E_{m,int}^{(0)}=\hbar^{2}\mathcal{C}_{m}/4\pi ma$. From the dynamic
virial theorem, the release energy is easily calculated, i.e.,

$E_{rel}\equiv E+\frac{N\epsilon_{b}}{2}\\\
=\left[\frac{1}{3b^{3}\left(t\right)}+\frac{2}{3}\right]E_{ho}^{(0)}-\frac{1}{2b^{3}\left(t\right)}E_{m,int}^{(0)},$
(16)

where $\epsilon_{b}=\hbar^{2}/ma^{2}$ is the binding energy of molecules. By
further using the equilibrium virial theorem for weakly-interacting bosons
initially in the trap $E_{m,int}^{(0)}=2E_{ho}^{(0)}/3$ (Pitaevskii2016B, ),
the release energy simply becomes $E_{rel}=2E_{ho}^{(0)}/3$, the well-known
result in the BEC limit (Giorgini2008T, ).

In the unitarity limit with divergent scattering length, the release energy is
rather easy to calculate, since the system obeys an exact scale-invariant
evolution, which yields $b\left(t\right)=\sqrt{1+\omega_{0}^{2}t^{2}}$
(Werner2006U, ; Maki2019Q, ). Besides, the second term on the right-hand side
of Eq.(1) vanishes in the unitarity limit. Consequently, the dynamic virial
theorem simply gives the well-known result $E_{rel}=E_{ho}^{(0)}$
(Giorgini2008T, ).

One another application of free expansion is to test the scale invariance of
strongly-interacting quantum gases as well as important consequences resulted
from the scale-symmetry breaking (Elliott2014O, ; Saint-Jalm2019D, ;
Maki2022D, ). Such scale-invariant dynamics could be identified according to
the evolution of cloud size, which is governed by the dynamic virial theorem.
If defining

$\tau^{2}\left(t\right)\equiv\frac{m}{2E_{ho}^{(0)}}\left(\left\langle
r^{2}\right\rangle-\left\langle r^{2}\right\rangle_{0}\right),$ (17)

we easily obtain the equation satisfied by $\tau^{2}\left(t\right)$

$\frac{d^{2}}{dt^{2}}\tau^{2}\left(t\right)=2+\frac{\hbar^{2}\left[\mathcal{C}\left(t\right)-\mathcal{C}_{0}\right]}{4\pi
maE_{ho}^{(0)}},$ (18)

where $\mathcal{C}_{0}$ is the initial contact before release. In
(Elliott2014O, ), $\tau^{2}\left(t\right)$, as an identification of the scale
invariance, is measured in the free expansion and obeys
$\tau^{2}\left(t\right)=t^{2}$ for a scale-invariant Fermi gas. In the weakly-
interacting limit, we obtain

$\frac{d^{2}}{dt^{2}}\tau^{2}\left(t\right)\approx
2+0.29278k_{FI}a\left[\left(1+\omega_{0}^{2}t^{2}\right)^{-3/2}-1\right],$
(19)

and

$\frac{d^{2}}{dt^{2}}\tau^{2}\left(t\right)\approx
2+2.4668\left(k_{FI}a\right)^{-1}\left[\left(1+\omega_{0}^{2}t^{2}\right)^{-1/2}-1\right]$
(20)

in the strongly-interacting limit, where $k_{FI}$ is the Fermi wave number at
trap center for an ideal gas. Generally, an accurate estimation of contact is
needed to depict the evolution of $\tau^{2}\left(t\right)$ in the free
expansion, for example, by using high-temperature virial expansion
(Elliott2014O, ; Liu2009V, ; Liu2013V, ).

##  IV Efimovian expansion

Scale-invariant quantum gases display a fancy scaling expansion dynamics in
time-dependent harmonic traps, i.e., $\omega\left(t\right)\sim
1/\sqrt{\lambda}t$ with variation rate $\lambda$. This phenomenon is termed as
“Efimovian expansion” (Deng2016O, ). Here, we show that such profound
expansion dynamics is inherently governed by the dynamic virial theorem. By
further taking the derivative of dynamic virial theorem with respect to $t$,
we obtain

$\frac{1}{4}\frac{d^{3}I}{dt^{3}}=\frac{dE}{dt}-2\frac{dE_{ho}}{dt}+\frac{\hbar^{2}}{8\pi
ma}\frac{d\mathcal{C}}{dt}.$ (21)

According to Hellmann-Feynman theorem, we have

$\frac{dE}{dt}=\frac{d}{dt}\left\langle\hat{H}\left(t\right)\right\rangle=\left\langle\frac{d}{dt}\hat{H}\left(t\right)\right\rangle=-\frac{I\left(t\right)}{\lambda
t^{3}}.$ (22)

Combining with $E_{ho}=I/2\lambda t^{2}$, we arrive at

$\frac{d^{3}I}{dt^{3}}+\frac{4}{\lambda t^{2}}\frac{dI}{dt}-\frac{4I}{\lambda
t^{3}}=\frac{\hbar^{2}}{2\pi ma}\frac{d\mathcal{C}}{dt}.$ (23)

For noninteracting and resonant-interacting (scale-invariant) Fermi gases, the
right-hand side of Eq.(23) vanishes, which recovers that of (Deng2016O, ; MoI,
). The evolution of cloud size demonstrates a temporal scaling expansion
behavior. Away from the scale-invariant point at finite interaction strength,
the expansion dynamics resulting from the scale-symmetry breaking is simply
characterized by the evolution of contact (Gharashi2016B, ; Qi2016T, ;
Shi2016E, ). In the strongly-interacting limit, the contact can approximately
be estimated as before, i.e.,
$\mathcal{C}\left(t\right)\approx\mathcal{C}_{0}/b\left(t\right)$ with
$b^{2}\left(t\right)=I\left(t\right)/I_{0}$. Here, $I_{0}$ is the initial
moment of inertia and $\mathcal{C}_{0}=256\pi\alpha Nk_{FI}/35\xi_{B}^{1/4}$
is the initial contact (Qu2016E, ; Pitaevskii2016B, ), where $N$ is the total
atom number, $\alpha\approx 0.12$ and $\xi_{B}\approx 0.37$ are universal
parameters. The evolution of cloud size (or moment of inertia) during
Efimovian expansion near resonant interaction is shown in Fig.1.

Figure 1: (Color online) Efimovian expansion of a two-component Fermi gas near
resonance. Here, the variation rate of trap is set to be $\lambda=0.01$,
$t_{0}$ is the initial time, $I_{0}$ is the initial moment of inertia (or
cloud size), and $k_{FI}$ is the initial Fermi wave number at trap center for
an ideal gas.

##  V Monopole oscillations

The study of low-energy elementary excitations is a subject of primary
importance in many-body physics. It is achieved, for example, by abruptly
disturbing the external trap and exerting the oscillation of system around its
equilibrium. In the follows, let us discuss the monopole oscillation of cold
atoms in a spherical trap, which we find is governed by the dynamic virial
theorem.

Let us firstly consider a cluster of classical particles in a harmonic trap
interacting according to short-range potentials. The moment of inertia of a
single particle is defined as $I\equiv mr^{2}$. Then we have $\dot{I}=2{\bf
p}\cdot{\bf r}$ with the momentum ${\bf p}$. By further taking the second-
order derivative with respect to $t$, we obtain $\ddot{I}=2\left(\dot{{\bf
p}}\cdot{\bf r}+{\bf p}\cdot\dot{{\bf r}}\right)$. As we have $\dot{{\bf
p}}={\bf F}$, the force acting on the particle, and the kinetic energy
$E_{kin}={\bf p}\cdot\dot{{\bf r}}/2$, we obtain $\ddot{I}=2\left({\bf
F}\cdot{\bf r}+2E_{kin}\right)$. In the harmonic trap, the force acting on the
particle is the negative gradient of the potential energy, i.e., ${\bf
F}=-\nabla E_{ho}$, which immediately gives ${\bf F}\cdot{\bf r}=-2E_{ho}$.
Finally, we obtain
$E\left(t\right)-2E_{ho}\left(t\right)=\ddot{I}\left(t\right)/4$. Since the
interactions between atoms are local, the positions of atoms as well as their
moment of inertia are not changed from the instant before to the instant after
collision. Therefore, the momentum and energy conversation implies that
(Lobser2015O, )

$E\left(t\right)-2E_{ho}\left(t\right)=\frac{1}{4}\frac{d^{2}I}{dt^{2}}$ (24)

holds for a cluster of classical particles. Eq.(24) is the classical version
of the dynamic virial theorem. The classical dynamic virial theorem simply
leads to a non-damping monopole oscillation with the frequency
$\omega_{M}=2\omega_{0}$, which has recently been observed in a Bose gas
(Lobser2015O, ).

In the low-temperature limit, quantum correlations resulting from interactions
come into play. Let us consider the small monopole oscillation of a two-
component Fermi gas initially prepared at equilibrium in the trap. Then the
total energy of the system is given by the virial theorem, and we have
$E=\left[2-\chi\left(g\right)/2\right]E_{ho}^{(0)}$ in the BCS limit,
$E=2E_{ho}^{(0)}$ in the unitarity limit, and
$E=5E_{ho}^{(0)}/3-N\epsilon_{b}/2$ in the BEC limit. Here,
$\chi\left(g\right)=E_{int}^{(0)}/E_{ho}^{(0)}$ is the ratio of the
interaction energy $E_{int}^{(0)}$ to the potential energy $E_{ho}^{(0)}$,
$\epsilon_{b}$ is the binding energy of molecules, and $N$ is the atom number.
At time $t=0$, the monopole oscillation is exerted due to a slight disturbing
on the trap. The evolution of contact is easily evaluated by using the
adiabatic ansatz as before. Then the evolution of the moment of inertia or the
cloud size is given by the dynamic virial theorem.

In the BCS limit, we obtain

$\frac{d^{2}y}{dt^{2}}+4\omega_{0}^{2}y-\omega_{0}^{2}\chi\left(g\right)y^{-3/2}-2\omega_{0}^{2}\left[2-\frac{\chi\left(g\right)}{2}\right]=0,$
(25)

where we have defined $y\equiv\left\langle
r^{2}\right\rangle\left(t\right)/\left\langle
r^{2}\right\rangle\left(0\right)$. By linearizing Eq.(25) around the
equilibrium, i.e., $y\approx 1+\delta y$, we obtain

$\frac{d^{2}\delta
y}{dt^{2}}+\left[4+\frac{3}{2}\chi\left(g\right)\right]\omega_{0}^{2}\delta
y=0.$ (26)

We find that the frequency of the monopole oscillation is shifted to
$\omega_{M}=\omega_{0}\sqrt{4+3\chi\left(g\right)/2}$ by interatomic
interactions. It further reduces to $\omega_{M}=2\omega_{0}$ in the
noninteracting limit. In the BEC limit, we similarly have

$\frac{d^{2}y}{dt^{2}}+4\omega_{0}^{2}y-\frac{2}{3}\omega_{0}^{2}y^{-3/2}-\frac{10}{3}\omega_{0}^{2}=0,$
(27)

which is linearized to

$\frac{d^{2}\delta y}{dt^{2}}+5\omega_{0}^{2}\delta y=0.$ (28)

The frequency of the monopole oscillation is then shifted to
$\omega_{M}=\sqrt{5}\omega_{0}$ in the BEC limit, consistent with the well-
known result of Bose gases in the presence of mean-field interactions. In the
unitarity limit, the dynamic virial theorem implies that a strongly
interacting Fermi gas follows exactly the same monopole oscillation as that of
an ideal gas with the frequency $\omega_{M}=2\omega_{0}$.

##  VI Maximum energy growth under increasing interactions

In this section, let us discuss the dynamic behavior of interacting many-body
systems with time-dependent interatomic interactions. Explicitly, we consider
how fast the energy could be pumped into a system by increasing interactions,
starting from the non-interacting limit. This fundamental issue has recently
been addressed (Qi2021M, ). Counterintuitively, it is not true that the faster
the interaction increases, the larger the rate of energy gain becomes. The
maximum energy growth theorem states that there exists an upper limit of
initial energy growth rate, which could be reached only when the scattering
length increases with time $t$ as $\sim\sqrt{t}$. However, the direct
measurement of the energy dynamics in experiments is difficult. Here, we are
going to show that the dynamic virial theorem provides an experimentally
accessible way to verify the maximum energy growth theorem according to the
evolution of the cloud size during expansion.

Let us consider a two-component Fermi gas initially prepared at equilibrium in
the trap. It is supposed that the initial interatomic interaction is so weak
that we may treat it as an ideal gas. Then the gas is released from the trap
for $t\geq 0$, accompanied by the increasing of interactions. To estimate the
_initial_ energy growth rate, it is sufficient to consider the dynamics of the
system in short time. Here, the short time is defined as the timescale much
smaller than the many-body timescale $t_{n}=\hbar/E_{F}$ related to the Fermi
energy $E_{F}$, and also larger than the timescale $t_{r}=mr_{0}^{2}/\hbar$
imposed by the range $r_{0}$ of the two-body potential. In this timescale, the
scattering length may be assumed to take a power-law form of

$a\left(t\right)=a_{ho}\beta\left(\omega_{0}t\right)^{\alpha},$ (29)

and the two-body physics dominates the expansion dynamics. Here,
$a_{ho}=\sqrt{\hbar/m\omega_{0}}$ is the harmonic length with respect to the
initial trap with frequency $\omega_{0}$, and $\alpha,\beta>0$ are parameters
characterize respectively the power and quench rate. Then the energy can be
obtained by simply integrating Tan’s sweep theorem over time $t$, i.e.,

$E\left(t\right)=E\left(0\right)+\int_{0}^{t}dt^{\prime}\frac{\hbar^{2}\mathcal{C}\left(t^{\prime}\right)}{4\pi
ma^{2}\left(t^{\prime}\right)}\frac{da}{dt^{\prime}},$ (30)

where $E\left(0\right)$ is the initial energy that is simply the potential
energy $E_{ho}\left(0\right)$ given by the virial theorem at equilibrium. The
evolution of the cloud size or the moment of inertia is related to the energy
characterized by the dynamic virial theorem in expansion,

$\frac{d^{2}I}{dt^{2}}=4E\left(t\right)+\frac{\hbar^{2}\mathcal{C}\left(t\right)}{2\pi
ma\left(t\right)}.$ (31)

Inserting Eq.(30) into (31) , we obtain

$\frac{d^{2}I}{dt^{2}}=2\omega_{0}^{2}I_{0}+\frac{\hbar^{2}\mathcal{C}\left(t\right)}{2\pi
ma\left(t\right)}+\frac{\hbar^{2}}{\pi
m}\int_{0}^{t}dt^{\prime}\frac{\mathcal{C}\left(t^{\prime}\right)}{a^{2}\left(t^{\prime}\right)}\frac{da}{dt^{\prime}},$
(32)

and $I_{0}=2E_{ho}\left(0\right)/\omega_{0}^{2}$ is the initial moment of
inertia. We find that the dynamics of Tan’s contact is crucial for evaluating
the energy as well as the cloud size, which is simply governed by the two-body
physics in short time as pointed out in (Qi2021M, ). There are totally three
typical behaviors of the evolution of contact in short time, depending on the
power parameter $\alpha$ (Qi2021M, ), i.e.,

$\mathcal{C}\left(t\right)=\begin{cases}16\pi^{2}g_{2}\left(0\right)a^{2}\left(t\right),&\alpha>1/2,\\\
\left|A\left(\beta\right)\right|^{2}g_{2}\left(0\right)\hbar
t/m,&\alpha=1/2,\\\
\left|A\left(\infty\right)\right|^{2}g_{2}\left(0\right)\hbar
t/m,&0<t<1/2,\end{cases}$ (33)

where $g_{2}\left({\bf r}\right)$ is defined as

$g_{2}\left({\bf r}\right)\equiv\int d{\bf R}\rho_{2}\left({\bf R}+{\bf
r}/2,{\bf R}-{\bf r}/2\right),$ (34)

and

$\rho_{2}\left({\bf r}_{1},{\bf
r}_{2}\right)=\left\langle\hat{\psi}_{\uparrow}^{\dagger}\left({\bf
r}_{1}\right)\hat{\psi}_{\downarrow}^{\dagger}\left({\bf
r}_{2}\right)\hat{\psi}_{\downarrow}\left({\bf
r}_{2}\right)\hat{\psi}_{\uparrow}\left({\bf r}_{1}\right)\right\rangle_{0}$
(35)

is the two-body density matrix for the initial state. Here, it is defined
$A\left(\beta\right)=\left[B\left(1/2\right)+1/4\pi\beta\right]^{-1}$ and
$B\left(\alpha\right)=i^{3/2}\Gamma\left(\alpha+1\right)/4\pi\Gamma\left(\alpha+1/2\right)$.
Subsequently, the energy as well as the evolution of the moment of inertia can
conveniently be evaluated as follows.

For $\alpha>1/2$, the contact takes the form of

$\mathcal{C}\left(t\right)=16\pi^{2}g_{2}\left(0\right)a^{2}\left(t\right)=16\pi^{2}g_{2}\left(0\right)a_{ho}^{2}\beta^{2}\left(\omega_{0}t\right)^{2\alpha}.$
(36)

The energy growth $\delta E\left(t\right)\equiv
E\left(t\right)-E\left(0\right)$ in short time is obtained according to
Eq.(30),

$\frac{\delta
E\left(t\right)}{g_{2}\left(0\right)a_{ho}^{3}\hbar\omega_{0}}=4\pi\beta\left(\omega_{0}t\right)^{\alpha}.$
(37)

By inserting Eqs.(29) and (36) into Eq.(32), the evolution of moment of
inertia in short time satisfies

$\frac{d^{2}I}{dt^{2}}=2\omega_{0}^{2}I_{0}+\frac{24\pi\hbar^{2}\beta}{m}g_{2}\left(0\right)a_{ho}\left(\omega_{0}t\right)^{\alpha}.$
(38)

Combining with the initial conditions $I\left(0\right)/I_{0}=1$, and
$\dot{I}\left(0\right)/I_{0}=0$, we obtain

$f_{0}^{-1}\left[\frac{I}{I_{0}}-\left(1+\left(\omega_{0}t\right)^{2}\right)\right]=\frac{12\pi\beta}{\left(1+\alpha\right)\left(2+\alpha\right)}\left(\omega_{0}t\right)^{2+\alpha}$
(39)

with
$f_{0}=g_{2}\left(0\right)a_{ho}^{3}\hbar\omega_{0}/E_{ho}\left(0\right)$. If
defining the reduced moment of inertia and energy growth as

$\displaystyle\tilde{I}\left(t\right)$ $\displaystyle\equiv$ $\displaystyle
f_{0}^{-1}\left[\frac{I}{I_{0}}-\left(1+\left(\omega_{0}t\right)^{2}\right)\right],$
(40) $\displaystyle\delta\tilde{E}\left(t\right)$ $\displaystyle\equiv$
$\displaystyle\frac{\delta
E\left(t\right)}{\hbar\omega_{0}g_{2}\left(0\right)a_{ho}^{3}},$ (41)

we easily find a one-on-one correspondence between the energy growth and the
moment of inertia

$\delta\tilde{E}\left(t\right)=4\pi\beta\left[\frac{\left(1+\alpha\right)\left(2+\alpha\right)}{12\pi\beta}\tilde{I}\left(t\right)\right]^{\alpha/\left(2+\alpha\right)}.$
(42)

At the critical point of $\alpha=1/2$, we have

$a\left(t\right)=\beta
a_{ho}\left(\omega_{0}t\right)^{1/2}=\beta\left(\frac{\hbar}{m}\right)^{1/2}t^{1/2},$
(43)

and the contact is linearly dependent on time, i.e.,
$\mathcal{C}\left(t\right)=\left|A\left(\beta\right)\right|^{2}g_{2}\left(0\right)\hbar
t/m$. Then the reduced initial energy growth takes the form of

$\delta\tilde{E}\left(t\right)=\frac{\left|A\left(\beta\right)\right|^{2}\left(\omega_{0}t\right)^{1/2}}{4\pi\beta}.$
(44)

Similarly, according to the dynamic virial theorem, the evolution of the
moment of inertia satisfies

$\frac{d^{2}I}{dt^{2}}=2\omega_{0}^{2}I_{0}+\frac{3\hbar^{2}}{2\pi
m\beta}\left|A\left(\beta\right)\right|^{2}g_{2}\left(0\right)a_{ho}\left(\omega_{0}t\right)^{1/2},$
(45)

which yields

$\tilde{I}\left(t\right)=\frac{\left|A\left(\beta\right)\right|^{2}\left(\omega_{0}t\right)^{5/2}}{5\pi\beta}.$
(46)

Then we arrive at the one-on-one correspondence between the reduced initial
energy growth and the moment of inertia

$\delta\tilde{E}\left(t\right)=\frac{\left|A\left(\beta\right)\right|^{8/5}}{4\pi\beta}\left[5\pi\beta\tilde{I}\left(t\right)\right]^{1/5}.$
(47)

For $0<\alpha<1/2$, the evolution of the contact linearly increases with time
as well, i.e.,
$\mathcal{C}\left(t\right)=\left|A\left(\infty\right)\right|^{2}g_{2}\left(0\right)\hbar
t/m$, and is independent of the parameters $\alpha$ and $\beta$. This means
that the contact is independent of how fast the scattering length varies in
time, even if the scattering length initially grows infinitely fast as in the
quench process. In this case, the corresponding reduced energy growth becomes

$\delta\tilde{E}\left(t\right)=\frac{16\alpha}{\beta\left(1-\alpha\right)}\left(\omega_{0}t\right)^{1-\alpha}.$
(48)

Again the evolution of the moment of inertia is given by the dynamic virial
theorem

$\frac{d^{2}I}{dt^{2}}=2\omega_{0}^{2}I_{0}+\frac{32\hbar^{2}a_{ho}}{m\beta}\frac{1+\alpha}{1-\alpha}g_{2}\left(0\right)\left(\omega_{0}t\right)^{1-\alpha},$
(49)

which yields

$\tilde{I}\left(t\right)=\frac{16\left(1+\alpha\right)}{\beta\left(1-\alpha\right)\left(2-\alpha\right)\left(3-\alpha\right)}\left(\omega_{0}t\right)^{3-\alpha}.$
(50)

Then we arrive at the relation

$\delta\tilde{E}\left(t\right)=\frac{16\alpha}{\beta\left(1-\alpha\right)}\\\
\times\left[\frac{\beta\left(1-\alpha\right)\left(2-\alpha\right)\left(3-\alpha\right)}{16\left(1+\alpha\right)}\tilde{I}\left(t\right)\right]^{\left(1-\alpha\right)/\left(3-\alpha\right)}.$
(51)

Till now, we have obtained the one-on-one correspondence between the
instantaneous energy gain and the moment of inertia (or cloud size). The
initial energy growth as a function of the atomic cloud size is plotted in
Fig.2 for several typical power parameters. The energy gain resulting from the
interaction increasing can conveniently be measured in experiments according
to the measurement of the cloud size in expansion. We find that expanding to
the same size, the energy gain of the system reaches the maximum when the
scattering length increases as $\sim\sqrt{t}$, which provides an
identification of the maximum energy growth theorem in experiments.

Figure 2: (Color online) The initial energy growth as a function of the atomic
cloud size during the expansion accompanied by an increasing of the scattering
length $a\left(t\right)/a_{ho}=\beta\left(\omega_{0}t\right){}^{\alpha}$ at
different power parameters. Here, we set $\beta=1$.

##  VII Thermodynamic relation

In the follows, we derive an out-of-equilibrium thermodynamic relation for
interacting quantum gases. Here, a two-fluid hydrodynamic theory is adopted to
study the dynamics at nonequilibrium, which is applicable in a wide range of
temperature. The basic idea of the two-fluid hydrodynamic theory lies in
considering the system as if it were a mixture of two different liquids at
finite temperature: a superfluid without viscosity and a viscous normal fluid.
Analogous to those for superfluids or normal fluids, the number density of
particles satisfies the equation of continuity,

$\frac{\partial n}{\partial t}+\nabla\cdot{\bf j}=0,$ (52)

where $n=n_{n}+n_{s}$ is the total number density, and ${\bf j}=n_{n}{\bf
v}_{n}+n_{s}{\bf v}_{s}$ is the current density with the number density
$n_{s(n)}$ for the superfluid (normal) component and the corresponding
velocity ${\bf v}_{s(n)}$. Besides, the current density ${\bf j}$ satisfies an
additional equation (Landau1987F, )

$\frac{\partial j_{i}}{\partial t}+\sum_{k}\frac{\partial\Pi_{ik}}{\partial
x_{k}}+\frac{n}{m}\frac{\partial V_{ext}}{\partial
x_{i}}=-\sum_{k}\frac{\partial\Pi_{ik}^{\prime}}{\partial x_{k}}$ (53)

for the $i$th component of ${\bf j}$, where $V_{ext}$ is the external
potential, and $\Pi_{ik}$ is the momentum current density tensor defined as

$\Pi_{ik}\equiv n_{n}v_{ni}v_{nk}+n_{s}v_{si}v_{sk}+\frac{p}{m}\delta_{ik},$
(54)

and $p$ is the pressure. The viscous normal fluid introduces an additional
dissipation described by

$\Pi_{ik}^{\prime}=-\frac{\eta}{m}\sigma_{ik}-\delta_{ik}\left(\frac{\zeta_{1}}{m}\sigma_{1}^{\prime}+\frac{\zeta_{2}}{m}\sigma_{2}^{\prime}\right),$
(55)

with

$\displaystyle\sigma_{ik}$ $\displaystyle\equiv$ $\displaystyle\frac{\partial
v_{ni}}{\partial x_{k}}+\frac{\partial v_{nk}}{\partial
x_{i}}-\frac{2}{3}\delta_{ik}\nabla\cdot{\bf v}_{n},$ (56)
$\displaystyle\sigma_{1}^{\prime}$ $\displaystyle\equiv$
$\displaystyle\nabla\cdot\left[mn_{s}\left({\bf v}_{n}-{\bf
v}_{s}\right)\right].$ (57) $\displaystyle\sigma_{2}^{\prime}$
$\displaystyle\equiv$ $\displaystyle\nabla\cdot{\bf v}_{n}.$ (58)

Here, $\eta$ and $\zeta_{1,2}$ are respectively the shear and bulk
viscosities. Eqs.(52) and (53) forms a set of basic equations for describing
the dynamics of interacting quantum gases at finite temperature.

At zero temperature, the system becomes a pure superfluid, and the number
density of the normal component vanishes as well as corresponding velocities.
Then Eq.(53) becomes

$\frac{\partial\left(nv_{i}\right)}{\partial
t}+\sum_{k}\frac{\partial}{\partial
x_{k}}\left(nv_{i}v_{k}\right)+\frac{1}{m}\frac{\partial p}{\partial
x_{i}}+\frac{n}{m}\frac{\partial V_{ext}}{\partial x_{i}}=0,$ (59)

which, combining with the equation of continuity, yields

$\frac{\partial{\bf v}}{\partial t}+\left({\bf v}\cdot\nabla\right){\bf
v}+\frac{\nabla p}{mn}+\frac{\nabla V_{ext}}{m}=0.$ (60)

By using the identity of

$\nabla\left(\frac{v^{2}}{2}\right)={\bf v}\times\left(\nabla\times{\bf
v}\right)+\left({\bf v}\cdot\nabla\right){\bf v},$ (61)

and concerning the fact that the superfluid corresponds to a potential flow
with $\nabla\times{\bf v}=0$, we arrive at

$\frac{\partial{\bf v}}{\partial
t}+\nabla\left(\frac{v^{2}}{2}\right)+\frac{\nabla p}{mn}+\frac{\nabla
V_{ext}}{m}=0.$ (62)

This is the well-known Euler equation for a superfluid (see for example in
(Pethick2008B, )). Here, all the quantities are those for the superfluid, so
that the subscript $s$ is dropped off without confusion.

Above the critical temperature, the system is entirely a normal fluid, and the
superfluid portion vanishes. Then Eq.(53) becomes the Euler equation for a
normal fluid in the presence of viscosity at finite temperature (Elliott2014O,
; Landau1987F, ), i.e.,

$\left[nm\frac{\partial}{\partial t}+nm\left({\bf
v}\cdot\nabla\right)\right]v_{i}+\frac{\partial p}{\partial
x_{i}}+n\frac{\partial V_{ext}}{\partial x_{i}}\\\
=\sum_{k}\frac{\partial}{\partial
x_{k}}\left(\eta\sigma_{ik}+\delta_{ik}\zeta_{2}\sigma_{2}^{\prime}\right).$
(63)

Consequently, we find that the two-fluid hydrodynamic theory provides a rather
general formulism to describe the dynamics of interacting many-body systems in
a wide range of temperature. Moreover, the quantum pressure comes into play at
zero temperature, which could formally be included in the definition of
pressure, i.e., $p=p_{c}+p_{q}$ and $p_{c\left(q\right)}$ stands for the
classical (quantum) pressure (Manfredi2001S, ).

Now we are ready for the discussion of the evolution of moment of inertia
$I\left(t\right)\equiv\left\langle mr^{2}\right\rangle$ based on the two-fluid
hydrodynamic formulism. The $i$th component of $\left\langle
r^{2}\right\rangle$ is defined as $\left\langle x_{i}^{2}\right\rangle=\int
d{\bf r}nx_{i}^{2}$ with $i=x,y,z$, which obeys

$\frac{d\left\langle x_{i}^{2}\right\rangle}{dt}=\int d{\bf r}\frac{\partial
n}{\partial t}x_{i}^{2}=-\int d{\bf r}\left(\nabla\cdot{\bf
j}\right)x_{i}^{2}=2\int d{\bf r}j_{i}x_{i}.$ (64)

By further taking the second-order derivative with respect to $t$, we have

$\frac{d^{2}\left\langle x_{i}^{2}\right\rangle}{dt^{2}}=2\int d{\bf
r}\frac{\partial j_{i}}{\partial t}x_{i}.$ (65)

According to Eq.(53) , we obtain

$\frac{d^{2}\left\langle x_{i}^{2}\right\rangle}{dt^{2}}=2\int d{\bf
r}\left(\Pi_{ii}+\Pi_{ii}^{\prime}\right)-2\int d{\bf
r}x_{i}\frac{n}{m}\frac{\partial V_{ext}}{\partial x_{i}}.$ (66)

By inserting the explicit form of $\Pi_{ii}^{(\prime)}$ into above equation
and assuming that the external potential is a harmonic trap, i.e.,
$V_{ext}\left({\bf r}\right)=m\omega^{2}r^{2}/2$, we arrive at

$\frac{1}{4}\frac{d^{2}\left\langle x_{i}^{2}\right\rangle}{dt^{2}}=\int d{\bf
r}\left[n_{n}\frac{v_{ni}^{2}}{2}+n_{s}\frac{v_{si}^{2}}{2}+\frac{p}{2m}\right.\\\
\left.-\frac{\eta}{2m}\sigma_{ii}-\frac{1}{2m}\left(\zeta_{1}\sigma_{1}^{\prime}+\zeta_{2}\sigma_{2}^{\prime}\right)\right]-\int
d{\bf r}\frac{n}{m}\left(\frac{1}{2}m\omega^{2}x_{i}^{2}\right).$ (67)

Summing the above equation over all three directions and noticing
$\sum_{i}\int d{\bf r}\eta\sigma_{ii}=0$, we finally obtain

$\frac{1}{4}\frac{d^{2}I}{dt^{2}}=E_{kin}-E_{ho}+\frac{3}{2}\mathcal{J}-\frac{3}{2}\int
d{\bf
r}\left(\zeta_{1}\sigma_{1}^{\prime}+\zeta_{2}\sigma_{2}^{\prime}\right),$
(68)

where $E_{kin}\equiv\int d{\bf
r}\left(n_{n}mv_{n}^{2}+n_{s}mv_{s}^{2}\right)/2$ corresponds to the total
kinetic energy of the normal fluid and superfluid, $E_{ho}=\int d{\bf
r}n\left(m\omega^{2}r^{2}/2\right)$ is the potential energy, and
$\mathcal{J}\equiv\int d{\bf r}p$ is the integral of the local pressure over
the whole space, namely the thermodynamic potential. Eq.(68) simply reduces to
that for a normal fluid with $\zeta_{1}=0$ above the critical temperature
(Elliott2014O, ).

Combining Eq.(68) and the dynamic virial theorem, we find the following simple
relation for interacting quantum gases

$\mathcal{J}=\frac{2}{3}E_{internal}\left(t\right)+\frac{\hbar^{2}\mathcal{C}\left(t\right)}{12\pi
ma\left(t\right)}+\Gamma\left(t\right),$ (69)

where we have defined the internal energy $E_{internal}\equiv
E\left(t\right)-E_{kin}\left(t\right)-E_{ho}\left(t\right)$ and the
dissipation energy resulting from viscosities $\Gamma\left(t\right)=\int d{\bf
r}\left(\zeta_{1}\sigma_{1}^{\prime}+\zeta_{2}\sigma_{2}^{\prime}\right)$.
This relation holds for both the normal fluid and superfluid as well as their
mixtures, and thus is valid in a wide range of temperature. Eq.(69) is a kind
of the out-of-equilibrium analog of Tan’s pressure relation, since it simply
reduces to the well-known form for uniform gases at equilibrium (Tan2008G, ),
i.e.,

$p=\frac{2}{3}\mathcal{E}_{internal}+\frac{\hbar^{2}\mathcal{I}}{12\pi ma}$
(70)

with the internal energy density $\mathcal{E}_{internal}$, the contact density
$\mathcal{I}=\mathcal{C}/V$ and the volume $V$.

##  VIII Conclusions

It is shown that a variety of out-of-equilibrium dynamics of interacting many-
body systems are elegantly governed by the dynamic virial theorem. Its
applications in several typical nonequilibrium dynamic processes of cold atoms
are presented, in which observable consequences in experiments are discussed.
Remarkably, the dynamic virial theorem provides an experimentally accessible
way to verify the maximum energy growth theorem according to the measurement
of the atomic cloud size in expansion. Besides, the dynamic virial theorem
gives rise to a simple thermodynamic relation, the analog of Tan’s pressure
relation at equilibrium, for interacting many-body systems in the framework of
two-fluid hydrodynamic theory. This thermodynamic relation holds in a wide
range of temperature. Our results provide a fundamental understanding of
generic behaviors of interacting many-body systems at nonequilibrium, and are
readily examined in future experiments with ultracold atoms.

###### Acknowledgements.

We would like to thank Ran Qi for inspiring discussion on the derivation of
dynamic virial theorem. We are particularly grateful to Xi-Wen Guan for his
helpful suggestions. This work is supported by the National Natural Science
Foundation of China under Grant No.11974384, the National Key R&D Program
under Grant No. 2022YFA1404102, K. C. Wong Education Foundation under Grant
No. GJTD- 2019-15 and the Natural Science Foundation of Hubei Province under
Grant No. 2021CFA027.

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