Thermalization in a Quantum Harmonic Oscillator with Random Disorder

Abstract

We propose a possible scheme to study the thermalization in a quantum harmonic oscillator with random disorder. Our numerical simulation shows that through the effect of random disorder, the system can undergo a transition from an initial nonequilibrium state to a equilibrium state. Unlike the classical damped harmonic oscillator where total energy is dissipated, total energy of the disordered quantum harmonic oscillator is conserved. In particular, at equilibrium the initial mechanical energy is transformed to the thermodynamic energy in which kinetic and potential energies are evenly distributed. Shannon entropy in different bases are shown to yield consistent results during the thermalization.

Full text

entropy
Article
Thermalization in a Quantum Harmonic Oscillator
with Random Disorder
Ya-Wei Hsueh 1, Che-Hsiu Hsueh 2,* and Wen-Chin Wu 2, *
1Department of Physics, National Central University, Jhong-li 32001, Taiwan; yhsueh@phy.ncu.edu.tw
2Department of Physics, National Taiwan Normal University, Taipei 11677, Taiwan
*Correspondence: chhsueh7@gmail.com (C.-H.H.); wu@ntnu.edu.tw (W.-C.W.)
Received: 12 June 2020; Accepted: 30 July 2020; Published: 31 July 2020


Abstract:
We propose a possible scheme to study the thermalization in a quantum harmonic oscillator
with random disorder. Our numerical simulation shows that through the effect of random disorder,
the system can undergo a transition from an initial nonequilibrium state to a equilibrium state.
Unlike the classical damped harmonic oscillator where total energy is dissipated, total energy of
the disordered quantum harmonic oscillator is conserved. In particular, at equilibrium the initial
mechanical energy is transformed to the thermodynamic energy in which kinetic and potential
energies are evenly distributed. Shannon entropy in different bases are shown to yield consistent
results during the thermalization.
Keywords: thermalization; quantum harmonic oscillator; random disorder
1. Introduction
Microscopic description of the thermalization has been a longstanding question in isolated
quantum systems. The biggest enigma is due to the discordance between reversible microscopic laws
and irreversible macroscopic phenomena. Among many fundamental issues, one common question
is whether an isolated quantum system can reach thermal equilibrium, i.e., the state with maximum
entropy [
1
–
8
]. Fortunately, ultracold quantum gases, which are both pure and controllable, provide an
excellent platform to study the nonequilibrium dynamics for isolated quantum systems. In atomic
Bose–Einstein condensate (BEC) experiments, Kinoshita et al. [
9
] showed no evidence of thermalization
by pairwise collision, from the Tonks–Girardeau limit to the intermediate coupling regime. However,
the dissipative motion of oscillating BEC in a disordered trap, done by Dries et al. [
10
], did manifest
the thermalization [11].
The phenomenon is even more interesting as it relates to Anderson localization (AL)—one of the
most important topics in condensed-matter physics [
12
]. AL has recently been observed in various
systems such as discrete-time quantum walks [
13
], Rydberg electrons [
14
], photonic lattices [
15
],
monolayer graphene [
16
], and even quantum chormodynamics [
17
,
18
]. Since 2003, AL has been
extensively studied and realized in BEC with random disorder both experimentally [
19
,
20
] and
theoretically [
11
,
19
–
34
]. These works considered AL in BEC in various regimes and setups.
More references can be found in the review article of Sanchez-Palencia et al. [35].
The quantum harmonic oscillator is one of the simplest systems in physics and plays a central
role in a wide variety of fields [
36
]. For example, quantum harmonic oscillator appears everywhere in
quantum optics and its properties have been seen regularly in experiments. In ultracold atoms, one can
use the Feshbach resonance technique to tune the
s
-wave scattering length to zero [
37
]. Inspired by
the experiment of an oscillating BEC in disordered trap [
10
], we aim to study the thermalization in a
disordered quantum harmonic oscillator. It will be shown in this system that the reversible microscopic
quantum mechanics actually conceals the irreversible macroscopic phenomena of thermodynamics.
Entropy 2020,22, 855; doi:10.3390/e22080855 www.mdpi.com/journal/entropy

Entropy 2020,22, 855 2 of 11
As mentioned earlier, works from over a decade ago have already discussed the related
phenomena in disordered BEC in the mean-field regime [
19
,
20
,
30
–
33
,
35
]. We nevertheless try to present
an exact result for a “noninteracting” quantum oscillator in a disorder trap. In principle, the exact
noninteracting harmonic oscillator could still be very different from the disordered dipole-oscillating
BEC in the mean-field regime, even in the noninteracting limit. Our results show that equilibrium
properties are very different between the current disordered noninteracting quantum oscillator and
the disordered dipole-oscillating BEC in the mean-field regime [
38
]. More interestingly, it will be
shown in the current quantum system that the equilibrium properties are very similar to those for
a classical harmonic oscillator in the microcanonical ensemble. The kind of quantum-to-classical
transition indicates strongly the “intrinsic decoherence” of the isolated system [39,40].
A possible scheme to this problem will first be proposed. In the theoretical simulation,
we alternatively prepare an initial coherent state with a centroid velocity
v0
, which is an
out-of-equilibrium state. Owing to the multiple scattering with the disorder, more and more (initial)
mechanical energy will transform to the thermodynamic one. Once the mechanical energy is fully
transformed to the thermodynamic one, the system reaches the equilibrium. During the whole process
from nonequilibrium to equilibrium, total energy is conserved. Moreover, at equilibrium, total energy is
evenly distributed between the kinetic energy and the potential energy associated with harmonic trap.
The paper is organized as follows. In Section 2, we propose the possible scheme to study the
thermalization in a quantum harmonic oscillator with random disorder. Theoretical approach for the
simulation is also introduced. In Section 3, we show the results of real-space density and momentum
distributions when the system reaches the equilibrium. In Section 4, due to the effect of disorder,
we show that mechanical energy is transformed to the thermodynamic one. In addition, at equilibrium
thermodynamic energy is evenly distributed between kinetic energy and harmonic potential energy.
In Section 5, Shannon entropy in different bases are studied for the thermalization process. Section 6is
a conclusion.
2. The Approach
To study the thermalization of a quantum harmonic oscillator with random disorder, our approach
is to solve the time-evolution wavefunction
ψ
of the particle for the whole process from nonequilibrium
to equilibrium. The corresponding properties can be accessed based on the solved ψ.
For simplicity, we consider a one-dimensional quantum harmonic oscillator in a disordered trap.
The time-dependent Schrödinger equation (TDSE) describing the system is
i¯h∂tψ(x,t)="−¯h2∂2
x
2m+1
2mω2x2+Vdis (x)#ψ(x,t), (1)
where
m
is the particle mass,
ω
is the trapping frequency, and
Vdis (x)
is the Gaussian correlated
disorder potential which satisfies the autocorrelation function
Z∞
−∞Vdis (x)Vdis (x+∆x)dx =V2
Dexp −2∆x2/σ2
D. (2)
where
VD
and
σD
correspond to the strength and correlation length of the random disorder, respectively.
Figure 1illustrates the set-up of a possible experiment. Before the disorder potential is turned on,
the system is in the ground state with the Gaussian wavefunction. At
t=
0, the trap is abruptly
displaced to the left, so the system gains energy and starts to move to the left. Afterwards, it oscillates.
Note that at the same (
t=
0), the disorder potential is turned on and due to the effect of it, the system
will eventually come to equilibrium.

Entropy 2020,22, 855 3 of 11
Figure 1.
Schematic of the setup for a quantum harmonic oscillator with random disorder. (
a
) At
t<
0,
the disorder is off and the system is in the ground state with a Gaussian wavefunction. (
b
) At
t=
0,
the trap is abruptly displaced to the left, so the system gains an energy to move to the left. Disorder is
on at this point. (
c
) At
t>
0, the system oscillates left and right until it comes to equilibrium due to the
disorder effect.
Alternatively, for convenience in simulation, at
t=
0 an initial coherent state with a velocity
v0
at
the center is prepared for the system,
ψ(x, 0) = ψg(x)exp (imv0x/ ¯h), (3)
where
ψg(x) = (mω/π¯h)1/4 exp(−mωx2/
2
¯h)
is the Gaussian ground state. After release, owing to
the multiple scattering from the disorder, the wave packet starts to redistribute. When evolution time
is long enough (
ttth
,
tth
is the thermalization time defined in Section 5), and the system approaches
the equilibrium. Together with (2) and (3), we have numerically solved (1) for a long enough time to see
the phenomenon of thermalization. In our simulation,
ε0≡¯hω
and the trapping length
l0≡√¯h/mω
are taken as the units of energy and length.
Here, we detail the simulation method to solve the TDSE (1). We performed the fast Fourier
transform (FFT) to calculate the integration and differentiation in space. The calculation domain is
fixed at
[−
64
π
, 64
π]
with a grid of
N=
12, 288 points. Moreover, the time integration is done by
adaptive Runge–Kutta method of orders 4 and 5, built-in in Matlab software. The random Gaussian
correlated disorder potential is taken as Vdis(x) = VDf(x), where
f(x) =
N
∑
i=1
Aiexp "−4(x−xi)2
σ2
D#(4)
with
xi
the grid points. The generated random numbers
Ai
are subject to
hAii=
0 and
hA2
ii=
1,
and the normalization
R∞
−∞f(x)2dx =
1 is finally made to satisfy Equation (2). We have been assured
that the wave function has not been contaminated by numerical noise (round-off error) even after
long-time computation. One can also consider the non-Gaussian correlated disorders such as the
sinc(x) = sin x/x
correlated function considered in [
30
,
34
]. We have also done the calculation based
on the
sinc
correlated disorder function and the results are found to be the same as the presented ones.
It is also worth noting that all results obtained are from a single (fixed) random potential. The result
does not average over noise at any points.
3. Equilibrium Distribution
It is important to note that “equilibrium” has a precise meaning in thermodynamics, namely,
the systems is described with a Boltzmann distribution when it is coupled with a reservoir at
temperature
T
. There is no “reservoir” in the current isolated system. Here, “equilibrium” means
a state of balance or a stable situation in which different parts of energies will no longer exchange
internally. As will be seen clearly, disorder plays the role to assist the energy exchange and lead to the
“equilibrium” of the system. In this regard, the thermalization discussed in the present paper can be
viewed as a kind of “intrinsic thermalization”.

Entropy 2020,22, 855 4 of 11
It is also important to note that a system connected to a random disorder cannot in general be
regarded as an isolated one. The isolated system under consideration refers to the combination of
the harmonic oscillator and the random disorder. The real and time-independent random Gaussian
correlated disorder considered does fulfill the time reversibility of Equation (1) and, at the same time,
results in the maximization of Shannon entropy (see Section 5).
With the solved dynamic wavefunction
ψ(x
,
t)
, one is ready to see the real-space density distribution
ρ(x,t) = |ψ(x,t)|2. (5)
At t=0 before release, the real-space density distribution is given by
ρ(x, 0) = |ψ(x, 0)|2= (mω/π¯h)1/2 exp(−mωx2/¯h). (6)
Once the system starts to oscillate, due to multiple scattering with the random disorder, the system
will eventually reach equilibrium from nonequilibrium.
Figure 2shows the equilibrium real-space density distributions at
ttth
. Three cases of disorder
strength
VD=
50
ε0
, 100
ε0
, and 200
ε0
are shown. The disorder correlation length
σD=
0.01
l0
and the
initial velocity
v0=
50
l0ω
are fixed for all three cases. As shown, the three equilibrium distributions
have almost overlapped with each other. It means that different finite disorder potential strengths
lead to the same equilibrium distribution for the isolated quantum oscillator. Of equal importance,
the distribution is very close to the microcanonical distribution for a classical harmonic oscillator. In a
microcanonical ensemble of a classical harmonic oscillator with a given energy
E0
, the probability
distribution in phase space is
ρc(x,p) = 1
2π¯hδ(H−E0);H=p2
2m+1
2mω2x2(7)
and thus the corresponding real-space density distribution is
ρc(x) = Z∞
−∞dp ρc(x,p) = 1
πl0r2E0
ε0−x2
l2
0
. (8)
The singularities at the classical turning points correspond to
E0=mv2
0/
2
=
1250
ε0
. The transition
from a quantum distribution to a classical one provides a clear evidence of quantum (intrinsic)
decoherence for the current system.
It is also of interest to study the momentum distribution
ρ(k
,
t)
. Momentum distribution is
given by
ρ(k,t) = |e
ψ(k,t)|2, (9)
where
e
ψ(k
,
t)
is the Fourier transform of
ψ(x
,
t)
. As at
t=
0 before release, the real-space wavefunction
is
ψ(x
, 0
) = ψg(x)exp (imv0x/¯h)
(see Equation (3)), and the corresponding initial momentum
distribution is then
ρ(k
, 0
) = |e
ψg(k−mv0/¯h)|2
, where
e
ψg(k−mv0/¯h)
is just a
k
-shift from
e
ψg(k)
,
the Fourier transformation of
ψg(x)
. In fact,
e
ψg(k)
is also a Gaussian function. The initial momentum
distribution is shown by the red curve in Figuer 3. Figure 3shows only the case with
VD=
50
ε0
.
When the system reaches equilibrium (
ttth
), the numerically solved momentum distribution is
shown by the blue curve in Figure 3, featuring two peaks at the high-
k
turning points. When comparing
the
t=
0 and
ttth
momentum distributions in Figure 3, it is important to note the following. (i) The
case of
t=
0 is asymmetric which implies a nonequilibrium state; the case of
ttth
is symmetric
which implies an equilibrium state. (ii) Through the effect of random disorder, initial mechanical energy
E0=mv2
0/
2 has been transformed to thermodynamic energy at equilibrium, which is distributed
among different kstates in a wide range up to the cut-off turning points.

Entropy 2020,22, 855 5 of 11
Figure 2.
Equilibrium real-space density distributions
ρ(x
,
ttth)
for a quantum harmonic oscillator
with random disorder. Three cases of
VD=
50
ε0
(solid blue curve), 100
ε0
(dotted red curve), and 200
ε0
(dot-dashed green curve) are shown which have almost overlapped with each other. Parameters
σD=
0.01
l0
and
v0=
50
l0ω
are fixed in all three cases. For comparison, dashed black curve corresponds
to the distribution of a classical harmonic oscillator in a microcanonical ensemble (see also Equation (8)).
Figure 3.
Equilibrium momentum distributions
ρ(k
,
t)
for a quantum harmonic oscillator with random
disorder. Red and blue curves correspond to the distribution at
t=
0 (nonequilibrium) and
ttth
(equilibrium), respectively. Parameters used are VD=50ε0,σD=0.01l0, and v0=50l0ω.
4. Energy Distribution
In the present system, total energy is the combination of three parts:
E(t) = K(t) + U(t) + Vdis (t)
,
where
K(t) = R|¯h∂xψ|2dx/(
2
m)
is the kinetic energy,
U=R(mω2x2/
2
)|ψ|2dx
is the potential energy
associated with harmonic trapping, and
Vdis =RVdis(x)|ψ|2dx
is the potential energy associated with
random potential. For random potential, the associated
Vdis
is typically small compared to
K
and
U
.
Figure 4shows the evolutions of
K
and
U
, as well as the total energy
E
for the case of
VD=
50
ε0
,
σD=0.01l0, and v0=50l0ω.
Unlike the classical damped harmonic oscillator where total energy is dissipated, the total energy
E
of the current disordered quantum harmonic oscillator is conserved. The disorder does not store
energy during the thermalization process. It plays the role only to redistribute the energy and result
in the final energy distribution. Both
K
and
U
are oscillating at the beginning. When
t
just passes
tth
,
the oscillation becomes relatively smaller, and when
ttth
, they approach the static limit. As a matter
of fact, energy distribution is another indication for the system going from a nonequilibrium to an
equilibrium state.

Entropy 2020,22, 855 6 of 11
One important feature shown in Figure 4is that when the system reaches equilibrium,
thermodynamic energy
E
is evenly distributed in
K
and
U
. Thus, at equilibrium the virial theorem
is satisfied:
K=U=E/
2. This is analogous to the case for a classical harmonic oscillator,
the time-averaged
hKi=hUi=E/
2. Moreover, for the present system, the equilibrium temperature
Teq
can be obtained as
kBTeq/
2
=K(ttth) = U(ttth )
. In view of the static value shown in
Figure 4,K(ttth)'U(ttth )'625ε0, it is identified that Teq '1250ε0/kB.
Figure 4.
Energy evolution of a quantum harmonic oscillator with random disorder. Total energy is
conserved in the whole process and kinetic energy and potential energy associated with harmonic
trapping are evenly distributed at equilibrium. The parameters used are
VD=
50
ε0
,
σD=
0.01
l0
,
and v0=50l0ω.
5. Entropy Evolution
Studies of thermalization often follow the von Neumann entropy
Sv(t) = −kBTr [ˆ
ρ(t)ln ˆ
ρ(t)], (10)
where ˆ
ρ=|ψ(t)ihψ(t)|is the density operator. Thus, the production rate of Svis
dSv(t)
dt =−kBTr ˙
ˆ
ρ(t)ln ˆ
ρ(t) + ˙
ˆ
ρ(t), (11)
where
˙
ˆ
ρ≡∂ˆ
ρ/∂t
and the second term in (11) vanishes due to the conservation of total states.
It is well known that
Sv(t)
as well as
dSv(t)/dt
are basis-independent, i.e., invariant under unitary
transformations. In terms of arbitrary bases,
Sv(t)=−kB∑
i,jhi|ˆ
ρ|jihj|ln ˆ
ρ|ii
dSv(t)
dt =−kB∑
i,jhi|˙
ˆ
ρ|jihj|ln ˆ
ρ|ii. (12)
However, if both basis |iiand |jiare chosen to be the eigenstates of ˆ
ρ, then
dSv(t)
dt =−kB∑
ihi|˙
ˆ
ρ|iihi|ln ˆ
ρ|ii
=ikB
¯h∑
ihi|[ˆ
H,ˆ
ρ]|iihi|ln ˆ
ρ|ii=0, (13)

Entropy 2020,22, 855 7 of 11
or
Sv(t)
is constant, which implies von Neumann entropy is not eligible to describe the time direction
of an isolated quantum system [41,42].
Alternatively we consider the Shannon entropy [43–45]
S(t) = −kB∑
ihi|ˆ
ρ(t)|iilnhi|ˆ
ρ(t)|ii, (14)
where the basis
|ii
is arbitrary but should rule out the eigenstates of either
ˆ
ρ
or
ˆ
H
. Unlike the
basis-independent von Neumann entropy, Shannon entropy is basis-dependent. To acknowledge
that the choice is arbitrary, we consider three choices. When the basis
|ii
is chosen to be the position
eigenstates, the corresponding Shannon entropy
Sx(t)=−kBZρ(x,t)ln [l0ρ(x,t)] dx, (15)
where
ρ(x
,
t) = |ψ(x
,
t)|2
is the real-space density distribution (see also (5)). Alternatively, if the basis
|iiis chosen to be the momentum eigenstates, the corresponding Shannon entropy is
Sp(t)=−kBZρ(k,t)ln ρ(k,t)
l0dk, (16)
where ρ(k,t) = |e
ψ(k,t)|2is the momentum distribution (see also (9)).
Figure 5shows the evolutions of both
Sx(t)
and
Sp(t)
. Taking into account all possible phase
spaces, the result of sum of the two,
Sx(t) + Sp(t)
, is also shown. While the parameters used,
VD=
50
ε0
and
v0=
50
l0ω
, are just a numerical trial, they are close to those of a real experimental set-up with
VD=
50.9
ε0
and
v0=
37.5
l0ω
[
10
]. In addition, we consider a shorter disorder correlation length
σD=
0.01
l0
. All three cases consistently show the increase of entropy for the thermalization process,
and they maximize (saturate) when the system approaches equilibrium. Therefore, the use of Shannon
entropy is suitable to describe the thermalization process in the current isolated quantum system.
In Figure 5, the black dotted line corresponds to the Shannon entropy for a classical harmonic oscillator
in a microcanonical ensemble. It is obtained by substituting the real-space density distribution
ρc(x)
in
Equation (8) into
Sx
in (15). In view of Figure 5, the thermalization time in this case after which the
system reaches equilibrium can be unambiguously identified to be tth =50(1/ω).
Figure 5.
Evolutions of Shannon entropy
Sx(t)
,
Sp(t)
, and the sum
Sp(t) + Sx(t)
for a quantum
harmonic oscillator with random disorder. Black dotted line corresponds to the Shannon entropy
for a classical harmonic oscillator in the microcanonical ensemble. Parameters used are
VD=
50
ε0
,
σD=0.01l0, and v0=50l0ω. The thermalization time is identified to be tth =50(1/ω)(see the text).

Entropy 2020,22, 855 8 of 11
To have a better understanding on how the thermalization time depends on the experimental
set-up, in Figure 6we study the effects of
VD
and
v0
on
S(t) = Sp(t) + Sx(t)
. Figure 6a shows the
evolution of
S(t)
for three disorder strengths:
VD=
50
ε0
, 100
ε0
, and 200
ε0
with fixed
v0=
50
l0ω
and
σD=
0.01
l0
. The thermalization times for the three cases are identified to be
tth =
50
(
1
/ω)
,
25
(
1
/ω)
, and 12.5
(
1
/ω)
, respectively. Therefore, the larger the
VD
is, the faster the system thermalizes.
More interestingly, the effect of
VD
on the thermalization rate (1
/tth
) is linear, 1
/tth ∝VD
. The linear
behavior seems to be quite general in all regimes (noninteracting and interacting) in an isolated
quantum system.
Figure 6.
Evolutions of Shannon entropy
S(t) = Sx(t) + Sp(t)
for a quantum harmonic oscillator
with random disorder. Frame (
a
) considers the effect of disorder potential strength
VD
with fixed
v0=
50
l0ω
and
σD=
0.01
l0
. Frame (
b
) considers the effect of initial velocity
v0
with fixed
VD=
50
ε0
and σD=0.01l0. The corresponding thermalization times are identified.
Figure 6b shows the evolution of
S(t)
for three initial velocities:
v0=
50
l0ω
, 25
l0ω
, and 12.5
l0ω
with fixed
VD=
50
ε0
and
σD=
0.01
l0
. The thermalization times for the three cases are found to
be
tth =
50
(
1
/ω)
, 25
(
1
/ω)
, and 17
(
1
/ω)
, respectively. Thus,
tth
goes monotonically with
v0
. In an
interacting Bose condensate with disorder, it has been found that when
v0>c
(
c
being the sound
velocity), thermalization will develop, while when
v0≤c
, thermalization is hardly developed or will
not develop [
46
]. It thus suggested in an interacting system that the thermalization is intimately related
to the generation of elementary excitations [
10
]. In the present noninteracting system, in contrast,
the situation is very different. There are only free real particles without the quasiparticles. Due to the
effect of disorder, the system tends to thermalize faster with a smaller
v0
as in this case lesser initial
mechanical energy needs to be transformed to the thermodynamic one.
To check the applicability of the Shannon entropy, we also consider the basis of the generalized
coherent states, |ii=|α,ni, where [47]
hx|α,ni=1
2n
2√n!π1
4√l0
exp "−(x−¯
x)2
2l2
0#Hnx−¯
x
l0exp i−n+1
2ωt+x¯
p
¯h−¯
x¯
p
2¯h. (17)
Here,
Hn(x)
are Hermite polynomials,
¯
x=hα
,
n|ˆ
x|α
,
ni=√2l0|α|cos(ωt−θ)
,
¯
p=hα,n|ˆ
p|α,ni=−(√2¯h|α|/l0)sin(ωt−θ)
, and
θ
is some arbitrary phase. The corresponding
Shannon entropy is given by
Sα(t)=−kB
∞
∑
n=0hψ(t)|α,nihα,n|ψ(t)iln [hψ(t)|α,nihα,n|ψ(t)i]. (18)
For simplicity, for this basis we choose the case with
t=
0,
¯
x=
0, and
¯
p=√2¯h|α|/l0≡mv0
.
In this case, Equation (17) results in

Entropy 2020,22, 855 9 of 11
hx|α=mv0l0/√2¯h,n=0i → 1
π1
4√l0
exp −x2
2l2
0!exp imv0x
¯h(19)
which is identical to our initial wavefunction shown in (3). Thus, the initial entropy is expected to
be zero in the case of a single basis function. Figure 7shows the simulation result of
Sα(t)
with
v0=
12.5
l0ω
. We only show the case of relatively smaller
v0
, as it takes much longer computing time
to calculate the cases of higher
v0
. For comparison, we also show
S(t)
, which was already shown in
Figure 6b.
While the Fermi–Pasta–Ulam–Tsingou (FPUT) recurrence effect [
48
,
49
] for the current oscillating
system is significant, the maximization process of
Sα(t)
is still evident. The key is to follow local
minima of
Sα(t)
. Of more importance, the thermalization time is predicted to be
tth '
17
(
1
/ω)
,
consistent with that predicted in
S(t)
. This concludes that Shannon entropy is an eligible one to study
the thermalization in an isolated quantum system.
Figure 7.
Evolutions of Shannon entropy
Sα(t)
for a quantum harmonic oscillator with random disorder
(red curve). The parameters are
v0=
12.5
l0ω
,
VD=
50
ε0
, and
σD=
0.01
l0
. As the FPUT recurrence
effect is significant, the key is to follow local minima of
Sα(t)
. For comparison,
S(t)
(blue curve) is also
shown for the same set of parameters (see also in Figure 6b). The thermalization time is identified to be
tth '17(1/ω), consistent in both cases.
6. Conclusions
In conclusion, we propose a simple, possible scheme to study the thermodynamics in a quantum
harmonic oscillator with random disorder. We have numerically shown that due to the effect of
random disorder, the system can undergo from nonequilibrium to equilibrium state, i.e., thermalization.
During the thermalization process, total energy of the system is conserved and at equilibrium the
kinetic and potential energies are evenly distributed. Shannon entropy in different bases are studied
and shown to maximize during the thermalization process.
Author Contributions:
Y.-W.H. performed the calculations, C.-H.H. and W.-C.W. proposed this work, and all
authors discussed the results and prepared the manuscript. All authors have read and agreed to the published
version of the manuscript.
Funding:
This research was funded by the Ministry of Science and Technology, Taiwan (under grant
No. 108-2112-M-003-006-MY2).
Conflicts of Interest: The authors declare no conflict of interest.

Entropy 2020,22, 855 10 of 11
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