††thanks: Equal contribution to this work††thanks: Equal contribution to this
work

# Understanding anti-parity-time symmetric systems with a conventional heat
transfer framework – comment on “Anti-parity-time symmetry in diffusive
systems”

Lin Zhao  Lenan Zhang  Bikram Bhatia  Evelyn N. Wang<EMAIL_ADDRESS>Department of Mechanical Engineering, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA

###### Abstract

Inspired by non-Hermitian physics, Li et al. (Science 364, 170-173)
theoretically predicted and experimentally demonstrated a stationary
temperature profile in a diffusive heat transfer system – seemingly indicating
that heat ”stops” diffusing. By analogy to the wave physics framework, the
motionless and moving temperature profiles are manifestations of the anti-
parity-time ($\mathcal{APT}$) symmetry and symmetry breaking states,
respectively. Their experimental setup consists of two thermally coupled rings
rotating in the opposite direction. At a particular rotation speed, known as
the exceptional point, the $\mathcal{APT}$ symmetry of the system changes,
resulting in the temperature profile switching between stationary and moving
states. In fact, this seemingly unusual and exotic behavior can be elegantly
captured and predicted using a conventional heat transfer framework with
similarity and scaling analysis. In this work, we show the system behavior can
be characterized into three zones by two widely-used dimensionless parameters
on a regime map. The exceptional point, discovered using wave physics, is
located precisely on the zone boundary on the regime map, indicating a balance
between the contribution of thermal coupling and mechanical motion.
Furthermore, the observed cessation of thermal diffusion is merely a result of
the long diffusion time constant of the experimental setup. The unfamiliarity
of concepts in another scientific field as well as the remarkable equivalence
of the two points of view prompts this in-depth discussion of the analogy
between wave physics and heat transfer. We believe that this work can help
bridge the gap and promote new developments in the two distinctly different
disciplines.

††preprint: APS/123-QED

In their report Li  _et al._ (2019), Li et al. studied the heat transfer in
counter-moving rings that exhibits anti-parity-time ($\mathcal{APT}$)
symmetry. Leveraging concepts from quantum physics and photonics, they
predicted and demonstrated a critical rotation speed at the exceptional point
resulting in spontaneous symmetry breaking that changes the temperature field
from a motionless state to a moving state. While analogies among different
scientific disciplines have led to several innovations, such as the
development of photonic crystals Joannopoulos  _et al._ (2008), vastly
different terminologies often make it challenging to appreciate the
equivalence among interdisciplinary fields such as wave physics and heat
transfer. Here we provide a corresponding heat transfer analysis of the same
system and more explicitly show the relationship between the two seemingly
distinct approaches.

The starting point of our heat transfer analysis is the thermal energy
equation Mills (1999),

$\rho c\frac{\partial T}{\partial t}=k\frac{\partial^{2}T}{\partial
x^{2}}-\rho cv\frac{\partial T}{\partial x}+\frac{h}{b}(T-T_{\infty})$ (1)

where $\rho$, $c$, and $k$ are the density, specific heat, and thermal
conductivity of the medium, $v$ is the velocity of the moving medium, $h$ is
the heat transfer coefficient (we follow the heat transfer convention of $h$,
which has the unit $\mathrm{W/m^{2}/K}$), $b$ is the thickness of the medium,
and $T_{\infty}$ is an arbitrary reference temperature. In the specific case
of this study, the medium is the polycaprolactam (PA6) ring and $T_{\infty}$
is the temperature of the other coupled ring. Note that $h/\rho cb$, defined
in this work, is equivalent to “$h$” in Li  _et al._ (2019).

Figure 1: System behavior and regime map based on two dimensionless
parameters: $\mathrm{Pe}$ and $\eta\mathrm{Nu}$. (A) Schematic showing three
competing heat transfer mechanisms (diffusion, coupling, and advection)
occurring simultaneously in the system. Diffusion, driven by the temperature
gradient in the individual ring, causes the temperature profile to smooth out
and eventually become isothermal at equilibrium. Coupling, driven by the
temperature difference between the two rings, induces inter-ring thermal
energy exchange and reduces their temperature difference. Advection, driven by
the mechanical motion of the ring, shifts the temperature profile on each ring
in the direction of movement. The resultant system behavior depends on the
relative importance of the three mechanisms. (B) System regime map as a
function $\mathrm{Pe}$ and $\eta\mathrm{Nu}$. Corresponding experimental
conditions reported by Li et al. are shown as open circles on the map. The
dashed diagonal line represents the boundary between the motionless (Zone II)
and moving (Zone III) temperature fields, which is equivalent to the
exceptional point showed by Li et al. By increasing the rotation speed
($\Omega$), Li et al. demonstrated the temperature fields changing from the
motionless (Inset: green circles in Zone II) to the moving state (Inset: blue
circle in Zone III). In Zone I, where $\mathrm{Pe}$ and $\eta\mathrm{Nu}$ are
smaller than unity, thermal diffusion becomes dominant and the temperature
field will always evolve to isothermal conditions first.

As in a typical heat transfer analysis, Eq. (1) is non-dimensionalized using
the following length, time, and temperature scales,

$x^{*}=\frac{x}{R},\>t^{*}=\frac{t}{t_{d}},\>T^{*}=\frac{T-T_{\infty}}{T_{\mathrm{max}}-T_{\infty}}$
(2)

where $R$ is the radius of the ring, $t_{d}={R^{2}}/{\alpha}$ is the thermal
diffusion time scale, $\alpha={k}/{\rho c}$ is the thermal diffusivity of the
ring, $T_{\mathrm{max}}$ is the maximum temperature in the system, and
quantities with asterisks are dimensionless variables. Substituting Eq. (2)
into Eq. (1), the dimensionless form of the thermal energy equation is given
by,

$\frac{\partial T^{*}}{\partial t^{*}}=\frac{\partial^{2}T^{*}}{\partial
t^{*2}}-\mathrm{Pe}\frac{\partial T^{*}}{\partial x^{*}}+\eta\mathrm{Nu}T^{*}$
(3)

where $\mathrm{Pe}=vR/\alpha=\Omega R^{2}/\alpha$ is the Peclet number,
$\mathrm{Nu}=hR/k$ is the Nusselt number, and $\eta=R/b$ is the ratio of
radius to thickness of the ring. These dimensionless numbers are commonly used
in heat transfer analysis since they show the similarity of different systems.
Specifically, the Peclet and Nusselt numbers describe the relative importance
of advection (mechanical motion) and thermal coupling (between ring 1 and ring
2) over thermal diffusion, respectively, which define the dynamic similarity.
$\eta$ is the aspect ratio of the ring, which defines the geometric
similarity. The three competing effects in the system, i.e., diffusion,
coupling, and advection, are depicted in Fig. 1A.

We show that the phase transition explained by the $\mathcal{APT}$ symmetry
viewpoint in Li  _et al._ (2019) can be well-predicted by dimensionless
parameter analysis. According to Eq. (3), the system characteristic is fully
determined by two dimensionless parameters: $\mathrm{Pe}$ and
$\eta\mathrm{Nu}$. Depending on the two dimensionless parameters, the system
behavior can be divided into three zones as shown in Fig. 1B. When
$\mathrm{Pe}<1$ and $\eta\mathrm{Nu}<1$ (Zone I), thermal diffusion is
dominant and the temperature field will first evolve to isothermal condition
regardless of thermal coupling and mechanical motion of the rings. When
$\mathrm{Pe}>1$ or $\eta\mathrm{Nu}>1$, the system behavior depends on the
relative magnitude of these two parameters with a transition at
$\mathrm{Pe}=\eta\mathrm{Nu}$ (the dashed diagonal line in Fig. 1B). When
$\mathrm{Pe}<\eta\mathrm{Nu}$ (Zone II), thermal coupling between the two
rings is dominant, which enables the “motionless” temperature field as
reported in Li  _et al._ (2019) at low rotation speeds (green circles in Fig.
1B inset). On the other hand when $\mathrm{Pe}>\eta\mathrm{Nu}$ (Zone III),
advection is dominant and the temperature field moves with the ring as
reported in Li  _et al._ (2019) at high rotation speed (blue circle in Fig. 1B
inset). The critical rotation speed $\Omega_{\mathrm{cr}}=1.27~{}\mathrm{rpm}$
corresponding to the exceptional point discovered in Li  _et al._ (2019) is
equivalent to the phase transition line between Zones II and III in Fig. 1B,
represented by $\mathrm{Pe}=\eta\mathrm{Nu}$. Therefore, this regime map
described by dimensionless parameters shows the same physics as the
$\mathcal{APT}$ symmetry analysis Li  _et al._ (2019).

The system behavior predicted by the regime map (Fig. 1B) can be further
validated by numerically solving Eq. (3), where the experimental results in
Figs. 4(G-I) in Li  _et al._ (2019) were also reproduced. Boundary conditions
were applied for both rings
($T^{*}_{1,2}(\theta=0)=T^{*}_{1,2}(\theta=2\pi)$), where $\theta$ is the
angular location. The initial dimensionless temperature profile,
$T^{*}_{1,2}(\theta,t=0)=(1+\sin{(\theta-\pi/2)})/2$, was adapted from Li  _et
al._ (2019) (Materials and Methods S3) to match the experimental conditions.
The temperature peak at $\theta=\pi$ represents the hottest point on the ring,
which is in contact with the high temperature copper block initially. The
temperature valley at $\theta=0$ and $\theta=2\pi$ represents the coldest
point on the ring, which is in contact with the low temperature copper block
initially.

Figure 2: Temperature profile evolution predicted by heat transfer analysis.
(A to C) Temperature profiles of ring 1 (upper) and ring 2 (lower) at three
rotation speeds, (A) 0.5 rpm, (B) 1 rpm, and (C) 1.5 rpm. Ten profiles equally
spaced between $t=0$ s and $t=30$ s are shown for each ring. (D to F)
Evolution of the temperature dipole angle and rotation angle of ring 1 at
three rotation speeds, (D) 0.5 rpm, (E) 1 rpm, and (F) 1.5 rpm. Red dots:
experimentally measured temperature dipole angles reported by Li et al. Red
curves: predicted temperature dipole angles in this work. Light gray line:
incorrect rotation angles of ring 1 reported by Li et al. Black line: correct
rotation angles of ring 1 in this work.

Figs. 2(A-C) show the temperature profile evolution of both rings (ten
profiles equally spaced between 0 s and 30 s) at three rotation speeds (0.5,
1, 1.5 rpm) used in the experiments reported in Li  _et al._ (2019). The
rotation angle $\phi$ of the temperature dipole (defined in Li  _et al._
(2019), Materials and Methods S4) from the initial state were then calculated
from the temperature profile evolution at the corresponding rotation speeds,
plotted as red curves in Figs. 2(D-F). Experimental temperature dipole angles
(red dots in Figs. 2(D-F)) and rotation angles of ring 1 (light gray lines in
Figs. 2(D-F)) reported in Figs. 4(G-I) in Li  _et al._ (2019) are also
included for comparison. We note an inconsistency between the rotation angles
of ring 1 reported in Li  _et al._ (2019) and in this work. We believe that
this is due to an error in Figs. 4(G-I) in Li  _et al._ (2019), since
$1~{}\mathrm{rpm}=6~{}\mathrm{deg/s}$. Overall, there is good agreement
between experimentally measured temperature results reported in Li  _et al._
(2019) and numerical results in this work, which demonstrates that heat
transfer analysis using dimensionless parameters is adequate to predict the
phase transition of the system.

To further demonstrate the underlying connection between the wave physics
point of view proposed in Li  _et al._ (2019) and the heat transfer analysis
in this work, we summarize the equivalence between the two approaches in Table
1. Although the two points of view appear vastly different, they can both
describe the same system behavior and yield identical results.

Table 1: Analogy between the wave physics point of view and conventional heat transfer  | Li et al. | This work
---|---|---
View of problem | Anti-parity-time symmetry | Similarity and scaling analysis
System characteristic | Hamiltonian: | Dimensionless parameters:
$H(k,D,h,v)$ | $f(\mathrm{Pe},\eta\mathrm{Nu})$
Motionless profile criteria |  $\mathcal{APT}$ symmetric: | Thermal-coupling dominant:
pure imaginary eigenvalues | $\mathrm{Pe}<\eta\mathrm{Nu},\eta\mathrm{Nu}>1$
Moving profile criteria |  $\mathcal{APT}$ symmetry broken: | Advection dominant:
non-zero real part eigenvalues | $\mathrm{Pe}>\eta\mathrm{Nu},\mathrm{Pe}>1$
Diffusing profile criteria | - | Diffusion dominant:
- | $\mathrm{Pe}<1,\eta\mathrm{Nu}<1$
Phase transition location | Exceptional point (EP): | Regime boundary:
$k^{2}v^{2}_{\mathrm{EP}}=h^{2}$ (motion) |  $\mathrm{Pe}=\eta\mathrm{Nu}$ (motion)
\- (diffusion) |  $\mathrm{Pe}=1$, $\eta\mathrm{Nu}=1$ (diffusion)

## References

  * Li  _et al._ (2019) Y. Li, Y.-G. Peng, L. Han, M.-A. Miri, W. Li, M. Xiao, X.-F. Zhu, J. Zhao, A. Alù, S. Fan, and C.-W. Qiu, Anti-parity-time symmetry in diffusive systems., Science 364, 170 (2019).
  * Joannopoulos  _et al._ (2008) J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade,  _Photonic crystals: Molding the Flow of Light_ , 2nd ed. (Princeton University Press, 2008).
  * Mills (1999) A. F. Mills,  _Heat Transfer_ , 2nd ed. (Prentice Hall, 1999).