# Quasiballistc Heat Conduction in Transient Grating Spectroscopy

Chengyun Hua  Austin J. Minnich<EMAIL_ADDRESS>Division of Engineering
and Applied Science
California Institute of Technology, Pasadena, California 91125,USA

###### Abstract

Transient grating (TG) spectroscopy is an important experimental technique to
measure mean free path (MFP) spectra using observations of quasiballistic heat
conduction. To obtain MFP spectra, the measurements must be interpreted within
the framework of the frequency-dependent Boltzmann transport equation (BTE),
but previous solutions have restricted validity due to simplifying
assumptions. Here, we analyze heat conduction in TG using a new analytical
solution of the frequency-dependent BTE that accurately describes thermal
transport from the diffusive to ballistic regimes. We demonstrate that our
result enables a more accurate measurement of MFP spectra and thus will lead
to an improved understanding of heat conduction in solids.

##  I Introduction

Quasiballistic heat conduction occurs if a temperature gradient exists over
length scales comparable to phonon mean free paths (MFPs)Chen (2005); Majumdar
(1993). In this regime, local thermal equilibrium does not exist and Fourier’s
law is no longer valid. Presently, quasiballistic transport is under
investigation due to its potential to infer information about the MFPs of
thermal phononsMinnich (2012), knowledge of which is crucial to engineer
thermal conductivity but remains unknown for most solidsDames and Chen (2005);
Minnich  _et al._ (2009).

Quasiballistic transport was originally observed in macroscopic samples using
heat pulse experimentsvon Gutfeld and Nethercot (1964) and later in silicon
membranes using a microfabricated platformSverdrup  _et al._ (2001). Nonlocal
theories of heat conduction based on the Boltzmann transport equation (BTE)
were introduced to describe the quasiballistic regime for phononsMahan and
Claro (1988); Claro and Mahan (1989); Chen (1996) and electronsEzzahri and
Shakouri (2009). Koh and Cahill reported modulation-frequency dependent
thermal conductivities in a time-domain transient thermoreflectance (TDTR)
experiment that they attributed to quasiballistic transport and suggested that
the meansurements could be used to measure MFPs. Recently, quasiballistic
transport has been observed in other experimental configurationsSiemens  _et
al._ (2010); Minnich  _et al._ (2011); Regner  _et al._ (2013); Johnson  _et
al._ (2013). Minnich introduced a reconstruction technique that described how
to quantitatively recover the MFP spectra from observations of quasiballistic
heat transferMinnich (2012).

One notable experimental method for observing quasiballistic transport is the
transient thermal grating (TTG) techniqueJohnson  _et al._ (2013); Maznev  _et
al._ (2011); Rogers  _et al._ (1994), in which the interference of two laser
pulses creates a sinusoidal initial temperature profile with wavelength
$\lambda$. The observed thermal decay yields information about the thermal
properties of the material. Recent work has demonstrated that these
measurements can also reveal MFPs if the grating wavelength is comparable to
MFPs, but interpreting measurements using the reconstruction method introduced
by Minnich requires a solution of the BTE. A previous work reported a modified
”two-channel” modelMaznev  _et al._ (2011), in which low and high frequency
phonons are described by the BTE and heat equation, respectively, but the
extent of the validity of this model is unclear. An analysis within the
framework of the BTECollins  _et al._ (2013) has been recently reported but
the analysis of the frequency-dependent BTE was solely numerical, complicating
its use for the reconstruction method.

Here, we analyze thermal transport in TTG using a new analytical solution to
the frequency-dependent BTE that is valid over the full range of the ballistic
to diffusive regimes. Our analysis demonstrates the existence of weakly and
strongly quasiballistic transport regimes that are distinguished by the
thermal decay time relative to the phonon relaxation times. We provide
theoretical justification for the use of a modified diffusion theory to
interpret observations of quasiballistic transport. Finally, we use our
solution to derive a corrected suppression function that enables phonon MFP
spectra to be measured more accurately. Our results will lead to a better
understanding of phonon heat conduction in solids like thermoelectrics.

##  II Modelling

Thermal transport in a TTG experiment, assuming only in-plane heat conduction,
is described by the 1D frequency-dependent BTEMajumdar (1993),

$\displaystyle\frac{\partial g_{\omega}}{\partial t}$ $\displaystyle+$
$\displaystyle\mu v_{\omega}\frac{\partial g_{\omega}}{\partial
x}=-\frac{g_{\omega}+f_{0}(T_{0})-f_{0}(T)}{\tau_{\omega}}$ (1) $\displaystyle
f_{0}(T)$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi}\hbar\omega
D(\omega)f_{BE}(T)\approx f_{0}(T_{0})+\frac{1}{4\pi}C_{\omega}\Delta T$ (2)

where $g_{\omega}=f_{\omega}(x,t,\mu)-f_{0}(T_{0})$ is the deviational
distribution function, $f_{0}=f_{0}(x,t)$ is the equilibrium distribution
function, $\mu=cos(\theta)$ is the directional cosine, $v_{\omega}$ is the
phonon group velocity, and $\tau_{\omega}$ is the phonon relaxation time.
Assuming a small temperature rise, $\Delta T=T-T_{0}$, relative to a reference
temperature, $T_{0}$, the equilibrium distribution is proportional to $\Delta
T$, as shown in Eq. (2). Here, $\hbar$ is the reduced Planck constant,
$\omega$ is the phonon frequency, $D(\omega)$ is the phonon density of states,
$f_{BE}$ is the Bose-Einstein distribution, and $C_{\omega}=\hbar\omega
D(\omega)\frac{\partial f_{BE}}{\partial T}$ is the mode specific heat. The
volumetric heat capacity is then given by
$C=\int_{0}^{\omega_{m}}C_{\omega}d\omega$ and the Fourier thermal
conductivity $k=\int_{0}^{\omega_{m}}k_{\omega}d\omega$, where
$k_{\omega}=\frac{1}{3}C_{\omega}v_{\omega}\Lambda_{\omega}$ and
$\Lambda_{\omega}=\tau_{\omega}v_{\omega}$ is the phonon MFP. To close the
problem, energy conservation is used to relate $g_{\omega}$ to $\Delta T$,
given by

$\int\int_{0}^{\omega_{m}}\left[\frac{g_{\omega}(x,t)}{\tau_{\omega}}-\frac{1}{4\pi}\frac{C_{\omega}}{\tau_{\omega}}\Delta
T(x,t)\right]d\omega d\Omega=0$ (3)

where $\Omega$ is the solid angle in spherical coordinates and $\omega_{m}$ is
the cut-off frequency. Note that summation over phonon branches is implied
without an explicit summation sign whenever an integration over phonon
frequency or MFP is performed.

Since the initial temperature profile in TTG is sinusoidal, we can assume that
both $g_{\omega}$ and $\Delta T$ are of the form $e^{iqx}$, where
$q=2\pi/\lambda$ is the grating wavevector. Substituting
$g_{\omega}=\widetilde{g}_{\omega}(t,\mu)e^{iqx}$ and $\Delta
T=\Delta\widetilde{T}(t)e^{iqx}$ into Eq. (1) leads to a first order ODE for
$\widetilde{g}_{\omega}(t)$, and its solution is given by

$\widetilde{g}_{\omega}=\frac{1}{4\pi}\frac{C_{\omega}}{\tau_{\omega}}\int_{0}^{t}e^{\gamma(t^{\prime}-t)}\Delta\widetilde{T}(t^{\prime})dt^{\prime}+\widetilde{g}_{\omega}(0)e^{-\gamma
t}$ (4)

where $\gamma=(1+iq\mu\Lambda_{\omega})/\tau_{\omega}$ and
$\widetilde{g}_{\omega}(0)$ is assumed to be independent of $\mu$.

Applying the initial condition,
$\widetilde{g}_{\omega}(0)=\Delta\widetilde{T}(0)C_{\omega}/(4\pi)$ and
substituting Eq. (4) into Eq. (3), we get an semi-analytical expression for
$\Delta\widetilde{T}(t)$:

$\Delta\widetilde{T}(t)=\frac{\Delta\widetilde{T}(0)}{2\int_{0}^{\omega_{m}}\frac{C_{\omega}}{\tau_{\omega}}d\omega}\int_{0}^{\omega_{m}}\int_{-1}^{1}\left[\frac{C_{\omega}}{\tau_{\omega}^{2}}\int_{0}^{t}e^{\gamma(t^{\prime}-t)}\frac{\Delta\widetilde{T}(t^{\prime})}{\Delta\widetilde{T}(0)}dt^{\prime}+\frac{C_{\omega}}{\tau_{\omega}}e^{-\gamma
t}\right]d\mu d\omega.$ (5)

Eq. (5) is typically solved numerically by discretizing the integrals and
solving a dense linear system, a computationally expensive task. Collins _et.
al._ Collins  _et al._ (2013) obtained an analytical solution by applying a
Fourier transform to the grey form of this equation. Here, we extend the
Fourier transform method to the frequency-dependent BTE by observing that the
time integral in Eq. (5) has the form of a convolution and thus can be
simplified in the frequency domain. By applying a Fourier transform, we are
able to decouple the nonlocal effects and obtain the following closed-form
expression for the unknown distribution function $\widetilde{g}_{\omega}$ and
transient temperature $\Delta\widetilde{T}$:

$\displaystyle\mathcal{F}[\widetilde{g}_{\omega}](\xi)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\frac{C_{\omega}}{\tau_{\omega}}\frac{\mathcal{F}[\Delta\widetilde{T}](\xi)}{\gamma-i\xi}+\frac{C_{\omega}}{4\pi}\frac{\Delta\widetilde{T}(0)}{\gamma-i\xi}$
(6) $\displaystyle\mathcal{F}[\Delta\widetilde{T}](\xi)$ $\displaystyle=$
$\displaystyle\frac{\Delta\widetilde{T}(0)\int^{\omega_{m}}_{0}C_{\omega}\mathcal{A}(\xi)d\omega}{\int^{\omega_{m}}_{0}\frac{C_{\omega}}{\tau_{\omega}}[1-\mathcal{A}(\xi)]d\omega}$
(7)
$\mathcal{A}(\xi)=\mathcal{F}[\tau_{\omega}^{-1}e^{-t/\tau_{\omega}}\text{sinc}(qv_{\omega}t)u(t)]=\frac{i}{2q\Lambda_{\omega}}\log\left(\frac{\tau_{\omega}\xi+q\Lambda_{\omega}+i}{\tau_{\omega}\xi-q\Lambda_{\omega}+i}\right)$
(8)

where $\mathcal{F}$ denotes Fourier transform, $u(t)$ is the step function and
$\xi$ is the Fourier variable. The time-domain solution is obtained by inverse
fast Fourier transform. Therefore, we have derived an analytical solution to
the frequency-dependent BTE that is valid from the ballistic to the diffusive
regimes, enabling a more rigorous understanding of thermal transport in TTG.

We can gain insight into which parameters determine the transport regime from
our solution. From Eq. (8), we identify two nondimensional parameters. One is
the familiar phonon Knudsen number Kn${}_{\omega}=q\Lambda_{\omega}$, which
compares the phonon MFP with a characteristic length, in this case $1/q$. To
identify the second parameter, we notice that $\xi^{-1}$ describes a time
scale that we assign to be the characteristic thermal decay time $\Gamma$. We
can therefore define a new non-dimensional parameter that we denote the
transient number, given by $\eta_{\omega}=\tau_{\omega}/\Gamma$, which
compares the phonon relaxation times with the thermal decay time $\Gamma$.

Note that the two parameters are not completely independent. For example, as
the grating wavelength decreases, the thermal decay time also decreases. In
the diffusion regime the relationship is trivial but in the quasiballistic and
ballistic regimes the relationship becomes much more complex. While the
Knudsen number can in principle completely distinguish the transport regime,
we find that the transient number is an additional convenient parameter by
which to specify the regime, particularly for quasiballistic transport where
the specific Knudsen number at which a transition occurs is not obvious.

Therefore, together, these two numbers completely specify the transport
regime. In the diffusive limit, length and time scales are much larger than
the phonon MFPs and relaxation times, respectively, corresponding to
Kn${}_{\omega}\ll 1$ and $\eta_{\omega}\ll 1$. In the ballistic regime,
lengths and times are much smaller than MFPs and relaxation times, or
Kn${}_{\omega}\gg 1$ and $\eta_{\omega}\gg 1$. The two regimes are well-
understood limits of the BTEChen (2005). Here, we focus on the intermediate
range of the two limits, the quasiballistic regime.

Figure 1: Temperature decay curves $\Delta\widetilde{T}$ in the (a) diffusive
limit, (b) ballistic limit, (c) weakly quasiballistic regime, and (d) strongly
quasiballistic regime. The BTE solutions are given by the solid lines, the
Fourier solution by the dashed lines, the ballistic conduction solution by the
dotted line, and the Modified Fourier solution by the dash-dotted lines.

##  III Results

To begin, we examine the transient temperature decay in the different regimes
as shown in Fig. 1. We perform our calculations for crystalline silicon, using
the experimental dispersion in the [100] direction and assuming the crystals
are isotropic. The numerical details concerning the dispersion and relaxation
times are given by Minnich’s recent workMinnich  _et al._ (2011).

###  III.1 Diffusive and ballistic Limits

We first confirm that our result correctly reproduces the diffusive and
ballistic limits. Examining the limit of Eq. (7) when both phonon relaxation
times and MFPs are much smaller than their corresponding characteristic scales
($\eta_{\omega}\ll 1$ & Kn${}_{\omega}^{2}\ll 1$), we find that the solution
reduces to the Fourier solution and the thermal decay time
$\Gamma=(q^{2}\alpha)^{-1}$, where $\alpha=k/C$ is the Fourier thermal
diffusivity. Fig. 1a demonstrates that the BTE solution agrees with the
Fourier solution in this limit. Similarly, at the ballistic limit
($\eta_{\omega}\gg 1$ & Kn${}_{\omega}^{2}\gg 1$) shown in Fig. 1b, the
transient temperature given by Eq. (7) agrees with the ballistic solution of
the BTE in which the relaxation times go to infinity.

###  III.2 Weakly Quasiballistic Regime

We now examine the intermediate quasiballistic regime by allowing the MFPs to
be comparable to or greater than the grating wavevector while requiring the
thermal decay time to be much longer than relaxation times,
Kn${}_{\omega}^{2}\sim 1$ but $\eta_{\omega}\ll 1$. We observe that the BTE
solution does not agree with the Fourier’s law solution, as shown in Fig. 1c.
However, we observe that the shape of the temperature decay remains
exponential, as in Fourier’s law, but with a smaller thermal conductivity. We
denote this regime the weakly quasiballistic regime, and the Fourier solution
with a modified thermal conductivity as the modified Fourier solution. So far,
the validity of the modified Fourier model to describe quasiballistic thermal
transport is largely based on experimental observationsJohnson  _et al._
(2013). The only theoretical approach to explain this observation was
developed by Maznev _et. al_. Their modified ”two-channel” model assumes that
the low-frequency phonons, which are analyzed by the BTE, only interact with
the thermal reservoir of high-frequency phonons, which are analyzed by the
diffusion equation. However, the extent of the validity of this assumption is
not clear.

Here, we give a more rigorous explanation using our solution. Under the
assumption of Kn${}_{\omega}^{2}\sim 1$ and $\eta_{\omega}\ll 1$, the Taylor
expansion of Eq. 8 around $\eta_{\omega}=0$ gives

$\mathcal{A}(\xi)=i\eta_{\omega}\xi+\frac{\text{tan}^{-1}(\text{Kn}_{\omega})}{\text{Kn}_{\omega}}\sim
1.$ (9)

We observe that in the denominator of Eq. (7),
$1-\mathcal{A}(\xi)\sim\tau_{\omega}$ and the full asymptotic expression of
$\mathcal{A}(\xi)$ should be used while in the numerator, $\mathcal{A}(\xi)$
can be approximated $1$. Therefore, Eq. (7) asymptotically approaches the
following form

$\mathcal{F}[\Delta\widetilde{T}](\xi)\approx\frac{\Delta\widetilde{T}(0)}{q^{2}\alpha_{mod}-i\xi}$
(10)
$k_{mod}=\int^{\omega_{m}}_{0}k_{\omega}\left\\{\frac{3}{(\text{Kn}_{\omega})^{2}}\left[1-\frac{\text{tan}^{-1}(\text{Kn}_{\omega})}{\text{Kn}_{\omega}}\right]\right\\}d\omega$
(11)

where $\alpha_{mod}=k_{mod}/C$ is the apparent thermal diffusivity and
$k_{mod}$ is the modified thermal conductivity. Recognizing that Eq. (10) is
simply the Fourier transform of an exponential decay, we find
$\Delta\widetilde{T}(t)\approx\Delta\widetilde{T}(0)\text{exp}\left(-q^{2}\alpha_{mod}t\right)$.
Thus, the formal solution of the BTE is equivalent to a modified diffusion
theory with a modified thermal conductivity given by Eq. (10). The thermal
decay time $\Gamma=(q^{2}\alpha_{mod})^{-1}$. We term this simplified solution
the weak solution to the BTE, valid in the weakly quasiballistic regime. The
modified thermal conductivity is the same expression given by Maznev _et. al_
Maznev  _et al._ (2011).

Most recent experimental observations of quasiballistic transport have
occurred in this weakly quasiballistic regime. For instance, in the TTG
measurement of silicon membranes reported by Johnson _et al._ Johnson  _et
al._ (2013), the typical Kn${}_{\omega}\approx 2.5$ and $\eta_{\omega}\sim
O(0.01)$, based on the median thermal phonon MFP at the room temperature.
Therefore, their measurements fall into the weakly quasiballistic regime and a
modified Fourier solution should explain the results, in agreement with the
experiment.

###  III.3 Strongly Quasiballistic Regime

As the grating wavelength decreases, eventually the thermal decay becomes so
fast that it is comparable to or greater than relaxation times such that
$\eta_{\omega}\sim 1$ and Kn${}_{\omega}^{2}\sim 1$. Here, the assumption made
in the modified ”two-channel” model is not valid because some phonons in the
thermal reservoir are now ballistic. For silicon, this regime occurs at small
grating wavelength ($\lesssim 0.5\ \mu m$) or at cryogenic temperatures. In
this case, the BTE solution deviates from the exponential decay and can no
longer be explained with any type of diffusion model as shown in Fig. 1d, and
a full solution given by Eq. (7) is necessary. We denote this regime the
strongly quasiballistic regime. The equivalent decay time $\Gamma$ is given by
$\int_{0}^{\infty}\Delta T(t)dt/\Delta T(0)$, which reduces to the
corresponding thermal decay times in the other two regimes above.

###  III.4 Suppression Function

We now seek to understand how the thermal length and time scales affect which
phonons conduct heat in each regime. From our model, we can calculate the
spectral thermal conductance, defined as the ratio of heat flux to the
temperature difference

$\sigma_{\omega}=\frac{\int\int q_{\omega}(x,t)dxdt}{\int\int\Delta
T(x,t)dxdt}$ (12)

where $q_{\omega}(x,t)=\int
g_{\omega}(x,t,\theta)v_{\omega}\cos(\theta)d\Omega$ is the spectral heat
flux. In this way, we remove any spatial and temporal factors and can directly
compare the heat flow induced by a unit temperature difference for each phonon
mode.

Substituting Eq. (4) into Eq. (12), we derive a general expression for the
spectral thermal conductance

$\sigma_{\omega}=\sigma_{f}\left\\{\frac{3}{(\text{Kn}_{\omega})^{2}}\left[1-\frac{\text{tan}^{-1}(\text{Kn}_{\omega})}{\text{Kn}_{\omega}}\right]\left[\eta_{\omega}+1\right]\right\\}$
(13)

where $\sigma_{f}=k_{\omega}/(\lambda/2)$ is the Fourier thermal conductance.
The term in the brackets, equal to the ratio of the BTE thermal conductance to
the Fourier thermal conductance, was previously termed the suppression
function $S(\text{Kn}_{\omega},\eta_{\omega})$ by MinnichMinnich (2012).

Figure 2: Spectral thermal conductance $\sigma_{\omega}$ in the (a) weakly
quasiballistic regime and (b) strongly quasiballistic regime. The BTE
solutions are given by the solid lines, the Fourier solution by the dashed
lines, and the weak BTE solution by the dash-dotted lines.

Now let us examine the thermal conductance in the two quasiballistic transport
regimes discussed above, shown in Fig. 2. We compare the thermal conductance
calculated by the Fourier’s law, weak BTE and full BTE solutions.

In weakly quasiballistic regime, where $\text{Kn}_{\omega}^{2}\sim 1$ but
$\eta_{\omega}\ll 1$ (Fig. 2a), $\eta_{\omega}+1\rightarrow 1$ and Eq. (13)
reduces to $\sigma_{\omega}=\sigma_{f}S_{weak}(\text{Kn}_{\omega})$, where
$S_{weak}(\text{Kn}_{\omega})$ is the same as the term in the brackets of Eq.
(11). In this regime, Fourier’s law overpredicts the heat flux but the weak
BTE solution still accurately describes the spectral heat distribution. From
Fig. 2b, we see that the heat contribution from low frequency phonons is
suppressed compared to the Fourier’s law prediction.

In the strongly quasiballistic regime (Fig. 2b), the weak BTE solution does
not accurately explain the spectral conductance and we must instead use Eq.
(13). The full BTE solution predicts a more gradual suppression than the weak
BTE solution for those low frequency phonons whose relaxation times are
comparable to or greater than the thermal decay time $\Gamma$. This
discrepancy is due to the correction term $\eta_{\omega}+1$, which approaches
its maximum value at low frequencies and reduces the suppression effects.
Rewriting $\eta_{\omega}$ into $\Lambda_{\omega}/(v_{\omega}\Gamma)$ in Eq.
(13), we find that our new suppression function decreases as
$1/\Lambda_{\omega}$ in the long MFP limit, in agreement with the ballistic
limit of the BTEChen (2005) while $S_{weak}(\text{Kn}_{\omega})$ predicts a
steeper slope, $1/\Lambda_{\omega}^{2}$, which is inconsistent with the
ballistic limit. Therefore, our new suppression function provides a more
accurate prediction of the heat flux suppression over the entire spectrum of
phonons compared to the approximate approaches in the literatureMaznev  _et
al._ (2011); Collins  _et al._ (2013).

Figure 3: Example MFP reconstructions for silicon at (a) 100 K and (b) 500 K
and (c) PbSe at 300 K using numerically simulated data. Plotted are the
analytical MFP distribution (solid line), the numerical apparent thermal
conductivities (squares), the reconstructed MFP distributions by the general
suppression function (triangles) and by the weak suppression function
(circles). The x-axis corresponds to the MFP for the distributions and to the
grating wavelength for the thermal conductivity data.

##  IV Application

We now show the utility of these insights by demonstrating how our new
suppression function may be used to more accurately measure MFP spectra. As
proposed by MinnichMinnich (2012), the apparent thermal conductivities can be
related to the MFP distribution by the equation
$k_{app}=\int_{0}^{\infty}S(\Lambda_{\omega})f(\Lambda_{\omega})d\Lambda_{\omega}$
where $S(\Lambda_{\omega})$ is the suppression function, the phonon MFP
$\Lambda_{\omega}$ is the independent variable and $f(\Lambda_{\omega})$ is
the desired MFP distribution. If the apparent thermal conductivities are
experimentally measured and the suppression function is known, then the MFP
distribution can be reconstructed by solving the integral equation as an
inverse problem.

From our analysis, we have already derived the necessary suppression function
$S(\text{Kn}_{\omega},\eta_{\omega})$ in Eq. (13). However, this suppression
function depends both on the independent variable, the phonon MFP
$\Lambda_{\omega}$, as well as the unknown relaxation time $\tau_{\omega}$. To
perform the reconstruction, $\Lambda_{\omega}$ should be the only unknown
variable.

To overcome this problem, we rewrite $\tau_{\omega}$ into
$\Lambda_{\omega}/v_{\omega}$ and assume that the phonon group velocity
$v_{\omega}$ is equal to the average sound velocity $v_{s}$. This assumption
is justified since for long MFP phonons for which the correction term is
important, the group velocity of phonons is close to the sound speed, while
for short MFP phonons this term is negligible and the choice of the velocity
is irrelevant. The reconstruction is also insensitive to the precise choice of
the value of $v_{s}$. For example, for PbSe, changing $v_{s}$ from 2000 m/s to
1000 m/s causes only a 10 % maximum error in the reconstructed MFP
distribution. Using this approximation, the suppression function is only a
function of the independent variable $\Lambda_{\omega}$.

To demonstrate the inversion procedure using this new suppression function, we
perform numerical experiments in which we obtain the modified thermal
conductivities of Silicon and PbSe at different temperatures for different
grating wavelengths from the temperature decay curves predicted by the BTE.
These modified thermal conductivities, along with the suppression function,
are then used as inputs for the reconstruction procedure. Fig. 3 shows the
results for the MFP reconstruction using the general and weak suppression
functions.

For materials with a MFP spectrum that is in the range of the experimental
length scales, such as Silicon at 100 K, the measurements of the apparent
thermal conductivities at different grating wavelengths span almost the entire
range of phonon MFP spectrum. In this case, as shown in Fig. 3(a), both the
weak and new suppression functions yield satisfactory results. However, phonon
MFPs vary by orders of magnitude and some part of the spectrum may be
inaccessible to experiment. For example, the smallest MFPs of Silicon at 500 K
are around 10 nm and the smallest MFPs of PbSe at room temperature are around
1 nm. These length scales are too small to be accessed with present
experimental methods, meaning the MFP distribution at small length scales must
be extrapolated from measurements at larger length scales. Such an
extrapolation requires evaluating the suppression function at large values of
the argument, precisely in the range where the correction term to the new
suppression function is important. As shown in Figs. 3(b) and (c), our new
suppression function yields more accurate results at short MFPs while the weak
suppression function overpredicts the MFP distribution.

##  V Summary

We have analyzed thermal transport in TTG using a new analytical solution to
the frequency-dependent BTE. We identify the thermal decay time relative to
the relaxation times as a key nondimensional parameter that separate two
quasiballistic transport regimes. If the thermal decay time is much larger
than relaxation times, a modified diffusion theory is the formal solution of
the BTE, providing theoretical justification for prior interpretations of
experimental observations of quasiballistic transport. Further, we demonstrate
how MFP spectra may be measured more accurately using our new suppression
function. Our results will lead to a better understanding of phonon heat
conduction in solids like thermoelectrics.

## Acknowledgement

The authors would like to thank Kimberlee Collins and Gang Chen for useful
discussions and for providing the PbSe data. This work was sponsored in part
by Robert Bosch LLC through Bosch Energy Research Network Grant no.
13.01.CC11, by the National Science Foundation under Grant no. CBET 1254213,
and by Boeing under the Boeing-Caltech Strategic Research & Development
Relationship Agreement.

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# Uniqueness of Equilibrium and redistributive policies: A Geometric Approach
to Efficiency

Andrea Loi  Andrea Loi, Dipartimento di Matematica e Informatica
Università di Cagliari, Italy<EMAIL_ADDRESS>,  Stefano Matta  Stefano Matta,
Dipartimento di Scienze economiche e Aziendali
Università di Cagliari, Italy<EMAIL_ADDRESS>and  Daria Uccheddu  Daria
Uccheddu, Dipartimento di Matematica e Informatica
Università di Cagliari, Italy<EMAIL_ADDRESS>
###### Abstract.

This paper examines the relationship between resource reallocation, uniqueness
of equilibrium and efficiency in economics. We explore the implications of
reallocation policies for stability, conflict, and decision-making by
analysing the existence of geodesic coordinate functions in the equilibrium
manifold.

Our main result shows that in an economy with $M=2$ consumers and $L$ goods,
if $L$ coordinate functions, representing policies, are geodesics on the
equilibrium manifold (a property that we call the finite geodesic property),
then the equilibrium is globally unique. The presence of geodesic variables
indicates optimization and efficiency in the economy, while non-geodesic
variables add complexity.

Finally, we establish a link between the existing results on curvature,
minimal entropy, geodesics and uniqueness in smooth exchange economies.

This study contributes to the understanding of the geometric and economic
properties of equilibria and offers potential applications in policy
considerations.

Keywords: Uniqueness of equilibrium, redistributive policies, geodesics,
equilibrium manifold, equilibrium selection, curvature, geodesics.

Subj.Class: C61, C65, D50, D51.

The first and third author were supported by INdAM. GNSAGA - Gruppo Nazionale
per le Strutture Algebriche, Geometriche e le loro Applicazioni and by GOACT
by Fondazione di Sardegna. The first author was also supported by PNRR e.INS
Ecosystem of Innovation for Next Generation Sardinia (CUP F53C22000430001,
codice MUR ECS00000038

##  1\. Introduction

In the absence of uniqueness of equilibrium, due to the role of belief
formation and expectation coordination in equilibrium selection,
redistributive policies may affect efficiency and social stability, by
increasing the likelihood of conflict and unpredictable change and by not
allowing decision-makers to make informed choices based on a well-defined
equilibrium point, which in turn may lead recursively to more unstable
outcomes.

The motivation of this paper is to establish a link between the resource
redistribution and uniqueness. Under what conditions can redistributive
policies take place under uniqueness? In our very general setting, a
redistributive policy can be thought of as a change of one or more economic
variables across the different states of the system representing the economy.
It can therefore be represented by a smooth curve on the equilibrium manifold,
which is a mathematical representation of all possible states of an economic
system in equilibrium. Each point corresponds to an equilibrium state where
all relevant economic variables are in balance, and there is no tendency for
the system to change unless external factors intervene.

Since a policy should be efficient, it is a natural assumption to consider
that it minimises an objective function, which can be represented by distance
with an appropriate metric.

A geodesic, roughly speaking, represents a straight line in a curved space or,
more precisely, a curve on a manifold that minimises distance locally. Its
behaviour is deeply related to the curvature of the manifold and its metric,
object of analysis in differential geometry. This concept is ubiquitous,
ranging from theoretical physics to games, information geometry, machine
learning, operations research, to name but a few.

This branch of mathematics has been applied to statistics and information
theory [1, 13] and subsequently to econometrics (see e.g. [12]). However, very
few results can be found in the economic literature, despite of the fact that
the minimisation property of the geodesic fits naturally into the economic
framework of optimality and efficiency and, being closely related to the
metrics of the space under analysis, allows for the development of powerful
methodologies that take into account several aspects considered relevant. As
an example, [6] studies the existence of Nash equilibria by embedding the non-
convex domains of the payoff functions in suitable Riemannian manifolds. In
general equilibrium theory, [7, 8] use geodesics to minimise the cost of
redistribution policies and catastrophes. But, more generally, lack of use of
geodesics had been observed by Balasko [2] with regard to planning theory and
general equilibrium theory, where, on the other hand, the differential
topology approach, following the ground-breaking article by [5], has been
prevalent and well suited to deal with global properties, multiplicity, and
uniqueness.

Our main result is Theorem 1, where we show that, in an economy with $M=2$
consumers and $L$ goods, if $L$ coordinate functions are geodesics on the
equilibrium manifold (a property that we call finite geodesic property), then
the equilibrium is globally unique.

Since each coordinate function acts on an economic good, the intuition behind
the result is that if all the economic resources follow optimal paths in the
equilibrium manifold, the equilibrium is globally unique. The interpretation
in terms of optimisation and efficiency is that when all the economic
variables are optimised and respond efficiently to changes in the economic
environment (e.g., policy distributions), this can lead to uniqueness, and
hence more efficient resource allocation, cost minimisation and welfare
maximisation in the economy.

The result lends itself to different interpretations and potential
applications. The existence of non-geodesic coordinate functions indicate that
some economic resources do not necessarily behave optimally in the equilibrium
manifold. These variables may respond less efficiently to changes in the
economic environment, leading to potential inefficiencies or suboptimal
outcomes. This can lead to new approaches in partial equilibrium analysis,
through a further analysis of specific sectors and industries or market
segments where optimization is more or less likely. Moreover, this perspective
can also pave the way for a different approach to complex economic dynamics:
the coexistence of geodesic and non-geodesic coordinate functions implies that
the economic system is not entirely homogenous in terms of optimising
behaviour. This can lead to interesting nonlinear dynamics, where different
sectors or variables may interact in different ways and respond differently to
shocks or policy changes. Finally, understanding which variables do and do not
follow geodesic paths and which do not can have important policy implications.
Policymakers can focus on optimising those variables that do not follow
geodesic paths in order to improve overall economic efficiency and stability.

In summary, the presence of a finite number of geodesic coordinate functions
in the equilibrium manifold indicates partial optimization and efficiency in
certain aspects of the economy. It also highlights areas where improvements
can be made to achieve better overall economic outcomes. However, the
coexistence of non-geodesic variables adds complexity to the economic dynamics
and requires careful analysis and policy considerations.

The technique employed is efficient, limiting the number of curves to be
studied. Furthermore, although the work is theoretical, the considerable
development of algorithms that allow the geodesics of even implicit functions
to be determined numerically makes this work potentially useful for
applications.

Finally, we believe that this result is interesting because it deepens our
knowledge of the connections between geometric and economic properties of
equilibria. In fact, it is deeply connected to the recent results on
uniqueness using the zero curvature [9, 11] and minimal entropy [10]
conditions. Corollary 4 summarises, to the best of our knowledge, the state of
the art on these relationships in smooth exchange economies.

In recent years the concepts of differential geometry have had several
applications in discrete spaces with economic properties. For example,
following a network approach, curvature has been associated with system
fragility (see, e.g., [14, 15]). There are still relatively few papers in the
economic literature that use differential geometry in its most natural domain,
that of manifolds. The aforementioned works highlight a link between
curvature, entropy and uniqueness. We hope that the present work will further
stimulate the reader’s interest in making connections with results obtained in
discrete spaces and exploring metrics with economic significance.

This paper is organized as follows. Section 2 recalls the main properties of
the equilibrium manifold. Section 3 presents the main concepts of differential
geometry used to prove our main result. Section 4 analyses the consequences of
the finite geodesic property in low dimension, when the equilibrium manifold
is a surface. Finally, in Section 5 we prove our main result, Theorem 1, and
explain in Corollary 4 the connections between the finite geodesic property
and curvature, minimal entropy and uniqueness on smooth economies.

##  2\. The economic setting

We consider a smooth pure exchange economy with $L$ goods and $M$ consumers.
The equilibrium manifold $E$ is defined as the set of pairs $(p,\omega)$ such
that the excess demand function is zero, where $p$ belongs to the set of
normalized prices
$S=\\{p=(p_{1},\ldots,p_{L})\in\mathbb{R}^{L}|p_{l}>0,l=1,\ldots,L,p_{L}=1\\}\cong\mathbb{R}^{L-1}$
and $\omega$ belongs to the space of endowments $\Omega=\mathbb{R}^{LM}$. The
equilibrium manifold enjoys very nice geometric properties, being a smooth
submanifold of $S\times\Omega$ globally diffeomorphic to $\mathbb{R}^{LM}$ [2,
Lemma 3.2.1].

If total resources $r\in\mathbb{R}^{L}$ are fixed, the equilibrium manifold,
denoted by $E(r)$, is a submanifold of $S\times\Omega(r)$ globally
diffeomorphic to $\mathbb{R}^{L(M-1)}$ [2, Corollary 5.2.5], that is,
$E(r)\cong B(r)\times\mathbb{R}^{(L-1)(M-1)}$, where $B(r)$ denotes the price-
income equilibria, a submanifold of $S\times\mathbb{R}^{M}$ diffeomorphic to
$\mathbb{R}^{M-1}$ [2, Corollary 5.2.4]. If we define this diffeomorphism as

$\varphi:\mathbb{R}^{M-1}\to B(r),$ (1)

$t=(t_{1},\ldots,t_{M-1})\mapsto(p(t),w_{1}(t),\ldots,w_{M-1}(t)),$

where $w_{i}$ denotes consumer $i$’s income, a parametrization of $E(r)$ (see
formulas (6), (7) and (10) in [9]) is given by

$\Phi:\mathbb{R}^{L(M-1)}\to E(r),$ (2)

$(t,\bar{\omega}_{1},\ldots,\bar{\omega}_{M-1})\mapsto(p(t),\bar{\omega}_{1},w_{1}(t)-p(t)\bar{\omega}_{1},\ldots,\bar{\omega}_{M-1},w_{M-1}(t)-p(t)\bar{\omega}_{M-1}),$

where $\bar{\omega}_{i}$ denotes the first $L-1$ components of $\omega_{i}$,
consumer $i$’s endowments vector.

##  3\. Mathematical Preliminaries

In this section we present some mathematical definitions that will be used
later. We refer readers unfamiliar with differential geometry to, e.g., [3,
4]. Consider a submanifold $M$ of $(\mathbb{R}^{n},g_{euclid})$ of dimension
$m$, where $g_{euclid}$ denotes the Euclidean “flat” metric. A metric is
naturally induced on $M$ by its ambient space as follows. If

$\begin{split}\varphi:&\>\mathbb{R}^{m}\longrightarrow
M\subset\mathbb{R}^{n},\\\
&(x_{1},\dots,x_{m})\mapsto(\varphi_{1},\dots\varphi_{n}),\end{split}$

is a parametrization of $M$, we can consider the vector fields
$X_{1}=(\frac{\partial\varphi_{1}}{\partial
x_{1}},\dots,\frac{\partial\varphi_{1}}{\partial x_{m}})$,
$X_{2}=(\frac{\partial\varphi_{2}}{\partial
x_{1}},\dots,\frac{\partial\varphi_{2}}{\partial x_{m}})$ and
$X_{n}=(\frac{\partial\varphi_{n}}{\partial
x_{1}},\dots,\frac{\partial\varphi_{n}}{\partial x_{m}})$. Moreover,
$\\{X_{1},\dots,X_{n}\\}$ is a basis of vector fields of $T_{p}M$ for $p\in
M$. The ambient space $\mathbb{R}^{n}$ induces on $M$ a metric given by

$ds^{2}=\displaystyle\sum_{i,j=1}^{m}g_{ij}dx_{i}dx_{j},$

where $g_{ij}=\langle X_{i},X_{j}\rangle_{g_{euclid}}$.
Let us denote with $\mathfrak{X}(M)$ the set of all vector fields of class
$C^{\infty}$ on $M$. There exists an affine connection111One can think of the
connection as an operator that enables to apply differentiation on an abstract
manifold $M$ intrinsically, that is without referring to its ambient space, by
extending the concept of directional derivative to vector fields defined on
$M$.

$\begin{split}\nabla:\mathfrak{X}(M)\times&\mathfrak{X}(M)\rightarrow\mathfrak{X}(M)\\\
(X,&Y)\mapsto\nabla_{X}Y\end{split}$

which satisfies the following properties:

  * •

$\nabla_{fX+gY}=f\nabla_{X}Z+g\nabla_{Y}Z$

  * •

$\nabla_{X}(Y+Z)=\nabla_{X}Y+\nabla_{X}Z$

  * •

$\nabla_{X}(fY)=f\nabla_{X}Y+X(f)Y$

where $f$ and $g$ are real-valued functions of class $C^{\infty}$ on $M$.

The Lie-braket, defined as $[X,Y]=XY-YX$, is a vector field belonging to
$\mathfrak{X}(M)$.

Theorem (Levi-Civita see p. 55 in [4]) Given a Riemannian manifold $M$, there
exists a unique affine connection $\nabla$ on $M$ satisfying the conditions:

  * •

$\nabla$ is symmetric, that is $[X,Y]=\nabla_{X}Y-\nabla_{Y}X$;

  * •

$\nabla$ is compatible with the Riemannian metric: $X\langle
Y,Z\rangle=\langle\nabla_{X}Y,Z\rangle+\langle Y,\nabla_{X}Z\rangle$ for
$X,Y,Z\in\mathfrak{X}(M)$.

In particular, the Levi-Civita connection can be written, in a coordinate
system $(U,x)$ as

$\displaystyle\nabla_{\frac{\partial}{\partial x_{i}}}\frac{\partial}{\partial
x_{j}}=\displaystyle\sum_{k=1}^{m}\Gamma_{ij}^{k}\frac{\partial}{\partial
x_{k}},$

where the coefficients $\Gamma_{ij}^{k}$ are called the Christoffell symbols
and can be computed using the following formula

$\Gamma_{ij}^{k}=\frac{1}{2}\displaystyle\sum_{h=1}^{m}g^{hk}\left(\frac{\partial
g_{jh}}{\partial x_{i}}+\frac{\partial g_{hi}}{\partial x_{j}}-\frac{\partial
g_{ij}}{\partial x_{h}}\right),$

where $g^{ij}$ is the inverse matrix of $g_{ij}=\langle X_{i},X_{j}\rangle$.

A parametrized curve $\gamma:I\to M$ is a geodesic if
$\nabla_{\dot{\gamma}}\dot{\gamma}=0$. This curve can be seen a generalization
of the concept of straight line, whose velocity vector has derivative equal to
zero.

The following proposition enables to study the geodesic on abstract Riemannian
manifolds in terms of ordinary differential equations.

Proposition (see [4]).

Let $M$ be a manifold endowed with a Levi-Civita connection $\nabla$. Choose
coordinates $x^{i}$ on some neighborhood $U$ of $p$. For any $p\in M$, any
$V\in T_{p}M$, and any $t_{0}\in\mathbb{R}$, there exists an open interval
$I\subset\mathbb{R}$ containing $t_{0}$ and a geodesic $\gamma:I\rightarrow M$
such that $\gamma(t_{0})=p$, and $\dot{\gamma}(t_{0})=V.$ If
$\gamma(t)=\left(x^{1}(t),\dots,x^{n}(t)\right)$, then its components satisfy

$\ddot{x}^{k}(t)+\Gamma^{k}_{ij}x(t)\dot{x}^{i}(t)\dot{x}^{j}(t)=0.$ (3)

Equation (3) is an ordinary differential equation with initial condition
$\gamma(t_{0})=p$ and $\dot{\gamma}(t_{0})=V$, hence, by the fundamental
theorem of differential equations, it has an unique solution.

Finally, we observe that a geodesic can also be characterized by the property
of having its acceleration vector in $\mathbb{R}^{n}$ everywhere parallel to
the normal vector to the surface, see e.g. [3, p.246].

##  4\. Basic Example with $L=2$ and $M=2$

In this section, we show that, with two consumers and two commodities, if the
coordinate curves $\Phi(t,0)$ and $\Phi(0,t)$ are geodesics, then the price is
constant. This is a special case we introduce here to provide the intuition
before presenting in the next section the proof of the more general case with
an arbitrary number of goods.

Observe that if $L=M=2$ the equilibrium manifold $E(r)$ is a surface, i.e. a
submanifold of dimension $2$ in a three-dimensional space 222 $E(r)\subset
S\times\Omega(r)$, where
$\Omega(r)=\\{\omega=(\omega_{1},\omega_{2}),\,|\,\omega_{i}\in\mathbb{R}^{2},\omega_{1}+\omega_{2}=r\\}=\\{(\omega_{1},r-\omega_{1})\\}\cong\mathbb{R}^{2}$..
An explicit parametrization of $E(r)$ is given by [9, Section 4]

$\Phi:\mathbb{R}^{2}\to E(r),$

where $(t,\alpha)\mapsto(p(t),\alpha,w(t)-p(t)\alpha).$

Consider a basis of vector fields in $T_{x}E(r)$ given by

$\Phi_{0}=\left(\frac{\partial p}{\partial t},0,\frac{\partial w}{\partial
t}-\frac{\partial p}{\partial t}\alpha\right)=(\dot{p},0,A)$
$\Phi_{1}=\left(0,1,-p\right),$

where $A:=\dot{w}-\dot{p}\>\alpha$.

Observe that $E(r)$ is a ruled surface (see [3, p.188]. In fact,
$\Phi(t,v)=\Phi(t,0)+v\>\Phi_{1}(t)$. The normal vector to the surface is the
orthogonal vector to the tangent space
$T_{x}E(r)=\mbox{span}\\{(\dot{p},0,\dot{w}-\dot{p}\alpha),(0,1,-p)\\}$, that
is $N=(\dot{p}\alpha-\dot{w},p\dot{p},\dot{p})$.

In particular, taking into account the last observation of Section 3, if we
suppose that the coordinate curve $\Phi(t,0)$ is a geodesic, then the second
derivative of the parametrization $\ddot{\Phi}$ is parallel to the normal
vector $N$ to the surface, that is $\ddot{\Phi}\wedge N=(0,0,0)$.

An explicit computation of $\ddot{\Phi}(t,0)$ gives
$\ddot{\Phi}(t,0)=(\ddot{p}(t),0,\ddot{w}(t))$ and hence, from
$\ddot{\Phi}\wedge
N=(-\dot{w}p\dot{p},\ddot{w}(\dot{p}\alpha-\dot{w})-\dot{p}\ddot{p},p\dot{p}\ddot{p})$,
we have that this vector is the null vector if and only if

$\left\\{\begin{array}[]{rrr}-\dot{w}p\dot{p}=0\\\
\ddot{w}(\dot{p}\alpha-\dot{w})-\dot{p}\ddot{p}=0\\\
p\dot{p}\ddot{p}=0\end{array}\right.$

Recalling that the price $p$ is different from zero, from the last equation we
have that $\ddot{p}=0$ and then $\dot{p}=b$ and $p(t)=b\>t+c$. Since the price
is positive, that is $p(t)>0$, the only possibility is that $p(t)$ is a
constant line always positive, that is a line parallel to the $t$ axis, and so
the price $p$ is constant. Clearly, the other coordinate curve, $\Phi(0,t)$,
is always a geodesic because we have $\ddot{\Phi}(0,t)=(0,0,0)$.

Now we present the general idea of the proof that will be used in the more
general case, using local coordinates through the Christoffel symbols.333For
an explicit and tedious calculation of the Christoffel symbols, we refer the
interested reader to [11]. We are going to prove that the price is constant if
the coordinate curves $\Phi(t,0)$ and $\Phi(0,t)$ are solutions of the ODE
(3). Curves will be parametrized by arc length.

The Christoffel symbols are given by

$\Gamma_{ij}^{k}=\frac{1}{2}\displaystyle\sum_{h=0}^{1}g^{hk}\left(\frac{\partial
g_{jh}}{\partial x_{i}}+\frac{\partial g_{hi}}{\partial x_{j}}-\frac{\partial
g_{ij}}{\partial x_{h}}\right).$

In particular, with our parametrization, they are (see [11, Appendix A])

$\begin{split}\Gamma_{00}^{0}=&\frac{[(1+p^{2})\dot{p}\ddot{p}+(\dot{w}-\dot{p}x_{1})(\ddot{w}-\ddot{p}x_{1})]}{(1+p^{2})(\dot{p})^{2}+A^{2}},\\\
\Gamma_{00}^{1}=&\frac{p\left[(\dot{w}-\dot{p}x_{1})\dot{p}\ddot{p}-(\ddot{w}-\ddot{p}x_{1})(\dot{p}^{2})\right]}{(1+p^{2})(\dot{p})^{2}+A^{2}},\\\
\Gamma_{10}^{0}=&\frac{-\dot{p}(\dot{w}-\dot{p}x_{1})}{(1+p^{2})(\dot{p})^{2}+A^{2}},\\\
\Gamma_{10}^{1}=&\frac{\dot{p}p(\dot{p})^{2}}{(1+p^{2})(\dot{p})^{2}+A^{2}},\\\
\Gamma^{0}_{11}=&\Gamma^{1}_{11}=0.\end{split}$

If $\gamma(t)=\left(\gamma_{0},\gamma_{1}\right)\subset E(r)=\Phi(t,\alpha)$,
Equation (3) becomes

$\left\\{\begin{array}[]{c}\ddot{\gamma}_{0}(t)+\Gamma^{0}_{00}\dot{\gamma}_{0}(t)^{2}+2\,\Gamma^{0}_{01}\dot{\gamma}_{0}(t)\dot{\gamma}_{1}(t)=0\\\
\ddot{\gamma}_{1}(t)+\Gamma^{1}_{00}\dot{\gamma}_{0}(t)^{2}+2\,\Gamma^{1}_{01}\dot{\gamma}_{0}(t)\dot{\gamma}_{1}(t)=0,\end{array}\right.$

which can be written, computing the Christoffel symbols,

$\left\\{\begin{array}[]{crrr}\ddot{\gamma}_{0}+\dfrac{[(1+p^{2})\dot{p}\ddot{p}+(\dot{w}-\dot{p}\gamma_{1})(\ddot{w}-\ddot{p}\gamma_{1})]}{(1+p^{2})(\dot{p})^{2}+A^{2}}\dot{\gamma}_{0}^{2}+2\dfrac{-\dot{p}(\dot{w}-\dot{p}\gamma_{1})}{(1+p^{2})(p^{\prime})^{2}+A^{2}})\dot{\gamma}_{0}\dot{\gamma}_{1}=0\\\
\\\
\ddot{\gamma}_{1}+\dfrac{p\left[(\dot{w}-\dot{p}\gamma_{1})\dot{p}\ddot{p}-(\ddot{w}-\ddot{p}\gamma_{1})(\dot{p}^{2})\right]}{(1+p^{2})(\dot{p})^{2}+A^{2}}\dot{\gamma}_{0}^{2}+2\dfrac{\dot{p}p(\dot{p})^{2}}{(1+p^{2})(\dot{p})^{2}+A^{2}}\dot{\gamma}_{0}\dot{\gamma}_{1}=0.\end{array}\right.$
(4)

In particular, if the curve is given by $\Phi(t,0)$, we have $\gamma_{1}=0$
and then we can write

$\left\\{\begin{array}[]{crr}\ddot{\gamma}_{0}+\dfrac{[(1+p^{2})\dot{p}\ddot{p}+\dot{w}\ddot{w}]}{(1+p^{2})(\dot{p})^{2}+A^{2}}\dot{\gamma}_{0}^{2}=0\\\
\\\
\dfrac{p\left[\dot{w}\dot{p}\ddot{p}-\ddot{w}(\dot{p}^{2})\right]}{(1+p^{2})(\dot{p})^{2}+A^{2}}\dot{\gamma}_{0}^{2}=0.\end{array}\right.$

If $\gamma_{0}=t$ then $\dot{\gamma}_{0}=1$ and $\ddot{\gamma}_{0}=0$, and
hence we obtain

$\left\\{\begin{array}[]{c}[(1+p^{2})\dot{p}\ddot{p}+\dot{w}\ddot{w}]=0\\\
p\left[\dot{w}\dot{p}\ddot{p}-\ddot{w}(\dot{p}^{2})\right]=0\end{array}\right.$

Moreover, by the definition of $\Phi$, we have $\Phi(t,0)=(p(t),0,w(t))$ and,
from the fact that $\gamma$ is parametrized by arc length, we have
$\|\dot{\gamma}(t)\|=1$, which implies $(\dot{p})^{2}+(\dot{w})^{2}=0$ and
hence $\dot{p}\ddot{p}+\dot{w}\ddot{w}=0$, that is
$\dot{w}\ddot{w}=-\dot{p}\ddot{p}$. Using this last result, we get

$\left\\{\begin{array}[]{c}p^{2}\dot{p}\ddot{p}=0\\\
p\left[\dot{w}\dot{p}\ddot{p}-\ddot{w}(\dot{p}^{2})\right]=0\end{array}\right.$

and we conclude that if $\gamma(t)=\Phi(t,0)$ is a geodesic, then the only
possibility is that $\dot{p}=0$ and hence the price is constant.

Finally, observe that the coordinate curve $\Phi(0,t)$ is always a geodesic,
because it is a solution of Equation (4) with $\gamma_{0}=0$ and
$\gamma_{1}=t$.

##  5\. The case $M=2$

Let us consider a more general case, which includes the previous one as a
special case, that is, an economy with two consumers and an arbitrary number
of goods. In this case $E(r)$ is a manifold of dimension $L$ in a space of
dimension $2L-1$. An explicit parametrization of $E(r)$ is given by [11]

$\begin{split}\Phi:\>\mathbb{R}^{L}&\overset{\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>}{\longrightarrow}E(r)\\\
(t,\alpha_{1},\dots,\alpha_{L-1})&\longmapsto(p_{1}(t),\dots,p_{L-1}(t),\alpha_{1},\dots,\alpha_{L-1},w(t)-p_{1}(t)\alpha_{1}-\dots-
p_{L-1}(t)\alpha_{L-1}),\\\ \end{split}$

where we set $x_{0}=t,\>\>x_{1}=\alpha_{1},\dots,x_{L-1}=\alpha_{L-1}$.

The following theorem shows that the finite geodesic property implies global
uniqueness.

######  Theorem 1.

Let $\overrightarrow{\alpha}_{j}$ be the $L-1$-entry vector consisting of
$L-2$ zeros and $1$ in the $j$-th position. If the $L$ coordinate curves
$\Phi(t,0)$ and $\Phi(0,\overrightarrow{\alpha}_{j})$ are geodesics, then the
price is constant.

Proof: Suppose that the curve $\gamma(t)=\Phi(t,0,\dots,0)$ is a geodesic
parametrized by arc length. We have that $\|\dot{\gamma}(t)\|=1$, so
$\sqrt{B+A^{2}}=1$ and then $C+AA^{\prime}=0$ where

$\framebox{$A=\dot{w}-\dot{p}_{1}x_{1}-\dots-\dot{p}_{L-1}x_{L-1}$}$
$\framebox{$B=(\dot{p}_{1})^{2}+\dots+(\dot{p}_{L-1})^{2}$}$
$\framebox{$C=\dot{p}_{1}\ddot{p}_{1}+\dot{p}_{2}\ddot{p}_{2}+\dots+\dot{p}_{L-1}\ddot{p}_{L-1}$}$

$\framebox{$\|p\|^{2}=(1+p_{1}^{2}+\dots+p_{L-1}^{2})$}$
$\framebox{$A^{\prime}=\ddot{w}-\ddot{p}_{1}x_{1}-\dots-\ddot{p}_{L-1}x_{L-1}$}.$

The Christoffel symbols are

$\begin{split}\Gamma_{00}^{0}=&\frac{[\|p\|^{2}C+AA^{\prime}]}{\|p\|^{2}B+A^{2}}\\\
\Gamma_{00}^{k}=&\frac{p_{k}\left[AC-A^{\prime}B\right]}{\|p\|^{2}B+A^{2}}\\\
\Gamma_{0j}^{0}=&\frac{-\dot{p}_{j}A}{\|p\|^{2}B+A^{2}}\\\
\Gamma_{0j}^{k}=&\frac{\dot{p}_{j}p_{k}B}{\|p\|^{2}B+A^{2}}\\\
\Gamma^{0}_{ij}=&\Gamma^{0}_{ii}=0\\\
\Gamma^{k}_{ij}=&\Gamma^{k}_{ii}=0\end{split}$

and the geodesic equation from Equation (3) becomes

$\left\\{\begin{array}[]{c}\ddot{\gamma}_{0}(t)+\Gamma^{0}_{00}\dot{\gamma}_{0}(t)^{2}+2\Gamma^{0}_{01}\dot{\gamma}_{0}(t)\dot{\gamma}_{1}(t)+2\Gamma^{0}_{02}\dot{\gamma}_{0}(t)\dot{\gamma}_{2}(t)+\dots+2\Gamma^{0}_{0(L-1)}\dot{\gamma}_{0}(t)\dot{\gamma}_{L-1}(t)+\mbox{Nulls
$\Gamma_{ij}$ }=0,\\\
\ddot{\gamma}_{1}(t)+\Gamma^{1}_{00}\dot{\gamma}_{0}(t)^{2}+2\Gamma^{1}_{01}\dot{\gamma}_{0}(t)\dot{\gamma}_{1}(t)+2\Gamma^{1}_{02}\dot{\gamma}_{0}(t)\dot{\gamma}_{2}(t)+\dots+2\Gamma^{0}_{0(L-1)}\dot{\gamma}_{0}(t)\dot{\gamma}_{L-1}(t)+\mbox{Nulls
$\Gamma_{ij}$ }=0,\\\ \dots\\\
\ddot{\gamma}_{L-1}(t)+\Gamma^{L-1}_{00}\dot{\gamma}_{0}(t)^{2}+2\Gamma^{L-1}_{01}\dot{\gamma}_{0}(t)\dot{\gamma}_{1}(t)+2\Gamma^{L-1}_{02}\dot{\gamma}_{0}(s)\dot{\gamma}_{2}(t)+\dots+2\Gamma^{L-1}_{0(L-1)}\dot{\gamma}_{0}(t)\dot{\gamma}_{L-1}(t)+\mbox{Nulls
$\Gamma_{ij}$}=0.\\\ \end{array}\right.$

If $\gamma(t)=\Phi(t,0,\dots,0)$, then the geodesic equations are

$\left\\{\begin{array}[]{c}\Gamma^{0}_{00}=0\\\ \Gamma^{1}_{00}=0\\\ \dots\\\
\Gamma^{L-1}_{00}=0,\\\ \end{array}\right.\>\>\>\>\mbox{ that is,
}\>\>\>\>\>\>\left\\{\begin{array}[]{c}\frac{[\|p\|^{2}C+AA^{\prime}]}{\|p\|^{2}B+A^{2}}=0\\\
\frac{p_{1}\left[AC-A^{\prime}B\right]}{\|p\|^{2}B+A^{2}}=0\\\
\dots\dots\dots\\\
\frac{p_{L-1}\left[AC-A^{\prime}B\right]}{\|p\|^{2}B+A^{2}}=0\\\
\end{array}\right.$

which can be simplified as follows

$\left\\{\begin{array}[]{c}[\|p\|^{2}C+AA^{\prime}]=0\\\
p_{1}\left[AC-A^{\prime}B\right]=0\\\ \dots\dots\dots\\\
p_{L-1}\left[AC-A^{\prime}B\right]=0\\\ \end{array}\right.$

On the other end, if we suppose that $-AA^{\prime}=C$, we get

$\left\\{\begin{array}[]{c}-(p_{1}^{2}+p_{2}^{2}+\dots+p_{L-1}^{2})AA^{\prime}=0\\\
p_{1}\left[-A^{2}A^{\prime}-A^{\prime}B\right]=0\\\ \dots\dots\dots\\\
p_{L-1}\left[-A^{2}A^{\prime}-A^{\prime}B\right]=0.\\\ \end{array}\right.$

This, applied to our curve, gives

$\left\\{\begin{array}[]{c}-(p_{1}^{2}+p_{2}^{2}+\dots+p_{L-1}^{2})AA^{\prime}=0\\\
-p_{1}\ddot{w}(A^{2}+B)=0\\\ \dots\dots\dots\\\
-p_{L-1}\ddot{w}(A^{2}+B)=0.\\\ \end{array}\right.$

which implies $\ddot{w}=0$.
Now, if we also assume that $\gamma(t)=\Phi(0,1,0,\dots,0)$ is a geodesic, we
obtain

$\left\\{\begin{array}[]{c}-(p_{1}^{2}+p_{2}^{2}+\dots+p_{L-1}^{2})AA^{\prime}=0\\\
-p_{1}(\ddot{w}-\ddot{p}_{1})(A^{2}+B)=0\\\ \dots\dots\dots\\\
-p_{L-1}(\ddot{w}-\ddot{p}_{1})(A^{2}+B)=0\\\ \end{array}\right.$

which implies $\ddot{p}_{1}=0$ and hence $\dot{p}_{1}=c$ and
$p_{1}(t)=c\,t+d$. But, since the $p_{j}$’s are positive, $p_{1}(t)$ must be
constant. The same reasoning can be applied to show that every $p_{j}$ is
constant, by letting $\gamma(t)=\Phi(0,0,\dots,1,\dots,0)$ with $1$ in the
$j$-position. ∎

The following two remarks highlight the role played by the economic
assumptions.

######  Remark 2.

The economic assumption of positive prices plays a crucial role as we can see
from the following counterexample. Consider the manifold with the following
parametrization, where $p_{1},\dots,p_{L-2}$ are constants and $p_{L-1}(t)=t$,

$\begin{split}\Phi:\>\mathbb{R}^{L}&\overset{\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>}{\longrightarrow}\mathbb{R}^{L-1}\\\
(t,\alpha_{1},\dots,\alpha_{L-1})&\longmapsto(p_{1},\dots,t,\alpha_{1},\dots,\alpha_{L-1},w(t)-p_{1}\alpha_{1}-\dots-t\alpha_{L-1}).\\\
\end{split}$

The curves $\Phi(t,0,\dots,0)$, $\Phi(0,1,\dots,0)$ and
$\Phi(0,0,\dots,1,\dots,0)$ are all geodesics of this manifold, but the price
is not constant.

######  Remark 3.

We provide an example of a ruled hypersurface
$F(t,\alpha)\subset\mathbb{R}^{L+1}$, $t\in\mathbb{R}$ and
$\alpha\in\mathbb{R}^{L-1}$, with zero sectional curvature that is not a
hyperplane, even if the set of functions $F(t,\bar{\alpha})$, for every
$\bar{\alpha}$ in $\mathbb{R}^{L-1}$, are geodesics. Let
$F(t,\alpha)=\gamma(t)+\alpha_{1}e_{1}+\dots+\alpha_{L-1}e_{L-1}$, where
$e_{1},\dots,e_{L-1}$ is the canonical orthonormal basis of $\mathbb{R}^{L-1}$
and $\gamma(t)=(0,\dots,0,\gamma_{1}(t),\gamma_{2}(t))\in\mathbb{R}^{L+1}$.
Moreover, the tangent space of $F(t,\alpha)$ is given by the span of $\partial
F_{t}=(0,0,\dots,\dot{\gamma}_{1}(t),\dot{\gamma}_{2}(t))$ and $\partial
F_{\alpha_{j}}=e_{j}=(0,0\dots,1,\dots,0)$ with $j=1,\dots,L-1$. Then the
curves $F(t,0)$ and $F(t,\bar{\alpha})$ are geodesics, in fact
$\ddot{F}(t,0)=(0,\dots,0,\ddot{\gamma}_{1}(t),\ddot{\gamma}_{2}(t))$ and
$\ddot{F}(t,\bar{\alpha})=(0,\dots,0,\ddot{\gamma}_{1}(t),\ddot{\gamma}_{2}(t))$
are both orthonormal to the tangent space and so parallel to the normal vector
of the hypersurface.

In the following corollary we summarise what, to the best of our knowledge, is
known about the relationship between curvature, minimal entropy, geodesics and
uniqueness on smooth exchange economies.

######  Corollary 4.

The following properties of the equilibrium manifold $E(r)$ are equivalent for
an economy with $M=2$ agents and an arbitrary number of goods:

  * (1)

constant equilibrium price for every economy;

  * (2)

uniqueness of equilibrium for every economy;

  * (3)

zero curvature property;

  * (4)

minimal entropy property;

  * (5)

the $L$ coordinate functions are geodesics.

[Proof]First observe that $(1)\implies(2)$ (obvious) and $(2)\implies(1)$ by
[2, p. 188 Theorem 7.3.9 part (2)], hence $(1)\iff(2)$. Moreover, property
$(1)$ means that the equilibrium correspondence is flat, so we have that
$(1)\implies(3)$ and $(1)\implies(5)$. On the other hand, $(3)\implies(2)$ by
[11, Corollary (4)] and $(5)\implies(2)$ by Theorem 1. Finally,
$(2)\implies(4)$ [10, Remark 2.2] and $(4)\implies(2)$ by [10, Theorem 3.1].

We conjecture that the equivalences in Corollary 4 hold for any number of
goods or agents. What it is known is that the equivalence $1\iff 2$ holds for
an arbitrary number of goods or agents. Moreover, for two-good economies $L=2$
with an arbitrary number of agents, $(3)\iff(2)$ by [9, Theorem 5.1]. On the
other hand, the known sufficient conditions used to extend the equivalence
$(4)\iff(2)$ to the same setting are quite strong [10, Theorem 4.1]. The
conjecture in the general case is still open.

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