# Thermodynamic bounds and general properties of optimal efficiency and power
in linear responses

Jian-Hua Jiang<EMAIL_ADDRESS>Department of Physics, University
of Toronto, Toronto, Ontario, M5S 1A7 Canada

###### Abstract

We study the optimal exergy efficiency and power for thermodynamic systems
with Onsager-type “current-force” relationship describing the linear-response
to external influences. We derive, in analytic forms, the maximum efficiency
and optimal efficiency for maximum power for a thermodynamic machine described
by a $N\times N$ symmetric Onsager matrix with arbitrary $N$. The figure of
merit is expressed in terms of the largest eigenvalue of the “coupling matrix”
which is solely determined by the Onsager matrix. Some simple but general
relationships between the power and efficiency at the conditions for (i)
maximum efficiency and (ii) optimal efficiency for maximum power are obtained.
We show how the second law of thermodynamics bounds the optimal efficiency and
the Onsager matrix, and relate those bounds together. The maximum power
theorem (Jacobi’s Law) is generalized to all thermodynamic machines with
symmetric Onsager matrix in the linear-response regime. We also discuss
systems with asymmetric Onsager matrix (such as systems under magnetic field)
for a particular situation and we show that the reversible limit of efficiency
can be reached at finite output power. Cooperative effects are found to
improve the figure of merit significantly in systems with multiply cross-
correlated responses. Application to example systems demonstrates that the
theory is helpful in guiding the search for high performance materials and
structures in energy researches.

###### pacs:

05.70.Ln

##  I Introduction

Under challenges imposed by increasing demand yet limited availability of
energy resources, improving energy efficiency becomes increasingly important
in technology developments. Historically, Carnot deduced that for a heat
engine operating between two reservoirs with temperatures $T_{h}$ and $T_{c}$
($T_{h}>T_{c}$), the energy conversion efficiency, $\eta=W/Q$ ($W$ is the
output work, $Q$ is the heat from the hot reservoir), has a maximum value,
namely the Carnot efficiency, $\eta_{C}=(T_{h}-T_{c})/T_{h}$callen . The
Carnot efficiency is only for ideal machines operating in the reversible
limit. Energy efficiency of realistic machines is reduced by unavoidable
irreversible entropy production. A way to count the reduction of energy
efficiency from the value at reversible limit is to use the exergy efficiency
(or “second-law efficiency”)odum ; 2ndlaw1 ; 2ndlaw2 ; 2ndlawbook ; caplan1 ;
caplan2

$\phi\equiv\frac{\dot{\cal A}_{out}}{\dot{\cal A}_{in}}$ (1)

where $\dot{\cal A}_{out}$ and $\dot{\cal A}_{in}$ are the output and input
exergy (i.e., the Gibbs free energy) per unit time. Exergy is defined as
${\cal A}={\cal U}-TS$ where ${\cal U}$ is the enthalpy (i.e., the total
energy), $T$ is the temperature, and $S$ is the entropy. Although the total
energy is conserved, the output exergy is reduced by entropy production,
$\dot{S}_{tot}$, as $\dot{\cal A}_{out}=\dot{\cal A}_{in}-T\dot{S}_{tot}$,
hence $\phi\leq 1$. Both $\phi\leq 1$ and $\eta\leq\eta_{C}$ are dictated by
the second law of thermodynamics. In fact for thermoelectric engine and
refrigerator the two are related byodum ; 2ndlawbook ; caplan1 ; ca-ex ;
Seifert-review

$\phi=\frac{\eta}{\eta_{C}}.$ (2)

For this reason, exergy efficiency is also called as “rational
efficiency”2ndlawbook . Using Onsager’s theory of irreversible thermodynamics
and the exergy efficiency, the study of efficiency of heat engines, chemical
engines, and other energy devices can be presented in an uniform mannerca-ex ;
Seifert-review ; odum ; 2ndlaw1 ; 2ndlaw2 ; 2ndlawbook ; caplan1 ; caplan2 .
Specifically, the efficiency of chemical engines, the output work divided by
the chemical work, is precisely Eq. (1), as the output work is equal to the
output exergy and the input chemical work is equal to the input (consumed)
exergyca-ex ; Seifert-review ; 2ndlaw1 . The exergy efficiency becomes
particularly convenient for machines with multiple forms of input (or output)
energycaplan2 . For example in a spin-thermoelectricbauer  refrigerator, both
electrical energy and magnetic energy are consumed to drive the cooling (see
Sec. VI.2).

A central issue in energy application is to find out the optimal efficiency
and maximum power of a machine and the conditions that realize themdevos ; ca
; nicolis . For example, Ioffe derived the optimal exergy efficiency for
isotropic thermoelectric materials in the linear-response regime asioffe

$\eta_{max}=\eta_{C}\frac{\sqrt{\xi+1}-1}{\sqrt{\xi+1}+1},\quad\quad\xi=\frac{\sigma
S^{2}T}{\kappa}.$ (3)

The figure of merit, $\xi$, is solely determined by the transport coefficients
of the material: the electrical conductivity $\sigma$, the Seebeck coefficient
$S$, and the thermal conductivity $\kappa$. This property is an important
guiding principle in the search of high performance thermoelectric
materialshonig ; ms ; review .

However, Eq. (3) was derived for isotropic systems, where, by choosing a
proper set of coordinate axes, the problem can be reduced to correlated
transport for two scalar currents: one heat current and one electric current.
Quite often in anisotropic materials, the complete description of
thermoelectric transport must involve six scalar currents as both the electric
and heat currents consist of three scalar components (e.g., the electrical
current $\vec{j}=(j_{x},j_{y},j_{z})$ with $j_{x}$, $j_{y}$, and $j_{z}$ being
the components in the $x$, $y$, and $z$ directions)honig ; aniso . For
piezoelectric energy conversion in an anisotropic material, the full
description of responses involves nine scalar “currents”: three of them are
electric displacements and the other six are strainspe . The description of
these cross-correlated responses can be simplified only for certain high
symmetry structures. Recent development of technologies for high-quality thin
film growth which allows precise control of composition, atomic arrangements
and interfaces provides the toolbox for functional nano-structured composite
materials which could have pronounced application values that does not share
by their compounds. Often these composite structures have lower symmetry and
the full description of cross-correlated responses cannot be simplified.
Besides, breaking time-reversal symmetry brings further complication to cross-
correlated responsestrb1 ; trb2 ; trb3 . Quite often Ioffe’s derivation of
optimal energy efficiency cannot be directly applied to those practical
systems. In those situations the (global) maximum efficiency is rather
difficult to find, although one can always easily find certain optimal
efficiencies under restrictionshonig ; aniso ; multi ; pe ; caplan2 .

Finding the optimal exergy efficiency and power for complex thermodynamic
systems has stimulated a number of studiesaniso ; bio ; caplan2 . It becomes
increasingly important as researches reveal more cross-correlated responses
and realize their applicationsbauer ; zlwang1 ; zlwang2 . Fast developing
nanotechnologies and material technologies offer a large number of materials
and structures of which complex cross-correlated responses are enhanced and
made available for practical applications. Examples are, spin-thermoelectric
effectbauer , piezopotential gatingzlwang1  and piezo-phototronicszlwang2 , to
name but just a few. Besides, biological systems are often characterized by
cross-correlated responses to density, temperature, and electrochemical
potential gradientsbio ; caplan2 . A typical example is transport across a
biological membrane: even for a single ionic solution, transmission through
the membrane must be described by three flows, the volume flow, the solute
flow, and the electrical flow, which are often cross-correlatedbio . Cross-
correlated responses enable energy conversion from one form to another, during
which the functions of a machine is realized (a “machine” is a system which
consumes input energy to achieve a practical goal by doing work to the
external). Caplan derived the analytic expression of the optimal exergy
efficiency for machines with only one flow for energy input but multiple flows
for output or vice versacaplan2 . However, general results on the optimal
efficiency and power are still absent, particularly in analytic forms.

In this work we derive analytic results for optimal efficiency and power under
general considerations that can be applied to a broad range of thermodynamic
systems. The requirements are only that there exists an Onsager-type “current-
force” relation that describes the responses to external influences
(“forces”)onsager  and that the system is operating at steady states in linear
responses. These requirements are often satisfied for physical systems with
forces not too stronggroot ; nicolis . The derived results can be connected
with realistic systems of which the output power is consumed by a device or by
large a power grid. We obtain some simple but general relationships that
connect the optimal power and efficiency for different optimization schemes.
These results are first obtained for systems with symmetric Onsager response
matrix and then extended to systems with asymmetric Onsager matrix (e.g.,
systems under magnetic field). We point out that cooperative effects can be
used to improve efficiency (figure of merit) for systems with multiple cross-
correlated responses. Such improvement, achieved via combining different input
(or output) forces rather than engineering materials, can be significant in
systems with multiple cross-correlated responses. Examples are given to
demonstrate how the theory is used to guide the search for high performance
energy applications.

This paper is organized as follows: in Sec. II we establish the basic
formalism by using Onsager’s theory of irreversible thermodynamic processes in
the linear-response regime. We derive the optimal efficiency and output power
for symmetric Onsager matrix in Sec. III. In Sec. IV the derivation is re-
interpreted with realistic considerations where parasitic dissipation and the
response of the device accepting the output energy are considered. We extend
the study to systems with asymmetric Onsager matrix in Sec. V. Examples that
illustrate the usefulness of the findings are presented in Sec. VI, and we
conclude in Sec. VII.

##  II Basic formalism

Under external influences (“forces”) a thermodynamic system develops motions
that deviate from their equilibrium values. These motions (“currents”) can be
described quantitatively by the rates of changes in thermodynamic state
variablesgroot ; landau . The relation between the forces $\vec{{\cal F}}$ and
currents $\vec{{\cal J}}$ is generally written asgroot ; landau

$\vec{{\cal J}}=\hat{{\cal M}}\vec{{\cal F}}\quad{\rm or}\quad{\cal
J}_{n}=\sum_{k}{\cal M}_{nk}{\cal F}_{k},$ (4)

where the index $n$ ($k$) numerates all currents (forces), and $\hat{{\cal
M}}$ is the Onsager matrix. When the forces are not too strong the dependence
of $\hat{{\cal M}}$ on the forces can be ignored. Cross-correlated responses
(e.g., thermoelectric effect) allow energy conversion from the input forms to
the output forms and realize functions of a machine. According to the theory
of irreversible thermodynamicsonsager ; groot , there are an equal number of
forces and currents. Each force ${\cal F}_{n}$ has a conjugated current ${\cal
J}_{n}$ such that the reduction of total exergy (Gibbs free energy) is given
by

$-\dot{\cal A}_{tot}=T\dot{S}_{tot}=\sum_{n}{\cal J}_{n}{\cal F}_{n}.$ (5)

The reduction of exergy $-\dot{\cal A}_{n}={\cal J}_{n}{\cal F}_{n}$
associated with the current ${\cal J}_{n}$ for exergy input is positive, while
for exergy output it is negative. Hence the input and output exergy arecaplan2

$\dot{\cal A}_{in}\equiv\sum_{n\in I}{\cal J}_{n}{\cal F}_{n},\quad\dot{\cal
A}_{out}\equiv-\sum_{k\in O}{\cal J}_{k}{\cal F}_{k},$ (6)

respectively. The sets $I$ and $O$ in the above refer to exergy input and
output, respectively. The output exergy is also the output work, i.e.,
$\dot{W}=\dot{\cal A}_{out}$ (Throughout this paper “work” is associated with
linear-response processes for given thermodynamic forces, i.e., work and
efficiency are functions of thermodynamic forces). For $\dot{\cal A}_{in}>0$
the exergy efficiency is

$\phi=\frac{-\sum_{k\in O}{\cal J}_{k}{\cal F}_{k}}{\sum_{n\in I}{\cal
J}_{n}{\cal F}_{n}}=\frac{\dot{\cal A}_{in}-T\dot{S}_{tot}}{\dot{\cal
A}_{in}}\leq 100\%.$ (7)

Only in the reversible limit, $\dot{S}_{tot}=0$, the exergy efficiency $\phi$
reaches its upper bound. The second law of thermodynamics requires
$\dot{S}_{tot}\geq 0$ for all possible values of forces. This is satisfied
only when all eigenvalues of the Onsager matrix $\hat{{\cal M}}$ are positive
[see Appendix A] (note that, as the reversible limit, $\dot{S}_{tot}=0$, does
not exist for realistic systems, we consider only situations with positive
entropy production. Zero entropy production is the limit when the entropy
production is extremely small. In this way, all eigenvalues of the Onsager
matrix must be greater than zero.) This property is briefly stated as that
Onsager matrix is positive.

##  III Optimizing efficiency and power for systems with symmetric Onsager
matrix

The maximum exergy efficiency is obtained by solving the differential equation

$\partial_{{\cal F}_{k}}\phi=0,\quad\forall k.$ (8)

Previous attempts of solving the above equationsaniso ; bio ; caplan2  have
ended up with very complicated calculations and discussions. This is because
for a $N\times N$ Onsager matrix, there are $N(N+1)/2$ independent response
coefficients (if the Onsager matrix is symmetric). Besides, there are $N-1$
coupled differential equations to solve (from Eqs. (4) and (7), scaling all
forces by a constant does not change $\phi$; this property reduces the number
of differential equations to be solved by one). Solving these equations
analytically becomes a formidable task when $N\geq 3$ (see, e.g., the rather
complicated discusssions in Ref. caplan2 ). In this work we manage to solve
the problem analytically in a particularly simple way.

We notice that the force-current relation can be rewritten as

$\left(\begin{array}[]{c}\vec{{\cal J}}_{O}\\\ \vec{{\cal
J}}_{I}\end{array}\right)=\left(\begin{array}[]{cccc}\hat{{\cal
M}}_{OO}&\hat{{\cal M}}_{OI}\\\ \hat{{\cal M}}_{IO}&\hat{{\cal
M}}_{II}\end{array}\right)\left(\begin{array}[]{c}\vec{{\cal F}}_{O}\\\
\vec{{\cal F}}_{I}\end{array}\right),$ (9)

where the symbols $O$ and $I$ are used to abbreviate the indices of forces and
currents for exergy output and input, respectively. E.g., $\vec{{\cal J}}_{O}$
is the vector of the output current and $\hat{{\cal M}}_{OO}$ is the matrix
relating the output force vector $\vec{{\cal F}}_{O}$ to the output current
vector $\vec{{\cal J}}_{O}$. Hence,

$\displaystyle\dot{\cal A}_{out}=-\vec{{\cal F}}_{O}^{T}\hat{{\cal
M}}_{OI}\vec{{\cal F}}_{I}-\vec{{\cal F}}_{O}^{T}\hat{{\cal M}}_{OO}\vec{{\cal
F}}_{O},$ (10a) $\displaystyle\dot{\cal A}_{in}=\vec{{\cal
F}}_{I}^{T}\hat{{\cal M}}_{IO}\vec{{\cal F}}_{O}+\vec{{\cal
F}}_{I}^{T}\hat{{\cal M}}_{II}\vec{{\cal F}}_{I},$ (10b)

where the superscript $T$ stands for matrix (vector) transpose. For symmetric
Onsager matrix, $\hat{{\cal M}}_{II}=\hat{{\cal M}}_{II}^{T}$, $\hat{{\cal
M}}_{OI}=\hat{{\cal M}}_{IO}^{T}$, and $\hat{{\cal M}}_{OO}=\hat{{\cal
M}}_{OO}^{T}$.

From Eqs. (7), (8), and (9), we find that

$\partial_{\vec{{\cal F}}_{O}}\dot{\cal
A}_{out}=\phi_{max}(\partial_{\vec{{\cal F}}_{O}}\dot{\cal A}_{in})$ (11)

which gives

$\vec{{\cal F}}_{O}=-\frac{1+\phi_{max}}{2}\hat{{\cal M}}_{OO}^{-1}\hat{{\cal
M}}_{OI}\vec{{\cal F}}_{I}.$ (12)

The inverse of the matrix $\hat{{\cal M}}_{OO}$ is justified as $\hat{{\cal
M}}_{OO}$ is a positive matrix. Inserting this into Eq. (1) we obtain

$\displaystyle\phi_{max}=\frac{\frac{1}{4}\left(1-\phi_{max}^{2}\right)\lambda}{1-\frac{1+\phi_{max}}{2}\lambda}$
(13)

where $\lambda\equiv{\rm max}{\langle\hat{\Lambda}\rangle}$ and
$\langle\hat{\Lambda}\rangle\equiv\vec{{g}}^{T}{\hat{\Lambda}}\vec{{g}}$ with
${\vec{g}}$ being a normalized vector (i.e., ${\vec{g}}^{T}{\vec{g}}=1$)
defined as

${\vec{g}}\equiv\left.\hat{{\cal M}}_{II}^{1/2}\vec{{\cal
F}}_{I}\right/\sqrt{\vec{{\cal F}}_{I}^{T}\hat{{\cal M}}_{II}\vec{{\cal
F}}_{I}},$ (14)

and

$\displaystyle\hat{\Lambda}\equiv\hat{{\cal M}}_{II}^{-1/2}\hat{{\cal
M}}_{IO}\hat{{\cal M}}_{OO}^{-1}\hat{{\cal M}}_{OI}\hat{{\cal
M}}_{II}^{-1/2}.$ (15)

The inverse square root of the matrix $\hat{{\cal M}}_{II}$ is well-defined
since $\hat{{\cal M}}_{II}$ is a positive matrix [see proof in Appendix A].

Eq. (13) is now a quadratic equation that can be solved analytically. The
physical solution with $\phi_{max}<1$ is

$\phi_{max}=\frac{\sqrt{\xi+1}-1}{\sqrt{\xi+1}+1},\quad\quad\xi\equiv\frac{\lambda}{1-\lambda}$
(16)

where $\xi$ is the figure of merit and $\lambda$ is called the “degree of
coupling”caplan1 . We call the matrix $\hat{\Lambda}$ as the “coupling
matrix”. Finally, $\vec{{\cal F}}_{I}$ or the normalized vector $\vec{{g}}$
must be tuned to maximize $\langle\hat{\Lambda}\rangle$. The maximum value is
achieved when $\vec{{g}}$ equals to the eigenvector of $\hat{\Lambda}$ which
corresponds to the largest eigenvalue, which gives

$\lambda={\rm largest\ eigenvalue\ of}\ \hat{\Lambda}.$ (17)

It is proven in Appendix B that $\lambda\leq 1$ as bounded by the second law
of thermodynamics. The $\lambda\to 1$ limit can be reached only in the
reversible limit when the determinant of the Onsager matrix is zeronote . Eq.
(17) represents one of the main results in this work which was not found in
Ref. caplan2  despite rather complicated treatment there.

The output power $\dot{W}=\dot{\cal A}_{out}$ at maximum exergy efficiency is

$\dot{W}(\phi_{max})=\frac{1}{4}(1-\phi_{max}^{2})\lambda\left(\vec{{\cal
F}}_{I}^{T}\hat{{\cal M}}_{II}\vec{{\cal F}}_{I}\right).$ (18)

We now study the exergy efficiency for maximum power. The physical concern is
to optimize the output power by tuning the output forces $\vec{{\cal F}}_{O}$
which corresponds to adjusting the response of the device accepting the output
energy to maximize the output power (as will be shown in the next section).
The output power is then optimized at $\partial_{\vec{{\cal F}}_{O}}\dot{\cal
A}_{out}=0$ which renders $\vec{{\cal F}}_{O}=-\frac{1}{2}\hat{{\cal
M}}_{OO}^{-1}\hat{{\cal M}}_{OI}\vec{{\cal F}}_{I}$. The equation for $\phi$
can be established by inserting the above into Eq. (1) which is then solved in
a way similar to the solution of Eq. (13). After that we optimize $\phi$ by
tuning the input forces $\vec{{\cal F}}_{I}$ and then obtain the optimal
exergy efficiency for maximum power as

$\phi_{opt}(\dot{W}_{max})=\frac{\xi}{2(\xi+2)}\leq 50\%,$ (19)

where $\xi$ is given in Eq. (16) and $\lambda$ is again the largest eigenvalue
of the coupling matrix $\hat{\Lambda}$. The above expression is consistent
with the well-known result that the upper limit of the exergy efficiency for
maximum power for systems with symmetric Onsager matrix is 50%ca-ex ; Seifert-
review ; odum ; maxpower . The above derivations also provide a solid proof of
the upper bound, 50%, for general thermodynamic systems in the linear-response
regime. The maximum output power is found to be

$\dot{W}_{max}=\frac{1}{4}\lambda\left(\vec{{\cal F}}_{I}^{T}\hat{{\cal
M}}_{II}\vec{{\cal F}}_{I}\right).$ (20)

Comparing the exergy efficiencies and output powers for the two optimization
schemes discussed in this section, we find that

$\displaystyle\frac{\phi_{opt}(\dot{W}_{max})}{\phi_{max}}=\frac{1}{1+\phi_{max}^{2}},$
(21a)
$\displaystyle\frac{\dot{W}(\phi_{max})}{\dot{W}_{max}}=1-\phi_{max}^{2}.$
(21b)

Remarkably, the above two simple relationships hold for all thermodynamic
machines with symmetric Onsager matrix in the linear-response regime
(thermodynamic systems with asymmetric Onsager matrix is discussed in Sec. V).
The above two relationships bear very important information on the optimal
efficiencies and powers which is one of the main results in the present work.
Fig. 1 represents them graphically. Particularly in the reversible limit
$\phi_{max}=1$, the output power at maximum efficiency vanisheslow-diss  while
the efficiency at maximum power reaches 50% (These properties were proven to
be general for time-reversal symmetric systems in Ref. Seifert-review  as
well). At low efficiency limit, $\phi_{max}\ll 100\%$, the power and
efficiency at the two optimal conditions are almost the same. Considerable
differences between the two optimal conditions appear only when
$\phi_{max}\gtrsim 20\%$, $\xi\gtrsim 1$, or $\lambda\gtrsim 0.5$.

Figure 1: (Color online) The ratio of optimal efficiency at maximum power to
the maximum efficiency $\frac{\phi_{opt}(\dot{W}_{max})}{\phi_{max}}$ (solid
curve) and the ratio of the power at maximum efficiency to the maximum power
$\frac{\dot{W}(\phi_{max})}{\dot{W}_{max}}$ (dashed curve) as functions of the
maximum efficiency $\phi_{max}$, as given by Eq. (21), for thermodynamic
machines with symmetric Onsager matrix.

We remark that the largest eigenvalue of $\hat{{\cal M}}_{II}^{-1/2}\hat{{\cal
M}}_{IO}\hat{{\cal M}}_{OO}^{-1}\hat{{\cal M}}_{OI}\hat{{\cal M}}_{II}^{-1/2}$
is the same as the largest eigenvalue of $\hat{{\cal M}}_{OO}^{-1/2}\hat{{\cal
M}}_{OI}\hat{{\cal M}}_{II}^{-1}\hat{{\cal M}}_{IO}\hat{{\cal M}}_{OO}^{-1/2}$
(proof is given in Appendix B). Particularly in thermoelectric energy
conversion, this means that the figures of merit for the engine, refrigerator,
and heat pump are the same. These properties can be used to simplify the
calculation of the figure of merit when one of the two is easier to calculate.

Figure 2: (Color online) Schematic of realistic thermodynamic machines. A
machine accepts input energy and convert it into output energy. The output can
be assigned to a huge reservoir (e.g., an electrical power grid with huge
capacity) (a), or to a finite device (b). In realistic situations there are
mechanisms that dissipate part of input energy and prevent it from been
converted into useful outputs, as well as mechanisms that consumes part of
output energy and reduce the amount of useful outputs. These mechanisms are
called “parasitic dissipation”.

##  IV Realistic considerations: output to a huge reservoir or to a finite
device

In realistic situations the input energy may pass through some parallel
channels without entering into the system which reduces the amount of useful
input energy. Besides, the output energy can also be dissipated into channels
parallel to the device accepting the output power. These mechanisms are called
as “parasitic dissipation”. The effect is described by the following
phenomenological equations

$\vec{{\cal J}}_{I}^{p}=\hat{{\cal M}}_{II}^{p}\vec{{\cal
F}}_{I},\quad\quad\vec{{\cal J}}_{O}^{p}=\hat{{\cal M}}_{OO}^{p}\vec{{\cal
F}}_{O}.$ (22)

Here the superscript $p$ stands for parasitic dissipation. The currents for
energy input into the operating system becomes $\vec{{\cal J}}_{I}+\vec{{\cal
J}}_{I}^{p}$. And the currents that load into the device becomes $\vec{{\cal
J}}_{O}+\vec{{\cal J}}_{O}^{p}$. The equivalent circuit is depicted in Fig. 2.
Taking into account of those parasitic currents modifies the response
coefficients as

$\hat{{\cal M}}_{II}\to\hat{{\cal M}}_{II}+\hat{{\cal
M}}_{II}^{p},\quad\hat{{\cal M}}_{OO}\to\hat{{\cal M}}_{OO}+\hat{{\cal
M}}_{OO}^{p}.$ (23)

Parasitic dissipation increases the eigenvalues of the matrices $\hat{{\cal
M}}_{II}$ and $\hat{{\cal M}}_{OO}$ because both $\hat{{\cal M}}_{II}^{p}$ and
$\hat{{\cal M}}_{OO}^{p}$ are positive matrices. As a consequence the degree
of coupling $\lambda$ and the figure of merit $\xi$ are reduced, according to
Eqs. (15) and (16). This is consistent with the physical picture that part of
the useful energy is consumed by the parasitic dissipation.

Energy from the operating system can be outputted to (i) a huge reservoir
(e.g., a power grid with huge capacity), or to (ii) a finite device. The
optimization presented in Sec. III is for option (i) where the output current
$\vec{{\cal J}}_{O}$ does not induce any observable effect on the huge
reservoir which in turn modifies the force $\vec{{\cal F}}_{O}$, so that
$\vec{{\cal J}}_{O}$ and $\vec{{\cal F}}_{O}$ are uncorrelated. In electrical
circuit analog, it is equivalent to using the output energy to charge a huge
capacitor where the charging current $\vec{{\cal J}}_{O}$ does not change the
voltage across the capacitor $\vec{{\cal F}}_{O}$. For option (ii) if the
response of the device is $\vec{{\cal J}}_{O}^{L}=\hat{{\cal M}}_{L}\vec{{\cal
F}}_{O}$, the Kirchhoff’s current law requires that $\vec{{\cal
J}}_{O}+\vec{{\cal J}}_{O}^{L}=0$. Therefore,

$\vec{{\cal F}}_{O}=-(\hat{{\cal M}}_{OO}+\hat{{\cal M}}_{L})^{-1}\hat{{\cal
M}}_{OI}\vec{{\cal F}}_{I}.$ (24)

The power consumed by the device is

$\displaystyle\vec{{\cal F}}_{O}^{T}\hat{{\cal M}}_{L}\vec{{\cal F}}_{O}$
$\displaystyle=$ $\displaystyle\vec{{\cal F}}_{I}^{T}\hat{{\cal
M}}_{IO}(\hat{{\cal M}}_{OO}+\hat{{\cal M}}_{L})^{-1}\hat{{\cal M}}_{L}$ (25)
$\displaystyle\ \ \times(\hat{{\cal M}}_{OO}+\hat{{\cal
M}}_{L})^{-1}\hat{{\cal M}}_{OI}\vec{{\cal F}}_{I}.$

The input exergy is

$\vec{{\cal F}}_{I}^{T}\vec{{\cal J}}_{I}=\vec{{\cal F}}_{I}^{T}(\hat{{\cal
M}}_{II}-\hat{{\cal M}}_{IO}(\hat{{\cal M}}_{OO}+\hat{{\cal
M}}_{L})^{-1}\hat{{\cal M}}_{OI})\vec{{\cal F}}_{I}.$ (26)

The exergy efficiency is then

$\phi=\frac{\vec{{\cal F}}_{O}^{T}\hat{{\cal M}}_{L}\vec{{\cal
F}}_{O}}{\vec{{\cal F}}_{I}^{T}\vec{{\cal J}}_{I}}.$ (27)

By varying $\hat{{\cal M}}_{L}$ of the device that receives power from the
operating system, we find that the maximum output power is reached at

$\hat{{\cal M}}_{L}=\hat{{\cal M}}_{OO},$ (28)

whereas the maximum exergy efficiency is reached when

$\hat{{\cal M}}_{L}=\sqrt{1-\lambda}\hat{{\cal M}}_{OO}.$ (29)

At these conditions we obtain again Eqs. (16), (18), (19), and (20). The above
results reflect the importance of matching between the response of the device
$\hat{{\cal M}}_{L}$ and that of the system $\hat{{\cal M}}_{OO}$ in
optimizing the efficiency and output powerimp-match . Particularly, Eq. (28)
generalizes the maximum power theorem (Jacobi’s Law for electrical circuits,
i.e., “Maximum power is transferred when the internal resistance of the source
equals the resistance of the load, when the external resistance can be varied,
and the internal resistance is constant”) to all thermodynamic machines with
symmetric Onsager matrix in the linear-response regime.

There are two possible schemes of adjusting the input forces, $\vec{{\cal
F}}_{I}$, to optimize the performance of the machine. The first scheme is to
optimize the efficiency, i.e., to optimize $\lambda$. This has been discussed
in Sec. III. This scheme reflects balance between optimizing output power and
efficiency which is relevant to some biological and ecological systemsodum .
The second scheme is to adjust $\vec{{\cal F}}_{I}$ for further optimization
of the output power. This will lead to efficiency smaller or equal to that in
Eq. (19). Hence the exergy efficiency for this scheme is also not larger than
50%. From Eqs. (15) and (20) one finds that
$\dot{W}_{max}=\frac{1}{4}\vec{{\cal F}}_{I}^{T}\hat{{\cal M}}_{IO}\hat{{\cal
M}}_{OO}^{-1}\hat{{\cal M}}_{OI}\vec{{\cal F}}_{I}$. The above can be
optimized to be $\dot{W}_{max}=\frac{1}{4}\Upsilon\left(\vec{{\cal
F}}_{I}^{T}\vec{{\cal F}}_{I}\right)$, with $\Upsilon$ being the largest
eigenvalue of the matrix $\hat{{\cal M}}_{IO}\hat{{\cal
M}}_{OO}^{-1}\hat{{\cal M}}_{OI}$. It can be shown that $\Upsilon$ is positive
[see Appendix B]. There is no obvious upper bound on it that is imposed by the
laws of thermodynamics (except maybe in the zero temperature limitjoe ). The
above derivation is meaningful only when all input thermodynamic forces ${\cal
F}_{n}$ ($\forall n\in I$) are measured in the same physical unit and scale.
This requirement is usually not satisfied for systems with more than one type
of input forces (e.g., if both mechanical and electrical forces are used for
energy input). Discussion on this scheme of performance optimization depends
on specific systems which is of little interest for our purpose.

##  V optimal exergy efficiency and power for systems with asymmetric Onsager
matrix

We now study systems with asymmetric Onsager matrix. We first note that
$\vec{{\cal F}}_{I}^{T}\hat{{\cal M}}_{II}\vec{{\cal F}}_{I}={\cal
F}_{I}^{T}\hat{{\cal M}}^{s}_{II}\vec{{\cal F}}_{I}$ and $\vec{{\cal
F}}_{O}^{T}\hat{{\cal M}}_{OO}{\cal F}_{O}=\vec{{\cal F}}_{O}^{T}\hat{{\cal
M}}^{s}_{OO}\vec{{\cal F}}_{O}$ where $\hat{{\cal
M}}^{s}_{II}=\frac{1}{2}\left(\hat{{\cal M}}_{II}+\hat{{\cal
M}}^{T}_{II}\right)$ and $\hat{{\cal M}}^{s}_{OO}=\frac{1}{2}\left(\hat{{\cal
M}}_{OO}+\hat{{\cal M}}^{T}_{OO}\right)$. This property is due to the symmetry
of the summation over indices of forces.

It is hard to derive the optimal exergy efficiency and power for general
systems with asymmetric Onsager matrix (see Appendix C). Here we focus on a
special situation where $\hat{{\cal M}}_{OI}=r\hat{{\cal M}}_{IO}^{T}$ with
$r$ being a real number. Such a simplification is for the convenience of
treatment instead of inspired by realistic physical systems. For this
particular situation, from Eq. (11), we find $\vec{{\cal
F}}_{O}=-\frac{1+r^{-1}\phi_{max}}{2}\left(\hat{{\cal
M}}_{OO}^{s}\right)^{-1}\hat{{\cal M}}_{OI}\vec{{\cal F}}_{I}$. Inserting this
into Eq. (1) and solving the equation for $\phi_{max}$, we obtain

$\phi_{max}=r\frac{\sqrt{\xi+1}-1}{\sqrt{\xi+1}+1}$ (30)

where $\xi$ is given by the same expression as in Eqs. (16) and (17) but with
$\hat{{\cal M}}_{OO}$ and $\hat{{\cal M}}_{II}$ replaced by their symmetric
counterparts $\hat{{\cal M}}_{OO}^{s}$ and $\hat{{\cal M}}_{II}^{s}$. The
exergy efficiency for maximum power is given by

$\phi_{opt}(\dot{W}_{max})=\frac{r\xi}{2(\xi+2)}.$ (31)

From the second law of thermodynamics the restriction on $\lambda$ is [see
Appendix B]

$\displaystyle\frac{4r}{(1+r)^{2}}\leq\lambda<0,\quad{\rm if}\quad r<0,$ (32a)
$\displaystyle 0\leq\lambda\leq\frac{4r}{(1+r)^{2}},\quad{\rm if}\quad r\geq
0.$ (32b)

The above restrictions give rise to $\xi+1=\frac{1}{1-\lambda}\geq 0$ and
$r(\sqrt{\xi+1}-1)>0$, so that the optimal exergy efficiency given in Eq. (30)
is positive and well-defined.

The maximum possible, i.e., the upper bound of exergy efficiency is reached at
$\lambda=\frac{4r}{(1+r)^{2}}$ as

$\displaystyle\phi_{bound}=r^{2},\quad{\rm if}\quad|r|<1,$ (33a)
$\displaystyle\phi_{bound}=1,\quad{\rm if}\quad|r|\geq 1.$ (33b)

The dissipation at the upper bound exergy efficiency is

$\displaystyle T\dot{S}_{tot}=(1-r)^{2}\left(\vec{{\cal F}}_{I}^{T}\hat{{\cal
M}}_{II}^{s}\vec{{\cal F}}_{I}\right),\quad{\rm if}\quad|r|<1,$ (34a)
$\displaystyle T\dot{S}_{tot}=0,\hskip 102.43008pt{\rm if}\quad|r|\geq 1.$
(34b)

The entropy production for $|r|<1$ is always positive hence the upper bound
efficiency is not 100%.

The upper bound of the exergy efficiency for maximum power is also reached at
$\lambda=\frac{4r}{(1+r)^{2}}$ with

$\displaystyle\left.\phi_{opt}(\dot{W}_{max})\right|_{bound}=\frac{r^{2}}{r^{2}+1}.$
(35)

From the above equation the Curzon-Ahlborn limit of exergy efficiencyca ; ca-
ex ; Seifert-review  $\phi_{CA}=50\%$ can be overcome when $|r|>1$. This is
first pointed out by Benenti et al. in the study of thermoelectric efficiency
in systems with broken time-reversal symmetrytrb1 .

The output power at maximum exergy efficiency is

$\dot{W}(\phi_{max})=\frac{1}{4}(1-r^{-2}\phi_{max}^{2})r\lambda\left(\vec{{\cal
F}}_{I}^{T}\hat{{\cal M}}_{II}^{s}\vec{{\cal F}}_{I}\right).$ (36)

Combining the above with Eq. (33), the upper bound of efficiency for $|r|<1$
is $\phi=r^{2}$ so that the output power is positive. For $|r|>1$ the maximum
efficiency can reach 100% without conflicting the requirement of positive
output power. The maximum output power is

$\dot{W}_{max}=\frac{1}{4}r\lambda\left(\vec{{\cal F}}_{I}^{T}\hat{{\cal
M}}_{II}^{s}\vec{{\cal F}}_{I}\right).$ (37)

We find that

$\displaystyle\frac{\phi_{max}}{\phi_{opt}(\dot{W}_{max})}=1+r^{-2}\phi_{max}^{2},$
(38a)
$\displaystyle\frac{\dot{W}(\phi_{max})}{\dot{W}_{max}}=1-r^{-2}\phi_{max}^{2}.$
(38b)

Eqs. (33b) and (38b) reveal that for systems with asymmetric Onsager matrix
with $|r|>1$, the output power is nonzero even when $\phi_{max}$ reaches the
value of 100% in the reversible limit. These results agree with the findings
of Benenti et al. on thermoelectric efficiency and power in time-reversal
symmetry broken systemstrb1 .

It is interesting to study the optimal exergy efficiency and power of the
reversed machine (i.e., the machine with output input reversed). The output
power of the reversed machine is $-\vec{{\cal F}}_{I}^{T}\vec{{\cal
J}}_{I}=-\dot{\cal A}_{in}$, while the input power becomes $\vec{{\cal
F}}_{O}^{T}\vec{{\cal J}}_{O}=-\dot{\cal A}_{out}$. The reversed machine is
working in the region with $\dot{\cal A}_{I}<0$. The efficiency of the
reversed machine is defined as

$\phi^{\prime}=\frac{\dot{\cal A}_{in}}{\dot{\cal A}_{out}}.$ (39)

We find that the optimal exergy efficiency and powers are similar but with $r$
replaced by $r^{-1}$. Therefore for $|r|>1$ the reversed machine can not reach
the efficiency of 100%, whereas for $|r|<1$ the reversed machine can have 100%
efficiency with finite power.

Figure 3:  (Color online) The exergy efficiency $\phi$ (solid curves), output
power $\dot{W}$ (dashed curves), and total entropy production $\dot{S}_{tot}$
(dotted curves) as functions of $x$ for $r=1$ (a), $r=0.6$ (b), and $r=1.2$
(c). For each figure the left region with positive efficiency is the operating
region of the machine, while the right region with positive efficiency is the
operating region for the reversed machine. The definitions of efficiency and
output power are different for the machine and the reversed machine.

To demonstrate this we plot the efficiency as a function of $x$ for
$\vec{{\cal F}}_{O}=-\frac{r+x}{2r}\left(\hat{{\cal
M}}_{OO}^{s}\right)^{-1}\hat{{\cal M}}_{OI}\vec{{\cal F}}_{I}$ in Fig. 3. At
the limit with $\lambda=\frac{4r}{(1+r)^{2}}$ the efficiency is

$\phi=\frac{r^{2}-x^{2}}{1+r^{2}-2x}.$ (40)

The output power $\dot{W}=\frac{r^{2}-x^{2}}{(1+r)^{2}}(\vec{{\cal
F}}_{I}^{T}\hat{{\cal M}}_{II}^{s}\vec{{\cal F}}_{I})$ is positive when
$|x|<|r|$. If $x>(1+r^{2})/2$, both the input and output exergy are negative
which indicates that the machine is operating at the reversed mode. The
efficiency of the reversed machine is then

$\phi^{\prime}=\frac{2x-1-r^{2}}{x^{2}-r^{2}}$ (41)

The output power is $\dot{W}=\frac{2x-1-r^{2}}{(1+r)^{2}}(\vec{{\cal
F}}_{I}^{T}\hat{{\cal M}}_{II}^{s}\vec{{\cal F}}_{I})$.

For all values of $r$ the reversible limit $T\dot{S}_{tot}=0$ is reached at
$x=1$. When $r=1$, 100% efficiency is reached by both the machine and the
reversed machine at $x=1$ where the input and output exergy as well as entropy
production vanish [see Fig. 3a]low-diss . For $|r|<1$, the machine cannot
reach to 100% efficiency, but the reversed machine can reach 100% efficiency
with finite output power, because at $x=1$ the machine is operating in the
reversed mode [see Fig. 3b]. For $|r|>1$, the output power of the machine is
positive at $x=1$, thus the machine can reach 100% efficiency with finite
output power [see Fig. 3c].

In systems with broken time-reversal symmetry, such as two-dimensional
electron systems under perpendicular magnetic field, Hall effect, and Nernst-
Ettingshausen effect give rise to asymmetric Onsager matrixchien ; trb3 . The
asymmetric Onsager matrix can be decomposed into the symmetric part and anti-
symmetric part. Specifically,

$\hat{{\cal M}}_{IO}=\hat{{\cal M}}_{IO}^{s}+\hat{{\cal
M}}_{IO}^{a},\quad\hat{{\cal M}}_{OI}=\hat{{\cal M}}_{OI}^{s}+\hat{{\cal
M}}_{OI}^{a}$ (42)

with $\hat{{\cal M}}_{IO}^{s}=(\hat{{\cal M}}_{OI}^{s})^{T}$ and $\hat{{\cal
M}}_{IO}^{a}=-(\hat{{\cal M}}_{OI}^{a})^{T}$. The symmetric part, $\hat{{\cal
M}}_{IO}^{s}$, is related to entropy production and is restricted by the
second law of thermodynamics. The anti-symmetric part, $\hat{{\cal
M}}_{IO}^{a}$, however, does not contribute to dissipation and is often
related to Berry phase effectsniu . The output and input exergy can be written
as

$\dot{\cal A}_{out}=\dot{\cal A}_{out}^{s}-\vec{{\cal F}}_{O}^{T}\hat{{\cal
M}}_{OI}^{a}\vec{{\cal F}}_{I},\quad\dot{\cal A}_{in}=\dot{\cal
A}_{in}^{s}-\vec{{\cal F}}_{O}^{T}\hat{{\cal M}}_{OI}^{a}\vec{{\cal F}}_{I}$
(43)

where $\dot{\cal A}_{out}^{s}$ and $\dot{\cal A}_{in}^{s}$ are the output and
input exergy for the symmetrized Onsager matrix with

$\displaystyle\dot{\cal A}_{out}^{s}=-\vec{{\cal F}}_{O}^{T}\hat{{\cal
M}}_{OI}^{s}\vec{{\cal F}}_{I}-\vec{{\cal F}}_{O}^{T}\hat{{\cal
M}}_{OO}^{s}\vec{{\cal F}}_{O},$ $\displaystyle\dot{\cal
A}_{in}^{s}=\vec{{\cal F}}_{I}^{T}\hat{{\cal M}}_{IO}^{s}\vec{{\cal
F}}_{O}+\vec{{\cal F}}_{I}^{T}\hat{{\cal M}}_{II}^{s}\vec{{\cal F}}_{I}.$

The additional term in Eq. (43), $\vec{{\cal F}}_{O}^{T}\hat{{\cal
M}}_{OI}^{a}\vec{{\cal F}}_{I}$, does not cause entropy production, but shift
the input and output powers by the same magnitude. In this way the reversible
limit is shifted from the boundary between the machine and the reversed
machine, into the operating region of the machine or the reversed machine,
whichever has positive output power in such limit.

It should be emphasized here that although potential advantages of systems
with asymmetric Onsager matrix have been predicted by Benenti et al.trb1  from
phenomenological theory (and extended in this work), no realistic physical
system has been shown to have finite power at 100% efficiencytrb2 ; trb3 . It
is very important to study efficiency and power of realistic physical systems
with asymmetric Onsager matrix to clarify whether breaking time reversal
symmetry could indeed improve the performance of a thermodynamic machinetrb2 ;
trb3 .

##  VI Application to realistic systems

###  VI.1 Example I: Thermoelectric energy conversion in isotropic systems

Thermoelectric transport equation for an isotropic system is given by

$\left(\begin{array}[]{c}\vec{j}\\\
\vec{j}_{q}\end{array}\right)=\left(\begin{array}[]{cccc}\sigma{\hat{\mathbf{1}}}&\sigma
ST{\hat{\mathbf{1}}}\\\ \sigma ST{\hat{\mathbf{1}}}&(\kappa T+\sigma
S^{2}T^{2}){\hat{\mathbf{1}}}\end{array}\right)\left(\begin{array}[]{c}\vec{{\cal
E}}\\\ -\vec{\mbox{\boldmath$\nabla$\unboldmath}}T/T\end{array}\right),$ (44)

where the electric field $\vec{{\cal E}}$ include both the external and
induced electric fields. Here $\sigma$ is the electrical conductivity, $S$ is
the Seebeck coefficient, $\kappa$ the thermal conductivity, and
$\hat{\mathbf{1}}$ is the $3\times 3$ identity matrix. The efficiency, or
coefficient of performance, of a thermoelectric refrigerator is

$\eta\equiv\frac{\dot{Q}}{\dot{W}}=\frac{T}{\Delta
T}\frac{\vec{j}_{q}\cdot\vec{\mbox{\boldmath$\nabla$\unboldmath}}T/T}{\vec{j}\cdot\vec{{\cal
E}}}=\eta_{C}\phi,\quad\eta_{C}\equiv\frac{T}{\Delta T}.$ (45)

For a slab of thickness $\ell_{z}$ with temperature gradient and electric
field along the direction $z$ which is perpendicular to the slab plane, the
temperature difference is $\Delta T=-\ell_{z}\frac{dT}{dz}>0$ for
$\frac{dT}{dz}<0$. The maximum coefficient of performance $\eta_{max}$ is
related to the maximum exergy efficiency by

$\eta_{max}=\eta_{C}\phi_{max}=\eta_{C}\frac{\sqrt{\xi+1}-1}{\sqrt{\xi+1}+1}.$
(46)

The figure of merit is related to the degree of coupling which, according to
Eq. (17), is the largest eigenvalue of the following coupling matrix

$\hat{\Lambda}=\frac{(\sigma ST)^{2}}{\sigma(\kappa T+\sigma
S^{2}T^{2})}\hat{\mathbf{1}}.$ (47)

Since $\hat{\Lambda}$ is proportional to an identity matrix, the largest
eigenvalue is just

$\lambda=\frac{\sigma S^{2}T}{\kappa+\sigma S^{2}T}.$ (48)

Therefore the figure of merit is

$\xi=\frac{\lambda}{1-\lambda}=\frac{\sigma S^{2}T}{\kappa},$ (49)

which recovers the well-known thermoelectric figure of merit as found by
Ioffe.

###  VI.2 Example II: Spin-thermoelectric effect

In conducting magnetic materials charge, spin, and thermal transports are
coupled together. There couplings are called spin-thermoelectric or spin-
caloric effectbauer . In isotropic materials spin-thermoelectric effect is
described by the following transport equationbauer

$\left(\begin{array}[]{c}\vec{j}\\\ \vec{j}_{s}\\\
\vec{j}_{q}\end{array}\right)=\left(\begin{array}[]{cccc}\sigma{\hat{\mathbf{1}}}&\sigma
P{\hat{\mathbf{1}}}&\sigma ST{\hat{\mathbf{1}}}\\\ \sigma
P{\hat{\mathbf{1}}}&\sigma{\hat{\mathbf{1}}}&P^{\prime}\sigma
ST{\hat{\mathbf{1}}}\\\ \sigma ST{\hat{\mathbf{1}}}&P^{\prime}\sigma
ST{\hat{\mathbf{1}}}&\kappa_{0}T{\hat{\mathbf{1}}}\end{array}\right)\left(\begin{array}[]{c}\vec{{\cal
E}}\\\ -\vec{\mbox{\boldmath$\nabla$\unboldmath}}m\\\
-\vec{\mbox{\boldmath$\nabla$\unboldmath}}T/T\end{array}\right).$ (50)

where $\vec{j}={\vec{j}}^{(\uparrow)}+{\vec{j}}^{(\downarrow)}$,
$\vec{j}_{s}={\vec{j}}^{(\uparrow)}-{\vec{j}}^{(\downarrow)}$ with
${\vec{j}}^{(\uparrow)}$ and ${\vec{j}}^{(\downarrow)}$ denoting the
electrical currents of the spin-up and spin-down electrons, respectively.
$\vec{{\cal E}}=-\vec{\mbox{\boldmath$\nabla$\unboldmath}}\mu/e$ with
$\mu\equiv(\mu_{\uparrow}+\mu_{\downarrow})/2$, and
$m\equiv(\mu_{\uparrow}-\mu_{\downarrow})/(2e)$ where $\mu_{\uparrow}$ and
$\mu_{\downarrow}$ are the electrochemical potentials for spin-up and spin-
down electrons, respectively, $e$ is the carrier charge. $\sigma$ is the
electrical conductivity, $S$ is the Seebeck coefficient, $P$ and $P^{\prime}$
are two dimensionless quantities describing spin polarization of carriers in
different transport channels, $\kappa_{0}$ is the heat conductivity at
$\vec{{\cal E}}=\vec{\mbox{\boldmath$\nabla$\unboldmath}}m=0$. Microscopically
they are given by

$\displaystyle\sigma=\int dE\left(-\frac{\partial n_{F}}{\partial
E}\right)\sum_{s}\sigma^{(s)}(E),$ (51a) $\displaystyle P=\langle
s_{z}\rangle,\quad S=\frac{\langle E\rangle}{eT},$ (51b) $\displaystyle
P^{\prime}=\frac{\langle Es_{z}\rangle}{\langle
E\rangle},\quad\kappa_{0}T=e^{-2}\sigma\langle E^{2}\rangle.$ (51c)

with $\sigma^{(s)}(E)$ ($s=\uparrow,\downarrow$) being spin- and energy-
dependent conductivity. We have set the energy zero to be at the (equilibrium)
chemical potential, i.e., $\mu\equiv 0$. $s_{z}=1$ or -1 for spin up and down,
respectively. $n_{F}=1/[\exp(\frac{E}{k_{B}T})+1]$ is the Fermi distribution
of the carrier. The averages in the above equations are defined as

$\langle{\cal O}\rangle\equiv\sigma^{-1}\int dE\left(-\frac{\partial
n_{F}}{\partial E}\right)\sum_{s}\sigma^{(s)}(E){\cal O}.$ (52)

The above equations can be viewed as Mott relationsMC  generalized to spin-
dependent transport. It assumes elastic transport (by which the energy
dependent conductivity is well-defined) and fails when inelastic transport
processes become important as pointed out by the author and
collaboratorsourworks .

We consider refrigeration driven by both the electric field $\vec{{\cal E}}$
and the spin density gradient $\vec{\mbox{\boldmath$\nabla$\unboldmath}}m$.
The coefficient of performance of the refrigerator is defined as

$\eta\equiv\frac{\dot{Q}}{\dot{W}}=\frac{T}{\Delta
T}\frac{\vec{j}_{q}\cdot\vec{\mbox{\boldmath$\nabla$\unboldmath}}T/T}{\vec{j}\cdot\vec{{\cal
E}}-\vec{j}_{s}\cdot\vec{\mbox{\boldmath$\nabla$\unboldmath}}m}=\eta_{C}\phi,\quad\eta_{C}=\frac{T}{\Delta
T}.$ (53)

Schematic of spin-thermoelectric cooling is shown in Fig. 4. Consider a slab
of thickness $\ell_{z}$ where the temperature gradient, electric field, and
spin density gradient are along the direction perpendicular to the slab plane,
i.e., the $z$ direction. The temperature difference is $\Delta
T=-\ell_{z}\frac{dT}{dz}>0$ for $\frac{dT}{dz}<0$. The maximum coefficient of
performance is again related to the maximum exergy efficiency as given in Eq.
(46). Using Eqs. (16) and (50) we obtain

$\xi=\frac{\sigma TS^{2}(1-2PP^{\prime}+P^{\prime
2})}{\kappa_{0}(1-P^{2})-\sigma TS^{2}(1-2PP^{\prime}+P^{\prime 2})}.$ (54)

Remarkably one can show that the above degree of coupling is greater than the
figure of merit for thermoelectric cooling,

$\xi_{TE}=\frac{\sigma TS^{2}}{\kappa_{0}-\sigma TS^{2}},$ (55)

and the figure of merit for spin-Peltier coolingbauer ; sp ,

$\xi_{SP}=\frac{\sigma TS^{2}P^{\prime 2}}{\kappa_{0}-\sigma TS^{2}P^{\prime
2}}.$ (56)

Figure 4:  (Color online) Spin-thermoelectric cooling. A spin-thermoelectric
(“spin-TE”) material (i.e., a conducting ferromagnetic material) sandwiched
between two ferromagnetic electrodes with different temperature $T$,
electrochemical potential $\mu\equiv(\mu_{\uparrow}+\mu_{\downarrow})/2$, and
spin accumulation $m\equiv(\mu_{\uparrow}-\mu_{\downarrow})/(2e)$ where
$\mu_{\uparrow}$ and $\mu_{\downarrow}$ are the electrochemical potentials for
spin-up and spin-down electrons, respectively, and $e$ is the carrier charge.
For a set-up with $T_{h}>T_{c}$, $\mu_{h}>\mu_{c}$, and $m_{h}>m_{c}$ (the
subscripts $h$ and $c$ denoting the hot and cold terminals, respectively),
cooling (heat flowing from the cold terminal to the hot terminal) is driven by
both the charge and spin flows.

This interesting phenomenon has a geometric origin which is understood as
follows. The electric field and the spin-density gradient can be parametrized
as

$\vec{{\cal E}}=\vec{{\cal
F}}_{0}\cos\theta,\quad-\vec{\mbox{\boldmath$\nabla$\unboldmath}}m=\vec{{\cal
F}}_{0}\sin\theta$ (57)

where $\vec{{\cal F}}_{0}=\vec{{\bf
e}}_{z}\sqrt{\frac{1}{2e^{2}}(|\vec{\mbox{\boldmath$\nabla$\unboldmath}}\mu_{\uparrow}|^{2}+|\vec{\mbox{\boldmath$\nabla$\unboldmath}}\mu_{\downarrow}|^{2})}$
with $\vec{{\bf e}}_{z}$ being the transport direction. $|\vec{{\cal F}}_{0}|$
is the total “magnitude” of the input force. The heat current,

$\vec{j}_{q}=\vec{j}_{q0}+\vec{j}_{q1}+\vec{j}_{q2},$ (58)

consists of three parts: thermal conduction
$\vec{j}_{q0}=-\kappa_{0}\vec{\mbox{\boldmath$\nabla$\unboldmath}}T$, Peltier
cooling $\vec{j}_{q1}=\sigma ST\vec{{\cal E}}$, and spin-Peltier cooling
$\vec{j}_{q2}=-P^{\prime}\sigma ST\vec{\mbox{\boldmath$\nabla$\unboldmath}}m$.
The cooling is achieved when the sum of the Peltier current $\vec{j}_{q1}$ and
the spin-Peltier current $\vec{j}_{q2}$ exceeds the thermal conduction current
$\vec{j}_{q0}$.

Tuning the angle $\theta$ changes the relative amplitude of the Peltier and
spin-Peltier heat currents, $\vec{j}_{q1}$ and $\vec{j}_{q2}$. These two
currents can be of the same sign, or the opposite sign, depending on $\theta$.
When $\vec{j}_{q1}$ and $\vec{j}_{q2}$ have the same sign, the cooling is
enhanced, leading to higher efficiency. However, when $\vec{j}_{q1}$ and
$\vec{j}_{q2}$ have opposite sign, the cooling is suppressed and the
efficiency is reduced. This is explicitly shown in Fig. 5. The underlying
physics is more complicated when the input work $\dot{W}$ is taken into
consideration as well. However, this simplified picture gives a snapshot that
the two cooling mechanisms can have cooperative effects.

We also calculated the figure of merit for spin-Peltier cooling $\xi_{SP}$ as
function of $P^{\prime}$ according to Eq. (56) as shown in Fig. 6a for $\sigma
S^{2}T/\kappa_{0}=0.1$. For the same parameter, we plot the enhancement factor
$\xi/{\rm max}(\xi_{TE},\xi_{SP})$ as function of $P$ and $P^{\prime}$ in Fig.
6b. Significant enhancement of figure of merit due to cooperative effect is
attainable when $P^{\prime}$ deviates from $P$ markedly.

Figure 5:  (Color online) Polar plot of $\xi$ vs. $\theta$. The parameters are
$P=0.2$, $P^{\prime}=0.8$, $S=50$ $\mu$V/K, and $T=300$ K. The heat
conductivity is $\kappa_{0}=\sigma LT$ with the Lorenz number of $L=2.5\times
10^{-8}$ W ${\rm\Omega}$ K-2. The arrows indicates the relative direction
between $\vec{j}_{q1}$ (red arrows) and $\vec{j}_{q2}$ (green arrows). The red
dots represent the thermoelectric figure of merit $\xi_{TE}$, the green
triangles represent the spin-Peltier figure of merit $\xi_{SP}$, and the blue
rhombuses denote the figure of merit $\xi$ of combined thermoelectric and
spin-Peltier cooling.

Efficient spin-thermoelectric cooling demands large Seebeck coefficient.
According to the literature, large Seebeck coefficient ranging from 100 to
45000 $\mu$V/K can be attained in magnetic or strongly-correlated
semiconductorsmag-sem  and magnetic tunnel junctionsmtj . Sizable figure of
merit, $\xi\sim 1$, however, is still to be achievedmag-sem .

The figure of merit at fixed $\theta$ is found as

$\xi(\theta)=\frac{\sigma
S^{2}T(P^{\prime}\sin\theta+\cos\theta)^{2}(1+2P\sin\theta\cos\theta)}{\kappa_{0}-\sigma
S^{2}T(P^{\prime}\sin\theta+\cos\theta)^{2}(1+2P\sin\theta\cos\theta)}$ (59)

The maximum exergy efficiency is achieved at $\theta=\theta_{M}$ with

$\tan\theta_{M}=\frac{P^{\prime}-P}{1-P^{\prime}P}.$ (60)

The figure of merit at $\theta=\theta_{M}$ is exactly the same as that given
in Eq. (54), which is greater than the figures of merit for thermoelectric and
spin-Peltier cooling, $\xi_{TE}$ and $\xi_{SP}$, unless $P=P^{\prime}$. Such
cooperative effect prevails in systems with multiple cross-correlated
responses, which can be exploited to improve the efficiency. The discussions
here can also be applied to the efficiency and figure of merit of spin-
thermoelectric power generatorstem-gen .

Figure 6:  (Color online) (a) The ratio of the figure of merit of spin-Peltier
cooling $\xi_{SP}$ to that of thermoelectric cooling $\xi_{TE}$ as a function
of $P^{\prime}$. (b) The enhancement of figure of merit due to cooperative
effect, $\xi/{\rm max}(\xi_{TE},\xi_{SP})$, as a function of $P$ and
$P^{\prime}$. The parameters are $S=50$ $\mu$V/K and $T=300$ K. The heat
conductivity is $\kappa_{0}=\sigma LT$ with the Lorenz number of $L=2.5\times
10^{-8}$ W ${\rm\Omega}$ K-2. The white region in (b) near $P=1$ is forbidden
by the second law of thermodynamics.

###  VI.3 Example III: Piezoelectric, piezomagnetic and magnetoelectric
effects

Piezoelectric energy harvest has been studied extensively and made into useful
devicespe-review . There is also the piezomagnetic effect where elastic strain
induces a magnetization or vice versatem1 . These two effects are common in
ferroelectric and ferromagnetic insulatorstem1 . Materials with simultaneous
ferroelectric and ferromagnetic properties, or more generally multiple
spontaneous electric and magnetic orderstem1 ; multi1 ; science , are called
multiferroics. An important technologically property of multiferroics is the
magnetoelectric effect which offers efficient conversion between electric and
magnetic energy in the radio frequency regimetem1 . Wood and Austinwood
suggested many possible applications of the magnetoelectric effect, among
which there are transducers which convert the microwave magnetic field into
microwave electric field, attenuators which are used to improve impedance
matching in circuits, and ultrasensitive magnetic field sensorstem1 .
Multiferroics with strong magnetoelectric response have been the aim of
extensive studiestem1 . Recently, strong magnetoelectric response were found
in both crystalline (such as CaMn7O12crys1 , TbMnO3crys2 , and HoMnO3crys3 )
and nano-composite (such as BiFeO3 thin film heterostructurescomp1  and
BaTiO3-CoFe2O4 nano-structurescomp2 ) materials. In many of these materials
the interplay of piezoelectric and piezomagnetic responses play an important
role. In fact, multiferroics can be made from nano-composites of ferroelectric
and ferromagnetic compounds where elastic strain at interfaces mediate
coupling between electric and magnetic polarizationsscience ; nan .

In these materials a full description of responses to external mechanical,
electric, and magnetic forces are given bynan ; tem1

$\left(\begin{array}[]{c}\hat{S}\\\ \vec{D}\\\
\vec{B}\end{array}\right)=\left(\begin{array}[]{cccc}\hat{s}&\hat{d}&\hat{q}\\\
\hat{d}^{T}&\hat{\epsilon}&\hat{\alpha}\\\
\hat{q}^{T}&\hat{\alpha}^{T}&\hat{\mu}_{m}\end{array}\right)\left(\begin{array}[]{c}\hat{T}\\\
\vec{E}\\\ \vec{H}\end{array}\right).$ (61)

where the forces are stress $\hat{T}$, electric field $\vec{E}$, and magnetic
field $\vec{H}$, the currents are strain $\hat{S}$, electric displacement
$\vec{D}$, and magnetic induction $\vec{B}$. Here $\vec{D}$ and $\vec{B}$
stand for the values deviate from the equilibrium ones (which could be nonzero
in materials with spontaneous polarization and magnetization). The response
matrix has the dimension of $12\times 12$. Specifically, $\hat{s}$ is the
$6\times 6$ compliance tensor, $\hat{\epsilon}$ is the $3\times 3$ dielectric
tensor, $\hat{\mu}_{m}$ is the ($3\times 3$) permeability tensor, $\hat{d}$
describes piezoelectric response, $\hat{q}$ describes piezomagnetic response,
and $\hat{\alpha}$ gives magnetoelectric response.

In general the response matrix is frequency dependent. Experiments have shown
resonance behavior in magnetoelectric responsereson . Without further
complication of specific circuits set-up for energy conversion at finite
frequenciespe1 ; pe2 , here we consider the low-frequency limit which is
sufficient to demonstrate the underlying principles. Extension of study to
finite frequency regimes will be achieved in future works. First, the coupling
matrix for piezoelectric energy conversion is

$\hat{\Lambda}_{pe}=\hat{\epsilon}^{-1/2}~{}\hat{d}^{T}~{}\hat{s}^{-1}~{}\hat{d}~{}\hat{\epsilon}^{-1/2},$
(62)

which coincides with the “electromechanical coupling tensor” introduced in
Ref. pe . The largest electromechanical coupling factor of a material is given
by the largest eigenvalue of the coupling matrix $\hat{\Lambda}_{pe}$.
Piezoelectric effect allows harvest of mechanical energy to power portable and
isolated electrical systems as well as small motors which have already found
applicationspe-review . Existing materials have already shown large
electromechanical coupling factors, reaching to $\gtrsim 0.5$pe1 ; ryu , which
allows efficient piezoelectric energy conversion. In realistic systems,
additional mechanical and electrical damping reduces the efficiencype1 ; pe2 .
Although further complication must be considered for a finite frequency set-up
with a mechanical oscillator, the efficiency is still an increasing function
of the electromechanical coupling factorpe1 ; pe2 . Piezomagnetic effect can
be used for magnetic field sensing, stress sensing, and mechanical generation
of spin-wavestem1 . The coupling matrix for piezomagnetic energy conversion is

$\hat{\Lambda}_{pm}=\hat{\mu}_{m}^{-1/2}~{}\hat{q}^{T}~{}\hat{s}^{-1}~{}\hat{q}~{}\hat{\mu}_{m}^{-1/2}.$
(63)

The largest piezomagnetic coupling factor is the largest eigenvalue of the
above matrix. Piezomagnetic coupling factor can be as large as 0.5 as welldong
. The coupling matrix for magnetoelectric energy conversion is

$\hat{\Lambda}_{em}=\hat{\mu}_{m}^{-1/2}~{}\hat{\alpha}^{T}~{}\hat{\epsilon}^{-1}~{}\hat{\alpha}~{}\hat{\mu}_{m}^{-1/2}.$
(64)

Experiments on laminated composites of rare-earth-iron alloys (Terfenol-D) and
lead-zirconate-titanate (PZT) achieved a magnetoelectric coefficient along the
stacking direction as high as $\alpha_{E}=\alpha/\epsilon=$10 V cm-1 Oe-1ryu .
Along this direction the relative dielectric constant is about 1000ryu  and
the relative permeability is about 4nan2 . According to these parameters, the
magnetoelectric coupling factor along the stacking direction is around 0.1.
The largest magnetoelectric coupling factor is given by the largest eigenvalue
of the matrix $\hat{\Lambda}_{em}$.

The system also allows multiple input or output energy forms. For example,
magnetic energy can be generated by simultaneously inputting electric and
mechanic energy. This yield the coupling matrix of

$\hat{\Lambda}_{m-pe}=\hat{\mu}_{m}^{-1/2}~{}\hat{q}_{pe}^{T}~{}\hat{h}_{pe}^{-1}~{}\hat{q}_{pe}~{}\hat{\mu}_{m}^{-1/2}$
(65)

where

$\hat{q}_{pe}=\left(\begin{array}[]{c}\hat{q}\\\
\hat{\alpha}\end{array}\right),\quad\hat{h}_{pe}=\left(\begin{array}[]{cccc}\hat{s}&\hat{d}\\\
\hat{d}^{T}&\hat{\epsilon}\end{array}\right).$ (66)

Similar to the results in Sec. VI.2, cooperative effect will lead to larger
degree of coupling from the above coupling matrix. That is, the exergy
efficiency is no less than those of piezomagnetic effect and magnetoelectric
effect. Significant improvement of efficiency could be possible by the
synergetic effect in systems with cross-correlated piezo-electric-magnetic
effect.

###  VI.4 Example IV: Biological energy conversion

Figure 7: (Color online) Energy conversion in biological reaction. Biological
reactions, $A+B\leftrightarrow C+energy$ and $Q+energy+E\leftrightarrow P$,
take place in the reaction center. The first reaction produces energy which is
stored in material $P$ via the second reaction. At steady states there are
continuous flows of materials across the membrane of the reaction center to
facilitate continuous reactions. The membrane keeps a density (chemical
potential) difference between the reaction center and the outside to control
reaction rates. Arrows in the figure indicate possible flows of materials when
energy is produced and stored in $P$.

Biological processes are driven by various energies: the internal energy
produced by oxidation and external energy from environments. Understanding of
bioenergetics is one of the most important and challenging task in biology.
Many of the processes can be described by Onsager’s linear-response theory
(although many others cannot)caplan1 ; caplan2 ; bio1 ; bio2 ; bio3 . One
example is transport across a membrane. The flows of various ions, such as
Na+, Ca2+, and H+ as well as other materials, such as phosphorylation, oxygen,
and sugars are all driven by their density gradients, chemical reaction and
other forcesbio . If, e.g., some of these materials involve in a chemical
reaction, flows of those materials will be correlated. Synergetic effects will
appear as multiple flows take place in coorperative ways. Biological systems,
may also utilize the cross-correlation of those flows to optimize energy
efficiency. There have been a lot of studies of bioenergetics using
irreversible thermodynamicsbio1 ; bio2 ; bio3 ; caplan1 ; caplan2 . However,
none of them have reached a simple analytic results as obtained in this work.

To demonstrate the usefulness of the theory, we consider a toy model describes
the reaction of

$A+B\leftrightarrow C+energy,\quad Q+energy+E\leftrightarrow P$ (67)

in a reaction center surrounded by a membrane. We assume the reactions are
reversible with the help of enzymes. In the former reaction $A$ and $B$ are
consumed to produce $C$ while some energy is generated which is absorbed by
$Q$ and $E$ to form $P$ (energy stored in $P$). We assume that all energy
generated in the former reaction is absorbed by the latter one. To describe
such a reaction, we use six flows, $J_{A}$, $J_{B}$, $J_{Q}$, and $J_{E}$ to
describe the rate of consumption of $A$, $B$, $Q$, and $E$, $-J_{C}$ and
$-J_{P}$ to describe the rate of production of $C$ and $P$. The flow and
reaction is illustrated in Fig. 7. The reaction is described by Eq. (9) in
linear-response regime with

$\displaystyle\vec{\cal J}_{I}=({\cal J}_{A},{\cal J}_{B},{\cal J}_{Q},{\cal
J}_{E})^{T},\quad\vec{\cal J}_{O}=({\cal J}_{C},{\cal J}_{P})^{T},$ (68a)
$\displaystyle\vec{\cal F}_{I}=({\cal F}_{A},{\cal F}_{B},{\cal F}_{Q},{\cal
F}_{E})^{T},\quad\vec{\cal F}_{O}=({\cal F}_{C},{\cal F}_{P})^{T}.$ (68b)

The forces can be written as ${\cal F}_{i}=\delta\mu_{i}+a_{i}$ where
$\delta\mu_{i}=\mu_{i}^{out}-\mu_{i}^{in}$ with $\mu_{i}^{out}$ and
$\mu_{i}^{in}$ being the chemical potential of $i$ outside and inside the
reaction center, respectively, $a_{i}$ is the affinity of material $i$ for the
reaction which is the free energy of $i$ per mole (if ${\cal J}_{i}$ is
measured in unit of mole per second). Biological systems can control those
flows and their correlations through chemical reaction processes (e.g., via
enzymes) as well as selective and tunable transmission of materials through
the membrane. The efficiency of the biological reaction is $\phi=-\vec{\cal
F}_{O}^{T}\vec{\cal J}_{O}/(\vec{\cal F}_{I}^{T}\vec{\cal J}_{I})$. The
optimal efficiency is then given by Eq. (16) where the degree of coupling is
given by the largest eigenvalue of the coupling matrix $\hat{\Lambda}$ given
by Eq. (15). This result is much simpler than that discussed in Ref. caplan2 .

##  VII Conclusion and discussions

We examined the important question of “what is the maximum efficiency of a
thermodynamic machine when its linear responses to the external is given?”.
This question has been answered in simple limits with two thermodynamic
currents. It becomes rather difficult to answer for a thermodynamic machine
with arbitrarily complex responses. Efforts on the problem in the literature
failed in yielding general and analytic results that are useful for material
and structure engineering in advanced energy technologies. Pushed by fast
developing nanotechnology and material technologies, complex systems with
advanced functions play more and more important roles. It becomes increasingly
demanded to extend the known, simple results on efficiency optimization with
two thermodynamic currents to those complex systems which is characterized by
a $N\times N$ Onsager matrix ($N>2$).

We derived the optimal efficiency and powers for general thermodynamic
machines with arbitrary linear-response coefficients. The results are written
in simple and analytic forms. Based on those results we establish two general
relationships between the optimal efficiency and powers for two realistic
optimization schemes: (i) maximum efficiency and (ii) optimal efficiency for
maximum power. We proved that the upper bound efficiency at maximum output
power is 50% for all thermodynamic systems with symmetric Onsager response
matrix. The results are confirmed by considering realistic energy systems
where the output power is consumed by a device of which the response
coefficients can be varied. We proved that the maximum output power is reached
when the response matrix of the device receiving the power, $\hat{{\cal
M}}_{L}$, is equal to that of the power-supplying machine in the output
sector, $\hat{{\cal M}}_{OO}$. This proof generalizes the maximum power
theorem (Jacobi’s Law) to all thermodynamic machines with symmetric Onsager
matrix in the linear-response regime. We also extend the studies to systems
with asymmetric Onsager matrix (for a particular class of systems) where the
efficiency at maximum output power can exceed 50%. Besides, in such systems
the second law of thermodynamics does not forbid the reversible limit of
efficiency, 100%, to be reached at finite output power. This phenomenon is
caused by redistribution of free energy between the input and output channels
induced by dissipationless responses (e.g., by magnetic field, geometric
phases, etc). We also show that such limit can only be reached in a machine by
its normal mode or reversed mode, but not by both of them.

Several examples are presented to demonstrate applications of the theory.
First for isotropic thermoelectric systems, we recover Ioffe’s well-known
results. We then consider refrigeration in spin-thermoelectric systems. It is
shown that driving cooling by both electrochemical potential and spin density
gradients yield maximum efficiency considerably higher than when only one of
the two gradients (forces) is applied to the system. Such enhancement of
maximum efficiency due to cooperative effects between different forces can be
significant in certain parameter regimes. We remark that such cooperative
effects prevail in systems with multiple cross-correlated responses and can be
used to improve energy efficiency for realistic machines. We also apply the
theory to discussions of piezoelectric, piezomagnetic, and magnetoelectric
energy conversion and their cooperative effects as well as biological energy
conversion. Studies in this work shed light on general properties of
optimization in energy applications and are helpful in guiding the search for
high performance energy materials and systems.

## Acknowledgements

I am greatly indebted to Rashmi C. Desai for a lot of discussions and
encouragements. I also wish to thank Yoseph Imry, Ora Entin-Wohlman, Sajeev
John, Christian van den Broeck, Baowen Li, Ming-Qi Weng, Gang Chen, Sidhartha
Goyal, Chushun Tian, and Daoyong Chen for illuminating discussions and
comments. This work was supported by the NSERC of Canada, the Canadian
Institute for Advanced Research, and the United States Department of Energy
Contract No. DE-FG02-10ER46754. Special thanks to CPTES at Tongji University
and IAS at Tsinghua University for hospitality where parts of this work were
completed.

##  Appendix A Positiveness of Onsager matrix and definition of inverse square
root of matrices

The second law of thermodynamics requires $\dot{S}_{tot}\geq 0$ for all
possible values of forces. That is

$\displaystyle T\dot{S}$ $\displaystyle=$ $\displaystyle\sum_{nk}{\cal
F}_{n}{\cal M}_{nk}{\cal F}_{k}\geq 0,\quad\forall\vec{{\cal F}},$ (69)
$\displaystyle=$ $\displaystyle\sum_{nk}{\cal F}_{n}{\cal M}_{nk}^{s}{\cal
F}_{k}\geq 0,\quad\forall\vec{{\cal F}},$

where ${\cal M}^{s}_{nk}=\frac{1}{2}\left({\cal M}_{nk}+{\cal M}_{kn}\right)$.
Since $\hat{{\cal M}}^{s}$ is a real symmetric matrix with dimension $N\times
N$, it has $N$ (real) eigenvectors and eigenvalues. For any vector $\vec{{\cal
F}}$ can be decomposed into the eigenvectors,

$\vec{{\cal F}}=\sum_{i=1}^{N}f_{i}\vec{{e}}_{i},$ (70)

with $\vec{{e}}_{i}$ corresponding to the eigenvalue $m_{i}$, then

$T\dot{S}=\sum_{i}m_{i}f_{i}^{2}.$ (71)

The above is positive definite only when $m_{i}\geq 0$ for all $i$. That is,
all eigenvalues of the matrix $\hat{{\cal M}}^{s}$ must be positive (In this
work we take the situation with $m_{i}=0$ as the limit that is approached from
the $m_{i}>0$ side, which has never been reached in realistic systems).

When $\hat{{\cal M}}_{II}$ is a real symmetric matrix there always exist an
orthogonal matrix $\hat{\Omega}_{I}$ such that $\hat{{\cal
M}}_{II}=\hat{\Omega}_{I}^{T}\hat{D}\hat{\Omega}_{I}$ where $\hat{D}$ is a
diagonal matrix. According to the second law of thermodynamics all the
eigenvalues of matrix $\hat{{\cal M}}_{II}$ are positive. Therefore all the
elements of the diagonal matrix $\hat{D}$ are positive. We can then define the
inverse square root of $\hat{{\cal M}}_{II}$ as

$\hat{{\cal
M}}_{II}^{-1/2}\equiv\hat{\Omega}_{I}^{T}\hat{D}^{-1/2}\hat{\Omega}_{I}.$ (72)

The inverse square root of $\hat{{\cal M}}_{OO}$ is defined similarly,

$\hat{{\cal
M}}_{OO}^{-1/2}\equiv\hat{\Omega}_{O}^{T}\hat{B}^{-1/2}\hat{\Omega}_{O}$ (73)

where $\hat{{\cal M}}_{OO}=\hat{\Omega}_{O}^{T}\hat{B}\hat{\Omega}_{O}$,
$\hat{\Omega}_{O}$ is orthogonal, and $\hat{B}$ is diagonal and positive.

##  Appendix B Prove that $\hat{\Lambda}$ is a positive matrix, $\lambda\leq
1$, and others

To simplify the proof, we perform an orthogonal transformation
$\hat{\Omega}_{O}\otimes\hat{\Omega}_{I}$ on the forces. To keep the currents
conjugated with forces, the same transformation must be exerted on the
currents. The transformation diagonalize the matrix $\hat{{\cal M}}_{II}$ and
$\hat{{\cal M}}_{OO}$. As both of them are positive matrix we can further
perform the following transformation

${\cal F}_{n}\to{\cal F}_{n}\sqrt{{\cal M}_{nn}},\quad{\cal
J}_{n}\to\left.{\cal J}_{n}\right/\sqrt{{\cal M}_{nn}}.$ (74)

This leads to

${\cal M}_{nk}\to\frac{{\cal M}_{nk}}{\sqrt{{\cal M}_{nn}{\cal M}_{kk}}}.$
(75)

After the above transformation the matrix $\hat{{\cal M}}_{II}$ and
$\hat{{\cal M}}_{OO}$ become identity matrix. Now for the real matrix
$\hat{{\cal M}}_{IO}$ there always exists a decomposition $\hat{{\cal
M}}_{IO}=\hat{\omega}_{I}^{T}\hat{C}\hat{\omega}_{O}$ where $\hat{\omega}_{I}$
and $\hat{\omega}_{O}$ are orthogonal matrices and $\hat{C}$ is a diagonal
matrix (but no need to be a square matrix) (see Ref. s-matrix ). Performing
the orthogonal transformation $\hat{\omega}_{O}\otimes\hat{\omega}_{I}$ on the
forces and currents and using Eq. (15), we obtain

$\hat{\Lambda}=\hat{{\cal M}}_{IO}\hat{{\cal M}}_{IO}^{T}=\hat{C}\hat{C}^{T}.$
(76)

Now $\hat{\Lambda}$ is a diagonal matrix with all diagonal elements greater
than or equal to zero. We thus proved that the coupling matrix $\Lambda$ is a
positive matrix. The largest eigenvalue of the coupling matrix $\hat{\Lambda}$
is also positive, i.e., $\lambda\geq 0$. Labeling the diagonal elements of
$\hat{C}$ as $y_{n}$ ($n=1,\dots N$ is integer if the dimension of the matrix
$\hat{C}$ is $N\times N^{\prime}$ with, say, $N\geq N^{\prime}$), the Onsager
matrix now becomes

${\cal M}=\left(\begin{array}[]{cccccccccc}1&&0&y_{1}&&0&&0\\\
&\ddots&&&\ddots&&\ddots&\\\ 0&&1&0&&y_{N}&&0\\\ y_{1}&&0&1&&0&&0\\\
&\ddots&&&\ddots&&\ddots&\\\ 0&&y_{N}&0&&1&&0\\\ &\ddots&&&\ddots&&\ddots&\\\
0&&0&0&&0&&1\\\ \end{array}\right).$ (77)

It follows from Eqs. (76) and (17) that

$\lambda={\rm max}\\{y_{n}^{2}\\}.$ (78)

According to the second law of thermodynamics all eigenvalues of the Onsager
matrix are positive, i.e.,

$1+y_{n}\geq 0,\quad 1-y_{n}\geq 0,\quad\forall n,$ (79)

according to Eq. (77). Therefore $0\leq\lambda\leq 1$ and the figure of merit
$\xi=\lambda/(1-\lambda)$ is positive definite.

At this point one can also show that when a machine is operating in a reverse
way, i.e., the output channels become input channels and vice versa. The
matrix $\hat{\Lambda}$ becomes $\hat{\Lambda}=\hat{C}^{T}\hat{C}$ which has
the same largest eigenvalue as before. In this way we proved that when a
machine is operated in a reverse way the degree of coupling $\lambda$ and the
figure of merit does not change.

Finally from Eq. (77) one can also directly show that $\hat{{\cal
M}}_{IO}\hat{{\cal M}}_{OO}^{-1}\hat{{\cal M}}_{OI}=\hat{C}\hat{C}^{T}$ is
positive matrix (i.e., all its eigenvalues are positive). Therefore the
largest eigenvalue of $\hat{{\cal M}}_{IO}\hat{{\cal M}}_{OO}^{-1}\hat{{\cal
M}}_{OI}$ is positive, i.e., $\Upsilon>0$.

##  Appendix C Thermodynamic bounds for systems with asymmetric Onsager matrix

We shall focus on the situation considered in the main text where $\hat{{\cal
M}}_{OI}=r\hat{{\cal M}}_{IO}^{T}$. For this situation one can perform the
same transformation as in previous section: symmetric matrices ${\cal
M}_{II}^{s}$ and ${\cal M}_{OO}^{s}$ can be diagonalized by orthogonal
transformations; after that performing the transformation (74) and another
orthogonal transformation ${\cal M}_{II}^{s}$ and ${\cal M}_{OO}^{s}$ become
identity matrices and ${\cal M}_{IO}\to\hat{C}$, ${\cal M}_{OI}\to
r\hat{C}^{T}$. The second law of thermodynamics requires that all eigenvalues
of $\hat{{\cal M}}^{s}$ are greater than or equal to zero. Therefore

$1-\frac{1}{2}(1+r)y_{n}\geq 0,\quad 1+\frac{1}{2}(1+r)y_{n}\geq
0,\quad\forall n.$ (80)

The degree of coupling is given by

$\lambda=r~{}{\rm max}\\{y_{n}^{2}\\}.$ (81)

Therefore

$0\leq\lambda\frac{(1+r)^{2}}{4r}\leq 1.$ (82)

The discussions in Sec. V can be generalized to the situation when ${\cal
M}_{OI}$ is not proportional to ${\cal M}_{IO}$ but they can still be
diagonalized simultaneously by an orthogonal transformation. The diagonal form
of $\hat{{\cal M}}_{IO}$ is $diag\\{y_{n}\\}$ while that of $\hat{{\cal
M}}_{OI}$ is $diag\\{r_{n}y_{n}\\}$. The optimal exergy efficiency is given by

$\phi_{max}={\rm
max}\left\\{r_{n}\frac{\sqrt{\xi_{n}+1}-1}{\sqrt{\xi_{n}+1}+1}\right\\},$ (83)

where

$\xi_{n}\equiv\frac{\lambda_{n}}{1-\lambda_{n}},\quad\lambda_{n}\equiv
r_{n}y_{n}^{2}.$ (84)

And the output power at maximum exergy efficiency is

$\dot{W}(\phi_{max})=\frac{1}{4}(1-r_{n}^{-2}\phi_{max}^{2})r_{n}\lambda_{n}\left(\vec{{\cal
F}}_{I}^{T}\hat{{\cal M}}_{II}\vec{{\cal F}}_{I}\right)$ (85)

for the $n$ that maximizes the efficiency. The maximum output power is

$\dot{W}_{max}=\frac{1}{4}{\rm
max}\left\\{r_{n}\lambda_{n}\right\\}\left(\vec{{\cal F}}_{I}^{T}\hat{{\cal
M}}_{II}\vec{{\cal F}}_{I}\right).$ (86)

The optimal exergy efficiency for maximum power is given by

$\phi_{opt}(\dot{W}_{max})=\frac{r_{n^{\prime}}\xi_{n^{\prime}}}{2(\xi_{n^{\prime}}+2)}$
(87)

for the $n^{\prime}$ that maximizes the output power (which may be different
from that maximizes the efficiency). As $n$ can be different from
$n^{\prime}$, the relationship between the two optimal efficiencies and powers
can be more complicated then we discussed in the main text.

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