  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 simple 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 in the linear-response regime. We also discuss systems with asymmetric
Onsager matrix (such as systems under magnetic field) where 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.
