Degree of coupling and efficiency of energy converters far-from-equilibrium

Abstract

In this paper, we introduce a real symmetric and positive semi-definite matrix, which we call the non-equilibrium conductance matrix, and which generalizes the Onsager response matrix for a system in a non-equilibrium stationary state. We then express the thermodynamic efficiency in terms of the coefficients of this matrix using a parametrization similar to the one used near equilibrium. This framework, then valid arbitrarily far from equilibrium allows to set bounds on the thermodynamic efficiency by a universal function depending only on the degree of coupling between input and output currents. It also leads to new general power-efficiency trade-offs valid for macroscopic machines that are compared to trade-offs previously obtained from uncertainty relations. We illustrate our results on an unicycle heat to heat converter and on a discrete model of molecular motor.

An ordered set of power-efficiency trade-offs

“…Let us also denote F i the affinity and J i the corresponding physical current of the process i = 1, 2 of the machine, then the partial entropy production σ i is simply σ i = F i J i . As explained above, we relate the physical currents to the affinities by a generalization of the Onsager matrix, which we call the non-equilibrium conductance matrix G, in such a way that J X = Y G X,Y F Y [18]. We then introduce a new parametrization of this matrix in terms of the degree of coupling ξ = G 12 / √ G 11 G 22 × sign (F 1 F 2 ) and the relative intrinsic dissipation ϕ = (G 22 F 2 2 )/(G 22 F 2 1 ).…”

“…(1)-(2), in this optimization, one can treat G 11 as constant, because there are only two independent parameters in the conductance matrix, so they can be chosen to be ϕ and ξ. In order to obtain a different bound now in terms of the efficiency η rather than the degree of coupling, one uses the expression of ϕ as a function of η and ξ [18] :…”

“…From the mean probability currents and edge affinities, we define an edge conductanceḠ (x,y) ≡ J (x,y) /F (x,y) which is a diagonal matrix in the space of edges. In [18], we derived a unique expression of the non-equilibrium conductance matrix by combining edge resistances (inverse of edge conductances) in series, and cycle conductance in parallel, leading to…”

“…In this section, we illustrate the above power-efficiency bounds using two simple models of thermodynamic autonomous machines studied in Ref. [18] : a unicyclic thermal engine and an isothermal molecular motor that has several cycles. We first describe these two models and then discuss our main results.…”

“…(A.6) by the following contraction: Since we are now minimizing quadratic functions, we can find the minimizer exactly as in Ref. [18]:…”