Onsager Coefficients in Periodically Driven Systems

We evaluate the Onsager matrix for a system under time-periodic driving by considering all its Fourier components. By application of the second law, we prove that all the fluxes converge to zero in the limit of zero dissipation. Reversible efficiency can never be reached at finite power. The implication for an Onsager matrix, describing reduced fluxes, is that its determinant has to vanish. In the particular case of only two fluxes, the corresponding Onsager matrix becomes symmetric.

Figure 1

A two-level system is modulated by two work sources, 1 and 2, each operating on a different energy level of the system, namely, ${\epsilon }_1(t)={\epsilon }_1+\sin (2\pi t/\mathfrak{T})+\cos (2\pi t/\mathfrak{T})$ and ${\epsilon }_2(t)={\epsilon }_2+\sin (2\pi t/\mathfrak{T})+\gamma \cos (2\pi t/\mathfrak{T})$, respectively. The parameters are chosen such that $p_1^{eq}=3/4$, $p_2^{eq}=1/4$, $T=1$, $\mathfrak{T}=1$, and the nonzero eigenvalue of ${\mathbf{W}}^{(0)}$ is equal to $-1$. Main figure: $J_1$ (blue) and $J_2$ (dark green) are the powers of engines 1 and 2, respectively, and $\dot{S}$ (red) the entropy production. $\gamma >1$ and $0.16<\gamma <1$ correspond to different operational regimes with 2, respectively 1, functioning as driving and 1, respectively 2, as load. The corresponding efficiencies are $\eta =-F_1{J}_1/(F_2{J}_2)$ and $\eta =-F_2{J}_2/(F_1{J}_1)$. For $\gamma <0.16$ both fluxes are larger than 0, and the efficiency is given by $\eta =0$. Upper and lower inset: Onsager asymmetry $L_{12}/L_{21}$ and efficiency $\eta$ as a function of the parameter $\gamma$. Zero entropy production with reversible efficiency $\eta =1$ is reached for $\gamma =1$, in which case the Onsager matrix becomes symmetric.