Quantifying the flow of energy within and through fluctuating nanoscale systems poses a significant challenge to understanding microscopic biological machines. A common approach involves coarse-graining, which allows simplified description such systems. This has side effect inducing so-called hidden contributions (due sub-resolution dynamics) that complicate resulting thermodynamics. Here we de...

We study the elliptic system \begin{equation*} \begin{cases} -\Delta u_1 - \kappa_1u_1 = \mu_1|u_1|^{p-2}u_1 + \lambda\alpha|u_1|^{\alpha-2}|u_2|^\beta u_1, \\ u_2 \kappa_2u_2 \mu_2|u_2|^{p-2}u_2 \lambda\beta|u_1|^\alpha|u_2|^{\beta-2}u_2, u_1,u_2\in D^{1,2}_0(\Omega), \end{cases} \end{equation*} where $\Omega$ is a bounded domain in $\mathbb R^N$, $N\geq 3$, $\kappa_1,\kappa_2\in\mathbb R$, $\...