The role of charge in thermodynamic uncertainty relations

Abstract

We demonstrate that the charge value of transport mechanisms heavily impacts the validity of thermodynamic uncertainty relations (TURs). Specifically, we show within the framework of full counting statistics, that the recently established quantum TUR can be violated by the presence of transport processes that carry more than one charge, like Andreev reflection processes in normal metal-superconductor junctions. We propose a modified quantum TUR, which incorporates the charge value and demonstrate that this charge-dependent quantum TUR can only be violated if the highest charge transport process exceeds this charge value. In particular, we establish that the breaking of the quantum TUR solely originates from the charge value of the highest charge transport process. Namely, our analytical considerations do not invoke the existence of superconductivity, and these considerations generally hold for non-interacting electronic transport which can be described by the scattering formalism.

Figures (2)

• Figure 1: Investigation of the charge dependent quantum TUR for the voltage-dependent current and noise in a normal metal-superconductor point contact with one channel of energy-independent transmission $\tau$ with temperature $k_{\rm B}T/\Delta(T)$ ($\Delta \equiv \Delta(T)$ is temperature dependent superconducting gap). (a) The minimal TUR-breaking coefficient for the quantum TUR [$\mathcal{N}=1$ in Eq. \ref{['nqTUR']}] for different temperatures $k_{\rm B}T$ and bare transmissions $\tau$. It is seen that the (charge-1) quantum TUR is broken for large temperatures and transmissions. (b) The minimal TUR-breaking coefficient for the charge-2 quantum TUR [$\mathcal{N}=2$ in Eq. \ref{['nqTUR']}] as in (a). It is seen that the coefficient is strictly positive.
• Figure 2: Investigation of the charge dependent quantum TUR for the voltage-dependent current and noise in a superconductor-superconductor point contact with one channel of energy-independent transmission $\tau$ with temperature $k_{\rm B}T/\Delta$ ($\Delta$ is superconducting gap). The minimal TUR-breaking coefficient $F_{\rm min}^{\mathcal{N}}$ for the charge-$\mathcal{N}$ quantum TUR [Eq. \ref{['nqTUR']}] for $\mathcal{N} = 1$ (quantum TUR) in panel (a), $\mathcal{N} = 2$ (charge-2 quantum TUR) in panel (b) and $\mathcal{N} = 5$ (charge-5 quantum TUR) in panel (c). It is seen that the larger the charge value $\mathcal{N}$, the smaller the region where the respective TUR is broken.