# Multitasking: what are the limits?

December 2, 2020

On the “first day of Christmas” (or at least the Christmas month), we had a paper out in Physical Review Research: https://journals.aps.org/prresearch/pdf/10.1103/PhysRevResearch.2.043302. I didn’t have time to write about when it was first posted on arXiv, but at least I get around to it now… It is about how to quantify the efficiency of thermal machines which perform more than one useful task at the same time.

Useful tasks, such as generating electric power, cooling stuff down in a refrigerator, or heating stuff in an oven, are all performed by machines which exchange energy with some energetic reservoirs. Classical thermodynamics places strict bounds on the efficiency of machines which consume heat to either produce work, cool, or heat. However, more complex machines which perform multiple tasks at once are also possible. In this work, we introduce efficiencies applicable to such hybrid machines and establish their thermodynamic limits.

A hybrid machine might, for example, simultaneously cool by removing heat from a cold reservoir while also producing work using heat from a hot reservoir — that is, simultaneously be a fridge and a power station. More generally, we use a very broad approach where the machines are allowed to exchange not only energy with the reservoirs, but also other conserved quantities such as particles, and to perform any number of tasks involving these quantities. This enables complex situations where one conserved quantity can be exchanged for another to favour one or the other task. The fundamental thermodynamic limits on performance are then governed by an generalised variant of the 2nd law of thermodynamics (that’s the one which says that disorder can never decrease and which gives us the arrow of time). Despite the complexity, by starting from the 2nd law, we are able to give simple expressions for the overall efficiency of general hybrid machines, as well as the efficiency of each individual tasks performed by the machines.

We also study the possibility to build hybrid machines in practise. We show that a minimal machine, with two conserved quantities and up to three useful tasks, can be implemented in tiny electronic circuits coupled to quantum dots – artifical, atom-like structures embedded in the electronics, where the energy is quantised. The device uses energy an particles (electrons) as the conserved quantities, and can cool, heat, and produce electrical work. It should be possible to realise such a setup with current technology, so our results can be tested in experiment. The also provide new insight into thermodynamics – in particular in the quantum regime – and could potentially be used to guide the design of new kinds of thermal machines.

Another nice thing about this work is that it resulted from a very international collaboration with colleagues at the Austrian Academy of Sciences, the Universidad Autónoma in Madrid, Spain, ETH Zürich and the University of Geneva in Switzerland, and the University of Lund in Sweden. Various subsets of us met and discussed these ideas in various workshops and conferences and scientific visits. But I am not sure we were ever all in one place at the same time. Were we? The internet is a wondrous thing…