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The thermodynamic uncertainty relations (TURs) provide lower bounds on theentropy production (EP) of a system in terms of the statistical precision of anarbitrary current in that system. All conventional TURs derived so far haveconcerned a single physical system, differing from one another in whatproperties they require the system to have. However, many physical scenarios ofinterest involve multiple interacting systems, e.g. organelles within abiological cell. Here we show how to extend the conventional TURs to thosescenarios. A common feature of these extended versions of the TURs is that theybound the global EP, jointly generated by the set of interacting systems, interms of a weighted sum of the precisions of the local currents generatedwithin those systems -- plus an information-theoretic correction term.Importantly, these extended TURs can bound the global EP even when the globalsystem does not meet any of the requirements of the conventional TURs. Afterderiving these extended TURs we use them to obtain bounds that do not involvethe global EP, but instead relate the local EPs of the individual systems andthe statistical coupling among the currents generated within those systems. Wederive such bounds for both scalar-valued and vector-valued currents withineach system. We illustrate our results with numerical experiments.

Thermodynamic Uncertainty Relations for Multipartite Processes