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Active particles that translate chemical energy into self-propulsion canmaintain a far-from-equilibrium steady state and perform work. The entropyproduction measures how far from equilibrium such a particle system operatesand serves as a proxy for the work performed. Field theory offers a promisingroute to calculating entropy production, as it allows for many interactingparticles to be considered simultaneously. Approximate field theories obtainedby coarse-graining or smoothing that draw on additive noise can capturedensities and correlations well, but they generally ignore the microscopicparticle nature of the constituents, thereby producing spurious results for theentropy production. As an alternative we demonstrate how to use Doi-Pelitifield theories, which capture the microscopic dynamics, including reactions andinteractions with external and pair potentials. Such field theories are inprinciple exact, while offering a systematic approximation scheme, in the formof diagrammatics. We demonstrate how to construct them from a Fokker-Planckequation (FPE) of the single-particle dynamics and show how to calculateentropy production of active matter from first principles. This framework iseasily extended to include interaction. We use it to derive exact, compact andefficient general expressions for the entropy production for a vast range ofinteracting particle systems. These expressions are independent of theunderlying field theory and can be interpreted as the spatial average of thelocal entropy production. They are readily applicable to numerical andexperimental data. In general, any pair interaction draws at most on the threepoint, equal time density and an n-point interaction on the (2n-1)-pointdensity. We illustrate the technique in a number of exact, tractable examples,including some with pair-interaction.

Field theories of active particle systems and their entropy production