# Fundamental Relation Between Entropy Production and Heat Current

## Abstract

We investigate the fundamental relation between entropy production rate and the speed of energy exchange between a system and baths in classical Markov processes. We establish the fact that quick energy exchange inevitably induces large entropy production in a quantitative form. More specifically, we prove two inequalities on instantaneous quantities: one is applicable to general Markov processes induced by heat baths, and the other is applicable only to systems with the local detailed-balance condition but is stronger than the former one. We demonstrate the physical meaning of our result by applying to some specific setups. In particular, we show that our inequality is tight in the linear response regime.

## Keywords

Heat engines Finite time thermodynamics Stochastic thermodynamics## Notes

### Acknowledgements

We are grateful to Hal Tasaki for fruitful discussion. He was a co-author in the joint work [41], and contributed to deriving several relations. NS was supported by Grant-in-Aid for JSPS Fellows JP17J00393. KS was supported by JSPS Grants-in-Aid for Scientific Research (No. JP25103003, JP16H02211 and JP17K05587).

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