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The stochastic thermodynamics of a dilute, well-stirred mixture ofchemically-reacting species is built on the stochastic trajectories of reactionevents obtained from the Chemical Master Equation. However, when the molecularpopulations are large, the discrete Chemical Master Equation can beapproximated with a continuous diffusion process, like the Chemical LangevinEquation or Low Noise Approximation. In this paper, we investigate to whatextent these diffusion approximations inherit the stochastic thermodynamics ofthe Chemical Master Equation. We find that a stochastic-thermodynamicdescription is only valid at a detailed-balanced, equilibrium steady state.Away from equilibrium, where there is no consistent stochastic thermodynamics,we show that one can still use the diffusive solutions to approximate theunderlying thermodynamics of the Chemical Master Equation.

Diffusion approximations to the chemical master equation only have a consistent stochastic thermodynamics at chemical equilibrium