Universal property of the housekeeping entropy production

The entropy production of a nonequilibrium system with broken detailed balance is a random variable whose mean value is nonnegative. The housekeeping entropy production, which is a part of total entropy production, is associated with the heat dissipation in maintaining a nonequilibrium steady state. We derive a Langevin-type stochastic differential equation for the housekeeping entropy production. The equation allows us to define a housekeeping entropic time tau. Remarkably it turns out that the probability distribution of the housekeeping entropy production at a fixed value of tau is given by the Gaussian distribution regardless of system details. The Gaussian distribution is universal for any systems, whether in the steady state or in the transient state and whether they are driven by time-independent or time-dependent driving forces. We demonstrate the universal distribution numerically for model systems.