Non-Markovian Entropy Dynamics in Living Systems from the Keldysh Formalism

Living systems are open nonequilibrium systems that continuously exchange energy, matter, and information with their environments, leading to stochastic dynamics with memory and active fluctuations. In this study, we derive a microscopic non-Markovian description of entropy dynamics in living systems using the Keldysh functional formalism, providing a quantitative foundation for the entropy bathtub picture. The approach naturally incorporates colored environmental noise, memory-dependent dissipation, and many-body interactions, yielding generalized Langevin dynamics and non-Markovian master equations. Within this framework we derive an exact frequency-domain expression for the entropy production rate and show that violations of the fluctuation-dissipation relation provide a direct thermodynamic signature of active biological fluctuations. We further demonstrate that environmental memory enhances low-frequency fluctuations and entropy production, leading to critical slowing down near dynamical instability. These results provide a microscopic physical foundation for the entropy "bathtub" picture of living systems and connect entropy evolution with development, aging, and death in nonequilibrium dynamics.