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

Abstract

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 develop a non-Markovian theoretical framework for the entropy dynamics of living systems based on the Keldysh functional formalism and stochastic thermodynamics. 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.

Figures (4)

• Figure 1: Time evolution of system entropy $S_{\mathrm{sys}}(t)$ for different memory times $\tau_c = 0$ (blue), $0.5$ (brown), $1$ (green), and $2$ (red). Gray bands indicate fluctuation ranges for $\tau_c=0$ and $\tau_c=2$. The vertical dotted lines mark the boundaries between life stages (development, maturity, aging, death). A reversible perturbation (disease event) is shown for $\tau_c=1$ and $2$, illustrating non-Markovian recovery. The horizontal dashed line represents the equilibrium entropy of the environment.
• Figure 2: Frequency-dependent fluctuation-dissipation ratio $X_{AB}(\omega)$ for the exponential memory kernel with $\tau_c=2$, computed from Eq. \ref{['X_Teff']} using effective temperature models representing different life stages. The blue dashed curve (development) shows a dip below unity around $\omega \sim 1/\tau_c$, indicating suppressed fluctuations. The green solid curve (maturity) remains close to $X=1$ (the gray dotted line marks the equilibrium value) with small oscillations reflecting residual memory effects. The red dash-dotted curve (aging) exhibits a pronounced enhancement at low frequencies, signaling amplified noise and reduced dissipation efficiency.
• Figure 3: Time evolution of the damage variable $D(t)$ from Eq. (\ref{['damage']}) with a power‑law memory kernel ($\theta = 0.5$). Below the critical point ($\mu = 0.8$, blue), damage decays to zero. At the critical point ($\mu = 1.0$, green), damage grows extremely slowly, exhibiting critical slowing down. Above the critical point ($\mu = 1.1$, purple; $\mu = 1.2$, red), damage accelerates and eventually saturates at a finite steady state. The gray dashed line marks $D=0$.
• Figure 4: Comparison of system entropy $S_{\mathrm{sys}}(t)$ (blue) and system–environment mutual information $I_{S:E}(t)$ (red) during the lifespan. The vertical dotted lines separate the developmental, maturity, aging, and death phases. During development, $S_{\mathrm{sys}}$ decreases while $I_{S:E}$ increases, reflecting the buildup of structured correlations with the environment. In the mature phase both quantities stabilize, while during aging the mutual information begins to decay before a significant rise in entropy. In the death phase $I_{S:E}$ rapidly drops while $S_{\mathrm{sys}}$ approaches the equilibrium entropy (horizontal dashed line).