Living systems sit far from equilibrium: they continuously exchange energy, matter, and information with their environment, and this exchange generates stochastic dynamics with memory. Nearby papers in the archive have already explored how collective fluctuations and chaos arise in biological systems, but mostly within the Markovian approximation, where the past is forgotten instantly. This article takes a step further: it places the living system inside the Keldysh formalism, where system and environment are treated on equal footing, and thereby brings colored noise and long memory into the picture. That makes it possible to move away from an idealized, history-free description and closer to real biology, where epigenetic marks and accumulated damage remember the past.
The authors use the Keldysh formalism of nonequilibrium field theory to describe an open living system interacting with a harmonic thermal reservoir. The environment is integrated out exactly, and its influence reduces to a nonlocal-in-time memory kernel that enters a generalized Langevin equation and a generalized master equation. To illustrate the entropy trajectory, they consider a one-dimensional chain of discrete states corresponding to life stages, with transitions only between neighbors. The memory kernel is chosen to be exponential, which in the high-temperature limit yields classical probabilistic dynamics; the generalized master equation is solved numerically with a rectangular external-driving protocol that mimics development and maturity.
The entropy of the system follows a bathtub shape: it falls during development, plateaus in maturity, and rises during aging, but environmental memory changes the details. At finite memory time, oscillations appear during development, along with slow drift and enhanced correlated fluctuations on the plateau, plus accelerated growth and a long tail during aging. The violation of the fluctuation-dissipation theorem, measured by a generalized ratio, reveals three regimes: suppressed fluctuations during development, nearly equilibrium behavior in maturity, and amplified low-frequency noise during aging. In addition, a damage variable is introduced whose dynamics exhibits critical slowing down near a critical point, interpreted as the approach to death.
The work stands out from nearby studies by building a single microscopic bridge between system-environment interaction, the memory kernel, generalized Langevin dynamics, entropy production, and violation of the fluctuation-dissipation theorem within one formalism. Where neighboring papers often focus on specific biological phenomena or Markovian models, this one offers a general framework that can accommodate colored noise, non-Gaussian fluctuations, and many-body interactions. It supplies a thermodynamic language for describing the life cycle and ties aging to a loss of dynamical stability, not merely to the accumulation of damage.
The formalism can be extended to realistic biochemical networks, spatially distributed systems, and active-matter models describing collective tissue behavior. It would be interesting to connect the predictions to measurable quantities, such as metabolic energy fluxes or fluctuation and response spectra, in order to test the theory against experimental data. Further development could incorporate quantum coherences and entanglement between system and environment, adding an information-theoretic dimension to the thermodynamic picture.
The results could influence the biophysics of aging, where thermodynamic constraints are becoming increasingly visible, as well as the stochastic thermodynamics of active systems. The proposed approach is useful for analyzing biological data in which memory and colored noise play a key role, for example in the dynamics of epigenetic clocks or cellular stress responses.
The next step will be to incorporate realistic biochemical networks and spatial structure into the model so that its predictions can be compared with experimental time series. It is also important to investigate how environmental heterogeneity and many-body interactions alter the critical properties of aging.
In the statistical physics of living systems, the question of how memory and long-range correlations affect thermodynamic irreversibility and phase transitions far from equilibrium remains open. This work directly links the unsolved problem of critical slowing down in biological systems to violation of the fluctuation-dissipation theorem, proposing a concrete mechanism through the memory kernel.
In the model, the entropy bathtub is not universal: as the memory time increases, the aging curve acquires a long tail, which means the organism retains traces of its past order for longer after decay sets in.