TDOS: Time-Scale Density of States

• TDOS is a framework that quantifies the distribution of collective relaxation rates, capturing both rapid and prolonged decay modes in complex systems.
• It employs linearization and field-theoretic methods to connect state-space geometry and potential curvature with temporal dynamics.
• TDOS informs our understanding of learning, memory formation, and quantum scattering by diagnosing emergent computation and collective excitation behaviors.

The time-scale density of states (TDOS) is a concept that organizes and quantifies the distribution of dynamical relaxation rates governing collective dynamics in high-dimensional stochastic systems. Unlike the conventional density of states, which enumerates energy eigenvalues, the TDOS encodes the spectrum of collective relaxation modes—specifically, the distribution of time scales ($\lambda_\alpha^{-1}$) over which perturbations to a high-dimensional system decay. This framework naturally arises both in field-theoretic descriptions of stochastic inference and learning processes and in quantum scattering problems as the spectral density associated with time-delay operators. As such, TDOS offers a physically grounded, model-independent diagnostic for the temporal organization of emergent computation, memory formation, and collective excitation dynamics across a variety of fields.

1. Stochastic Dynamical Origins and MSRJD Formalism

The TDOS is defined within the context of stochastic collective dynamics. The dynamics are specified by a Langevin equation in a high-dimensional system ($x(t) \in \mathbb{R}^N$): $\dot x = -G^{-1}(x)\nabla_x\Phi(x) + R(x) + \xi(t)$ where $\Phi(x)$ is the effective potential, $G(x)$ is a state-dependent Riemannian metric on the collective state space, $R(x)$ is a solenoidal (divergence-free) flow, and $\xi(t)$ is Gaussian white noise. The Martin–Siggia–Rose–Janssen–de Dominicis (MSRJD) path integral formalism is used to construct a field-theoretic description amenable to saddle-point and fluctuation (loop) analyses. The noiseless limit of the theory recovers deterministic inference as saddle-point trajectories, while fluctuation-induced corrections generate and organize the spectrum of collective modes (Chae, 15 Jan 2026).

2. Definition and Spectral Construction of TDOS

The TDOS emerges from linearizing the dynamics around a fixed-point or inference trajectory, yielding a stability (relaxation) matrix: $M = G^{-1}\nabla^2\Phi|_{x^*} - \nabla R|_{x^*}$ The (generally complex) eigenvalues $\mu_\alpha$ of $M$ possess real parts $\lambda_\alpha = \Re\,\mu_\alpha$, interpreted as collective relaxation rates. The TDOS is the normalized distribution of these rates: $$\rho(\lambda) = \frac{1}{N} \sum_{\alpha=1}^N \delta(\lambda-\lambda_\alpha)$$ Empirically or numerically, a smoothed form is often used. Alternatively, in the large-$N$ limit, the resolvent: $G_z(z) = \frac{1}{N} \mathrm{Tr}(zI - M)^{-1}$ provides a connection between TDOS and the imaginary part of the resolvent: $$\rho(\lambda) = \frac{1}{\pi} \lim_{\epsilon \to 0^+} \Im G_z(\lambda + i\epsilon)$$ This spectral measure captures both the multiplicity and the relative weight of slow and fast collective processes (Chae, 15 Jan 2026).

3. Structural Determinants and Typical Profiles

The TDOS is highly sensitive to both the geometry of state space (encoded in $G(x)$) and the curvature of the effective potential ($\nabla^2\Phi$). Specifically:

• Curvatures of $\nabla^2\Phi$: Steep potential directions yield large $\lambda$ (fast decay), while soft directions contribute to small $\lambda$ (slow modes).
• Metric anisotropy ($G$): Highly anisotropic metrics can compress certain directions (e.g., radial) and leave others (e.g., tangential) soft, leading to complex or bimodal TDOS profiles characteristic of systems with homeostatic shell dynamics.
• Non-conservative flows ($R$): Antisymmetric components broaden and skew the spectrum, though the principal effect on real parts is subleading at the quadratic (Gaussian) level.

Three canonical TDOS profiles are observed:

• Fast-decay: A sharp peak at high $\lambda$, little weight at small $\lambda$ (rapid relaxation).
• Critical/sloppy: Power-law tails $\rho(\lambda) \sim \lambda^{-\alpha}$ at small $\lambda$ (scale-free collective modes).
• Homeostatic shell: Bimodal distribution with peaks at both small and large $\lambda$ (coexistence of slow and fast collective directions).

The area under $\rho(\lambda)$ near $\lambda \to 0$ captures the density of long-lived, slow modes responsible for persistent memory and context integration (Chae, 15 Jan 2026).

4. TDOS in Learning, Homeostasis, and Emergence

Learning and homeostatic adaptation are interpreted as processes that reshape both the effective potential $\Phi(x)$ and state-space geometry $G(x)$, thereby reorganizing the TDOS. The spectral picture is as follows:

• Learning: Deepens selected minima in $\Phi(x)$, increasing curvatures and the associated $\lambda$, which stabilizes memory and inference along desired directions.
• Homeostasis: Dynamically tunes $G(x)$ to concentrate stability (high $\lambda$) in some collective coordinates, while preserving soft (slow) fluctuations in others. For instance, feedback homeostatic recurrent networks (FHRN) generate radial stiffness and tangential softness, yielding bimodal TDOS.
• Nonlinear fluctuation corrections: Loop-induced mass renormalization dynamically generates an infrared cutoff, further shaping $\rho(\lambda)$ near zero as per Dyson-type self-consistency relations.

This evolving TDOS is directly linked to the functional capacity for stable inference, noise resilience, and flexible context-dependent computation, offering a unified dynamical substrate for memory and cognition as emergent collective phenomena (Chae, 15 Jan 2026).

5. Physical and Computational Significance

The TDOS governs the system's entire temporal architecture:

• Inference and Power Spectra: Each collective mode $\psi_\alpha$ contributes a Lorentzian $(-i\omega+\lambda_\alpha)^{-1}$ to the causal response; thus, $\rho(\lambda)$ sets both the fluctuation spectrum and memory kernel.
• Memory and Integration: Modes with $\lambda \ll 1$ lead to persistent, long-lived excitations; the density of such modes determines both bandwidth and duration of memory integration.
• Cognitive Function: The emergent distribution of time scales, not the microstates or specific patterns, mediates cognitive computation. Scale-free TDOS enhance dynamic range and adaptability, while networks with sharply separated time scales can segregate memory and rapid sensory integration.

The TDOS framework thus enables a compact summary of a high-dimensional system's recurrent temporal organization, is computable from the linearized stability matrix, and is empirically accessible via power spectral observation (Chae, 15 Jan 2026).

6. Connections to Quantum TDOS and Time Delay

In quantum systems, particularly scattering settings, the TDOS manifests as the spectral density of time delays induced by disorder or boundaries. In such frameworks, the trace per unit volume of the time-delay operator $T(E) = -iS_E^* \partial_E S_E$ (with $S_E$ the on-shell scattering matrix) is equated to the density of surface states: $$\rho_{\text{surf}}(E) = \tau(E) = \frac{1}{2\pi i} \mathcal{T}_1 \operatorname{Tr}_2(S_E^* \partial_E S_E)$$ A generalized Levinson theorem establishes that the total time delay (integrated over energy) counts the total number of surface-localized states: $$\int_{E_-}^{E_+} \rho_{\rm surf}(E) dE = \int_{E_-}^{E_+} \tau(E) dE$$ Thus, in scattering theory, the TDOS quantifies the distribution of temporal trapping times (surface time scales), which is a spectral diagnostic analogous to the relaxation-mode TDOS of nonlinear stochastic field theory (Schulz-Baldes, 2013).

7. Summary Table: TDOS in Different Contexts

Context	TDOS: Definition and Role	Reference
Stochastic inference, learning	Density of collective relaxation rates $\lambda_\alpha$	(Chae, 15 Jan 2026)
Quantum scattering (surface states)	Density of time delays per unit energy/volume	(Schulz-Baldes, 2013)
Tunneling (1D Luttinger, Anderson)	Energy-resolved local DOS / time scales near junctions	(Roy et al., 2019, Burmistrov et al., 2013)

While the technical instantiations differ, all TDOS frameworks encode the overcomplete spectrum of relevant dynamical or quantum time scales, providing a unifying structural summary for both collective excitations and emergent macroscopic phenomena.