Type-I Aging in Nonequilibrium Systems

• Type-I aging is an irreversible, age-dependent degradation in nonequilibrium systems marked by increasing relaxation time with waiting time.
• It manifests in diverse domains such as glass-forming materials, cyclically driven amorphous matter, and biological tissues, each with distinct operational definitions.
• Underlying mechanisms include deepening trap effects, logarithmic dissipation, and record dynamics, challenging renewal models and unifying age-dependent nonstationarity.

Searching arXiv for the specified paper and closely related uses of “Type-I aging” across domains. Type-I aging is a domain-dependent label for irreversible, age-dependent slowing or degradation in nonequilibrium systems. In the most explicit recent usage, for glass-forming materials after a temperature jump, it denotes the regime in which the relaxation time increases with waiting time $t_w$ for both temperature increase and temperature decrease; under cyclic strain in amorphous matter, it denotes logarithmic decay of dissipation per cycle rather than rapid convergence to a periodic steady state; and in record-dynamics treatments of jammed matter, it denotes decelerating, history-dependent relaxation governed by a logarithmic clock rather than a renewal process (Ueno et al., 3 Sep 2025, Shohat et al., 10 Jun 2025, Boettcher et al., 2018). The available literature also applies closely related “type-I-like” ideas to tissue aging, population dynamics, social dynamics, holography, and detector degradation, but it does not present a single universal taxonomy across those fields.

1. Domain-specific definitions and diagnostic signatures

The most explicit definitions in the cited literature are operational rather than purely semantic. In glassy temperature-jump protocols, Type-I aging is defined by direction-independent slowing with age. In cyclically driven amorphous materials, it is identified by long-time logarithmic relaxation under repeated loading. In several biological and systems papers, the exact label is absent, but the discussed regimes are age-dependent, history-dependent, and nonstationary in a comparable sense (Ueno et al., 3 Sep 2025, Shohat et al., 10 Jun 2025, Privman et al., 2015).

Domain	Operational meaning of Type-I aging	Principal signature
Glass-forming materials after $T$-jump	Relaxation time grows with waiting time for both $T$-up and $T$-down	$\tau$ increases with $t_w$
Amorphous materials under cyclic strain	Persistent cyclic aging rather than rapid settling	$A_n \sim A_0 + \tilde C \log n$
Tissue-level statistical mechanics	Aging of tissues/organs composed of interacting cells	Loss of homeostasis and connectivity
Biological evolution models	Age-dependent increase in death and decrease in reproduction	Rising mortality, declining fertility
RPC detector studies	Performance degradation from accumulated irradiation and charge deposition	Efficiency loss and surface damage

A common misconception is that “Type-I aging” has a single standardized definition across condensed matter, biology, and engineering. The surveyed literature does not support that. A more accurate statement is that the term is anchored locally by each modeling tradition. What recurs across those traditions is broken time-translation invariance, age dependence of observables, and slow or irreversible evolution of an internal state.

2. Trap-dominated Type-I aging in glass-forming materials

In the free-energy-landscape treatment of aging after a temperature jump, structural relaxation is represented as a random walk of a representative point hopping among basins. The bath temperature is changed from $T_i$ to $T_f$ at $t=0$, and the landscape may respond with delay through an internal temperature

$T$0

with $T$1, $T$2, and in the simulations

$T$3

The jump rate from basin $T$4 is

$T$5

with exponentially distributed basin-depth variable

$T$6

and the occupation probability obeys

$T$7

The corresponding two-time observable is constructed from the self-intermediate scattering function,

$T$8

$T$9

with temporal relaxation time

$T$0

Within this framework, the trapping diffusion model without FEL delay, $T$1, gives Type-I aging: for both $T$2-down ($T$3) and $T$4-up ($T$5), the relaxation time increases with waiting time $T$6 (Ueno et al., 3 Sep 2025).

The physical reason is trap hierarchy. As time proceeds, the system statistically migrates toward deeper traps; escape rates become smaller; and later-time relaxation becomes slower. The waiting-time effect therefore does not depend on the sign of the temperature jump. The same paper notes that the SISF is only locally KWW-like,

$T$7

and that for KWW relaxation the temporal relaxation time increases with observation time. In the trap picture, this qualitative increase is not imposed phenomenologically but arises from the broad escape-rate distribution and progressive occupation of longer-lived basins. The resulting Type-I aging is thus the landscape’s trap-dominated “deepening with age” effect.

3. Distinction from Type-II aging and coexistence mechanisms

The same temperature-jump study introduces a sharp contrast with Type-II aging. To isolate delayed FEL response, all basin depths are set equal, $T$8, so that the only nontrivial mechanism is the time-dependent internal temperature $T$9. In that reduced model, relaxation time increases with $T$0 for $T$1-down but decreases with $T$2 for $T$3-up. Type-II aging is therefore defined by a sign change of the waiting-time dependence under reversal of the temperature-jump direction, whereas Type-I is direction-independent (Ueno et al., 3 Sep 2025).

When both trapping and delayed FEL response coexist, the overall behavior need not be purely monotone. For $T$4-up and intermediate $T$5, the temporal relaxation time $T$6 is non-monotonic: it first decreases, reaches a minimum, and then increases. The paper interprets that minimum as a crossover between early-time dynamics dominated by delayed FEL response and later-time dynamics dominated by trapping. If $T$7 is too small, the delayed-response effect is negligible and the minimum disappears; if $T$8 is too large, the effect is smeared out and the minimum also disappears. The short-time behavior of $T$9 is therefore a discriminator between trap-dominated Type-I aging, delayed-response Type-II aging, and mixed regimes.

This distinction is conceptually important because it separates two different sources of nonstationarity. One is statistical migration toward deeper traps; the other is delayed reconfiguration of the landscape itself. The paper further argues that the material time or internal clock and the fictive temperature introduced phenomenologically can be understood as concepts describing delayed FEL response. In the appendix, for the simplified trapping random walk with $\tau$0, the scaled time

$\tau$1

is identified with the material time, and the exact SISF becomes

$\tau$2

That identification links the aging clock directly to an evolving landscape-controlled timescale rather than to an external parametrization.

4. Cyclic Type-I aging and logarithmic dissipation

Under slow cyclic strain, Type-I aging is formulated in terms of repeated driving rather than a single quench. Experiments on crumpled Mylar sheets, amorphous steel wool bundles, and Nitinol wire in the martensitic phase compare static aging with cyclic aging by monitoring hysteresis loops $\tau$3 or $\tau$4, force or torque versus cycle number $\tau$5, hysteresis-loop area, and, in the crumpled sheet, acoustic emissions as a proxy for local instabilities. The central observation is

$\tau$6

where $\tau$7 is the strain- and history-dependent material creep function, and the dissipated energy per cycle

$\tau$8

shows the long-time trend

$\tau$9

Type-I aging under cyclic drive is thus marked by logarithmic decay of dissipation per cycle, not by an exponential approach to a steady state (Shohat et al., 10 Jun 2025).

The logarithm is not presented as a fit of convenience. Defining the cycle-to-cycle increment

$t_w$0

and assuming

$t_w$1

with $t_w$2 at long times, a generic Taylor expansion near the fixed point gives

$t_w$3

If $t_w$4, then asymptotically $t_w$5, and therefore $t_w$6. Because $t_w$7 is the loop integral of the force response, the same large-$t_w$8 logic yields $t_w$9.

The paper evaluates three mesoscopic models against these data. A broad distribution of independent relaxation rates,

$A_n \sim A_0 + \tilde C \log n$0

reproduces static logarithmic aging but under cyclic driving gives roughly $A_n \sim A_0 + \tilde C \log n$1 and does not generate a closed-loop creep function. A noisy interacting hysteron model with switching thresholds

$A_n \sim A_0 + \tilde C \log n$2

falls into one or a few limit cycles and gives $A_n \sim A_0 + \tilde C \log n$3 with $A_n \sim A_0 + \tilde C \log n$4. Only the structural random-network model with bistable elastic bonds,

$A_n \sim A_0 + \tilde C \log n$5

reproduces the full experimental package: static logarithmic aging, cyclic logarithmic decay of dissipation, closed-loop creep function $A_n \sim A_0 + \tilde C \log n$6, slow decay in the number of active instabilities, and persistent state-space exploration. The associated isoconfigurational-ensemble analysis of inter-copy distances

$A_n \sim A_0 + \tilde C \log n$7

is interpreted as showing a richer landscape structure in the network model than in the hysteron model.

5. Logarithmic clocks, quakes, and the non-renewal interpretation

A different condensed-matter formulation argues that aging in jammed disordered materials is better described as a log-Poisson process than as a renewal process. In this view, aging follows a rapid quench into a non-equilibrium jammed state and is governed by intermittent, irreversible “quakes” generated by record-breaking fluctuations. The renewal alternative is the continuous-time random walk with

$A_n \sim A_0 + \tilde C \log n$8

which yields

$A_n \sim A_0 + \tilde C \log n$9

By contrast, the log-Poisson description treats the number of quakes as Poisson distributed in logarithmic time, with inter-event logarithmic spacings

$T_i$0

that are approximately exponential,

$T_i$1

and with hyperbolic event rate

$T_i$2

Two-time observables subordinated to the quake count then collapse as functions of $T_i$3 (Boettcher et al., 2018).

The conceptual dispute here is not only statistical but mechanistic. Renewal models reset history after each jump. Record dynamics does not: each quake irreversibly reorganizes the system into a more stable metastable valley, so future dynamics depend on prior quakes. The paper argues that the observed exponential statistics of $T_i$4, the $T_i$5 event rate, and the extensivity of the quake rate with system size are incompatible with a renewal process except in singular or physically artificial limits. This places one influential interpretation of Type-I aging in direct tension with trap-based renewal pictures, even though both address slow, age-dependent relaxation in disordered matter.

6. Biological, tissue, ecological, and evolutionary reformulations

In tissue-level statistical mechanics, Type-I aging is explicitly defined as the aging of tissues or organs composed of interacting cells rather than the intracellular aging of a single cell. A dynamic lattice percolation model represents healthy, senescent, dead, and vacant sites on a regular lattice, with stochastic rules for division, senescence, apoptosis, and neighborhood effects. Key observables include the fraction of healthy live cells $T_i$6, the senescent fraction $T_i$7, and the fraction $T_i$8 of healthy live cells belonging to the largest connected cluster. In the simplified simulations, healthy cells die with probability $T_i$9, divide with probability $T_f$0, and otherwise remain unchanged; the reported parameter values include $T_f$1 and $T_f$2. A central result is that introducing long-lived senescent cells shortens the lifespan of regular healthy live cells, consistent with the reported observation that inhibition of senescence can extend lifespan (Privman et al., 2015).

Related biological papers do not uniformly employ the Type-I label, but they develop closely aligned notions of systemic, irreversible aging. In a tissue-engineered model of failing tissues, dense PEG-RGD hydrogel constructs of primary neonatal rat cardiac fibroblasts show strong density dependence: at 100K density, young tissues live about 25 days whereas pre-aged tissues live about 8–10 days, while at 10K and 1K the difference becomes much smaller or disappears. Mortality rate increases over time in dense tissues and is nearly constant in sparse tissues, supporting a cascading-failure interpretation of tissue aging driven by interdependence rather than by isolated cell-autonomous failure (Acun et al., 2017). In Misrepair-accumulation theory, aging is framed as accumulation of “incorrect reconstruction of an injured living structure,” with the strongest claim placed at tissue level: organismal aging is attributed primarily to irreversible changes in the spatial relationship between cells and extracellular matrix rather than to unrepaired damage persisting indefinitely (Wang-Michelitsch et al., 2015).

Evolutionary and ecological models introduce another reformulation. An in silico genome-evolution model treats biological aging as an age-dependent increase in the probability of death together with a decrease in reproductive capacity. Genomes contain age-specific survival modules $T_f$3, reproduction modules $T_f$4, and a neutral region $T_f$5; reproduction begins only at stage 16. Under constant resources, sexual populations evolve the canonical pattern of high early-life survival followed by rising mortality and declining reproduction after maturation, whereas asexual populations under constant resources do not develop the same age-dependent pattern (Šajina et al., 2016). A separate lattice model proposes senescence as adaptive under environmental change: mutation and environmental decline can make pruning of older individuals beneficial at lineage level, because it removes less well adapted survivors and accelerates adaptation of descendants (Martins, 2011). In many-variable interacting populations, aging arises through repeated near-extinctions in the model

$T_f$6

with long-time two-time correlations collapsing as

$T_f$7

There the authors emphasize that the system ages near unstable fixed points rather than marginal ones (Pirey et al., 2022).

7. Other technical usages and cross-disciplinary commonalities

Several additional fields use “aging” in ways that overlap structurally with Type-I aging while differing in ontology. In the noisy voter model, aging means that an agent becomes less prone to state change the longer it has stayed in its current state: the internal time $T_f$8 counts the time since the last change, pairwise interaction requires activation with probability $T_f$9, and state change resets age to $t=0$0. This aging mechanism transforms the model’s finite-size discontinuous transition into a continuous transition with mean-field critical point

$t=0$1

on the complete graph, with Ising-class critical behavior (Artime et al., 2018). In holography, time-dependent Lifshitz-type solutions in type IIB supergravity are argued to realize aging-like dynamics through broken time-translation invariance, dynamical scaling with $t=0$2, and slow nonexponential relaxation from a Lifshitz nonequilibrium regime toward an AdS equilibrium regime (Uzawa et al., 2013). In detector physics, Type-I aging in RPCs denotes performance degradation caused by accumulated irradiation and charge deposition, evidenced by efficiency loss in irradiated regions, sparking marks on Bakelite electrodes, and mitigation by Linseed oil coating or oil-embedded Bakelite (Lu et al., 2010).

These usages are not interchangeable, but several cross-disciplinary motifs recur. One is an internal clock: material time in the delayed-FEL picture, logarithmic time in record dynamics, internal age in voter dynamics, or accumulated dose in RPCs. Another is irreversibility: deeper-trap occupation, quakes, Misrepairs, tissue disorganization, cascading failures, or microscopic surface damage. A third is the failure of simple exponential equilibration. Depending on the domain, the observable consequence is growth of relaxation time with waiting time, $t=0$3 decay of cyclic dissipation, $t=0$4 quake rates, rising late-life mortality, continuous phase transitions induced by age dependence, or monotone detector-efficiency loss. The term “Type-I aging” therefore denotes a family resemblance of age-dependent nonequilibrium phenomena rather than a single transdisciplinary law.