Evolution Factor in Dynamic Systems

• Evolution factor is a context-dependent parameter that quantifies dynamic changes, representing exponents, ratios, or state variables in various fields.
• In astrophysics, it appears as the redshift exponent in blazar Doppler evolution and the Lorentz factor in GRB jets, derived from spectral cutoffs and Bayesian methods.
• Other applications include gas depletion in galaxy clusters, AGN covering factors, tokamak safety profiles, industrial dynamics, and ecological models, highlighting methodological challenges.

“Evolution factor” is a context-dependent term rather than a single standardized quantity. In the literature surveyed here, it denotes either a parameter that quantifies how an observable changes with time or redshift, a dynamical variable whose profile evolves under a governing equation, or a mechanism that selects among competing states. Astrophysical uses include the redshift exponent in blazar Doppler-factor evolution, the Lorentz-factor history of gamma-ray-burst jets, the gas depletion factor in galaxy clusters, the covering factor of dusty AGN tori, and cosmological scale factors; other uses include the safety-factor profile in tokamak sawteeth, the order and control parameters of industrial evolution, energy-limited natural selection, and evolutionary refinement of latent-factor models (Yan et al., 26 Jan 2025, Lin et al., 2019, Li et al., 2022, Holanda et al., 2021, Gu, 2013, Rałowski et al., 2023, Chakraborty et al., 2017, Li et al., 2014, Abadi et al., 2020, Chen et al., 2022).

1. Terminological scope and formal meanings

Across the cited works, the term is attached to different mathematical objects. In some cases it is an explicit fitted exponent or redshift slope; in others it is the evolving physical quantity itself. This usage pattern is especially clear in astrophysics, where the same phrase can refer to the exponent $m$ in $\delta \propto (1+z)^m$, the slope $g_1$ in $g(z)=g_0(1+g_1 z)$, or the empirical ratio $C_f=L_{\rm IR}/L_{\rm bol}$ or $L_{\rm IR}/L_{\rm agn}$ whose possible redshift dependence is then tested (Yan et al., 26 Jan 2025, Holanda et al., 2021, Gu, 2013, Rałowski et al., 2023).

Domain	Quantity called or serving as an evolution factor	Formal expression
Blazar jets	Redshift exponent of Doppler-factor evolution	$\delta \propto (1+z)^m$
Galaxy clusters	Gas depletion factor with redshift slope	$g(z)=g_0(1+g_1 z)$
Quasars/AGNs	Dusty torus covering factor	$C_f=L_{\rm IR}/L_{\rm bol}$, $\mathrm{CF}=L_{\rm IR}/L_{\rm agn}$
GRB jets	Bulk Lorentz factor evolution	$\delta \propto (1+z)^m$0 inferred from $\delta \propto (1+z)^m$1
Tokamak sawteeth	Safety-factor profile	$\delta \propto (1+z)^m$2 or $\delta \propto (1+z)^m$3
Industry dynamics	Order/control variables of system evolution	$\delta \propto (1+z)^m$4 in the Haken model
Evolutionary ecology	Energy-availability constraint under competition	$\delta \propto (1+z)^m$5
Cosmology	Scale factor governing global evolution	$\delta \propto (1+z)^m$6 in several ansätze

A recurring methodological distinction is between a quantity that evolves and a parameter that measures the strength of that evolution. The blazar paper explicitly defines the “Evolution Factor” as the exponent $\delta \propto (1+z)^m$7 in the cosmological scaling of the Doppler factor (Yan et al., 26 Jan 2025). By contrast, the GRB papers focus on the Lorentz factor $\delta \propto (1+z)^m$8 itself and reconstruct its time dependence from high-energy cutoffs (Lin et al., 2019, Li et al., 2022). In cluster and AGN studies, the evolving object is often a phenomenological ratio—depletion factor or covering factor—whose constancy versus redshift is the scientific issue (Holanda et al., 2021, Gu, 2013, Rałowski et al., 2023).

2. Relativistic jets: Lorentz and Doppler factors

In gamma-ray bursts, the evolution factor is tied to the bulk Lorentz factor of the emitting shell. For GRB 160625B, a clear, smooth high-energy spectral cutoff is detected in the first pulse of the second emission episode, at observer times $\delta \propto (1+z)^m$9–$g_1$0 s, and is fit by a Band spectrum multiplied by an exponential cutoff. Interpreting the cutoff as $g_1$1 opacity with $g_1$2, the analysis derives $g_1$3 and $g_1$4. The central result is that the radiation location increases with time while $g_1$5 remains approximately constant, with the best-fit parametrization $g_1$6 giving $g_1$7, $g_1$8, and $g_1$9, effectively a coasting solution (Lin et al., 2019).

A broader pulse-resolved study extends this logic to nine \textit{Fermi} bursts—090323, 090926A, 100724B, 120226A, 130821A, 160509A, 160625B, 170405A, and 180720B. Out of 70 identified pulses, 34 show high-energy spectral cutoffs and yield $g(z)=g_0(1+g_1 z)$0 estimates spanning roughly $g(z)=g_0(1+g_1 z)$1 to $g(z)=g_0(1+g_1 z)$2. The main observational result is not a universal monotonic law but burst-dependent variability: within an individual GRB, $g(z)=g_0(1+g_1 z)$3 generally fluctuates within a certain range and without a general trend. Three cases—GRBs 130821A, 160509A, and 160625B—show an apparent increase with time across successive pulses (Li et al., 2022).

For blazars, the relevant evolution factor is not the Lorentz factor history within one event but the redshift exponent $g(z)=g_0(1+g_1 z)$4 in the Doppler-factor scaling

$g(z)=g_0(1+g_1 z)$5

Using 141 \textit{Fermi}-detected bright $g(z)=g_0(1+g_1 z)$6-ray blazars and characteristic energies from log-parabolic fits, the measured relation is $g(z)=g_0(1+g_1 z)$7, corresponding to $g(z)=g_0(1+g_1 z)$8. Using optical damping timescales from Gaussian-process/DRW modeling of 89 blazars, the result is $g(z)=g_0(1+g_1 z)$9, i.e. $C_f=L_{\rm IR}/L_{\rm bol}$0, with larger scatter. The paper stresses that these are sample-average indices and further notes that low-luminosity sources show stronger evolution, with $C_f=L_{\rm IR}/L_{\rm bol}$1 in the low-luminosity $C_f=L_{\rm IR}/L_{\rm bol}$2-ray subsample (Yan et al., 26 Jan 2025).

These jet studies collectively separate three notions that are often conflated: instantaneous shell dynamics in a single GRB, pulse-to-pulse evolution within one prompt-emission episode, and population-level cosmological evolution across redshift. A plausible implication is that “evolution factor” in relativistic outflow studies should be read only relative to the temporal or ensemble scale being analyzed.

3. Redshift-dependent ratios: depletion and covering factors

In galaxy-cluster cosmology, the evolution factor is the gas depletion factor $C_f=L_{\rm IR}/L_{\rm bol}$3, which calibrates how the cluster gas mass fraction is depleted relative to the cosmic baryon fraction. The paper writes

$C_f=L_{\rm IR}/L_{\rm bol}$4

and combines 40 Chandra $C_f=L_{\rm IR}/L_{\rm bol}$5 measurements at $C_f=L_{\rm IR}/L_{\rm bol}$6 with strong-lensing systems from SLACS, BELLS, and SL2S. The reported constraints are $C_f=L_{\rm IR}/L_{\rm bol}$7 for low-mass lenses, $C_f=L_{\rm IR}/L_{\rm bol}$8 for intermediate-mass lenses, and $C_f=L_{\rm IR}/L_{\rm bol}$9 for high-mass lenses. The error-weighted average is $L_{\rm IR}/L_{\rm agn}$0, interpreted as a $L_{\rm IR}/L_{\rm agn}$1 detection of a mildly decreasing depletion factor with redshift (Holanda et al., 2021).

In quasar studies, the analogous observable is the dusty torus covering factor. One large SDSS-based sample of 5,996 quasars defines

$L_{\rm IR}/L_{\rm agn}$2

with $L_{\rm IR}/L_{\rm agn}$3 integrated over rest-frame $L_{\rm IR}/L_{\rm agn}$4 and $L_{\rm IR}/L_{\rm agn}$5 over $L_{\rm IR}/L_{\rm agn}$6. Both the high-$L_{\rm IR}/L_{\rm agn}$7 ($L_{\rm IR}/L_{\rm agn}$8) and low-$L_{\rm IR}/L_{\rm agn}$9 ($\delta \propto (1+z)^m$0) samples show strong anti-correlations between $\delta \propto (1+z)^m$1 and $\delta \propto (1+z)^m$2, but the fitted tracks differ. In the overlapping luminosity range $\delta \propto (1+z)^m$3, the median values are $\delta \propto (1+z)^m$4 for 778 low-$\delta \propto (1+z)^m$5 quasars and $\delta \propto (1+z)^m$6 for 966 high-$\delta \propto (1+z)^m$7 quasars, with KS statistic $\delta \propto (1+z)^m$8 and $\delta \propto (1+z)^m$9, implying systematically larger covering factors at high redshift (Gu, 2013).

A later study re-examines this issue with a different photometric treatment and stronger emphasis on selection effects. It defines

$g(z)=g_0(1+g_1 z)$0

uses cross-matched SDSS, GALEX, UKIDSS, WISE, and SPITZER data, and argues that WISE W4 is problematic because of calibration issues, redleak, and low-SNR behavior. With SPITZER MIPS $g(z)=g_0(1+g_1 z)$1 data, the reported medians are $g(z)=g_0(1+g_1 z)$2 and $g(z)=g_0(1+g_1 z)$3, while the Efron–Petrosian test still finds strong luminosity evolution in both $g(z)=g_0(1+g_1 z)$4 and $g(z)=g_0(1+g_1 z)$5. In the high-$g(z)=g_0(1+g_1 z)$6 SPITZER-selected comparison, the one-dimensional KS test gives $g(z)=g_0(1+g_1 z)$7, and the study concludes that no compelling evidence for genuine covering-factor evolution remains once selection effects and mid-IR systematics are controlled (Rałowski et al., 2023).

Taken together, these papers make covering-factor evolution a live controversy rather than a settled fact. The earlier result attributes the effect to genuinely larger obscuring structures at high $g(z)=g_0(1+g_1 z)$8, whereas the later analysis argues that much of the apparent trend can be reproduced by flux limits, mid-IR calibration systematics, sample imbalance, and contamination from additional dust components (Gu, 2013, Rałowski et al., 2023). This suggests that, for luminosity-ratio evolution factors, robustness against truncation and photometric systematics is as important as the nominal redshift trend itself.

4. Cosmological scale factors as engines of evolution

In cosmology, the relevant “factor” is frequently the scale factor $g(z)=g_0(1+g_1 z)$9, which directly encodes the expansion history. One model adopts a Gaussian-type ansatz,

$C_f=L_{\rm IR}/L_{\rm bol}$0

with $C_f=L_{\rm IR}/L_{\rm bol}$1, $C_f=L_{\rm IR}/L_{\rm bol}$2, and $C_f=L_{\rm IR}/L_{\rm bol}$3. This yields $C_f=L_{\rm IR}/L_{\rm bol}$4 as $C_f=L_{\rm IR}/L_{\rm bol}$5, a maximum $C_f=L_{\rm IR}/L_{\rm bol}$6, and a Hubble parameter

$C_f=L_{\rm IR}/L_{\rm bol}$7

The model therefore describes expansion for $C_f=L_{\rm IR}/L_{\rm bol}$8, contraction for $C_f=L_{\rm IR}/L_{\rm bol}$9, asymptotically static phases in the infinite past and future, and no type I, II, or III strong singularities (Chakraborty et al., 2017).

A different proposal studies fast fluctuations around a slow FLRW background,

$\mathrm{CF}=L_{\rm IR}/L_{\rm agn}$0

and derives modified averaged Friedmann equations,

$\mathrm{CF}=L_{\rm IR}/L_{\rm agn}$1

$\mathrm{CF}=L_{\rm IR}/L_{\rm agn}$2

The added positive terms arise from averaging the kinetic energy of rapid oscillations and are interpreted as producing late-time accelerated expansion and a Kapitza-like modification of the effective universe potential (Smolyaninov, 2021).

A more recent unified construction proposes

$\mathrm{CF}=L_{\rm IR}/L_{\rm agn}$3

with $\mathrm{CF}=L_{\rm IR}/L_{\rm agn}$4, $\mathrm{CF}=L_{\rm IR}/L_{\rm agn}$5, and $\mathrm{CF}=L_{\rm IR}/L_{\rm agn}$6 chosen so as to interpolate smoothly between inflation, radiation domination, matter domination, and late-time acceleration. In the stated limits, the model recovers $\mathrm{CF}=L_{\rm IR}/L_{\rm agn}$7, $\mathrm{CF}=L_{\rm IR}/L_{\rm agn}$8, $\mathrm{CF}=L_{\rm IR}/L_{\rm agn}$9, and $\delta \propto (1+z)^m$00. Sigmoid functions are used to smooth the transitions, and the paper interprets $\delta \propto (1+z)^m$01 as a proxy for $\delta \propto (1+z)^m$02 in a brane-world setting, while identifying Bose–Einstein-like structures in the effective Hubble parameter through terms such as $\delta \propto (1+z)^m$03 and $\delta \propto (1+z)^m$04 (Safarzadeh-Maleki, 30 May 2025).

These models all treat the scale factor as the primary carrier of cosmic evolution, but they operationalize that role differently: through a bounded non-singular history, through fast-fluctuation corrections to standard dynamics, or through a single analytic interpolation across all major epochs. A plausible implication is that “evolution factor” in cosmology often collapses into the choice of dynamical ansatz itself.

5. Dynamical-profile evolution and coordinate dependence

In tokamak sawteeth, the relevant evolving factor is the safety-factor profile $\delta \propto (1+z)^m$05. The paper contrasts the traditional definition,

$\delta \propto (1+z)^m$06

with a new definition based on the twisted magnetic axis,

$\delta \propto (1+z)^m$07

The central claim is that the old definition becomes inconsistent with Poincaré plots once the magnetic axis is twisted by the $\delta \propto (1+z)^m$08 kink instability, because it measures poloidal winding relative to the original axis rather than the actual helical core structure (Zhang et al., 2020).

The physical consequence is that the apparent evolution of $\delta \propto (1+z)^m$09 depends strongly on the coordinate choice. For normal sawteeth, the new definition finds $\delta \propto (1+z)^m$10 initially, nearly unchanged for most of the cycle, and jumping to $\delta \propto (1+z)^m$11 only near the end of reconnection. In the non-axisymmetric stationary state, the new definition keeps $\delta \propto (1+z)^m$12 below unity, with examples $\delta \propto (1+z)^m$13 and $\delta \propto (1+z)^m$14, while in incomplete reconnection it remains fixed at the initial value $\delta \propto (1+z)^m$15 throughout. The old definition, by contrast, tends to report premature flattening to $\delta \propto (1+z)^m$16 near the core (Zhang et al., 2020).

This case is methodologically significant because it shows that an apparent evolution factor can be an artifact of the reference frame used to define it. The same caution is visible, in different form, in covering-factor studies that distinguish genuine redshift evolution from survey selection effects (Rałowski et al., 2023).

6. Evolutionary drivers in industry, biology, and optimization

In industrial dynamics, the evolution factor is framed through synergetics. Using self-organization theory and the Haken model, the ICT-industry study introduces two endogenous variables obeying

$\delta \propto (1+z)^m$17

with discrete-time econometric forms used for estimation. Technology progress is operationalized as TPL via the DEA-based Malmquist index, and industry convergence as TCR via patent-transfer matrices. Over 2002–2012, TPL rises from $\delta \propto (1+z)^m$18 in 2002h1 to $\delta \propto (1+z)^m$19 in 2012h2, while TCR rises from $\delta \propto (1+z)^m$20 to $\delta \propto (1+z)^m$21, peaking at $\delta \propto (1+z)^m$22 in 2012h1. The supported configuration is that TPL is the order variable and TCR the control variable, with estimated equations

$\delta \propto (1+z)^m$23

$\delta \propto (1+z)^m$24

The study therefore concludes that technology progress dominates industry evolution and that industry convergence is largely its outcome (Li et al., 2014).

In evolutionary ecology, the organizing factor is energy availability under competition for a renewable resource. The model places asexual individuals on an $\delta \propto (1+z)^m$25 grid with logistic resource growth

$\delta \propto (1+z)^m$26

individual energy balance

$\delta \propto (1+z)^m$27

and genotype $\delta \propto (1+z)^m$28. Selection arises because traits alter energy intake, metabolic cost, movement cost, and growth cost. Reproduction requires adulthood and sustained energy above $\delta \propto (1+z)^m$29 for $\delta \propto (1+z)^m$30 steps, while death occurs through starvation or an age-dependent probability with intrinsic lifespan $\delta \propto (1+z)^m$31 steps. In practice, starvation dominates, and the realized lifetime is around $\delta \propto (1+z)^m$32 steps. From these rules, the simulations exhibit speciation, competitive exclusion, punctuated equilibrium, and altruistic behavior emerging from selfish rules (Abadi et al., 2020).

In high-dimensional sparse-data modeling, evolutionary dynamics are repurposed algorithmically rather than biologically. The SGDE-PLFA model first learns latent factors with PLFA and then refines them through Sequential-Group Differential Evolution, optimizing row groups $\delta \propto (1+z)^m$33 and column groups $\delta \propto (1+z)^m$34 without crossover. On four datasets—ML10M, ExtEpinion, Flixter, and Douban—the method reports the best predictive performance on RMSE and MAE; for example, on D3 it achieves RMSE $\delta \propto (1+z)^m$35. The Friedman statistic is $\delta \propto (1+z)^m$36, above the critical value $\delta \propto (1+z)^m$37 at significance level $\delta \propto (1+z)^m$38, and reported runtimes include $\delta \propto (1+z)^m$39 s on D2 and $\delta \propto (1+z)^m$40 s on D3 (Chen et al., 2022).

These examples broaden the meaning of evolution factor beyond physics. In one case it is an order parameter that dominates macroscopic structural change; in another it is the resource-mediated energy constraint that turns mutation into selection; in a third it is not a measured factor at all but an evolutionary search mechanism for latent-factor refinement. The shared theme is that evolution is represented through low-dimensional control variables or rules that compress a more complex underlying system.

7. Comparative interpretation and recurrent issues

Several recurring themes cut across these otherwise unrelated literatures. First, many evolution factors are not directly observed; they are inferred through forward models. GRB Lorentz factors are derived from pair-opacity cutoffs via $\delta \propto (1+z)^m$41 (Lin et al., 2019, Li et al., 2022). Blazar Doppler evolution is inferred from characteristic $\delta \propto (1+z)^m$42-ray energies and optical DRW timescales via hierarchical Bayesian linear regression (Yan et al., 26 Jan 2025). Cluster depletion evolution is obtained by combining reconstructed $\delta \propto (1+z)^m$43, strong-lensing distance ratios, CDDR, GPR, and MCMC (Holanda et al., 2021). Industry evolution is inferred from simultaneous-equation estimation under Haken-model restrictions (Li et al., 2014).

Second, apparent evolution can be produced or erased by definitional and selection choices. The tokamak safety-factor study shows that a change in reference axis alters whether a $\delta \propto (1+z)^m$44 core is inferred at all (Zhang et al., 2020). The AGN covering-factor debate shows that WISE W4 systematics, SPITZER substitution, SED-integration method, and truncated-sample treatment materially affect whether redshift evolution appears significant (Gu, 2013, Rałowski et al., 2023). This suggests that, in encyclopedia usage, an evolution factor should not be understood independently of the measurement protocol that defines it.

Third, some studies distinguish explicitly between average and object-specific evolution. The blazar Doppler-factor index $\delta \propto (1+z)^m$45 is an average over the full sample and “not a universal constant for every blazar” (Yan et al., 26 Jan 2025). The nine-GRB study finds no general monotonic prompt-emission $\delta \propto (1+z)^m$46 trend across bursts, even though specific bursts can show increasing $\delta \propto (1+z)^m$47 with time (Li et al., 2022). The same tension appears in AGN work, where a broad high-$\delta \propto (1+z)^m$48/low-$\delta \propto (1+z)^m$49 contrast can disappear after matching in $\delta \propto (1+z)^m$50, $\delta \propto (1+z)^m$51, or $\delta \propto (1+z)^m$52 (Rałowski et al., 2023).

In this comparative sense, “evolution factor” is best understood as a family of domain-specific constructs for compressing dynamical change into a parameter, profile, or governing mechanism. Sometimes it is an exponent such as $\delta \propto (1+z)^m$53 or $\delta \propto (1+z)^m$54; sometimes it is a physical ratio such as $\delta \propto (1+z)^m$55; sometimes it is a directly evolving state variable such as $\delta \propto (1+z)^m$56, $\delta \propto (1+z)^m$57, or $\delta \propto (1+z)^m$58; and sometimes it is the causal structure behind the evolution itself, as with technology progress or energy-limited competition. What unifies these usages is not a common formula but a common function: to make evolution inferable, comparable, and mathematically tractable.