Human population dynamics as a Bayesian inverse transport problem

Chong Qi

REVIEW 4 major objections 5 minor 50 references

Human population dynamics as a Bayesian inverse transport problem

T0 review · 4 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read A population's age structure can be inferred and forecast as a Bayesian inverse transport problem, with neural networks learning fertility and mortality inside the exact conservation equation so every posterior sample conserves mass.

desk verdict A novel BNN-in-PDE method with real demographic applications, but the 'exact' mass-conservation claim is undercut by omitted migration; still solid enough to referee. read the letter →

arxiv 2607.13171 v1 pith:QXU25SHJ submitted 2026-07-14 physics.soc-ph nlin.CDphysics.data-an

Chong Qi This is my paper · ORCID

classification physics.soc-phnlin.CDphysics.data-an

keywords demographyage-structuredpopulationstransportequationBayesianinverseproblemneuralnetworksmassconservationpopulationforecastingnon-equilibriumdynamics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper is trying to establish that the age-time density of a human population obeys an exact first-order transport equation—age acts as a spatial coordinate, births as a boundary source, mortality as an internal sink—and that the unknown, time-varying fertility and mortality laws can be learned from sparse, noisy demographic observations by representing them as Bayesian neural networks and sampling their posterior. If correct, this unifies historical reconstruction, missing-data imputation, and uncertainty forecasting in a single framework where every posterior sample automatically respects mass conservation and cohort advection. Applied to China, Japan, and South Korea, the framework reconstructs the full age grid over 75 years for China from aggregate totals and three coarse censuses, and over five decades for Japan and South Korea, then projects to 2070, showing a structural contraction in which South Korea's elderly share reaches about half of the population and its potential support ratio falls below one. The paper also introduces thermodynamic-style diagnostics—a total demographic entropy and a divergence from the stable age distribution—as quantitative markers of how far these systems are from demographic equilibrium.

What carries the argument

The load-bearing object is the age-time transport PDE ∂tρ + ∂aρ = −μρ, treated as a hard equality inside a Bayesian inverse problem rather than as a regression target. The constitutive relations—the fertility kernel f(a,t) and the survival probability s(a,t)—are outputs of Bayesian neural networks, and the discrete annual transport map is compiled as a loop so the exact cohort advection (including terminal-age aggregation and the birth boundary condition) is embedded in the likelihood. What this machinery achieves is that the posterior is defined on a manifold of mass-conserving density fields, so uncertainty propagates along cohort characteristics and cannot produce the unphysical crossings

What would settle it

A synthetic-data experiment would settle this: generate age-structured populations from the transport equation with a known time-varying migration flux M(a,t), fit the paper's migration-free model to those data, and check whether the posterior for fertility and mortality moves systematically away from the true generating functions. A complementary empirical test is a holdout calibration check on a country with high net migration, asking whether the 95% credible intervals for age counts still cover independently observed census values beyond the training window.

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Extended reading notes

Core claim

The central claim is that the age-structured population density ρ(a,t) is the solution of the exact transport equation ∂tρ + ∂aρ = −μ(a,t)ρ with boundary condition ρ(0,t) = ∫ f(a,t)ρ(a,t) da, and that the latent constitutive laws f and μ can be inferred directly from observations via Bayesian inversion when they are parameterized by Bayesian neural networks. Because the transport step is applied exactly in discrete annual form, every posterior sample of the network weights and the initial profile yields a density field that conserves mass by construction. The paper demonstrates this by reconstructing Japan (1970–2024) and South Korea (2003–2024) from complete age-specific grids, and China (1

Load-bearing premise

The load-bearing premise is that net migration is negligible over the modeled periods, so the migration-free transport equation is exact; if substantial unmodeled migration occurred, it would be absorbed into the inferred fertility and mortality and could distort the reconstructed age structure and forecasts.

Editorial extensions

If this is right

The full historical age grid for China is reconstructed from only totals, births, deaths, and sparse census brackets, demonstrating that missing age structure can be recovered by transport inversion rather than interpolation.
Forecast uncertainty is structurally coherent: a shock to fertility in year t propagates downstream along the advection characteristics, so credible intervals for future age shares are correlated across ages in a way that unconstrained time-series projections do not capture.
The model recovers historical TFR trajectories without ever being shown TFR data, matching external vital records for all three countries and confirming that fertility is identifiable from the age structure alone.
The posterior yields timing estimates for structural milestones: South Korea's potential support ratio falls below 2.0 by 2037 with narrow credible intervals, China follows near 2049 with wider uncertainty, and all three countries cross or have crossed the point where elderly outnumber children.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper asserts the transport equation is exact but omits migration; an inference worth testing is that, for countries with non-negligible net migration, the inferred fertility and mortality schedules silently absorb the migration flux, which would compress or stretch the reconstructed pyramid and bias long-horizon forecasts even though the internal logic of the PDE is correct.
The toy model's 'demographic half-life' of 26 years and its delayed dependency-ratio crisis at roughly 65 years are generalizable predictions; one could compare them against the posterior forecasts for countries with different dip durations to see if the scaling holds across transition profiles.
The demographic entropy metric is presented as a leading indicator of decline; since the paper only demonstrates it retrospectively, a prospective test on other low-fertility populations would clarify whether the entropy turning point reliably precedes working-age and total population peaks by about a decade.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

This paper is worth a serious look. It embeds Bayesian neural networks as latent time-varying fertility and mortality laws inside the exact McKendrick PDE, and samples the posterior with NUTS. That specific combination is not in the demographic forecasting literature it cites, and the age-resolved inverse problem for China — reconstructing 75 years of age structure from sparse census and vital statistics — is a genuine test of the idea. The conservation-enforced transport gives you calibrated-looking uncertainty, and the sensitivity checks (prior scaling, hidden layer size) are decent.

The main soft spot is the one the stress-test flagged: migration is absent from the empirical model. The PDE in Eq. (1) is exact only for a closed population. Japan and South Korea have net migration that is not negligible over the modeled periods, so the inferred fertility and mortality are 'effective' rates that absorb migration. The paper flags migration as a future extension (Eq. 12) but still calls the mass conservation 'exact.' That's an overstatement, and the 2070 forecasts inherit the misspecification. It's addressable — add M(a,t), or at least defend its size — but it should not go to press without a fix or a caveat.

The validation also has circular elements. For Japan/Korea, the full age grid is the likelihood, so reproducing births and deaths is a consistency check, not an independent test. The China backtest is in-sample by construction. The TFR comparison is more interesting, but TFR is inferred from the same transport dynamics that produced the age structure, so it's not fully external. The 'Status Quo random walk scenario' is never defined — that's a clear gap.

The entropy and KL metrics are descriptive, not independent physical laws; the 'leading indicator' claim is plausible but not a rigorous finding.

That said, the core method is sound and the paper deserves a real referee rather than a desk reject. I'd send it out, with the expectation of a major revision that deals with migration, defines the forecast scenario, and ideally releases code and posterior samples.

Referee Report

4 major / 5 minor

Summary. The manuscript proposes a Bayesian inverse transport framework for age-structured population dynamics. Fertility and mortality are parameterized by Bayesian neural networks embedded inside the McKendrick--von Foerster transport equation, and the posterior over network weights and the initial age profile is sampled with NUTS. The framework is applied to China, Japan, and South Korea, where it reconstructs historical age densities from sparse observations and produces forecasts to 2070. The paper also introduces information-theoretic metrics such as 'total demographic entropy' and the KL divergence from Lotka equilibrium, and compares its China projections with UN WPP, YuWa, IHME, and other institutional models.

Significance. The central idea is attractive and timely: enforcing an exact advection constraint while learning constitutive laws with BNNs is a principled way to combine physical conservation with flexible data-driven modeling. If the empirical results were fully credible, the framework would provide a useful addition to demographic forecasting, particularly for age-structured outputs and uncertainty quantification. The manuscript also contains some genuine strengths: the discrete transport update is standard and correctly written, the posterior formulation is coherent, and the comparison of inferred TFR with World Bank data and the China census back-testing provide independent checks. However, the empirical claims currently rest on a closed-population model that omits migration, and several validation statements are partly circular. These issues are fixable but require substantial revision before the advertised 'exact' conservation claim can be accepted for the real countries studied.

major comments (4)

[II.A, III.B, Eq. (12)] The governing equations (1)--(5) contain no migration flux; Eq. (12) explicitly defers M(a,t) to future work. For Japan (1970--2024) and South Korea (2003--2024), the HMD populations are open to international migration, and net migration is not negligible relative to birth/death flows. In the actual fitted model, rho(a+1,t+1)=s(a,t)rho(a,t), so all net migration is absorbed into the inferred survival s(a,t) and into the birth boundary. This biases the reconstructed mortality/fertility and, through the transport dynamics, the 2070 forecasts. The paper's claim that every posterior sample satisfies exact mass conservation is therefore exact only for a closed population, not for the observed data. Please either include a migration term with real data or priors, or explicitly reposition the empirical studies as closed-population illustrations and provide a quantitative sensitivity analysis bo
[III.B.1, Eq. (14), Fig. 1--2] For Japan and South Korea the likelihood includes the complete annual age grid (Eq. 14). Births and deaths are not entered into the likelihood; they are deterministic outputs of the fitted age grid through Eqs. (2)--(5). Thus the statement in the Fig. 1/2 captions that accurate reconstruction of births/deaths 'confirms that the transport PDE conserves mass' is close to tautological: any model that fits rho(a,t) at all ages and years will, by construction, reproduce the implied fluxes. This is not an independent validation. The real independent checks are the external TFR comparison (Fig. 5) and the China census comparison (Fig. 4). I recommend adding an out-of-sample or holdout validation, for example training on Japan 1970--2010 and validating 2011--2024, or moving births/deaths into the likelihood and reporting posterior predictive checks.
[IV.C, Table I, Fig. 7] The forecast mechanism is underspecified. The text says trajectories are propagated to 2070 'under the Status Quo random walk scenario,' but no equations are given for how the fertility/mortality laws are extrapolated beyond the training years, what random walk is applied to TFR(t) or s(a,t), or how the posterior over W is mapped to future constitutive laws. Because the medium-horizon forecasts and PSR milestone dates in Table I and Fig. 7 are central outputs, this omission makes the forecasts unreproducible and prevents evaluation of whether the credible intervals are calibrated. Please provide the exact forecasting equations, scenario priors, and the relationship to the posterior samples.
[II.D] The posterior space contains on the order of several hundred BNN weights plus the initial age profile. The manuscript states only that 'all three models converged with zero divergences' and gives no split-R-hat values, trace plots, or effective sample sizes for the quantities reported in Tables I and III. With 4 chains x 1000 post-warmup iterations, the posterior sample size is small for a high-dimensional problem, and the reported credible intervals are not yet substantiated. Please provide convergence diagnostics and effective sample sizes for the key reported quantities, and consider longer runs if needed.

minor comments (5)

[III.B.1, Eq. (14)] The text says the dataset provides Ypop, YB, and YD, but the displayed likelihood only uses Ypop. Clarify whether births/deaths are used in the likelihood or only as posterior predictive quantities.
[III.B.3] The baseline hazard mu0(a)=0.005+8e-5 exp(0.082a) and the stable initial profile S_base(a) are introduced without source or sensitivity discussion. Please document their origin or cite a demographic standard, and state how sensitive the China reconstruction is to this choice.
[IV.D.1, Eq. (20), Table III] The quantity S_total mixes Shannon entropy with an expected log 'phase volume'; it is not a standard thermodynamic or information entropy and can take negative values (as in Table III). Please rename it or provide a clearer justification for calling it entropy.
[IV.A, Eq. (17)] The toy-model expression P(D) ~ P0 * 2^{-D/26} should state explicitly that D is the duration in years and clarify the range of validity; as written the exponent appears dimensionally unusual.
[Global] No data or code availability statement is provided. For a methods paper with forecasts and credible intervals, releasing code and data-processing scripts (or a detailed pseudocode) is important for reproducibility.

Circularity Check

2 steps flagged · score 6.0 of 10

Central Bayesian inversion is self-contained, but two validation claims reduce to fitted likelihood targets: Japan/Korea births/deaths reconstruction and China's 'independent' census backtest.

fitted input called prediction [Sec. III.B.1 (Japan), Fig. 1 caption; Eq. (4)]
"Because the total births and deaths were not explicitly provided in the likelihood function (which relies solely on the cross-sectional age grid), their accurate reconstruction confirms that the transport PDE conserves mass while advecting the population cohort backward in time."

By Eq. (4), B(t)=ρ(0,t+1), and the age-0 bin is part of the cross-sectional age grid Ypop(a,t) used in the Eq. (14) likelihood. Deaths are likewise cohort differences between adjacent age cells in that same grid. Hence the 'reconstructed' births and deaths are deterministic functions of the fitted state, not independent confirmations; the validation reduces to checking that the model fits the age grid it was trained on.
fitted input called prediction [Sec. III.B.3 (China), Eq. (16)]
"In addition, the reconstructed age distributions match the independent census observations with high precision, successfully capturing the major historical cohort bulges and troughs (Fig. 4)."

The 'independent census observations' are the same coarse census age groups C_g(t) that Eq. (16) explicitly places in the likelihood ('under a log-normal likelihood with a 5% coefficient of variation'). The model is fitted to these census brackets, so presenting their alignment as an independent backtest is a check of the training target, not an out-of-sample prediction.

full rationale

The core Bayesian inverse step (Eq. 7) is not circular: the posterior over BNN weights and initial profile is conditioned on observed demographic data through the transport PDE, and the forecasts are generated by propagating posterior samples forward. No parameter is defined in terms of its own forecast, and no load-bearing result rests on a self-citation (refs [12] and [19] are incidental). The circularity is confined to the validation narrative. For Japan/Korea, Eq. (4) sets births equal to the age-0 density, which is part of the age-grid likelihood, and deaths are cohort differences within the same grid; hence the 'reconstruction' of vital events is a consistency identity, not an independent confirmation. For China, the coarse census age groups are put into the likelihood (Eq. 16) and then the match to them is described as 'independent'; that is a fit, not a backtest. The TFR comparison to World Bank data is not circular because those external estimates were not used in the likelihood. The omitted migration flux (Eq. 12 defers M(a,t)) is a model misspecification risk, not a circularity.

Assumptions & free parameters 5 free parameters · 7 assumptions · 2 invented entities

The framework leans on standard PDE transport, but several load-bearing modeling choices are ad hoc: the no-migration assumption, the Gaussian fertility kernel, and the China baseline hazard. The BNN weights and initial population states are posterior parameters rather than externally constrained constants. The invented entropy metrics add interpretation but have no independent falsifiable handle.

free parameters (5)

BNN weights and biases W_f, W_s = Posterior samples (no explicit values reported)
Central unknown constitutive laws; priors Wij ~ N(0,1/n_in), biases ~ N(0,1); Sec II.B.
Initial age profile rho0(a), including China's growth rate r and scale N_scale = N_scale ~ N(5.52,0.27) scaled by 1e8 for China
Initial condition of the transport PDE sampled in posterior; Eq. (15).
Observation noise sigma_p = HalfNormal posterior
Log-normal likelihood on population grids and census brackets; Eqs. (14), Sec III.B.
Baseline hazard constants in mu0(a) = 0.005 + 8e-5 exp(0.082a) = 0.005, 8e-5, 0.082
Baseline survival model for China, Eq. (11); constants are asserted without a cited source.
Logistic transition parameters Bmax, Bmin, t0, k = k≈0.35 (South Korea), 0.28 (China), 0.12 (Japan)
Macroscopic empirical classification fitted to historical fertility transition; Eq. (13), Sec III.A.

assumptions (7)

domain assumption Age-structured population follows the McKendrick PDE with unit advection and no migration
Eq. (1)-(5); migration appears only as a future extension in Eq. (12), not in empirical reconstructions.
ad hoc to paper Fertility kernel is a TFR-scaled Gaussian density over maternal age, masked to 15-49
Eq. (8): f(a,t)=0.5*TFR_t*g(a;nu_t,sigma_t); a strong parametric form chosen for convenience, not derived from data or biological mechanism.
ad hoc to paper Survival baseline for China mu0(a)=0.005+8e-5 exp(0.082a)
Eq. (11); introduced to stabilize China inference, but the constants are not justified by cited data.
domain assumption Normal priors with variance 1/n_in enforce smooth, slowly-varying schedules
Sec II.B: the priors act as L2 regularization; sensitivity checks are reported but no formal justification that the true schedule lies in this class.
domain assumption Log-normal likelihood with independent noise and 5% coefficient of variation for China census brackets
Sec III.B: likelihood specification; no validation of the independence or variance assumptions.
standard math Terminal age 100 aggregation closes the age grid
Eq. (3): standard finite-age boundary treatment in age-structured models.
standard math Lotka stable age distribution is the appropriate equilibrium attractor for D_KL
Eq. (21)-(22): classical stable population theory used to define non-equilibrium divergence.

invented entities (2)

Demographic phase volume Omega(a) = A_max - a
purpose: Defines a per-capita remaining-life potential used to build total demographic entropy S_total
New construct in Sec IV.D.1; no external calibration, purely a modeling choice.
Total demographic entropy S_total
purpose: Quantify structural contraction and act as a leading indicator of population decline
Sec IV.D.1: a composite metric whose 'leading indicator' claim is demonstrated only qualitatively against known historical peaks, not by a predictive out-of-sample test.

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Cite this review

Pith. "Pith review of Human population dynamics as a Bayesian inverse transport problem." pith.science (2026). https://pith.science/paper/QXU25SHJ

@misc{pith2026260713171,
  author       = {Pith},
  title        = {Pith review of: Human population dynamics as a Bayesian inverse transport problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QXU25SHJ}},
  note         = {Machine review of arXiv:2607.13171}
}

read the original abstract

Many open problems across physical, biological, and engineered systems involve non-equilibrium transport processes where the governing conservation laws are known, but the underlying constitutive relations remain latent and time-varying. Conventional data-driven approaches like deep neural networks capture statistical patterns but routinely violate fundamental mass conservation. Here, we introduce a unified Bayesian inverse transport framework that resolves this by embedding Bayesian Neural Networks (BNNs) directly within exact partial differential equations in age-time space. By evaluating this framework on complex, real-world human cohort advection across China, Japan, and South Korea, we demonstrate that this physical constraint enables consistent uncertainty propagation and missing-data reconstruction from sparse observations. Beyond demography, this framework provides a generalizable foundation for observing and forecasting non-equilibrium boundary dynamics across various fields.

Figures

Figures reproduced from arXiv: 2607.13171 by the authors.

**Figure 1.** FIG. 1. Validation of the model for Japan. The left panel compares the inferred total population (blue) against observed data [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** FIG. 2. Validation of the model for South Korea. The left and middle panels show the model’s ability to infer macroscopic [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. Posterior predictive check for China’s macroscopic vital events. The model’s inferred total population, annual births, [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: FIG. 4. Historical backtesting of China’s age structure. The model’s reconstructed continuous age density aggregated into [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5. Model-inferred historical (solid lines) and projected [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6. (A) Comparative 2050 population pyramids (mean and 95% credible intervals) for China (red), Japan (blue), and [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: FIG. 7. EDR for China, Japan, and South Korea. Solid lines [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: FIG. 8. Reconstructed and projected age-structured popu [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: FIG. 9. Total demographic entropy [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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The likeli- hood is constructed using log-normal distributions: lnY pop(a, t)∼ N lnρ(a, t), σ2 p (14) with HalfNormal priors on observation noise parameters

Japan (1970-2024) Japan’s dataset from the Human Mortality Database (HMD) [32] provides complete annual population grids Ypop(a, t), birthsYB(t), and deathsY D(a, t). The likeli- hood is constructed using log-normal distributions: lnY pop(a, t)∼ N lnρ(a, t), σ2 p (14) with HalfNormal priors on observation noise parameters. As shown in Fig. 1, the model su...

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South Korea (2003-2024) South Korea’s HMD dataset [32] is aligned from 2003 to 2024. The training setup matches Japan’s complete grid likelihood. Fig. 2 demonstrates the model’s ability to accurately infer South Korea’s macroscopic vital events and age density profiles directly from the inversion of the age- structure PDE. The model also infers a world-re...

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Comparison with existing models and scenario projections for China To contextualize our results, we use our framework to predict China’s future age density (Fig. 8) and compare these projections to existing independent models and in- stitutional reports (summarized in Table II). These alter- native models yield significantly different results, reflect- in...

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TABLE III

The rapid fertility collapse in East Asia drives a steep structural contraction phase, rapidly stripping the popula- tions of their internal potential as they transition into hyper- aged demographic states. TABLE III. Non-Equilibrium Physics and Aging Metrics (Mean±SD) across ...

2000

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [1]

The likeli- hood is constructed using log-normal distributions: lnY pop(a, t)∼ N lnρ(a, t), σ2 p (14) with HalfNormal priors on observation noise parameters

Japan (1970-2024) Japan’s dataset from the Human Mortality Database (HMD) [32] provides complete annual population grids Ypop(a, t), birthsYB(t), and deathsY D(a, t). The likeli- hood is constructed using log-normal distributions: lnY pop(a, t)∼ N lnρ(a, t), σ2 p (14) with HalfNormal priors on observation noise parameters. As shown in Fig. 1, the model su...

1970

[2] [2]

The training setup matches Japan’s complete grid likelihood

South Korea (2003-2024) South Korea’s HMD dataset [32] is aligned from 2003 to 2024. The training setup matches Japan’s complete grid likelihood. Fig. 2 demonstrates the model’s ability to accurately infer South Korea’s macroscopic vital events and age density profiles directly from the inversion of the age- structure PDE. The model also infers a world-re...

2003

[3] [3]

No annual age-structured population grid is avail- able

China (1950-2024): Reconstructing incomplete data China represents a highly challenging inverse prob- lem. No annual age-structured population grid is avail- able. We only observe annual births, annual deaths, and total population size aggregated from multiple conflict- ing sources (specifically, manually compiled records from Chinese vital statistics and...

1950

[4] [4]

8) and compare these projections to existing independent models and in- stitutional reports (summarized in Table II)

Comparison with existing models and scenario projections for China To contextualize our results, we use our framework to predict China’s future age density (Fig. 8) and compare these projections to existing independent models and in- stitutional reports (summarized in Table II). These alter- native models yield significantly different results, reflect- in...

2024

[5] [5]

Total demographic entropy and phase volume While standard Shannon entropyS=− P p(a) lnp(a) quantifies the dispersion of a distribution, it is struc- turally symmetric: a sharply peaked young population and a sharply peaked elderly population yield identical values, ignoring the biological arrow of time. To physi- cally distinguish these states, we introdu...

1960

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Distance from Lotka equilibrium In classical mathematical demography [4], if the fertil- ityf(a, t) and mortality schedules were frozen at their instantaneous values at yeart, the population would asymptotically converge to the Lotka stable age distri- bution: p∗ t (a) =C te−rtal(a, t),(21) wherel(a, t) = Qa−1 i=0 s(i, t) is the survival probability to ag...

2024

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Its Aging Index is projected to reach 860.9% by 2050 (representing over 8.6 elderly citizens for ev- ery child), with its PSR dropping to 1.34

Labor scarcity and social support dynamics To link these physical state variables to macro-social constraints, we report the Aging Index in Table III: AgeIdx(t) = PAmax a=65 ρ(a, t)P14 a=0 ρ(a, t) .(23) Under the Status Quo forecast, South Korea shows a transition into a hyper-aged, highly ordered state with declining entropy and extremely low potential s...