Universal Exponential Scaling Law

• Universal exponential scaling law is a framework where rescaling exponential processes yields invariant relationships across diverse systems in growth, decay, and transport.
• It utilizes minimal autocatalytic models that predict time-invariant distributions and latent power-law behaviors emerging from exponential averaging.
• The law robustly explains collapsed observables in stochastic growth, spatial statistics, and thermoelectric transport, linking empirical data with theoretical scaling.

The universal exponential scaling law designates a set of mathematical phenomena where rescaling or averaging over simple exponential processes yields invariant—or "collapsed"—scaling relationships across diverse systems and observables. In contexts ranging from stochastic single-cell growth and spatial distributions to thermoelectric transport in correlated electron systems, universally recurring scaling forms emerge due to minimal model structures, characteristic dimensionalities, or kernel-induced exponential suppression. This unification underlying exponential scaling laws allows distinct observables, apparent power laws, and statistical distributions to be systematically traced to underlying exponential dynamics or bandwidths.

1. Stochastic Autocatalytic Growth and Distribution Collapse

In stochastic growth processes, particularly at the single-cell level, universality arises from minimal microscopic autocatalytic cycles, encapsulated by the stochastic Hinshelwood cycle (SHC) framework (Iyer-Biswas et al., 2014). The SHC is governed by an $N$-step autocatalytic network, where each molecular species $X_{i-1}$ catalyzes the production of $X_i$ at rate $k_i x_{i-1}$, and the chemical Master Equation formally describes the probabilistic kinetics: $$\frac{\partial P({\bf x},t)}{\partial t} = \sum_{i,j=1}^{N}{\mathbb K}_{ij}\,x_{j}\, [ P(\ldots,x_{i}-1,\ldots) - P(\ldots,x_{i},\ldots) ],$$ with ${\mathbb K}_{ij} = k_{i}\,\delta_{i-1,j}$.

The exact analytical solution shows the mean copy-number vector grows as

$\langle x_{q}(t)\rangle \sim C_q e^{\kappa t},$

where $\kappa$ is the geometric mean of the cycle rates. Asymptotically, all molecular species share both the same exponential rate and fixed ratios, independent of the initial state. The covariance and higher cumulants also grow exponentially at rate $2\kappa$, preserving proportionality, and yielding a constant coefficient of variation (COV) in the long-time limit. This analysis establishes that exponential growth and perfectly correlated size fluctuations are universal consequences of the cyclic autocatalytic architecture.

2. Universal Mean-Rescaled Distributions and Scaling Collapse

The explicit construction of time-dependent cell-size distributions in the SHC reveals a negative-binomial form converging for large size and time to a gamma distribution: $$P_\infty(s\mid s_0) = \frac{s^{\,s_0-1} e^{-s_0\,s}}{\Gamma(s_0)} s_0^{\,s_0},$$ where $s$ is the total copy number and $s_0$ the initial size. Importantly, by defining the scaled variable $y \equiv s/\langle s \rangle$,

$$P(s, t)\,ds = \langle s(t) \rangle^{-1} f(y)\,dy, \quad f(y) = \frac{(y s_0)^{s_0-1}}{\Gamma(s_0)} e^{-s_0 y},$$

the cell-size distributions at arbitrary times collapse onto a time-invariant gamma scaling function when plotted versus $s/\langle s\rangle$. This distributional collapse is a hallmark of universal exponential scaling, appearing in single-cell growth but also in analogous contexts, as it results solely from the properties of the underlying master equation and the structure of the autocatalytic network.

3. First-Passage-Time Distributions and Temporal Scaling

The law extends to the statistics of division times or first-passage times with respect to a size threshold $\theta$. The first-passage density is given by

$$\mathcal P(\tau) = -\frac{d}{d\tau} \int_0^{\theta} ds\, P(s, \tau \mid s_0),$$

which, substituting the gamma form, leads to a beta-exponential distribution: $$\mathcal P(\tau) = \kappa\, e^{-\kappa s_0 \tau} (1 - e^{-\kappa \tau})^{\theta-s_0} \frac{B(s_0, 1+\theta-s_0)}{B(s_0, \theta-s_0)},$$ where $B$ denotes the beta function. Its dependence on time appears only via the scaling variable $\kappa \tau$, so by rescaling time with the mean division time ($\langle\tau\rangle \propto \kappa^{-1}$), all division-time distributions likewise collapse onto a universal curve. Empirically this is verified in single-cell studies and is a direct quantitative prediction of the universal exponential scaling law.

4. Latent Power-Law Scaling via Exponential Averaging

A distinct but intimately related manifestation of exponential scaling arises in spatial statistics and network analysis (Chen, 2013). Here, averaging (or cumulatively summing) a 1D or 2D exponential decay produces asymptotic power-law behavior, with exponents determined by dimensionality rather than the presence of self-similarity:

• 1D: $y(r) = y_0 e^{-r/r_0}$ yields, upon averaging over $[0,r]$,

$$F(r) \approx \frac{y_0\, r_0}{r} \quad\text{as}\quad r\gg r_0,$$

i.e., a $r^{-1}$ tail.

• 2D: For $p(r) = p_0 e^{-r/r_0}$, the cumulative-averaged surface density over a disk scales as

$$F(r) \approx \frac{2 p_0 r_0^2}{r^2} \quad\text{as}\quad r\gg r_0,$$

i.e., a $r^{-2}$ tail.

Thus, these power laws are "spurious," arising solely from averaging over the exponential kernel and always featuring scaling exponents that match the Euclidean dimension, rather than any true scale-free or fractal underlying structure. This means apparent Zipf or Pareto laws may be artifacts of underlying exponential statistics and should be rigorously distinguished from genuine self-similar or critical distributions by means of inverse reduction tests.

5. Universal Exponential Scaling in Condensed Matter Transport

In strongly correlated electron systems, the universal exponential scaling law also emerges in the temperature dependence of transverse transport coefficients, notably in the quasiparticle Nernst effect of underdoped cuprates (Yang, 2023). In this regime, the quasiparticle Nernst coefficient per temperature follows: $\frac{\nu^{\rm qp}}{T} = A + B \exp(-T/T_0),$ where $T_0$ is set by the effective "Berry-curvature bandwidth" $D_h$ as $T_0 \approx 0.8\,D_h$, and $A \ll B$ in relevant windows. The derivation traces this scaling to the general transport formula: $$\frac{\sigma_{xy}^{\,\alpha}}{H} = \frac{4\pi^2}{3} \int_{-\infty}^{\infty} d\omega \left(-\frac{\partial f(\omega)}{\partial\omega}\right) \left(\frac{\omega}{T}\right)^{\alpha} \mathcal{B}(\omega, T),$$ where the energy dependence of $\mathcal{B}(\omega, T)$ is sharply constrained to a narrow bandwidth $[\pm D_h]$, and the thermal kernel imparts exponential temperature suppression. The universality is reflected in the fact that, after accounting for trivial multiplicative factors and observable-specific algebraic prefactors (e.g., $t^{-(2-\beta)}$), all such transport coefficients in this regime collapse onto a single $\exp(-T/T_0)$ law, as confirmed experimentally in YBa$_2$Cu$_3$O$_y$. The observed scaling constants differ systematically (e.g., $T_0$ for Nernst and Hall coefficients versus Hall resistivity), consistent with the derived theoretical structure.

6. Independence from Microscopic Details and General Implications

Both reaction‐cycle stochastic models and narrow-bandwidth transport frameworks reveal that universal exponential scaling is robust to underlying complexity. In autocatalytic networks, any complex topology with a dominant autocatalytic loop reduces asymptotically to SHC behavior with a unique exponential rate and scaling structure. Similarly, in condensed matter, any system where the response kernel is effectively windowed (by bandwidth or selection function) will exhibit exponential scaling in the thermal regime, regardless of detailed band structure or scattering mechanisms.

A key implication is that whenever a characteristic length, time, or energy scale exists—realized either by network topology or spectral kernel—the exponential function and its transformational properties dictate the scaling collapse, the scaling exponents, and the shape invariance of distributions. Apparent power laws, time- or size-invariant statistical forms, and collapsed observables can all emerge without underlying criticality or fractality, calling for stringent tests (such as inverse reduction) to distinguish true self-similar phenomena from artifacts of exponential processes.

7. Practical Criteria and Tests for Universal Exponential Scaling

These scaling laws have direct utility for empirical data analysis. In spatial, network, and rank-size statistics, running-average power laws or Zipf plots with integer exponents may be artifacts of underlying exponential decay and should be tested via inverse averaging reductions. In dynamical and transport contexts, temperature or time collapse of distributions or observables—after rescaling by the mean or an empirically fitted exponential scale—is a fingerprint for underlying exponential scaling. Disparities between "onset" scales inferred from different observables (as in the cuprate pseudogap) are resolved by the algebraic prefactors in the unified scaling theory. Broadly, universal exponential scaling provides a minimal, predictive, and falsifiable framework for diverse systems exhibiting exponential relaxation, growth, or suppression, and is widely applicable across stochastic processes, spatial statistics, and transport phenomena.