Expectation-Level Decompression Law

• Expectation-Level Decompression Law is a framework that quantifies the transition from compressed states to expanded equilibrium across geodynamics, thermodynamics, and stochastic systems.
• In geoscience and thermodynamics, the law enables prediction of tectonic activity and entropy production through explicit energy and volume change formulas.
• In stochastic analysis, the law underpins g-expectations that transform dynamic, path-dependent problems into tractable, distribution-based optimization models.

The Expectation-level Decompression Law encompasses a set of principles observing how decompression—across geodynamic, thermodynamic, and stochastic systems—exhibits quantifiable, often predictable relations between system-level change and observable physical or informational outputs. The term surfaces in disparate but thematically linked disciplines: geoscience (Whole-Earth Decompression Dynamics), stochastic analysis (law-invariant $g$-expectations), and non-equilibrium thermodynamics (irreversible entropy production during sudden expansion). Across these domains, the “expectation-level decompression law” characterizes the systematic or optimal adjustment of state variables, often from an initial, compressed condition to an expanded, decompressed equilibrium, with specific, analyzable laws governing the cost, efficiency, or entropy production of the process.

1. Geodynamic Interpretation: Whole-Earth Decompression Dynamics

Within geoscience, the principle of Earth decompression is formalized in Whole-Earth Decompression Dynamics (WEDD) (Herndon, 2013). Here, decompression is not merely a passive increase in volume, but an active, energy-coupled driver of planet-scale tectonic evolution.

• Initial State: The Earth begins as a Jupiter-like gas giant, with a rocky kernel under extreme compression, surrounded by $\sim$300 Earth-masses of hydrogen, helium, and ices.
• Trigger: Loss of this massive envelope (e.g., via T-Tauri wind) initiates planetary decompression, setting the stage for long-term geodynamic evolution.
• Quantitative Energy Law: The energy released through decompression is

$E_{\text{decomp}} = \int_{V_i}^{V_f} P\,dV,$

where $V_i$ and $V_f$ are the initial and final planetary volumes, corresponding to radii $R_i$ and $R_f$, with

$\Delta V = \frac{4\pi}{3}(R_f^3 - R_i^3).$

• Curvature-driven Stress and Folding: As $R$ increases, the original continental crust area exceeds that required by the new surface curvature. The resultant misfit strain,

$$\epsilon \approx \frac{L_{\text{old}} - L_{\text{expected}}}{L_{\text{expected}}},$$

leads to buckling and fold-mountain formation, even in the absence of plate collision.

WEDD thus provides a geophysically explicit “expectation-level decompression law”: the magnitude and type of surface deformation, continental breakup, and mountain formation can be calculated or anticipated directly from the decompression-induced changes in volume and curvature.

2. Thermodynamic Asymmetry: Entropy Production in Sudden Expansion

In non-equilibrium thermodynamics, the expectation-level decompression law governs the asymmetry of entropy production in sudden expansions (decompression) versus compressions in ideal gases (Vallejo et al., 2023).

• Isothermal Sudden Compression: For a perfect gas coupled to a thermal reservoir and subjected to a sudden pressure increase from $P_1$ to $P_2$, the entropy produced in the universe (system + reservoir) is:

$\Delta S_{\text{univ}}^{1\to2} = R[x - \ln(1+x)],$

where $x = (P_2 - P_1)/P_1$.

• Isothermal Sudden Expansion: When pressure drops from $P_2$ to $P_1$,

$$\Delta S_{\text{univ}}^{2\to1} = R\left[ \ln(1+x) - \frac{x}{1+x} \right],$$

with $x$ as above.

• Law Characteristics:
  • For compression, entropy production grows linearly with $x$ for large pressure jumps.
  • For decompression (expansion), the growth is logarithmic.
• Implications: For small perturbations ($x \ll 1$), both modes are symmetric—entropy production grows quadratically. However, for large decompressions, the entropy penalty is substantially lower than for large compressions.

The expectation-level decompression law here denotes that decompression (expansion) processes during sudden pressure changes are inherently “closer to reversible” than compressions, a law arising from the geometry of entropy versus pressure change and with practical implications for thermodynamic cycle design.

3. Distribution-level Decompression in Nonlinear Expectation

In stochastic analysis and mathematical finance, the expectation-level decompression law is realized in the context of $g$-expectations of distributions, as defined through backward stochastic differential equations (BSDEs) (Xu et al., 2022).

• $g$-Expectation via BSDE: For a BSDE with driver $g(t,y,z)$ and terminal random variable $X$, the $g$-expectation is $E^g[X] = Y_0$, where $(Y_t, Z_t)$ solves

$$dY_t = g(t, Y_t, Z_t)\,dt + Z_t\,dB_t, \quad Y_T = X.$$

• Expectation at the Distribution (Law) Level: Given all $X$ with law $\mu$,

$E^g[\mu] := \inf\{E^g[X]: X \sim \mu\}.$

This operation “decompresses” the expectation from full sample-path data to a law—or expectation—over distribution.

• Special Cases:
  • For linear drivers or homogeneous nonlinearities in $z$, $E^g[\mu]$ reduces to $E[X]$.
  • For more general nonlinearities (e.g., market models with distinct lending and borrowing rates), explicit formulas relate $E^g[\mu]$ to quantile functions or state-price densities.
• Law-invariance: When the $g$-expectation depends only on the terminal distribution and not the representation, the decompression is complete—sample path details are “compressed” into the expectation associated with $\mu$.

This construction provides an expectation-level decompression law by translating the replication problem in dynamic control (such as portfolio optimization) into a static, distributional minimization problem.

4. Analytical Formulations and Quantitative Expressions

A recurrent theme is the emergence of explicit, system-level expressions connecting decompression (increase in accessible state space, surface area, portfolio distribution, etc.) to expectation values, energy release, or entropy production.

Domain	Law/Expression	Interpretation
Geodynamics	$E_{\text{decomp}} = \int_{V_i}^{V_f} P\,dV$<br>$\sigma \propto E (\Delta R)/R$	Energy, strain, and fold-mountain formation under decompression
Thermodynamics	$$\Delta S_{\text{univ}}^{2\to1} = R\left[ \ln(1+x) - x/(1+x) \right]$$	Entropy production scaling logarithmically in decompression
Stochastics	$E^g[\mu] = \inf\{E^g[X]: X \sim \mu\}$	Minimal expectation cost for replicating a terminal law

These laws formalize how decompression “propagates” through system constraints, enabling the conversion of microscopic (sample path, energetic, or structural) data into effective macroscopic or systemic laws.

5. Theoretical Implications and Practical Applications

The expectation-level decompression law organizes the relationship between driving forces (energy, probabilistic law, or pressure) and system response in several applied contexts:

• Geoscience: Predicts the genesis and morphology of major crustal features (mountain belts, rift zones, basaltic plateaus) as systematic mechanical responses to decompression, obviating the need for mantle convection (Herndon, 2013).
• Thermodynamics: Quantifies the irreversibility gap between expansion and compression in practical gas-handling or heat engine cycles, guiding process optimization where low entropy production is critical (Vallejo et al., 2023).
• Finance and Stochastics: Facilitates the solution of optimal investment and replication problems by reducing dynamic, path-wise stochastic optimization to tractable static quantile optimization at the distributional level, especially where preferences are law-invariant (Xu et al., 2022).

6. Extensions and Open Questions

Several avenues remain for deepening the framework suggested by the expectation-level decompression law:

• General Law-Invariance: Full conditions for $g$-expectation law-invariance beyond sufficient criteria remain open, especially for multidimensional or path-dependent drivers (Xu et al., 2022).
• Irreversibility Scaling in Complex Systems: While the entropy production scaling law for perfect gases is precise, extensions to more complex, interacting systems (e.g., real gases, planetesimal accretion) could yield new decompression asymmetries (Vallejo et al., 2023).
• Nonlinearities and Robustness in Finance: Supreme $g$-expectations and their decompression at the law level offer robust control tools for worst-case (rather than average-case) calibration (Xu et al., 2022).

7. Conceptual Unification and Cross-Disciplinary Significance

The expectation-level decompression law, appearing variously in geodynamics, thermodynamics, and probabilistic analysis, embodies the principle that decompression processes—whether physical, stochastic, or informational—admit system-level descriptions in terms of expectation values, state functions, or minimal costs that can often be made explicit and predictive. The law highlights both the predictability and the asymmetry inherent in decompression phenomena and offers a unifying template for analyzing expansion-driven transitions, whether in planetary evolution, physical entropy production, or informational optimization.