Patent tokenization aims to unlock illiquid IP under uncertainty. Static release plans and simplified pricing cannot adapt to shocks (litigation, regulation, and demand), and they rarely encode risk preferences, which is precisely where value is at stake in SDG-relevant domains.

We treat patent tokenization as a risk-sensitive control and use a thermodynamic change of variables to make the problem linear for fixed controls so that optimal schedules follow from a simple threshold structure and can be computed in milliseconds.

• Theory. The risk-sensitive HJB with the correct sign convention linearizes under Φ = e − γ V and implies threshold (bang–bang) policies; a small quadratic penalty yields smooth, implementable schedules.

Section 3 derives the transform and control structure. Section 4 reports estimates and simulations, and Section 5 interprets the robustness–efficiency trade-off and SDG implications; the Supplementary Information documents sensitivity and implementation details.

The global patent system contains vast reserves of unused value (World Intellectual Property Organization WIPO, 2024; European Patent Office EPO, 2023; United States Patent and Trademark Office USPTO, 2024). Although patents are designed to secure innovation, empirical studies show that between 90% and 95% never generate licensing income (Gambardella et al., 2007; Arora et al., 2008).

One reason is the rigid all-or-nothing structure of conventional transactions: patent holders either retain their rights completely, leaving them without immediate funding, or sell them outright and forgo future upsides (Lemley and Shapiro, 2005; Lanjouw and Schankerman, 2004; Bessen and Meurer, 2008; Boldrin and Levine, 2008; Scotchmer, 2004). Distributed ledger technology offers an alternative by enabling fractional and tradable ownership through tokenization. Digital rights can be created and exchanged on blockchain networks, opening new channels for liquidity (Catalini and Gans, 2018; Howell et al., 2020; Benedetti and Kostovetsky, 2021; Chen, 2018; Cong and He, 2019; Yermack, 2017; Malinova and Park, 2017). Nevertheless, deciding when and how to release tokens under uncertainty remains difficult since poorly chosen schedules can materially erode value (Akcigit and Kerr, 2018; Bloom et al., 2020; Jones and Williams, 2000; Kortum and Lerner, 2000).

This article is aligned with the Frontiers research topic Blockchain and Tokenomics for Sustainable Development (Volume II) and follows its call for sustainability-driven tokenomics and governance models (Frontiers, 2025).

Practical experiments have already demonstrated feasibility. IPwe tokenized more than 500 patents using Black–Scholes pricing and recorded approximately 25 million USD in transaction volume. IBM placed over 800 patents on Hyperledger with fixed release plans and reached roughly 31 million USD. WIPO ran a pilot with more than 200 patents. Ocean Protocol applied bonding curves to intellectual property and data, generating close to 6 million USD (IPwe, 2024; IBM, 2022; Deloitte, 2023).

These cases show what is possible but also highlight structural weaknesses. Existing platforms do not incorporate risk preferences, they rely on rigid schedules that cannot adapt to shocks such as litigation, and they charge fees in the percent range instead of technically attainable near-negligible levels. Their dynamics follow known patterns of multi-sided platforms (Hagiu and Wright, 2015; Rochet and Tirole, 2006; Evans and Schmalensee, 2008; Rysman, 2009).

This study embeds risk-aware dynamic optimization into tokenization using a thermodynamic perspective. The key step is transforming the nonlinear Hamilton–Jacobi–Bellman equation into a linear partial differential equation via Φ = e − γ V , exploiting the negative sign of the quadratic gradient term from risk-sensitive control (Fleming and Soner, 2006). Consistent with path-integral control (Kappen, 2005; Todorov, 2009), this turns an otherwise intractable problem into one solvable in milliseconds. In data from 45 tokenized assets (real estate, art, and patents), the empirical estimate of risk aversion is γ = 2.1 ± 0.38 , while a composite objective calibrated to market-like parameters places the optimum near γ * ≈ 3.0 .

Simulations confirm the gap: the empirical value yields the highest survival rate, and the theoretical benchmark maximizes efficiency. Beyond the immediate application, links to the free-energy principle (Friston, 2010), maximum entropy (Jaynes, 1957), and large deviations (Touchette, 2009) point to deeper structural parallels between economic decision-making and adaptive systems. Together, these elements show how a thermodynamic approach can make tokenization more flexible, risk-sensitive, and efficient.

Our analysis begins with the risk-sensitive Hamilton–Jacobi–Bellman (HJB) equation. To facilitate readability, we reproduce it here once more in the version adapted to our framework: − ∂ V ∂ t = max u ∈ 0 , u max x u + μ x ∂ V ∂ x − u ∂ V ∂ f + σ 2 x 2 2 ∂ 2 V ∂ x 2 − γ σ 2 x 2 2 ∂ V ∂ x 2 . (3.1)

It is critical to note the negative sign in front of the quadratic gradient term. Without this sign, the thermodynamic transformation would fail because the nonlinear contribution would survive. Based on this structure, as shown in risk-sensitive control theory (Fleming and Soner, 2006), the transformation becomes straightforward. We define the substitution, Φ t , x , f = exp − γ V t , x , f .

By direct differentiation, we obtain the relations collected in Equation 3.2-3.6. ∂ Φ ∂ t = − γ Φ ∂ V ∂ t , (3.2) ∂ Φ ∂ x = − γ Φ ∂ V ∂ x , (3.3) ∂ V ∂ x = − 1 γ Φ ∂ Φ ∂ x , (3.4) ∂ Φ ∂ f = − γ Φ ∂ V ∂ f ⇒ ∂ V ∂ f = − 1 γ Φ ∂ Φ ∂ f , (3.5) ∂ 2 Φ ∂ x 2 = − γ Φ ∂ 2 V ∂ x 2 + γ 2 Φ ∂ V ∂ x 2 . (3.6)

One key observation is that the nonlinear terms cancel exactly when substituted into Equation 3.1. The second derivative contributes a positive quadratic term, which is offset by the negative quadratic term already present in the HJB. The algebra looks heavy at first, but the cancellation is straightforward once written out. What matters in practice is that the nonlinear term drops out and we end up with a linear PDE. This makes the difference between a system that is computationally intractable and one that can be solved quickly enough for actual tokenization platforms: ∂ Φ ∂ t + μ x ∂ Φ ∂ x + σ 2 x 2 2 ∂ 2 Φ ∂ x 2 = max u ∈ 0 , u max γ x u Φ + u ∂ Φ ∂ f . (3.7)

The optimal control emerges from maximizing the Hamiltonian in Equation 3.8. H u = x u − u ∂ V ∂ f . (3.8) Since H is linear in u (not quadratic), we have ∂ 2 H ∂ u 2 = 0 . Therefore, the control structure is not determined by phase transitions but by the sign of ∂ H ∂ u = x − V f combined with the box constraints: u * = u max , if  x > V f bang - bang high , 0 , if  x < V f bang - bang low , any  u ∈ 0 , u max , if  x = V f singular . (3.9)

If we add a small penalty − κ 2 u 2 to the cost function, we obtain H ( u ) = x u − V f u − κ 2 u 2 and thus the interior point u ⋆ ( t , x , f ) = Π [ 0 , u max ] ( ( x − V f ) / κ ) . For κ → 0 , the resulting bang-bang structure shown in Equation 3.9 for small κ > 0 , smooth controls arise. The apparent smoothness in certain parameter regimes arises from

• numerical regularization/discretization in implementation;

These are implementation choices or practical frictions, not intrinsic phase transitions of the Hamiltonian itself.

To determine the optimal risk parameter γ * , we define the multi-objective function F ( γ ) in Equation 3.10 that balances three critical performance dimensions of a tokenization strategy. The optimal parameter γ * is found by maximizing the function γ * = a r g   m a x γ > 0 F γ = a r g   m a x γ > 0 S γ ⋅ K γ ⋅ B γ . (3.10)

The components are defined as follows. These functional forms are modeling choices inspired by, but not identical to, standard financial metrics.

• The modified Sharpe component S ( γ ) is given in Equation 3.11. This term is inspired by the Sharpe ratio (Sharpe, 1994; Markowitz, 1952) but adapted for our context. It is estimated through Monte Carlo simulation (Metropolis and Ulam, 1949; Glasserman, 2003; Boyle et al., 1997) with 10,000 paths:

The product form S ⋅ K ⋅ B ensures that a strategy is only considered optimal if it performs well across all three dimensions simultaneously. This value is calibration-specific. Different plausible parameter choices ( μ , σ , r ) would shift γ * . Calibrations use equity-like orders of magnitude (Campbell et al., 1997; Fama and French, 2002). Sensitivity over ( μ , σ , r ) grids is reported in Supplementary Information section B.1.

To understand the deviation in our sample, we simulated ten heterogeneous agents with risk-aversion levels ranging from γ = 0.5 to γ = 4.5 . Each agent executed 2,000 Monte Carlo paths, giving 20,000 trajectories in total. The outcomes fall into three distinct behavioral regimes (not phase transitions). For γ < 1.5 , agents follow predominantly bang–bang strategies that swing between full tokenization and complete restraint. Between γ = 1.5 and γ = 4.5 , token release becomes smoother due to numerical regularization and implementation choices. At the highest values, above γ = 4.5 , strategies become very conservative, with minimal tokenization. Performance metrics highlight these differences. The detailed results are summarized in Table 1. Agent 1 ( γ = 0.5 ) earns the largest average revenue but suffers drawdowns above 60% and a success rate of only 43%. Agent 5 ( γ = 2.1 ) achieves strong survival outcomes, with 92% of trajectories meeting the revenue threshold. Agent 7 ( γ ≈ 3.02 ) approaches the multi-objective optimum under our calibration, although its success rate is 89%. Finally, Agent 10 ( γ = 4.5 ) represents a very conservative state, where returns are minimal and growth effectively stops.

The simulation results also highlight why market behavior gravitates toward γ ≈ 2.1 . At that point, strategies were not only effective in terms of returns but also survivable under stress. By contrast, the formal optimum at γ ≈ 3.0 gave slightly better scores on the composite metric, but at the cost of narrower margins. The data thus suggest that human decision-makers prefer robustness over squeezing out the last bit of efficiency.

We next examined how quickly different starting conditions approach the long-term estimate. Each agent began from a different initial risk preference and then simulated 10,000 Monte Carlo paths. Table 2 shows that, regardless of whether the process starts from γ 0 = 0.5 , 1.0, 3.0, 5.0, or even 10.0, the trajectories converge toward the same neighborhood at approximately γ = 2.1 .

The numerical patterns in the table suggest that convergence is rapid, although the speed depends on the starting point. The Ornstein-Uhlenbeck formula and probability calculation for convergence are given in Equation 4.2 (Øksendal, 2003). d γ t = θ γ empirical − γ t d t + ξ d W t , (4.2) where θ = 0.38 describes the reversion speed, ξ = 0.15 is the volatility, and the stationary mean is γ empirical = 2.1 (the empirical estimate from our sample).

The behavioral transitions (not phase transitions) occur approximately at:

• γ 1 = 1.5 ± 0.1 : transition from predominantly bang–bang to smoother control (due to regularization);

These values emerge from implementation choices and numerical regularization, not from bifurcations in the Hamiltonian (which remains linear in u ).

Note that Figure 3 visualizes only the real estate subset used in our study. The empirical estimate γ = 2.1 ± 0.38 is obtained from the full multi-asset dataset (real estate, artworks, and patents) and therefore does not assume that patent trajectories coincide with γ = 2.1 .

To demonstrate the practical advantages of our thermodynamic framework, we compare tokenization trajectories from existing platforms against our model’s output at both the empirical estimate from our sample ( γ = 2.1 ) and mathematical optimum under our calibration ( γ * ≈ 3.02 ) .

To assess robustness, we perform comprehensive sensitivity analysis on key parameters using Monte Carlo simulations (n = 10,000 paths; see Table 3) (Metropolis and Ulam, 1949; Glasserman, 2003; Boyle et al., 1997). See Supplementary Information for detailed results.

The analysis underscores that both the empirical estimate from our sample ( γ = 2.1 ) and mathematical optimum under our calibration ( γ * ≈ 3.02 ) represent stable operating points, with minimal sensitivity to moderate parameter variations.

Implementation involves three components: (i) valuation oracles estimating patent value using gradient boosting and transformers, (ii) HJB solvers using the thermodynamic transformation achieving 47 ms latency on NVIDIA RTX 3090, and (iii) blockchain infrastructure for token management with sub-10s total latency. See Supplementary Information for full algorithm with ten-agent verification.

Monte Carlo simulations (n = 10,000 paths per agent) quantify the framework’s effectiveness (see Table 4) (Metropolis and Ulam, 1949; Glasserman, 2003; Boyle et al., 1997). Risk-neutral strategies yield highest expected returns but face losses below 50% of the initial value in 42% of scenarios. The empirical estimate from our sample ( γ = 2.1 ) reduces this probability to 10% while achieving 92% success rate. The mathematical optimum under our specific calibration ( γ * ≈ 3.02 ) further reduces downside to 5% but at the cost of lower returns. The Sharpe ratio peaks near γ = 0.5 due to high returns, but practical success rate peaks at γ = 2.1 , explaining the empirical estimate in our sample (Markowitz, 1952; Sharpe, 1994; Merton, 1972).

Our results are consistent with a simple rule-of-thumb: robust tokenization (empirical γ ≈ 2.1 ) maximizes survival and stabilizes revenues; the benchmark ( γ * ≈ 3.0 ) optimizes a composite efficiency score under our calibration. That gap is not a flaw of the math but an adaptation to frictions and uncertainty.

Risk-aware schedules reduce volatility, tighten downside, and lower effective fees. In practical terms, this widens access to IP monetization and supports SDG-aligned innovation (e.g., resilient infrastructure and climate-related technologies) without relying on fixed, brittle plans. Linearization makes such schedules operational at platform latencies. The framework developed in this study provides a new way to optimize patent tokenization.

By linking economic risk aversion to the inverse temperature in statistical physics, the nonlinear Hamilton–Jacobi–Bellman equation, normally nonlinear and hard to solve, could be linearized through the transformation Φ = e − γ V . This step made it possible to compute optimal tokenization rates efficiently, even under uncertainty.

The empirical analysis of 45 tokenized assets shows that in our sample, the maximum likelihood estimate is γ = 2.1 ± 0.38 . In contrast, mathematical analysis of the composite function F ( γ ) = S ( γ ) ⋅ K ( γ ) ⋅ B ( γ ) places the optimum at γ * ∈ [ 2.9 , 3.2 ] (baseline γ * ≈ 3.02 ) under our specific calibration with parameters from equity-like markets (Campbell et al., 1997; Fama and French, 2002).

This gap amounts to roughly 25.395% in performance terms. Rather than a flaw, it highlights behavioral factors specific to our dataset. The highest success rate of 92% occurs at γ = 2.1 in our sample, even though the theoretical maximum under our calibration is located elsewhere. Our multi-agent simulations support this interpretation.

With ten heterogeneous agents and 20,000 simulated paths, the system repeatedly converged to the empirical estimate.

Agent 5, positioned at γ = 2.1 , achieved strong survival outcomes, with 92% of trajectories meeting the revenue threshold. Agent 7, at γ * ≈ 3.02 , maximized the composite function under our calibration but with slightly different trade-offs.

This pattern is consistent with behavioral finance insights: loss aversion and evolutionary fitness matter as much as formal optimization (Kahneman and Tversky, 1979; Barberis, 2013).

Computational benchmarks show an ≈ 850 × speedup (from 40 s to 47 ms on NVIDIA RTX 3090), making real-time applications feasible. The broader implication is that mathematical optima under specific calibrations are not always the points observed in empirical samples. Instead, systems under uncertainty tend to evolve toward strategies that balance multiple objectives in practice.

In our case, the estimate at γ = 2.1 reflects such balance in our specific dataset.

For future models, explicitly incorporating behavioral anchors and regulatory frictions will be essential if predictions are to match practice. Finally, the approach suggests a wider lesson.

The structures uncovered here are not unique to patents: they mirror principles found in physics, biology, and information theory.

The optimal behavior in complex systems appears to follow universal variational principles, balancing efficiency with survival.

From a practical angle, the numbers are clear: revenue volatility went down substantially, fees dropped to almost negligible levels, and survival rates improved. While these outcomes are encouraging, they are not presented as universal laws. They simply illustrate what can be achieved in our dataset under the chosen calibration. Future work may well find different balances once larger samples or other asset classes are considered. This makes it a candidate for broader application beyond patents to other illiquid assets such as real estate or fine art and positions thermodynamics as a useful lens for understanding financial decision-making under uncertainty.

Endowment effects and loss aversion lead to conservative tokenization. Patent holders may overvalue their IP, preferring to retain more control than mathematically optimal. This aligns with prospect theory, where losses loom larger than gains (Kahneman and Tversky, 1979; Barberis, 2013). The deviation in our sample may reflect these biases, suggesting that behavioral adjustments to the model could better match empirical data.