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# A simplified thermodynamically-consistent single-step mechanism for hydrogen combustion

Kevin Turgut, Ashwin Chinnayya, Pierre Boivin, Omar Dounia

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Kevin Turgut, Ashwin Chinnayya, Pierre Boivin, Omar Dounia. A simplified thermodynamically-consistent single-step mechanism for hydrogen combustion. 2025. hal-05343706

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Distributed under a Creative Commons Attribution 4.0 International License
A simplified thermodynamically-consistent single-step
mechanism for hydrogen combustion

Kevin Turgut$^{a,b,*}$, Ashwin Chinnayya$^b$, Pierre Boivin$^c$, Omar Dounia$^a$

$^a$CERFACS, 42 avenue Gaspard Coriolis, Toulouse, 31100, France

$^b$Institut PPRIME - UPR 3346 - CNRS - ISAE-ENSMA - Université de Poitiers, 1
Avenue Clément Ader, Futuroscope-Chasseneuil Cedex, 86961, France

$^c$Aix Marseille Univ, CNRS, Centrale Med, M2P2, Marseille, France

**Abstract**

Effective combustion modeling relies on the precise estimation of hydrogen-
air combustion characteristics. In this sense, Millán-Merino and Boivin [1]
designed a single-step mechanism for hydrogen combustion that accurately
recovers the adiabatic flame temperature using a variable stoichiometric coef-
ficient formalism. The present study proposes a drastic simplification of this
approach (further reducing computational cost and complexity) and formally
clarifies the contribution of the variable stoichiometric coefficients and their
evolution across the flame internal structure.

Keywords: Hydrogen, Combustion, Reduced chemistry

1. Context

In recent years, significant efforts have been dedicated to developing a re-
duced single-step formalism for hydrogen combustion, leveraging analytical
Quasi-Steady State (QSS) analysis [1–4]. These schemes have several advan-
tages over more detailed mechanisms even for the case of H₂/air chemistry
[5]: (i) a simple reaction rate which does not suffer from the stiffness associ-
ated with some elementary reactions in the detailed chemistry; (ii) reduced
computational cost and enhanced interpretability; (iii) particularly suited
for theoretical derivations. However, a remaining limitation of these reduced
schemes lies in their prediction of the adiabatic flame temperature, which

*Corresponding author at: CERFACS, Toulouse, France.
Email address: turgut@cerfacs.fr (Kevin Turgut)

Preprint submitted to International Journal of Hydrogen Energy

November 3, 2025
tends to be overestimated, by approximately 200 K, near stoichiometric conditions in H₂/air flames.

Millán-Merino and Boivin [1] demonstrated that a one-step model incorporating seven species can accurately predict the adiabatic flame temperature, T<sub>ad</sub>, and laminar flame speed, S<sub>L</sub>, for hydrogen-air combustion:

$$ \nu'_{\text{H}}\text{H}_2 + \nu'_{\text{O}}\text{O}_2 \xrightarrow{\omega} \nu'_{\text{H}_2\text{O}}\text{H}_2\text{O} + \nu'_{\text{OH}}\text{OH} + \nu'_{\text{H}}\text{H} + \nu'_{\text{O}}\text{O} \quad (1) $$

A distinguishing feature of this model is its use of variable stoichiometric coefficients $\nu'_i$ to ensure an accurate prediction of the adiabatic temperature, considering the contribution of three dissociation reactions (R2-R4):

$$ (\text{R1}) \quad \text{H}_2 + \frac{1}{2}\text{O}_2 \xrightarrow{\xi_1} \text{H}_2\text{O} $$

$$ (\text{R2}) \quad \text{H}_2\text{O} \xrightarrow{\xi_2} \frac{1}{2}\text{H}_2 + \text{OH}, \quad (\text{R3}) \quad \frac{1}{2}\text{H}_2 \xrightarrow{\xi_3} \text{H}, \quad (\text{R4}) \quad \frac{1}{2}\text{O}_2 \xrightarrow{\xi_4} \text{O} $$

The novelty in [1] lies in the fact that the model removes the need for the rate of reactions (R2-R4) (and any associated need for adjustment) instead only requires knowledge about their equilibrium constant, which is taken into account in the coefficients $\nu'_i$:

$$ \nu'_{i} = 2 \sum_{k=1}^{4} \frac{\xi_{k}}{\xi_{1}^{m}} \nu_{k,i}, \quad (2) $$

where $\nu_{k,i}$ represents the stoichiometric coefficient for species i in the reaction (Rk) and $\xi_1^m = \min(\phi_Z, 1)$ is the molar advancement of the irreversible reaction (R1)<sub>f</sub>. $\phi_Z$ is the element-based equivalence ratio given by:

$$ \phi_Z = \frac{\frac{1}{4}Z_{\text{mole,H}}}{\frac{1}{2}Z_{\text{mole,O}}} \quad (3) $$

where $Z_{\text{mole,e}}$ is the elemental mole fraction of element e.

The molar advancement $\xi_1$ for reaction (R1) is obtained by solving the following cubic equation:

$$ K_1 = \frac{n_{\text{H}_2\text{O}}^0 + \xi_1}{\phi - \xi_1} \sqrt{\frac{2N^0 - \xi_1}{1 - \xi_1}} \quad (4) $$
with $K_1$ the associated equilibrium constant, $n_i^0$ the reduced initial number of moles of the species $i$ at the beginning of each iteration, and $N^0 = \sum_i n_i^0$ the total number of reduced moles. We further clarify here that $\phi$ denotes the local equivalence ratio given by $2n_{H_2}^0/n_{O_2}^0$. The molar advancements of the other reactions are computed independently under the assumption that major-species concentrations (H$_2$, O$_2$, and H$_2$O) remain unchanged by the dissociation reactions. This assumption is highlighted by the superscript 0 on $\xi_k$ in Eqs. (5–7)

$$ \xi_2^0 = K_2 (n_{H_2O}^0 + \xi_1) \sqrt{\frac{N^0 - \frac{\xi_1}{2}}{\phi - \xi_1}} - n_{OH}^0, \quad (5) $$

$$ \xi_3^0 = K_3 \sqrt{(\phi - \xi_1) \left(N^0 - \frac{\xi_1}{2}\right)} - n_H^0, \quad (6) $$

$$ \xi_4^0 = \frac{K_4}{2} \sqrt{(\phi - \xi_1) (2N^0 - \xi_1)} - n_O^0. \quad (7) $$

where $K_k$ is the equilibrium constant for reaction (Rk). $K_2 - K_4$ depend on the equilibrium temperature $T_{eq}$, which is a-priori unknown. An iterative method is proposed in [1] to estimate $T_{eq}$ for each point on the flame profile using a gradient descent algorithm (here, Newton-Raphson).

## 2. Methodology and results

In the original paper of [1], all the species H, OH and O are assumed to contribute to equilibrium and are therefore introduced in the single-step formulation. The paper is motivated by the realization that O and H species contribute negligibly to equilibrium near stoichiometry, as highlighted in Fig. 1, which shows a difference of nearly one order of magnitude in the equilibrium mole fractions X of the intermediate species O, H, and OH in the considered zone, suggesting that they can be neglected in Eq. (1). Of course, this reasoning is only valid in the context of the single-step formulation adopted here and does not imply that these species have a negligible contribution to the internal structure of detailed-chemistry H$_2$ flames, especially in the chain-branching zone. In the present formulation, the addition of these intermediates therefore only contributes to equilibrium via the inclusion of slow dissociation reactions (and not fast chain-branching ones). The motivation of the paper also comes from the still heavy computational
<figure><img src="image_3.png" /><figcaption>Figure 1: Comparison of the equilibrium molar fractions X as a function of equivalence ratio φ between S7R1 and the San Diego detailed chemistry S9R21 [6], for H₂/air mixture with pressure P = 1 atm.</figcaption></figure>

time of the proposed model in [1], which is only, in average, 5% faster than the detailed 9 species 21 reactions formalism (this result is obtained with the chemical solver Cantera [7] and the following initial condition: pressure: 101325 Pa, initial temperature: 300 K). The contribution of O, H and OH to the system is studied by considering the following alternative mechanisms:

$$ (S7R1: \xi_k \neq 0) \quad \nu'_H H_2 + \nu'_O O_2 \xrightarrow{\omega} \nu'_{H_2O} H_2O + \nu'_{OH} OH + \nu'_H H + \nu'_O O \quad (8) $$

$$ (S6R1: \xi_4 = 0) \quad \nu'_H H_2 + \nu'_O O_2 \xrightarrow{\omega} \nu'_{H_2O} H_2O + \nu'_{OH} OH + \nu'_H H \quad (9) $$

$$ (S5R1: \xi_3 = \xi_4 = 0) \quad \nu'_H H_2 + \nu'_O O_2 \xrightarrow{\omega} \nu'_{H_2O} H_2O + \nu'_{OH} OH \quad (10) $$

$$ (S4R1: \xi_2 = \xi_3 = \xi_4 = 0) \quad \nu'_H H_2 + \nu'_O O_2 \xrightarrow{\omega} \nu'_{H_2O} H_2O. \quad (11) $$

S7R1 corresponds to the original proposition [1], and $\nu'_i$ still given by (2).

The global reaction rate $\omega$ associated to each mechanism is identical to the work of [1] for its validity for the whole range of flammable H₂/air mixtures. It is of the form $\omega = \lambda\zeta$, where $\lambda$ is a characteristic inverse branching time associated with the chemistry of premixed flame. $\zeta$ is a concentration accounting for H, O and OH species growth. In this study, all results are obtained using mixture-averaged transport [7].

Fig. 2 illustrates the relative importance of the additional reactions (R2)-(R4) in terms of adiabatic flame temperature predictions. Fig. 2 also includes
<figure><img src="image_4.png" /><figcaption>Figure 2: Comparison of the adiabatic temperature $T_{ad}$ as a function of equivalence ratio $\phi$ between S7R1, S6R1, S5R1, S4R1, S4R1 irreversible and the San Diego detailed chemistry S9R21 [6], for H$_2$/air mixture with pressure $\mathbf{P} = 1$ atm and $T_0 = 300$ K.</figcaption></figure>

the irreversible S4R1 scheme $2\text{H}_2 + \text{O}_2 \xrightarrow{\omega} 2\text{H}_2\text{O}$ and is equivalent to S4R1 with the constraint $\xi_1 = \xi_1^m$. This simple scheme results in serious overestimation of the adiabatic flame temperature around stoichiometry (by more than 100 K). On the other hand, S4R1, which accounts for the reversibility of reaction (R1), improves the accuracy of the results. Finally, the additional consideration of reaction (R2) in S5R1 suffices to retrieve the correct adiabatic flame temperature of hydrogen-air flames at atmospheric pressure, for all equivalence ratios $\phi$. For rich and lean mixtures, the temperature is low enough for the impact of the dissociation processes to be negligible. As shown in Tab. 1, S5R1 corresponds to a considerable reduction in computational cost compared to the mechanism proposed in [1] in addition to the simplification it provides to it. S5R1 is on average 50% faster and even faster in lean and rich conditions as suggested in the variance associated to the computational times.

Furthermore, the S5R1 model is able to reproduce the correct adiabatic flame temperature and flame speed over a wide pressure range ([0.5, 20] atm), as shown in Fig. 3.

However, the definition of $\omega$ still requires improvement, since the laminar flame speed $S_L$ predicted by S5R1 departs from detailed chemistry results when the initial conditions (pressure, temperature, and dilution) strongly differ from those tested in [1], as illustrated in Fig 3 and in the Supplementary
<table><caption>Table 1: Computation time of the chemical Cantera solver using S7R1 and S5R1 models. The empirical mean and variance are calculated through a range of $\phi$ in [0.4, 5] with 540 samples, at isentropic compression with $T_0 = 300$ K at $P = 1$ atm.</caption><thead><tr><th>Pressure</th><th>Model</th><th>Empirical mean</th><th>Empirical variance</th></tr></thead><tbody><tr><td rowspan="2">1 atm</td><td>S7R1</td><td>0.423116 s</td><td>0.05124092 s</td></tr><tr><td>S5R1</td><td>0.27683646 s</td><td>0.03168761 s</td></tr><tr><td rowspan="2">10 atm</td><td>S7R1</td><td>0.70783529 s</td><td>0.48815777 s</td></tr><tr><td>S5R1</td><td>0.29929146 s</td><td>0.04432407 s</td></tr><tr><td rowspan="2">30 atm</td><td>S7R1</td><td>0.67850577 s</td><td>0.20812965 s</td></tr><tr><td>S5R1</td><td>0.33511044 s</td><td>0.04289359 s</td></tr><tr><td rowspan="2">0.5 atm</td><td>S7R1</td><td>0.56822927 s</td><td>0.21442538 s</td></tr><tr><td>S5R1</td><td>0.31241129 s</td><td>0.0846491 s</td></tr></tbody></table>

<figure><img src="image_5.png" /><figcaption>Figure 3: Comparison of the adiabatic temperature $T_{ad}$ and laminar flame speed $S_L$ as a function of the equivalence ratio $\phi$ between S5R1 and the San Diego detailed chemistry S9R21 [6]. Bottom: **P = 0.5 atm** with $T_0 = 300$ K. Top: H$_2$/air mixture with pressure **P = 1 atm** (black), **P = 10 atm** (blue) and **P = 20 atm** (red), initial temperature given by an isentropic compression from initial conditions of 1 bar and 300 K.</figcaption></figure>
Material, where the validity range of the reduced mechanism are further analyzed. This leads to the following validity ranges: (i) pressure in the range [1, 20] atm with $T_0$ obtained by isentropic compression, (ii) pressure in the range [0.4, 1] atm with fixed initial temperature $T_0 = 300$ K, (iii) temperature in the range [280 - 400] K with fixed pressure $P = 1$ atm, (iv) dilution with N₂ in the range $X_{N_2}/X_{O_2} \in [3, 4]$, (v) dilution with H₂O in the range $X_{H_2O}/X_{H_2} \in [0, 0.1]$ and (vi) dilution with CO₂ in the range $X_{CO_2}/X_{H_2} \in [0, 0.2]$.

The discrepancies appearing outside these validity ranges mainly concern the flame speed and arise from the formulation of the chemical source term $\omega = \lambda\zeta$ in [1, 3] where a $\zeta$ is formulated as a geometrical average of its analytic formulations which are valid far away from stoichiometry. This hinders the model's adaptability to varying initial conditions when both $S_L$ and $T_{ad}$ must be accurately captured. Correction functions could be applied to the $\omega$ term to solve this problem, not done here for brevity. The present paper focuses exclusively on the issue of adiabatic temperature and thermodynamics, as outlined in the Context section.

### 3. Formal analysis and numerical improvement

We also clarify here the contribution of the dissociation reaction (R2) in the variable stoichiometric-coefficient formulation of (S5R1). Eq. (2) is expanded for the (S5R1) scheme:

$$ \nu'_{i} = 2 \frac{\xi_{1}}{\xi_{1}^{m}} \nu_{1,i} + 2 \frac{\xi_{2}}{\xi_{1}^{m}} \nu_{2,i}, \qquad (12) $$

where $\xi_1/\xi_1^m$ and $\xi_2/\xi_1^m$ represent the contributions of reaction (R1) and reaction (R2) respectively. The exact formula for $\xi_2$ is given by the root of a cubic polynomial:

$$ K_2 = \frac{(n_{OH}^0 + \xi_2) \sqrt{n_{H_2}^0 - \xi_1 + \frac{1}{2}\xi_2}}{(n_{H_2O}^0 + \xi_1 - \xi_2) \sqrt{N^0 - \frac{1}{2}\xi_1 + \frac{1}{2}\xi_2}}. \qquad (13) $$

Instead of solving Eq. (13) exactly for $\xi_2$, one can use a Taylor series approximation of $K_2$ using $\xi_2$ as a small parameter, or more precisely:

$$ \epsilon_1 = \frac{\xi_2}{n_{H_2}^0 - \xi_1} \ll 1, \quad \epsilon_2 = \frac{\xi_2}{n_{H_2O}^0 + \xi_1} \ll 1, \quad \epsilon_3 = \frac{\xi_2}{2N^0 - \xi_1} \ll 1. \qquad (14) $$
The zero-th order approximation of $\xi_2$ in the limit $\epsilon_k \to 0$ is exactly the formula $\xi_2^0$ provided by Eqs (17–19), for which a more rigorous justification is provided here.

Eq. (13) can be expressed in cubic-polynomial form:

$$ \sum_{j=0}^{3} C_j \xi_2^j = 0 \qquad (15) $$

where the coefficients $C_j$ are given by:

$$ C_0 = (n_{\text{OH}}^0)^2 - (K'_2)^2 \qquad (16) $$

$$ C_1 = \frac{(n_{\text{OH}}^0)^2}{2(n_{\text{H}_2}^0 - \xi_1)} + 2n_{\text{OH}}^0 - (K'_2)^2 \left( \frac{1}{2N^0 - \xi_1} - \frac{2}{n_{\text{H}_2\text{O}}^0 + \xi_1} \right) \qquad (17) $$

$$ C_2 = 1 + \frac{2n_{\text{OH}}^0}{2(n_{\text{H}_2}^0 - \xi_1)} - (K'_2)^2 \left( -\frac{2}{(n_{\text{H}_2\text{O}}^0 + \xi_1)(2N^0 - \xi_1)} + \frac{1}{(n_{\text{H}_2\text{O}}^0 + \xi_1)^2} \right) \qquad (18) $$

$$ C_3 = \frac{1}{2(n_{\text{H}_2}^0 - \xi_1)} - (K'_2)^2 \left( \frac{1}{2N^0 - \xi_1}(n_{\text{H}_2\text{O}}^0 + \xi_1)^2} \right) \qquad (19) $$

$K'_2$ depends on $K_2$ and $\xi_1$ and is given by:

$$ K'_2 = K_2 \frac{(n_{\text{H}_2\text{O}}^0 + \xi_1) \sqrt{N^0 - \xi_1/2}}{\sqrt{n_{\text{H}_2}^0 - \xi_1}} \qquad (20) $$

$$ = \frac{n_{\text{OH}}^0 + \xi_1}{1 - \frac{\xi_2}{n_{\text{H}_2\text{O}}^0} + \xi_1} \frac{\sqrt{1 + \frac{\xi_2}{2(n_{\text{H}_2}^0 - \xi_1)}}}{\sqrt{1 + \frac{\xi_2}{2(N^0 - \frac{1}{2}\xi_1)}}} \qquad (21) $$

Using the small parameters around which a Taylor series expansion is performed (Eq (14)), the equation is rewritten as:

$$ C_1 \xi_2 = 2 n_{\text{OH}}^0 \xi_2 + \frac{(n_{\text{OH}}^0)^2}{2} \xi_1 - (K'_2)^2 (\epsilon_3 - 2\epsilon_2) \qquad (22) $$

$$ C_2 \xi_2^2 = \xi_2^2 + n_{\text{OH}}^0 \xi_2 \epsilon_1 - (K'_2)^2 (-2\epsilon_2 \epsilon_3 + \epsilon_2^2) \qquad (23) $$

$$ C_3 \xi_2^3 = \frac{1}{2} \xi_2^2 \epsilon_1 - (K'_2)^2 \epsilon_3 \xi_2^2 \qquad (24) $$

$$ C_0 = (n_{\text{OH}}^0)^2 - (K'_2)^2 \qquad (25) $$
The zeroth-order solution of this equation in the limit $\epsilon_k \to 0$ corresponds to Eq. (5), i.e $\xi_2^0$:

$$ \sum_{j=0}^{3} C_j \xi_2^j = 0 \iff (\xi_2 + n_{\text{OH}}^0)^2 = (K_2')^2 + \mathcal{O}(\epsilon_1, \epsilon_2, \epsilon_3) \quad (26) $$

$\xi_2^0$ corresponds to the positive solution to Eq. (26). A first-order solution to Eq. (13) can also be obtained by retaining the first order terms in Eqs. (22)-(25). The solution is denoted by $\xi_2'$ and is expressed as:

$$ \xi_2' = \frac{\xi_2^0}{1 + K_2' \left[ \frac{1}{4(n_{H_2}^0 - \xi_1)} - \frac{1}{4(N^0 - \xi_1/2)} + \frac{1}{(n_{H_2O}^0 + \xi_1)} \right]} + \mathcal{O}(\epsilon_1^2, \epsilon_2^2, \epsilon_3^2) \quad (27) $$

The spatial profile of normalized molar advancement terms $\xi_1/\xi_1^m$ and $\xi_2/\xi_1^m$, appearing in Eq. (12), are displayed in Fig. 4 for a lean and stoichiometric H₂/air flames. Fig. 4 highlights the capacity of the formalism proposed here to automatically adapt to the local conditions to ensure an accurate prediction of the equilibrium state without the need to adjust the reaction rate of the additional dissociation reactions: (i) the contribution of the dissociation reaction (R2) and the reverse reaction (R1) are both negligible in the first part of the flame ($T \le 1600$ K), i.e. $\xi_2 \ll \xi_1^m$ and $\xi_1 \approx \xi_1^m$, and the scheme essentially reduces to the irreversible reaction $2H_2 + O_2 \xrightarrow{\xi_1^m} 2H_2O$, thereby confirming their negligible impact in H₂/air flames sufficiently away from stoichiometry. (ii) for higher temperature, the contribution of the first reaction progressively vanishes $\xi_1 \to 0$ as the end gas region is approached and the (S5R1) scheme smoothly reduces to the reversible dissociation reaction $H_2O \xrightarrow{\xi_2} \frac{1}{2}H_2 + OH$, thereby providing the OH species necessary to accurately predict equilibrium. Finally, $\xi_2^0$ is confirmed to be a good approximation for $\xi_2$, the first order approximation negligibly deviating from $\xi_2^0$, except in the vicinity of the equilibrium state for mixtures out of stoichiometry (see Fig. 4-bottom). The first-order approximation mitigates the sharp variations in $\xi_2$ that were previously observed with $\xi_2^0$ before reaching equilibrium ($\xi_k = 0$). This improves numerical stability and slightly speeds up the algorithm. This can be shown by measuring the stiffness of the associated chemical kinetics ODE system, evaluated by measuring the lowest chemical timescale associated to each model. This is expressed through the following formulation:

$$ S(x) = \min_k \tau_k(x) = \min_k C_k(x)/\dot{\omega} \quad (28) $$
<figure><img src="image_6.png" /><figcaption>Figure 4: Evolution of the terms $\xi_1/\xi_1^m$ (black) and $\xi_2/\xi_1^m$ (blue) in the variable stoichio-metric coefficients given by Eq. (12) across a H$_2$/air flame at atmospheric conditions. Top: $\phi = 1$, bottom: $\phi = 0.6$. The zero-th (Eq. (5)), first (Eq. (27)) order approximations for $\xi_2/\xi_1^m$ are displayed in dashed and solid blue curves, respectively. The exact solution for $\xi_2/\xi_1^m$ (Eq. (15)) is displayed in red dot-dashed lines.</figcaption></figure>
where $\tau_k \equiv C_k/\dot{\omega}_k$ denotes the characteristic timescale of species k. The minimum characteristic timescale over the flame,

$$S_{\min} = \min_{x} S(x) \qquad (29)$$

quantifies the needs in terms of temporal resolution for each model. Using this approach for the case $P = 1$ atm, $T_0 = 300$ K, and $\phi = 1$, $S_{\min}$ is found to be approximately 2.56 $\mu$s for the S7R1 model, 2.63 $\mu$s for the S5R1 model using $\xi_2^0$ and 38.3 $\mu$s for the S5R1 model using $\xi_2'$. The first order approximation for $\xi_2$ therefore reduces the need for temporal resolution in the chemical integration step by one order of magnitude.

## 4. Conclusions

Overall, the paper highlights the capacity of the formalism proposed in [1] to retrieve the correct adiabatic flame temperature, and drastically simplified the formalism by reducing the number of dissociation reactions needed to do so. Finally, the role of the variable stoichiometric coefficient formulation of (S5R1) was also clarified, with particular emphasis on the capacity of S5R1 to recover the behavior of simplified schemes as well-defined limits for $\xi_1$ and $\xi_2$. Note that the mechanism proposed remains compatible with the correction presented in [1] to correctly account for the auto-ignition chemistry.

### CRediT authorship contribution statement

**Kevin Turgut:** Conceptualization. Software. Investigation. Writing - Original Draft. Writing - Review & Editing. **Ashwin Chinnayya:** Writing - Review & Editing. Funding acquisition. **Pierre Boivin:** Writing - Review & Editing. **Omar Dounia:** Conceptualization. Formal analysis. Writing - Original Draft. Writing - Review & Editing. Supervision. Funding acquisition.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments

This work was supported by a grant from the French National Research Agency (ANR) as part of the AAPG 2023 call (ANR-23-CE05-0021). For the purpose of Open Access, a CC-BY public copyright license has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission.

Supplementary Material

Additional results are provided in a Supplementary Material. The code corresponding to the mechanisms described in this paper will be available upon request.

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