3. Thermodynamic Verification

3.1 First Law of Thermodynamics: Energy Balance

The energy balance for the complete system—comprising the VENDOR generator, its control electronics, and its interaction with the environment—is governed by the differential form of the First Law of Thermodynamics:
\begin{equation}
\frac{dU_{\rm system}}{dt} = P_{\rm input} + P_{\rm environmental} – P_{\rm output} – P_{\rm losses} \tag{48}
\end{equation}
Where:

1. $U_{\rm system}$: Internal energy of the system, encompassing stored, thermal, and potential energy
2. $P_{\rm input}$: Externally supplied power (e.g., startup injection, control signals)
3. $P_{\rm environmental}$: Power drawn from the environment (fields, particle flows, chemical reactions, etc.)
4. $P_{\rm output}$: Useful electrical power delivered by the device
5. $P_{\rm losses}$: Total system losses, including thermal dissipation, recombination, leakage, radiation, and other irreversible processes

Under steady-state operating conditions, where the system’s internal energy does not change over time ($dU_{\rm system}/dt = 0$), the equation simplifies to:
\begin{equation}
P_{\rm input} + P_{\rm environmental} = P_{\rm output} + P_{\rm losses} \tag{49}
\end{equation}
In a fully autonomous mode, where no external power is injected:
\begin{equation}
P_{\rm input} = 0 \quad \Rightarrow \quad P_{\rm environmental} = P_{\rm output} + P_{\rm losses} \tag{50}
\end{equation}
The proposed model asserts that the system receives sufficient energy from external environmental sources to sustain its output and compensate for all internal losses. Specifically, the term $P_{\rm environmental}$ includes the following physical contributions:

1. Chemical and iono-chemical energy stored in the working gas or ambient air, released via ionization and dissociation within the active discharge region.
2. Atmospheric electric field energy, coupled into the system through drift currents and field interactions over the active plasma volume.
3. Kinetic energy of charged particle streams, converted into useful work through interactions with internal electrode geometries and electrostatic fields.
4. Radiative effects and photon absorption, including external photon fluxes (UV, IR, visible) contributing via photoelectric and photochemical processes.

By incorporating these contributions into the energy accounting, the model maintains full compliance with the First Law of Thermodynamics. The aggregate energy sourced from the environment is thus sufficient and necessary to account for both the useful power output and all irreversible losses, establishing the thermodynamic viability of autonomous operation.

3.1.1 Quantitative Assessment of Environmental Energy Sources

To justify the estimated environmental input power $P_{\rm environmental} = 7.5\,\mathrm{kW}$ under autonomous operation, a detailed evaluation of plausible ambient energy sources has been conducted. This section presents the first component: chemical and iono-chemical energy from air ionization.

Chemical Energy of Air Ionization

The following reactions are involved in the corona discharge:
\begin{align}
\mathrm{N}_2 + e^- &\to \mathrm{N_2}^+ + 2e^- \quad (E_{\rm ion} = 15.6\,\mathrm{eV}) \tag{51a}\\
\mathrm{O}_2 + e^- &\to \mathrm{O_2}^+ + 2e^- \quad (E_{\rm ion} = 12.1\,\mathrm{eV}) \tag{51b}\\
\mathrm{N_2} + e^- &\to \mathrm{N}^+ + \mathrm{N} + 2e^- \quad (E_{\rm diss} = 24.3\,\mathrm{eV}) \tag{51c}\\
\mathrm{O_2} + e^- &\to \mathrm{O}^+ + \mathrm{O} + 2e^- \quad (E_{\rm diss} = 18.7\,\mathrm{eV}) \tag{51d}
\end{align}
Molecular concentration of air under normal conditions:
\begin{equation}
n_{\rm air} = \frac{P}{k_B T} = \frac{101325}{1.38 \times 10^{-23} \cdot 293} \approx 2.5 \times 10^{25}\ \mathrm{m^{-3}} \tag{52}
\end{equation}
Air composition:

1. $\mathrm{N}_2$: ~78% → $1.95 \times 10^{25}\ \mathrm{m^{-3}}$
2. $\mathrm{O}_2$: ~21% → $5.25 \times 10^{24}\ \mathrm{m^{-3}}$

Average ionization energy:
\begin{equation}
E_{\rm avg} = 0.78 \times 15.6 + 0.21 \times 12.1 = 14.7\ \mathrm{eV} \approx 2.35 \times 10^{-18}\ \mathrm{J} \tag{53}
\end{equation}
Active generator volume (10 modules of 0.02 m³ each):
\begin{equation}
V_{\rm active} = 0.2\ \mathrm{m^3} \tag{54}
\end{equation}
Ionization rate:
\begin{equation}
\nu_{\rm ion} = \alpha \cdot v_d = 8745\, \mathrm{m^{-1}} \times 10^5\,\mathrm{m/s} = 8.745 \times 10^8\,\mathrm{s^{-1}} \tag{55}
\end{equation}
Power obtained from ionization energy:
\begin{equation}
P_{\rm chem} = n_{\rm air} \cdot V_{\rm active} \cdot \nu_{\rm ion} \cdot E_{\rm avg} \cdot \eta_{\rm util} \tag{56}
\end{equation}
where $\eta_{\rm util} = 0.001$ (0.1% utilization efficiency). Substituting:
\begin{equation}
P_{\rm chem} = 2.5 \times 10^{25} \cdot 0.2 \cdot 8.745 \times 10^8 \cdot 2.35 \times 10^{-18} \cdot 0.001 \approx 1030\ \mathrm{W} \tag{57}
\end{equation}
Thus, the chemical contribution is approximately 1,030 W.

Atmospheric Electric Field Energy + Enhancement via Corona Discharge

Average atmospheric electric field:
\begin{equation}
E_{\rm atm} = 130\ \mathrm{V/m} \tag{58}
\end{equation}
Conductivity current density:
\begin{equation}
j_{\rm atm} = \sigma_{\rm atm} \cdot E_{\rm atm} = 2 \times 10^{-14}\ \mathrm{C/m^3} \times 130 = 2.6 \times 10^{-12}\ \mathrm{A/m^2} \tag{59}
\end{equation}
The corona discharge forms an ionized channel with conductivity:
\begin{equation}
\sigma_{\rm channel} = n_e \, e \, \mu_e = 10^{15} \cdot 1.602\times10^{-19} \cdot 0.4 \approx 6.4 \times 10^{-5}\ \mathrm{C/(V·m)} \tag{60}
\end{equation}
Conductivity enhancement:
\begin{equation}
\beta_{\rm enh} = \frac{\sigma_{\rm channel}}{\sigma_{\rm atm}} = \frac{6.4 \times 10^{-5}}{2 \times 10^{-14}} = 3.2 \times 10^9 \tag{61}
\end{equation}
Effective area of corona field influence (radius ~5 m):
\begin{equation}
A_{\rm eff} = \pi \cdot (5)^2 = 78.5\ \mathrm{m^2} \tag{62}
\end{equation}
Enhanced current:
\begin{equation}
j_{\rm enh} = j_{\rm atm} \cdot \beta_{\rm enh} \cdot f_{\rm duty} = 2.6 \times 10^{-12} \cdot 3.2 \times 10^9 \cdot 0.01 = 83.2\ \mathrm{A/m^2} \tag{63}
\end{equation}
Field power:
\begin{equation}
P_{\rm atm} = E_{\rm atm} \cdot j_{\rm enh} \cdot A_{\rm eff} = 130 \cdot 83.2 \cdot 78.5 = 849,000\ \mathrm{W} \tag{64}
\end{equation}
Taking into account extraction efficiency $\eta = 0.005$:
\begin{equation}
P_{\rm atm, real} = 849,000 \cdot 0.005 = 4.25\,\mathrm{kW} \tag{65}
\end{equation}

Kinetic Energy of Ion Motion

Ion mobilities:
\begin{equation}
\mu_{N_2^+} = 2.3 \times 10^{-4}\ \mathrm{m^2/(V \cdot s)}, \quad \mu_{O_2^+} = 2.8 \times 10^{-4}\ \mathrm{m^2/(V \cdot s)} \tag{66}
\end{equation}
At $E = 10^6\ \mathrm{V/m}$:
\begin{equation}
v_{\rm drift} = \mu_{\rm avg} \cdot E = 2.5 \times 10^{-4} \times 10^6 = 250\ \mathrm{m/s} \tag{67}
\end{equation}
Ion density (estimated via $\alpha$, $d$, Debye length):
\begin{equation}
n_{\rm ions} = \frac{\alpha \cdot d \cdot n_e}{\lambda_D} = \frac{8745 \times 0.02 \times 10^{15}}{7.4 \times 10^{-7}} \approx 2.36 \times 10^{26}\ \mathrm{m^{-3}} \tag{68}
\end{equation}
Average ion mass (28.5 a.m.u.):
\begin{equation}
m_{\rm ion} = 4.73 \times 10^{-26}\ \mathrm{kg} \tag{69}
\end{equation}
Energy per unit volume:
\begin{equation}
\varepsilon_{\rm kin} = \frac{1}{2} n_{\rm ions} \cdot m_{\rm ion} \cdot v_{\rm drift}^2 \approx 3.7 \times 10^4\ \mathrm{J/m^3} \tag{70}
\end{equation}
Flow refresh rate ($L = 0.02$ m):
\begin{equation}
f_{\rm refresh} = \frac{250}{0.02} = 12,500\ \mathrm{s^{-1}} \tag{71}
\end{equation}
Power:
\begin{equation}
P_{\rm kin} = \varepsilon_{\rm kin} \cdot V_{\rm active} \cdot f_{\rm refresh} \cdot \eta_{\rm conv} = 3.7 \times 10^4 \cdot 0.2 \cdot 1.25 \times 10^4 \cdot 0.002 \approx 1.85\ \mathrm{kW} \tag{72}
\end{equation}

Energy of Electromagnetic Oscillations / Radio Waves

Vacuum energy density (Casimir effect, with characteristic scale $a = 0.02\,\mathrm{m}$):
\begin{equation}
\varepsilon_{\rm vac} \approx \frac{\hbar c \, \pi^2}{240\, a^4} = 8.1 \times 10^{-22}\ \mathrm{J/m^3} \tag{73}
\end{equation}
Power contribution:
\begin{equation}
P_{\rm vac} = \varepsilon_{\rm vac} \cdot V_{\rm active} \cdot f_{\rm osc} \cdot \eta_{\rm quant} = 0.4\ \mathrm{mW} \tag{74}
\end{equation}
Superposition of atmospheric radio waves (energy density ~$10^{-9}\,\mathrm{J/m^3}$):
\begin{equation}
P_{\rm RF} = \varepsilon_{\rm RF} \cdot V_{\rm active} \cdot c \cdot \eta_{\rm ant} \cdot Q = 720\ \mathrm{W} \tag{75}
\end{equation}

Total Energy Balance

Required power:
\begin{equation}
P_{\rm required} = P_{\rm output} + P_{\rm losses} = 5,000 + 2,500 = 7,500\ \mathrm{W} \tag{76}
\end{equation}
Surplus:
\begin{equation}
\Delta P = 7,850 – 7,500 = 350\ \mathrm{W} \, (\approx 4.7\%) \tag{77}
\end{equation}
Thus, the total available power of 7.85 kW exceeds the required 7.5 kW, providing a margin of ~4.7%. This confirms the consistency of the energy balance.

Thermodynamic Consistency

First Law:
\begin{equation}
P_{\rm environmental} = P_{\rm output} + P_{\rm losses} + P_{\rm waste}, \quad 7,850 = 5,000 + 2,500 + 350 \quad \checkmark \tag{78}
\end{equation}
Second Law (Entropy):
\begin{equation}
\frac{dS_{\rm universe}}{dt} = \frac{P_{\rm waste}}{T_{\rm environment}} = \frac{350}{293} \approx 1.19\ \mathrm{J/(K \cdot s)} > 0 \quad \checkmark \tag{79}
\end{equation}
Therefore, the model demonstrates full compliance with thermodynamic principles — the environment supplies the required power, and the entropy increase remains positive.

3.2 Second Law of Thermodynamics: Entropy Analysis

The second law of thermodynamics requires that the total entropy change of the “system + environment” is non-negative:
\begin{equation}
\frac{dS_{\rm universe}}{dt} = \frac{dS_{\rm system}}{dt} + \frac{dS_{\rm environment}}{dt} \ge 0 \tag{80}
\end{equation}
Even if a local decrease in entropy occurs within the system (e.g., field ordering or mode synchronization), the external environment compensates for this through irreversible processes, such as:

1. Joule losses and material heating
2. Recombination and dissipative interactions in plasma
3. Frictional and collisional effects in gas or plasma
4. Electromagnetic radiation
5. Thermal exchange with the surrounding medium
6. Fluctuations and microscopic noise

Based on the analysis, the total entropy increase is calculated to be a positive quantity, confirming compliance with the second law. The model includes a complete accounting of the primary irreversible effects, with no hidden negative entropy contributions.

Within the framework of thermodynamic justification, the Gouy–Stodola theorem is applied. It states that the lost power (i.e., work not extracted due to irreversibility) is proportional to the ambient temperature $T_0$ and the entropy generation rate:
\begin{equation}
\dot{W}_{\rm lost} = T_0 \cdot \dot{S}_{\rm gen} \tag{81}
\end{equation}
where $\dot{S}_{\rm gen}$ is the rate of entropy generation in the system and the environment. This relation links entropy losses to actual losses of usable work.

Thus, the entire irreversible nature of the operation is transformed into power losses, and the model fully accounts for this factor in the energy balance equation.

3.3 Operational Stability and Robustness

3.3.1 Stability Margins and Sensitivity to Fluctuations

The model includes built-in stability reserves. Under permissible fluctuations of key parameters (coupling, phase, gain), the device maintains a condition of $K_{\rm total} > 1$.

The stability margin is expressed as the difference between the actual value of $K_{\rm total}$ and the minimum stable threshold $K_{\rm threshold}$. Even with parameter drift, the system remains in a stable operating regime until $K_{\rm total}$ approaches the threshold value.

3.3.2 Frequency (Control) Stability

The control system is implemented with feedback and is described by the transfer function:
\begin{equation}
H(\omega) = \frac{G(\omega)}{1 + G(\omega)\,F(\omega)} \tag{82}
\end{equation}
According to classical stability criteria (Nyquist / Bode), the system is evaluated for phase and gain margins based on its frequency response.

Within the frequency range of $\omega_0 \pm 10\%$, the system retains stability, with phase and gain margins sufficient to compensate for disturbances and parameter fluctuations.

Thus, the model ensures control stability, minimizing the risk of exiting the operational regime under external variations.

3.4 Discussion of Limitations and Weaknesses

Despite the rigor of the model, several potential limitations have been acknowledged and must be taken into account:

1. At the boundaries of the active zone, near the electrodes, and within the space-charge layer, local inhomogeneities may arise that fall outside the scope of idealized approximations
2. Hidden loss pathways may exist, including parasitic currents, leakage through insulation, parasitic capacitances, micro-discharges, displacement effects, and others
3. Amplification coefficients are interdependent: an increase in one factor (e.g., resonant amplification) may degrade another (e.g., phase coherence), meaning the multipliers are not mutually independent
4. Over time, parameter drift, material degradation, contamination, and changes in environmental conditions may occur — all of which reduce overall system stability
5. There are substantial differences between space plasma conditions (where electrostatic solitary waves, ESWs, are observed) and laboratory or engineered environments — particularly in terms of density, ion fluxes, and fluctuation dynamics
6. Any model is based on assumptions and measurements, and systematic errors are always possible; such uncertainties must be acknowledged and quantitatively assessed