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Physico-Mathematical Justification of the Feasibility of the Autonomous Energy Generator VENDOR: Rigorous Validation Based on Spaceborne Observations of Electrostatic Solitons

Authors: O.Krishevich, V.Peretyachenko

Abstract

This paper presents a physico-mathematical framework for evaluating the feasibility of the VENDOR autonomous operating regime within a multimodule nonlinear electrodynamic system (patent WO2024209235). The methodology is informed by space-based studies of electrostatic solitary waves (ESWs / ES structures) in the Earth’s magnetosphere (Leonenko et al., JETP Letters, 2025) and is applied here strictly as an analogical reference for nonlinear stability, wave persistence, and low-dissipation transport in plasma environments. The framework comprises the following stages:
  1. Mathematical modeling of avalanche ionization in gaseous or rarefied media based on the Townsend mechanism, incorporating space charge effects and the Raether limit constraint.
  2. Derivation of resonance phenomena and parametric amplification, including nonlinear components, mode coupling, and saturation-resilience analysis.
  3. Analysis of multimodule synchronization, involving phase locking of oscillatory modes, field superposition effects, and dynamic phase-shift compensation.
  4. Rigorous thermodynamic verification, including energy balance, continuity, conservation laws (energy and entropy), and comprehensive evaluation of loss channels (thermal, radiative, recombinative, etc.).
Within the proposed model, it is shown that under specific configurations — including gas/plasma density, electrode geometry, field topology, and coherent phase alignment — the system may enter a stable autonomous oscillatory regime characterized by an internal closed-loop gain exceeding unity (in the sense of nonlinear feedback and resonance). This gain must not be interpreted as energy creation and does not imply any violation of conservation laws.

1. Introduction

Contemporary science is faced with a foundational question: Is it possible to engineer autonomous operating regimes in nonlinear electrodynamic systems where small control inputs organize large internal circulating energy flows, while remaining fully consistent with the laws of thermodynamics and energy conservation? A key requirement in this context is rigorous control over all energy-exchange processes, including loss mechanisms, nonlinear feedback, saturation effects, and fluctuations. In recent years, the Magnetospheric Multiscale Mission (MMS) has provided high-resolution data on electromagnetic and electrostatic disturbances within the Earth’s magnetosphere (e.g., Hansel et al., Mapping MMS Observations of Solitary Waves, 2021). Notably, Leonenko et al. (2025) reported intense electrostatic solitary waves (ESWs) in the Central Plasma Sheet (CPS) of the magnetotail, with electric field amplitudes reaching ~100 mV/m. These structures are stable nonlinear waveforms capable of transporting and redistributing energy within plasma environments with minimal dissipative losses. Such naturally occurring phenomena motivate a careful engineering question: If the physical mechanisms associated with stable nonlinear electrostatic structures can be translated into an engineered context, they may inform design principles for regime stability, low-loss transport, and robust oscillatory dynamics — without implying any new energy source beyond the externally sustained electrodynamic conditions of operation. However, significant differences exist between spaceborne plasma environments and terrestrial devices (e.g., density, scale, boundary conditions, inhomogeneity, dissipative losses, and instabilities). This necessitates a rigorous physico-mathematical translation and validation of the underlying principles. This work presents a stepwise, internally coherent justification for the feasibility of the VENDOR autonomous energy generator, structured as follows:
  1. Mathematical modeling of avalanche and corona ionization in gas/plasma media, accounting for space charge buildup and the Raether limit.
  2. Analysis of resonance phenomena, parametric amplification, nonlinear interactions, and saturation dynamics.
  3. Multimodal phase synchronization within a modular system architecture, including field alignment and active phase compensation.
  4. Thermodynamic validation, covering complete energy balance, dissipation mechanisms, system stability, and compliance with conservation laws.
We demonstrate that, under carefully tuned physical parameters (geometry, medium density, field intensities), it is possible to achieve a stable autonomous regime characterized by an internal closed-loop gain \begin{equation} K_{\rm total} > 1 \tag{1} \end{equation} where \(K_{\rm total}\) denotes a composite feedback-and-resonance gain of the oscillatory system. This criterion is used here as a regime-stability condition in nonlinear dynamics and must not be interpreted as net energy creation or a violation of fundamental physical laws. The following sections provide theoretical derivations, numerical evaluations, and experimental observations consistent with the proposed model.

2. Theoretical Foundations

2.1 Parameters of Electrostatic Solitons and Technical Analog

In the study by Leonenko et al. (2025) — along with related investigations into electrostatic structures in the Earth’s magnetosphere — the following average and peak parameters of electrostatic solitary waves (ESWs) in the central plasma sheet (CPS) of the magnetotail have been documented. For clarity and precision, we report the values with their stated uncertainties:

Temporal Characteristics

Duration of a single solitonic pulse: \begin{equation} \tau = (15 \pm 5)\times 10^{-3}\ \mathrm{s} \tag{2} \end{equation} Interaction / coherence time (i.e., the duration over which the structure remains spatially localized): \begin{equation} \Delta t = (12 \pm 3)\times 10^{-3}\ \mathrm{s} \tag{3} \end{equation}

Electrical Characteristics

Average electric field amplitude:
\begin{equation} E = (25 \pm 8)\times 10^{-3}\ \mathrm{V/m}, \quad \text{with peaks up to } 100\times10^{-3}\ \mathrm{V/m} \tag{4} \end{equation}
Longitudinal propagation velocity of the soliton: \begin{equation} v = (650 \pm 350)\ \mathrm{km/s} \tag{5} \end{equation}

Beam Energetic Parameters

Change in kinetic energy per electron:
\begin{equation} \Delta E_{\rm beam} = (1.0 \pm 0.1)\ \mathrm{keV} = (1.602 \pm 0.016)\times10^{-16}\ \mathrm{J} \tag{6} \end{equation}
Electron beam density (possibly off-peak):
\begin{equation} n_{\rm beam} = (0.15 \pm 0.02)\ \mathrm{cm}^{-3} = (1.5 \pm 0.2)\times10^{5}\ \mathrm{m}^{-3} \tag{7} \end{equation}
Observed power density:
\begin{equation} P_{\rm obs} = j \cdot E’ \approx (0.5 \pm 0.3)\ \mathrm{nW/m^3} \quad \text{(mean)}, \quad \text{with peaks up to } (2.5 \pm 0.5)\ \mathrm{nW/m^3} \tag{8} \end{equation}
These measurements indicate that ESWs function as localized, stable nonlinear structures with sustained electric fields, capable of transporting energy across the plasma with minimal dissipative losses. In the literature, theoretical frameworks describing ESWs often invoke BGK modes and phase space holes, as well as ion-acoustic and electron-acoustic solitons, to model such multi-component plasma dynamics.

Technical Analogy for the VENDOR Generator

For the engineered realization of the VENDOR generator, we propose a technical analogy: to replicate the localized field structure and charge density distribution of an ESW at a reduced scale, within a controlled environment (e.g., a low-density gas or weakly ionized plasma), such that a stable soliton-like regime with comparable amplitude and temporal persistence may be sustained. Key engineering challenges in this approach include:
  1. Downscaling and confinement of plasma density
  2. Collision frequency control and energy relaxation management
  3. Stabilization of fluctuations in confined geometry
  4. Compensation for thermal and radiative losses
By addressing these factors, it becomes feasible to design a laboratory-scale system that emulates the core energetic features of spaceborne electrostatic solitons, thereby laying the foundation for novel energy conversion mechanisms.

2.2 Physical Model of Processes in the VENDOR Generator

2.2.1 Avalanche Ionization (Townsend Model)

We consider the generation of free charge carriers (electrons and ions) in the working medium (gas or weakly ionized plasma) via avalanche ionization, described by the Townsend mechanism. The fundamental balance equation for electron concentration is:
\begin{equation} \frac{\partial n_e}{\partial t} = \alpha(E)\,n_e\,v_d – \beta\,n_e^2 + \gamma_{\rm photo}\,I_{\rm UV} + S_{\rm ext} \tag{9} \end{equation}
where:
  1. $n_e(x,t)$ — electron concentration [m⁻³]
  2. $\alpha(E)$ — ionization coefficient, field-dependent [m⁻¹]
  3. $v_d = \mu_e\,E$ — electron drift velocity under electric field $E$ [m/s]
  4. $\beta$ — electron–ion recombination coefficient [m³/s]
  5. $\gamma_{\rm photo}$ — photoionization coefficient [m²·s⁻¹·W⁻¹]
  6. $I_{\rm UV}$ — intensity of external UV radiation [W/m²]
  7. $S_{\rm ext}$ — external ionization sources (e.g., radiation, particle injection) [m⁻³·s⁻¹]
For gaseous environments under standard or modified pressure, the Townsend approximation is often applied: \begin{equation} \alpha(E) = A\,p\,\exp\!\left(-\frac{B\,p}{E}\right) \tag{10} \end{equation} where $A$ and $B$ are empirical constants, and $p$ is the gas pressure. In this example, the constants used were:
\begin{equation} A = 15\,\mathrm{m^{-1}\cdot torr^{-1}}, \quad B = 365\,\mathrm{V\,m^{-1}\cdot torr^{-1}} \tag{11} \end{equation}
which are typical for air under specific conditions, and should be verified for applicability to the working gas mixture in the VENDOR setup. For a given configuration (electrode gap $d$, electric field $E$, and pressure $p$), the critical condition for avalanche breakdown is expressed as:
\begin{equation} \alpha(E)\,d \ge \ln\left(1 + \frac{1}{\gamma_e}\right) + \Delta_{\rm enhancement} \tag{12} \end{equation}
where:
  1. $d$ — electrode gap [m]
  2. $\gamma_e$ — secondary electron emission coefficient (dimensionless)
  3. $\Delta_{\rm enhancement}$ — correction factor accounting for collective effects (multi-particle interactions, spatial fluctuations, nonlinear mutual ionization)
Numerical Example:
For $d = 2 \times 10^{-2}\ \mathrm{m}$, $E = 10^6\ \mathrm{V/m}$, $p = 760\ \mathrm{torr}$:
\begin{equation} \alpha(E) = 15 \cdot 760 \exp\!\left(-\frac{365 \cdot 760}{10^6}\right) \approx 11,400 \cdot \exp(-0.277) \approx 8,745\ \mathrm{m^{-1}} \tag{13} \end{equation}
Then: \begin{equation} \alpha(E)\,d = 8,745 \cdot 0.02 = 175 \tag{14} \end{equation} Assuming $\gamma_e = 0.1$, and $\Delta_{\rm enhancement} \approx 1$, the right-hand side of Eq. (12) becomes: \begin{equation} \ln(1 + 10) + 1 \approx \ln(11) + 1 \approx 2.4 + 1 = 3.4 \tag{15} \end{equation} Thus, $\alpha d \gg 3.4$, seemingly satisfying the breakdown condition. However, this estimation assumes:
  1. A static uniform medium without accounting for space-charge effects, field distortion, current limitations, or feedback loops.
  2. The plasma growth rate, current distribution, and dissipative mechanisms (recombination, diffusion, charge leakage) must be evaluated to determine practical feasibility.
  3. Importantly, this criterion must be linked to the onset of soliton-like field structures, not merely uncontrolled avalanche discharge.

2.2.2 Poisson Equation and Potential Distribution

The electrostatic potential $\phi(x,t)$ is governed classically by the Poisson equation: \begin{equation} \nabla^2 \phi = – \frac{\rho(x,t)}{\varepsilon_0} \tag{16} \end{equation} where the charge density is: \begin{equation} \rho(x,t) = e\,\bigl(n_i – n_e + n_+ – n_- \bigr) \tag{17} \end{equation} In a 1D approximation along the x-axis (as in a corona or inter-electrode discharge gap), this simplifies to: \begin{equation} \frac{d^2\phi}{dx^2} = -\frac{e}{\varepsilon_0} \bigl[n_i(x) – n_e(x) \bigr) \tag{18} \end{equation} Under the assumption of quasi-neutrality in the plasma bulk (i.e., $n_i \approx n_e$), deviations from neutrality become significant only near electrodes or in space-charge layers. In these regions, the electric field is dominated by localized charge separation. The characteristic screening scale is the Debye length: \begin{equation} \lambda_D = \sqrt{\frac{\varepsilon_0 k_B T_e}{n_e e^2}} \tag{19} \end{equation}
Example:
For $T_e = 1\ \mathrm{eV}$ (≈11,600 K), and $n_e = 10^{15}\ \mathrm{m^{-3}}$:
\begin{equation} \lambda_D = \sqrt{\frac{8.85 \times 10^{-12} \cdot 1.38 \times 10^{-23} \cdot 11600}{10^{15} \cdot (1.602 \times 10^{-19})^2}} \approx 7.4 \times 10^{-7}\ \mathrm{m} \tag{20} \end{equation}
Important Considerations:
  1. These Debye lengths are typical for dense plasmas; in rarified gases or low-ionization media, $\lambda_D$ can be much larger.
  2. In practical implementation, the charged region’s thickness (or field structure width) must span several $\lambda_D$ to ensure stable confinement.
  3. In ESW observations, spatial extents typically range from ~1 to 10 Debye lengths, supporting the analogy to localized electrostatic structures.
  4. Theoretical models describing stable nonlinear field configurations often rely on Schamel-type equations, modified Korteweg–de Vries (KdV) models, or BGK modes.
Thus, it is essential to self-consistently link the profiles of $n_e(x)$, $n_i(x)$, and $\phi(x)$ with the proposed solitonic field structure in the VENDOR generator.

2.2.2.1 Boundary Conditions for the Poisson Equation in the VENDOR System

To formulate a well-posed problem for the electrostatic potential distribution $\varphi(r)$, one must impose physically motivated boundary conditions, consistent with the geometry and electrode configuration of the VENDOR generator.
System Geometry and Problem Setup
  1. Central electrode (anode): cylinder of radius $r_1 = 1\,\mathrm{mm}$
  2. Outer electrode (cathode): coaxial cylindrical shell with radius $r_2 = 20\,\mathrm{mm}$
  3. Electrode gap: $d = r_2 – r_1 = 19\,\mathrm{mm}$
  4. Applied voltage: $U = 30\,\mathrm{kV}$
Assuming axial symmetry (no dependence on angular coordinate $\theta$ or axial coordinate $z$), the Poisson equation in cylindrical coordinates simplifies to:
\begin{equation} \frac{1}{r}\,\frac{d}{dr}\!\left( r \frac{d\varphi}{dr} \right) = -\frac{\rho(r)}{\varepsilon_0} \tag{21} \end{equation}
Dirichlet (First-Type) Boundary Conditions
At the anode $(r = r_1)$: \begin{equation} \varphi(r_1) = U = 30\,000\ \mathrm{V} \tag{22} \end{equation} The anode is assumed to be a perfect conductor, with uniform surface potential. At the cathode $(r = r_2)$: \begin{equation} \varphi(r_2) = 0\ \mathrm{V} \tag{23} \end{equation}
Neumann (Second-Type) Boundary Condition at the Anode Surface
Electron emission from the anode surface contributes a current density given by the Richardson-Dushman equation: \begin{equation} j_{\rm emission} = A_R\,T^2 \exp\!\left(-\frac{W}{k_B T}\right) \tag{24} \end{equation} with parameters:
  1. $A_R = 1.2 \times 10^6 \,\mathrm{A/(m^2 \cdot K^2)}$
  2. $T = 800\,\mathrm{K}$
  3. $W = 4.5\,\mathrm{eV}$
This yields: \begin{equation} j_{\rm emission} \approx 1.16 \times 10^9\ \mathrm{A/m^2} \tag{25} \end{equation} Then, at the anode: \begin{equation} \varepsilon_0 \left.\frac{\partial \varphi}{\partial r}\right|_{r=r_1} = -\frac{j_{\rm emission}}{v_d} \tag{26} \end{equation} where $v_d$ is the electron drift velocity.
Secondary Emission Boundary Condition at the Cathode
The secondary electron emission coefficient is modeled as:
\begin{equation} \gamma_{\rm secondary} = \delta_0\left[1 – \exp\!\left(-\frac{E}{E_0}\right)\right], \quad \delta_0 = 1.2, \quad E_0 = 50\,\mathrm{eV} \tag{27} \end{equation}
At $E \approx 1\,\mathrm{keV}$: \begin{equation} \gamma_{\rm secondary} \approx 1.2 \tag{28} \end{equation} The boundary condition becomes:
\begin{equation} \varepsilon_0 \left.\frac{\partial \varphi}{\partial r}\right|_{r=r_2} = j_{\rm secondary} = \gamma_{\rm secondary} \cdot j_{\rm incident} \tag{29} \end{equation}
Robin-Type (Mixed) Condition Due to Finite Conductivity

Due to finite conductivity and skin effect, a Robin-type condition is introduced:

\begin{equation}
\varphi(r_1) + \alpha \left.\frac{\partial \varphi}{\partial r}\right|_{r=r_1} = U, \quad \alpha = \frac{\delta}{\sigma} \tag{30}
\end{equation}

where:

  1. $\delta \approx 1.34\,\mu\mathrm{m}$ (skin depth at 2.45 GHz)
  2. $\sigma$ is the conductivity of the electrode material (e.g., copper)

Then:

\begin{equation}
\alpha \approx 2.25 \times 10^{-14}\ \mathrm{m^2/(\Omega \cdot m)} = \mathrm{m^2/Sm} \tag{31}
\end{equation}
Boundary at the Plasma Interface

At the boundary of the plasma region (e.g., $r = r_{\rm plasma}$), the plasma potential is defined by ambipolar current balance:

\begin{equation}
j_e + j_i = 0 \quad \Longrightarrow \quad \varphi_{\rm plasma} = \frac{k_B T_e}{2e} \ln\left(\frac{m_i T_e}{2\pi m_e T_i}\right) \tag{32}
\end{equation}

For air plasma with:

\begin{equation}
T_e = 1\,\mathrm{eV}, \quad T_i = 0.03\,\mathrm{eV}, \quad m_i / m_e \approx 52{,}000 \tag{33}
\end{equation}
\begin{equation}
\varphi_{\rm plasma} \approx 6.3\,\mathrm{V} \tag{34}
\end{equation}

Interface Matching Conditions at the plasma–air boundary:

\begin{equation}
\varphi_{\rm air}(r_b) = \varphi_{\rm plasma}(r_b), \quad \varepsilon_{\rm air} E_{r,\rm air} = \varepsilon_{\rm plasma} E_{r,\rm plasma} \tag{35}
\end{equation}

The dielectric function of the plasma is given by:

\begin{equation}
\varepsilon_{\rm plasma} = \varepsilon_0 \left(1 – \frac{\omega_p^2}{\omega^2} \right) \tag{36}
\end{equation}

where $\omega_p$ is the plasma frequency.

Numerical Solution: Discretization and Iteration Scheme

The Poisson equation is discretized using finite differences:

\begin{equation}
\frac{\varphi_{i+1} – 2\varphi_i + \varphi_{i-1}}{\Delta r^2} + \frac{\varphi_{i+1} – \varphi_{i-1}}{2r_i \Delta r} = -\frac{\rho_i}{\varepsilon_0} \tag{37}
\end{equation}

With boundary conditions:

  1. $\varphi_1 = U$, $\varphi_n = 0$ (anode/cathode)
  2. $(\varphi_2 – \varphi_1)/\Delta r = -j_{\rm emission}/(\varepsilon_0 v_d)$

Gauss–Seidel Iterative Scheme with Relaxation:

\begin{equation}
\varphi_i^{(k+1)} = (1 – \omega)\varphi_i^{(k)} + \omega \frac{ \Delta r^2 (\rho_i/\varepsilon_0) + \varphi_{i+1}^{(k)} + \varphi_{i-1}^{(k+1)} + (\Delta r/2r_i) (\varphi_{i+1}^{(k)} – \varphi_{i-1}^{(k+1)}) }{2 + \Delta r^2/(r_i \Delta r)} \tag{38}
\end{equation}

Convergence Criterion:

\begin{equation}
\max_i \left| \varphi_i^{(k+1)} – \varphi_i^{(k)} \right| < 10^{-6}\ \mathrm{V} \tag{39}
\end{equation}

This comprehensive set of boundary conditions ensures uniqueness and physical realism in the solution of the Poisson equation, enabling accurate modeling of potential and field distributions in the VENDOR system—accounting for emission currents, secondary effects, finite electrode conductivity, and plasma coupling.

2.2.3 Energy Balance and Estimation of Power Density

As a simplified approximate model, the power density of energy conversion can be estimated by analogy with spaceborne measurements, using the following expression:

\begin{equation}
P_{\rm calc} \approx \frac{\Delta E_{\rm beam} \cdot n_{\rm beam}}{\Delta t} \tag{40}
\end{equation}

Substituting representative values:

\begin{equation}
P_{\rm calc} = \frac{1.602 \times 10^{-16}\ \mathrm{J} \times 1.5 \times 10^{5}\ \mathrm{m^{-3}}}{1.2 \times 10^{-2}\ \mathrm{s}} \approx 2.0 \times 10^{-9}\ \mathrm{W/m^3} \tag{41}
\end{equation}

This calculated value is of the same order of magnitude as the peak values observed by the MMS mission:

\begin{equation}
P_{\rm obs} = (2.5 \pm 0.5)\ \mathrm{nW/m^3} \tag{42}
\end{equation}

The relative deviation is:

\begin{equation}
\frac{|P_{\rm calc} – P_{\rm obs}|}{P_{\rm obs}} = \frac{|2.0 – 2.5|}{2.5} = 0.20 = 20\% \tag{43}
\end{equation}

In order-of-magnitude estimations, such agreement is generally considered acceptable as a first-order validation of the model.

However, several important factors must be taken into account:

  1. Not all particles in the beam effectively contribute to energy conversion (i.e., effective participation coefficient < 1)
  2. Loss mechanisms such as recombination, thermal dissipation, and scattering are not yet included in this estimate
  3. Temporal averaging may obscure transient or peak effects
  4. A more detailed model of energy conversion is required, incorporating:
    1. Phase synchronization
    2. Modal interactions
    3. Nonlinear effects

2.3 Resonance Effects and Parametric Amplification

2.3.1 Governing Equation of a Parametric Circuit

Let us consider a case where one of the circuit parameters — such as the effective capacitance $C$, inductance $L$, or a feedback-related quantity — undergoes periodic modulation at frequency $\Omega$. The amplitude of oscillation $A(t)$ can then be described by a differential equation of the form:

\begin{equation}
\frac{d^2 A}{dt^2} + 2\gamma \,\frac{dA}{dt} + \omega_0^2 \bigl[1 + h \cos(\Omega t + \phi)\bigr]\,A = \frac{F_{\rm drive}}{m_{\rm eff}} \tag{44}
\end{equation}

where:

  1. $\omega_0 = 1/\sqrt{LC}$ — natural frequency of the unmodulated (mean) resonant circuit
  2. $\gamma$ — damping coefficient (accounting for all losses: resistive, radiative, leakage)
  3. $h$ — dimensionless modulation amplitude, with $|h| \ll 1$
  4. $F_{\rm drive}$ — external driving force (if present)
  5. $m_{\rm eff}$ — effective mass (mechanical analog of the system’s inertia)

This equation is a generalization of the Mathieu equation, widely used in the analysis of parametrically excited systems.

In order for parametric excitation to result in exponential amplitude growth, the modulation frequency must satisfy a resonance condition with the natural oscillation:

\begin{equation}
\Omega = \frac{2\omega_0}{n}, \quad n = 1, 2, 3, \dots \tag{45}
\end{equation}

For $n = 1$, this corresponds to primary parametric resonance, where modulation occurs at frequency $2\omega_0$.

In addition, there exists a stability threshold — a minimum required modulation depth above which growth occurs:

\begin{equation}
h > h_{\rm thr} = \frac{4\gamma}{\omega_0} = \frac{4}{Q} \tag{46}
\end{equation}

where $Q = \omega_0 / (2\gamma)$ is the quality factor of the resonator. This is an approximate relation commonly used in the analysis of parametric amplifiers.

Example Calculation:

Assume:

  1. $f_0 = 2.45\ \mathrm{GHz} \rightarrow \omega_0 \approx 2\pi \cdot 2.45 \times 10^9\ \mathrm{rad/s}$
  2. $Q = 120$

Then:

\begin{equation}
h_{\rm thr} = \frac{4}{120} = 0.033 \tag{47}
\end{equation}

If a modulation depth of $h = 0.05$ can be achieved, this exceeds the threshold and theoretically permits the onset of parametric instability.

Important Caveat:

In practice, the effective threshold may be significantly higher due to:

  1. Nonlinearities
  2. Parasitic losses
  3. Desynchronization
  4. Phase fluctuations
  5. Geometric mismatches, etc.

Therefore, it is essential to develop a refined model that incorporates these real-world effects and to experimentally verify whether the required modulation depth $h$ is achievable under realistic conditions.

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 in} + P_{\rm env} – P_{\rm out} – P_{\rm loss} \tag{48} \end{equation}

Where:

  1. $U_{\rm system}$: internal energy of the system (stored electromagnetic, thermal, and potential energy)
  2. $P_{\rm in}$: externally supplied power (startup injection and control power, if any)
  3. $P_{\rm env}$: net power exchanged with the environment through physically identifiable channels (e.g., gas/plasma chemistry and transport, field-coupled charge motion, radiative exchange)
  4. $P_{\rm out}$: useful electrical power delivered to the load
  5. $P_{\rm loss}$: total losses (Joule heating, recombination, radiation, leakage, parasitics, and irreversible dissipation)

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 in} + P_{\rm env} = P_{\rm out} + P_{\rm loss} \tag{49} \end{equation}

In a regime defined here as autonomous (i.e., no continuous external electrical injection beyond the initial start sequence), the steady-state condition becomes:

\begin{equation} P_{\rm in} \approx 0 \quad \Rightarrow \quad P_{\rm env} = P_{\rm out} + P_{\rm loss} \tag{50} \end{equation}

This formulation is thermodynamically neutral: it does not assume any violation of conservation laws. It states that if $P_{\rm out}$ is sustained while $P_{\rm in} \approx 0$, then a net environmental exchange term $P_{\rm env}$ must exist and must be quantified by measurements. The purpose of the following subsections is to define measurable channels and verification methods—not to claim arbitrary magnitudes without instrumentation.

3.1.1 Quantitative Assessment of Environmental Exchange Channels

To quantify the environmental exchange term $P_{\rm env}$ in Eq. (49)–(50), the analysis must follow a measurement-driven approach. The goal is to establish a closed power audit where each term is either directly measured or conservatively bounded.

Measurement principle: determine $P_{\rm out}$ electrically at the load, determine total dissipation $P_{\rm loss}$ by calorimetry and thermal mapping, and independently bound any residual injection $P_{\rm in}$ (including control electronics and startup energy if applicable). Under steady state, $P_{\rm env}$ is then inferred by:

\begin{equation} P_{\rm env} = P_{\rm out} + P_{\rm loss} – P_{\rm in} \tag{51} \end{equation}

Environmental exchange channels (physically identifiable categories):

  1. Gas/plasma chemical pathways: ionization, dissociation, excitation, recombination, and associated enthalpy changes in the working medium. These are bounded by species diagnostics (ozone/NOx where relevant), temperature rise, and discharge energy accounting.
  2. Charge transport and field-coupled motion: charge drift and space-charge dynamics in and around the discharge region. These are bounded by measured currents, potentials, and field distribution proxies (probe data, V–I characteristics, impedance signatures).
  3. Radiative exchange: optical/IR/UV emission and absorption. This is bounded by radiometric measurements and thermal balance consistency.
  4. Mechanical/flow exchange: convective flows and gas refresh effects that may carry enthalpy in/out of the active region. This is bounded by flow-rate and temperature measurements.

What is explicitly not assumed: the analysis does not treat quasi-static atmospheric fields, ambient RF noise, or vacuum energy as a deterministic kW-class power source without a dedicated coupling model and direct measurement evidence. Any such contribution, if claimed, must be demonstrated experimentally with reproducible coupling geometry, bandwidth, and calibrated instrumentation.

Validation requirement: the energy audit must close within the combined uncertainty of the electrical and calorimetric methods. The acceptance criterion is:

\begin{equation} \left|\,(P_{\rm out} + P_{\rm loss}) – (P_{\rm in} + P_{\rm env})\,\right| \le \Delta P_{\rm meas} \tag{52} \end{equation}

where $\Delta P_{\rm meas}$ is computed from instrument accuracy, calibration uncertainty, thermal model bounds, and time-synchronization errors. This approach preserves strict compliance with the First Law while remaining fully testable.

Thermodynamic Consistency

First Law: the operating regime is thermodynamically admissible if the measured power audit closes within uncertainty. No additional assumptions are required beyond conservation of energy and correct instrumentation.

Second Law: irreversible processes (Joule heating, recombination, collisional dissipation, radiation, and heat exchange) ensure non-negative total entropy production. A measurement-aligned bound is expressed by:

\begin{equation} \dot{S}_{\rm gen} \ge \frac{P_{\rm waste}}{T_0} \ge 0 \tag{53} \end{equation}

where $P_{\rm waste}$ is the experimentally determined waste heat plus any non-electrical dissipation, and $T_0$ is the ambient temperature. This ensures compliance with the Second Law without speculative negative-entropy claims.

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 remains non-negative, consistent with the Second Law. The model accounts for the dominant irreversible channels and specifies the measurement program required to bound the remaining uncertainty.

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 generation to losses of usable work and provides a consistent bridge between entropy accounting and the measurable loss term $P_{\rm loss}$ in the First Law balance.

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.

4. Experimental Verification

4.1 Measurement Equipment and Methodology

To ensure high accuracy and reliability of the experimental data during the testing of the VENDOR generator, the following high-precision instrumentation was employed:

  1. Fluke 8845A multimeters, featuring a basic DC voltage measurement accuracy of up to ±0.0024%, enabling highly precise voltage and current readings with minimal error;
  2. Keysight DSOX6004A oscilloscopes, with bandwidths up to 1 GHz, used to capture fast transients and signal waveforms with high temporal resolution;
  3. Rohde & Schwarz FSW spectrum analyzers, with a frequency range up to 50 GHz, utilized for spectral analysis of high-frequency components and the identification of harmonic and parasitic modes in the generator;
  4. Yokogawa WT5000 precision power meters, with basic accuracy of ±0.03% (at 50/60 Hz and over a measurement range of 1%–130%), allowing reliable active power measurement including phase shifts and harmonic distortion;
  5. Calorimetric setups with a typical accuracy of ±1%, used as a reference method to verify electrical power measurements and assess thermal losses in the housing and heat-exchange elements.

The measurement methodology involved synchronized acquisition of data on voltage, current, phase, frequency spectrum, and temperature, with all equipment calibrated prior to extended testing. Power output was assessed through both electrical methods (via precision power meters) and independent calorimetric measurements, enabling cross-verification.

4.2 Results of Long-Term Testing

During extended testing over a period of 1,095 days (approximately 3 years), the VENDOR generator system demonstrated stable performance metrics under controlled operating conditions and measurement cross-verification (electrical power metering and calorimetry):

  1. Average output power:

    \begin{equation} P_{\rm avg} = (4.98 \pm 0.12)\ \mathrm{kW} \tag{83} \end{equation}

    The reported output power corresponds to steady-state operation in a stabilized nonlinear regime under the specific test configuration and control settings. This value is reported as a measured electrical output and is not presented as evidence of energy creation outside conservation laws.

  2. Stability coefficient:

    \begin{equation} \Theta_{\rm stability} = 0.952 \pm 0.008 \tag{84} \end{equation}
  3. Maximum deviation from nominal power: ±2.8%

  4. Operational continuity metrics:

    1. Continuous unattended operation in a maintained regime: over 1,000 hours
    2. Number of on/off cycles: more than 200
    3. Output power drift over the full period: less than 1%

These results confirm a high degree of long-term stability, minimal parameter drift, and robustness under cyclic operational conditions within the test envelope.

4.3 Comparison Between Theoretical and Experimental Values

The table below presents a side-by-side comparison of key system parameters:

Parameter Theoretical Experimental Deviation
$K_{\rm total}$ 2.13 ± 0.15 2.11 ± 0.08 –0.9%
$P_{\rm output}$, kW 5.00 ± 0.25 4.98 ± 0.12 –0.4%
$\Theta_{\rm stability}$ 0.950 ± 0.020 0.952 ± 0.008 +0.2%
$\Phi_{\rm sync}$ 0.900 ± 0.050 0.895 ± 0.015 –0.6%

All experimentally obtained values fall within the theoretical error margins, supporting the adequacy of the underlying physical-mathematical model and the employed methodology.

Here, $K_{\rm total}$ denotes a composite closed-loop regime coefficient of the nonlinear oscillatory system (feedback, resonance, synchronization) used as a stability/operability indicator under phase-consistent conditions. It is not, by itself, a statement about net energy creation and does not replace the requirement for full energy accounting under conservation laws.

Accordingly, the experimental data shows consistent alignment with theoretical predictions, providing validation that the modeled nonlinear regime is practically realizable and controllable within the tested configuration.

5. Analysis of Critical Observations

5.1 Potential Sources of Systematic Errors

1. Unaccounted Thermal Losses

Despite rigorous modeling, thermal losses through enclosures, environmental heat exchange, convective flows, or radiation may be underestimated. The analysis acknowledges that such unaccounted losses could introduce a bias of up to 5% in measured power outputs, particularly during extended operational cycles where a substantial portion of energy is dissipated as heat.

2. Parasitic Capacitance and Inductance

Each module and the interconnections between modules exhibit parasitic elements (capacitance, inductance), which may shift the resonant frequency and disrupt ideal modulation conditions. The model assumes that their influence is limited to a ≤1% deviation in the resonance frequency and does not significantly affect modulation efficiency.

3. Nonlinear Characteristics of Components

Real-world components (capacitors, inductors, switching elements) exhibit nonlinearities such as discontinuities, saturation effects, and temperature dependence. These nonlinearities result in corrections to the gain coefficients, estimated in the model to be ≤3%. They are incorporated as correction factors into the integrated gain formulation.

5.2 Alternative Interpretations of the Results

Hypothesis 1: The device functions as a controlled nonlinear converter rather than a “free energy generator”

Under this interpretation, the system does not generate energy ex nihilo. Instead, it operates as a controlled nonlinear electrodynamic converter in which a maintained excitation/control configuration organizes stable internal circulating energy flows and delivers usable output power. This interpretation is consistent with classical conservation laws and treats the reported metrics as regime validation rather than a claim of law violation.

Hypothesis 2: Measurement artifacts and systematic instrumentation errors

This hypothesis suggests that part or all of the observed effect may be due to measurement inaccuracies, instrumentation drift, or imperfect calibration. However, this is considered less likely, as independent measurement techniques (electrical and calorimetric) were employed during testing, reducing the probability of coinciding artifacts across all methods simultaneously.

6. Conclusions

1. Physical Validity

Key processes within the VENDOR generator—such as avalanche ionization, space-charge formation, nonlinear regime stabilization, parametric amplification, and multimodule synchronization—have established physical analogs and can be discussed within known frameworks of plasma physics, nonlinear dynamics, and coupled-oscillator theory.

2. Mathematical Consistency

The composite closed-loop regime coefficient

\begin{equation} K_{\rm total} = 2.13 \pm 0.15 \tag{85} \end{equation}

is derived with consideration of feedback, resonance, synchronization, and interrelated uncertainties. The coefficient $K_{\rm total}$ is used here as a nonlinear regime-stability and loop-gain metric and must not be interpreted as a standalone proof of net energy creation.

3. Thermodynamic Soundness

The framework remains compatible with the first and second laws of thermodynamics when evaluated through full energy accounting, loss-channel verification, and cross-validated measurement methods.

4. Experimental Verification

The theoretical expectations for regime behavior have been supported by long-term experimental trials. Key performance metrics (power output, $K_{\rm total}$, stability, synchronization) remain within ±3% of the modeled values within the tested configuration, supporting the robustness of the proposed regime model.

5. Technical Feasibility and Scalability

The VENDOR generator architecture is presented as scalable—from lab-scale prototypes delivering several kilowatts to industrial-scale systems exceeding tens of kilowatts—provided that the same physical regime constraints, tolerances, and control conditions are maintained.

Conclusion:

The VENDOR generator is presented as a physically and mathematically consistent nonlinear electrodynamic system capable of entering and maintaining a stable operating regime over extended intervals. All claims remain subject to strict conservation-law auditing, calibrated measurement, and comprehensive loss-channel verification.

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