`batch_duration` Stability and Steady Value

This section organizes the theoretical relationship between sim.batch_duration in the BEACH batch loop, or the one-batch physical time determined from sim.batch_duration_step, and the validity and stability of the converged wall-charge distribution. In the current implementation, when batch_duration_step is used, sim.batch_duration = sim.dt * sim.batch_duration_step; in reservoir_face injection, the physical inflow per batch determines the macro-particle count or weight.

Implementation entry points:

• Batch procedure: BEACH Algorithm Overview and Particle Tracking and Charge Accumulation
• Parameter definitions: Parameters for sim.batch_duration / sim.batch_duration_step
• Injection usage: src/particles/bem_injection.f90 (reservoir_face / photo_raycast)
• Batch generation and weight resolution: src/config/bem_app_config_runtime.f90

Practical Guide

For first runs, choose batch_duration empirically before relying on the theory below.

1. Start with a conservative, small sim.batch_duration_step.
2. Inspect charge_history.csv, last_rel_change in summary.txt, and absorbed / escaped counts.
3. Run 2x and 1/2x batch_duration_step comparisons and compare final charge distributions and history shapes.
4. If the final charge and history are nearly unchanged and no oscillation or divergence appears, keep that value.
5. If fluctuations dominate, tune target_macro_particles_per_batch, w_particle, batch_count, and history_stride in addition to batch_duration.

Symptom	Check	Typical response
Charge history oscillates strongly by batch	charge_history.csv	Lower batch_duration_step
Final charge changes strongly with step size	1/2x and 2x comparison	Recompute with smaller batch_duration_step
History is too noisy to read	target_macro_particles_per_batch, w_particle	Adjust macro-particle count or weight
Run stops before settling	batches in summary.txt	Increase batch_count

Treat batch_duration as the deterministic explicit-update time step, and particle count / weight as the Monte Carlo noise controls.

1. Reduction to a continuous-time model

Let $q_j(t)$ be the accumulated charge of insulator wall element j, and let $J_j(\mathbf q)$ be the incident charge flux per unit wall area at that charge state. The absorption-only model becomes:

$$\frac{dq_j}{dt} \;=\; J_j(\mathbf q)\, A_j$$

where $A_j$ is the element area. Since $J$ depends on the field created by wall charge, it is generally nonlinear.

One BEACH batch can be viewed as an explicit update that freezes the field at the start of the batch. In expectation:

$$\mathbf q^{n+1} \;=\; \mathbf q^n \;+\; \Delta t_b \cdot \mathbf J(\mathbf q^n)\,\mathbf A \;+\; \boldsymbol\eta^n$$

where:

• $\Delta t_b =$ sim.batch_duration
• $\mathbf A$ is the element-area vector
• $\boldsymbol\eta^n$ is Monte Carlo sampling error within the batch

The implementation follows this picture: particles in a batch see the same field $E(\mathbf q^n)$, and the charge delta is applied to the wall at the end of the batch. Thus batch_duration is the time step of this explicit update.

2. Validity of the steady value

Write the mean update map as:

$$F_{\Delta t_b}(\mathbf q) \;=\; \mathbf q \;+\; \Delta t_b\, \mathbf J(\mathbf q)\,\mathbf A$$

Its fixed point $\mathbf q^{\ast}$ satisfies:

$$F_{\Delta t_b}(\mathbf q^{\ast}) = \mathbf q^{\ast} \quad\Longleftrightarrow\quad \mathbf J(\mathbf q^{\ast}) = 0$$

Therefore, the fixed point of the mean model itself does not depend on $\Delta t_b$.

In this sense, if the iteration converges stably and Monte Carlo error is sufficiently averaged, changing batch_duration does not change the targeted continuous-time steady solution.

However, this statement applies only to the fixed point of the mean model. Actual runs include:

• finite-sample error per batch
• fluctuation in monitoring quantities used to judge convergence
• residual error from stopping at a finite batch count

So the observed converged value can retain weak batch_duration dependence. The safe statement is:

The mean fixed point of the iteration is independent of batch_duration, but finite-sample and finite-time calculations can show small step-size dependence.

3. Linear stability

Near a fixed point $\mathbf q^{\ast}$, define perturbations $\delta\mathbf q^n = \mathbf q^n - \mathbf q^{\ast}$. The linearized mean update is:

$$\delta \mathbf q^{n+1} \;=\; \bigl(I + \Delta t_b\, M\bigr)\,\delta \mathbf q^n, \qquad M_{ij} \;\equiv\; \frac{\partial (J_i A_i)}{\partial q_j}\bigg|_{\mathbf q^{\ast}}$$

The stability condition for a general multi-degree-of-freedom system is the spectral-radius condition:

$$\rho\!\left(I + \Delta t_b\, M\right) < 1$$

For each eigenvalue $\lambda_k$:

$$|1 + \Delta t_b\, \lambda_k| < 1$$

This is the essential BEACH stability condition.

As an insulator wall accumulates charge, it tends to attract fewer particles of the same sign, so the dominant eigenvalues of $M$ are expected to be real negative, $\mathrm{Re}(\lambda_k) < 0$. Only under this real-negative dominant mode assumption, using response time scale $\tau_k \equiv 1/|\lambda_k|$, the fastest mode avoids divergence when:

$$0 \;<\; \Delta t_b \;<\; \frac{2}{|\lambda_{\max}|} \;=\; 2\,\tau_{\min}$$

and converges monotonically, or overdamped, when:

$$0 \;<\; \Delta t_b \;<\; \frac{1}{|\lambda_{\max}|} \;=\; \tau_{\min}$$

Practical interpretation:

• $\Delta t_b < 2\,\tau_{\min}$: non-divergence under the real-negative dominant mode assumption
• $\Delta t_b < \tau_{\min}$: monotone convergence under the same assumption
• in a general coupled system, the precise condition is $\rho(I + \Delta t_b\, M) < 1$

Thus the $2\tau$ / $\tau$ rule is better described as an explicit-Euler stability guide under a real-negative dominant mode assumption, not as a strict BEACH CFL condition.

4. Relation to Monte Carlo noise

For a one-mode approximation with noise:

$$\delta q^{n+1} \;=\; \left(1 - \frac{\Delta t_b}{\tau}\right)\,\delta q^n \;+\; \xi^n$$

the steady variance depends on the variance of $\xi^n$. The key point is that the $\Delta t_b$ dependence of $\mathrm{Var}(\xi^n)$ depends on injection normalization. BEACH reservoir_face has two modes.

4.1 Fixed w_particle

When w_particle is specified directly, the physical inflow count changes in proportion to $\Delta t_b$, so the expected macro-particle count per batch follows:

$$N_\text{macro} \;\propto\; \Delta t_b$$

The shot-noise variance of the batch charge increment can be regarded roughly as proportional to $\Delta t_b$:

$$\mathrm{Var}(\xi^n) \;\approx\; \alpha\, \Delta t_b$$

In the limit $\Delta t_b \ll \tau$, the steady variance does not depend strongly on batch_duration.

4.2 Fixed target_macro_particles_per_batch

When w_particle is solved from target_macro_particles_per_batch, the weight is determined as in src/config/bem_app_config_runtime.f90:644:

$$w_\text{particle} \;\propto\; \frac{\Gamma\, A\, \Delta t_b}{N_\text{target}}$$

so the noise dependence on $\Delta t_b$ differs from the simple $\mathrm{Var}(\xi^n) \propto \Delta t_b$ of §4.1. The macro-particle count is fixed, while the contribution per particle is proportional to $\Delta t_b$.

4.3 Practical interpretation

The useful separation is:

• batch_duration mainly controls deterministic stability
• the main knobs for statistical noise are w_particle or target_macro_particles_per_batch

In particular, neither of these is generally true:

Making batch_duration smaller always lowers noise. Making batch_duration larger leaves noise almost unchanged.

The answer depends on injection normalization.

5. Physical estimate of $\tau_{\min}$

$\tau_{\min}$ is the fastest effective response time that controls numerical stability. It is hard to express with one general physical formula because it depends on geometry, potential distribution, upstream distribution function, and injection model. In practice, estimate two different quantities.

5.1 Charging / sheath relaxation time

A natural estimate uses an effective capacitance $C_\text{eff}$ and effective conductance $G_\text{eff}$:

$$\tau_\text{charge} \;\sim\; \frac{C_\text{eff}}{G_\text{eff}}$$

or a typical potential change $\Delta\phi$ and effective current $I_\text{eff}$:

$$\tau_\text{charge} \;\sim\; \frac{C_\text{eff}\,\Delta\phi}{I_\text{eff}}$$

This is a relatively slow charging timescale affected by geometry and shielding.

5.2 Inverse plasma frequency

Another fast reference is:

$$\tau_{pe} \;=\; \omega_{pe}^{-1} \;=\; \sqrt{\frac{\varepsilon_0\, m_e}{n_e\, e^2}}$$

This is the microscopic fast timescale of an electron plasma and is useful as a reference for how sharply the system can respond.

However, treating $\omega_{pe}^{-1}$ directly as an upper bound on $\tau_{\min}$ is too strong. It is better viewed as a fast-side physical reference. The effective time constant that limits batch_duration often comes from $\tau_\text{charge}`, including geometry and incoming-flux limits.

5.3 Practical choice

For $\tau_{\min}$, estimate:

• $\omega_{pe}^{-1}$: microscopic fast reference
• $\tau_\text{charge}$: system-specific charging / sheath relaxation timescale

Then refine with numerical experiments.

$\omega_{pe}^{-1}$ is only a fast reference; the actual stability limit is set by an effective response time that often includes $\tau_\text{charge}$.

6. Practical usage

1. Estimate both $\omega_{pe}^{-1}$ and $\tau_\text{charge}$ as physical scales.
2. Start with a conservative, smaller batch_duration if oscillation should be avoided.
3. Compare charge history and monitoring quantities with batch_duration multiplied by 1/2 and 2, as a step-size sensitivity check.
4. If the converged values nearly agree and no oscillation or divergence is visible, the batch_duration is practically adequate.
5. If noise is large, first adjust w_particle or target_macro_particles_per_batch. Do not try to solve noise only by changing batch_duration.
6. Oscillation in charge_history.csv last_rel_change, or jitter in element charge time series, is a useful diagnostic. This is better called a step-size sensitivity check than strict Richardson extrapolation, because it does not assume a power law of the error.

7. Summary

Item	Conclusion
Validity of steady value	The fixed point of the mean update does not depend on batch_duration
Exact stability condition	$\rho(I + \Delta t_b\, M) < 1$
$2\tau$, $\tau$ rules	Explicit-Euler approximate guide under real-negative dominant modes
Role of $\omega_{pe}^{-1}$	Microscopic fast reference, not generally a direct stability upper bound
Noise and batch_duration	Dependence is set by injection normalization
Main noise-reduction knobs	Adjust w_particle or target_macro_particles_per_batch
Practical check	Step-size sensitivity check by varying batch_duration

It is theoretically clean to say that the steady value of the mean model does not depend on how batch_duration is chosen. The general stability condition $\rho(I + \Delta t_b M) < 1$ follows directly from classical stability analysis. The remaining uncertainty is the value of $\tau_{\min}$ itself, which must be narrowed down with both case-specific physical estimates and numerical experiments.

Related documents

• Fortran parameter file specification — how to set sim.batch_duration / sim.batch_duration_step
• Fortran-centered workflow — batch-loop execution control
• BEACH Algorithm Overview — one-batch computation procedure and stop conditions
• Particle Tracking and Charge Accumulation — particle generation, tracking, and charge commit details