systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor

systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor(
    *args,
    **kwargs,
)

Continuous-time stochastic batch reactor with process noise.

Extends the deterministic batch reactor to include stochastic effects from process disturbances, parameter uncertainty, and inherent randomness. This model is essential for robust control design, risk analysis, and state estimation under uncertainty.

Stochastic Differential Equations

The reactor dynamics with additive noise:

dC_A = (-r₁)·dt + σ_A·dW_A
dC_B = (r₁ - r₂)·dt + σ_B·dW_B
dT = (Q - α·(T - T_amb))·dt + σ_T·dW_T

where: - r₁ = k₁·C_A·exp(-E₁/T): Reaction 1 rate (A → B) - r₂ = k₂·C_B·exp(-E₂/T): Reaction 2 rate (B → C) - σ_A, σ_B: Concentration noise intensities [mol/(L·√s)] - σ_T: Temperature noise intensity [K/√s] - W_A, W_B, W_T: Independent Wiener processes

Physical Interpretation

Noise Sources:

Concentration Noise (σ_A, σ_B):
- Feed composition variability
- Mixing imperfections
- Sampling uncertainty
- Kinetic parameter fluctuations
Temperature Noise (σ_T):
- Heat transfer variations
- Ambient temperature changes
- Control system imperfections
- Measurement noise

Why Three Independent Sources? - Different physical mechanisms - Spatial separation (bulk vs jacket) - Independent control loops

State Space

State: x = [C_A, C_B, T] Same as deterministic model but now stochastic: - C_A(t): Random process, not deterministic function - C_B(t): Random process - T(t): Random process

Each trajectory is one realization from probability distribution.

Control: u = [Q] - Q: Heating/cooling rate [K/s] - Same as deterministic (no noise in control)

Noise: w = [w_A, w_B, w_T] - Three independent Wiener processes - Dimension: nw = 3 - Structure: Diagonal (uncorrelated)

Key Properties

Stochastic Nature: - Multiple runs give different trajectories - Statistics (mean, variance) evolve over time - Need ensemble analysis (Monte Carlo)

Mean Behavior: For additive noise: E[X(t)] follows deterministic dynamics approximately.

Variance Growth: Variance increases with time due to noise accumulation: Var[C_A(t)] ≈ Var[C_A(0)] + σ_A²·t

Non-Gaussian: Even with Gaussian noise, nonlinear dynamics create non-Gaussian distributions (except in linear approximation).

Parameters

Name	Type	Description	Default
k1	float	Pre-exponential factor for A→B reaction [1/s]	required
k2	float	Pre-exponential factor for B→C reaction [1/s]	required
E1	float	Activation energy for reaction 1 [K]	required
E2	float	Activation energy for reaction 2 [K]	required
alpha	float	Heat transfer coefficient [1/s]	required
T_amb	float	Ambient temperature [K]	required
C_A0	Optional[float]	Initial concentration of A for equilibrium setup [mol/L]	required
T0	Optional[float]	Initial temperature for equilibrium setup [K]	required
sigma_A	float	Concentration noise for A [mol/(L·√s)] - Controls C_A fluctuations - Typical: 0.001-0.1 - Should be << C_A0 for realistic noise	`0.01`
sigma_B	float	Concentration noise for B [mol/(L·√s)] - Controls C_B fluctuations - Typical: 0.001-0.1	`0.01`
sigma_T	float	Temperature noise [K/√s] - Controls T fluctuations - Typical: 0.1-5.0 - More critical than concentration noise (exponential effect via Arrhenius)	`1.0`

Applications

1. Robust Optimal Control: Design controllers accounting for uncertainty: - Stochastic MPC with chance constraints - Risk-sensitive control - Robust trajectory optimization

2. State Estimation: Kalman filter for noisy measurements: - Extended Kalman Filter (EKF) - Unscented Kalman Filter (UKF) - Particle filter

3. Risk Analysis: Assess probability of constraint violation: - Monte Carlo simulation - Rare event estimation - Safety verification

4. Parameter Identification: Estimate parameters from noisy data: - Maximum likelihood - Bayesian inference - Sequential Monte Carlo

5. Process Design: Design for robustness: - Worst-case analysis - Six Sigma methodology - Process capability studies

Numerical Simulation

Recommended Methods: - Euler-Maruyama: Simple, robust, dt ~ 0.01-0.1 s - Milstein: Higher order, dt ~ 0.001-0.01 s - SRK: Even higher order, more cost

Monte Carlo Ensemble: Run N = 100-10,000 simulations to characterize: - Mean trajectory: E[X(t)] - Variance: Var[X(t)] - Confidence bands: μ ± 2σ - Probability distributions

Noise Structure: - Type: ADDITIVE (state-independent) - Dimension: nw = 3 (three Wiener processes) - Correlation: DIAGONAL (independent noise sources)

Comparison with Deterministic

Deterministic: - Single trajectory - Perfect prediction - Nominal design

Stochastic: - Ensemble of trajectories - Probabilistic prediction - Robust design

When Stochastic is Necessary: - Real process has significant noise - Robust control needed - Risk assessment required - Parameter uncertainty significant

Limitations

Additive noise only (not multiplicative)
Constant noise intensity (not state-dependent)
Independent noise sources (no correlation)
No jumps (only continuous Brownian paths)

Extensions

Multiplicative noise: g(x) = diag(σ_A·C_A, σ_B·C_B, σ_T·T)
Correlated noise: Full covariance matrix
Jump diffusion: Add Poisson jumps
Parameter uncertainty: Random walk parameters

Methods

Name	Description
compute_signal_to_noise_ratio	Compute signal-to-noise ratio for each state.
define_system	Define stochastic batch reactor dynamics.
estimate_variance_growth	Estimate variance growth for additive noise (approximate).
get_noise_intensities	Get current noise intensity parameters.
setup_equilibria	Set up equilibrium points.

compute_signal_to_noise_ratio

systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor.compute_signal_to_noise_ratio(
    x,
    t,
)

Compute signal-to-noise ratio for each state.

SNR = |x| / (σ·√t)

Higher SNR → noise less significant.

Parameters

Name	Type	Description	Default
x	np.ndarray	Current state [C_A, C_B, T]	required
t	float	Time since start [s]	required

Returns

Name	Type	Description
	np.ndarray	SNR for [C_A, C_B, T]

Notes

SNR > 10: Noise negligible SNR ~ 1: Noise significant SNR < 0.1: Noise dominates

Examples

>>> reactor = ContinuousStochasticBatchReactor()
>>> x = np.array([0.5, 0.3, 360.0])
>>> snr = reactor.compute_signal_to_noise_ratio(x, t=10)
>>> print(f"SNR: C_A={snr[0]:.1f}, C_B={snr[1]:.1f}, T={snr[2]:.1f}")

define_system

systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor.define_system(
    k1_val=0.5,
    k2_val=0.3,
    E1_val=1000.0,
    E2_val=1500.0,
    alpha_val=0.1,
    T_amb_val=300.0,
    sigma_A=0.01,
    sigma_B=0.01,
    sigma_T=1.0,
    C_A0=None,
    T0=None,
)

Define stochastic batch reactor dynamics.

Parameters

Name	Type	Description	Default
k1_val	float	Pre-exponential factor for A→B reaction [1/s]	`0.5`
k2_val	float	Pre-exponential factor for B→C reaction [1/s]	`0.3`
E1_val	float	Activation energy for reaction 1 [K]	`1000.0`
E2_val	float	Activation energy for reaction 2 [K]	`1500.0`
alpha_val	float	Heat transfer coefficient [1/s]	`0.1`
T_amb_val	float	Ambient temperature [K]	`300.0`
sigma_A	float	Concentration noise intensity for A [mol/(L·√s)] - Typical: 0.001-0.1 mol/(L·√s) - Should be << C_A0 for realistic noise - Rule of thumb: σ_A ~ 0.01·C_A0	`0.01`
sigma_B	float	Concentration noise intensity for B [mol/(L·√s)] - Same magnitude as σ_A typically - Represents mixing and kinetic uncertainty	`0.01`
sigma_T	float	Temperature noise intensity [K/√s] - Typical: 0.1-5.0 K/√s - More critical than concentration noise - Affects reaction rates exponentially - Rule of thumb: σ_T ~ 1 K/√s	`1.0`
C_A0	Optional[float]	Initial conditions for equilibrium setup	`None`
T0	Optional[float]	Initial conditions for equilibrium setup	`None`

Notes

Drift (Deterministic Part): Identical to deterministic reactor: f(x, u) = [-r₁, r₁ - r₂, Q - α·(T - T_amb)]ᵀ

Diffusion (Stochastic Part): Diagonal matrix (additive, independent noise): g(x, u) = diag(σ_A, σ_B, σ_T)

This gives three independent Wiener processes driving concentration and temperature fluctuations.

Noise Intensity Guidelines:

For 1% relative noise at C_A0 = 1.0 mol/L over 1 second: σ_A ~ 0.01 mol/(L·√s)

For 1 K standard deviation over 1 second: σ_T ~ 1.0 K/√s

Physical Justification:

Additive noise models: - External disturbances (feed, ambient) - Measurement errors (sensors) - Control system imperfections - Unmodeled dynamics (lumped effects)

Alternative: Multiplicative noise would be: g(x) = diag(σ_A·C_A, σ_B·C_B, σ_T·T)

This models relative errors scaling with state magnitude.

Validation:

Check noise levels are reasonable: 1. Run deterministic + stochastic simulations 2. Compare final states 3. Coefficient of variation should be 5-20%: CV = std(C_B_final) / mean(C_B_final) 4. If CV > 30%: Noise too large 5. If CV < 1%: Noise negligible

estimate_variance_growth

systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor.estimate_variance_growth(
    t,
)

Estimate variance growth for additive noise (approximate).

For additive noise with independent sources: Var[X(t)] ≈ Var[X(0)] + diag(σ²)·t

This is exact for linear systems, approximate for nonlinear.

Parameters

Name	Type	Description	Default
t	float	Time [s]	required

Returns

Name	Type	Description
	np.ndarray	Estimated variance [Var(C_A), Var(C_B), Var(T)]

Notes

This is a rough estimate. For accurate statistics, run Monte Carlo.

Examples

>>> reactor = ContinuousStochasticBatchReactor(
...     sigma_A=0.01, sigma_B=0.01, sigma_T=1.0
... )
>>> var_100s = reactor.estimate_variance_growth(t=100)
>>> std_100s = np.sqrt(var_100s)
>>> print(f"Estimated std after 100s: C_A={std_100s[0]:.3f}, "
...       f"C_B={std_100s[1]:.3f}, T={std_100s[2]:.3f}")

get_noise_intensities

systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor.get_noise_intensities(
)

Get current noise intensity parameters.

Returns

Name	Type	Description
	dict	Dictionary with keys ‘sigma_A’, ‘sigma_B’, ‘sigma_T’

Examples

>>> reactor = ContinuousStochasticBatchReactor(
...     sigma_A=0.01, sigma_B=0.01, sigma_T=1.0
... )
>>> noise = reactor.get_noise_intensities()
>>> print(f"Temperature noise: {noise['sigma_T']} K/√s")

setup_equilibria

systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor.setup_equilibria(
)

Set up equilibrium points.

Note: For stochastic systems, “equilibrium” refers to deterministic part only. Actual trajectories fluctuate around this point.

# systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor { #cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor } ```python systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor( *args, **kwargs, ) ``` Continuous-time stochastic batch reactor with process noise. Extends the deterministic batch reactor to include stochastic effects from process disturbances, parameter uncertainty, and inherent randomness. This model is essential for robust control design, risk analysis, and state estimation under uncertainty. ## Stochastic Differential Equations {.doc-section .doc-section-stochastic-differential-equations} The reactor dynamics with additive noise: dC_A = (-r₁)·dt + σ_A·dW_A dC_B = (r₁ - r₂)·dt + σ_B·dW_B dT = (Q - α·(T - T_amb))·dt + σ_T·dW_T where: - r₁ = k₁·C_A·exp(-E₁/T): Reaction 1 rate (A → B) - r₂ = k₂·C_B·exp(-E₂/T): Reaction 2 rate (B → C) - σ_A, σ_B: Concentration noise intensities [mol/(L·√s)] - σ_T: Temperature noise intensity [K/√s] - W_A, W_B, W_T: Independent Wiener processes ## Physical Interpretation {.doc-section .doc-section-physical-interpretation} **Noise Sources:** 1. **Concentration Noise (σ_A, σ_B):** - Feed composition variability - Mixing imperfections - Sampling uncertainty - Kinetic parameter fluctuations 2. **Temperature Noise (σ_T):** - Heat transfer variations - Ambient temperature changes - Control system imperfections - Measurement noise **Why Three Independent Sources?** - Different physical mechanisms - Spatial separation (bulk vs jacket) - Independent control loops ## State Space {.doc-section .doc-section-state-space} State: x = [C_A, C_B, T] Same as deterministic model but now stochastic: - C_A(t): Random process, not deterministic function - C_B(t): Random process - T(t): Random process Each trajectory is one realization from probability distribution. Control: u = [Q] - Q: Heating/cooling rate [K/s] - Same as deterministic (no noise in control) Noise: w = [w_A, w_B, w_T] - Three independent Wiener processes - Dimension: nw = 3 - Structure: Diagonal (uncorrelated) ## Key Properties {.doc-section .doc-section-key-properties} **Stochastic Nature:** - Multiple runs give different trajectories - Statistics (mean, variance) evolve over time - Need ensemble analysis (Monte Carlo) **Mean Behavior:** For additive noise: E[X(t)] follows deterministic dynamics approximately. **Variance Growth:** Variance increases with time due to noise accumulation: Var[C_A(t)] ≈ Var[C_A(0)] + σ_A²·t **Non-Gaussian:** Even with Gaussian noise, nonlinear dynamics create non-Gaussian distributions (except in linear approximation). ## Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |---------|-------------------|-----------------------------------------------------------------------------------------------------------------------------------------------------|------------| | k1 | float | Pre-exponential factor for A→B reaction [1/s] | _required_ | | k2 | float | Pre-exponential factor for B→C reaction [1/s] | _required_ | | E1 | float | Activation energy for reaction 1 [K] | _required_ | | E2 | float | Activation energy for reaction 2 [K] | _required_ | | alpha | float | Heat transfer coefficient [1/s] | _required_ | | T_amb | float | Ambient temperature [K] | _required_ | | C_A0 | Optional\[float\] | Initial concentration of A for equilibrium setup [mol/L] | _required_ | | T0 | Optional\[float\] | Initial temperature for equilibrium setup [K] | _required_ | | sigma_A | float | Concentration noise for A [mol/(L·√s)] - Controls C_A fluctuations - Typical: 0.001-0.1 - Should be << C_A0 for realistic noise | `0.01` | | sigma_B | float | Concentration noise for B [mol/(L·√s)] - Controls C_B fluctuations - Typical: 0.001-0.1 | `0.01` | | sigma_T | float | Temperature noise [K/√s] - Controls T fluctuations - Typical: 0.1-5.0 - More critical than concentration noise (exponential effect via Arrhenius) | `1.0` | ## Applications {.doc-section .doc-section-applications} **1. Robust Optimal Control:** Design controllers accounting for uncertainty: - Stochastic MPC with chance constraints - Risk-sensitive control - Robust trajectory optimization **2. State Estimation:** Kalman filter for noisy measurements: - Extended Kalman Filter (EKF) - Unscented Kalman Filter (UKF) - Particle filter **3. Risk Analysis:** Assess probability of constraint violation: - Monte Carlo simulation - Rare event estimation - Safety verification **4. Parameter Identification:** Estimate parameters from noisy data: - Maximum likelihood - Bayesian inference - Sequential Monte Carlo **5. Process Design:** Design for robustness: - Worst-case analysis - Six Sigma methodology - Process capability studies ## Numerical Simulation {.doc-section .doc-section-numerical-simulation} **Recommended Methods:** - Euler-Maruyama: Simple, robust, dt ~ 0.01-0.1 s - Milstein: Higher order, dt ~ 0.001-0.01 s - SRK: Even higher order, more cost **Monte Carlo Ensemble:** Run N = 100-10,000 simulations to characterize: - Mean trajectory: E[X(t)] - Variance: Var[X(t)] - Confidence bands: μ ± 2σ - Probability distributions **Noise Structure:** - Type: ADDITIVE (state-independent) - Dimension: nw = 3 (three Wiener processes) - Correlation: DIAGONAL (independent noise sources) ## Comparison with Deterministic {.doc-section .doc-section-comparison-with-deterministic} **Deterministic:** - Single trajectory - Perfect prediction - Nominal design **Stochastic:** - Ensemble of trajectories - Probabilistic prediction - Robust design **When Stochastic is Necessary:** - Real process has significant noise - Robust control needed - Risk assessment required - Parameter uncertainty significant ## Limitations {.doc-section .doc-section-limitations} - Additive noise only (not multiplicative) - Constant noise intensity (not state-dependent) - Independent noise sources (no correlation) - No jumps (only continuous Brownian paths) ## Extensions {.doc-section .doc-section-extensions} - Multiplicative noise: g(x) = diag(σ_A·C_A, σ_B·C_B, σ_T·T) - Correlated noise: Full covariance matrix - Jump diffusion: Add Poisson jumps - Parameter uncertainty: Random walk parameters ## See Also {.doc-section .doc-section-see-also} ContinuousBatchReactor : Deterministic version OrnsteinUhlenbeck : Mean-reverting stochastic process GeometricBrownianMotion : Multiplicative noise example ## Methods | Name | Description | | --- | --- | | [compute_signal_to_noise_ratio](#cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor.compute_signal_to_noise_ratio) | Compute signal-to-noise ratio for each state. | | [define_system](#cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor.define_system) | Define stochastic batch reactor dynamics. | | [estimate_variance_growth](#cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor.estimate_variance_growth) | Estimate variance growth for additive noise (approximate). | | [get_noise_intensities](#cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor.get_noise_intensities) | Get current noise intensity parameters. | | [setup_equilibria](#cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor.setup_equilibria) | Set up equilibrium points. | ### compute_signal_to_noise_ratio { #cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor.compute_signal_to_noise_ratio } ```python systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor.compute_signal_to_noise_ratio( x, t, ) ``` Compute signal-to-noise ratio for each state. SNR = |x| / (σ·√t) Higher SNR → noise less significant. #### Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |--------|------------|-----------------------------|------------| | x | np.ndarray | Current state [C_A, C_B, T] | _required_ | | t | float | Time since start [s] | _required_ | #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|------------|-----------------------| | | np.ndarray | SNR for [C_A, C_B, T] | #### Notes {.doc-section .doc-section-notes} SNR > 10: Noise negligible SNR ~ 1: Noise significant SNR < 0.1: Noise dominates #### Examples {.doc-section .doc-section-examples} ```python >>> reactor = ContinuousStochasticBatchReactor() >>> x = np.array([0.5, 0.3, 360.0]) >>> snr = reactor.compute_signal_to_noise_ratio(x, t=10) >>> print(f"SNR: C_A={snr[0]:.1f}, C_B={snr[1]:.1f}, T={snr[2]:.1f}") ``` ### define_system { #cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor.define_system } ```python systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor.define_system( k1_val=0.5, k2_val=0.3, E1_val=1000.0, E2_val=1500.0, alpha_val=0.1, T_amb_val=300.0, sigma_A=0.01, sigma_B=0.01, sigma_T=1.0, C_A0=None, T0=None, ) ``` Define stochastic batch reactor dynamics. #### Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |-----------|-------------------|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------|-----------| | k1_val | float | Pre-exponential factor for A→B reaction [1/s] | `0.5` | | k2_val | float | Pre-exponential factor for B→C reaction [1/s] | `0.3` | | E1_val | float | Activation energy for reaction 1 [K] | `1000.0` | | E2_val | float | Activation energy for reaction 2 [K] | `1500.0` | | alpha_val | float | Heat transfer coefficient [1/s] | `0.1` | | T_amb_val | float | Ambient temperature [K] | `300.0` | | sigma_A | float | Concentration noise intensity for A [mol/(L·√s)] - Typical: 0.001-0.1 mol/(L·√s) - Should be << C_A0 for realistic noise - Rule of thumb: σ_A ~ 0.01·C_A0 | `0.01` | | sigma_B | float | Concentration noise intensity for B [mol/(L·√s)] - Same magnitude as σ_A typically - Represents mixing and kinetic uncertainty | `0.01` | | sigma_T | float | Temperature noise intensity [K/√s] - Typical: 0.1-5.0 K/√s - More critical than concentration noise - Affects reaction rates exponentially - Rule of thumb: σ_T ~ 1 K/√s | `1.0` | | C_A0 | Optional\[float\] | Initial conditions for equilibrium setup | `None` | | T0 | Optional\[float\] | Initial conditions for equilibrium setup | `None` | #### Notes {.doc-section .doc-section-notes} **Drift (Deterministic Part):** Identical to deterministic reactor: f(x, u) = [-r₁, r₁ - r₂, Q - α·(T - T_amb)]ᵀ **Diffusion (Stochastic Part):** Diagonal matrix (additive, independent noise): g(x, u) = diag(σ_A, σ_B, σ_T) This gives three independent Wiener processes driving concentration and temperature fluctuations. **Noise Intensity Guidelines:** For 1% relative noise at C_A0 = 1.0 mol/L over 1 second: σ_A ~ 0.01 mol/(L·√s) For 1 K standard deviation over 1 second: σ_T ~ 1.0 K/√s **Physical Justification:** Additive noise models: - External disturbances (feed, ambient) - Measurement errors (sensors) - Control system imperfections - Unmodeled dynamics (lumped effects) Alternative: Multiplicative noise would be: g(x) = diag(σ_A·C_A, σ_B·C_B, σ_T·T) This models relative errors scaling with state magnitude. **Validation:** Check noise levels are reasonable: 1. Run deterministic + stochastic simulations 2. Compare final states 3. Coefficient of variation should be 5-20%: CV = std(C_B_final) / mean(C_B_final) 4. If CV > 30%: Noise too large 5. If CV < 1%: Noise negligible ### estimate_variance_growth { #cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor.estimate_variance_growth } ```python systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor.estimate_variance_growth( t, ) ``` Estimate variance growth for additive noise (approximate). For additive noise with independent sources: Var[X(t)] ≈ Var[X(0)] + diag(σ²)·t This is exact for linear systems, approximate for nonlinear. #### Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |--------|--------|---------------|------------| | t | float | Time [s] | _required_ | #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|------------|-------------------------------------------------| | | np.ndarray | Estimated variance [Var(C_A), Var(C_B), Var(T)] | #### Notes {.doc-section .doc-section-notes} This is a rough estimate. For accurate statistics, run Monte Carlo. #### Examples {.doc-section .doc-section-examples} ```python >>> reactor = ContinuousStochasticBatchReactor( ... sigma_A=0.01, sigma_B=0.01, sigma_T=1.0 ... ) >>> var_100s = reactor.estimate_variance_growth(t=100) >>> std_100s = np.sqrt(var_100s) >>> print(f"Estimated std after 100s: C_A={std_100s[0]:.3f}, " ... f"C_B={std_100s[1]:.3f}, T={std_100s[2]:.3f}") ``` ### get_noise_intensities { #cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor.get_noise_intensities } ```python systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor.get_noise_intensities( ) ``` Get current noise intensity parameters. #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|--------|------------------------------------------------------| | | dict | Dictionary with keys 'sigma_A', 'sigma_B', 'sigma_T' | #### Examples {.doc-section .doc-section-examples} ```python >>> reactor = ContinuousStochasticBatchReactor( ... sigma_A=0.01, sigma_B=0.01, sigma_T=1.0 ... ) >>> noise = reactor.get_noise_intensities() >>> print(f"Temperature noise: {noise['sigma_T']} K/√s") ``` ### setup_equilibria { #cdesym.systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor.setup_equilibria } ```python systems.builtin.stochastic.continuous.ContinuousStochasticBatchReactor.setup_equilibria( ) ``` Set up equilibrium points. Note: For stochastic systems, "equilibrium" refers to deterministic part only. Actual trajectories fluctuate around this point.