systems.builtin.stochastic.continuous.LangevinDynamics

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

Langevin dynamics - particle in viscous fluid with thermal noise.

The fundamental equation of statistical mechanics, combining Newton’s law with friction and thermal fluctuations. Provides the microscopic foundation for Brownian motion and the bridge between mechanics and thermodynamics.

Stochastic Differential Equations

Full Langevin equation (position + velocity):

dx = v·dt
dv = -(γ/m)·v·dt + (F/m)·dt + √(2·k_B·T/m)·dW

where: - x: Position [m] - v: Velocity [m/s] - m: Particle mass [kg] - γ: Friction coefficient [kg/s] - F: External force (control, potential gradient) [N] - k_B: Boltzmann constant = 1.380649×10⁻²³ J/K - T: Temperature [K] - W: Standard Wiener process

This is OU Process for Velocity: Velocity component is Ornstein-Uhlenbeck process.

Physical Interpretation

Newton’s Law with Fluctuations:

m·dv/dt = F_total = F_external - γ·v + F_random

Components: 1. F_external: Applied force (gravity, fields, control) 2. -γ·v: Friction (Stokes drag, dissipation) 3. F_random: Thermal fluctuations (molecular collisions)

Friction: For sphere in fluid (Stokes’ law): γ = 6π·η·r

where η is viscosity, r is radius.

Thermal Noise: Fluctuation-dissipation theorem: ⟨F_random(t)·F_random(s)⟩ = 2·γ·k_B·T·δ(t-s)

White noise with intensity √(2·γ·k_B·T).

Temperature: Sets equilibrium energy scale: ⟨(1/2)·m·v²⟩ = (1/2)·k_B·T (equipartition)

Key Features

Fluctuation-Dissipation: Friction γ and noise intensity related by temperature. Cannot choose independently!

Thermal Equilibrium: Velocity distribution: Maxwell-Boltzmann p(v) ∝ exp(-m·v²/2k_B·T)

Two Time Scales: - Fast: Velocity relaxation τ_v = m/γ - Slow: Position diffusion (no natural scale)

Markov Property: Future depends only on current (x, v), not history.

Ergodicity: Long trajectory → Boltzmann distribution.

Mathematical Properties

Velocity (OU Process):

For F = 0 (free particle): Mean: E[v(t)] = v₀·exp(-γ·t/m) Variance: Var[v(t)] = (k_B·T/m)·(1 - exp(-2γ·t/m))

Asymptotic: Var[v(∞)] = k_B·T/m (equipartition)

Position (Diffusion):

Mean: E[x(t)] = x₀ + (m/γ)·v₀·(1 - exp(-γ·t/m)) Variance: Var[x(t)] = 2·D·t - 2·D·(m/γ)·(1 - exp(-γ·t/m))

Long time: Var[x] ≈ 2·D·t (Einstein diffusion)

Diffusion Coefficient: D = k_B·T/γ (Einstein relation)

Physical Interpretation

Mass m: - Particle inertia - Heavier → slower velocity changes - Time scale: m/γ

Friction γ: - Energy dissipation rate - Larger → faster equilibration - From Stokes: γ = 6π·η·r

Temperature T: - Thermal energy reservoir - Higher T → larger fluctuations - Sets equilibrium distribution

Boltzmann k_B: - Fundamental constant - Connects microscopic and macroscopic - Value: 1.380649×10⁻²³ J/K

State Space

State: X = [x, v] ∈ ℝ² - x: Position (unbounded, diffuses) - v: Velocity (Boltzmann distributed at equilibrium)

Control: u = F/m ∈ ℝ (optional) - Acceleration input - Can be potential gradient: -∇U/m

Noise: w ∈ ℝ - Single Wiener process - Enters velocity equation (thermal fluctuations)

Parameters

Name	Type	Description	Default
m	float	Particle mass [kg] - Colloidal particle: ~10⁻¹⁵ kg - Protein: ~10⁻²⁰ kg - Affects time scale m/γ	`1e-15`
gamma	float	Friction coefficient [kg/s] - From Stokes: 6π·η·r - Water, 1μm sphere: ~10⁻⁸ kg/s	`1e-8`
T	float	Temperature [K] - Room temperature: 300 K - Physiological: 310 K - Sets thermal energy k_B·T	`300.0`
k_B	float	Boltzmann constant [J/K] - Fundamental constant (fixed)	`1.380649e-23`

Stochastic Properties

System Type: LINEAR (no external force)
Noise Type: ADDITIVE (constant)
SDE Type: Itô
Noise Dimension: nw = 1
Stationary: Velocity (yes), Position (no)
Ergodic: Yes (for velocity distribution)
Physical: Yes (fluctuation-dissipation satisfied)

Applications

1. Molecular Dynamics: - Thermostat (constant T simulations) - Protein folding simulations - Material properties

2. Bayesian Sampling: - Langevin MCMC - Posterior sampling - Optimization with noise

3. Colloidal Physics: - Optical tweezers experiments - Rheology measurements - Single-molecule biophysics

4. Machine Learning: - Stochastic gradient Langevin dynamics - Bayesian neural networks - Exploration in RL

5. Statistical Mechanics: - Equilibrium sampling - Free energy calculations - Rare event simulation

Numerical Integration

Time Step Selection:

Must resolve velocity relaxation: Δt < m/γ (typically 0.1·m/γ)

For colloidal particle (m/γ ~ 10⁻⁷ s): Δt ~ 10⁻⁸ s

Overdamped Approximation:

If observation time >> m/γ, use: dx = (F/γ)·dt + √(2·D)·dW

Much larger Δt possible (no velocity constraint).

Validation

Physical Checks:

Equipartition: Var[v] = k_B·T/m (after equilibration)
Einstein Relation: D_measured = Var[x]/2t should equal k_B·T/γ
Maxwell-Boltzmann: Velocity histogram should be Gaussian with variance k_B·T/m
Autocorrelation: ⟨v(0)·v(t)⟩ = (k_B·T/m)·exp(-γ·t/m)

Comparison with Other Models

vs. Brownian Motion: - BM: Position only, no velocity - Langevin: Position + velocity (more complete)

vs. Ornstein-Uhlenbeck: - OU: Velocity equation only - Langevin: Position + velocity

vs. Stochastic Double Integrator: - DI: No friction (conservative) - Langevin: With friction (dissipative, equilibrium)

Limitations

Linear (no potential in base class)
1D space (extend to 3D for realism)
Constant γ, T (no hydrodynamic interactions)
Additive noise (no multiplicative)
Markov (memoryless, no viscoelasticity)

Extensions

Potential: Add U(x) via control F = -∇U
3D: Three spatial dimensions
Generalized Langevin: Memory kernel (viscoelasticity)
Active particles: Self-propulsion
Hydrodynamic interactions: Many-body γ_ij

Methods

Name	Description
check_fluctuation_dissipation	Verify fluctuation-dissipation theorem is satisfied.
define_system	Define Langevin dynamics (position + velocity with thermal noise).
get_diffusion_coefficient	Get diffusion coefficient D = k_B·T/γ (Einstein relation).
get_relaxation_time	Get velocity relaxation time τ_v = m/γ.
get_thermal_velocity_std	Get thermal velocity standard deviation from equipartition.
setup_equilibria	Set up equilibrium points.

check_fluctuation_dissipation

systems.builtin.stochastic.continuous.LangevinDynamics.check_fluctuation_dissipation(
)

Verify fluctuation-dissipation theorem is satisfied.

Returns

Name	Type	Description
	dict	Expected vs implemented noise intensity

Examples

>>> langevin = LangevinDynamics()
>>> check = langevin.check_fluctuation_dissipation()
>>> print(f"FDT satisfied: {check}")

define_system

systems.builtin.stochastic.continuous.LangevinDynamics.define_system(
    m=1e-15,
    gamma=1e-08,
    T=300.0,
    k_B=1.380649e-23,
)

Define Langevin dynamics (position + velocity with thermal noise).

Parameters

Name	Type	Description	Default
m	float	Particle mass [kg] - Colloidal particle: ~10⁻¹⁵ kg - Protein: ~10⁻²⁰ kg - Determines time scale m/γ	`1e-15`
gamma	float	Friction coefficient [kg/s] - From Stokes: γ = 6π·η·r - Water, 1μm sphere: ~10⁻⁸ kg/s - Controls relaxation rate	`1e-8`
T	float	Temperature [K] - Room temperature: 300 K - Physiological: 310 K - Sets thermal energy k_B·T	`300.0`
k_B	float	Boltzmann constant [J/K] - Physical constant (don’t change)	`1.380649e-23`

Raises

Name	Type	Description
	ValueError	If m, gamma, T, or k_B are non-positive

Notes

Fluctuation-Dissipation:

Noise intensity MUST be: σ = √(2·γ·k_B·T/m)

This is not a free parameter! Determined by γ, T.

Time Scales:

Velocity relaxation: τ_v = m/γ - Fast equilibration of velocity - After ~5·τ_v: Velocity is Maxwellian

Position diffusion: No characteristic time - Variance grows linearly: 2·D·t - Diffusion coefficient: D = k_B·T/γ

Regime Classification:

Underdamped: m/γ >> observation time - Ballistic short-time: ⟨x²⟩ ~ t² (free flight) - Diffusive long-time: ⟨x²⟩ ~ t (Brownian)

Overdamped: m/γ << observation time - Pure diffusion: ⟨x²⟩ ~ t always - Can use simplified equation (position only)

Physical Examples:

Colloidal particle (1 μm, water, 300 K): - m ~ 10⁻¹⁵ kg - γ ~ 10⁻⁸ kg/s - τ_v ~ 10⁻⁷ s (100 ns) - D ~ 4×10⁻¹³ m²/s

Protein in water: - m ~ 10⁻²⁰ kg - γ ~ 10⁻¹¹ kg/s - τ_v ~ 10⁻⁹ s (1 ns) - D ~ 10⁻¹¹ m²/s

Validation:

Check equilibrium properties: 1. Velocity variance: k_B·T/m 2. Diffusion coefficient: k_B·T/γ 3. Velocity autocorrelation: exp(-γ·t/m)

get_diffusion_coefficient

systems.builtin.stochastic.continuous.LangevinDynamics.get_diffusion_coefficient(
)

Get diffusion coefficient D = k_B·T/γ (Einstein relation).

Returns

Name	Type	Description
	float	Diffusion coefficient [m²/s]

Examples

>>> langevin = LangevinDynamics(gamma=1e-8, T=300)
>>> D = langevin.get_diffusion_coefficient()
>>> print(f"Diffusion: {D:.3e} m²/s")

get_relaxation_time

systems.builtin.stochastic.continuous.LangevinDynamics.get_relaxation_time()

Get velocity relaxation time τ_v = m/γ.

Returns

Name	Type	Description
	float	Relaxation time [s]

Examples

>>> langevin = LangevinDynamics(m=1e-15, gamma=1e-8)
>>> tau = langevin.get_relaxation_time()
>>> print(f"Relaxation: {tau:.3e} s")

get_thermal_velocity_std

systems.builtin.stochastic.continuous.LangevinDynamics.get_thermal_velocity_std(
)

Get thermal velocity standard deviation from equipartition.

σ_v = √(k_B·T/m)

Returns

Name	Type	Description
	float	Thermal velocity std [m/s]

Notes

From equipartition: ⟨(1/2)·m·v²⟩ = (1/2)·k_B·T

Examples

>>> langevin = LangevinDynamics(m=1e-15, T=300)
>>> sigma_v = langevin.get_thermal_velocity_std()
>>> print(f"Thermal velocity: {sigma_v:.3e} m/s")

setup_equilibria

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

Set up equilibrium points.

For free Langevin (u=0), equilibrium is origin with thermal velocity.

# systems.builtin.stochastic.continuous.LangevinDynamics { #cdesym.systems.builtin.stochastic.continuous.LangevinDynamics } ```python systems.builtin.stochastic.continuous.LangevinDynamics(*args, **kwargs) ``` Langevin dynamics - particle in viscous fluid with thermal noise. The fundamental equation of statistical mechanics, combining Newton's law with friction and thermal fluctuations. Provides the microscopic foundation for Brownian motion and the bridge between mechanics and thermodynamics. ## Stochastic Differential Equations {.doc-section .doc-section-stochastic-differential-equations} Full Langevin equation (position + velocity): dx = v·dt dv = -(γ/m)·v·dt + (F/m)·dt + √(2·k_B·T/m)·dW where: - x: Position [m] - v: Velocity [m/s] - m: Particle mass [kg] - γ: Friction coefficient [kg/s] - F: External force (control, potential gradient) [N] - k_B: Boltzmann constant = 1.380649×10⁻²³ J/K - T: Temperature [K] - W: Standard Wiener process **This is OU Process for Velocity:** Velocity component is Ornstein-Uhlenbeck process. ## Physical Interpretation {.doc-section .doc-section-physical-interpretation} **Newton's Law with Fluctuations:** m·dv/dt = F_total = F_external - γ·v + F_random Components: 1. F_external: Applied force (gravity, fields, control) 2. -γ·v: Friction (Stokes drag, dissipation) 3. F_random: Thermal fluctuations (molecular collisions) **Friction:** For sphere in fluid (Stokes' law): γ = 6π·η·r where η is viscosity, r is radius. **Thermal Noise:** Fluctuation-dissipation theorem: ⟨F_random(t)·F_random(s)⟩ = 2·γ·k_B·T·δ(t-s) White noise with intensity √(2·γ·k_B·T). **Temperature:** Sets equilibrium energy scale: ⟨(1/2)·m·v²⟩ = (1/2)·k_B·T (equipartition) ## Key Features {.doc-section .doc-section-key-features} **Fluctuation-Dissipation:** Friction γ and noise intensity related by temperature. Cannot choose independently! **Thermal Equilibrium:** Velocity distribution: Maxwell-Boltzmann p(v) ∝ exp(-m·v²/2k_B·T) **Two Time Scales:** - Fast: Velocity relaxation τ_v = m/γ - Slow: Position diffusion (no natural scale) **Markov Property:** Future depends only on current (x, v), not history. **Ergodicity:** Long trajectory → Boltzmann distribution. ## Mathematical Properties {.doc-section .doc-section-mathematical-properties} **Velocity (OU Process):** For F = 0 (free particle): Mean: E[v(t)] = v₀·exp(-γ·t/m) Variance: Var[v(t)] = (k_B·T/m)·(1 - exp(-2γ·t/m)) Asymptotic: Var[v(∞)] = k_B·T/m (equipartition) **Position (Diffusion):** Mean: E[x(t)] = x₀ + (m/γ)·v₀·(1 - exp(-γ·t/m)) Variance: Var[x(t)] = 2·D·t - 2·D·(m/γ)·(1 - exp(-γ·t/m)) Long time: Var[x] ≈ 2·D·t (Einstein diffusion) **Diffusion Coefficient:** D = k_B·T/γ (Einstein relation) ## Physical Interpretation {.doc-section .doc-section-physical-interpretation} **Mass m:** - Particle inertia - Heavier → slower velocity changes - Time scale: m/γ **Friction γ:** - Energy dissipation rate - Larger → faster equilibration - From Stokes: γ = 6π·η·r **Temperature T:** - Thermal energy reservoir - Higher T → larger fluctuations - Sets equilibrium distribution **Boltzmann k_B:** - Fundamental constant - Connects microscopic and macroscopic - Value: 1.380649×10⁻²³ J/K ## State Space {.doc-section .doc-section-state-space} State: X = [x, v] ∈ ℝ² - x: Position (unbounded, diffuses) - v: Velocity (Boltzmann distributed at equilibrium) Control: u = F/m ∈ ℝ (optional) - Acceleration input - Can be potential gradient: -∇U/m Noise: w ∈ ℝ - Single Wiener process - Enters velocity equation (thermal fluctuations) ## Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |--------|--------|--------------------------------------------------------------------------------------------------|----------------| | m | float | Particle mass [kg] - Colloidal particle: ~10⁻¹⁵ kg - Protein: ~10⁻²⁰ kg - Affects time scale m/γ | `1e-15` | | gamma | float | Friction coefficient [kg/s] - From Stokes: 6π·η·r - Water, 1μm sphere: ~10⁻⁸ kg/s | `1e-8` | | T | float | Temperature [K] - Room temperature: 300 K - Physiological: 310 K - Sets thermal energy k_B·T | `300.0` | | k_B | float | Boltzmann constant [J/K] - Fundamental constant (fixed) | `1.380649e-23` | ## Stochastic Properties {.doc-section .doc-section-stochastic-properties} - System Type: LINEAR (no external force) - Noise Type: ADDITIVE (constant) - SDE Type: Itô - Noise Dimension: nw = 1 - Stationary: Velocity (yes), Position (no) - Ergodic: Yes (for velocity distribution) - Physical: Yes (fluctuation-dissipation satisfied) ## Applications {.doc-section .doc-section-applications} **1. Molecular Dynamics:** - Thermostat (constant T simulations) - Protein folding simulations - Material properties **2. Bayesian Sampling:** - Langevin MCMC - Posterior sampling - Optimization with noise **3. Colloidal Physics:** - Optical tweezers experiments - Rheology measurements - Single-molecule biophysics **4. Machine Learning:** - Stochastic gradient Langevin dynamics - Bayesian neural networks - Exploration in RL **5. Statistical Mechanics:** - Equilibrium sampling - Free energy calculations - Rare event simulation ## Numerical Integration {.doc-section .doc-section-numerical-integration} **Time Step Selection:** Must resolve velocity relaxation: Δt < m/γ (typically 0.1·m/γ) For colloidal particle (m/γ ~ 10⁻⁷ s): Δt ~ 10⁻⁸ s **Overdamped Approximation:** If observation time >> m/γ, use: dx = (F/γ)·dt + √(2·D)·dW Much larger Δt possible (no velocity constraint). ## Validation {.doc-section .doc-section-validation} **Physical Checks:** 1. **Equipartition:** Var[v] = k_B·T/m (after equilibration) 2. **Einstein Relation:** D_measured = Var[x]/2t should equal k_B·T/γ 3. **Maxwell-Boltzmann:** Velocity histogram should be Gaussian with variance k_B·T/m 4. **Autocorrelation:** ⟨v(0)·v(t)⟩ = (k_B·T/m)·exp(-γ·t/m) ## Comparison with Other Models {.doc-section .doc-section-comparison-with-other-models} **vs. Brownian Motion:** - BM: Position only, no velocity - Langevin: Position + velocity (more complete) **vs. Ornstein-Uhlenbeck:** - OU: Velocity equation only - Langevin: Position + velocity **vs. Stochastic Double Integrator:** - DI: No friction (conservative) - Langevin: With friction (dissipative, equilibrium) ## Limitations {.doc-section .doc-section-limitations} - Linear (no potential in base class) - 1D space (extend to 3D for realism) - Constant γ, T (no hydrodynamic interactions) - Additive noise (no multiplicative) - Markov (memoryless, no viscoelasticity) ## Extensions {.doc-section .doc-section-extensions} - Potential: Add U(x) via control F = -∇U - 3D: Three spatial dimensions - Generalized Langevin: Memory kernel (viscoelasticity) - Active particles: Self-propulsion - Hydrodynamic interactions: Many-body γ_ij ## See Also {.doc-section .doc-section-see-also} OrnsteinUhlenbeck : Velocity-only version StochasticDoubleIntegrator : No friction version ## Methods | Name | Description | | --- | --- | | [check_fluctuation_dissipation](#cdesym.systems.builtin.stochastic.continuous.LangevinDynamics.check_fluctuation_dissipation) | Verify fluctuation-dissipation theorem is satisfied. | | [define_system](#cdesym.systems.builtin.stochastic.continuous.LangevinDynamics.define_system) | Define Langevin dynamics (position + velocity with thermal noise). | | [get_diffusion_coefficient](#cdesym.systems.builtin.stochastic.continuous.LangevinDynamics.get_diffusion_coefficient) | Get diffusion coefficient D = k_B·T/γ (Einstein relation). | | [get_relaxation_time](#cdesym.systems.builtin.stochastic.continuous.LangevinDynamics.get_relaxation_time) | Get velocity relaxation time τ_v = m/γ. | | [get_thermal_velocity_std](#cdesym.systems.builtin.stochastic.continuous.LangevinDynamics.get_thermal_velocity_std) | Get thermal velocity standard deviation from equipartition. | | [setup_equilibria](#cdesym.systems.builtin.stochastic.continuous.LangevinDynamics.setup_equilibria) | Set up equilibrium points. | ### check_fluctuation_dissipation { #cdesym.systems.builtin.stochastic.continuous.LangevinDynamics.check_fluctuation_dissipation } ```python systems.builtin.stochastic.continuous.LangevinDynamics.check_fluctuation_dissipation( ) ``` Verify fluctuation-dissipation theorem is satisfied. #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|--------|-----------------------------------------| | | dict | Expected vs implemented noise intensity | #### Examples {.doc-section .doc-section-examples} ```python >>> langevin = LangevinDynamics() >>> check = langevin.check_fluctuation_dissipation() >>> print(f"FDT satisfied: {check}") ``` ### define_system { #cdesym.systems.builtin.stochastic.continuous.LangevinDynamics.define_system } ```python systems.builtin.stochastic.continuous.LangevinDynamics.define_system( m=1e-15, gamma=1e-08, T=300.0, k_B=1.380649e-23, ) ``` Define Langevin dynamics (position + velocity with thermal noise). #### Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |--------|--------|------------------------------------------------------------------------------------------------------------------|----------------| | m | float | Particle mass [kg] - Colloidal particle: ~10⁻¹⁵ kg - Protein: ~10⁻²⁰ kg - Determines time scale m/γ | `1e-15` | | gamma | float | Friction coefficient [kg/s] - From Stokes: γ = 6π·η·r - Water, 1μm sphere: ~10⁻⁸ kg/s - Controls relaxation rate | `1e-8` | | T | float | Temperature [K] - Room temperature: 300 K - Physiological: 310 K - Sets thermal energy k_B·T | `300.0` | | k_B | float | Boltzmann constant [J/K] - Physical constant (don't change) | `1.380649e-23` | #### Raises {.doc-section .doc-section-raises} | Name | Type | Description | |--------|------------|-----------------------------------------| | | ValueError | If m, gamma, T, or k_B are non-positive | #### Notes {.doc-section .doc-section-notes} **Fluctuation-Dissipation:** Noise intensity MUST be: σ = √(2·γ·k_B·T/m) This is not a free parameter! Determined by γ, T. **Time Scales:** Velocity relaxation: τ_v = m/γ - Fast equilibration of velocity - After ~5·τ_v: Velocity is Maxwellian Position diffusion: No characteristic time - Variance grows linearly: 2·D·t - Diffusion coefficient: D = k_B·T/γ **Regime Classification:** Underdamped: m/γ >> observation time - Ballistic short-time: ⟨x²⟩ ~ t² (free flight) - Diffusive long-time: ⟨x²⟩ ~ t (Brownian) Overdamped: m/γ << observation time - Pure diffusion: ⟨x²⟩ ~ t always - Can use simplified equation (position only) **Physical Examples:** Colloidal particle (1 μm, water, 300 K): - m ~ 10⁻¹⁵ kg - γ ~ 10⁻⁸ kg/s - τ_v ~ 10⁻⁷ s (100 ns) - D ~ 4×10⁻¹³ m²/s Protein in water: - m ~ 10⁻²⁰ kg - γ ~ 10⁻¹¹ kg/s - τ_v ~ 10⁻⁹ s (1 ns) - D ~ 10⁻¹¹ m²/s **Validation:** Check equilibrium properties: 1. Velocity variance: k_B·T/m 2. Diffusion coefficient: k_B·T/γ 3. Velocity autocorrelation: exp(-γ·t/m) ### get_diffusion_coefficient { #cdesym.systems.builtin.stochastic.continuous.LangevinDynamics.get_diffusion_coefficient } ```python systems.builtin.stochastic.continuous.LangevinDynamics.get_diffusion_coefficient( ) ``` Get diffusion coefficient D = k_B·T/γ (Einstein relation). #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|--------|------------------------------| | | float | Diffusion coefficient [m²/s] | #### Examples {.doc-section .doc-section-examples} ```python >>> langevin = LangevinDynamics(gamma=1e-8, T=300) >>> D = langevin.get_diffusion_coefficient() >>> print(f"Diffusion: {D:.3e} m²/s") ``` ### get_relaxation_time { #cdesym.systems.builtin.stochastic.continuous.LangevinDynamics.get_relaxation_time } ```python systems.builtin.stochastic.continuous.LangevinDynamics.get_relaxation_time() ``` Get velocity relaxation time τ_v = m/γ. #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|--------|---------------------| | | float | Relaxation time [s] | #### Examples {.doc-section .doc-section-examples} ```python >>> langevin = LangevinDynamics(m=1e-15, gamma=1e-8) >>> tau = langevin.get_relaxation_time() >>> print(f"Relaxation: {tau:.3e} s") ``` ### get_thermal_velocity_std { #cdesym.systems.builtin.stochastic.continuous.LangevinDynamics.get_thermal_velocity_std } ```python systems.builtin.stochastic.continuous.LangevinDynamics.get_thermal_velocity_std( ) ``` Get thermal velocity standard deviation from equipartition. σ_v = √(k_B·T/m) #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|--------|----------------------------| | | float | Thermal velocity std [m/s] | #### Notes {.doc-section .doc-section-notes} From equipartition: ⟨(1/2)·m·v²⟩ = (1/2)·k_B·T #### Examples {.doc-section .doc-section-examples} ```python >>> langevin = LangevinDynamics(m=1e-15, T=300) >>> sigma_v = langevin.get_thermal_velocity_std() >>> print(f"Thermal velocity: {sigma_v:.3e} m/s") ``` ### setup_equilibria { #cdesym.systems.builtin.stochastic.continuous.LangevinDynamics.setup_equilibria } ```python systems.builtin.stochastic.continuous.LangevinDynamics.setup_equilibria() ``` Set up equilibrium points. For free Langevin (u=0), equilibrium is origin with thermal velocity.