---
title: Stochastic Schrödinger–Langevin Equation
url: https://www.emergentmind.com/topics/stochastic-schrodinger-langevin-equation
type: topic
---

# Stochastic Schrödinger–Langevin Equation

The stochastic Schrödinger–Langevin equation (SLE) is a class of nonlinear, stochastic partial differential equations for the quantum wave function of a system coupled to a thermal bath. SLEs generalize the traditional Schrödinger equation by incorporating both dissipation—typically via nonlinear friction functionals—and thermal fluctuations, most commonly represented as additive or multiplicative Gaussian noise. These equations provide a phenomenological framework for modeling open quantum systems undergoing decoherence, relaxation, and quantum-to-classical transitions under the influence of an environment.

## 1. Mathematical Formulation and Representations

The canonical SLE augments the unitary quantum evolution with explicit friction and noise terms. In wave-function form, the SLE is typically expressed as
\[
i\hbar \frac{\partial\psi(x,t)}{\partial t}
= \left[ H_0 + V_{\text{diss}} + V_{\text{noise}} \right] \psi(x,t)
\]
where:
- \( H_0 = -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{ext}}(x) \) is the subsystem's isolated Hamiltonian,
- \( V_{\text{diss}} \) is a nonlinear functional encoding dissipation (e.g., phase-based friction),
- \( V_{\text{noise}} \) is a stochastic potential, often coupling linearly to a classical Gaussian random process.

For the Kostin/Langevin form,
\[
i\hbar \frac{\partial\psi(x,t)}{\partial t} =
\left[ -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}
+ V(x, t)
+ x F_r(t)
+ \frac{\gamma\hbar}{2i} \left(\ln\frac{\psi}{\psi^*}-\langle \ln\frac{\psi}{\psi^*}\rangle\right)
\right]\psi(x, t)
\]
where \( \gamma \) is the friction coefficient and \( F_r(t) \) is real Gaussian noise [1504.08087, 1401.4405, 1902.02147].

In the Madelung (hydrodynamic) representation, the SLE decomposes into equations for the probability density and velocity field:
\[
\psi(x,t) = R(x,t)\,e^{i\,S(x,t)/\hbar}
\]
\[
\begin{cases}
\partial_t \rho + \nabla \cdot (\rho v) = 0 \\
\partial_t S + \frac{(\nabla S)^2}{2m} + V_{\text{ext}} + Q + \text{friction} - x F_r(t) = 0
\end{cases}
\]
where \( Q \) is the Bohmian quantum potential [1504.08087, 2008.00796].

## 2. Dissipation, Noise, and Fluctuation–Dissipation Structure

SLEs encode both deterministic and stochastic effects:

- **Dissipation:** Nonlinear friction may be formulated as \( V_{\text{diss}} = \hbar A \left(S(x, t) - \langle S \rangle\right) \) (phase friction), or more generally as an integral over the current for state-dependent friction [1504.08087, 1401.4405]. In the generalized (Kostin) framework, the friction potential can be nonlocal and history-dependent to capture nonlinear bath couplings.

- **Noise:** The random force \( F_r(t) \) is a zero-mean Gaussian process with autocorrelation \( \langle F_r(t) F_r(t') \rangle \). Both white (Markovian) and colored (finite correlated, non-Markovian) noise models are used:
    - **White noise:** \( \langle F_r(t)F_r(t')\rangle = B\delta(t-t') \), \( B \) set by a fluctuation–dissipation relation.
    - **Colored noise:** \( \langle F_r(t)F_r(t')\rangle \) is a function with spectrum matching the Bose–Einstein distribution at temperature \( T_{\text{bath}} \), yielding non-Markovian effects [1504.08087].

The fluctuation–dissipation theorem constrains the noise amplitude in terms of dissipation, ensuring thermal equilibrium is approached at long times [1504.08087, 1401.1787].

## 3. Eigenfunction Expansion, Functional Integral, and Fokker–Planck Approach

The eigenfunction expansion approach rewrites \( \psi(x, t) \) as \( \psi(x, t) = \sum_{n} a_n(t) \phi_n(x) \), where \( \phi_n(x) \) diagonalize \( H_0 \). The mode amplitudes \( a_n(t) \) obey coupled stochastic differential equations:
\[
y \frac{d a_n}{dt} = -\frac{\partial E}{\partial a_n^*} + \eta_n(t)
\]
\( E[ \{ a_n \} ] \) is the mode-space energy, and \( \eta_n(t) \) are independent complex Gaussian noises [1506.01787].

Functional integral methods allow analytical derivations of the corresponding Fokker–Planck equation for the probability distribution over \( \{ a_n \} \), leading to explicit solutions in certain regimes, such as the independent-mode (Boltzmann) and semiclassical limits [1506.01787].

## 4. Hydrodynamical and Bohmian Trajectory Analysis

In the Bohmian or Madelung form, the SLE induces quantum stochastic trajectories. For a Gaussian density ansatz, the dynamics separate into a classical stochastic equation for the centroid,
\[
\ddot{q}(t) + \gamma \dot{q}(t) + \frac{1}{m} V'(q(t)) = \frac{F_r(t)}{m}
\]
and a deterministic nonlinear equation for the width \( \sigma(t) \):
\[
\ddot{\sigma}(t) + \gamma \dot{\sigma}(t) - \frac{\hbar^2}{4m^2 \sigma^3(t)} + \frac{\sigma(t)}{m} V''(q(t)) = 0
\]
The diffusion coefficient, arrival times, and dwell times can be calculated exactly for solvable cases, and stochasticity enters only through the centroid equation [1902.02147].

## 5. Thermalization, Equilibrium, and Limitations

Ensembles of stochastic wave-function evolutions allow computation of long-time equilibrium statistics:
- In the harmonic oscillator with white noise, SLEs yield exact canonical Boltzmann weights for the eigenstate populations \( p_n \propto \exp(-E_n/(k_B T_{\text{bath}})) \) at the bath temperature, for all friction strengths [1504.08087].
- For colored noise, Gibbs equilibrium is precisely recovered only in the weak coupling and/or high-temperature limits; otherwise, the system thermalizes at a temperature \( T_{\text{sub}} \), which can deviate from the bath temperature.
- Non-harmonic or strongly non-Markovian regimes exhibit departures from ideal Boltzmann distributions, with partial thermalization or effective temperatures [1504.08087].

Key limitations:
- Standard SLEs are nonlinear and not of Lindblad type, generally admitting only phenomenological correspondence to quantum master equations.
- The c-number noise approximation can lead to failure of positivity and improper thermalization for strong non-Markovian couplings or arbitrary potentials.
- Extensions to spatially correlated (colored) noise are nontrivial and model-dependent; strict fluctuation–dissipation relationships may only hold in special cases [1504.08087, 1401.4405].

## 6. Generalizations, State-Dependent Friction, and Measurement Back-Action

Generalized SLEs permit nonlinear, state-dependent friction and multiplicative noise by considering nontrivial system–bath couplings such as \( H_{sb} \sim f(x) x_i \) and introducing functional friction operators and nonlinear noise couplings \( V_d[\Psi] \), \( V_r[\Psi] \). These frameworks support modeling of quantum measurement-induced decoherence, weak and continuous measurement back-action, and connections to the generalized uncertainty principle (GUP):
\[
i\hbar \frac{\partial\Psi}{\partial t}
= \left[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x)\right]\Psi 
- m\gamma \int^x dy [f'(y)]^2 \frac{J(y, t)}{\rho(y, t)}\Psi
- f(x)\xi(t)\Psi
\]
where \( J \) and \( \rho \) denote the current and density, respectively [1401.4405].

Measurement-induced nonlinearities (e.g., continuous monitoring via log-amplitude terms) can be incorporated to reflect observation back-action in open quantum systems [1401.4405].

## 7. Physical Regimes, Quantum–Classical Transition, and Large-Scale Limits

In the stochastic quantum hydrodynamic model (SQHM), SLEs arise from Madelung fluid equations driven by stochastic mass-density fluctuations of "dark" vacuum origin. For microscopic systems (\( L \ll \lambda_c \), the thermal De Broglie length), SLEs provide the appropriate open-system description, with a finite friction coefficient and noise amplitude determined via system-bath energy scales. In the macroscopic limit (\( L \gg \lambda_c \)), quantum coherence is lost, and the dynamics cross over to classical Fokker–Planck or Smoluchowski evolution, consistent with the emergence of classical statistical mechanics from stochastic coarse-grained quantum evolution [2008.00796].

The transition from quantum to classical is thus governed by the interplay between the Bohmian quantum potential, the vacuum-induced stochastic noise, and the scale separation between system dimensions and intrinsic bath correlation lengths.

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**References**  
- Katz & Gossiaux, "The Schrödinger-Langevin equation with and without thermal fluctuations" [1504.08087]  
- Tsuchida & Kuratsuji, "Stochastic approach to generalized Schrödinger equation: A method of eigenfunction expansion" [1506.01787]  
- Mousavi & Miret-Artés, "Stochastic Bohmian mechanics within the Schrödinger–Langevin framework: A trajectory analysis of wave-packet dynamics..." [1902.02147]  
- Chiarelli & Chiarelli, "Stochastic quantum hydrodynamic model..." [2008.00796]  
- Bargueño & Miret-Artés, "The Generalized Schrödinger–Langevin equation" [1401.4405]  
- Attard, "Quantum Statistical Mechanics. II. Stochastic Schrodinger Equation" [1401.1787]

Source: https://www.emergentmind.com/topics/stochastic-schrodinger-langevin-equation