Keldysh-Langevin Molecular Dynamics

• Keldysh-Langevin molecular dynamics is a framework that combines non-equilibrium Green’s function techniques with generalized Langevin equations to model open quantum-classical systems.
• It employs advanced splitting schemes and rigorous error control to integrate non-Markovian dynamics with memory kernels and colored, non-Gaussian noise.
• The approach incorporates quantum-classical coupling and nonconservative forces to simulate complex transport, dissipation, and reaction phenomena in various systems.

Keldysh-Langevin molecular dynamics refers to a multifaceted theoretical and computational framework for simulating open quantum or quantum-classical systems far from equilibrium, where nuclear, electronic, and environmental degrees of freedom interact with significant memory effects and nontrivial stochastic fluctuations. Combining elements from the Keldysh non-equilibrium Green’s function formalism, generalized (often non-Markovian) Langevin equations, and advanced numerical integration protocols, these approaches rigorously incorporate non-Gaussian noise, memory kernels, quantum dissipation, and non-adiabatic feedback, providing a foundational toolset for simulating transport, dissipation, and response phenomena in molecular, solid-state, and mesoscopic systems.

1. Theoretical Foundation: From Microscopic Dynamics to Non-Markovian Langevin Equations

A central principle in Keldysh-Langevin molecular dynamics is the derivation of a generalized Langevin equation from a microscopically faithful description of the system (often via quantum molecular dynamics or projection operator techniques). In "Non-Gaussian Fluctuation and Non-Markovian Effect in the Nuclear Fusion Process: Langevin Dynamics Emerging from Quantum Molecular Dynamics Simulations" (Wen et al., 2013), the fusion process between two nuclei is simulated using improved quantum molecular dynamics (ImQMD), with nucleon wave packets and the extraction of the collective coordinate (relative motion between centers of mass). By analyzing event-resolved forces, the fluctuating (random) force $\delta F$ is found to possess non-Gaussian statistics and strong time correlation near the Coulomb barrier, motivating the generalized Langevin equation: $$\frac{du(t)}{dt} = -\int_{-\infty}^t \gamma(t-t')u(t')dt' + \frac{1}{\mu}\delta F(t) - \frac{1}{\mu} \frac{dV(R)}{dR}$$ where $\gamma(t-t')$ is a non-Markovian friction kernel and $\delta F(t)$ is drawn from the full event-wise force distribution. The friction $\gamma_0(R)$ and force correlation $\sigma(R, \tau)$ are computed numerically, leading to nontrivial memory effects and establishing a generalized fluctuation-dissipation relation: $$\langle \delta F(t) \delta F(t-t') \rangle = \mu T \gamma(t-t')$$ with an effective non-Markovian temperature $T_\text{non-Markov}(R)$ constructed explicitly from simulation data. This formulation provides a microscopic origin for non-Markovian Langevin equations relevant to Keldysh-Langevin frameworks.

2. Numerical Integration and Error Control: Splitting Schemes for Stochastic Equations

Accurate and stable integration of generalized Langevin equations—especially in the non-equilibrium, overdamped, or memory-dominated regimes—demands splitting-based numerical schemes with rigorous error analysis. In "The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics" (Leimkuhler et al., 2013), splitting methods decompose the generator of Langevin-Keldysh dynamics into exactly propagable parts (e.g., Hamiltonian evolution, dissipation, Keldysh driving), allowing construction of operators such as

$$P_{\Delta t}^{(\text{Keldysh})} = \exp(\Delta t \cdot \gamma C) \exp(\Delta t \cdot (B + \eta D)) \exp(\Delta t \cdot A)$$

The bias in sampling observables is quantified via Talay-Tubaro expansions: $$\int \psi d\mu_n = \int \psi d\mu + \Delta t^a \int \psi f_a d\mu + O(\Delta t^{a+1})$$ where $a$ is set by integration order. This machinery is crucial when simulating Keldysh-Langevin systems: the error in computed averages and transport coefficients can be bounded and systematically corrected via Poisson equation-based extrapolation, and overdamped regimes (large friction) admit controlled projections to position-only steady states, vital for long-time sampling in driven quantum-classical environments.

3. Memory Kernels, Colored Noise, and the Generalized Langevin Equation

The non-Markovian structure of Keldysh-Langevin dynamics mandates explicit treatment of time-dependent friction and non-white noise. In "The Generalized Langevin Equation: An efficient approach to non-equilibrium molecular dynamics of open systems" (Stella et al., 2013), the system-bath coupling is derived via projection (Mori-Zwanzig), leading to a GLE: $$m_i \ddot{r}_{i\alpha} = -\frac{\partial \bar{V}(r)}{\partial r_{i\alpha}} - \int_{-\infty}^t dt' \sum_{i'\alpha'} K_{i\alpha,i'\alpha'}(t, t'; r) \dot{r}_{i'\alpha'}(t') + \eta_{i\alpha}(t; r)$$ Realistic environments often produce memory kernels $K(t-t')$ with non-exponential decay, and enforcing the fluctuation-dissipation theorem requires $$\langle \eta_{i\alpha}(t) \eta_{i'\alpha'}(t') \rangle = k_B T K_{i\alpha,i'\alpha'}(t-t'; r)$$. Efficient algorithms map this problem to a finite set of auxiliary variables (embedding in extended phase space), where the colored noise is dynamically generated and memory integrals avoided, making non-equilibrium simulation computationally feasible.

4. Quantum and Semi-Classical Extensions: Nuclear Quantum Effects and Electronic Coupling

Keldysh-Langevin approaches have been extended to treat nuclear quantum effects and coupling to electronic reservoirs. The semi-classical generalized Langevin equation (SGLE) framework (Lü et al., 2017) uses influence functional techniques (Feynman-Vernon) and adiabatic expansions to derive equations of the form

$$\ddot{X} = F^\text{I}(X) + F_e(X) - \Gamma_e \dot{X} + \chi_e$$

with electronic friction $\Gamma_e$ and colored quantum noise obeying $$\langle \chi_e(\omega) \chi_e^*(\omega) \rangle_{eq} = \omega \Gamma_e \coth\left(\omega / 2 k_B T\right)$$. Nonequilibrium conditions (e.g., applied electronic biases) induce nonconservative and Berry-phase forces, leading to vibrational instabilities, current-induced heating, and nontrivial dynamical behavior in molecular junctions and adsorbate-surface systems.

5. Self-Consistent Quantum Transport and Nuclear Feedback: Complete Keldysh-Langevin Modeling

In "Non-Adiabatic Effects of Nuclear Motion in Quantum Transport of Electrons: A Self-Consistent Keldysh-Langevin Study" (Kershaw et al., 2020), quantum transport in molecular junctions is treated by coupling nuclear motion to NEGF-based electron dynamics. The nuclear coordinates obey

$$dp_\mu/dt = -\partial_\mu U(x) + F_\mu(x) - \sum_\nu \xi_{\mu\nu}(x) \dot{x}_\nu + \delta f_\mu(t)$$

where forces $F_\mu(x)$, friction $\xi_{\mu\nu}(x)$, and stochastic amplitudes are computed directly from electronic Green's functions, including non-adiabatic corrections in nuclear velocities and accelerations via explicit Wigner time-scale separation. Non-equilibrium nuclear distributions lead to effective potentials exhibiting the Landauer blowtorch effect (multi-minima induced by non-uniform local heating), which in turn produce switching dynamics, modified current characteristics, and enhanced current noise (Fano factor).

6. Recent Advances: Hierarchical Equations of Motion and Position-Dependent Couplings

Hierarchically coupled system-bath treatments are further merged with Langevin protocols in "Nonadiabatic Dynamics of Molecules Interacting with Metal Surfaces: Extending the Hierarchical Equations of Motion and Langevin Dynamics Approach to Position-Dependent Metal-Molecule Couplings" (Mäck et al., 5 Jun 2024). Partial Wigner transforms enable a separation of quantum and classical degrees of freedom, with the HEOM used to compute electronic steady states and friction tensors. When the metal-molecule couplings $g_\alpha(x)$ depend explicitly on nuclear positions, their gradients introduce new terms in both the adiabatic force and friction tensor: $$F_i^\text{ad}(x) = F_i^\text{ad,mol}(x) + F_i^\text{ad,met-mol}(x)$$ and

$$\gamma_{ij}(x) = \gamma_{ij,\text{mol}}(x) + \gamma_{ij,\text{met-mol}}(x)$$

Such features are essential for simulating surface scattering, desorption, shuttling, and reaction dynamics where spatially varying couplings induce regimes of enhanced or negative friction and nontrivial dissipation.

7. Implications, Applications, and Future Directions

Keldysh-Langevin molecular dynamics frameworks provide a rigorous, self-consistent pathway from microscopic quantum dynamics and non-equilibrium statistical mechanics to macroscopic observables in systems characterized by non-Gaussian fluctuations, non-Markovian dissipation, and quantum-classical coupling. Applications range from nuclear fusion modeling (dissipation governed by transferred nucleons and non-Gaussian force statistics (Wen et al., 2013)), quantum transport in molecular junctions with Landauer blowtorch phenomena (Kershaw et al., 2020), to nano-scale friction and reaction dynamics on surfaces, and vibrational stability under nonconservative electronic forces (Lü et al., 2017, Rudge et al., 20 Feb 2024, Mäck et al., 5 Jun 2024). Kernel-controlled complex Langevin implementations for quantum real-time dynamics on the Schwinger-Keldysh contour offer convergence improvements via drift term modification and cost-function-based kernel optimization (Alvestad et al., 2022, Alvestad et al., 2022).

The explicit, data-driven fitting of memory kernels and noise statistics—via likelihood maximization and auxiliary variable embeddings (Vroylandt et al., 2021)—finally bridges the gap between all-atom molecular simulation and reduced non-Markovian models capable of efficiently sampling long-time statistical properties and rare event statistics. Systematic error control, self-consistent feedback mechanisms, and algorithmic advances make these techniques key to predictive simulation and fundamental paper of open quantum and quantum-classical systems far from equilibrium.