Wigner-Fokker-Planck Phase-Space Formulation

• The topic defines a framework that maps density matrices to real quasi-probability functions, integrating quantum coherence with dissipative dynamics.
• It employs multi-state and spin system generalizations to capture nonadiabatic transitions and electronic coherences without resorting to ad hoc rules.
• Efficient numerical schemes, such as spectral-splitting methods, demonstrate exponential convergence and robust simulation of open quantum systems.

The Wigner-Fokker-Planck (WFP) phase-space formulation provides a unified quantum kinetic and dissipative framework for the dynamics of open quantum systems. Combining the Wigner representation—mapping quantum states (density matrices) to real-valued quasi-probability functions in phase space—with the differential-operator structure of Fokker-Planck equations, the WFP approach enables the explicit treatment of both quantum coherence and environmental dissipation. Multi-state generalizations (MSQFPE) account for electronic coherences and nonadiabatic transitions, such as those at conical intersections, while generalizations exist for spin systems and self-consistent (e.g., Poisson-coupled) multi-particle dynamics. The formulation underpins both mathematical results on existence and long-time behavior and efficient numerical schemes for wavepacket dynamics.

1. Mathematical Structure of the Wigner-Fokker-Planck Equation

The fundamental object is the Wigner function $W(x, \xi, t)$, the Weyl transform of the quantum density operator, defined by

$$W(x, \xi, t) = (2\pi)^{-d} \int_{\mathbb{R}^d} p\left(t, x + \frac{y}{2}, x - \frac{y}{2}\right) e^{-i \xi \cdot y} \,dy$$

where $p(t, x, y)$ is the density-matrix kernel, $x$ is position, and $\xi$ the conjugate momentum.

The generic WFP equation for a single degree of freedom reads

$$\partial_t W(x, \xi, t) + \xi \, \partial_x W + \Theta[V] W = D_{pp} \partial^2_{\xi\xi} W + 2\gamma \, \partial_\xi (\xi W) + D_{qq} \partial^2_{xx} W + 2 D_{pq} \partial^2_{x\xi} W$$

where $V(x)$ is the potential, $\gamma$ is the friction coefficient, $D_{pp}$, $D_{qq}$, $D_{pq}$ are diffusion constants, and $\Theta[V]$ is a nonlocal pseudo-differential quantum potential operator: $$(\Theta[V] W)(x, \xi) = \frac{1}{(\pi \varepsilon)^d} \int_{\mathbb{R}^d}\int_{\mathbb{R}^d} \delta V(x, y) \, W(x, \xi')\, e^{\frac{2i}{\varepsilon} y \cdot (\xi' - \xi)}\, d\xi'\, dy$$ with $$\delta V(x, y) = \frac{i}{\varepsilon}[V(x+y) - V(x-y)]$$.

In the classical limit, $\Theta[V] W \to -\nabla_x V \cdot \nabla_\xi W$, reducing the equation to the classical Fokker-Planck (Kramers) form (Yi et al., 14 Sep 2025, Arnold et al., 2010, Colmenares, 2017).

2. Multi-State Quantum Fokker-Planck Formulation and Nonadiabatic Coupling

For molecular systems with several electronic states, the density matrix $\rho_{ij}(z, z', t)$ is mapped to an $N \times N$ matrix Wigner function,

$$W_{ij}(p, q, t) = \frac{1}{(2\pi)^N}\int dr\, e^{-i p \cdot r }\, \rho_{ij}\left(q + \frac{r}{2}, q - \frac{r}{2}, t\right)$$

where $i,j$ index electronic states, and $(q, p)$ are nuclear coordinates and momenta.

The time evolution is then governed by the multi-state quantum Fokker-Planck equation (MSQFPE): $$\begin{aligned} \frac{\partial}{\partial t} W(p, q, t) &= -\sum_{s} \omega_s p_s \partial_{q_s} W - \frac{i}{\hbar} \left[ U(q) \star W - W \star U(q) \right] \ &+ \sum_s \zeta_s \partial_{p_s} \left( p_s + D_s \partial_{p_s} \right) W \end{aligned}$$ where $U(q)$ is the diabatic potential matrix, $\star$ denotes the Moyal star product for phase-space operator ordering, and the Fokker-Planck operator encodes system-bath interactions with Ohmic spectral density under the Markovian, high-temperature approximation (Ikeda et al., 2018).

Multi-electronic-state coupling appears through the matrix star-commutator, which couples populations ($i=j$) and coherences ($i\neq j$) and incorporates nonadiabatic (e.g., conical intersection) effects without the need for ad hoc transition rules.

3. Key Approximations and Physical Assumptions

The Wigner-Fokker-Planck and MSQFPE formulations rest on several key approximations:

• Ohmic dissipation: Linear (frequency-proportional) spectral density of the system-bath coupling, yielding Markovian damping and diffusion terms.
• High-temperature limit: $$\coth(\beta \hbar \omega / 2) \approx 2/(\beta \hbar \omega)$$, rendering the Fokker-Planck operator local in time for phase-space diffusion.
• Truncated Moyal expansion: In numerical practice, the star product is expanded to finite order in $\hbar$; higher-order terms are retained only if wavepackets wander far from Gaussian profiles or potentials are strongly anharmonic.
• Smooth potentials and perturbations: For existence and uniqueness results, the external potential is often assumed to be a smooth (at least $C^m$) perturbation of a confining quadratic (harmonic) potential (Arnold et al., 2010).

These assumptions ensure well-posedness of the evolution, mass conservation, and contractivity in weighted Sobolev (Hilbert) spaces.

4. Stationary States, Long-Time Dynamics, and Return to Equilibrium

Under confining and suitably regular $V(x)$, and nondegenerate diffusion, the WFP semigroup is hypocoercive with spectral gap, implying exponential convergence to a unique stationary state $w_\infty(x, \xi)$ in polynomially weighted $L^2$ or Sobolev spaces: $$\| w(t) - w_\infty \|_{\mathcal{H}_m} \leq S_m e^{-\kappa_m t} \|w_0 - w_\infty\|_{\mathcal{H}_m}$$ where the decay rate $\kappa_m = Y_m - |\lambda| I_m > 0$ depends on the norm of the perturbation (Arnold et al., 2010). The stationary state corresponds under the inverse Wigner transform to a positive, trace-one density operator.

For more general, including far-from-harmonic, potentials, numerical results provide evidence for existence and uniqueness of stationary states, with relaxation to equilibrium observed even in cases not covered by analyticity-based proofs (Yi et al., 14 Sep 2025).

Theoretical and numerical relaxation criteria include monitoring the decay of $L^2$-distances between iterates and the stabilization of conserved observables such as particle number, current, and energy.

5. Numerical Methods for the Wigner-Fokker-Planck Equation

Efficient simulation is enabled by spectral schemes exploiting phase-space periodicity. A second-order Strang-splitting (time-splitting) pseudospectral method decomposes the WFP evolution into exactly-solvable substeps:

• Free streaming: Solved in Fourier-$x$ space.
• Potential: Advanced via explicit exponentials involving the pseudo-differential $\Theta[V]$ (implemented in Fourier-$\xi$).
• Diffusion and friction: Solved by matrix exponentiation in Fourier/collocation space.

The full Strang update applies a seven-stage sequence: $$W^{n+1} \approx e^{\frac{\Delta t}{2} \mathcal{L}_1} e^{\frac{\Delta t}{2} \mathcal{L}_2} e^{\frac{\Delta t}{2} \mathcal{L}_3} e^{\Delta t \mathcal{L}_4} e^{\frac{\Delta t}{2} \mathcal{L}_3} e^{\frac{\Delta t}{2} \mathcal{L}_2} e^{\frac{\Delta t}{2} \mathcal{L}_1} W^n$$ with each step either diagonal in Fourier space or reduced to banded linear ODEs (Yi et al., 14 Sep 2025). The method achieves $O(\Delta t^2)$ time accuracy and spectral convergence in phase-space discretization.

Numerical studies confirm exponential approach to stationary states for both harmonic and non-harmonic potentials, substantiating hypocoercivity and uncovering steady-state formation beyond analytic guarantees.

6. Wigner-Fokker-Planck Formulation for Spin Systems

For spins, the Wigner—Stratonovich map defines a quasiprobability distribution $W_s(\Omega, t)$ over the sphere, corresponding to the density matrix in Hilbert space. The evolution is governed by a Wigner-Fokker-Planck equation of the form

$$\partial_t W(\Omega, t) = -\nabla_\Omega \cdot [A(\Omega) W] + \nabla_\Omega \nabla_\Omega{:}[D(\Omega) W]$$

where $A$ is a drift vector and $D$ a diffusion tensor determined by the spin Hamiltonian and bath parameters (Kalmykov et al., 2016). This equation formally parallels the classical rotational Fokker-Planck equation, admitting expansion in spherical harmonics and matrix continued-fraction solutions. In the classical ($S \to \infty$) limit, the quantum equation reduces to its classical counterpart.

The phase-space formulation facilitates direct calculation of both quantum and semiclassical observables and underpins extensions of classical computation techniques—such as mean first-passage times and mode reduction—to quantum relaxation processes.

7. Applications and Contexts

Internal Conversion and Conical Intersections

The MSQFPE has been instrumental in simulating internal conversion processes in photoexcited molecules subject to nonadiabatic effects at conical intersections. The explicit representation of coherences and populations in phase space allows for direct calculation of transition yields and their dependence on the energetic profile and geometry of the conical intersection (Ikeda et al., 2018). Nonadiabatic transitions manifest without the need for surface-hopping schemes; dissipation and relaxation are naturally included.

Open Quantum Systems and Quantum Master Equations

The WFP equation accommodates the open-system dynamics of quantum Brownian motion (Caldeira-Leggett, Schramm-Jung-Grabert models), mapping operator-based master equations to partial differential equations for the Wigner function. Solution in this formalism elucidates the crossover between quantum and classical regimes and provides explicit time-dependent drift and diffusion under environmental coupling (Colmenares, 2017).

Mathematical Analysis and Functional Frameworks

Rigorous results establish stationary states and large-time returned equilibrium for the WFP equation with smooth confining potentials, connecting phase-space solutions to Lindblad-type operators and their spectral properties (Arnold et al., 2010). Key inequalities (Poincaré, spectral gap) in weighted $L^2$-spaces underpin contractivity and exponential decay.

Numerical Exploration of Strongly Nonlinear Regimes

Validated spectral-splitting schemes enable the investigation of strongly anharmonic systems and self-consistent (Poisson-coupled) quantum kinetic models (Wigner–Poisson–Fokker–Planck/ WPFP), revealing new stationary-state phenomena not yet theoretically classified (Yi et al., 14 Sep 2025).

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References:

• "Phase-space wavepacket dynamics of internal conversion via conical intersection: Multi-state quantum Fokker-Planck equation approach" (Ikeda et al., 2018)
• "A time-splitting Fourier pseudospectral method for the Wigner(-Poisson)-Fokker-Planck equations" (Yi et al., 14 Sep 2025)
• "The Wigner-Fokker-Planck equation: Stationary states and large-time behavior" (Arnold et al., 2010)
• "Fokker-Planck equation of the reduced Wigner function associated to an Ohmic quantum Langevin dynamics" (Colmenares, 2017)
• "Spin relaxation in phase space" (Kalmykov et al., 2016)