Space Inhomogeneous Quantum Fokker-Planck Equations

• Space inhomogeneous quantum Fokker-Planck equations are kinetic models that describe the evolution of quantum probability distributions with spatial variations and nonlocal dissipative effects.
• They integrate quantum statistical features such as nonlinear drift and diffusion with time-dependent memory kernels to model open, non-Markovian quantum systems.
• These equations provide analytical and numerical frameworks to study relaxation, stability, and convergence toward quantum Maxwellian equilibria in complex environments.

A space inhomogeneous quantum Fokker-Planck equation (QFPE) describes the time evolution of (quasi-)probability distributions—either classical densities, quantum phase-space distributions, or Wigner functions—in systems where quantum statistics, spatial inhomogeneity, and dissipative or diffusive dynamics are all relevant. These equations provide a fundamental framework for open quantum systems, quantum statistical mechanics, relaxation in finite or infinite systems, and for modeling memory effects inherent to non-Markovian quantum environments.

1. Mathematical Formulation of Space-Inhomogeneous Quantum Fokker-Planck Equations

The space inhomogeneous quantum Fokker-Planck equation is typically a kinetic equation for a distribution $f=f(t,x,p)$ or a Wigner function $w=w(t,x,\xi)$, with $x$ the spatial coordinate and $p,\xi$ a momentum variable. For quantum kinetic models of bosons and fermions, a canonical example is

$$\partial_t f(t,x,p) + p\cdot\nabla_x\left[f + \kappa f^2\right] = \nabla_p\cdot\left[\nabla_p f + p\left(f + \kappa f^2\right)\right],$$

where $\kappa=1$ (bosons) or $\kappa=-1$ (fermions), and the nonlinear term $(1+\kappa f)$ encodes the quantum inclusion or exclusion principle. The equation can be posed on the flat torus $T^d$ in $x$ and $\mathbb{R}^d$ in $p$ (Arnold et al., 24 Jan 2026).

For quantum models with Wigner functions in $\mathbb{R}^d$, equations of the form

$$\partial_t w + \xi \cdot \nabla_x w - \Omega_0^2 (x \cdot \nabla_\xi w) - 2\gamma \nabla_\xi \cdot (\xi w) + \Theta[V]w = c(t) \Delta_\xi w - d(t) \nabla_x \cdot \nabla_\xi w$$

appear, where $w(t,x,\xi)$ is the Wigner function, $\Theta[V]$ is a (pseudo)differential Hartree term for self-consistent potential, and $c(t), d(t)$ are time-dependent transport coefficients reflecting non-Markovianity (Alejo et al., 2018).

For Wigner space models with arbitrary space-dependent potential $V(x)$ and non-Markovian system-bath coupling, the quantum hierarchical Fokker-Planck (QHFP) equations in phase space take the form of coupled hierarchies for auxiliary Wigner functions, as in (Tanimura, 2015).

2. Physical Origins and Derivation Strategies

Space-inhomogeneous QFPEs arise from either:

• Quantum statistical exclusion/inclusion principles (for bosons/fermions), leading to nonlinear drift and diffusion operators in phase space (Sakhnovich et al., 2013, Arnold et al., 24 Jan 2026).
• The reduction of the unitary evolution of quantum many-body systems (such as spin ladders) to effective FPEs for macroscopic variables via projection, time-convolutionless (TCL) expansions, or coarse-graining procedures (Niemeyer et al., 2012).
• Open quantum system models, describing the evolution of a reduced density matrix or a Wigner function under the influence of a dissipative environment (e.g., Caldeira-Leggett or Hu–Paz–Zhang models), either in Markovian or non-Markovian regimes (Tanimura, 2015, Alejo et al., 2018).

Nonlinear QFPEs such as those describing quantum gases or plasmas encode quantum statistics via density-dependent drift coefficients. In open quantum systems with memory, non-Markovianity is captured via time-dependent coefficients, convolutions, and hierarchical structures.

3. Analytical Structure: Stationary States, Entropy, and Functional Inequalities

The stationary solutions of nonlinear quantum Fokker-Planck equations are quantum Maxwellians of the form: $$M_k(v) = \frac{C_k\,e^{-\frac{1}{2}|v|^2}}{1 - k C_k e^{-\frac{1}{2}|v|^2}},$$ where $k>0$ denotes bosons, $k<0$ fermions, and $C_k$ is determined by normalization (total mass). This form interpolates between Maxwell–Boltzmann, Bose–Einstein, and Fermi–Dirac statistics (Sakhnovich et al., 2013, Arnold et al., 24 Jan 2026).

To quantify relaxation to equilibrium and stability, generalized entropy (relative entropy, free energy) and Lyapunov functionals are constructed:

• Relative entropy: for $f$ and $g$ (with same mass) and $\kappa\ne 0$,

$$H[f|g] = \int \left[f\ln\frac{f}{g} - \kappa(1+\kappa f)\ln\frac{1+\kappa f}{1+\kappa g}\right]\,dp\,dx.$$

• Generalized divergence $D_k(f\|M_k)$ plays a role analogous to the Kullback–Leibler divergence in the classical case and is used to establish local (Lyapunov) stability of quantum Maxwellians (Sakhnovich et al., 2013).

Entropy dissipation inequalities, log-Sobolev inequalities, and micro-macro decompositions (projection onto local equilibrium manifolds) are central to the hypocoercivity approach, ensuring exponential convergence to equilibrium even far from equilibrium (Arnold et al., 24 Jan 2026).

4. Non-Markovian Quantum Kinetic Models

In non-Markovian quantum Fokker-Planck models, such as the Unruh–Zurek (UZ) and Hu–Paz–Zhang (HPZ) equations, the dissipative and diffusive coefficients become nonlocal in time, represented by memory integrals. Examples: $$\partial_t w + \xi\cdot\nabla_x w - [\Omega^2 - 2a(t)] (x \cdot \nabla_\xi w) - 2b(t)\nabla_\xi\cdot(\xi w) + \Theta[V]w = c(t)\Delta_\xi w - d(t)\nabla_x\cdot\nabla_\xi w,$$ with $a(t), b(t), c(t), d(t)$ given by explicit history integrals over drude-Lorentz cutoff spectral densities. The non-Markovian models typically lack the semigroup property and exhibit strong temporal singularities in their propagators, requiring kinetic energy estimates and Duhamel-formulation for global well-posedness (Alejo et al., 2018).

5. Emergence and Numerical Verification in Quantum Many-body Systems

Space inhomogeneous Fokker–Planck dynamics emerge in closed quantum many-body systems by projecting the microscopic quantum evolution onto distributions of macroscopic observables, followed by Kramers–Moyal expansions. In the case of a 16-spin ladder, the relaxation of magnetization inhomogeneity is explicitly described by

$$\partial_t p(z,t) = -\partial_z [A(z)p(z,t)] + \partial_z^2 [B(z)p(z,t)],$$

with $A(z) = -2\gamma\kappa^2 z$, $B(z) = (\gamma\kappa^2/2N)[1/4 + 4z^2]$, and $z$ normalized magnetization difference. This description quantitatively matches exact quantum dynamics for moderate system sizes and weak inter-chain couplings (Niemeyer et al., 2012).

6. Quantum Hierarchical Fokker-Planck Equations for Wigner Functions

The quantum hierarchical Fokker-Planck (QHFP) equations extend the phase-space approach to open quantum systems with arbitrary space dependence and non-Markovianity. In Wigner space, the real-time QHFP equations read: $$\partial_t W^{(n)}_j(x,p;t) = -\Big[L_W + n\gamma + \sum_k j_k \nu_k + \Xi'\Big] W^{(n)}_j + \Phi\Big(W^{(n+1)}_j + \sum_k W^{(n)}_{j_1,...,j_k+1,...,j_K}\Big) + \cdots$$ where $L_W$ is the quantum Liouvillian including arbitrary $V(x)$, $\Phi$ and $\Theta$ encode bath-induced momentum drift and diffusion, and the auxiliary indices $n, j_k$ account for Drude and Matsubara memory modes. These equations support both real-time and imaginary-time propagation for thermodynamic calculations, and are applicable to arbitrary $V(x)$ (Tanimura, 2015).

7. Stability, Relaxation, and Physical Consequences

For nonlinear boson/fermion QFPEs, global-in-time well-posedness, exponential decay to equilibrium, and Lyapunov stability have been rigorously established using hypocoercivity methods and entropy functionals—even for large deviations from equilibrium (Arnold et al., 24 Jan 2026, Sakhnovich et al., 2013). The energy and entropy strictly distinguish between quantum and classical statistics: at fixed density, bosonic energy is lower and entropy is suppressed relative to the classical case, whereas for fermions, both are enhanced, reflecting condensation and exclusion effects respectively.

For systems governed by non-Markovian QFPEs, existence and uniqueness of global mild solutions have been established under finite energy and mass constraints, despite weaker Sobolev regularity due to time-dependent memory kernels (Alejo et al., 2018). In practical terms, the QHFP equations provide computationally tractable schemes for arbitrary external potentials and are used to extract thermodynamic quantities and response functions (Tanimura, 2015).