Markov-Fokker-Planck description of inertial particles in turbulence: Stokes-number dependence of the Einstein-Markov coherence time, the large-Stokes limit, and fluctuation theorems under preferential concentration

The Friedrich-Peinke program describes turbulent cascade statistics by a Fokker-Planck equation in scale, resting on an empirically verified Markov property of velocity increments above a minimum step size, the Einstein-Markov coherence scale. Fuchs et al. (EPL 137, 53001, 2022) extended this to the Lagrangian frame: in direct numerical simulations of homogeneous isotropic turbulence at Taylor-scale Reynolds number $\mathrm{Re}_\lambda\approx 240$ with point-like inertial particles (St = 0, 0.5, 3, 6), velocity increments along particle trajectories are Markovian in the time-separation variable above a Stokes-number-dependent Einstein-Markov coherence time, and the associated trajectory entropies satisfy the integral and detailed fluctuation theorems of stochastic thermodynamics. The unresolved problem is to establish the validity domain and the physical mechanisms of this description: (i) the origin and generality of the opposite Stokes-number dependence of the coherence time for the particle velocity (increasing with St) versus the carrier-fluid velocity sampled along the trajectory (decreasing with St); (ii) the range of validity in St and the breakdown mechanism as St grows so large that particles decouple from the flow; and (iii) a theoretical explanation of why preferential concentration, the non-random sampling of flow regions by inertial particles, does not break the fluctuation theorems, and under what conditions it would. A 2026 follow-up extended the framework to inhomogeneous, anisotropic swirling flows and finite-size spherical particles ($\mathrm{St}$ up to about 9), confirming the increasing trend and explicitly leaving the remaining cores open.

Background

A small particle suspended in a turbulent flow responds to the fluid with a lag set by its inertia. For a point-like heavy particle this is modeled by the Maxey-Riley-Gatignol equation, $\dot{\mathbf{v}}(t) = [\mathbf{u}(\mathbf{x}_p,t)-\mathbf{v}(t)]/\tau_p$, where $\mathbf{v}$ is the particle velocity, $\mathbf{u}(\mathbf{x}_p,t)$ the carrier-fluid velocity at the particle position $\mathbf{x}_p$, and $\tau_p$ the particle relaxation (Stokes) time. The Stokes number $\mathrm{St}=\tau_p/\tau_\eta$, the ratio of $\tau_p$ to the Kolmogorov time $\tau_\eta=(\nu/\epsilon)^{1/2}$ (the turnover time of the smallest eddies), measures inertia relative to the smallest flow scales. Tracers ($\mathrm{St}=0$) follow fluid elements exactly; inertial particles ($\mathrm{St}>0$) do not, and because they are centrifuged out of vortical regions they accumulate in strain-dominated regions, sampling the flow inhomogeneously, a phenomenon called preferential concentration.

The Friedrich-Peinke approach describes the multi-scale statistics of turbulence by treating velocity increments as a Markov process evolving across scales (reviewed by Peinke, Tabar and Wächter, 2019). In its Eulerian version, increments $u_r=[\mathbf{u}(\mathbf{x}+\mathbf{r})-\mathbf{u}(\mathbf{x})]\cdot\mathbf{r}/|\mathbf{r}|$ were shown (Friedrich and Peinke, 1997; Renner, Peinke and Friedrich, 2001) to be approximately Markovian in the separation $r$ above a minimum step, so that their scale evolution obeys a Fokker-Planck equation, the evolution equation for the probability density of a drift-diffusion process, whose Kramers-Moyal coefficients, the drift $D^{(1)}(u_r,r)$ and diffusion $D^{(2)}(u_r,r)$ (the conditional mean and variance rates of the increment process), can be estimated directly from data. The minimum step above which the Markov property sets in is the Einstein-Markov coherence length (Lück, Renner, Peinke and Friedrich, 2006; Stresing, Kleinhans, Friedrich and Peinke, 2011), in analogy to the mean free path that separates independent steps of Brownian motion; it is conventionally detected with a parameter-free Wilcoxon test comparing single-conditioned and double-conditioned increment probability density functions.

Fuchs, Obligado, Bourgoin, Gibert, Mininni and Peinke (EPL 137, 53001, 2022) transferred this framework to the Lagrangian frame. For particles in direct numerical simulations (DNS) of forced homogeneous isotropic turbulence (HIT) at Taylor-scale Reynolds number $\mathrm{Re}_\lambda\approx 240$ (the Reynolds number based on the Taylor microscale, the intermediate length scale between the energy-containing and dissipation ranges), with point particles and one-way coupling (the particles are transported by the flow without back-reacting on it), the time-increments $u_\tau=[\mathbf{v}(t+\tau)-\mathbf{v}(t)]\cdot\mathbf{e}_i$ form a Markov process in the time-separation $\tau$ (a cascade from large to small time lags along each trajectory) once $\tau$ exceeds an Einstein-Markov coherence time $\Delta_{EM}$ that grows with inertia: $\Delta_{EM}\approx 7, 10, 13, 16\,\tau_\eta$ for $\mathrm{St}=0, 0.5, 3, 6$. In striking contrast, the same analysis applied to the carrier-fluid velocity sampled along the particle trajectory, $\mathbf{u}(\mathbf{x}_p,t)$, gives a coherence time that decreases with $\mathrm{St}$ (supplementary material of that letter, Appendix F). Note that at $\mathrm{St}=0$ the two processes coincide, since for tracers $\mathbf{v}(t)=\mathbf{u}(\mathbf{x}_p,t)$; the two coherence times are therefore necessarily equal there, and the opposite $\mathrm{St}$-dependence concerns $\mathrm{St}>0$. The Markov property licenses a Fokker-Planck equation for the multi-scale statistics, which opens a bridge to stochastic thermodynamics (Seifert, 2012): each cascade trajectory $[u(\cdot)]$ can be assigned a total entropy variation $\Delta S_{\mathrm{tot}}=\Delta S_{\mathrm{sys}}+\Delta S_{\mathrm{med}}$ built from the empirically estimated drift and diffusion coefficients, and the ensemble must then satisfy the integral fluctuation theorem (IFT) $\langle e^{-\Delta S_{\mathrm{tot}}}\rangle=1$ and, under stronger conditions, the detailed fluctuation theorem (DFT) $\ln[p(\Delta S_{\mathrm{tot}})/p(-\Delta S_{\mathrm{tot}})]=\Delta S_{\mathrm{tot}}$. Both were verified empirically for all four Stokes numbers. Related fluctuation-theorem analyses of the Eulerian cascade include Nickelsen and Engel (2013), Reinke, Fuchs, Nickelsen and Peinke (2018), and Fuchs et al. (Physical Review Fluids 5, 034602, 2020); earlier Lagrangian irreversibility measurements (Xu et al., 2014) had found a Jarzynski-like equality for particle energetics whose slope was not the DFT value.

The letter explicitly acknowledges what remains unexplained: the physical origin of the opposite $\mathrm{St}$-dependence of $\Delta_{EM}$ for $\mathbf{v}(t)$ versus $\mathbf{u}(\mathbf{x}_p,t)$ (offering only the speculation that larger drag at higher $\mathrm{St}$ dissipates memory faster for the sampled fluid velocity); the fate of the description as $\mathrm{St}\to\infty$, where particles decouple from the flow and the relations should not hold; and why preferential concentration, which makes the particles' sampling of the Eulerian velocity non-random, does not break the IFT and DFT. In addition, the letter refers to general open questions concerning inhomogeneous flows and finite-size and non-spherical particles, and suggests that the coherence time can be used to quantify effective parameters of such particles.

Problem Statement

Establish the range of validity and the physical mechanisms of the Markov-Fokker-Planck description of inertial-particle velocity increments in turbulent flow, together with the fluctuation theorems it implies, as introduced by Fuchs et al. (EPL 137, 53001, 2022). Concretely, determine the following jointly linked questions, which together fix when and why the stochastic-thermodynamic description of particle-laden turbulence holds.

1. Coherence-time mechanisms. The origin and generality of the opposite Stokes-number dependence of the Einstein-Markov coherence time: why $\Delta_{EM}$ increases with $\mathrm{St}$ for the particle velocity $\mathbf{v}(t)$ while it decreases with $\mathrm{St}$ for the carrier-fluid velocity sampled at the particle position $\mathbf{u}(\mathbf{x}_p,t)$; whether these trends are universal, Reynolds-number dependent, or specific to particular particle models or coupling approximations; and a quantitative theory predicting $\Delta_{EM}(\mathrm{St})$ for both processes, which is desired but not required for this component.

2. Large-inertia limit. The range of $\mathrm{St}$ over which the Markov property, the Fokker-Planck closure, and the integral and detailed fluctuation theorems remain valid for inertial particles; the critical Stokes number, if finite, beyond which they break down as particles decouple from the flow; and the mechanism of that breakdown.

3. Fluctuation theorems under preferential concentration. A theoretical account of why inertial particles, which sample the flow non-randomly through preferential concentration, nevertheless satisfy the integral and detailed fluctuation theorems; the conditions (particle inertia, density ratio, flow configuration) under which this consistency would fail; and how particle clustering and trajectory time-irreversibility are reflected in the Kramers-Moyal coefficients and the entropy statistics. This is a separate deliverable from component 2, not a restatement of it: component 2 fixes the Stokes-number threshold at which the Markov property, the Fokker-Planck closure, and the fluctuation theorems themselves break down, whereas this component fixes why the fluctuation theorems hold despite preferential concentration and when they fail for an otherwise applicable description; addressing one leaves the other uncovered.

4. Generality across particle classes and regimes. Whether the same description extends to the classes and regimes not yet validated: (a) non-spherical particles; (b) two-way coupling; (c) particle-particle interactions; and (d) inertial particles (not only tracers) at Reynolds numbers substantially above $\mathrm{Re}_\lambda\approx 240$-$350$. These are independent sub-questions: an answer must address each of (a)-(d), stating for each whether the description extends, does not extend, or remains undetermined, with the reason; any of (a)-(d) left unaddressed leaves this component uncovered.

5. Small-entropy deviation from the detailed fluctuation theorem. The origin of the systematic small-entropy deviation ('dip') from the detailed fluctuation theorem reported in the 2026 follow-up for anisotropic flows with a mean large-scale flow, where removing the mean flow improves the agreement with the theorem but introduces filtering artifacts.

An answer must address all five components above, resolving each to the extent the evidence allows and stating explicitly any part that is left open; the parts already settled by later work (extension to inhomogeneous and anisotropic swirling flows, finite-size spherical particles, the tracer Reynolds-number trend, and the finding, in the 2026 follow-up's comparison against the von Kármán swirling-flow experiment, that a nonlinear-drag model reproduces the increase of $\Delta_{EM}$ for the particle velocity at $\mathrm{St}\ge 2$ while the Maxey-Riley point-particle model does not) are already established and need not be redone.

References

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