An official website of the United States government
Wasserstein gradient flows provide a powerful means of understanding and solving many diffusion equations. Specifically, Fokker-Planck equations, which model the diffusion of probability measures, can be understood as gradient descent over entropy functionals in Wasserstein space. This equivalence, introduced by Jordan, Kinderlehrer and Otto, inspired the so-called JKO scheme to approximate these diffusion processes via an implicit discretization of the gradient flow in Wasserstein space. Solving the optimization problem associated with each JKO step, however, presents serious computational challenges. We introduce a scalable method to approximate Wasserstein gradient flows, targeted to machine learning applications. Our approach relies on input-convex neural networks (ICNNs) to discretize the JKO steps, which can be optimized by stochastic gradient descent. Contrarily to previous work, our method does not require domain discretization or particle simulation. As a result, we can sample from the measure at each time step of the diffusion and compute its probability density. We demonstrate the performance of our algorithm by computing diffusions following the Fokker-Planck equation and apply it to unnormalized density sampling as well as nonlinear filtering.
more »
« less
This content will become publicly available on August 1, 2026
Effective Long-Time Diffusivity of Particles of Arbitrary Shape in an External Orienting Field
Abstract Pertaining to the motion of a rigid particle in a flow, several distinct “centers” of the rigid particle can be identified, including the geometric center (centroid), center of mass, hydrodynamic center, and center of diffusion. In this work, we elucidate the relevance of these centers in Brownian motion and diffusion. Starting from the microscopic stochastic equations of motions, we systematically derive the coarse-grained Fokker–Planck equations that govern the evolution of the probability distribution function (PDF) in phase space and in configurational space. For consistency with the equilibrium statistical mechanics, we determine the unknown Brownian forces and torques. Next, we analyze the Fokker–Planck equation for the PDF in the position and orientation space. Through a multiscale analysis, we find the unit cell problem for defining the effective long-time translational diffusivity of a particle of arbitrary shape in an external orienting field. We also show some fundamental properties of the effective long-time translational diffusivity, including rigorous variational bounds for effective long-time diffusivity and invariance of effective diffusivity with respect to change of reference or tracking points. Exact results are obtained in the absence of an orienting field and in the presence of a strong orienting field. These fundamental results hold significant potential for applications in biophysics, colloidal science, and micro-swimmers design.
more »
« less
- Award ID(s):
- 2306254
- PAR ID:
- 10596178
- Publisher / Repository:
- ASME
- Date Published:
- Journal Name:
- Journal of Applied Mechanics
- Volume:
- 92
- Issue:
- 8
- ISSN:
- 0021-8936
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
The relative velocities and positions of monodisperse high-inertia particle pairs in isotropic turbulence are studied using direct numerical simulations (DNS), as well as Langevin simulations (LS) based on a probability density function (PDF) kinetic model for pair relative motion. In a prior study (Rani et al. , J. Fluid Mech. , vol. 756, 2014, pp. 870–902), the authors developed a stochastic theory that involved deriving closures in the limit of high Stokes number for the diffusivity tensor in the PDF equation for monodisperse particle pairs. The diffusivity contained the time integral of the Eulerian two-time correlation of fluid relative velocities seen by pairs that are nearly stationary. The two-time correlation was analytically resolved through the approximation that the temporal change in the fluid relative velocities seen by a pair occurs principally due to the advection of smaller eddies past the pair by large-scale eddies. Accordingly, two diffusivity expressions were obtained based on whether the pair centre of mass remained fixed during flow time scales, or moved in response to integral-scale eddies. In the current study, a quantitative analysis of the (Rani et al. 2014) stochastic theory is performed through a comparison of the pair statistics obtained using LS with those from DNS. LS consist of evolving the Langevin equations for pair separation and relative velocity, which is statistically equivalent to solving the classical Fokker–Planck form of the pair PDF equation. Langevin simulations of particle-pair dispersion were performed using three closure forms of the diffusivity – i.e. the one containing the time integral of the Eulerian two-time correlation of the seen fluid relative velocities and the two analytical diffusivity expressions. In the first closure form, the two-time correlation was computed using DNS of forced isotropic turbulence laden with stationary particles. The two analytical closure forms have the advantage that they can be evaluated using a model for the turbulence energy spectrum that closely matched the DNS spectrum. The three diffusivities are analysed to quantify the effects of the approximations made in deriving them. Pair relative-motion statistics obtained from the three sets of Langevin simulations are compared with the results from the DNS of (moving) particle-laden forced isotropic turbulence for $$St_{\unicode[STIX]{x1D702}}=10,20,40,80$$ and $$Re_{\unicode[STIX]{x1D706}}=76,131$$ . Here, $$St_{\unicode[STIX]{x1D702}}$$ is the particle Stokes number based on the Kolmogorov time scale and $$Re_{\unicode[STIX]{x1D706}}$$ is the Taylor micro-scale Reynolds number. Statistics such as the radial distribution function (RDF), the variance and kurtosis of particle-pair relative velocities and the particle collision kernel were computed using both Langevin and DNS runs, and compared. The RDFs from the stochastic runs were in good agreement with those from the DNS. Also computed were the PDFs $$\unicode[STIX]{x1D6FA}(U|r)$$ and $$\unicode[STIX]{x1D6FA}(U_{r}|r)$$ of relative velocity $$U$$ and of the radial component of relative velocity $$U_{r}$$ respectively, both PDFs conditioned on separation $$r$$ . The first closure form, involving the Eulerian two-time correlation of fluid relative velocities, showed the best agreement with the DNS results for the PDFs.more » « less
-
As a counterpoint to classical stochastic particle methods for linear diffusion equations, such as Langevin dynamics for the Fokker-Planck equation, we develop a deterministic particle method for the weighted porous medium equation and prove its convergence on bounded time intervals. This generalizes related work on blob methods for unweighted porous medium equations. From a numerical analysis perspective, our method has several advantages: it is meshfree, preserves the gradient flow structure of the underlying PDE, converges in arbitrary dimension, and captures the correct asymptotic behavior in simulations.more » « less
-
null (Ed.)Abstract We rigorously justify the mean-field limit of an N -particle system subject to Brownian motions and interacting through the Newtonian potential in $${\mathbb {R}}^3$$ R 3 . Our result leads to a derivation of the Vlasov–Poisson–Fokker–Planck (VPFP) equations from the regularized microscopic N -particle system. More precisely, we show that the maximal distance between the exact microscopic trajectories and the mean-field trajectories is bounded by $$N^{-\frac{1}{3}+\varepsilon }$$ N - 1 3 + ε ( $$\frac{1}{63}\le \varepsilon <\frac{1}{36}$$ 1 63 ≤ ε < 1 36 ) with a blob size of $$N^{-\delta }$$ N - δ ( $$\frac{1}{3}\le \delta <\frac{19}{54}-\frac{2\varepsilon }{3}$$ 1 3 ≤ δ < 19 54 - 2 ε 3 ) up to a probability of $$1-N^{-\alpha }$$ 1 - N - α for any $$\alpha >0$$ α > 0 . Moreover, we prove the convergence rate between the empirical measure associated to the regularized particle system and the solution of the VPFP equations. The technical novelty of this paper is that our estimates rely on the randomness coming from the initial data and from the Brownian motions.more » « less
-
The Langevin equations (LE) and the Fokker–Planck (FP) equations are widely used to describe fluid behavior based on coarse-grained approximations of microstructure evolution. In this manuscript, we describe the relation between LE and FP as related to particle motion within a fluid. The manuscript introduces undergraduate students to two LEs, their corresponding FP equations, and their solutions and physical interpretation.more » « less
