A Kinetic Theory Approach To Ordered Fluids: Direct Simulation Monte Carlo Methods
Abstract
We present a direct simulation Monte Carlo (DSMC) method for the kinetic theory of ordered fluids recently introduced by the authors. The microscopic constituents of these fluids carry an order parameter belonging to a manifold. The method applies Strang splitting to couple a Monte Carlo treatment of the binary collisions with a symplectic integration of the mean-field Vlasov transport on the order-parameter manifold. We use the Nanbu-Babovsky approach for Maxwellian collision kernels, and Bird's acceptance-rejection method for general ones. The mean-field Vlasov force is reconstructed from the particle ensemble at each time step. We also include a Bhatnagar-Gross-Krook (BGK) / Andersen thermostat to control the system temperature. We formulate the method for a generic order-parameter manifold, and specialize it to two-dimensional calamitic (rod-like) molecules. Numerical tests validate the method. We confirm the relaxation to the Maxwellian distribution predicted by the H-theorem, and simulate the mean-field alignment driven by quadratic, Kuramoto, and Onsager interaction potentials. With the Onsager potential we successfully capture the isotropic-nematic phase transition at the correct critical temperature.
Related
- Interior-point proximal methods for nonsmooth optimization in Hilbert spaces with cone-ordered constraints
- Methods for Deriving Nonlinear Kinetic Equations from the Dynamics of Quantum Systems
- Convergence analysis of moment-based approximations in kinetic theory
- Kinetic theory for Transformers and the lost-in-the-middle phenomenon
- Fluctuating Kinetic Theory: A Poissonian Stochastic Boltzmann Equation
BibTeX
@misc{carrillo2026kinetic,
title = {{A Kinetic Theory Approach To Ordered Fluids: Direct Simulation Monte Carlo Methods}},
author = {José A. Carrillo and Patrick E. Farrell and Andrea Medaglia and Umberto Zerbinati},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31072},
eprint = {2609.31072},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31072},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Kinetic Theory Approach To Ordered Fluids: Direct Simulation Monte Carlo Methods
AU - José A. Carrillo
AU - Patrick E. Farrell
AU - Andrea Medaglia
AU - Umberto Zerbinati
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31072
UR - https://arxiv.org/abs/2609.31072
AB - We present a direct simulation Monte Carlo (DSMC) method for the kinetic theory of ordered fluids recently introduced by the authors. The microscopic constituents of these fluids carry an order parameter belonging to a manifold. The method applies Strang splitting to couple a Monte Carlo treatment of the binary collisions with a symplectic integration of the mean-field Vlasov transport on the order-parameter manifold. We use the Nanbu-Babovsky approach for Maxwellian collision kernels, and Bird's acceptance-rejection method for general ones. The mean-field Vlasov force is reconstructed from the particle ensemble at each time step. We also include a Bhatnagar-Gross-Krook (BGK) / Andersen thermostat to control the system temperature. We formulate the method for a generic order-parameter manifold, and specialize it to two-dimensional calamitic (rod-like) molecules. Numerical tests validate the method. We confirm the relaxation to the Maxwellian distribution predicted by the H-theorem, and simulate the mean-field alignment driven by quadratic, Kuramoto, and Onsager interaction potentials. With the Onsager potential we successfully capture the isotropic-nematic phase transition at the correct critical temperature.
ER -