The Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy provides a time-reversal-symmetric framework for describing the nonequilibrium evolution of many-body systems. Despite the success of Boltzmann-based numerical approaches, systematically extending beyond this lowest-order truncation to the full nonlinear BBGKY hierarchy remains a major challenge. Moreover, even at the Boltzmann level, accurately treating the nonlinear collision term still presents significant difficulties. Here we propose the spectral BBGKY hierarchy, an analytically equivalent and numerically tractable reformulation of the conventional BBGKY hierarchy. The spectral formulation reduces the original $6n$-dimensional phase-space problem to the evolution of spectral coefficients over the $3n$-dimensional coordinate space, thereby eliminating the need for momentum-space discretization. In addition, we develop an analytic scheme for computing the collision integrals, which achieves high accuracy and removes the need for ensemble averaging over repeated stochastic evolutions from the same initial state. The scheme evaluates the full eight-dimensional integral exactly for massless particles, and reduces it to a three-dimensional one for massive particles. The validity of the spectral BBGKY hierarchy is verified through conservation law analysis, comparison with an analytical solution, convergence tests, and analysis of spectral coefficient leakage. At minimal truncation, the spectral BBGKY yields a spectral nonlinear Boltzmann equation that captures nonlinear dynamics with a computational cost comparable to that of conventional linearized approaches. When extended to higher-order truncations ($n\ge 2$), the spectral BBGKY hierarchy provides a flexible framework for studying multiparticle correlations across a wide range of systems—from ultracold atomic gases and quark-gluon plasma produced in relativistic heavy-ion collisions to the primordial plasma of the early universe. This framework advances our ability to investigate the early thermalization puzzle in relativistic heavy-ion collisions and to elucidate the applicability of hydrodynamics at remarkably early stages of quark-gluon plasma evolution.

Close

PrevNext

View figure in article

• Quark-gluon plasma
• Relativistic heavy-ion collisions
• Transport in heavy-ion collisions