BBGKY Hierarrchy for N D0-Branes

TL;DR

The paper addresses deriving an exact BBGKY hierarchy for a system of $N$ D0-branes described by matrix quantum mechanics. It constructs the full phase-space distribution $\rho_N(\Phi,\Pi,t)$ obeying Liouville dynamics and defines reduced $\rho_n$ via integration over the remaining degrees of freedom to obtain an exact BBGKY chain linking $\rho_n$ to $\rho_{n+1}$ through an integro-differential operator $\hat{L}$. It provides the explicit Hamiltonian form $H_N = \frac{g_s l_s}{2} \mathrm{Tr} \Pi_I \delta^{IJ} \Pi_J - \frac{1}{4 g_s l_s} \mathrm{Tr} [\Phi^I,\Phi^J][\Phi_I,\Phi_J]$ and discusses prospects for kinetic closures and hydrodynamics of D0-branes, with potential extensions to supersymmetric matrix mechanics.

Abstract

We study statistical description of N D0-branes system that is defined by matrix mechanics. We determine BBGKY hierarchy for collection of distribution functions that gives exact statistical description of this system.