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Introduction to the Theory of Stochastic Processes
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93-particle analysis approximation arbitrary assume assumption average behavior binary collisions Boltzmann equation Brownian motion calculation Chapter collision operator collision term consider correlation function corresponding defined definition denoted depends derivation dilute gas Dirac delta function distribution function divergence dynamics eigenfunctions eigenvalues Enskog equilibrium example exponential expression external force factor finite fluid Fokker-Planck equation formal formula Fourier transform Gaussian given Green-Kubo hard spheres hydrodynamic initial condition integral integrand interaction introduced involving kinetic equation Langevin equation Laplace transform large number leads Liouville equation macroscopic Markovian master equation mathematical molecules Moreover neutron nonequilibrium nonlinear obtain one-particle parameters particles physical potential probability density problem properties quantity random variable result right-hand side scattering Section self-diffusion simple solution statistical mechanics stochastic theorem theory thermodynamic limit tion transport coefficients valid vanishes vector velocity wave numbers write zero