Abstract—In this paper, we present a quantum statistical mechanical formulation from our recently analytical expressions for partial-wave transition matrix of a three-particle system. We report the quantum reactive cross sections for three-body scattering processes 1+(2,3)→1+(2,3) as well as recombination 1+(2,3)→1+(3,1) between one atom and a weakly-bound dimer. The analytical expressions of three-particle transition matrices and their corresponding cross-sections were obtained from the three- dimensional Faddeev equations subjected to the rank-two non-local separable potentials of the generalized Yamaguchi form. The equilibrium quantum statistical mechanical properties such partition function and equation of state as well as non-equilibrium quantum statistical properties such as transport cross-sections and their corresponding transport collision integrals were formulated analytically. This leads to obtain the transport properties, such as viscosity and diffusion coefficient of a moderate dense gas. Keywords—Statistical mechanics, Nonlocal separable potential, three-body interaction, Faddeev equations. I. INTRODUCTION HE three-particle problems have been extensively proved in a wide variety of problems in all area of physics, especially in quantum statistical mechanics of moderately dense gases. In the quantum theory of three-body systems, Faddeev [1] introduced a set of equations that is analogous to the Lippmann-Schwinger (LS) equation for two-body scattering. Faddeev showed that a well-behaved set of three- body equations involves the two-body T-matrix. In a recent paper, we solved analytically the Faddeev equations for three-body scattering at arbitrary angular momentum and obtained the transition matrices for some transition processes, including scattering and recombination channels in terms of free-particle resolvent matrix. We used a generalized Yamaguchi rank-two nonlocal separable potential (NLSP) to obtain the analytical expressions for partial wave scattering properties of a three-particle system. The NLSPs have been widely used in many branches of physics, because of their extreme simplicity and yield algebraic solution in the LS equation [2]-[8]. Because of their extreme simplicity, these NLPS have been extensively used to theoretically describe the multiparticle problems, particularly in determination of three- body scattering properties using a two-body separable potential. A. Maghari, Department of Chemical Physics, University of Tehran, Tehran, Iran (Corresponding author, phone: +98-21-61113307; fax: +98-21- 66405141; e-mail: maghari@ ut.ac.ir). V. M. Maleki, Department of Chemical Physics, University of Tehran, Tehran, Iran (e-mail: vahdat@khayam.ut.ac.ir). The NLSP model can generally be written as       n i i i i ; ; v V ˆ 1 12 1 2       (1) where n is the rank of the potential operator 12 V ˆ , i v is the attractive (or repulsive) coupling strength and  ; i  is state of the system with angular momentum quantum number  , which is a real number in the unitary case. The momentum representation of such potential is         p p p p p p ) ( i * ) ( i n i i v V ˆ , V ˆ              1 12 12 1 2 (2) where     ; i ) ( i   p p  is the momentum representation of form factor. In the present work, our previous formulations of three- particle scattering properties [4] were used to obtain a new formulation for both equilibrium and non-equilibrium statistical mechanical properties of moderately dense gases. We formulated an analytic expression for equilibrium partition function of two and three-particle correlated states via NLSP. Moreover, in the framework of the non-equilibrium quantum- statistical mechanics and in the corresponding kinetic theory, we obtained the analytical expressions for three-particle collision cross-sections and their corresponding collision integrals, which leads to obtain the transport properties, such as viscosity and diffusion coefficient of a moderate dense gas. II. FADDEEV EQUATIONS AND TRANSITION MATRICES Let us consider three-particle system with the total Hamiltonian V ˆ H ˆ H ˆ   0 , where 0 H ˆ is the total kinetic energy operator and V ˆ is the sum of pair interactions i V ˆ ( jk i V ˆ V ˆ  ) of the three-body system, which treated on an equal footing. The kinetic energy 0 H ˆ in the Jacobi coordinates may be written as a c , b a bc bc p p H ˆ      2 2 2 2 0 (3) where c b c b b c bc m m m m    k k p (4) c b a c b a a c b a m m m m m m       ) ( ) ( k k k p (5) Quantum Statistical Mechanical Formulations of Three-Body Problems via Non-Local Potentials A. Maghari, V. H. Maleki T World Academy of Science, Engineering and Technology International Journal of Nuclear and Quantum Engineering Vol:9, No:10, 2015 610 International Scholarly and Scientific Research & Innovation 9(10) 2015 scholar.waset.org/1307-6892/10002507 International Science Index, Nuclear and Quantum Engineering Vol:9, No:10, 2015 waset.org/Publication/10002507