Abstract
The Wigner equation is considered for a system of a large numberN of identical particles with interaction factor of the order of 1/N. In both the Bose and the Fermi cases, we construct the asymptotics of the solution of the Cauchy problem for this equation with regard to the exchange effect for the case in which the Planck constant is of the order ofN −1/d, whered is the space dimension. This asymptotics is interpreted in terms of the theory of the complex germ on a curved phase space equivalent to the space of functions with values on the Riemann sphere in the Fermi case and on the Lobachevskii plane in the Bose case. The classical equations of motion in both cases are reduced to the Vlasov equation; since the phase space is infinite-dimensional, the complex germ is subjected to additional conditions depending on the type of statistics.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. P. Maslov and O. Yu. Shvedov, “Asymptotic solutions to the Wigner equation for systems of a large number of particles,”Russian J. Math. Phys.,3, No. 1, 65–80 (1995).
V. P. Maslov,The Complex WKB Method for Nonlinear Equations [in Russian], Nauka, Moscow (1977).
V. P. Maslov and O. Yu. Shvedov, “The complex germ method in the Fock space. I. Asymtotics of the wave packet type,”Teoret. Mat. Fiz. [Theoret. and Math. Phys.],104, No. 2, 310–329 (1995).
V. P. Maslov and O. Yu. Shvedov, “The complex germ method in the Fock space. II. Asymtotics corresponding to finite-dimensional isotropic manifolds,”Teoret. Mat. Fiz. [Theoret. and Math. Phys.],104, No. 3, 479–506 (1995).
F. A. Berezin,The Method of Second Quantization [in Russian], Nauka, Moscow (1986).
M. V. Karasev and V. P. Maslov,Nonlinear Poisson Brackets. Geometry and Quantization, Nauka, Moscow [in Russian] (1991).
V. P. Maslov and O. Yu. Shvedov, “An asymptotic formula for theN-particle density function asN→∞ and violation of the chaos hypothesis,”Russian J. Math. Phys.,2, No. 2, 217–234 (1994).
V. P. Maslov and O. Yu. Shvedov, “On the approximation for the quantum large canonical distribution,”Russian J. Math. Phys.,6, No. 1, 80–89 (1998).
A. I. Kostrikin,An Introduction to Algebra [in Russian], Nauka, Moscow (1977).
V. P. Maslov,Operator Methods [in Russian], Nauka, Moscow (1973).
V. P. Maslov,Perturbation Theory and Asymptotic Methods [in Russian], Izd. Moskov. Univ., Moscow (1965).
F. A. Berezin, “Models of Gross-Neveu type are quantization of a classical mechanics with nonlinear phase space,”Comm. Math. Phys.,63, 131–153 (1978).
Author information
Authors and Affiliations
Additional information
Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 84–106, January, 1999.
Rights and permissions
About this article
Cite this article
Maslov, V.P., Shvedov, O.Y. Asymptotics of the density matrix for a system of a large number of identical particles. Math Notes 65, 70–88 (1999). https://doi.org/10.1007/BF02675012
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02675012