Cumulant expansions for solution of quantum BBGKY hierarchy

Cumulant expansions for solution of quantum BBGKY hierarchy

V.I. Gerasimenko  and V.O. Shtyk

Institute of Mathematics of NAS of Ukraine
3 Tereshchenkivs’ka Str., 01601 Kyiv-4, Ukraine

Abstract. We construct a new representation of the solution of the initial-value problem to the quantum BBGKY hierarchy of equations as an expansion over particle clusters whose evolution are described by the corresponding-order cumulant (semi-invariant) of the evolution operators of finitely many particle quantum systems. For the initial data from the space of sequences of the trace operators the existence and uniqueness theorem is proved. On the basis of the cluster expansions of the evolution operators of finitely many particle quantum systems, we give the classification of possible solution representations of the quantum BBGKY hierarchy in the case of Maxwell-Boltzmann statistics. We discuss the problem of the construction of a solution in the space of sequences of bounded operators. For the initial data from this space the stated cumulant nature of the solution expansion guarantees the compensation of divergent traces in each its term.

Existence and uniqueness theorem

The series (6), that represents the solution of the BBGKY hierarchy (1)-(2), converges in the sense of the norm of the space L1a (FH ) if a > e. Indeed, according to the inequality [5]

For the interaction potential F satisfying the Kato conditions [1], which guarantee the self-adjointness of the Hamiltonian, the existence and uniqueness theorem of the cumulant representation (6), (8) of a solution of the initial-value problem to the quantum BBGKY hierarchy (1)-(2) is valid

This solution is a strong solution for F (0) 2 D(N ) L1a (FH ) and a weak one for arbitrary initial data from the space L1a (FH ).

The theorem is proved in the standard way [1-4].

Expression (9) is defined the one-parametric mapping t ! F (t) that defines a group of class C0 in the space L1a (FH ). This group preserves Hermicity and the cone of sequences of positivity trace operators.

Conclusion

It should be noted that different representations for the solution of the BBGKY hierarchy for the initial data from the space L1a (FH ) are equivalent in mentioned above sense. For description of the states of infinitely particle systems we have to construct the solution for the initial data belonging to the functional spaces different from the space of trace class operators. For instance, in the capacity of such space it can be chosen the space of sequences of bounded operators, in particular, the sequence of n – particle equilibrium operators [1] belongs to this space. For initial data from this space every term of the solution expansion (6) contains the expressions with divergent traces. The cumulant representation for the solution constructed above makes it possible to prove that expressions with divergent traces in each term of the expansion for the solution are mutually compensated.

ACKNOWLEDGMENTS

This work was supported by INTAS and by NAS of Ukraine through grant No. 0105U005666 for young scientists.

REFERENCES

Petrina D.Ya. Mathematical Foundations of Quantum Statistical Mechanics. Continuous Systems. – Amsterdam: Kluwer, 1995.

Cercignani C., Gerasimenko V.I., Petrina D.Ya. Many-Particle Dynamics and Kinetic Equations. – Kluwer Acad. Publ., 1997. – p. 256 .

Petrina D.Ya., Gerasimenko V.I., Malyshev P.V. Mathematical foundations of classical statistical mechanics. Continuous systems, Second edition, Francis and Taylor, 2002.

Dautray R., Lions J.L. Mathematical Analysis and Numerical Methods for Science and Technology. – v.5. – Berlin, N.Y.: Springer-Verlag, 1992.

Gerasimenko V.I. and Ryabukha T.V., Cumulant representation of solutions of the BBGKY hierarchy of equations, Ukr. Math. J.. – 2002 – v. 54, N 10. — pp. 1583–1601.

 


 

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