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Bulletin de la Classe des sciences

On some basic theorems in subdynamics of correlations

Claude D. George

Published Jan 1, 1973• 0 citations
Publisher

Abstract

The possibility of describing the evolution of large dynamical systems in terms of a superposition of subdynamics, and of formulating a transformation theory of the Liouville-von Neumann equation for such systems rests on a set of relations between dynamical operators. We present a proof of these relations.
DOI
10.3406/barb.1973.60652

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It is shown, using only elementary operator algebra, that an open quantum system coupled to its environment will have a subdynamics (reduced dynamics) as an exact consequence of the reversible dynamics of the composite system only when the states of system and environment are uncorrelated. Furthermore, it is proved that for a finite temperature the KMS condition for the lowest-order correlation function cannot be reproduced by any type of linear subdynamics except the reversible Hamiltonian one of a closed system. The first statement can be seen as a particular case of a more general theorem of Takesaki on the properties of conditional expectations in von Neumann algebras. The concept of subdynamics used here allows for memory effects, no assumption is made of a Markov property. For dynamical systems based on commutative algebras of observables the subdynamics always exists as a stochastic process in the random variable defining the open subsystem.
Previously developed ``stochastic representation of deterministic interactions`` enables exact treatment of an open system without leaving its native phase space (Hilbert space) due to peculiar stochastic extension of the Liouville (von Neumann) equation for its statistical operator. Can one reformulate the theory in terms of stochastic ``Langevin equations'' for its variables? Here it is shown that in case of classical Hamiltonian underlying dynamics the answer is principally positive, and general explicit method of constructing such equations is described.
One of us (C.G) has developed a transformation theory in nonequilibrium statistical mechanics that generalizes the usual transformation theory of quantum mechanics in the framework of the resolvent formalism of the Liouville-von Neumann operator extensively used by the Brussels school. Prigogine, George, and Henin have shown that the evolution of a dissipative system can be decomposed into separate subdynamics in orthogonal subspaces. The π subspace, containing all thermodynamics, can be seen as an extension of the subspace of the invariants in the absence of dissipation. In this subspace a suitable contracted description of the system can be given through a dressing operator χ, which has the property that HR = χ−1H0 can be considered as the renormalized energy. A differential equation that determines χ has been proposed by Mandel. It is useful to see, even for a nondissipative system, what relation the transformation theory in the Liouville-von Neumann formalism has to the standard transformations used in quantum mechanics. In this paper we consider a system that can be completely diagonalized by the Bogoliubov transformation. It is shown that in this case the Mandel equation can be solved and that χ−1H0 = HD, where HD is the diagonalized Hamiltonian, obtained by the Bogoliubov transformation.