Transport Coefficients of Quantum-Classical Systems Kapral and G. Ciccotti

Kapral and G. Ciccotti Transport Coefficients of Quantum-Classical Systems, Lect. Notes Phys. 703, 519-551 (2006) 

Quantum mechanics provides us with the most fundamental description of natural phenomena. In many instances classical mechanics constitutes an adequate approximation and it is widely used in simulations of both static and d3mamic properties of many-body systems. Often, however, quantum effects cannot be neglected and one is faced with the task of devising methods to simulate the behavior of the quantum system. 

The computation of the equilibrium properties of quantum systems is a challenging problem. The simulation of dynamical properties, such as transport coefficients, presents additional problems since the solution of the quantum equations of motion for many-body systems is even more difficult. This fact has prompted the development of approximate methods for dealing with such problems. 

The topic of this chapter is the description of a quantum-classical approach to compute transport coefficients. Transport coefficients are most often expressed in terms of time correlation functions whose evaluation involves two aspects sampling initial conditions from suitable equilibrium distributions and evolution of dynamical variables or operators representing observables of the system. The schemes we describe for the computation of transport properties pertain to quantum many-body systems that can usefully be partitioned into two subsystems, a quantum subsystem S and its environment . We shall be interested in the limiting situation where the dynamics of the environmental degrees of freedom, in isolation from the quantum subsystem S, obey classical mechanics. 

We show how the quantum-classical evolution equations of motion can be obtained as an approximation to the full quantum evolution and point out some of the difficulties that arise because of the lack of a Lie algebraic structure. The computation of transport properties is discussed from two different perspectives. Transport coefficient formulas may be derived by starting from an approximate quantum-classical description of the system. Alternatively, the exact quantum transport coefficients may be taken as the starting point of the computation with quantum-classical approximations made only to the dynamics while retaining the full quantum equilibrium structure. The utility of quantum-classical Liouville methods is illustrated by considering the computation of the rate constants of quantum chemical reactions in the condensed phase. 

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