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By I. Prigogine, Stuart A. Rice

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The original approach is similar to the more widely known stationary-phase Monte Carlo (SPMC) method later put forward by Doll and co-workers [15,16]. A conceptually similar variant has also been proposed by Makri and Miller [17]. The basic idea in the SPMC method is to synthesize part of the phase interference analytically by a stationary-phase-like approximation instead of putting all the load on the stochastic sampling. The result of this is a filtering function that biases the Metropolis trajectory [18] toward regions where the action is nearly stationary [16].

J. Reynolds, and W. A . , Pkys. Rev. Lett. 61, 2312 (1988). 38. J. Carlson, J. W. Moskowitz, and K. E. Schmidt, J . Ckem. Pkys. 90, 1003 (1989). 39. D. R. Hamann, M. Schliiter, and C. Chiang, Pkys. Rev. Lett. 43, 1494 (1979). 40. W. E . Pickett, Comput. Pkys. Rep. 9, 115 (1989): M. Krauss and W. J. Stevens. Ann. Rev. Pkys. Ckem. 35, 357 (1984). 41. M. Dolg, U. Wedig, H. Stoll, and H. Preuss, J . Ckem. Pkys. 86, 866 (1987). 42. E. L. Shirley, L. Mitas, and R. M. Martin, Pkys. Rev. B 44,3395 (1991).

5”becomes “=” in Eq. 8)]. The same arguments carry over to the real-time case where @ is an arbitrary complex phase factor [25,31]. Along a similar line of reasoning, a partial path summation scheme for numerical real-time simulations was suggested by Winterstetter and Domcke [32]. For historical reasons, let us relate these ideas to earlier approaches to the dynamical sign problem. H. MAK AND R. EGGER discrete) coordinate x : where S is the usual action. (ri) = x i and X ( t f ) = x ~ In .

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