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Journal of Statistical Physics

Mapping of Stochastic dynamics onto associated quantum models and (d+l)-dimensional classical and static systems

T. Schneider

Published Jun 1, 1986• 0 citations
Publisher
DOI
10.1007/bf02628316

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Physical review. B, Condensed matter•Mar 1, 1985•37 citations
We propose the transformation of the Langevin equation into an eigenvalue problem as the framework of the stochastic simulation of quantum systems. The potential of the method is illustrated by numerical results for the low-lying energy eigenvalues of one-particle systems. In the case of many degrees of freedom, the critical dynamics of a classical ddd-dimensional system is mapped onto the statics of an associated quantum system and of its classical (d+1d+1d+1)-dimensional counterpart. These mappings yield an equality relating the dynamical critical exponent $\ensuremath{\Delta}$ and the static susceptibility exponents of three models.
We propose transformation of the Langevin equation into an eigenvalue problem as the framework of the stochastic simulation of quantum systems. The potential of the method is illustrated by numerical results for the low-lying energy levels of one-particle systems and coupled harmonic oscillators. Another application is the relationship between the dynamics of a d-dimensional system evolving according to the Langevin equation, an associated d-dimensional quantum system, and its (d+1)-dimensional classical counterpart. On this basis we relate the critical dynamics to the static properties of an associated quantum system and its (d+1)-dimensional classical analog. This then leads to equalities and scaling relations between the static and dynamic critical exponents. As a specific example, we treat the time-dependent Ginzburg-Landau model without conservation laws. In the large-n limit, the critical dynamics is traced back to Gaussian (tricritical Lifshitz) behavior of the associated (d+1)-dimensional classical model. The static exponents describing the critical dynamics are determined, however, by the crossover from Lifshitz to Gaussian (tricritical Lifshitz) behavior.