We investigate a system of harmonically coupled identical nonlinear constituents subject to noise in different spatial arrangements. For global coupling, we find for infinitely many constituents the coexistence of several ergodic components and a bifurcation behavior like in first-order phase transitions. These results are compared with simulations for finite systems both for global coupling and for nearest-neighbor coupling on two-and three-dimensional cubic lattices. The mean-field-type results for global coupling provide a better understanding of the more complex behavior in the latter case.

We describe nonequilibrium phase transitions in arrays of dynamical systems with cubic nonlinearity driven by multiplicative Gaussian white noise. Depending on the sign of the spatial coupling we observe transitions to ferromagnetic or antiferromagnetic ordered states. We discuss the phase diagram, the order of the transitions, and the critical behavior. For global coupling we show analytically that the critical exponent of the magnetization exhibits a transition from the value 1/2 to a nonuniversal behavior depending on the ratio of noise strength to the magnitude of the spatial coupling.

The Ising model in a transverse field is treated by means of Green function technique to calculate the dynamic structure factor above and below T,. The soft mode is overdamped in the disordered phase and near T,, while it is nearly undamped a t low temperatures. The coupling between the transverse soft mode and the relaxing longitudinal mode produces a central mode in the dynamic structure factor. The temperature dependence of the dynamic structure factor for zero wave vector is calculated numerically.

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The diagram technique previously developed by one of the authors for the Green's function of a Heisenberg ferromagnet with spin 1/2 is generalized to arbitrary spin. The summation of diagrams is the main problem in the present paper. The diagrams split into two classes; one of them may be summed with help of Dyson's equation, the other cannot be treated as a contribution to the mass operator. The latter class of diagrams is found to be the series expansion of (8") and (Sf St). The resulting spin wave energy corresponds in successive approximations to the results of MFA, of Tyablikov's decoupling, of Dyson, and of the decoupling due to Mubayi and Lange, and Kenan, respectively. The low temperature magnetization is calculated directly from the Green's function and agrees with Callen's result in the highest approximation worked out.Die friiher von einem der Verfasser vorgeschlagene Diagrammtechnik fur die Greensche Funktion eines Heisenberg-Ferromagnetikums mit Spin 1/2 wird fur beliebigen Spin verallgemeinert. Beziiglich der Summation zerfallen die Graphen in zwei Klassen: Eine Klasse kann mit Hilfe der Dyson-Gleichung summiert werden, die andere kann nicht als Beitrag zum Massenoperator behandelt werden. Letztere Diagramme ergeben die Entwicklungen von (8') und (Sf Sk). Die resultierenden Spinwellenenergien entsprechen in sukzessiven Naherungen den Ergebnissen von MFA, Tyablikov, Dyson bzw. Mubayi und Lange sowie von Kenan. Die Tieftemperaturentwicklung der Magnetisierung wird direkt aus der Greenschen Funktion berechnet und stimmt in der hochsten betrachteten Naherung mit dem Ergebnis von Callen uberein. " 1) Present address: Technische Hochschule Ilmenau, Sektion PHYTEB, 63 Ilmenau, Weimarer Str.. 40 physica (b) GO/Z