We introduce a general class of generating functionals for the calculation of quantum-mechanical expectation values of arbitrary functionals of fluctuating paths with fixed end points in configuration or momentum space. The generating functionals are calculated explicitly for harmonic oscillators with time-dependent frequency, and used to derive a smearing formulas for correlation functions of polynomial and nonpolynomials functions of time-dependent positions and momenta. These formulas summarize the effect of thermal and quantum fluctuations, and serve to derive generalized Wick rules and Feyn-man diagrams for perturbation expansions of nonpolynomial interactions.
A broad class of (N+1) -species ratio-dependent predator-prey stochastic models, which consist of one predator population and N prey populations, is considered. The effect of a fluctuating environment on the carrying capacities of prey populations is taken into account as colored noise. In the framework of the mean-field theory, approximate self-consistency equations for prey-populations mean density and for predator-population density are derived (to the first order in the noise variance). In some cases, the mean field exhibits Hopf bifurcations as a function of noise correlation time. The corresponding transitions are found to be reentrant, e.g., the periodic orbit appears above a critical value of the noise correlation time, but disappears again at a higher value of the noise correlation time. The nonmonotonous dependence of the critical control parameter on the noise correlation time is found, and the conditions for the occurrence of Hopf bifurcations are presented. Our results provide a possible scenario for environmental-fluctuations-induced transitions between the oscillatory regime and equilibrium state of population sizes observed in nature.
An N-species Lotka-Volterra stochastic model of a symbiotic ecological system with the Verhulst self-regulation mechanism is considered. The effect of fluctuating environment on the carrying capacity of a population is modeled as the colored three-level Markovian (trichotomous) noise. In the framework of the mean-field theory an explicit self-consistency equation for stationary states is presented. Stability and instability conditions and colored-noise-induced discontinuous transitions (catastrophic shifts) in the model are investigated. In some cases the mean field exhibits hysteresis as a function of the noise parameters. It is shown that the occurrence of catastrophic shifts can be controlled by noise parameters, such as correlation time, amplitude, and flatness. The dependence of the critical coupling strengths on the noise parameters is found and illustrated by phase diagrams. Implications of the results on some modifications of the model are discussed.
The colored three-level Markovian noise-driven nonequilibrium dynamics of overdamped Brownian particles in a spatially periodic asymmetric potential (ratchet) is investigated. An explicit second-order linear ordinary differential equation for the stationary probability density distribution is obtained for the process. In the case of a piecewise linear potential with an additive three-level (trichotomous) noise the exact formula for the stationary current is presented. The dependence of the current reversals on the noise parameters is investigated in detail and illustrated by a phase diagram. Asymptotic formulas for the current for various limits of the noise parameters are found and compared with the results of other authors. Applications to the fluctuation-induced separation of particles are also discussed.
A nonlinear one-dimensional process driven by a multiplicative exponentially correlated three-level Markovian noise (trichotomous noise) is considered. An explicit second-order linear ordinary differential equation for the stationary probability density distribution is obtained for the process. In the case of a linear process with an additive trichotomous noise the exact formula for the steady-state distribution is obtained. The well-known dichotomous noise can be regarded as a special case of the trichotomous noise. As a rule, the system variable has three specific values where the probability density distribution can be singular. For the case of the Hongler model the dependence of the behavior of the stationary probability density on the noise parameters is investigated in detail and illustrated by a phase diagram. Applications to the Gompertz and Verhulst models are also discussed.
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