Dissipative contributions of internal multiplicative noise: I. mechanical oscillator

Langevin equations for closed systems with multiplicative fluctuations must also include appropriate dissipative terms that ensure eventual equilibration of the system. We consider an oscillator coupled to a heat bath and show that a particular nonlinear coupling to a harmonic heat bath leads to a fluctuating frequency and to nonlinear dissipative terms. We also analyze the effects of the multiplicative fluctuations and of the corresponding nonlinear dissipation on the temporal evolution of the average oscillator energy. We find that the rate of equilibration of this system can be significantly different from that of an oscillator with only additive fluctuations and linear dissipation.     

Lagrangian approach and dissipative magnetic systems

A Lagrangian is introduced which includes the coupling between magnetic moments m and the degrees of freedom σ of a reservoir. In case the system-reservoir coupling breaks the time reversal symmetry the magnetic moments perform a damped precession around an effective field which is self-organized by the mutual interaction of the moments. The resulting evolution equation has the form of the Landau-Lifshitz-Gilbert equation. In case the bath variables are constant vector fields the moments m fulfill the reversible Landau-Lifshitz equation. Applying Noether?s theorem we find conserved quantities under rotation in space and within the configuration space of the moments.     

Quantum Dissipative Threshold and Its Effect on Decay of Metastable State

The coupling between system and reservoir is considered to be linear in the coordinates of the bath but nonlinear in the system＇s coordinate. A dissipative threshold is observed at finite temperatures due to nonlinear dissipation. The quantum decay rate of a metastable state including higher-order expanded terms of the coupling form function is proposed, which can be strongly decreased at finite temperatures when the quantum dissipative threshold is added to the saddle point of the potential.     

Nonlinear quantum dynamical semigroups for many-body open systems

The notion of a nonlinear quantum dynamical semigroup is introduced, and the existence and uniqueness of solutions of the corresponding nonlinear evolution equations are studied in a more abstract framework. The construction of nonlinear quantum dynamical semigroups is carried out for two different mean-field models. First a mean-field coupling between a system of noninteracting subsystems and the bath is investigated. As examples, a nonlinear frictional Schrödinger equation and a model for a quantum Boltzmann equation are discussed. Second, a many-body system with mean-field interaction coupled to a bath is considered. Here, again, the form of the generator is derived; however, it cannot be obtained rigorously, except for some particular examples. Finally, the quantum Ising-Weiss model is briefly studied.     

Fokker-Planck equation approach to fluctuations about nonequilibrium steady states

We examine the properties of steady states in systems which interact at the boundary with a nonequilibrium environment. The examination is based on a nonlinear Fokker-Planck equation, the structure of which is determined by the fact that it also governs the time evolution of the equilibrium fluctuations of the system. The nonlinearities in the Fokker-Planck equation may have two origins: thermodynamic nonlinearities which arise if the thermodynamic potential is not a bilinear function of the state variables, and nonlinear mode coupling which arises if the transport coefficients depend on the state. While these nonlinearities have only a small effect on the equilibrium fluctuations of a system away from critical points, they are shown to be important for the determination of fluctuations about nonequilibrium steady states. In particular the state dependence of the transport coefficients may lead to deviations from local equilibrium and to a breakdown of detail balance. An explicit formula for the time correlations of fluctuations about the nonequilibrium steady state is obtained. The formula leads to long-range correlations in fluids in the presence of a temperature gradient. The result is compared with earlier approaches to the same problem. Finally, we study the linear response to external forces and obtain a generalization of the fluctuation-dissipation formula relating the response functions with the nonequilibrium correlation functions.     

Beam-core evolution equation and space-charge limit

The nonlinear beam-core evolution equation approach is proposed as a powerful tool to estimate the acceptable beam current in a given circular accelerator. The approach is justified by the macroparticle simulation over a wide beam current parameter region. Space-charge effects on the beam-core evolution are discussed by Poincaré mapping on the beam-core phase space (σ, dσ/ds). The instability seen in the beam-core evolution is rigorously analyzed as an eigenvalue problem in the coupled linear system derived from the linearized beam-core evolution equations. A threshold current resulting in the instability is given by both the nonlinear beam-core evolution equation approach and the coupled linear system approach. As an example, a fast cycling induction synchrotron is evaluated in which the space-charge effects are significant because it is injector-free.     

Inhomogeneous quantum diffusion and decay of a meta-stable state

We consider the quantum stochastic dynamics of a system whose interaction with the reservoir is considered to be linear in bath co-ordinates but nonlinear in system co-ordinates. The role of the space-dependent friction and diffusion has been examined in the decay rate of a particle from a meta-stable well. We show how the decay rate can be hindered by inhomogeneous dissipation due to nonlinear system–bath coupling strength.     

Fluctuations in nonlinear Markovian systems

We study nonlinear irreversible processes by statistical mechanical methods from a general point of view. Assuming that the macroscopic variables behave approximatively Markovian we derive evolution equations for the mean values as well as for the fluctuations about the mean. The mean values obey nonlinear transport equations and the fluctuations obey linear nonstationary Langevin equations. The equations of motion are completely specified by the entropy and the transport coefficients as functions of the macroscopic state. The theory provides a statistical mechanical basis for some phenomenological approaches put forward recently.     

Microscopic theory of brownian motion: III. The nonlinear Fokker-Planck equation

The nonlinear Fokker-Planck equation for the momentum distribution of a brownian particle of mass M in a bath of particles of mass m is derived. The contribution to this equation arising from initial deviation from bath equilibrium is analysed. This contribution is free of slow M-dependent decays and with certain restrictions leads to an effective shift in the initial value of the B particle momentum. The nonlinear Fokker-Planck equation for an initial bath equilibrium state is analyzed in terms of its predictions for momentum relaxation and mode coupling effects. It is found that in addition to nonlinear renormalization of the type previously found for the momentum correlation function, mode coupling leads to long-lived memory of the initial momentum state.     

Heating and fluctuations in elastoplastic systems

Starting from the equations of motion of a simple system possessing the properties of elastic and plastic bodies, we construct its Lagrangian and Hamiltonian functions and also the Rayleigh dissipation function. This allows us to find the rate of heating of the system and to analyze the fluctuations of basic observables. Introducing into the Hamilton-Rayleigh equation of motion a random force producing on average the same effects as a dissipation function, we arrive first at the Langevin equations describing the fluctuations and then at a kinetic equation for the distribution function defined in the space of the collective variables. In this way a rather general scheme is established for solving dynamical problems in different and more complex elastoplastic systems, in nuclear physics and maybe even in physics of molecules and atomic clusters. In a preliminary study, the model is applied to estimate the probability of the quasi-fission process coming from the thermal fluctuations of the nuclear shape.     

Hamiltonian Structure for Dispersive and Dissipative Dynamical Systems

We develop a Hamiltonian theory of a time dispersive and dissipative inhomogeneous medium, as described by a linear response equation respecting causality and power dissipation. The proposed Hamiltonian couples the given system to auxiliary fields, in the universal form of a so-called canonical heat bath. After integrating out the heat bath the original dissipative evolution is exactly reproduced. Furthermore, we show that the dynamics associated to a minimal Hamiltonian are essentially unique, up to a natural class of isomorphisms. Using this formalism, we obtain closed form expressions for the energy density, energy flux, momentum density, and stress tensor involving the auxiliary fields, from which we derive an approximate, “Brillouin-type,” formula for the time averaged energy density and stress tensor associated to an almost mono-chromatic wave.     

External dissipation in driven two-dimensional turbulence

Turbulence in a freely suspended soap film is created by electromagnetic forcing and measured by particle tracking. The velocity fluctuations are shown to be adequately described by the forced Navier-Stokes equation for an incompressible two-dimensional fluid with a linear drag term to model the frictional coupling to the surrounding air. Using this equation, the energy dissipation rates due to air friction and the film's internal viscosity are measured, as is the rate of energy injection from the electromagnetic forcing. Comparison of these rates demonstrates that the air friction is a significant energy dissipation mechanism in the system.     

Stability of Decoherence-Free Subspaces under Stochastic Phase Fluctuations

We study the stability of decoherence-free subspaces under stochastic phase fluctuations by analytically and numerically evaluating the fidelity of the corresponding decoherence-free subspace bases with stochastic phase fluctuations under the evolution of environment. The environment is modeled by a bath of oscillators with infinite degrees of freedom and the register-bath coupling is chosen to be a general dissipation-decoherence form. It is found that the decoherence-free subspaces take on good stability in the case of small dissipation and small phase fluctuations.     

Current in open quantum systems

We show that a dissipative current component is present in the dynamics generated by a Liouville-master equation, in addition to the usual component associated with Hamiltonian evolution. The dissipative component originates from coarse graining in time, implicit in a master equation, and needs to be included to preserve current continuity. We derive an explicit expression for the dissipative current in the context of the Markov approximation. Finally, we illustrate our approach with a simple numerical example, in which a quantum particle is coupled to a harmonic phonon bath and dissipation is described by the Pauli master equation.     

Stability of Decoherence-Free Subspaces under Stochastic Phase Fluctuations

We study the stability of decoherence-free subspaces under stochastic phase fluctuations by analytically and numerically evaluating the fidelity of the corresponding decoherence-free subspace bases with stochastic phase fluctuations under the evolution of environment. The environment is modeled by a bath of oscillators with infinite degrees of freedom and the register-bath coupling is chosen to be a general dissipation-decoherence form. It is found that the decoherence-free subspaces take on good stability in the case of small dissipation and small phase fluctuations.     

Generation of macroscopic superpositions of quantum states by linear coupling to a bath

We demonstrate through an exactly solvable model that collective coupling to any thermal bath induces effectively nonlinear couplings in a quantum many-body (multispin) system. The resulting evolution can drive an uncorrelated large-spin system with high probability into a macroscopic quantum-superposition state. We discuss possible experimental realizations.