Stochastic Theory of Relativistic Particles Moving in a Quantum Field: II. Scalar Abraham-Lorentz-Dirac-Langevin Equation, Radiation Reaction and Vacuum Fluctuations

We apply the open systems concept and the influence functional formalism introduced in Paper I to establish a stochastic theory of relativistic moving spinless particles in a quantum scalar field. The stochastic regime resting between the quantum and semi-classical captures the statistical mechanical attributes of the full theory. Applying the particle-centric world-line quantization formulation to the quantum field theory of scalar QED we derive a time-dependent (scalar) Abraham-Lorentz-Dirac (ALD) equation and show that it is the correct semiclassical limit for nonlinear particle-field systems without the need of making the dipole or non-relativistic approximations. Progressing to the stochastic regime, we derive multiparticle ALD-Langevin equations for nonlinearly coupled particle-field systems. With these equations we show how to address time-dependent dissipation/noise/renormalization in the semiclassical and stochastic limits of QED. We clarify the the relation of radiation reaction, quantum dissipation and vacuum fluctuations and the role that initial conditions may play in producing non-Lorentz invariant noise. We emphasize the fundamental role of decoherence in reaching the semiclassical limit, which also suggests the correct way to think about the issues of runaway solutions and preacceleration from the presence of third derivative terms in the ALD equation. We show that the semiclassical self-consistent solutions obtained in this way are ``paradox'' and pathology free both technically and conceptually. This self-consistent treatment serves as a new platform for investigations into problems related to relativistic moving charges.Comment: RevTex; 20 pages, 3 figures, Replaced version has corrected typos, slightly modified derivation, improved discussion including new section with comparisons to related work, and expanded reference

Metastability in Interacting Nonlinear Stochastic Differential Equations II: Large-N Behaviour

We consider the dynamics of a periodic chain of N coupled overdamped particles under the influence of noise, in the limit of large N. Each particle is subjected to a bistable local potential, to a linear coupling with its nearest neighbours, and to an independent source of white noise. For strong coupling (of the order N^2), the system synchronises, in the sense that all oscillators assume almost the same position in their respective local potential most of the time. In a previous paper, we showed that the transition from strong to weak coupling involves a sequence of symmetry-breaking bifurcations of the system's stationary configurations, and analysed in particular the behaviour for coupling intensities slightly below the synchronisation threshold, for arbitrary N. Here we describe the behaviour for any positive coupling intensity \gamma of order N^2, provided the particle number N is sufficiently large (as a function of \gamma/N^2). In particular, we determine the transition time between synchronised states, as well as the shape of the "critical droplet", to leading order in 1/N. Our techniques involve the control of the exact number of periodic orbits of a near-integrable twist map, allowing us to give a detailed description of the system's potential landscape, in which the metastable behaviour is encoded

Conservation laws and their associated symmetries for stochastic differential equations

The modelling power of Itˆo integrals has a far reaching impact on a spectrum of diverse fields. For example, in mathematics of finance, its use has given insights into the relationship between call options and their non-deterministic underlying stock prices; in the study of blood clotting dynamics, its utility has helped provide an understanding of the behaviour of platelets in the blood stream; and in the investigation of experimental psychology, it has been used to build random fluctuations into deterministic models which model the dynamics of repetitive movements in humans. Finding the quadrature for these integrals using continuous groups or Lie groups has to take families of time indexed random variables, known as Wiener processes, into consideration. Adaptations of Sophus Lie’s work to stochastic ordinary differential equations (SODEs) have been done by Gaeta and Quintero [1], Wafo Soh and Mahomed [2], ¨Unal [3], Meleshko et al. [4], Fredericks and Mahomed [5], and Fredericks and Mahomed [6]. The seminal work [1] was extended in Gaeta [7]; the differential methodology of [2] and [3] were reconciled in [5]; and the integral methodology of [4] was corrected and reconciled in [5] via [6]. Symmetries of SODEs are analysed. This work focuses on maintaining the properties of the Weiner processes after the application of infinitesimal transformations. The determining equations for first-order SODEs are derived in an Itˆo calculus context. These determining equations are non-stochastic. Many methods of deriving Lie point-symmetries for Itˆo SODEs have surfaced. In the Itˆo calculus context both the formal and intuitive understanding of how to construct these symmetries has led to seemingly disparate results. The impact of Lie point-symmetries on the stock market, population growth and weather SODE models, for example, will not be understood until these different results are reconciled as has been attempted here. Extending the symmetry generator to include the infinitesimal transformation of the Wiener process for Itˆo stochastic differential equations (SDEs), has successfully been done in this thesis. The impact of this work leads to an intuitive understanding of the random time change formulae in the context of Lie point symmetries without having to consult much of the intense Itˆo calculus theory needed to derive it formerly (see Øksendal [8, 9]). Symmetries of nth-order SODEs are studied. The determining equations of these SODEs are derived in an Itˆo calculus context. These determining equations are not stochastic in nature. SODEs of this nature are normally used to model nature (e.g. earthquakes) or for testing the safety and reliability of models in construction engineering when looking at the impact of random perturbations. The symmetries of high-order multi-dimensional SODEs are found using form invariance arguments on both the instantaneous drift and diffusion properties of the SODEs. We then apply this to a generalised approximation analysis algorithm. The determining equations of SODEs are derived in an It¨o calculus context. A methodology for constructing conserved quantities with Lie symmetry infinitesimals in an Itˆo integral context is pursued as well. The basis of this construction relies on Lie bracket relations on both the instantaneous drift and diffusion operators

Nonintegrability, Chaos, and Complexity

Two-dimensional driven dissipative flows are generally integrable via a conservation law that is singular at equilibria. Nonintegrable dynamical systems are confined to n*3 dimensions. Even driven-dissipative deterministic dynamical systems that are critical, chaotic or complex have n-1 local time-independent conservation laws that can be used to simplify the geometric picture of the flow over as many consecutive time intervals as one likes. Those conserevation laws generally have either branch cuts, phase singularities, or both. The consequence of the existence of singular conservation laws for experimental data analysis, and also for the search for scale-invariant critical states via uncontrolled approximations in deterministic dynamical systems, is discussed. Finally, the expectation of ubiquity of scaling laws and universality classes in dynamics is contrasted with the possibility that the most interesting dynamics in nature may be nonscaling, nonuniversal, and to some degree computationally complex