Prequantum Classical Statistical Field Theory: Complex Representation, Hamilton-Schrödinger Equation, and Interpretation of Stationary States

We develop a prequantum classical statistical model in that the role of hidden variables is played by classical (vector) fields. We call this model Prequantum Classical Statistical Field Theory (PCSFT). The correspondence between classical and quantum quantities is asymptotic, so we call our approach asymptotic dequantization. We construct the complex representation of PCSFT. In particular, the conventional Schrödinger equation is obtained as the complex representation of the system of Hamilton equations on the infinite-dimensional phase space. In this note we pay the main attention to interpretation of so called pure quantum states (wave functions) in PCSFT, especially stationary states. We show, see Theorem 2, that pure states of QM can be considered as labels for Gaussian measures concentrated on one dimensional complex subspaces of phase space that are invariant with respect to the Schrödinger dynamics. “A quantum system in a stationary state ψ” in PCSFT is nothing else than a Gaussian ensemble of classical fields (fluctuations of the vacuum field of a very small magnitude) which is not changed in the process of Schrödinger's evolution. We interpret in this way the problem of stability of hydrogen atom. One of unexpected consequences of PCSFT is the infinite dimension of physical space on the prequantum scale.     

Nonequilibrium quantum impurities: from entropy production to information theory

Nonequilibrium steady-state currents, unlike their equilibrium counterparts, continuously dissipate energy into their physical surroundings leading to entropy production and time-reversal symmetry breaking. This Letter discusses these issues in the context of quantum impurity models. We use simple thermodynamic arguments to define the rate of entropy production sigma and show that sigma has a simple information-theoretic interpretation in terms of nonequilibrium distribution functions. This allows us to show that the entropy production is strictly positive for any nonequilibrium steady state. We conclude by applying these ideas to the resonance level model and the Kondo model.     

A Reader’s Comment on: “Hysteresis Model of Unconscious-Conscious Interconnection: Exploring Dynamics on m-Adic Trees”

p-Adic Numbers, Ultrametric Analysis and Applications - This comment is aimed to point out that the recent work due to H. Kim, J-Y. Moon, G. A. Mashour and U. Lee ([22]), in which the clinical and...     

On the Problem of Initial Conditions for Inflation

I review the present status of the problem of initial conditions for inflation and describe several ways to solve this problem for many popular inflationary models, including the recent generation of the models with plateau potentials favored by cosmological observations.     

Asymptotic exactness of an a posteriori error estimator based on the equilibrated residual method

In this paper we analyze an a posteriori error estimator based on the equilibrated residual method. We prove that this estimator is asymptotically exact in the energy norm for regular solutions and meshes. Numerical examples are included to illustrate the theoretical results.     

Statistical limit theorems for suspension flows

In dynamical systems theory, a standard method for passing from discrete time to continuous time is to construct the suspension flow under a roof function. In this paper, we give conditions under which statistical laws, such as the central limit theorem and almost sure invariance principle, for the underlying discrete time system are inherited by the suspension flow. As a consequence, we give a simpler proof of the results of Ratner (1973) and recover the results of Denker and Philipp (1984) for Axiom A flows. Morcover, we obtain several new results for nonuniformly and partially hyperbolic flows, including frame flows on negatively curved manifolds satisfying a pinching condition.     

On Performance of Methods with Third- and Fifth-Order Compact Upwind Differencing

The difference schemes for fluid dynamics type of equations based on third- and fifth-order Compact Upwind Differencing (CUD) are considered. To validate their properties following from a linear analysis, calculations were carried out using the inviscid and viscous Burgers' equation as well as the compressible Navier–Stokes equation written in the conservative form for curvilinear coordinates. In the latter case, transonic cascade flow was chosen as a representative example. The performance of the CUD methods was estimated by investigating mesh convergence of the solutions and comparing with the results of second-order schemes. It is demonstrated that the oscillation-free steep gradients solutions obtained without using smoothing techniques can provide considerable increase of accuracy even when exploiting coarse meshes.     

The Role of Information in Managing Interactions from a Multifractal Perspective

In the framework of the multifractal hydrodynamic model, the correlations informational entropy–cross-entropy manages attractive and repulsive interactions through a multifractal specific potential. The classical dynamics associated with them imply Hubble-type effects, Galilei-type effects, and dependences of interaction constants with multifractal degrees at various scale resolutions, while the insertion of the relativistic amendments in the same dynamics imply multifractal transformations of a generalized Lorentz-type, multifractal metrics invariant to these transformations, and an estimation of the dimension of the multifractal Universe. In such a context, some correspondences with standard cosmologies are analyzed. Since the same types of interactions can also be obtained as harmonics mapping between the usual space and the hyperbolic plane, two measures with uniform and non-uniform temporal flows become functional, temporal measures analogous with Milne’s temporal measures in a more general manner. This work furthers the analysis published recently by our group in “Towards Interactions through Information in a Multifractal Paradigm”.