The Proceedings of Mechanical Engineering Congress Japan

Definition of stochastic differential for distributed time differences and its relationship to tensor products and renormalization groups

Wako Takanashi

Published Jan 1, 2024• 0 citations

Abstract

Stochastic differentials in stochastic differential equations are defined by Ito integrals, however if stochastic differentials and stochastic variations are defined in a mechanical engineering field more better.In this paper, stochastic differentiation as a difference due to distributed time is defined, and clarify its relationship with tensor products and renormalization groups in engineering.The examine the meaning of the Rough path as tensor product , and are obtained the Uncertainty principle in quantum mechanics due to the Rough path based on the renormalization group due to this rough path. Furthermore, the quantum gravity equation is obtained from the rough pass theory.

DOI: 10.1299/jsmemecj.2024.j012-02

Related papers

Quantum Mechanical versus Stochastic Processes in Path Integration

Marco Patriarca

By using path integrals, the stochastic process associated to the time evolution of the quantum probability density is formally rewritten in terms of a stochastic differential equation, given by Newton's equation of motion with an additional multiplicative stochastic force. However, the term playing the role of the stochastic force is defined by a non-positive-definite probability functional, providing a clear example of the negative (or "extended") probabilities characteristic of quantum mechanics.

arXiv:1801.00510Jan 1, 20180 citations

Stochastic metric space and quantum mechanics

Y. Kurihara

A new idea for the quantization of dynamic systems, as well as space time itself, using a stochastic metric is proposed. The quantum mechanics of a mass point is constructed on a space time manifold using a stochastic metric. A stochastic metric space is, in brief, a metric space whose metric tensor is given stochastically according to some appropriate distribution function. A mathematically consistent model of a space time manifold equipping a stochastic metric is proposed in this work. The quantum theory in the local Minkowski space can be recognized as a classical theory on the stochastic Lorentz-metric-space. A stochastic calculus on the space time manifold is performed using white noise functional analysis. A path-integral quantization is introduced as a stochastic integration of a function of the action integral, and it is shown that path-integrals on the stochastic metric space are mathematically well-defined for large variety of potential functions. The Newton–Nelson equation of motion can also be obtained from the Newtonian equation of motion on the stochastic metric space. It is also shown that the commutation relation required under the canonical quantization is consistent with the stochastic quantization introduced in this work. The quantum effects of general relativity are also analyzed through natural use of the stochastic metrics. Some example of quantum effects on the Universe is discussed.

arXiv:1612.04228Journal of Physics CommunicationsJan 17, 201810 citations

Sensitivity analysis in stochastic mechanics

Marvin McClendon, HERSCHEL A. RABITZ

The formalism of sensitivity analysis for stochastic differential equations is adapted and extended to the framework of stochastic mechanics. Thus a methodology is provided for analyzing the dependence of expectations of functionals of the stochastic mechanical diffusion process on parameters defining the system. This should prove useful in numerical simulations employing the latter theory. We take into account the fact that in stochastic mechanics typically a probability distribution of initial positions (rather than the initial position itself) is prescribed. The expression for a sensitivity density with respect to a drift variation depending only on time is shown to follow from the Ito-Girsanov formula, which is also employed to develop new formulas for sensitivity densities with respect to position-dependent drift variations. As an example to verify these techniques, the sensitivity coefficients of the harmonic-oscillator correlation function with respect to the angular frequency and the diffusion constant are calculated; the results coincide with those obtained by direct differentiation.

Physical Review AMay 1, 19882 citations

Stochastic path integrals can be derived like quantum mechanical path integrals

John J. Vastola, William R. Holmes

Stochastic mechanics---the study of classical stochastic systems governed by things like master equations and Fokker-Planck equations---exhibits striking mathematical parallels to quantum mechanics. In this article, we make those parallels more transparent by presenting a quantum mechanics-like formalism for deriving a path integral description of systems described by stochastic differential equations. Our formalism expediently recovers the usual path integrals (the Martin-Siggia-Rose-Janssen-De Dominicis and Onsager-Machlup forms) and is flexible enough to account for different variable domains (e.g. real line versus compact interval), stochastic interpretations, arbitrary numbers of variables, explicit time-dependence, dimensionful control parameters, and more. We discuss the implications of our formalism for stochastic biology.

arXiv:1909.12990Sep 28, 201910 citations

Measurement in stochastic mechanics

David S. Shucker

Stochastic mechanics is an explanation of nonrelativistic quantum phenomena in terms of stochastic differential equations. In this note a simple example of a measurement is constructed and the behavior of the sample paths of the corresponding stochastic differential equation is examined. The sample demonstrates that stochastic mechanics provides a natural explanation of the ’’reduction of the wave packet.’’

Journal of Mathematical PhysicsMar 1, 19817 citations

Stochastic dynamics of open quantum systems: Derivation of the differential Chapman-Kolmogorov equation

Heinz‐Peter Breuer, Francesco Petruccione

A formulation of quantum statistical ensembles in terms of probability distributions on a projective Hilbert space is developed. The combination of statistically independent systems and the reduction of a system to one of its subsystems are described by means of a tensor product of probability distributions and a reduction formula for the reduced probability distribution. Within this framework the dynamics of open quantum systems is investigated starting form a microscopic system-plus-reservoir model. Employing the Markov approximation of the classical theory of stochastic processes the short time behavior of the conditional transition probability is derived and shown to yield a differential Chapman-Kolmogorov equation. The latter has the form of a Liouville master equation for the reduced probability distribution of the open quantum system. Thus it is shown that the time-dependent wave function of an open quantum system represents a well-defined and unique stochastic process in the space of rays of the underlying Hilbert space. This stochastic process consists of a continuous time evolution generated by a nonlinear Schr\"odinger equation and a discontinuous jump process; the realizations of the processes correspond to those of the Monte Carlo wave function simulation method and to those of the jump-type evolution of the quantum trajectory method. The quantum master equation for the statistical operator is derived as the equation of motion for the two-point correlation function of the stochastic process.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsMay 1, 199523 citations