A Stochastic Problem in Physics

Massimo Cencini, Andrea Puglisi B, Davide Vergni +1 more

John W. Negele, Henri Orland

This chapter describes the range of techniques available for the stochastic evaluation of path integrals and how this flexibility may be effectively exploited to incorporate one's understanding of the underlying physics in the method. It provides precise calculations of physical observables which are exact within controllable systematic and stochastic sampling errors. A special problem frequently arising in stochastic solutions of many-body problems is sampling of the inverse of a matrix. The principal intellectual challenge is to exploit the freedom and to utilize physical insight to incorporate as much of the essential physics of the problem as possible into the stochastic algorithm. It is natural to ask whether any of the functional integral representations of the many-body evolution operator using fields are preferable to the particle coordinate path integral for stochastic calculations. One alternative to evaluate the evolution operator for spin systems is to use the Trotter formula for other path integrals.

Gian Fabrizio De Angelis

John S. Baras

In this paper, we consider multi-agent stochastic optimization and control problems, with partial information. The agent can operate in a distributed and asynchronous fashion. We investigate new problems that arise out of the interaction between observations and control actions by the agent. We show that new non-classical and non-commutative probability models are needed in order to properly formulate such problems. The models we develop here are inspired by models developed for dynamical physics problems. We establish a series of fundamental results for the trade-off between information and control patterns in distributed stochastic control, detection and estimation.

Paul C Abbott

Sami Calvo

This paper aims to first explain, somewhat more clearly, the Stochastic-Quantum correspondence put forward in by Barandes in 2023. Specifically, the quantum-mechanical bra-ket notation is used, illuminating some results of previous results. With this, we prove the six axioms of textbook quantum mechanics from a single axiom: every physical system evolves according to a, generally indivisible, stochastic law. Afterwards, we generalise the treatment to continuous bases, which showcases a problem with them, indicating that space (and other physical variables) may be discrete in nature. Some concrete examples are also given, including the generalisation to classical and quantum fields. Then, we treat some practical issues of this new stochastic approach, regarding the solving of problems in physics, which turns out to still be most tractable in the traditional way. Finally, we explain the classical limit, where a system of many particles is found to behave classically according to Newton's second law. Along with that, we present a way of solving the measurement problem, characterising what is an environment and a measuring device and explaining how the wavefunction collapse comes about. Specifically, it is found that what distinguishes an environment is its number of degrees of freedom, while a measuring device is a low-entropy type of environment.