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Random Walks Through Fock States of a Graph

Manfred Harringer

Published Apr 4, 2024• 0 citations
Publisher

Abstract

Abstract I present process models for motion on a graph Γ(V,E) along its edges, motivated by examples. I build random sequences of edges and I derive sequences of Fock states. Therefore, I can distinguish two kinds of process steps, acceptance and rejection, comparable to stimulated and spontaneous emission. There is always at least one vertex, which represents the vacuum of the system. Rejections and vacuum are not visible in a quantum grand canonical ensemble.

DOI
10.21203/rs.3.rs-3372215/v6

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Abstract I build random walks on finite graphs Γ(V,E) with vertices V and directed edges E by sequences of randomly selected edges. Edge selection may also be organized by a curve on a torus, observing circuits in both directions of the course at a single circle. The edges trigger state changes of the vertices for particles, bosons or fermions, due to the state spaces at the vertices. As a statistical result, I obtain a quantum canonical ensemble together with a vacuum. There are two kinds of process steps, acceptions and rejections, corresponding to stimulated and spontaneous emissions. Such processes are asynchronous versions of time evolution in quantum mechanics.
Abstract I build random walks on finite graphs Γ(V,E) with vertices V and directed edges E by sequences of randomly selected edges. I obtain quantum canonical ensembles with vacuum. Single steps of these walks are acceptions and rejections, corresponding to stimulated and spontaneous emission. Split into small steps, such random walks are asynchronous versions of time evolution in quantum mechanics. As a result, for a fixed weight function of edges I obtain a family of quantum canonical ensembles, where the members are parametrized by the number of elements N, with a common balance condition, fulfilled by the first eigenvector of the weight matrix. The family connects a Markov chain with a quantum grand canonical ensemble by a passage from N = 1 to large N-values.
Abstract I build random walks on finite graphs Γ(V,E) with vertices V and directed edges E by sequences of randomly selected edges. I obtain quantum canonical ensembles with vacuum. Single steps of these walks are acceptions and rejections, corresponding to stimulated and spontaneous emission. Split into small steps, such random walks are asynchronous versions of time evolution in quantum mechanics. As a result, for a fixed weight function of edges I obtain a family of quantum canonical ensembles, where the members are parametrized by the number of elements N, with a common balance condition, fulfilled by the first eigenvector of the weight matrix. The family connects a Markov chain with a quantum grand canonical ensemble by a passage from N = 1 to large N-values.
Abstract We build probabilistic process models of motions on a finite graph. For particle statistics, at first we select a particle. Then we build a path of the particle along consecutive edges. Particles are distinguished. Single particles perform Markov chains. For Bose statistics, at first we select an edge as request for a path of a particle along the edge. Particles are not distinguished. The edges are randomly selected by probability weights, or they are triggered by circuits of courses in small steps on circles. We present examples for these process models. We obtain some results, which follow from the balance equation, a refinement of the corresponding balance equation for Markov chains. Quantum mechanics inhibits these process models, because there the annihilation of the zero state is zero.