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Asynchronous Quantum Random Walks

Manfred Harringer

Published Jan 8, 2024• 0 citations
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Abstract

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.

DOI
10.21203/rs.3.rs-3372215/v4

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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. 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 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. 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. 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.
The paper dealt with quantum canonical ensembles by random walks, where state transitions are triggered by the connections between labels, not by elements, which are transferred. The balance conditions of such walks lead to emission rates of the labels. The labels with emission rates definitely lower than 1 are like modes. For labels with emission rates very close to 1, the quantum numbers are concentrated around a mean value. As an application I consider the role of the zero label in a quantum gas in equilibrium.