<p>Traditionally, the evolution of interacting particle systems has been based on an assumption that only the individual components undergo state changes. However, this rigid assumption is not the only possibility. This research explored a class of one-dimensional random processes that evolved in discrete time. During each time step, components in the state zero exhibited the following transitions. First, they could change to one with a probability <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="23_2025_1554_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta _0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>β</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>. Second, they could be replaced by a sequence of <i>k</i> consecutive zeros with a probability <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="23_2025_1554_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta _k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>β</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> (where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="23_2025_1554_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="91" /> </InlineMediaObject> <EquationSource Format="TEX">$k=1,\ldots ,n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>). Moreover, these transitions occurred independent of the events occurring elsewhere in the involved system. Notably, this study revealed an unexpected phenomenon—the occurrence of a first-order phase transition between ergodic and non-ergodic behaviors within this system. Furthermore, in the non-ergodic regime, the existence of an invariant measure distinct from the trivial one was demonstrated.</p>

Ergodic and Non-Ergodic Phenomena in One-Dimensional Random Processes: Exploring Unconventional State Transitions

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发表日期：

2025年2月28日

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A. D. Ramos,
C. S. Sousa,
L. P. Cavalcanti

摘要

Traditionally, the evolution of interacting particle systems has been based on an assumption that only the individual components undergo state changes. However, this rigid assumption is not the only possibility. This research explored a class of one-dimensional random processes that evolved in discrete time. During each time step, components in the state zero exhibited the following transitions. First, they could change to one with a probability $\beta _0$ $β_{0}$ . Second, they could be replaced by a sequence of k consecutive zeros with a probability $\beta _k$ $β_{k}$ (where $k=1,\ldots ,n$ $k = 1, \dots, n$ ). Moreover, these transitions occurred independent of the events occurring elsewhere in the involved system. Notably, this study revealed an unexpected phenomenon—the occurrence of a first-order phase transition between ergodic and non-ergodic behaviors within this system. Furthermore, in the non-ergodic regime, the existence of an invariant measure distinct from the trivial one was demonstrated.