Abstract
We investigate time-dependent properties of a single-particle model in which a random walker moves on a triangle and is subjected to nonfocal boundary conditions. This model exhibits spontaneous breaking of a Z 2 symmetry. The reduced size of the configuration space (compared to related many-particle models that also show spontaneous symmetry breaking) allows us to study the spectrum of the time evolution operator. We break the symmetry explicitly and find a stable phase, and a metastable phase which vanishes at a spinodal point. At this point, the spectrum of the time evolution operator has a gapless and universal band of excitations with a dynamical critical exponent z=1. Surprisingly, the imaginary parts of the eigenvalues E j(L) are equally spaced, following the rule \(\mathcal{J}E_j (L) \propto j/L\). Away from the spinodal point, we find two time scales in the spectrum. These results are related to scaling functions for the mean path of the random walker and to first passage times. For the spinodal point, we find universal scaling behavior. A simplified version of the model which can be handled analytically is also presented.
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Arndt, P.F., Heinzel, T. Metastability and Spinodal Points for a Random Walker on a Triangle. Journal of Statistical Physics 92, 837–864 (1998). https://doi.org/10.1023/A:1023036408873
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DOI: https://doi.org/10.1023/A:1023036408873