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Fractional Brownian Motion (FBM) is a stochastic process with long-time correlations which has been used to model anomalous diffusion in numerous biological systems. Recently, it has been used to study the distribution of serotonergic fibers in the brain [1,2]. To better represent the biological process we are trying to simulate, we introduce the concept of branching FBM (bFBM). In this stochastic process, individual particles perform FBM but may randomly split into two. Here, we study bFBM in one space dimension in the subdiffusive and superdiffusive regimes, both in free space and on finite intervals with reflecting boundaries. We examine three possible types of behavior of the correlations (memory) at a branching event: both particles keep the memory of the previous steps, only one particle keeps the memory, and no particles keep the memory. We calculate the mean-square particle displacement, the corresponding probability distribution, and displacement correlation function. We find that the qualitative features of the bFBM process strongly depend on the type of branching event. We also discuss implications of our results for the distribution of serotonergic fibers, and we discuss possible future refinements of the model, including interactions between different fibers. [1] T. Vojta, S. Halladay, S. Skinner, S. Janusonis, T. Guggenberger, R. Metzler, Phys. Rev. E 102, 032108 (2020). [2] S. Janusonis, N. Detering, R. Metzler, T. Vojta, Front. Comput. Neurosci. 14, 56 (2020). This work was supported in part by the NSF under grant no. IIS-2112862 and by a Cottrell SEED award from Research Corporation.
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Interplay between superconductivity and non-Fermi liquid behavior at a quantum critical point in a metal. III. The γ model and its phase diagram across γ=1
In this paper we continue our analysis of the interplay between the pairing and the non-Fermi liquid behavior in a metal for a set of quantum-critical models with an effective dynamical electron-electron interaction V(Ωm)∝1/|Ωm|γ (the γ model). We analyze both the original model and its extension, in which we introduce an extra parameter N to account for nonequal interactions in the particle-hole and particle-particle channel. In two previous papers [A. Abanov and A. V. Chubukov, Phys. Rev. B 102, 024524 (2020) and Y. Wu et al. Phys. Rev. B 102, 024525 (2020)] we considered the case 0<γ<1 and argued that (i) at T=0, there exists an infinite discrete set of topologically different gap functions Δn(ωm), all with the same spatial symmetry, and (ii) each Δn evolves with temperature and terminates at a particular Tp,n. In this paper we analyze how the system behavior changes between γ<1 and γ>1, both at T=0 and a finite T. The limit γ→1 is singular due to infrared divergence of ∫dωmV(Ωm), and the system behavior is highly sensitive to how this limit is taken. We show that for N=1, the divergencies in the gap equation cancel out, and Δn(ωm) gradually evolve through γ=1 both at T=0 and a finite T. For N≠1, divergent terms do not cancel, and a qualitatively new behavior emerges for γ>1. Namely, the form of Δn(ωm) changes qualitatively, and the spectrum of condensation energies Ec,n becomes continuous at T=0. We introduce different extension of the model, which is free from singularities for γ>1.
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- Award ID(s):
- 1834856
- PAR ID:
- 10228600
- Date Published:
- Journal Name:
- Physical review
- Volume:
- 102
- ISSN:
- 1550-235X
- Page Range / eLocation ID:
- 094516 1-22
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript
- Journal Article:
- https://doi.org/10.1103/PhysRevB.102.094516
