Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0708.0034

Analytic Formulation, Exact Solutions, and Generalizations of the Elephant and the Alzheimer Random Walks

V. M. Kenkre
View original source ↗Read source document ↗Export metadata ↓

Abstract

An analytic formulation of memory-possessing random walks introduced recently [Cressoni et al., Phys. Rev. Lett. 98, 070603 (2007) and Schütz and Trimper, Phys. Rev. E 70, 045101 (2004)] for Alzheimer behavior and related phenomena is provided along with exact solutions on the basis of Fokker-Planck equations. The solution of a delay-differential equation derived for the purpose is shown to produce log-periodic oscillations and to coincide rather accurately with previously published computer simulation results. Generalizations along several directions are also constructed on the basis of the formalism.

Explore related subjects

cond-mat.stat-mech

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

V. M. Kenkre. 2007-08-06. Analytic Formulation, Exact Solutions, and Generalizations of the Elephant and the Alzheimer Random Walks. https://arxiv.org/abs/0708.0034

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quantum Many-Body Mpemba Effect through Ruelle-Pollicott Resonances

Relaxation towards equilibrium is often assumed to be slower when a system starts farther from equilibrium, but this intuition fails in the Mpemba effect. Recent advances in controllable quantum platforms have enabled the exploration of its quantum analogue, the quantum Mpemba effect (QME), yet its microscopic origin remains largely unclear. Here we provide a general framework for understanding the QME in closed quantum many-body chaotic systems by reformulating the equilibration process of local subsystems in terms of Ruelle-Pollicott (RP) resonances. We show that suppressing the initial-state overlap with the dominant RP resonant mode accelerates subsystem equilibration and thereby yields the QME. We further uncover that a novel type of strong QME can occur via complete translation-symmetry breaking of initial states. We substantiate our predictions using the prototypical kicked Ising chain and a quantum automaton circuit. These findings cast the QME in closed many-body systems into a unified framework with open-system analogues and provide experimentally accessible signatures on state-of-the-art quantum platforms.

cond-mat.stat-mech↗
arXiv2026

Protected domains and cooperative invasion on weighted networks

Reciprocal contacts can favor cooperation by protecting a lineage until its descendants compete as a group. An individual is chosen to reproduce from the whole population, with a probability increasing with its averaged cooperation payoff. Its offspring replaces a neighbor through contacts weighted equally in both directions. We compare a single cooperator taking over defectors with the reverse invasion, using the same distribution of introduction sites. When payoffs weakly affect reproduction, every network with at least six sites and a path visiting them all admits weights favoring cooperation at any benefit-cost ratio $b/c>1$. This holds for mutants placed uniformly or arising in newborn offspring. Unequal contacts sustain aligned domains around two protected sites, converting individual reproduction into collective competition. Cooperation cannot have an advantage when its benefit does not exceed its cost, at any positive selection strength with positive reproduction rates. Where mutants arise changes which contact patterns can use this mechanism. On a five-site path, the weak-selection threshold has the sharp lower bound seven under uniform placement; newborn introduction never favors cooperation at this order. We classify the contact patterns with at most five sites that can favor cooperation under either rule; newborn introduction requires a four- or five-site cycle. Tracing copies backward to their ancestors separates domain competition from the probability of reaching it. Exact polynomial inequalities prove sharp threshold bounds for paths and the four- and five-site cycles. The same ancestry reconstructs evolution without selection on weighted paths and cycles and fixes the slowest relaxation rate on every connected reciprocal network. Cooperation thus depends on both the collective competitors created by contacts and the route by which a new lineage enters them.

cond-mat.stat-mech↗
arXiv2026

Dynamical phase transition for a logarithm observable in chaotic systems

We consider the full distribution $P_N(A)$ of the dynamical observable $A = - \sum_{i=1}^N \ln x_i$ where $x_1, \dots, x_N$ is a sequence generated by a chaotic map $f(x)$ via $x_{n+1} = f(x_n)$. We study the particular cases where $f(x)$ is the doubling, tent or logistic map, and focus on the limit $N \gg 1$. Typical fluctuations of $A$ are Gaussian with mean $\left\langle A\right\rangle$ and variance which are both of order $\sim N$. We find that the left tail of the distribution ($A < \left\langle A\right\rangle$) scales as $P_{N}(A) \sim e^{-NI(A/N)}$, and calculate the rate function $I(a)$ which depends on the map $f(x)$. Interestingly, we find that the right tail of the distribution ($A > \left\langle A\right\rangle$) follows an anomalous scaling: $P_{N}(A) \sim e^{-N^{1/3} F((A-\left\langle A\right\rangle) / N^{2/3})}$. The large-deviation function $F(y)$ has a universal form for the three maps, differing only in the scaling constants. At a critical value $y=y_c$, it exhibits a corner singularity which has the character of a condensation transition. The transition separates between a subcritical phase $0< y y_c$ in which a small fraction of the $x_i$'s give a macroscopic contribution to the fluctuation. Despite this anomalous behavior of the right tail, we find that the cumulants of the distribution $P_N(A)$ grow linearly with $N$ at large $N$, and calculate the associated coefficients.

cond-mat.stat-mech↗