Dynamical phase transition for a logarithm observable in chaotic systems
Abstract
We consider the full distribution PN(A) of the dynamical observable A = − ∑i=1N ln xi where x1, …, xN is a sequence generated by a chaotic map f(x) via xn+1 = f(xn). We study the particular cases where f(x) is the doubling, tent or logistic map, and focus on the limit N ≫ 1. Typical fluctuations of A are Gaussian with mean ⟨ A⟩ and variance which are both of order ∼ N. We find that the left tail of the distribution (A < ⟨ A⟩) scales as PN(A) ∼ 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 > ⟨ A⟩) follows an anomalous scaling: PN(A) ∼ e−N1/3 F((A−⟨ A⟩) / N2/3). 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=yc, it exhibits a corner singularity which has the character of a condensation transition. The transition separates between a subcritical phase 0< y<yc in which the fluctuations are Gaussian and arise homogeneously from all of the xi's, and a supercritical phase y>yc in which a small fraction of the xi's give a macroscopic contribution to the fluctuation. Despite this anomalous behavior of the right tail, we find that the cumulants of the distribution PN(A) grow linearly with N at large N, and calculate the associated coefficients.
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BibTeX
@misc{defaveri2026dynamical,
title = {{Dynamical phase transition for a logarithm observable in chaotic systems}},
author = {Lucianno Defaveri and Naftali R. Smith},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21288},
eprint = {2610.21288},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21288},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dynamical phase transition for a logarithm observable in chaotic systems
AU - Lucianno Defaveri
AU - Naftali R. Smith
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21288
UR - https://arxiv.org/abs/2610.21288
AB - We consider the full distribution P_N(A) of the dynamical observable A = − ∑_i=1^N ln x_i where x_1, …, 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 ≫ 1. Typical fluctuations of A are Gaussian with mean ⟨ A⟩ and variance which are both of order ∼ N. We find that the left tail of the distribution (A < ⟨ A⟩) scales as P_N(A) ∼ 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 > ⟨ A⟩) follows an anomalous scaling: P_N(A) ∼ e^−N^1/3 F((A−⟨ A⟩) / 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 the fluctuations are Gaussian and arise homogeneously from all of the x_i's, and a supercritical phase 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.
ER -