Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1010.2657

Harmonic oscillator under Levy noise: Unexpected properties in the phase space

Igor M. Sokolov·Bartlomiej Dybiec·Werner Ebelling
View original source ↗Read source document ↗Export metadata ↓

Abstract

A harmonic oscillator under influence of the noise is a basic model of various physical phenomena. Under Gaussian white noise the position and velocity of the oscillator are independent random variables which are distributed according to the bivariate Gaussian distribution with elliptic level lines. The distribution of phase is homogeneous. None of these properties hold in the general Lévy case. Thus, the level lines of the joint probability density are not elliptic. The coordinate and the velocity of the oscillator are strongly dependent, and this dependence is quantified by introducing the corresponding parameter ("width deficit"). The distribution of the phase is inhomogeneous and highly nontrivial.

Explore related subjects

cond-mat.stat-mechmath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Igor M. Sokolov, Bartlomiej Dybiec, Werner Ebelling. 2011-02-22. Harmonic oscillator under Levy noise: Unexpected properties in the phase space. https://doi.org/10.1103/physreve.83.041118

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

KEEP EXPLORING

Related papers

arXiv2026

Far tails of the biased CTRW model under the short time limit

It has been observed in numerous experiments, simulations, and various theoretical studies that the spreading of particles can be modeled by the continuous-time random walk. We consider two widely used cases, namely Gaussian and discrete displacements, to compute the position distribution and demonstrate the emergence of exponential decay in the far tails when a bias is introduced. We further analyze the temporal rate function and the positional rate function to examine the convergence of the theoretical predictions. For Gaussian displacements, we also discuss the relationship between the position distributions with and without bias in different asymptotic limits.

cond-mat.stat-mech↗
arXiv2026

Stochastic Thermodynamics for Autoregressive Generative Models: A Non-Markovian Perspective

Autoregressive generative models -- including Transformers, recurrent neural networks, classical Kalman filters, state space models, and Mamba -- all generate sequences by sampling each output from a deterministic summary of the past, producing genuinely non-Markovian observed processes. We develop a general theoretical framework based on stochastic thermodynamics for this class of architectures and introduce the entropy production, which can be efficiently estimated from sampled trajectories without exponential sampling cost, despite the non-Markovian nature of the observed dynamics. As a proof-of-concept experiment with a large language model (LLM), we evaluate the entropy production for a pre-trained Transformer-based model, GPT-2. We find that the token-level entropy production is dominated by a syntactic artifact, while the sentence-level entropy production tends to be larger for causally ordered than for non-causal text sets. This observation is supported by a re-evaluation with a substantially larger model, Qwen3-4B-Base. We also demonstrate the framework in the linear Gaussian case, where the model reduces to the Kalman innovation representation and the entropy production admits an analytical expression. We also show that the entropy production decomposes exactly into non-negative per-step contributions in terms of retrospective inference, and each of those terms further splits into information-theoretically meaningful terms: a compression loss and a model mismatch. Our results establish a bridge between stochastic thermodynamics and modern generative models, and provide a starting point for using irreversibility as a quantitative probe of the highly non-Markovian processes generated by models such as LLMs.

cond-mat.stat-mech↗
arXiv2026

Quantum tunneling Mpemba effect

We investigate a quantum tunneling Mpemba effect for a particle in a continuous one-dimensional symmetric double-well potential subject to irreversible boundary loss. The dissipation is described by a complex absorbing potential (CAP) and the corresponding conditional no-jump evolution. Using a biorthogonal spectral representation, we derive the exact non-Hermitian decomposition of the subnormalized density matrix and analyze the corresponding relaxation dynamics. We show that the off-diagonal coefficients exhibit a parity selection rule, leaving same-parity interference terms in general. To isolate the interplay between thermal spectral preparation and mode-dependent decay rates, we introduce a diagonal spectral relaxation measure $S(t,T_i)$. For a quartic double well, numerical calculations reveal finite-time crossings of $S(t,T_i)$ prepared at different initial temperatures, as well as a non-monotonic temperature dependence of selected diagonal spectral weights. These features are explained by the enhanced thermal population of spatially extended, rapidly leaking excited states. In contrast, the trace distance of the normalized conditional state to the asymptotic quasi-stationary state exhibits no crossing, demonstrating that the anomalous ordering is specifically a spectral Mpemba effect rather than a universal acceleration of the complete conditional state. Our continuous-space formulation provides a real-space perspective on anomalous quantum relaxation without invoking universal nodal or topological theorems.

cond-mat.stat-mech↗