Ergodic observables in non-ergodic systems: The example of the harmonic chain

Baldovin, Marco; Marino, Raffaele; Vulpiani, Angelo

doi:10.1016/j.physa.2023.129273

Cited by 3 publications

(2 citation statements)

References 29 publications

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“…Even though neural networks are well analyzed from statistical learning, they have become an active field of research in statistical mechanics. Statistical mechanics predicts the properties of a macroscopic system from the laws of its microscopic dynamics [2][3][4]. In this area, a major role is played by phase transitions that regulate what is achievable in principle (information theoretical thresholds) and what is achievable in practice (algorithmic thresholds) [5][6][7][8][9][10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Phase transitions in the mini-batch size for sparse and dense two-layer neural networks

Marino,

Ricci-Tersenghi

2024

Mach. Learn.: Sci. Technol.

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The use of mini-batches of data in training artificial neural networks is nowadays very common. Despite its broad usage, theories explaining quantitatively how large or small the optimal mini-batch size should be are missing. This work presents a systematic attempt at understanding the role of the mini-batch size in training two-layer neural networks.Working in the teacher-student scenario, with a sparse teacher, and focusing on tasks of different complexity, we quantify the effects of changing the mini-batch size $m$.We find that often the generalization performances of the student strongly depend on $m$ and may undergo sharp phase transitions at a critical value $m_c$, such that for $m<m_c$ the training process fails, while for $m>m_c$ the student learns perfectly or generalizes very well the teacher. Phase transitions are induced by collective phenomena firstly discovered in statistical mechanics and later observed in many fields of science. Observing a phase transition by varying the mini-batch size across different architectures raises several questions about the role of this hyperparameter in the neural network learning process.

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Section: Introductionmentioning

confidence: 99%

Phase transitions in the mini-batch size for sparse and dense two-layer neural networks

Marino,

Ricci-Tersenghi

2024

Mach. Learn.: Sci. Technol.

Self Cite

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“…Coupled oscillators, in which particles are connected with springs, have been used to investigate the statistical properties of various interacting many-body systems, such as ergodicity [1][2][3][4], transport phenomena [5][6][7][8], and synchronization [9,10]. The time evolution of a chain of coupled harmonic oscillators is deterministic and can be analytically expressed.…”

Section: Introductionmentioning

confidence: 99%

Statistical analysis of coupled oscillations with a single fixed endpoint and its application to carbon nanomaterials

Inui

2024

Phys. Scr.

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Squared deviations from the equilibrium positions of one-dimensional coupled harmonic oscillators with fixed and free endpoints are calculated, and the time averages are expressed as a function of the initial displacements and velocities. Furthermore, we consider the averages of squared deviations over an ensemble of initial displacements and velocities, which distribute based on a product of the same distribution functions with variances of the initial coordinates σx 2 and velocities σv 2, respectively. We demonstrate that the mean squared deviation linearly increases as the oscillator separates further from the fixed endpoint because of the asymmetrical boundary conditions, and that the increase rate depends only on σv 2 and not on σx 2. This simple statistical property of harmonic oscillation is similarly observed in the oscillations of a graphene sheet and carbon nanotube in molecular dynamics simulations, in which the interacting forces are nonlinear.

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Random Pure Gaussian States and Hawking Radiation

Aurell,

Hackl,

Horodecki

et al. 2024

Phys. Rev. Lett.

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A black hole evaporates by Hawking radiation. Each mode of that radiation is thermal. If the total state is nevertheless to be pure, modes must be entangled. Estimating the minimum size of this entanglement has been an important outstanding issue. We develop a new theory of constrained random symplectic transformations, based on the assumptions that the total state is pure and Gaussian with given marginals. In the random constrained symplectic model we then compute the distribution of mode-mode correlations, from which we bound mode-mode entanglement. Modes of frequency much larger than [kBTH(t)/ℏ] are not populated at time t and drop out of the analysis. Among other relatively thinly populated modes (early-time high-frequency modes and/or late modes of any frequency), we find correlations and hence entanglement to be strongly suppressed. Relatively highly populated modes (early-time low-frequency modes) can, on the other hand, be strongly correlated, but a detailed analysis reveals that they are nevertheless very unlikely to be entangled. Our analysis hence establishes that restoring unitarity after a complete evaporation of a black hole does not require any significant quantum entanglement between any pair of Hawking modes. Our analysis further gives exact general expressions for the distribution of mode-mode correlations in random, pure, Gaussian states with given marginals, which may have applications beyond black hole physics. Published by the American Physical Society 2024

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Ergodic observables in non-ergodic systems: The example of the harmonic chain

Cited by 3 publications

References 29 publications

Phase transitions in the mini-batch size for sparse and dense two-layer neural networks

Phase transitions in the mini-batch size for sparse and dense two-layer neural networks

Statistical analysis of coupled oscillations with a single fixed endpoint and its application to carbon nanomaterials

Random Pure Gaussian States and Hawking Radiation

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