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Deriving Neural Scaling Laws from the statistics of natural language
Authors:
Francesco Cagnetta,
Allan Raventós,
Surya Ganguli,
Matthieu Wyart
Abstract:
Despite the fact that experimental neural scaling laws have substantially guided empirical progress in large-scale machine learning, no existing theory can quantitatively predict the exponents of these important laws for any modern LLM trained on any natural language dataset. We provide the first such theory in the case of data-limited scaling laws. We isolate two key statistical properties of lan…
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Despite the fact that experimental neural scaling laws have substantially guided empirical progress in large-scale machine learning, no existing theory can quantitatively predict the exponents of these important laws for any modern LLM trained on any natural language dataset. We provide the first such theory in the case of data-limited scaling laws. We isolate two key statistical properties of language that alone can predict neural scaling exponents: (i) the decay of pairwise token correlations with time separation between token pairs, and (ii) the decay of the next-token conditional entropy with the length of the conditioning context. We further derive a simple formula in terms of these statistics that predicts data-limited neural scaling exponents from first principles without any free parameters or synthetic data models. Our theory exhibits a remarkable match with experimentally measured neural scaling laws obtained from training GPT-2 and LLaMA style models from scratch on two qualitatively different benchmarks, TinyStories and WikiText.
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Submitted 2 July, 2026; v1 submitted 7 February, 2026;
originally announced February 2026.
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Deep networks learn to parse uniform-depth context-free languages from local statistics
Authors:
Jack T. Parley,
Francesco Cagnetta,
Matthieu Wyart
Abstract:
Understanding how the structure of language can be learned from sentences alone is a central question in both cognitive science and machine learning. Studies of the internal representations of Large Language Models (LLMs) support their ability to parse text when predicting the next word, while representing semantic notions independently of surface form. Yet, which data statistics make these feats…
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Understanding how the structure of language can be learned from sentences alone is a central question in both cognitive science and machine learning. Studies of the internal representations of Large Language Models (LLMs) support their ability to parse text when predicting the next word, while representing semantic notions independently of surface form. Yet, which data statistics make these feats possible, and how much data is required, remain largely unknown. Probabilistic context-free grammars (PCFGs) provide a tractable testbed for studying these questions. However, prior work has focused either on the post-hoc characterization of the parsing-like algorithms used by trained networks; or on the learnability of PCFGs with fixed syntax, where parsing is unnecessary. Here, we (i) introduce a tunable class of PCFGs in which both the degree of ambiguity and the correlation structure across scales can be controlled; (ii) provide a learning mechanism -- an inference algorithm inspired by the structure of deep convolutional networks -- that links learnability and sample complexity to specific language statistics; and (iii) validate our predictions empirically across deep convolutional and transformer-based architectures. Overall, we propose a unifying framework where correlations at different scales lift local ambiguities, enabling the emergence of hierarchical representations of the data.
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Submitted 1 June, 2026; v1 submitted 31 January, 2026;
originally announced February 2026.
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Scaling Laws and Representation Learning in Simple Hierarchical Languages: Transformers vs. Convolutional Architectures
Authors:
Francesco Cagnetta,
Alessandro Favero,
Antonio Sclocchi,
Matthieu Wyart
Abstract:
How do neural language models acquire a language's structure when trained for next-token prediction? We address this question by deriving theoretical scaling laws for neural network performance on synthetic datasets generated by the Random Hierarchy Model (RHM) -- an ensemble of probabilistic context-free grammars designed to capture the hierarchical structure of natural language while remaining a…
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How do neural language models acquire a language's structure when trained for next-token prediction? We address this question by deriving theoretical scaling laws for neural network performance on synthetic datasets generated by the Random Hierarchy Model (RHM) -- an ensemble of probabilistic context-free grammars designed to capture the hierarchical structure of natural language while remaining analytically tractable. Previously, we developed a theory of representation learning based on data correlations that explains how deep learning models capture the hierarchical structure of the data sequentially, one layer at a time. Here, we extend our theoretical framework to account for architectural differences. In particular, we predict and empirically validate that convolutional networks, whose structure aligns with that of the generative process through locality and weight sharing, enjoy a faster scaling of performance compared to transformer models, which rely on global self-attention mechanisms. This finding clarifies the architectural biases underlying neural scaling laws and highlights how representation learning is shaped by the interaction between model architecture and the statistical properties of data.
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Submitted 11 May, 2025;
originally announced May 2025.
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Learning curves theory for hierarchically compositional data with power-law distributed features
Authors:
Francesco Cagnetta,
Hyunmo Kang,
Matthieu Wyart
Abstract:
Recent theories suggest that Neural Scaling Laws arise whenever the task is linearly decomposed into power-law distributed units. Alternatively, scaling laws also emerge when data exhibit a hierarchically compositional structure, as is thought to occur in language and images. To unify these views, we consider classification and next-token prediction tasks based on probabilistic context-free gramma…
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Recent theories suggest that Neural Scaling Laws arise whenever the task is linearly decomposed into power-law distributed units. Alternatively, scaling laws also emerge when data exhibit a hierarchically compositional structure, as is thought to occur in language and images. To unify these views, we consider classification and next-token prediction tasks based on probabilistic context-free grammars -- probabilistic models that generate data via a hierarchy of production rules. For classification, we show that having power-law distributed production rules results in a power-law learning curve with an exponent depending on the rules' distribution and a large multiplicative constant that depends on the hierarchical structure. By contrast, for next-token prediction, the distribution of production rules controls the local details of the learning curve, but not the exponent describing the large-scale behaviour.
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Submitted 11 May, 2025;
originally announced May 2025.
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How Compositional Generalization and Creativity Improve as Diffusion Models are Trained
Authors:
Alessandro Favero,
Antonio Sclocchi,
Francesco Cagnetta,
Pascal Frossard,
Matthieu Wyart
Abstract:
Natural data is often organized as a hierarchical composition of features. How many samples do generative models need in order to learn the composition rules, so as to produce a combinatorially large number of novel data? What signal in the data is exploited to learn those rules? We investigate these questions in the context of diffusion models both theoretically and empirically. Theoretically, we…
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Natural data is often organized as a hierarchical composition of features. How many samples do generative models need in order to learn the composition rules, so as to produce a combinatorially large number of novel data? What signal in the data is exploited to learn those rules? We investigate these questions in the context of diffusion models both theoretically and empirically. Theoretically, we consider a simple probabilistic context-free grammar - a tree-like graphical model used to represent the hierarchical and compositional structure of data such as language and images. We demonstrate that diffusion models learn the grammar's composition rules with the sample complexity required for clustering features with statistically similar context, a process similar to the word2vec algorithm. However, this clustering emerges hierarchically: higher-level features associated with longer contexts require more data to be identified. This mechanism leads to a sample complexity that scales polynomially with the said context size. As a result, diffusion models trained on an intermediate dataset size generate data coherent up to a certain scale, but lacking global coherence. We test these predictions across different domains and find remarkable agreement: both generated texts and images achieve progressively larger coherence lengths as the training time or dataset size grows. We discuss connections between the hierarchical clustering mechanism we introduce here and the renormalization group in physics.
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Submitted 4 June, 2025; v1 submitted 17 February, 2025;
originally announced February 2025.
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Towards a theory of how the structure of language is acquired by deep neural networks
Authors:
Francesco Cagnetta,
Matthieu Wyart
Abstract:
How much data is required to learn the structure of a language via next-token prediction? We study this question for synthetic datasets generated via a Probabilistic Context-Free Grammar (PCFG) -- a tree-like generative model that captures many of the hierarchical structures found in natural languages. We determine token-token correlations analytically in our model and show that they can be used t…
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How much data is required to learn the structure of a language via next-token prediction? We study this question for synthetic datasets generated via a Probabilistic Context-Free Grammar (PCFG) -- a tree-like generative model that captures many of the hierarchical structures found in natural languages. We determine token-token correlations analytically in our model and show that they can be used to build a representation of the grammar's hidden variables, the longer the range the deeper the variable. In addition, a finite training set limits the resolution of correlations to an effective range, whose size grows with that of the training set. As a result, a Language Model trained with increasingly many examples can build a deeper representation of the grammar's structure, thus reaching good performance despite the high dimensionality of the problem. We conjecture that the relationship between training set size and effective range of correlations holds beyond our synthetic datasets. In particular, our conjecture predicts how the scaling law for the test loss behaviour with training set size depends on the length of the context window, which we confirm empirically in Shakespeare's plays and Wikipedia articles.
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Submitted 29 October, 2024; v1 submitted 28 May, 2024;
originally announced June 2024.
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Kernels, Data & Physics
Authors:
Francesco Cagnetta,
Deborah Oliveira,
Mahalakshmi Sabanayagam,
Nikolaos Tsilivis,
Julia Kempe
Abstract:
Lecture notes from the course given by Professor Julia Kempe at the summer school "Statistical physics of Machine Learning" in Les Houches. The notes discuss the so-called NTK approach to problems in machine learning, which consists of gaining an understanding of generally unsolvable problems by finding a tractable kernel formulation. The notes are mainly focused on practical applications such as…
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Lecture notes from the course given by Professor Julia Kempe at the summer school "Statistical physics of Machine Learning" in Les Houches. The notes discuss the so-called NTK approach to problems in machine learning, which consists of gaining an understanding of generally unsolvable problems by finding a tractable kernel formulation. The notes are mainly focused on practical applications such as data distillation and adversarial robustness, examples of inductive bias are also discussed.
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Submitted 5 July, 2023;
originally announced July 2023.
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How Deep Neural Networks Learn Compositional Data: The Random Hierarchy Model
Authors:
Francesco Cagnetta,
Leonardo Petrini,
Umberto M. Tomasini,
Alessandro Favero,
Matthieu Wyart
Abstract:
Deep learning algorithms demonstrate a surprising ability to learn high-dimensional tasks from limited examples. This is commonly attributed to the depth of neural networks, enabling them to build a hierarchy of abstract, low-dimensional data representations. However, how many training examples are required to learn such representations remains unknown. To quantitatively study this question, we in…
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Deep learning algorithms demonstrate a surprising ability to learn high-dimensional tasks from limited examples. This is commonly attributed to the depth of neural networks, enabling them to build a hierarchy of abstract, low-dimensional data representations. However, how many training examples are required to learn such representations remains unknown. To quantitatively study this question, we introduce the Random Hierarchy Model: a family of synthetic tasks inspired by the hierarchical structure of language and images. The model is a classification task where each class corresponds to a group of high-level features, chosen among several equivalent groups associated with the same class. In turn, each feature corresponds to a group of sub-features chosen among several equivalent ones and so on, following a hierarchy of composition rules. We find that deep networks learn the task by developing internal representations invariant to exchanging equivalent groups. Moreover, the number of data required corresponds to the point where correlations between low-level features and classes become detectable. Overall, our results indicate how deep networks overcome the curse of dimensionality by building invariant representations, and provide an estimate of the number of data required to learn a hierarchical task.
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Submitted 3 July, 2024; v1 submitted 5 July, 2023;
originally announced July 2023.
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How deep convolutional neural networks lose spatial information with training
Authors:
Umberto M. Tomasini,
Leonardo Petrini,
Francesco Cagnetta,
Matthieu Wyart
Abstract:
A central question of machine learning is how deep nets manage to learn tasks in high dimensions. An appealing hypothesis is that they achieve this feat by building a representation of the data where information irrelevant to the task is lost. For image datasets, this view is supported by the observation that after (and not before) training, the neural representation becomes less and less sensitiv…
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A central question of machine learning is how deep nets manage to learn tasks in high dimensions. An appealing hypothesis is that they achieve this feat by building a representation of the data where information irrelevant to the task is lost. For image datasets, this view is supported by the observation that after (and not before) training, the neural representation becomes less and less sensitive to diffeomorphisms acting on images as the signal propagates through the net. This loss of sensitivity correlates with performance, and surprisingly correlates with a gain of sensitivity to white noise acquired during training. These facts are unexplained, and as we demonstrate still hold when white noise is added to the images of the training set. Here, we (i) show empirically for various architectures that stability to image diffeomorphisms is achieved by both spatial and channel pooling, (ii) introduce a model scale-detection task which reproduces our empirical observations on spatial pooling and (iii) compute analitically how the sensitivity to diffeomorphisms and noise scales with depth due to spatial pooling. The scalings are found to depend on the presence of strides in the net architecture. We find that the increased sensitivity to noise is due to the perturbing noise piling up during pooling, after being rectified by ReLU units.
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Submitted 23 November, 2022; v1 submitted 4 October, 2022;
originally announced October 2022.
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What Can Be Learnt With Wide Convolutional Neural Networks?
Authors:
Francesco Cagnetta,
Alessandro Favero,
Matthieu Wyart
Abstract:
Understanding how convolutional neural networks (CNNs) can efficiently learn high-dimensional functions remains a fundamental challenge. A popular belief is that these models harness the local and hierarchical structure of natural data such as images. Yet, we lack a quantitative understanding of how such structure affects performance, e.g., the rate of decay of the generalisation error with the nu…
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Understanding how convolutional neural networks (CNNs) can efficiently learn high-dimensional functions remains a fundamental challenge. A popular belief is that these models harness the local and hierarchical structure of natural data such as images. Yet, we lack a quantitative understanding of how such structure affects performance, e.g., the rate of decay of the generalisation error with the number of training samples. In this paper, we study infinitely-wide deep CNNs in the kernel regime. First, we show that the spectrum of the corresponding kernel inherits the hierarchical structure of the network, and we characterise its asymptotics. Then, we use this result together with generalisation bounds to prove that deep CNNs adapt to the spatial scale of the target function. In particular, we find that if the target function depends on low-dimensional subsets of adjacent input variables, then the decay of the error is controlled by the effective dimensionality of these subsets. Conversely, if the target function depends on the full set of input variables, then the error decay is controlled by the input dimension. We conclude by computing the generalisation error of a deep CNN trained on the output of another deep CNN with randomly-initialised parameters. Interestingly, we find that, despite their hierarchical structure, the functions generated by infinitely-wide deep CNNs are too rich to be efficiently learnable in high dimension.
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Submitted 31 May, 2023; v1 submitted 1 August, 2022;
originally announced August 2022.
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Learning sparse features can lead to overfitting in neural networks
Authors:
Leonardo Petrini,
Francesco Cagnetta,
Eric Vanden-Eijnden,
Matthieu Wyart
Abstract:
It is widely believed that the success of deep networks lies in their ability to learn a meaningful representation of the features of the data. Yet, understanding when and how this feature learning improves performance remains a challenge: for example, it is beneficial for modern architectures trained to classify images, whereas it is detrimental for fully-connected networks trained for the same t…
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It is widely believed that the success of deep networks lies in their ability to learn a meaningful representation of the features of the data. Yet, understanding when and how this feature learning improves performance remains a challenge: for example, it is beneficial for modern architectures trained to classify images, whereas it is detrimental for fully-connected networks trained for the same task on the same data. Here we propose an explanation for this puzzle, by showing that feature learning can perform worse than lazy training (via random feature kernel or the NTK) as the former can lead to a sparser neural representation. Although sparsity is known to be essential for learning anisotropic data, it is detrimental when the target function is constant or smooth along certain directions of input space. We illustrate this phenomenon in two settings: (i) regression of Gaussian random functions on the d-dimensional unit sphere and (ii) classification of benchmark datasets of images. For (i), we compute the scaling of the generalization error with number of training points, and show that methods that do not learn features generalize better, even when the dimension of the input space is large. For (ii), we show empirically that learning features can indeed lead to sparse and thereby less smooth representations of the image predictors. This fact is plausibly responsible for deteriorating the performance, which is known to be correlated with smoothness along diffeomorphisms.
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Submitted 12 October, 2022; v1 submitted 24 June, 2022;
originally announced June 2022.
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Locality defeats the curse of dimensionality in convolutional teacher-student scenarios
Authors:
Alessandro Favero,
Francesco Cagnetta,
Matthieu Wyart
Abstract:
Convolutional neural networks perform a local and translationally-invariant treatment of the data: quantifying which of these two aspects is central to their success remains a challenge. We study this problem within a teacher-student framework for kernel regression, using `convolutional' kernels inspired by the neural tangent kernel of simple convolutional architectures of given filter size. Using…
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Convolutional neural networks perform a local and translationally-invariant treatment of the data: quantifying which of these two aspects is central to their success remains a challenge. We study this problem within a teacher-student framework for kernel regression, using `convolutional' kernels inspired by the neural tangent kernel of simple convolutional architectures of given filter size. Using heuristic methods from physics, we find in the ridgeless case that locality is key in determining the learning curve exponent $β$ (that relates the test error $ε_t\sim P^{-β}$ to the size of the training set $P$), whereas translational invariance is not. In particular, if the filter size of the teacher $t$ is smaller than that of the student $s$, $β$ is a function of $s$ only and does not depend on the input dimension. We confirm our predictions on $β$ empirically. We conclude by proving, using a natural universality assumption, that performing kernel regression with a ridge that decreases with the size of the training set leads to similar learning curve exponents to those we obtain in the ridgeless case.
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Submitted 12 November, 2021; v1 submitted 16 June, 2021;
originally announced June 2021.
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Universal properties of active membranes
Authors:
Francesco Cagnetta,
Viktor Skultety,
Martin R. Evans,
Davide Marenduzzo
Abstract:
We put forward a general field theory for membranes with embedded activators and analyse their critical properties using renormalization group techniques. Depending on the membrane-activator coupling, we find a crossover between acoustic and diffusive scaling regimes, with mean-field dynamical critical exponents z = 1 and 2 respectively. We argue that the acoustic scaling, which is exact in all sp…
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We put forward a general field theory for membranes with embedded activators and analyse their critical properties using renormalization group techniques. Depending on the membrane-activator coupling, we find a crossover between acoustic and diffusive scaling regimes, with mean-field dynamical critical exponents z = 1 and 2 respectively. We argue that the acoustic scaling, which is exact in all spatial dimensions, is a suitable candidate for the universal description of the spatiotemporal patterns observed at the leading edge of motile cells. Furthermore, one-loop corrections to the diffusive mean-field exponents reveal universal behaviour distinct from the Kardar-Parisi-Zhang scaling of passive interfaces and signs of strong-coupling behaviour.
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Submitted 16 November, 2021; v1 submitted 14 April, 2021;
originally announced April 2021.
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A renormalization group study of the dynamics of active membranes: universality classes and scaling laws
Authors:
Francesco Cagnetta,
Viktor Skultety,
Martin R. Evans,
Davide Marenduzzo
Abstract:
Motivated by experimental observations of patterning at the leading edge of motile eukaryotic cells, we introduce a general model for the dynamics of nearly-flat fluid membranes driven from within by an ensemble of activators. We include, in particular, a kinematic coupling between activator density and membrane slope which generically arises whenever the membrane has a non-vanishing normal speed.…
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Motivated by experimental observations of patterning at the leading edge of motile eukaryotic cells, we introduce a general model for the dynamics of nearly-flat fluid membranes driven from within by an ensemble of activators. We include, in particular, a kinematic coupling between activator density and membrane slope which generically arises whenever the membrane has a non-vanishing normal speed. We unveil the phase diagram of the model by means of a perturbative field-theoretical renormalization group analysis. Due to the aforementioned kinematic coupling the natural dynamical scaling is acoustic, that is the dynamical critical exponent is 1. However, as soon as the the normal velocity of the membrane is tuned to zero, the system crosses over to diffusive dynamic scaling in mean field. Distinct critical points can be reached depending on how the limit of vanishing velocity is realised: in each of them corrections to scaling due to nonlinear coupling terms must be taken into accounts. The detailed analysis of these critical points reveals novel scaling regimes wich can be accessed with perturbative methods, together with signs of strong coupling behaviour, which establishes a promising ground for further non-perturbative calculations. Our results unify several previous studies on the dynamics of active membrane, while also identifying nontrivial scaling regimes which cannot be captured by passive theories of fluctuating interfaces and are relevant for the physics of living membranes.
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Submitted 16 November, 2021; v1 submitted 14 April, 2021;
originally announced April 2021.
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Work Fluctuations in the Active Ornstein- Uhlenbeck Particle model
Authors:
Massimiliano Semeraro,
Antonio Suma,
Isabella Petrelli,
Francesco Cagnetta,
Giuseppe Gonnella
Abstract:
We study the large deviations of the power injected by the active force for an Active Ornstein-Uhlenbeck Particle (AOUP), free or in a confining potential. For the free-particle case, we compute the rate function analytically in d-dimensions from a saddle-point expansion, and numerically in two dimensions by it a) direct sampling of the active work in numerical solutions of the AOUP equations and…
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We study the large deviations of the power injected by the active force for an Active Ornstein-Uhlenbeck Particle (AOUP), free or in a confining potential. For the free-particle case, we compute the rate function analytically in d-dimensions from a saddle-point expansion, and numerically in two dimensions by it a) direct sampling of the active work in numerical solutions of the AOUP equations and b) Legendre-Fenchel transform of the scaled cumulant generating function obtained via a cloning algorithm. The rate function presents asymptotically linear branches on both sides and it is independent of the system's dimensionality, apart from a multiplicative factor. For the confining potential case, we focus on two-dimensional systems and obtain the rate function numerically using both methods a) and b). We find a different scenario for harmonic and anharmonic potentials: in the former case, the phenomenology of fluctuations is analogous to that of a free particle, but the rate function might be non-analytic; in the latter case the rate functions are analytic, but fluctuations are realised by entirely different means, which rely strongly on the particle-potential interaction. Finally, we check the validity of a fluctuation relation for the active work distribution. In the free-particle case, the relation is satisfied with a slope proportional to the bath temperature. The same slope is found for the harmonic potential, regardless of activity, and for an anharmonic potential with low activity. In the anharmonic case with high activity, instead, we find a different slope which is equal to an effective temperature obtained from the fluctuation-dissipation theorem.
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Submitted 30 September, 2021; v1 submitted 21 March, 2021;
originally announced March 2021.
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Work fluctuations of self-propelled particles in the phase separated state
Authors:
P. Chiarantoni,
F. Cagnetta,
F. Corberi,
G. Gonnella,
A. Suma
Abstract:
We study the large deviations of the distribution P(W_τ) of the work associated with the propulsion of individual active brownian particles in a time interval τ, in the region of the phase diagram where macroscopic phase separation takes place. P(W_τ) is characterised by two peaks, associated to particles in the gaseous and in the clusterised phases, and two separate non-convex branches. According…
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We study the large deviations of the distribution P(W_τ) of the work associated with the propulsion of individual active brownian particles in a time interval τ, in the region of the phase diagram where macroscopic phase separation takes place. P(W_τ) is characterised by two peaks, associated to particles in the gaseous and in the clusterised phases, and two separate non-convex branches. Accordingly, the generating function of W_τcumulants displays a double singularity. We discuss the origin of such non-convex branches in terms of the peculiar dynamics of the system phases, and the relation between the observation time τand the typical persistence times of the particles in the two phases.
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Submitted 18 May, 2020; v1 submitted 23 January, 2020;
originally announced January 2020.
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Kinetic roughening in active interfaces
Authors:
Francesco Cagnetta,
Martin R. Evans,
Davide Marenduzzo
Abstract:
The essential features of many interfaces driven out of equilibrium are described by the same equation---the Kardar-Parisi-Zhang (KPZ) equation. How do living interfaces, such as the cell membrane, fit into this picture? In an endeavour to answer such a question, we proposed in [F. Cagnetta, M. R. Evans, D. Marenduzzo, PRL 120, 258001 (2018)] an idealised model for the membrane of a moving cell. H…
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The essential features of many interfaces driven out of equilibrium are described by the same equation---the Kardar-Parisi-Zhang (KPZ) equation. How do living interfaces, such as the cell membrane, fit into this picture? In an endeavour to answer such a question, we proposed in [F. Cagnetta, M. R. Evans, D. Marenduzzo, PRL 120, 258001 (2018)] an idealised model for the membrane of a moving cell. Here we discuss how the addition of simple ingredients inspired by the dynamics of the membrane of moving cells affects common kinetic roughening theories such as the KPZ and Edwards-Wilkinson equations.
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Submitted 16 December, 2019;
originally announced December 2019.
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A nonequilibrium strategy for fast target search on the genome
Authors:
F. Cagnetta,
D. Michieletto,
D. Marenduzzo
Abstract:
Vital biological processes such as genome repair require fast and efficient binding of selected proteins to specific target sites on DNA. Here we propose an active target search mechanism based on "chromophoresis", the dynamics of DNA-binding proteins up or down gradients in the density of epigenetic marks, or colours (biochemical tags on the genome). We focus on a set of proteins that deposit mar…
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Vital biological processes such as genome repair require fast and efficient binding of selected proteins to specific target sites on DNA. Here we propose an active target search mechanism based on "chromophoresis", the dynamics of DNA-binding proteins up or down gradients in the density of epigenetic marks, or colours (biochemical tags on the genome). We focus on a set of proteins that deposit marks from which they are repelled---a case which is only encountered away from thermodynamic equilibrium. For suitable ranges of kinetic parameter values, chromophoretic proteins can perform unidirectional motion and are optimally redistributed along the genome. Importantly, they can also locally unravel a region of the genome which is collapsed due to self-interactions and "dive" deep into its core, for a striking enhancement of the efficiency of target search on such an inaccessible substrate. We discuss the potential relevance of chromophoresis for the location of DNA lesions.
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Submitted 23 April, 2020; v1 submitted 19 August, 2019;
originally announced August 2019.
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Efficiency of one-dimensional active transport conditioned on motility
Authors:
Francesco Cagnetta,
Emil Mallmin
Abstract:
By conditioning a stochastic process on the value of an observable, one obtains a new stochastic process with different properties. We apply this idea in the context of active matter, and condition interacting self-propelled particles on their individual motility. Using the effective process formalism from dynamical large deviations theory, we derive the interactions that actuate the imposed mobil…
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By conditioning a stochastic process on the value of an observable, one obtains a new stochastic process with different properties. We apply this idea in the context of active matter, and condition interacting self-propelled particles on their individual motility. Using the effective process formalism from dynamical large deviations theory, we derive the interactions that actuate the imposed mobility against jamming interactions in two toy models---the totally asymmetric exclusion process and run-and-tumble particles, \emil{in the case of two or three particles}. We provide a framework which takes into account the energy-consumption required for self-propulsion, and address the question of how energy-efficient the emergent interactions are. Upon conditioning, run-and-tumble particles develop an alignment interaction and achieve a higher gain in efficiency than TASEP particles. A point of diminishing returns in efficiency is reached beyond a certain level of conditioning. With recourse to a general formula for the change in energy efficiency upon conditioning, we conclude that the most significant gains occur when there are large fluctuations in mobility to exploit. From a detailed comparison of the emergent effective interaction in a two- versus a three-body scenario, we discover evidence of a screening effect which suggests that conditioning can produce topological rather than metric interactions.
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Submitted 12 February, 2020; v1 submitted 2 July, 2019;
originally announced July 2019.
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Inviscid limit of the active interface equations
Authors:
Francesco Cagnetta,
Martin R. Evans
Abstract:
We present a detailed solution of the active interface equations in the inviscid limit. The active interface equations were previously introduced as a toy model of membrane-protein systems: they describe a stochastic interface where growth is stimulated by inclusions which themselves move on the interface. In the inviscid limit, the equations reduce to a pair of coupled conservation laws. After di…
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We present a detailed solution of the active interface equations in the inviscid limit. The active interface equations were previously introduced as a toy model of membrane-protein systems: they describe a stochastic interface where growth is stimulated by inclusions which themselves move on the interface. In the inviscid limit, the equations reduce to a pair of coupled conservation laws. After discussing how the inviscid limit is obtained, we turn to the corresponding Riemann problem: the solution of the set of conservation laws with discontinuous initial condition. In particular, by considering two physically meaningful initial conditions, a giant trough and a giant peak in the interface, we elucidate the generation of shock waves and rarefaction fans in the system. Then, by combining several Riemann problems, we construct an oscillating solution of the active interface with periodic boundaries conditions. The existence of this oscillating state reflects the reciprocal coupling between the two conserved quantities in our system.
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Submitted 28 August, 2019; v1 submitted 18 April, 2019;
originally announced April 2019.
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Statistical mechanics of a single active slider on a fluctuating interface
Authors:
Francesco Cagnetta,
Martin R. Evans,
Davide Marenduzzo
Abstract:
We study the statistical mechanics of a single active slider on a fluctuating interface, by means of numerical simulations and theoretical arguments. The slider, which moves by definition towards the interface minima, is active as it also stimulates growth of the interface. Even though such a particle has no counterpart in thermodynamic systems, active sliders may provide a simple model for ATP-de…
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We study the statistical mechanics of a single active slider on a fluctuating interface, by means of numerical simulations and theoretical arguments. The slider, which moves by definition towards the interface minima, is active as it also stimulates growth of the interface. Even though such a particle has no counterpart in thermodynamic systems, active sliders may provide a simple model for ATP-dependent membrane proteins that activate cytoskeletal growth. We find a wide range of dynamical regimes according to the ratio between the timescales associated with the slider motion and the interface relaxation. If the interface dynamics is slow, the slider behaves like a random walker in a random envinronment which, furthermore, is able to escape environmental troughs by making them grow. This results in different dynamic exponens to the interface and the particle: the former behaves as an Edward-Wilkinson surface with dynamic exponent 2 whereas the latter has dynamic exponent 3/2. When the interface is fast, we get sustained ballistic motion with the particle surfing a membrane wave created by itself. However, if the interface relaxes immediately (i.e., it is infinitely fast), particle motion becomes symmetric and goes back to diffusive. Due to such a rich phenomenology, we propose the active slider as a toy model of fundamental interest in the field of active membranes and, generally, whenever the system constituent can alter the environment by spending energy.
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Submitted 17 April, 2019; v1 submitted 30 November, 2018;
originally announced November 2018.
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Active interface growth and pattern formation in membrane-protein systems
Authors:
F. Cagnetta,
M. R. Evans,
D. Marenduzzo
Abstract:
Inspired by recent experimental observation of patterning at the membrane of a living cell, we propose a generic model for the dynamics of a fluctuating interface driven by particle-like inclusions which stimulate its growth. We find that the coupling between interfacial and inclusions dynam- ics yields microphase separation and the self-organisation of travelling waves. These patterns are strikin…
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Inspired by recent experimental observation of patterning at the membrane of a living cell, we propose a generic model for the dynamics of a fluctuating interface driven by particle-like inclusions which stimulate its growth. We find that the coupling between interfacial and inclusions dynam- ics yields microphase separation and the self-organisation of travelling waves. These patterns are strikingly similar to those detected in the aforementioned experiments on actin-protein systems. Our results further show that the active growth kinetics does not fall into the Kardar-Parisi-Zhang universality class for growing interfaces, displaying instead a novel superposition of equilibrium-like scaling and sustained oscillations.
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Submitted 11 May, 2018; v1 submitted 15 December, 2017;
originally announced December 2017.
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Large fluctuations and dynamic phase transition in a system of self-propelled particles
Authors:
Francesco Cagnetta,
Federico Corberi,
Giuseppe Gonnella,
Antonio Suma
Abstract:
We study the statistics, in stationary conditions, of the work $W_τ$ done by the active force in different systems of self-propelled particles in a time $τ$. We show the existence of a critical value $W_τ^†$ such that fluctuations with $W_τ>W_τ^†$ correspond to configurations where interaction between particles plays a minor role whereas those with $W_τ< W_τ^†$ represent states with single particl…
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We study the statistics, in stationary conditions, of the work $W_τ$ done by the active force in different systems of self-propelled particles in a time $τ$. We show the existence of a critical value $W_τ^†$ such that fluctuations with $W_τ>W_τ^†$ correspond to configurations where interaction between particles plays a minor role whereas those with $W_τ< W_τ^†$ represent states with single particles dragged by clusters. This two-fold behavior is fully mirrored by the probability distribution $P(W_τ)$ of the work, which does not obey the large-deviation principle for $W_τ<W_τ^†$. This pattern of behavior can be interpreted as due to a phase transition occurring at the level of fluctuating quantities and an order parameter is correspondingly identified.
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Submitted 15 October, 2017;
originally announced October 2017.
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Strong anomalous diffusion of the phase of a chaotic pendulum
Authors:
Francesco Cagnetta,
Giuseppe Gonnella,
Alessandro Mossa,
Stefano Ruffo
Abstract:
In this letter we consider the phase diffusion of a harmonically driven undamped pendulum and show that it is anomalous in the strong sense. The role played by the fractal properties of the phase space is highlighted, providing an illustration of the link between deterministic chaos and anomalous transport. Finally, we build a stochastic model which reproduces most properties of the original Hamil…
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In this letter we consider the phase diffusion of a harmonically driven undamped pendulum and show that it is anomalous in the strong sense. The role played by the fractal properties of the phase space is highlighted, providing an illustration of the link between deterministic chaos and anomalous transport. Finally, we build a stochastic model which reproduces most properties of the original Hamiltonian system by alternating ballistic flights and random diffusion.
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Submitted 21 April, 2015;
originally announced April 2015.