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The same system can exhibit a completely different dynamical behavior when it evolves in equilibrium conditions or when it is driven out-of-equilibrium by, e.g., connecting some of its components to heat baths kept at different temperatures. Here we concentrate on an analytically solvable and experimentally-relevant model of such a system -- the so-called Brownian gyrator -- a two-dimensional nanomachine that performs a systematic, on average, rotation around the origin under non-equilibrium conditions, while no net rotation takes place in equilibrium. On this example, we discuss a question whether it is possible to distinguish between two types of a behavior judging not upon the statistical properties of the trajectories of components, but rather upon their respective spectral densities. The latter are widely used to characterize diverse dynamical systems and are routinely calculated from the data using standard built-in packages. From such a perspective, we inquire whether the power spectral densities possess some "fingerprint" properties specific to the behavior in non-equilibrium. We show that indeed one can conclusively distinguish between equilibrium and non-equilibrium dynamics by analyzing the cross-correlations between the spectral densities of both components in the short frequency limit, or from the spectral densities of both components evaluated at zero frequency. Our analytical predictions, corroborated by experimental and numerical results, open a new direction for the analysis of a non-equilibrium dynamics., 23 pages, 6 figures

The Hebbian unlearning algorithm, i.e. an unsupervised local procedure used to improve the retrieval properties in Hopfield-like neural networks, is numerically compared to a supervised algorithm to train a linear symmetric perceptron. We analyze the stability of the stored memories: basins of attraction obtained by the Hebbian unlearning technique are found to be comparable in size to those obtained in the symmetric perceptron, while the two algorithms are found to converge in the same region of Gardner's space of interactions, having followed similar learning paths. A geometric interpretation of Hebbian unlearning is proposed to explain its optimal performances. Because the Hopfield model is also a prototypical model of disordered magnetic system, it might be possible to translate our results to other models of interest for memory storage in materials., Comment: 11 pages, 13 figures, 43 references

We discuss the statistical properties of a single-trajectory power spectral density $S(\omega,\mathcal{T})$ of an arbitrary real-valued centered Gaussian process $X(t)$, where $\omega$ is the angular frequency and $\mathcal{T}$ the observation time. We derive a double-sided inequality for its noise-to-signal ratio and obtain the full probability density function of $S(\omega,\mathcal{T})$. Our findings imply that the fluctuations of $S(\omega,\mathcal{T})$ exceed its average value $\mu(\omega,\mathcal{T})$. This implies that using $\mu(\omega,\mathcal{T})$ to describe the behavior of these processes can be problematic. We finally evaluate the typical behavior of $S(\omega,\mathcal{T})$ and find that it deviates markedly from the average $\mu(\omega,\mathcal{T})$ in most cases., Comment: 10 pages

We study the recognition capabilities of the Hopfield model with auxiliary hidden layers, which emerge naturally upon a Hubbard-Stratonovich transformation. We show that the recognition capabilities of such a model at zero temperature outperform those of the original Hopfield model, due to a substantial increase of the storage capacity and the lack of a naturally defined basin of attraction. The modified model does not fall abruptly into the regime of complete confusion when memory load exceeds a sharp threshold. This latter circumstance, together with an increase of the storage capacity, renders such a modified Hopfield model a promising candidate for further research, with possible diverse applications.

The correlation length $\xi$, a key quantity in glassy dynamics, can now be precisely measured for spin glasses both in experiments and in simulations. However, known analysis methods lead to discrepancies either for large external fields or close to the glass temperature. We solve this problem by introducing a scaling law that takes into account both the magnetic field and the time-dependent spin-glass correlation length. The scaling law is successfully tested against experimental measurements in a CuMn single crystal and against large-scale simulations on the Janus II dedicated computer., Comment: Revised version, including supplemental material

The question how the extremal values of a stochastic process achieved on different time intervals are correlated to each other has been discussed within the last few years on examples of the running maximum of a Brownian motion, of a Brownian Bridge and of a Slepian process. Here, we focus on the two-time correlations of the running range of Brownian motion - the maximal extent of a Brownian trajectory on a finite time interval. We calculate exactly the covariance function of the running range and analyse its asymptotic behaviour. Our analysis reveals non-trivial correlations between the value of the largest descent (rise) of a BM from the top to a bottom on some time interval, and the value of this property on a larger time interval., 13 pages, 3 figures

The Mpemba effect occurs when a hot system cools faster than an initially colder one, when both are refrigerated in the same thermal reservoir. Using the custom built supercomputer Janus II, we study the Mpemba effect in spin glasses and show that it is a non-equilibrium process, governed by the coherence length \xi of the system. The effect occurs when the bath temperature lies in the glassy phase, but it is not necessary for the thermal protocol to cross the critical temperature. In fact, the Mpemba effect follows from a strong relationship between the internal energy and \xi that turns out to be a sure-tell sign of being in the glassy phase. Thus, the Mpemba effect presents itself as an intriguing new avenue for the experimental study of the coherence length in supercooled liquids and other glass formers., Comment: Version accepted for publication in PNAS. 6 pages, 7 figures

Time-dependent processes are often analysed using the power spectral density (PSD), calculated by taking an appropriate Fourier transform of individual trajectories and finding the associated ensemble-average. Frequently, the available experimental data sets are too small for such ensemble averages, and hence it is of a great conceptual and practical importance to understand to which extent relevant information can be gained from $S(f,T)$, the PSD of a single trajectory. Here we focus on the behavior of this random, realization-dependent variable, parametrized by frequency $f$ and observation-time $T$, for a broad family of anomalous diffusions---fractional Brownian motion (fBm) with Hurst-index $H$---and derive exactly its probability density function. We show that $S(f,T)$ is proportional---up to a random numerical factor whose universal distribution we determine---to the ensemble-averaged PSD. For subdiffusion ($H1/2)$ $S(f,T)\sim BT^{2H-1}/f^2$ with random amplitude $B$. Remarkably, for $H>1/2$ the PSD exhibits the same frequency-dependence as Brownian motion, a deceptive property that may lead to false conclusions when interpreting experimental data. Notably, for $H>1/2$ the PSD is ageing and is dependent on $T$. Our predictions for both sub- and superdiffusion are confirmed by experiments in live cells and in agarose hydrogels, and by extensive simulations., 13 pages, 5 figures, Supplemental Material can be found at https://journals.aps.org/prx/supplemental/10.1103/PhysRevX.9.011019/prx_SM_final.pdf

microRNAs (miRNAs) regulate gene expression at post-transcriptional level by repressing target RNA molecules. Competition to bind miRNAs tends in turn to correlate their targets, establishing effective RNA-RNA interactions that can influence expression levels, buffer fluctuations and promote signal propagation. Such a potential has been characterized mathematically for small motifs both at steady state and \red{away from stationarity}. Experimental evidence, on the other hand, suggests that competing endogenous RNA (ceRNA) crosstalk is rather weak. Extended miRNA-RNA networks could however favour the integration of many crosstalk interactions, leading to significant large-scale effects in spite of the weakness of individual links. To clarify the extent to which crosstalk is sustained by the miRNA interactome, we have studied its emergent systemic features in silico in large-scale miRNA-RNA network reconstructions. We show that, although generically weak, system-level crosstalk patterns (i) are enhanced by transcriptional heterogeneities, (ii) can achieve high-intensity even for RNAs that are not co-regulated, (iii) are robust to variability in transcription rates, and (iv) are significantly non-local, i.e. correlate weakly with miRNA-RNA interaction parameters. Furthermore, RNA levels are generically more stable when crosstalk is strongest. As some of these features appear to be encoded in the network's topology, crosstalk may functionally be favoured by natural selection. These results suggest that, besides their repressive role, miRNAs mediate a weak but resilient and context-independent network of cross-regulatory interactions that interconnect the transcriptome, stabilize expression levels and support system-level responses., 25 pages, includes Supporting Information; to appear in PLoS Comp Biol

Recent high precision experimental results on spin-glass films ask for a detailed understanding of the domain-growth dynamics of two-dimensional spin glasses. To achieve this goal, we numerically simulate the out-equilibrium dynamics of the Ising spin glass for a time that spans close to twelve orders of magnitude (from picoseconds to order of a second), in systems large enough to avoid finite-size effects. We find that the time-growth of the size of the glassy domains is excellently described by a single scaling function. A single time-scale $\tau(T)$ controls the dynamics. $\tau(T)$ diverges upon approaching the $T=0$ critical point. The divergence of $\tau(T\to 0)$ is Arrhenius-like, with a barrier height that depends very mildly on temperature. The growth of this barrier-height is best described by critical dynamics. As a side product we obtain an impressive confirmation of universality of the equilibrium behavior of two-dimensional spin-glasses., Comment: 21 pages, 9 figures. Updated references. Added DOI and Journal ref

Giorgio Parisi belongs to a rare class of universal scientists. For fifty years, his large breadth of interests and creative intuition has led to new seminal ideas in far-away areas of science. In his scientific career he addressed topics as diverse as particle physics, field theory, statistical mechanics, fluid dynamics, condensed matter, numerical simulations and the constructions of scientific computers. The trans-disciplinary approach of statistical physics to complex systems has been one of the long lasting interests of Parisi, who contributed to the theory of neural networks and the immune system, surface growth, optimization and computational complexity and the collective movement of groups of animals. His work has had a big impact in all the fields he touched. A single work that represents a genuine breakthrough, is the replica symmetry breaking approach to disordered systems and its twin formulation, the cavity method. These methods led to a new understanding in the whole field of complex systems and keep generating new knowledge and stimulating multidisciplinary research. In 1979 Parisi found the exact solution of the Sherrington-Kirkpatrick spin glass through an astonishing analysis of the permutation group of n elements in the limit n -> 0. The effort to interpret his replica symmetry breaking (RSB) led him and his collaborators M´ezard and Virasoro to predict the existence of a new surprising phase, with broken ergodicity, hierarchically organised pure states and non-trivial fluctuations of intensive thermodynamic quantities. The mathematisation of these pioneering results required 30 years of efforts and mathematicians are now discovering vast extensions of these concepts. The impact in physics and other areas of science was immediate. The newly gained confidence in the replica method allowed in the 80's to solve Hopfield model of a neural network and a myriad of applications in neural networks and machine learning appeared. In the same years, the RSB based random first order transition theory (RFOT) for structural glasses was proposed. From the 90's the deeply innovative extensions of the cavity and replica method to models on Bethe lattices, led to spectacular advances in computer science. Parisi with M´ezard and Zecchina could exactly find the satisfiability threshold in the celebrated random K-Sat model, and, most interestingly, exploited this exact solution to build a new family of algorithms that improved the state of the art by several orders of magnitude. More recently the RSB approach allowed the exact description of glassy phases of particles in the limit of high dimension. This is the last step in Parisi's long quest for a first principle theory of structural glasses. This solution provides the microscopic foundation of RFOT theory. At the same time it proposes a picture that goes much beyond, predicting the existence of a new fundamental glass-to-glass transition at low temperature where the glass becomes marginally 1751-8121/20/500301+3$33.00 © 2020 IOP Publishing Ltd Printed in the UK 1 J. Phys. A: Math. Theor. 53 (2020) 500301 Preface stable and ungapped excitations appear. Numerical evidence in favour of the presence of this transition in finite dimension has been found by Parisi and collaborators. The most spectacular tested consequence of the theory is a universal description of jammed states of hard spheres, which accounts for the behaviour found in numerical simulations (also by the group of Parisi) in spatial dimension D spanning from 2 to 8. While the infinite dimensional solution will remain a cornerstone in the theory of the glasses the derivation of all its implication is only at the beginning. The present special issue celebrates Giorgio Parisi's 70th birthday, and tries to give an overview of the current state of research in the fields of statistical mechanics and interdisciplinary applications which have been marked by Giorgio Parisi's seminal contributions. The manuscripts accepted for publication in this special issue do indeed cover a broad range of subjects, but they share the same theoretical approach based on the techniques pioneered by Giorgio Parisi

We have introduced a variational method to improve the computation of integrated correlation times in the Parallel Tempering Dynamics, obtaining a better estimate (a lower bound, at least) of the exponential correlation time. Using this determination of the correlation times, we revisited the problem of the characterization of the chaos in temperature in finite dimensional spin glasses by means of the study of correlations between different chaos indicators computed in the static and the correlation times of the Parallel Tempering dynamics. The sample-distribution of the characteristic time for the Parallel Tempering dynamics turns out to be fat-tailed and it obeys finite-size scaling., Comment: New appendix added. Version to appear in JSTAT

We consider the spatial correlation function of the two-dimensional Ising spin glass under out-equilibrium conditions. We pay special attention to the scaling limit reached upon approaching zero temperature. The field-theory of a non-interacting field makes a surprisingly good job at describing the spatial shape of the correlation function of the out-equilibrium Edwards-Anderson Ising model in two dimensions., 20 pages + 5 Figures