In this article we study the noncommutative transport distance introduced by Carlen and Maas and its entropic regularization defined by Becker and Li. We prove a duality formula that can be understood as a quantum version of the dual Benamou-Brenier formulation of the Wasserstein distance in terms of subsolutions of a Hamilton-Jacobi-Bellmann equation.

Using publicly available data from the football database , it is possible to construct a trade network between football clubs. This work regards the network of the flow of transfer fees between European top league clubs from eight countries between 1992 and 2020 to analyse the network of each year's transfer market. With the transfer fees as weights, the market can be represented as a weighted network in addition to the classic binary network approach. This opens up the possibility to study various topological quantities of the network, such as the degree and disparity distributions, the small-world property and different clustering measures. This article shows that these quantities stayed rather constant during the almost three decades of transfer market activity, even despite massive changes in the overall market volume.

We study the BCS energy gap in the high-density limit and derive an asymptotic formula, which strongly depends on the strength of the interaction potential on the Fermi surface. In combination with the recent result by one of us (Math. Phys. Anal. Geom. 25, 3, 2022) on the critical temperature at high densities, we prove the universality of the ratio of the energy gap and the critical temperature.

We study Gaussian random functions on the complex plane whose stochastics are invariant under the Weyl-Heisenberg group (twisted stationarity). The theory is modeled on translation invariant Gaussian entire functions, but allows for non-analytic examples, in which case winding numbers can be either positive or negative. We calculate the first intensity of zero sets of such functions, both when considered as points on the plane, or as charges according to their phase winding. In the latter case, charges are shown to be in a certain average equilibrium independently of the particular covariance structure (universal screening). We investigate the corresponding fluctuations, and show that in many cases they are suppressed at large scales (hyperuniformity). This means that universal screening is empirically observable at large scales. We also derive an asymptotic expression for the charge variance. As a main application, we obtain statistics for the zero sets of the short-time Fourier transform of complex white noise with general windows, and also prove the following uncertainty principle: the expected number of zeros per unit area is minimized, among all window functions, exactly by generalized Gaussians. Further applications include poly-entire functions such as covariant derivatives of Gaussian entire functions.

We study atypical behavior in bootstrap percolation on the Erdős-Rényi random graph. Initially a set is infected. Other vertices are infected once at least of their neighbors become infected. Janson et al. (Ann Appl Probab 22(5):1989-2047, 2012) locates the critical size of , above which it is likely that the infection will spread almost everywhere. Below this threshold, a central limit theorem is proved for the size of the eventually infected set. In this work, we calculate the rate function for the event that a small set eventually infects an unexpected number of vertices, and identify the least-cost trajectory realizing such a large deviation.

A dry frictional interface loaded in shear often displays stick-slip. The amplitude of this cycle depends on the probability that a microscopic event nucleates a rupture and on the rate at which microscopic events are triggered. The latter is determined by the distribution of soft spots, (), which is the density of microscopic regions that yield if the shear load is increased by some amount . In minimal models of a frictional interface-that include disorder, inertia and long-range elasticity-we discovered an 'armouring' mechanism by which the interface is greatly stabilised after a large slip event: () then vanishes at small argument as (de Geus et al., Proc Natl Acad Sci USA 116(48):23977-23983, 2019. 10.1073/pnas.1906551116). The exponent is non-zero only in the presence of inertia (otherwise ). It was found to depend on the statistics of the disorder in the model, a phenomenon that was not explained. Here, we show that a single-particle toy model with inertia and disorder captures the existence of a non-trivial exponent , which we can analytically relate to the statistics of the disorder.

We prove a factorization formula for the point-to-point partition function associated with a model of directed polymers on the space-time lattice . The polymers are subject to a random potential induced by independent identically distributed random variables and we consider the regime of weak disorder, where polymers behave diffusively. We show that when writing the quotient of the point-to-point partition function and the transition probability for the underlying random walk as the product of two point-to-line partition functions plus an error term, then, for large time intervals [0, ], the error term is small uniformly over starting points and endpoints in the sub-ballistic regime , where can be arbitrarily close to 1. This extends a result of Sinai, who proved smallness of the error term in the diffusive regime . We also derive asymptotics for spatial and temporal correlations of the field of limiting partition functions.

Statistical inference can be seen as information processing involving input information and output information that updates belief about some unknown parameters. We consider the Bayesian framework for making inferences about dynamical systems from ergodic observations, where the Bayesian procedure is based on the Gibbs posterior inference, a decision process generalization of standard Bayesian inference (see [7, 37]) where the likelihood is replaced by the exponential of a loss function. In the case of direct observation and almost-additive loss functions, we prove an exponential convergence of the a posteriori measures to a limit measure. Our estimates on the Bayes posterior convergence for direct observation are related and extend those in [47] to a context where loss functions are almost-additive. Our approach makes use of non-additive thermodynamic formalism and large deviation properties [39, 40, 57] instead of joinings.

This paper develops approximate message passing algorithms to optimize multi-species spherical spin glasses. We first show how to efficiently achieve the algorithmic threshold energy identified in our companion work (Huang and Sellke in arXiv preprint, 2023. arXiv:2303.12172), thus confirming that the Lipschitz hardness result proved therein is tight. Next we give two generalized algorithms which produce multiple outputs and show all of them are approximate critical points. Namely, in an -species model we construct approximate critical points when the external field is stronger than a "topological trivialization" phase boundary, and exponentially many such points in the complementary regime. We also compute the local behavior of the Hamiltonian around each. These extensions are relevant for another companion work (Huang and Sellke in arXiv preprint, 2023. arXiv:2308.09677) on topological trivialization of the landscape.

The aim of this paper is to obtain polynomial decay of correlations of a Lorenz-like flow where the hyperbolic saddle at the origin is replaced by a neutral saddle. To do that, we take the construction of the geometrical Lorenz flow and proceed by changing the nature of the saddle fixed point at the origin by a neutral fixed point. This modification is accomplished by changing the linearised vector field in a neighbourhood of the origin for a neutral vector field. This change in the nature of the fixed point will produce polynomial tails for the Dulac times, and combined with methods of Araújo and Melbourne (used to prove exponential mixing for the classical Lorenz flow) this will ultimately lead to polynomial upper bounds of the decay of correlations for the modified flow.

For continuous-time Markov chains we prove that, depending on the notion of effective affinity , the probability of an edge current to ever become negative is either 1 if else . The result generalizes a "noria" formula to multicyclic networks. We give operational insights on the effective affinity and compare several estimators, arguing that stopping problems may be more accurate in assessing the nonequilibrium nature of a system according to a local observer. Finally we elaborate on the similarity with the Boltzmann formula. The results are based on a constructive first-transition approach.

This paper is concerned with a modified entropy method to establish the large-time convergence towards the (unique) steady state, for kinetic Fokker-Planck equations with non-quadratic confinement potentials in whole space. We extend previous approaches by analyzing Lyapunov functionals with non-constant weight matrices in the dissipation functional (a generalized Fisher information). We establish exponential convergence in a weighted -norm with rates that become sharp in the case of quadratic potentials. In the defective case for quadratic potentials, i.e. when the drift matrix has non-trivial Jordan blocks, the weighted -distance between a Fokker-Planck-solution and the steady state has always a sharp decay estimate of the order , with the friction parameter. The presented method also gives new hypoelliptic regularization results for kinetic Fokker-Planck equations (from a weighted -space to a weighted -space).

Optimal control theory deals with finding protocols to steer a system between assigned initial and final states, such that a trajectory-dependent cost function is minimized. The application of optimal control to stochastic systems is an open and challenging research frontier, with a spectrum of applications ranging from stochastic thermodynamics to biophysics and data science. Among these, the design of nanoscale electronic components motivates the study of underdamped dynamics, leading to practical and conceptual difficulties. In this work, we develop analytic techniques to determine protocols steering finite time transitions at a minimum thermodynamic cost for stochastic underdamped dynamics. As cost functions, we consider two paradigmatic thermodynamic indicators. The first is the Kullback-Leibler divergence between the probability measure of the controlled process and that of a reference process. The corresponding optimization problem is the underdamped version of the Schrödinger diffusion problem that has been widely studied in the overdamped regime. The second is the mean entropy production during the transition, corresponding to the second law of modern stochastic thermodynamics. For transitions between Gaussian states, we show that optimal protocols satisfy a Lyapunov equation, a central tool in stability analysis of dynamical systems. For transitions between states described by general Maxwell-Boltzmann distributions, we introduce an infinite-dimensional version of the Poincaré-Lindstedt multiscale perturbation theory around the overdamped limit. This technique fundamentally improves the standard multiscale expansion. Indeed, it enables the explicit computation of momentum cumulants, whose variation in time is a distinctive trait of underdamped dynamics and is directly accessible to experimental observation. Our results allow us to numerically study cost asymmetries in expansion and compression processes and make predictions for inertial corrections to optimal protocols in the Landauer erasure problem at the nanoscale.

Landauer's Principle states that the energy cost of information processing must exceed the product of the temperature, Boltzmann's constant, and the change in Shannon entropy of the information-bearing degrees of freedom. However, this lower bound is achievable only for quasistatic, near-equilibrium computations-that is, only over infinite time. In practice, information processing takes place in finite time, resulting in dissipation and potentially unreliable logical outcomes. For overdamped Langevin dynamics, we show that counterdiabatic potentials can be crafted to guide systems rapidly and accurately along desired computational paths, providing shortcuts that allow for the precise design of finite-time computations. Such shortcuts require additional work, beyond Landauer's bound, that is irretrievably dissipated into the environment. We show that this dissipated work is proportional to the computation rate as well as the square of the information-storing system's length scale. As a paradigmatic example, we design shortcuts to create, erase, and transfer a bit of information metastably stored in a double-well potential. Though dissipated work generally increases with operation fidelity, we show that it is possible to compute with perfect fidelity in finite time with finite work. We also show that the robustness of information storage affects an operation's energetic cost-specifically, the dissipated work scales as the information lifetime of the bistable system. Our analysis exposes a rich and nuanced relationship between work, speed, size of the information-bearing degrees of freedom, storage robustness, and the difference between initial and final informational statistics.

We show that the topological entropy of the billiard map in a Bunimovich stadium is at most .

Based on the recognition of the huge change of the transport properties for diffusion particles in non-static media, we consider a Lévy walk model subjected to an external constant force in non-static media. Since the physical and comoving coordinates of non-static media are related by scale factor, we equivalently transfer the process from physical coordinate into comoving coordinate and derive the master equation governing the probability density function of the position of the particles in comoving coordinate. Utilizing the Hermite orthogonal polynomial expansions, some statistical properties are obtained, including the asymptotic behaviors of the first two moments in both coordinates and kurtosis. For some representative types of non-static media and Lévy walks, the striking and interesting phenomena originating from the interplay between non-static media, external force, and intrinsic stochastic motion are observed. The stationary distribution are also analyzed for some cases through numerical simulations.

In this work, we study the stochastic entropy production in open quantum systems whose time evolution is described by a class of non-unital quantum maps. In particular, as in Phys Rev E 92:032129 (2015), we consider Kraus operators that can be related to a nonequilibrium potential. This class accounts for both thermalization and equilibration to a non-thermal state. Unlike unital quantum maps, non-unitality is responsible for an unbalance of the forward and backward dynamics of the open quantum system under scrutiny. Here, concentrating on observables that commute with the invariant state of the evolution, we show how the non-equilibrium potential enters the statistics of the stochastic entropy production. In particular, we prove a fluctuation relation for the latter and we find a convenient way of expressing its average solely in terms of relative entropies. Then, the theoretical results are applied to the thermalization of a qubit with non-Markovian transient, and the phenomenon of irreversibility mitigation, introduced in Phys Rev Res 2:033250 (2020), is analyzed in this context.

We present a novel mathematical model that seeks to capture the key design feature of generative adversarial networks (GANs). Our model consists of two interacting spin glasses, and we conduct an extensive theoretical analysis of the complexity of the model's critical points using techniques from Random Matrix Theory. The result is insights into the loss surfaces of large GANs that build upon prior insights for simpler networks, but also reveal new structure unique to this setting which explains the greater difficulty of training GANs.

We derive, through subordination techniques, a generalized Feynman-Kac equation in the form of a time fractional Schrödinger equation. We relate such equation to a functional which we name the subordinated local time. We demonstrate through a stochastic treatment how this generalized Feynman-Kac equation describes subdiffusive processes with reactions. In this interpretation, the subordinated local time represents the number of times a specific spatial point is reached, with the amount of time spent there being immaterial. This distinction provides a practical advance due to the potential long waiting time nature of subdiffusive processes. The subordinated local time is used to formulate a probabilistic understanding of subdiffusion with reactions, leading to the well known radiation boundary condition. We demonstrate the equivalence between the generalized Feynman-Kac equation with a reflecting boundary and the fractional diffusion equation with a radiation boundary. We solve the former and find the first-reaction probability density in analytic form in the time domain, in terms of the Wright function. We are also able to find the survival probability and subordinated local time density analytically. These results are validated by stochastic simulations that use the subordinated local time description of subdiffusion in the presence of reactions.

We derive a multi-species BGK model with velocity-dependent collision frequency for a non-reactive, multi-component gas mixture. The model is derived by minimizing a weighted entropy under the constraint that the number of particles of each species, total momentum, and total energy are conserved. We prove that this minimization problem admits a unique solution for very general collision frequencies. Moreover, we prove that the model satisfies an H-Theorem and characterize the form of equilibrium.

In this paper, we prove the existence of a crystallization transition for a family of hard-core particle models on periodic graphs in dimension . We consider only models featuring a single species of particles, which in particular forbids the particles from rotation and reflection, and establish a criterion under which crystallization occurs at sufficiently high densities. The criterion is more general than that in Jauslin and Lebowitz (Commun Math Phys 364:655-682, 2018), as it allows models in which particles do not tile the space in the close-packing configurations, such as discrete hard-disk models. To prove crystallization, we prove that the pressure is analytic in the inverse of the fugacity for large enough complex fugacities, using Pirogov-Sinai theory. One of the main new tools used for this result is the definition of a local density, based on a discrete generalization of Voronoi cells. We illustrate the criterion by proving that it applies to three examples: staircase models and the radius 2.5 hard-disk model on , and a heptacube model on .