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.

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.

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 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.

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 define infinite tensor product spaces that extend Fock space, and allow for implementing Bogoliubov transformations which violate the Shale or Shale-Stinespring condition. So an implementation on the usual Fock space would not be possible. Both the bosonic and fermionic case are covered. Conditions for implementability in an extended sense are stated and proved. From these, we derive conditions for a quadratic Hamiltonian to be diagonalizable by a Bogoliubov transformation that is implementable in the extended sense. We apply our results to Bogoliubov transformations from quadratic bosonic interactions and BCS models, where the Shale or Shale-Stinespring condition is violated, but an extended implementation nevertheless works.

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 .

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 consider a classical model of non-equilibrium statistical mechanics accounting for non-Markovian effects, which is referred to as the Generalized Langevin Equation in the literature. We derive reduced Markovian descriptions obtained through the neglection of inertial terms and/or heat bath variables. The adopted reduction scheme relies on the framework of the Invariant Manifold method, which allows to retain the slow degrees of freedom from a multiscale dynamical system. Our approach is also rooted on the Fluctuation-Dissipation Theorem, which helps preserve the proper dissipative structure of the reduced dynamics. We highlight the appropriate time scalings introduced within our procedure, and also prove the commutativity of selected reduction paths.

For any multi-fractal dynamical system, a precise estimate of the local dimension is essential to infer variations in its number of degrees of freedom. Following extreme value theory, a local dimension may be estimated from the distributions of pairwise distances within the dataset. For absolutely continuous random variables and in the absence of zeros and singularities, the theoretical value of this local dimension is constant and equals the phase-space dimension. However, due to uneven sampling across the dataset, practical estimations of the local dimension may diverge from this theoretical value, depending on both the phase-space dimension and the position at which the dimension is estimated. To explore such variations of the estimated local dimension of absolutely continuous random variables, approximate analytical expressions are derived and further assessed in numerical experiments. These variations are expressed as a function of 1. the random variables' probability density function, 2. the threshold used to compute the local dimension, and 3. the phase-space dimension. Largest deviations are anticipated when the probability density function has a low absolute value, and a high absolute value of its Laplacian. Numerical simulations of random variables of dimension 1 to 30 allow to assess the validity of the approximate analytical expressions. These effects may become important for systems of moderately-high dimension and in case of limited-size datasets. We suggest to take into account this source of local variation of dimension estimates in future studies of empirical data. Implications for weather regimes are discussed.

We introduce new probabilistic arguments to derive optimal-order central moment bounds in planar directed last-passage percolation. Our technique is based on couplings with the increment-stationary variants of the model, and is presented in the context of i.i.d. exponential weights for both zero and near-stationary boundary conditions. A main technical novelty in our approach is a new proof of the left-tail fluctuation upper bound with exponent 3/2 for the last-passage times.

We prove that Villain interaction applied to lattice gauge theory can be obtained as the limit of both Wilson and Manton interactions on a larger graph which we call the . This is the lattice gauge theory analog of a well-known property for spin () models where Villain type interactions are the limit of spin systems defined on a . Perhaps surprisingly in the setting of lattice gauge theory, our proof also applies to non-Abelian lattice theory such as (3)-lattice gauge theory and its limiting Villain interaction. In the particular case of an Abelian lattice gauge theory, this allows us to extend the validity of Ginibre inequality to the case of the Villain interaction.

We perform a systematic symmetry classification of the Markov generators of classical stochastic processes. Our classification scheme is based on the action of involutive symmetry transformations of a real Markov generator, extending the Bernard-LeClair scheme to the arena of classical stochastic processes and leading to a set of up to fifteen allowed symmetry classes. We construct families of solutions of arbitrary matrix dimensions for five of these classes with a simple physical interpretation of particles hopping on multipartite graphs. In the remaining classes, such a simple construction is prevented by the positivity of entries of the generator particular to classical stochastic processes, which imposes a further requirement beyond the usual symmetry classification constraints. We partially overcome this difficulty by resorting to a stochastic optimization algorithm, finding specific examples of generators of small matrix dimensions in six further classes, leaving the existence of the final four allowed classes an open problem. Our symmetry-based results unveil new possibilities in the dynamics of classical stochastic processes: Kramers degeneracy of eigenvalue pairs, dihedral symmetry of the spectra of Markov generators, and time reversal properties of stochastic trajectories and correlation functions.

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

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.

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.

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.

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.

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).

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.

We consider an abstract non-inertial model of aggregation under the influence of a Gaussian white noise with prescribed space-covariance, and prove a formula for the mean collision rate , per unit of time and volume. Specializing the abstract theory to a non-inertial model obtained by an inertial one, with physical constants, in the limit of infinitesimal relaxation time of the particles, and the white noise obtained as an approximation of a Gaussian noise with correlation time , up to approximations the formula reads where is the particle number per unit of volume and is the square-average of the increment of random velocity field between points at distance , the particle radius. If we choose the Kolmogorov time scale and we assume that is in the dissipative range where , we get Saffman-Turner formula for the collision rate .