Table of contents

We revise the classical problem of characterizing first exit times of a harmonically trapped particle whose motion is described by a one- or multidimensional Ornstein–Uhlenbeck process. We start by recalling the main derivation steps of a propagator using Langevin and Fokker–Planck equations. The mean exit time, the moment-generating function and the survival probability are then expressed through confluent hypergeometric functions and thoroughly analyzed. We also present a rapidly converging series representation of confluent hypergeometric functions that is particularly well suited for numerical computation of eigenvalues and eigenfunctions of the governing Fokker–Planck operator. We discuss several applications of first exit times, such as the detection of time intervals during which motor proteins exert a constant force onto a tracer in optical tweezers single-particle tracking experiments; adhesion bond dissociation under mechanical stress; characterization of active periods of trend-following and mean-reverting strategies in algorithmic trading on stock markets; relation to the distribution of first crossing times of a moving boundary by Brownian motion. Some extensions are described, including diffusion under quadratic double-well potential and anomalous diffusion.

In this paper, we introduce and investigate the statistical mechanics of hierarchical neural networks. First, we approach these systems à la Mattis, by thinking of the Dyson model as a single-pattern hierarchical neural network. We also discuss the stability of different retrievable states as predicted by the related self-consistencies obtained both from a mean-field bound and from a bound that bypasses the mean-field limitation. The latter is worked out by properly reabsorbing the magnetization fluctuations related to higher levels of the hierarchy into effective fields for the lower levels. Remarkably, mixing Amitʼs ansatz technique for selecting candidate-retrievable states with the interpolation procedure for solving for the free energy of these states, we prove that, due to gauge symmetry, the Dyson model accomplishes both serial and parallel processing. We extend this scenario to multiple stored patterns by implementing the Hebb prescription for learning within the couplings. This results in Hopfield-like networks constrained on a hierarchical topology, for which, by restricting to the low-storage regime where the number of patterns grows at its most logarithmical with the amount of neurons, we prove the existence of the thermodynamic limit for the free energy, and we give an explicit expression of its mean-field bound and of its related improved bound. We studied the resulting self-consistencies for the Mattis magnetizations, which act as order parameters, are studied and the stability of solutions is analyzed to get a picture of the overall retrieval capabilities of the system according to both mean-field and non-mean-field scenarios. Our main finding is that embedding the Hebbian rule on a hierarchical topology allows the network to accomplish both serial and parallel processing. By tuning the level of fast noise affecting it or triggering the decay of the interactions with the distance among neurons, the system may switch from sequential retrieval to multitasking features, and vice versa. However, since these multitasking capabilities are basically due to the vanishing 'dialogue' between spins at long distance, this effective penury of links strongly penalizes the networkʼs capacity, with results bounded by low storage.

Quantum criticality of strongly attractive Fermi gas with $SU(3)$ symmetry in one dimension is studied via the thermodynamic Bethe ansatz (TBA) equations. The phase transitions driven by the chemical potential $\mu$, effective magnetic field H₁, H₂ (chemical potential biases) are analyzed at the quantum criticality. The phase diagram and critical fields are analytically determined by the TBA equations in the zero temperature limit. High accurate equations of state, scaling functions are also obtained analytically for the strong interacting gases. The dynamic exponent $z=2$ and correlation length exponent $\nu =1/2$ read off the universal scaling form. It turns out that the quantum criticality of the three-component gases involves a sudden change of density of states of one cluster state, two or three cluster states. In general, this method can be adapted to deal with the quantum criticality of multicomponent Fermi gases with $SU(N)$ symmetry.

We look at the equilibrium of a Brownian particle in an inhomogeneous space following the alternative approach proposed in (Bhattacharyay 2013 Physica A 392 4265). We consider a coordinate dependent damping resulting in a stochastic problem with multiplicative noise. We convert this stochastic problem with multiplicative noise to additive noise form using our alternative approach. The resulting equilibrium velocity distribution is locally Maxwellian with a zero mean. The position distribution is of a modified Boltzmann form where that of a free particle is a function of damping constant.

We develop a superoperator coupled cluster method for nonequilibrium open many-body quantum systems described by the Lindblad master equation. The method is universal and applicable to systems of interacting fermions, bosons or their mixtures. We present a general theory and consider its application to the problem of quantum transport through the system with electron–phonon correlations. The results are assessed against the perturbation theory and nonequilibrium configuration interaction theory calculations.

We consider an Ising ferromagnet endowed with zero-temperature spin-flip dynamics and examine the evolution of the Ising quadrant, namely the spin configuration when the minority phase initially occupies one quadrant while the majority phase occupies the three remaining quadrants. The two phases are then always separated by a single interface, which generically recedes into the minority phase in a self-similar diffusive manner. The area of the invaded region grows (on average) linearly with time and exhibits non-trivial fluctuations. We map the interface separating the two phases onto the one-dimensional symmetric simple exclusion process and utilize this isomorphism to compute basic cumulants of the area. First, we determine the variance via an exact microscopic analysis (the Bethe ansatz). Then we turn to a continuum treatment by recasting the underlying exclusion process into the framework of the macroscopic fluctuation theory. This provides a systematic way of analyzing the statistics of the invaded area and allows us to determine the asymptotic behaviors of the first four cumulants of the area.

We consider the logarithmic negativity of a finite interval embedded in an infinite one dimensional system at finite temperature. We focus on conformal invariant systems and we show that the naive approach based on the calculation of a two-point function of twist fields in a cylindrical geometry yields a wrong result. The correct result is obtained through a four-point function of twist fields in which two auxiliary fields are inserted far away from the interval, and they are sent to infinity only after having taken the replica limit. In this way, we find a universal scaling form for the finite temperature negativity which depends on the full operator content of the theory and not only on the central charge. In the limit of low and high temperatures, the expansion of this universal form can be obtained by means of the operator product expansion. We check our results against exact numerical computations for the critical harmonic chain.

We study an asymmetric exclusion model with one dynamic roadblock particle. The roadblock particle is allowed to move diffusively as well as by long-range jumps mimicking an unbinding/rebinding process. Using Monte Carlo simulations and analytical arguments, the two types of roadblock moves are considered both separately and in combination. Several interesting dynamic phenomena are observed. The long-range jumps of the roadblock lead to a current that depends on the number of particles in the system rather than on the particle density, and thus scales linearly with the system size (up to a critical size). This behavior can be explained by a collective motion of all particles following the roadblock. The diffusive roadblock movements on the other hand lead to a ratcheting motion with the active (driven) particles pushing the roadblock forward.

Thermal contact is the archetype of non-equilibrium processes driven by constant non-equilibrium constraints when the latter are enforced by reservoirs exchanging conserved microscopic quantities. At a mesoscopic scale only the energies of the macroscopic bodies are accessible together with the configurations of the contact system. We consider a class of models where the contact system, as well as macroscopic bodies, have a finite number of possible configurations. The global system, with only discrete degrees of freedom, has no microscopic Hamiltonian dynamics, but it is shown that, if the microscopic dynamics is assumed to be deterministic and ergodic and to conserve energy according to some specific pattern, and if the mesoscopic evolution of the global system is approximated by a Markov process as closely as possible, then the mesoscopic transition rates obey three constraints. In the limit where macroscopic bodies can be considered as reservoirs at thermodynamic equilibrium (but with different intensive parameters), the mesoscopic transition rates turn into transition rates for the contact system and the third constraint becomes local detailed balance; the latter is generically expressed in terms of the microscopic exchange entropy variation, namely the opposite of the variation of the thermodynamic entropy of the reservoir involved in a given microscopic jump of the contact system configuration. For a finite-time evolution after contact has been switched on, we derive a fluctuation relation for the joint probability of the heat amounts received from the various reservoirs. The generalization to systems exchanging energy, volume and matter with several reservoirs, with a possible conservative external force acting on the contact system, is given explicitly.

We discuss statistical properties of vehicular headways measured on signalized crossroads. On the basis of mathematical approaches, we formulate theoretical and empirically inspired criteria for the acceptability of theoretical headway distributions. Sequentially, the multifarious families of statistical distributions (commonly used to fit real-road headway statistics) are confronted with these criteria, and with original empirical time clearances gauged among neighboring vehicles leaving signal-controlled crossroads after a green signal appears. Using three different numerical schemes, we demonstrate that an arrangement of vehicles on an intersection is a consequence of the general stochastic nature of queueing systems, rather than a consequence of traffic rules, driver estimation processes, or decision-making procedures.

Nonlinear wave interaction processes for a quasilinear hyperbolic homogeneous system of first-order partial differential equations multicomponent chromatography are investigated. The wave analysis is worked out by extending to the present multicomponent case the leading ideas of a well-established method of approach that was developed for solving initial value-wave problems in terms of exact solutions to 2 × 2 hyperbolic homogeneous systems. These interaction processes may model different situations concerning the separation of a mixture into its chemical components. Several numerical plots are also given in order to illustrate the behavior of the exact wave solutions arising from the analysis that is accomplished.

Twisted symmetries, widely studied in the last decade, have proved to be as effective as standard ones in the analysis and reduction of nonlinear equations. We explain this effectiveness in terms of a Lie–Frobenius reduction; this requires focus not just on the prolonged (symmetry) vector fields, but on the distributions spanned by these and on systems of vector fields in involution in the Frobenius sense, not necessarily spanning a Lie algebra.

We investigate the invariants of the 25-dimensional real representation of the group ${\bf SO}(3)\;\wr \;{{{\bf Z}}_{2}}$ given by the left and right actions of ${\bf SO}(3)$ on 5 × 5 matrices together with matrix transposition; the action on column vectors is the irreducible five-dimensional representation of ${\bf SO}(3)$. The 25-dimensional representation arises naturally in the study of nematic liquid crystals, where the second-rank orientational order parameters of a molecule are represented by a symmetric 3 × 3 traceless symmetric matrix, and where a rigid rotation in ${\bf R}$³ induces a linear transformation of this space of matrices. The entropy contribution to a free energy density function in this context turns out to have ${\bf SO}(3)\;\wr \;{{{\bf Z}}_{2}}$ symmetry. Although it is unrealistic to expect to describe the complete algebraic structure of the ring of invariants, we are able to calculate as a rational function the Molien series that gives the number of linearly independent invariants at each homogeneous degree. The form of the function indicates a basis of 19 primary invariants and suggests there are $N=1{\mkern 1mu} 453{\mkern 1mu} 926{\mkern 1mu} 048$ linearly independent secondary invariants; we prove that their number is an integer multiple of $N/4$. The algebraic structure of invariants up to degree 4 is investigated in detail.

By introducing some special bi-orthogonal polynomials, we derive the so-called discrete hungry quotient-difference (dhQD) algorithm and a system related to the QD-type discrete hungry Lotka–Volterra (QD-type dhLV) system, together with their Lax pairs. These two known equations can be regarded as extensions of the QD algorithm. When this idea is applied to a higher analogue of the discrete-time Toda (HADT) equation and the quotient–quotient-difference (QQD) scheme proposed by Spicer, Nijhoff and van der Kamp, two extended systems are constructed. We call these systems the hungry forms of the higher analogue discrete-time Toda (hHADT) equation and the quotient-quotient-difference (hQQD) scheme, respectively. In addition, the corresponding Lax pairs are provided.

Simulation of quantum dynamics is a grand challenge of computational physics. In this work we investigate methods for reducing the demands of such simulation by identifying reduced-order models for dynamics generated by parameterized quantum Hamiltonians. In particular, we first formulate an algebraic condition that certifies the existence of invariant subspaces for a model defined by a parameterized Hamiltonian and an initial state. Following this we develop and analyze two methods to explicitly construct a reduced-order model, if one exists. In addition to general results characterizing invariant subspaces of arbitrary finite dimensional Hamiltonians, by exploiting properties of the generalized Pauli group we develop practical tools to speed up simulation of dynamics generated by certain spin Hamiltonians. To illustrate the methods developed we apply them to several paradigmatic spin models.

Vasilievʼs higher-spin (HS) theories in various dimensions are uniformly represented as a simple system of equations. These equations and their gauge invariances are based on two superalgebras and have a transparent algebraic meaning. For a given HS theory these algebras can be inferred from the vacuum HS symmetries. The proposed system of equations admits a concise AKSZ formulation. We also discuss novel HS systems including partially-massless and massive fields in AdS, as well as conformal and massless off-shell fields.