The paper deals with the asymptotic behavior of a widely used correlation characteristic in large quantum systems. The correlation is quantum entanglement, the characteristic is entanglement entropy, and the system is an ideal gas of lattice fermions. If the one-body Hamiltonian of fermions is an ergodic finite difference operator with an exponentially decaying spectral projection, then the large-block form of the entanglement entropy is the so-called area law. However, the only class of one-body Hamiltonians for which this spectral condition was verified consists of discrete Schrödinger operators with random potential. In this paper, we prove the area law for several classes of Schrödinger operators whose potentials are ergodic but not random. We begin with quasiperiodic and limit-periodic operators and then move to a highly non-trivial case of potentials generated by subshifts of finite type. These arose in the theory of dynamical systems when studying chaotic phenomena. The corresponding asymptotic study requires involved spectral analysis, which therefore constitutes the bulk of the paper. Specifically, we prove uniform localisation of the eigenfunctions for the Maryland model and exponential decay of the eigenfunction correlator for various models. We believe these properties are of significant independent interest.

We consider the stochastic dynamics of a particle on a plane in the presence of two noises and a confining parabolic potential-an analog of the experimentally relevant Brownian gyrator (BG) model. In contrast to the standard BG model, we suppose here that the time evolution of the position components is driven not by Gaussian white noise, but by two statistically independent dichotomous noises. We calculate analytically the position variances and cross-correlations, as well as the mean angular momentum, which permits us to establish the conditions in which a spontaneous rotational motion of the particle around the origin takes place. We also present a numerical analysis of the mean angular velocity. Finally, we analytically calculate some marginal position probability density functions, revealing a remarkably rich behavior that emerges in such a system of two coupled linear stochastic differential equations. We show that, depending on the values of the parameters characterizing the noise, these distributions approach the steady-state forms defined on a finite support, having very unusual shapes, possessing multiple maxima and minima, plateaus, and exhibiting a discontinuous behavior.

The turbulent wake behind a flat-back Ahmed body is investigated using stacked stereoscopic particle image velocimetry. The wake is disturbed by a steady jet from the centre of the base and the effects are quantified for key blowing rates. The unactuated wake exhibits bistable dynamics in the horizontal plane that are completely subdued for the optimal blowing case, yielding a base drag reduction of 9 %. The three-dimensional mean wake is reconstructed and used to evaluate the wake mass fluxes whose equilibrium determines the recirculation length. The results for the unactuated wake show that up to 80 % of replenishment fluid flux entering the recirculation bubble from the free-stream flow is provided through the low-pressure side of the base, where the symmetry-breaking shear layer roll-up occurs near the base. For the optimal blowing configuration, where the wake becomes symmetric, the flux of wake replenishment is severely reduced. This flow configuration results in elongated shear layers on all sides, which terminate the bubble with a roll-up of reduced intensity at a further downstream location. The dominant cause of bubble growth and the accompanying drag reduction is attributed to the momentum of the base blowing, and the new regime is referred to as the 'favourable momentum regime'. Similar trends are observed when the model is at 5 degrees yaw where a reduction of drag and yaw-induced asymmetry are obtained. Proper orthogonal decomposition of the wake reveals the coherent structures related to the bistable flow and the symmetric wake under optimal blowing coefficient.

We consider a quantum system of large size N and its subsystem of size L, assuming that N is much larger than L, which can also be sufficiently large, i.e., 1≪L≲N. A widely accepted mathematical version of this inequality is the asymptotic regime of successive limits: first the macroscopic limit N→∞, then an asymptotic analysis of the entanglement entropy as L→∞. In this paper, we consider another version of the above inequality: the regime of asymptotically proportional L and N, i.e., the simultaneous limits L→∞,N→∞,L/N→λ>0. Specifically, we consider a system of free fermions that is in its ground state, and such that its one-body Hamiltonian is a large random matrix, which is often used to model long-range hopping. By using random matrix theory, we show that in this case, the entanglement entropy obeys the volume law known for systems with short-range hopping but described either by a mixed state or a pure strongly excited state of the Hamiltonian. We also give streamlined proof of Page’s formula for the entanglement entropy of black hole radiation for a wide class of typical ground states, thereby proving the universality and the typicality of the formula.

We study the stochastic dynamics of a two-dimensional particle whose coordinates are described by two coupled one-dimensional random-acceleration processes, that evolve in a confining parabolic potential and are subject to independent Gaussian white noises with different amplitudes (temperatures). We first determine standard characteristics: the mixed moments of positions and velocities, as well as the position-velocity probability density function (PDF) and those of its kinetic and potential energies. Going then beyond these standard characteristics, we consider the emerging rotational motion of the particle around the origin: We show that if the amplitudes of the noises are not equal, the particle experiences a non-zero (on average) torque, such that the angular momentum L and the angular velocity W have non-zero mean values which both are (irregularly) oscillating with time t. We evaluate the PDF-s of L and W and show that the former has exponential tails for any fixed t, and hence, all moments. In the large-time limit this PDF converges to a uniform distribution with a diverging variance. The PDF of W possesses heavy power-law tails such that the mean W is the only existing moment. However, this PDF converges to a well-defined large-time limit manifesting the possibility of stabilizing phenomenon even in frictionless driven systems. Surprisingly, the limit is independent of the amplitudes of noises.

The paper deals with the eigenvalue distribution of the product of two n x n positive definite matrices B tau, tau = +/- 1, rotated with respect to each other by the random orthogonal and Haar distributed matrix. The prob-lem has been considered in several works by using various techniques. We propose a streamlined approach based on the random matrix theory tech-niques and a certain symmetry of the problem. We prove the convergence with probability 1 as n tends to infinity of the Normalized Counting Mea-sure (NCM) of eigenvalues of the product to a non-random limit, derive a functional equation that determines the Stieltjes transform of the limiting NCM of the product in terms of limiting NCMs of the factors B tau, tau = +/- 1, and consider an interesting example.

Другого сiчня 2023 р. виповнилося вiсiмдесят п'ять рокiв вiд дня народження всесвiтньо вiдомого українського математика в галузi диференцiальних рiвнянь

We study the distribution of singular values of product of random matrices pertinent to the analysis of deep neural networks. The matrices resemble the product of the sample covariance matrices, however, an important difference is that the population covariance matrices assumed to be non-random or random but independent of the random data matrix in statistics and random matrix theory are now certain functions of random data matrices (synaptic weight matrices in the deep neural network terminology). The problem has been treated in recent work [25, 13] by using the techniques of free probability theory. Since, however, free probability theory deals with population covariance matrices which are independent of the data matrices, its applicability has to be justified. The justification has been given in [22] for Gaussian data matrices with independent entries, a standard analytical model of free probability, by using a version of the techniques of random matrix theory. In this paper we use another, more streamlined, version of the techniques of random matrix theory to generalize the results of [22] to the case where the entries of the synaptic weight matrices are just independent identically distributed random variables with zero mean and finite fourth moment. This, in particular, extends the property of the so-called macroscopic universality on the considered random matrices.

This paper deals with the distribution of singular values of the input–output Jacobian of deep untrained neural networks in the limit of their infinite width. The Jacobian is the product of random matrices where the independent weight matrices alternate with diagonal matrices whose entries depend on the corresponding column of the nearest neighbor weight matrix. The problem has been considered in the several recent studies of the field for the Gaussian weights and biases and also for the weights that are Haar distributed orthogonal matrices and Gaussian biases. Based on a free probability argument, it was claimed in those papers that, in the limit of infinite width (matrix size), the singular value distribution of the Jacobian coincides with that of the analog of the Jacobian with special random but weight independent diagonal matrices, the case well known in random matrix theory. In this paper, we justify the claim for random Haar distributed weight matrices and Gaussian biases. This, in particular, justifies the validity of the mean field approximation in the infinite width limit for the deep untrained neural networks and extends the macroscopic universality of random matrix theory to this new class of random matrices.

The dynamics of the wake behind a flat-back Ahmed body are modified using a large-scale fluidic oscillator, aiming at drag reduction and the reflectional symmetry breaking (RSB) mode suppression. In the present experiment, the sweeping jet (SWJ) actuator is integrated into the base of the bluff body such that its nozzle width corresponds to about 2/3 of the bluff body base width. The jet is sweeping in the horizontal plane, which coincides with the plane of the wake flow state switching due to the RSB mode. The impact of the SWJ actuator is evaluated for three different locations along the base's height, and for a range of blowing coefficients. The base suction coefficient is recorded from four pressure measurements at the base, while the drag coefficient is obtained from direct force and moment measurements. Particle image velocimetry of the near wake reveals the modifications of the mean flow, which elucidate on the changes in the base suction and drag coefficients. Both drag reduction and RSB mode suppression are achieved, however, not for the same blowing coefficient. The symmetrized wake yields a near Gaussian distribution of the base pressure gradients around zero in both gy and gz directions. This result is corroborated by the joint probability distributions of pitching and yawing moment fluctuations from force balance measurements.

Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter Facebook Reddit LinkedIn Tools Icon Tools Reprints and Permissions Cite Icon Cite Search Site Citation S. A. Gredeskul, L. A. Pastur, Yu. A. Freiman, V. M. Kontorovich, V. G. Peschansky; Mark Yakovlevich Azbel (May 12, 1932–March 31, 2020). Low Temp. Phys. 1 May 2022; 48 (5): 359–362. https://doi.org/10.1063/10.0010198 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAIP Publishing PortfolioLow Temperature Physics Search Advanced Search |Citation Search

The paper describes the decisive role by B. I. Verkin, the founder of the Institute for Low Temperature Physics and Engineering (ILTPE) and the journal Low Temperature Physics, in the return from the near 50-year oblivion the name of L. V. Shubnikov, an outstanding Soviet physicist arrested and executed in 1937. We begin with a brief survey of Shubnikov’s works, several of them are of the Nobel prize level. Then we briefly describe the foundation by B. I. Verkin of the Institute for Low Temperature Physics and Engineering. Finally, we outline the history of writing the book about Shubnikov and the outstanding efforts by B. I. Verkin in the realization of this last, according to him, achievement of his life.

We consider the entanglement evolution of two qubits embedded into disordered multiconnected environment. We model the environment and its interaction with qubits by large random matrices allowing for a possibility to describe environments of meso- and even nanosize. We obtain general formulas for the time dependent reduced density matrix of the qubits corresponding to several cases of the qubit-environment interaction and initial condition. We then work out an analog of the Born-Markov approximation to find the evolution of the widely used entanglement quantifiers: the concurrence, the negativity and the quantum discord. We show that even in this approximation the time evolution of the reduced density matrix can be non-Markovian, thereby describing certain memory effects due to the backaction of the environment on qubits. In particular, we find the vanishing of the entanglement (Entanglement Sudden Death) at finite moments and its revivals (Entanglement Sudden Birth). Our results, partly known and partly new, can be viewed as a manifestation of the universality of certain properties of decoherent qubit evolution which have been found previously in various versions of bosonic macroscopic environment.

We review one-dimensional lattice models and the corresponding results that describe the low-temperature properties of quasi-one-dimensional lattice systems with long-range interaction. A widely known example is narrow-band low-dimensional conductors with long-range interelectron repulsion. The models deal with particles that live on the one-dimensional host lattice (chain), translation invariant or disordered, and interact via the long-range repulsive potential. The results are presented concerning the translation invariant host chain, in particular: the low-temperature thermodynamics incommensurable ground states and related devil-staircase form of various characteristics as functions of pertinent parameters, the self-localization of particles, a new branch of elementary excitations, etc. In the disordered case, where the sites of the host chain fluctuate randomly around the sites of the periodic chain, the low-temperature thermodynamics and the structure of the ground state are discussed in the framework of a certain model, which we call the cluster model and which seems to be a fairly reasonable approximation for low temperatures and small concentration of particles. Using analytical and numerical tools we analyze the thermodynamics and the ground state of the model. The latter proves to be a sequence of random domains and we study in detail their distribution.

The dynamics of an oscillating shear layer when confined is enriched by retarded actions whose physical modeling is not trivial. We present a nonlinear delayed saturation feedback model, which allows us to correctly reproduce the complex shear layer spectra observed experimentally in open cavity flows in the incompressible limit. The model describes the evolution of the amplitude of the shear layer instabilities and considers two hydrodynamic feedback mechanisms directly related to the confinement introduced by the walls. One is associated with reflections of instability waves on the vertical cavity walls and the other to intracavity recirculation flow. These feedback mechanisms provide retarded actions with time lags that are used in the delay differential equation and allow the computation of the model parameters on physical grounds. The frequency components of six experimental cases in different flow regimes are well recovered by the dynamical model. The results show that the model with a single feedback mechanism produces monoperiodic oscillations of the amplitude, while the interplay of two purely hydrodynamic feedback mechanisms allow quasiperiodicity to develop.

The use of rear cavities at the base of a square-back Ahmed body has been experimentally evaluated as a passive control device under cross-wind conditions with yaw angles beta <= 10 degrees, by means of pressure, force and velocity measurements. A comparative study has been performed at a Reynolds number Re = 10(5), considering the reference square-back body (i.e. the body without any passive control device), and the same body implementing both straight and curved cavities as add-on devices. It is shown that the performance of a straight cavity, which is widely acknowledged as a robust drag reduction device for car models, is hindered under moderate cross-wind conditions, and does not constitute an efficient control strategy, especially when compared with a curved cavity. In particular, when the freestream is aligned with the body, the curved cavity provides a stronger attenuation of the fluctuating nature and the bi-stable dynamics of the wake (characteristic of the wake behind a square-back Ahmed body) than the straight one. Besides, the reduced size of the near wake, which is provoked by flow reorientation and the reduced span between the rear edges of the curved cavity, leads to an important base pressure recovery, that translates into relative reductions of the drag of 9.1% in comparison with the reference case (i.e. 2.6%, with respect to the straight cavity). The results are considerably improved under cross-wind conditions, since the increase with the yaw angle of the force is particularly intense for the body with the straight cavity and attenuated for the model with the curved cavity. Thus, the relative reduction of the drag coefficient with respect to the reference body becomes negligible for the straight device at a yaw angle of 10 degrees, while it still represents approximately a 10% for the curved cavity. Furthermore, flow visualizations show that the wake is deflected as the incident flow is increasingly yawed, leading to the formation of a single leeward vortex core that approaches progressively the body, decreasing the base pressure. This phenomenon is minored when a curved cavity is implemented, increasing the low pressure induced at the body base.

Spin valves based on materials in which the spin-flip is suppressed by the spatial separation of charge carriers, while maintaining electric neutrality in the valve volume, are considered. The possibility of using these valves as electric batteries is discussed. It is shown that if the potential difference across the valve is controlled, incommensurability effects such as the "devil's staircase" may be expected, which are associated with the Coulomb interaction and redistribution of electrons occurring while the battery is charged and discharged. The effects of the emergence and vanishing of spontaneous spin polarization of conduction electrons with a change in the Fermi level in the valve are predicted. Such spin valves can also be used in implementing spintronic memory cells, supercapacitors, and similar devices.