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Abstract: We continue to work on the study of possibilities to describe flows of rarefied gas by higher (compared to the Navier–Stokes level) equations of hydrodynamics. The main question is how to make the next step with respect to Knudsen number and to derive the well-posed equations at the Burnett level. Our method allows to construct well-posed equations at this level. However, they are not unique, since they include some indefinite parameters. In our previous paper on this topic we have studied the group properties of these equations. In the present work we show that invariant solutions (travelling waves and bounded in the half-space stationary solutions) of our equations clearly indicate some disadvantages of the Navier–Stokes equations. These solutions allow to choose the optimal set of parameters in Generalized Burnett Equations (GBEs). This system was already used in our studies on the basis of earlier numerical results, but we present in the paper the full theoretical proof. It is proved that there is a unique set of parameters in GBEs such that (a) main properties (related to dimensions of stable and unstable manifolds for large x) of the kinetic and hydrodynamic description are qualitatively similar and (b) the number of third derivatives in the equations has a minimal possible value \( n = 1 \) . All considerations are made for monoatomic gases with arbitrary intermolecular potential (with standard restrictions). PubDate: 2022-05-18
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Abstract: Consider the nearest-neighbor Ising model on \(\Lambda _n:=[-n,n]^d\cap {\mathbb {Z}}^d\) at inverse temperature \(\beta \ge 0\) with free boundary conditions, and let \(Y_n(\sigma ):=\sum _{u\in \Lambda _n}\sigma _u\) be its total magnetization. Let \(X_n\) be the total magnetization perturbed by a critical Curie–Weiss interaction, i.e., $$\begin{aligned} \frac{d F_{X_n}}{d F_{Y_n}}(x):=\frac{\exp [x^2/\left( 2\langle Y_n^2 \rangle _{\Lambda _n,\beta }\right) ]}{\left\langle \exp [Y_n^2/\left( 2\langle Y_n^2\rangle _{\Lambda _n,\beta }\right) ]\right\rangle _{\Lambda _n,\beta }}, \end{aligned}$$ where \(F_{X_n}\) and \(F_{Y_n}\) are the distribution functions for \(X_n\) and \(Y_n\) respectively. We prove that for any \(d\ge 4\) and \(\beta \in [0,\beta _c(d)]\) where \(\beta _c(d)\) is the critical inverse temperature, any subsequential limit (in distribution) of \(\{X_n/\sqrt{{\mathbb {E}}\left( X_n^2\right) }:n\in {\mathbb {N}}\}\) has an analytic density (say, \(f_X\) ) all of whose zeros are pure imaginary, and \(f_X\) has an explicit expression in terms of the asymptotic behavior of zeros for the moment generating function of \(Y_n\) . We also prove that for any \(d\ge 1\) and then for \(\beta \) small, $$\begin{aligned} f_X(x)=K\exp (-C^4x^4), \end{aligned}$$ where \(C=\sqrt{\Gamma (3/4)/\Gamma (1/4)}\) and \(K=\sqrt{\Gamma (3/4)}/(4\Gamma (5/4)^{3/2})\) . Possible connections between \(f_X\) and the high-dimensional critical Ising model with periodic boundary conditions are discussed. PubDate: 2022-05-16
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Abstract: We consider a stochastic model describing the spiking activity of a countable set of neurons spatially organized into a homogeneous tree of degree d, \(d \ge 2\) ; the degree of a neuron is just the number of connections it has. Roughly, the model is as follows. Each neuron is represented by its membrane potential, which takes non-negative integer values. Neurons spike at Poisson rate 1, provided they have strictly positive membrane potential. When a spike occurs, the potential of the spiking neuron changes to 0, and all neurons connected to it receive a positive amount of potential. Moreover, between successive spikes and without receiving any spiking inputs from other neurons, each neuron’s potential behaves independently as a pure death process with death rate \(\gamma \ge 0\) . In this article, we show that if the number d of connections is large enough, then the process exhibits at least two phase transitions depending on the choice of rate \(\gamma \) : For large values of \(\gamma \) , the neural spiking activity almost surely goes extinct; For small values of \(\gamma \) , a fixed neuron spikes infinitely many times with a positive probability, and for “intermediate” values of \(\gamma \) , the system has a positive probability of always presenting spiking activity, but, individually, each neuron eventually stops spiking and remains at rest forever. PubDate: 2022-05-13
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Abstract: We study a model of multi-excited random walk with non-nearest neighbour steps on \(\mathbb {Z}\) , in which the walk can jump from a vertex x to either \(x+1\) or \(x-i\) with \(i\in \{1,2,\dots ,L\}\) , \(L\ge 1\) . We first point out the multi-type branching structure of this random walk and then prove a limit theorem for a related multi-type Galton–Watson process with emigration, which is of independent interest. Combining this result and the method introduced by Basdevant and Singh (Probab Theory Relat Fields 141:3–4, 2008), we extend their result (w.r.t. the case \(L=1\) ) to our model. More specifically, we show that in the regime of transience to the right, the walk has positive speed if and only if the expected total drift \(\delta >2\) . This confirms a special case of a conjecture proposed by Davis and Peterson. PubDate: 2022-05-09
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Abstract: In this work, we first prove the existence of the classic solutions to the dimensionless bipolar Vlasov–Poisson–Boltzmann equations by employing hypocoercive properties of the linear Boltzmann operators. Based on the uniform estimates and employing the Ohm’s law, two fluids Navier–Stokes–Poisson system is derived from the dimensionless Vlasov–Poisson–Boltzmann equations. PubDate: 2022-05-07
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Abstract: This paper is about the rate of convergence to equilibrium for hypocoercive linear kinetic equations. We look for the spatially dependent jump rate which yields the fastest decay rate of perturbations. For the Goldstein–Taylor model, we show (i) that for a locally optimal jump rate the spectral bound is determined by multiple, possibly degenerate, eigenvectors and (ii) that globally the fastest decay is obtained with a spatially homogeneous jump rate. Our proofs rely on a connection to damped wave equations and a relationship to the spectral theory of Schrödinger operators. PubDate: 2022-05-06
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Abstract: Abstract We study a renormalization group (RG) map for tensor networks that include two-dimensional lattice spin systems such as the Ising model. Numerical studies of such RG maps have been quite successful at reproducing the known critical behavior. In those numerical studies the RG map must be truncated to keep the dimension of the legs of the tensors bounded. Our tensors act on an infinite-dimensional Hilbert space, and our RG map does not involve any truncations. Our RG map has a trivial fixed point which represents the high-temperature fixed point. We prove that if we start with a tensor that is close to this fixed point tensor, then the iterates of the RG map converge in the Hilbert-Schmidt norm to the fixed point tensor. It is important to emphasize that this statement is not true for the simplest tensor network RG map in which one simply contracts four copies of the tensor to define the renormalized tensor. The linearization of this simple RG map about the fixed point is not a contraction due to the presence of so-called CDL tensors. Our work provides a first step towards the important problem of the rigorous study of RG maps for tensor networks in a neighborhood of the critical point. PubDate: 2022-05-02
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Abstract: Abstract This work is devoted to the asymptotic behavior of eigenvalues of an elliptic operator with rapidly oscillating random coefficients on a bounded domain with Dirichlet boundary conditions. A sharp convergence rate is obtained for eigenvalues towards those of the homogenized problem, as well as a quantitative two-scale expansion result for eigenfunctions. Next, a quantitative central limit theorem is established for fluctuations of isolated eigenvalues; more precisely, a pathwise characterization of eigenvalue fluctuations is obtained in terms of the so-called homogenization commutator, in parallel with the recent fluctuation theory for the solution operator. PubDate: 2022-04-28
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Abstract: Abstract We study crosscap states in integrable field theories and spin chains in \(1+1\) dimensions. We derive an exact formula for overlaps between the crosscap state and any excited state in integrable field theories with diagonal scattering. We then compute the crosscap entropy, i.e. the overlap for the ground state, in some examples. In the examples we analyzed, the result turns out to decrease monotonically along the renormalization group flow except in cases where the discrete symmetry is spontaneously broken in the infrared. We next introduce crosscap states in integrable spin chains, and obtain determinant expressions for the overlaps with energy eigenstates. These states are long-range entangled and their entanglement entropy grows linearly until the size of the subregion reaches half the system size. This property is reminiscent of pure-state black holes in holography and makes them interesting for use as initial conditions of quantum quench. As side results, we propose a generalization of Zamolodchikov’s staircase model to flows between D-series minimal models, and discuss the relation to fermionic minimal models and the GSO projection. PubDate: 2022-04-22
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Abstract: Abstract We propose locally stable sparse hard-disk packings, as introduced by Böröczky, as a model for the analysis and benchmarking of Markov-chain Monte Carlo (MCMC) algorithms. We first generate such Böröczky packings in a square box with periodic boundary conditions and analyze their properties. We then study how local MCMC algorithms, namely the Metropolis algorithm and several versions of event-chain Monte Carlo (ECMC), escape from configurations that are obtained from the packings by slightly reducing all disk radii by a relaxation parameter. We obtain two classes of ECMC, one in which the escape time varies algebraically with the relaxation parameter (as for the local Metropolis algorithm) and another in which the escape time scales as the logarithm of the relaxation parameter. A scaling analysis is confirmed by simulation results. We discuss the connectivity of the hard-disk sample space, the ergodicity of local MCMC algorithms, as well as the meaning of packings in the context of the NPT ensemble. Our work is accompanied by open-source, arbitrary-precision software for Böröczky packings (in Python) and for straight, reflective, forward, and Newtonian ECMC (in Go). PubDate: 2022-04-22
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Abstract: Abstract The fourth virial coefficient is calculated exactly for a fluid of hard spheres in even dimensions. For this purpose the complete star cluster integral is expressed as the sum of two three-folded integrals only involving spherical angular coordinates. These integrals are solved analytically for any even dimension d, and working with existing expressions for the other terms of the fourth cluster integral, we obtain an expression for the fourth virial coefficient \(B_{4}(d)\) for even d. It reduces to the sum of a finite number of simple terms that increases with d. PubDate: 2022-04-21
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Abstract: Abstract In this paper, we give an overview of the results established in Alonso (http://arxiv.org/org/abs/2008.05173, 2020) which provides the first rigorous derivation of hydrodynamic equations from the Boltzmann equation for inelastic hard spheres in 3D. In particular, we obtain a new system of hydrodynamic equations describing granular flows and prove existence of classical solutions to the aforementioned system. One of the main issue is to identify the correct relation between the restitution coefficient (which quantifies the rate of energy loss at the microscopic level) and the Knudsen number which allows us to obtain non trivial hydrodynamic behavior. In such a regime, we construct strong solutions to the inelastic Boltzmann equation, near thermal equilibrium whose role is played by the so-called homogeneous cooling state. We prove then the uniform exponential stability with respect to the Knudsen number of such solutions, using a spectral analysis of the linearized problem combined with technical a priori nonlinear estimates. Finally, we prove that such solutions converge, in a specific weak sense, towards some hydrodynamic limit that depends on time and space variables only through macroscopic quantities that satisfy a suitable modification of the incompressible Navier–Stokes–Fourier system. PubDate: 2022-04-20
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Abstract: Abstract We study the local eigenvalue statistics \(\xi _{\omega ,E}^N\) associated with the eigenvalues of one-dimensional, \((2N+1) \times (2N+1)\) random band matrices with independent, identically distributed, real random variables and band width growing as \(N^\alpha \) , for \(0< \alpha < \frac{1}{2}\) . We consider the limit points associated with the random variables \(\xi _{\omega ,E}^N [I]\) , for \(I \subset \mathbb {R}\) , and \(E \in (-2,2)\) . For random band matrices with Gaussian distributed random variables and for \(0 \le \alpha < \frac{1}{7}\) , we prove that this family of random variables has nontrivial limit points for almost every \(E \in (-2,2)\) , and that these limit points are Poisson distributed with positive intensities. The proof is based on an analysis of the characteristic functions of the random variables \(\xi _{\omega ,E}^N [I]\) and associated quantities related to the intensities, as N tends towards infinity, and employs known localization bounds of (Peled et al. in Int. Math. Res. Not. IMRN 4:1030–1058, 2019, Schenker in Commun Math Phys 290:1065–1097, 2009), and the strong Wegner and Minami estimates (Peled et al. in Int. Math. Res. Not. IMRN 4:1030–1058, 2019). Our more general result applies to random band matrices with random variables having absolutely continuous distributions with bounded densities. Under the hypothesis that the localization bounds hold for \(0< \alpha < \frac{1}{2}\) , we prove that any nontrivial limit points of the random variables \(\xi _{\omega ,E}^N [I]\) are distributed according to Poisson distributions. PubDate: 2022-04-19
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Abstract: Abstract Using publicly available data from the football database transfermarkt.co.uk, 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. PubDate: 2022-04-19
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Abstract: Abstract We investigate a class of nonequilibrium media described by Langevin dynamics that satisfies the local detailed balance. For the effective dynamics of a probe immersed in the medium, we derive an inequality that bounds the violation of the second fluctuation-dissipation relation (FDR). We also discuss the validity of the effective dynamics. In particular, we show that the effective dynamics obtained from nonequilibrium linear response theory is consistent with that obtained from a singular perturbation method. As an example of these results, we propose a simple model for a nonequilibrium medium in which the particles are subjected to potentials that switch stochastically. For this model, we show that the second FDR is recovered in the fast switching limit, although the particles are out of equilibrium. PubDate: 2022-04-18
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Abstract: Abstract A form of time series path integral expansion is provided that enables both analytic and numerical temporal effect calculations for a range of stochastic processes. All methods rely on finding a suitable reproducing kernel associated with an underlying representative algebra to perform the expansion. Birth–death processes can be analysed with these techniques, using either standard Doi-Peliti coherent states, or the \({\mathfrak {s}}{\mathfrak {u}}(1,1)\) Lie algebra. These result in simplest expansions for processes with linear or quadratic rates, respectively. The techniques are also adapted to diffusion processes. The resulting series differ from those found in standard Dyson time series field theory techniques. PubDate: 2022-04-16
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Abstract: Abstract 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. PubDate: 2022-04-15
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Abstract: Abstract In this paper we study ergodic optimization and multifractal behavior of Lyapunov exponents for matrix cocycles. We show that the restricted variational principle holds for generic cocycles [in the sense of (Bonatti and Viana in Ergod Theory Dyn Syst 24(5):1295–1330, 2004)] over mixing subshifts of finite type. We also show that the Lyapunov spectrum is equal to the closure of the set where the entropy spectrum is positive for such cocycles. Moreover, we show the continuity of the entropy spectrum at boundary of Lyapunov spectrum in the sense that \(h_{top}(E(\alpha _{t}))\ \rightarrow h_{top}(E(\beta ({\mathcal {A}}))\) , where \(E(\alpha )=\{x\in X: \lim _{n\rightarrow \infty }\frac{1}{n}\log \Vert {\mathcal {A}}^{n}(x)\Vert =\alpha \}\) , for such cocycles. We prove the continuity of the lower joint spectral radius for linear cocycles under the assumption that linear cocycles satisfy a cone condition. PubDate: 2022-04-15
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Abstract: Abstract We establish new connections between percolation, bootstrap percolation, probabilistic cellular automata and deterministic ones. Surprisingly, by juggling with these in various directions, we effortlessly obtain a number of new results in these fields. In particular, we prove the sharpness of the phase transition of attractive absorbing probabilistic cellular automata, a class of bootstrap percolation models and kinetically constrained models. We further show how to recover a classical result of Toom on the stability of cellular automata w.r.t. noise and, inversely, how to deduce new results in bootstrap percolation universality from his work. PubDate: 2022-04-12
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Abstract: Abstract This paper is concerned with the probability of consensus in a multivariate, socially structured version of the Hegselmann–Krause model for the dynamics of opinions. Individuals are located on the vertices of a finite connected graph representing a social network, and are characterized by their opinion, with the set of opinions \(\Delta \) being a general bounded convex subset of a finite dimensional normed vector space. Having a confidence threshold \(\tau \) , two individuals are said to be compatible if the distance (induced by the norm) between their opinions does not exceed the threshold \(\tau \) . Each vertex x updates its opinion at rate the number of its compatible neighbors on the social network, which results in the opinion at x to be replaced by a convex combination of the opinion at x and the nearby opinions: \(\alpha \) times the opinion at x plus \((1 - \alpha )\) times the average opinion of its compatible neighbors. The main objective is to derive a lower bound for the probability of consensus when the opinions are initially independent and identically distributed with values in the opinion set \(\Delta \) . PubDate: 2022-04-11
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