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In this paper, the existence and uniqueness of initial value problems for nonlinear Langevin equation involving three fractional orders are discussed. We use a new norm that is convenient for the fractional and singular differential equations. This norm and the contraction mapping principles are the main tools for investigating the existence and uniqueness of the desired issue. The fractional derivatives are described in Caputo sense. [ABSTRACT FROM AUTHOR]

In this paper we present a simple microscopic stochastic model describing short term plasticity within a large homogeneous network of interacting neurons. Each neuron is represented by its membrane potential and by the residual calcium concentration within the cell at a given time. Neurons spike at a rate depending on their membrane potential. When spiking, the residual calcium concentration of the spiking neuron increases by one unit. Moreover, an additional amount of potential is given to all other neurons in the system. This amount depends linearly on the current residual calcium concentration within the cell of the spiking neuron. In between successive spikes, the potentials and the residual calcium concentrations of each neuron decrease at a constant rate. We show that in this framework, short time memory can be described as the tendency of the system to keep track of an initial stimulus by staying within a certain region of the space of configurations during a short but macroscopic amount of time before finally being kicked out of this region and relaxing to equilibrium. The main technical tool is a rigorous justification of the passage to a large population limit system and a thorough study of the limit equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence problem for a stationary relaxational models of the quantum Boltzmann equation. More precisely, we establish the existence of mild solution to the fermionic or bosonic quantum BGK model in a slab with inflow boundary data. Unlike the classical case, it is necessary to verify that the quantum local equilibrium state is well-defined, and the transition from the non-condensed state to the condensated state (bosons), or from the non-saturated state to the saturated state (fermions) does not arise in our solution space. [ABSTRACT FROM AUTHOR]

The Kuramoto model is an important model for studying the onset of phase-locking in an ensemble of nonlinearly coupled phase oscillators. Each oscillator has a natural frequency (often taken to be random) and interacts with the other oscillators through the phase difference. It is known that, as the coupling strength is increased, there is a bifurcation in which the incoherent state becomes unstable and a stable phase-locked solution is born. Beginning with the work of Lohe there have been a number of paper that have generalized the Kuramoto model for phase-locking to a non-commuting setting. Here we propose and analyze another such model. We consider a collection of matrix-valued variables that evolve in such a way as to try to align their eigenvector frames. The phase-locked state is one where the eigenframes all align, and thus the matrices all commute. We analyze the stability of the equal frequency phase-locked state and show that it is stable. We also analyze (for the case of symmetric matrices) a dynamic analog of the twist states arising in the standard Kuramoto model, and show that these twist states are dynamically unstable. [ABSTRACT FROM AUTHOR]

The derivation of effective equations for interacting many body systems has seen a lot of progress in the recent years. While dealing with classical systems, singular potentials are quite challenging (Hauray and Jabin in Annales scientifiques de l'École Normale Supérieure, 2013, Lazarovici and Pickl in Arch Ration Mech Anal 225(3):1201–1231, 2017) comparably strong results are known to hold for quantum systems (Knowles and Pickl in Comm Math Phys 298:101–139, 2010). In this paper, we wish to show how techniques developed for the derivation of effective descriptions of quantum systems can be used for classical ones. While our future goal is to use these ideas to treat singularities in the interaction, the focus here is to present how quantum mechanical techniques can be used for a classical system and we restrict ourselves to regular two-body interaction potentials. In particular we compute a mean field limit for the Hamilton Vlasov system in the sense of (Fröhlich et al. in Comm Math Phys 288:1023–1058, 2009; Neiss in Arch Ration Mech Anal. 10.1007/s00205-018-1275-8) that arises from classical dynamics. The structure reveals strong analogy to the Bosonic quantum mechanical ensemble of the many-particle Schrödinger equation and the Hartree equation as its mean field limit (Pickl in arXiv:0808.1178v1, 2008). [ABSTRACT FROM AUTHOR]

This paper is concerned with establishing global asymptotic stability results for a class of non-linear PDE, in which the nonlinearity enters as a scalar multiple which is non-local. The method of proof does not involve a Lyapounov function. It is shown that stability for the PDE is equivalent to stability for a differential delay equation. Stability for the delay equation is proven by exploiting certain monotonicity properties. These are established by using the methods of optimal control theory. [ABSTRACT FROM AUTHOR]

We consider the two-dimensional one-component plasma (jellium) of mobile pointlike particles with the same charge e, interacting pairwisely by the logarithmic Coulomb potential and immersed in a fixed neutralizing background charge density. Particles are in thermal equilibrium at the inverse temperature β , the only relevant dimensionless parameter is the coupling constant Γ ≡ β e 2 . In the bulk fluid regime and for any value of the coupling constant Γ = 2 × integer , Šamaj and Percus (J Stat Phys 80:811–824, 1995) have derived an infinite sequence of sum rules for the coefficients of the short-distance expansion of particle pair correlation function. In the context of the equivalent fractional quantum Hall effect, by using specific methods of quantum geometry Haldane (Phys Rev Lett 107:116801, 2011; arXiv:1112.0990v2, 2011) derived a self-dual relation for the Landau-level guiding-center structure factor. In this paper, we establish the relation between the guiding-center structure factor and the pair correlation function of jellium particles. It is shown that the self-dual formula, which provides an exact relation between the pair correlation function and its Fourier component, comes directly from the short-distance symmetry of the bulk jellium. The short-distant symmetry of pair correlations is extended to the semi-infinite geometry of a rectilinear plain hard wall with a fixed surface charge density, constraining particles to a half-space. The symmetry is derived for the original jellium model as well as its simplified version with no background charge (charged wall surface with "counter-ions only"). The obtained results are checked at the exactly solvable free-fermion coupling Γ = 2 . [ABSTRACT FROM AUTHOR]

The basic κ -color box-ball (BBS) system is an integrable cellular automaton on one dimensional lattice whose local states take { 0 , 1 , ... , κ } with 0 regarded as an empty box. The time evolution is defined by a combinatorial rule of quantum group theoretical origin, and the complete set of conserved quantities is given by a κ -tuple of Young diagrams. In the randomized BBS, a probability distribution on { 0 , 1 , ... , κ } to independently fill the consecutive n sites in the initial state induces a highly nontrivial probability measure on the κ -tuple of those invariant Young diagrams. In a recent work Kuniba et al. (Nucl Phys B 937:240–271, 2018), their large n 'equilibrium shape' has been determined in terms of Schur polynomials by a Markov chain method and also by a very different approach of thermodynamic Bethe ansatz (TBA). In this paper, we establish a large deviations principle for the row lengths of the invariant Young diagrams. As a corollary, they are shown to converge almost surely to the equilibrium shape at an exponential rate. We also refine the TBA analysis and obtain the exact scaling form of the vacancy, the row length and the column multiplicity, which exhibit nontrivial factorization in a one-parameter specialization. [ABSTRACT FROM AUTHOR]

We show that for the anti-ferromagnetic Ising model on the Bethe lattice, weak spatial mixing implies strong spatial mixing. As a by-product of our analysis, we obtain what is to the best of our knowledge the first rigorous proof of the uniqueness threshold for the anti-ferromagnetic Ising model (with non-zero external field) on the Bethe lattice. Following a method due to Weitz [], we then use the equivalence between weak and strong spatial mixing to give a deterministic fully polynomial time approximation scheme for the partition function of the anti-ferromagnetic Ising model with arbitrary field on graphs of degree at most $$d$$ , throughout the uniqueness region of the Gibbs measure on the infinite $$d$$ -regular tree. By a standard correspondence, our results translate to arbitrary two-state anti-ferromagnetic spin systems with soft constraints. Subsequent to a preliminary version of this paper, Sly and Sun [] have shown that our results are optimal in the sense that, under standard complexity theoretic assumptions, there does not exist a fully polynomial time approximation scheme for the partition function of such spin systems on graphs of maximum degree $$d$$ for parameters outside the uniqueness region. Taken together, the results of [] and of this paper therefore indicate a tight relationship between complexity theory and phase transition phenomena in two-state anti-ferromagnetic spin systems. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to correct errors presented recently in the paper [Lv et al. J Stat Phys 149:619-628 ()], where the authors analyzed Fractional Fokker-Planck equation (FFPE) with space-time dependent drift $$F(x,t)=F(x)f(t)$$ and diffusion $$D(x,t)=D(x)\tilde{d}(t)$$ coefficients in the factorized form. We show an important drawback in the derivation of the stochastic representation of FFPE presented in the aforementioned paper, which makes the whole proof wrong. Moreover, we present a correct proof of their result in even more general case, when both drift and diffusion can have any, not necessarily factorized, form. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stability and asymptotic stability at large time of solutions to a system of equations, which includes the Lifschitz-Slyozov-Wagner (LSW) system in the case when the initial data has compact support. The main result of the paper is a proof of weak global asymptotic stability for LSW like systems. Previously strong local asymptotic stability results were obtained by Niethammer and Velázquez for the LSW system with initial data of compact support. Comparison to a quadratic model plays an important part in the proof of the main theorem when the initial data is critical. The quadratic model extends the linear model of Carr and Penrose, and has a time invariant solution which decays exponentially at the edge of its support in the same way as the infinitely differentiable self-similar solution of the LSW model. [ABSTRACT FROM AUTHOR]

Let $$\{\mu (\cdot ,t):t\ge 0\}$$ be the family of probability measures corresponding to the solution of the inelastic Kac model introduced in Pulvirenti and Toscani (J Stat Phys 114:1453-1480, ). It has been proved by Gabetta and Regazzini (J Stat Phys 147:1007-1019, ) that the solution converges weakly to equilibrium if and only if a suitable symmetrized form of the initial data belongs to the standard domain of attraction of a specific stable law. In the present paper it is shown that, for initial data which are heavier-tailed than the aforementioned ones, the limiting distribution is improper in the sense that it has probability $$1/2$$ 'adherent' to $$-\infty $$ and probability $$1/2$$ 'adherent' to $$+\infty $$ . It is explained in which sense this phenomenon is amenable to a sort of explosion, and the main result consists in an explicit expression of the rate of such an explosion. The presentation of these statements is preceded by a discussion about the necessity of the assumption under which their validity is proved. This gives the chance to make an adjustment to a portion of a proof contained in the above-mentioned paper by Gabetta and Regazzini. [ABSTRACT FROM AUTHOR]

A bosonic analogue of the fractional quantum Hall effect occurs in rapidly rotating trapped Bose gases: There is a transition from uncorrelated Hartree states to strongly correlated states such as the Laughlin wave function. This physics may be described by effective Hamiltonians with delta interactions acting on a bosonic N-body Bargmann space of analytic functions. In a previous paper (Rougerie et al. in Phys. Rev. A 87:023618, ) we studied the case of a quadratic plus quartic trapping potential and derived conditions on the parameters of the model for its ground state to be asymptotically strongly correlated. This relied essentially on energy upper bounds using quantum Hall trial states, incorporating the correlations of the Bose-Laughlin state in addition to a multiply quantized vortex pinned at the origin. In this paper we investigate in more details the density of these trial states, thereby substantiating further the physical picture described in (Rougerie et al. in Phys. Rev. A 87:023618, ), improving our energy estimates and allowing to consider more general trapping potentials. Our analysis is based on the interpretation of the densities of quantum Hall trial states as Gibbs measures of classical 2D Coulomb gases (plasma analogy). New estimates on the mean-field limit of such systems are presented. [ABSTRACT FROM AUTHOR]

Through this paper we analyze the ergodic properties of continuous time Markov chains with values on the one-dimensional spin lattice $\{1,\dots,d\}^{{\mathbb{N}}}$ (also known as the Bernoulli space). Initially, we consider as the infinitesimal generator the operator [InlineEquation not available: see fulltext.], where [InlineEquation not available: see fulltext.] is a discrete time Ruelle operator (transfer operator), and $A:\{1,\dots,d\}^{{\mathbb{N}}}\to\mathbb{R}$ is a given fixed Lipschitz function. The associated continuous time stationary Markov chain will define the a priori probability. Given a Lipschitz interaction $V:\{1,\dots,d\}^{{\mathbb{N}}}\to\mathbb{R}$, we are interested in Gibbs (equilibrium) state for such V. This will be another continuous time stationary Markov chain. In order to analyze this problem we will use a continuous time Ruelle operator (transfer operator) naturally associated to V. Among other things we will show that a continuous time Perron-Frobenius Theorem is true in the case V is a Lipschitz function. We also introduce an entropy, which is negative (see also Lopes et al. in Entropy and Variational Principle for one-dimensional Lattice Systems with a general a-priori probability: positive and zero temperature. Arxiv, 2012), and we consider a variational principle of pressure. Finally, we analyze large deviations properties for the empirical measure in the continuous time setting using results by Y. Kifer (Tamsui Oxf. J. Manag. Sci. 321(2):505-524, ). In the last appendix of the paper we explain why the techniques we develop here have the capability to be applied to the analysis of convergence of a certain version of the Metropolis algorithm. [ABSTRACT FROM AUTHOR]

The drastic effect of local alliances in three-party competition is investigated in democratic hierarchical bottom-up voting. The results are obtained analytically using a model which extends a sociophysics frame introduced in 1986 (Galam in J. Math. Phys. 30:426, ) and 1990 (Galam in J. Stat. Phys. 61:943, ) to study two-party systems and the spontaneous formation of democratic dictatorship. It is worth stressing that the 1990 paper was published in the Journal of Statistical Physics, the first paper of its kind in this journal. It was shown how a minority in power can preserve its leadership using bottom-up democratic elections. However such a bias holds only down to some critical value of minimum support. The results were used latter to explain the sudden collapse of European communist parties in the nineties. The extension to three-party competition reveals the mechanisms by which a very small minority party can get a substantial representation at higher levels of the hierarchy when the other two competing parties are big. Additional surprising results are obtained, which enlighten the complexity of three-party democratic bottom-up voting. In particular, the unexpected outcomes of local voting alliances are singled out. Unbalanced democratic situations are exhibited with strong asymmetries between the actual bottom support of a party and its associated share of power at the top leadership. Subtle strategies are identified for a party to maximize its hold on the top leadership. The results are also valid to describe opinion dynamics with three competing opinions. [ABSTRACT FROM AUTHOR]

This is the third in a series of three papers in which we study a two-dimensional lattice gas consisting of two types of particles subject to Kawasaki dynamics at low temperature in a large finite box with an open boundary. Each pair of particles occupying neighboring sites has a negative binding energy provided their types are different, while each particle has a positive activation energy that depends on its type. There is no binding energy between particles of the same type. At the boundary of the box particles are created and annihilated in a way that represents the presence of an infinite gas reservoir. We start the dynamics from the empty box and are interested in the transition time to the full box. This transition is triggered by a critical droplet appearing somewhere in the box. In the first paper we identified the parameter range for which the system is metastable, showed that the first entrance distribution on the set of critical droplets is uniform, computed the expected transition time up to and including a multiplicative factor of order one, and proved that the nucleation time divided by its expectation is exponentially distributed, all in the limit of low temperature. These results were proved under three hypotheses, and involved three model-dependent quantities: the energy, the shape and the number of critical droplets. In the second paper we proved the first and the second hypothesis and identified the energy of critical droplets. In the third paper we prove the third hypothesis and identify the shape and the number of critical droplets, thereby completing our analysis. Both the second and the third paper deal with understanding the geometric properties of subcritical, critical and supercritical droplets, which are crucial in determining the metastable behavior of the system, as explained in the first paper. The geometry turns out to be considerably more complex than for Kawasaki dynamics with one type of particle, for which an extensive literature exists. The main motivation behind our work is to understand metastability of multi-type particle systems. [ABSTRACT FROM AUTHOR]

In 2011 I reviewed the calculation by Onsager and Kaufman of the spontaneous magnetization of the square-lattice Ising model, which Onsager announced in 1949 but never published. I have recently been alerted to further original papers that bear on the subject. It is quite clear that the draft paper on which I relied was indeed written by Onsager, who was working on the problem with Kaufman, and that they had two derivations of the result. [ABSTRACT FROM AUTHOR]

Kac's d dimensional model gives a linear, many particle, binary collision model from which, under suitable conditions, the celebrated Boltzmann equation, in its spatially homogeneous form, arise as a mean field limit. The ergodicity of the evolution equation leads to questions about the relaxation rate, in hope that such a rate would pass on the Boltzmann equation as the number of particles goes to infinity. This program, starting with Kac and his one dimensional 'Spectral Gap Conjecture' at 1956, finally reached its conclusion in the Maxwellian case in a series of papers by authors such as Janvresse, Maslen, Carlen, Carvalho, Loss and Geronimo, but the hope to get a limiting relaxation rate for the Boltzmann equation with this linear method was already shown to be unrealistic (although the problem is still important and interesting due to its connection with the linearized Boltzmann operator). A less linear approach, via a many particle version of Cercignani's conjecture, is the grounds for this paper. In our paper, we extend recent results by the author from the one dimensional Kac model to the d dimensional one, showing that the entropy-entropy production ratio, Γ, still yields a very strong dependency in the number of particles of the problem when we consider the general case. [ABSTRACT FROM AUTHOR]

In this paper we consider the relation between random sums and compositions of different processes. In particular, for independent Poisson processes N( t), N( t), t>0, we have that $N_{\alpha}(N_{\beta}(t)) \stackrel{\mathrm{d}}{=} \sum_{j=1}^{N_{\beta}(t)} X_{j}$, where the Xs are Poisson random variables. We present a series of similar cases, where the outer process is Poisson with different inner processes. We highlight generalisations of these results where the external process is infinitely divisible. A section of the paper concerns compositions of the form $N_{\alpha}(\tau_{k}^{\nu})$, ν∈(0,1], where $\tau_{k}^{\nu}$ is the inverse of the fractional Poisson process, and we show how these compositions can be represented as random sums. Furthermore we study compositions of the form Θ( N( t)), t>0, which can be represented as random products. The last section is devoted to studying continued fractions of Cauchy random variables with a Poisson number of levels. We evaluate the exact distribution and derive the scale parameter in terms of ratios of Fibonacci numbers. [ABSTRACT FROM AUTHOR]

This paper deals with a new class of random flights in ℝ, d≥2, characterized by non-uniform probability distributions on the multidimensional sphere. These random motions differ from similar models appeared in literature where the directions are taken according to the uniform law. The family of angular probability distributions introduced in this paper depends on a parameter ν≥0, which gives the anisotropy of the motion. Furthermore, we assume that the number of changes of direction performed by the random flight is fixed. The time lengths between two consecutive changes of orientation have joint probability distribution given by a Dirichlet density function. The analysis of the position $\underline{\mathbf{X}}_{d}(t)$, t>0, is not an easy task, because it involves integrals which are not always computable. Therefore, we get the probability distribution of the process $\underline{\mathbf{X}}_{m}^{d}(t)$, t>0, obtained as projection onto the lower space ℝ, m< d, of the original random motion in ℝ. In its general framework, the analysis of $\underline{\mathbf{X}}_{d}(t)$, t>0, is very complicated; nevertheless for some values of ν, we provide some explicit results about the process. Indeed, for ν=1 we get the characteristic function of the random flight moving in ℝ. By inverting the obtained characteristic function, we derive the analytic form (up to some constants) of the probability distribution of $\underline{\mathbf{X}}_{d}(t)$, t>0. It is worth to mention that the stochastic processes considered in this paper belong to the class of the non-isotropic random walks, which has several applications in the mechanical statistics. [ABSTRACT FROM AUTHOR]

This paper is the continuation of our earlier paper (Balázs et al. in Ann. Inst. Henri Poincaré Probab. Stat. 48(1):151-187, ), where we proved t-order of current fluctuations across the characteristics in a class of one dimensional interacting systems with one conserved quantity. We also claimed two models with concave hydrodynamic flux which satisfied the assumptions which made our proof work. In the present note we show that the totally asymmetric exponential bricklayers process also satisfies these assumptions. Hence this is the first example with convex hydrodynamics of a model with t-order current fluctuations across the characteristics. As such, it further supports the idea of universality regarding this scaling. [ABSTRACT FROM AUTHOR]

In this paper we study the metastability problem for a stochastic dynamics with a parallel updating rule; in particular we consider a finite volume Probabilistic Cellular Automaton (PCA) in a small external field at low temperature regime. We are interested in the nucleation of the system, i.e., the typical excursion from the metastable phase (the configuration with all minuses) to the stable phase (the configuration with all pluses), triggered by the formation of a critical droplet. The main result of the paper is the sharp estimate of the nucleation time: we show that the nucleation time divided by its average converges to an exponential random variable and that the rate of the exponential random variable is an exponential function of the inverse temperature β times a prefactor that does not scale with β. Our approach combines geometric and potential theoretic arguments. [ABSTRACT FROM AUTHOR]

We consider the limiting location and limiting distribution of the largest eigenvalue in real symmetric ( β=1), Hermitian ( β=2), and Hermitian self-dual ( β=4) random matrix models with rank 1 external source. They are analyzed in a uniform way by a contour integral representation of the joint probability density function of eigenvalues. Assuming the 'one-band' condition and certain regularities of the potential function, we obtain the limiting location of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is not the critical value, and further obtain the limiting distribution of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is greater than the critical value. When the nonzero eigenvalue of the external source matrix is less than or equal to the critical value, the limiting distribution of the largest eigenvalue will be analyzed in a subsequent paper. In this paper we also give a definition of the external source model for all β>0. [ABSTRACT FROM AUTHOR]

In this paper we study the threshold voter models on homogeneous trees. We give the definition of the 'critical density' of the models and obtain an estimation of it. Furthermore, we study the exponential rate at which the process converges to an absorbed state δ in the subcritical case. At the end of the paper, we propose two conjectures about the phase transition phenomenons of our models. [ABSTRACT FROM AUTHOR]

A molecular network can be characterized by several different ways, like a matrix, a polynomial, a drawing or a topological descriptor. A topological descriptor is a numeric quantity associated with a network or a graph that characterizes its whole structural properties. Analyzing and determining the topological indices and structural properties of a network or a graph have been a worthy studied topic in the field of chemistry, networks analysis, etc. In this paper, we consider several types of the generalized Sierpiński networks and investigate the explicit expressions of some well-known valency-based topological indices. Taking into account the other structural property of the generalized Sierpiński networks, the average degree is determined. [ABSTRACT FROM AUTHOR]

In this paper we obtain a decoupling feature of the random interlacements process I u ⊂ Z d , at level u, d ≥ 3 . More precisely, we show that the trace of the random interlacements process on two disjoint finite sets, F and its translated F + x , can be coupled with high probability of success, when ‖ x ‖ is large, with the trace of a process of independent excursions, which we call the noodle soup process. As a consequence, we obtain an upper bound on the covariance between two [0, 1]-valued functions depending on the configuration of the random interlacements on F and F + x , respectively. This improves a previous bound obtained by Sznitman (Ann Math 2(171):2039–2087, 2010). [ABSTRACT FROM AUTHOR]

In 1987, Affleck, Kennedy, Lieb, and Tasaki introduced the AKLT spin chain and proved that it has a spectral gap above the ground state. Their concurrent conjecture that the two-dimensional AKLT model on the hexagonal lattice is also gapped remains open. In this paper, we show that the AKLT Hamiltonian restricted to an arbitrarily long chain of hexagons is gapped. The argument is based on explicitly verifying a finite-size criterion which is tailor-made for the system at hand. We also discuss generalizations of the method to the full hexagonal lattice. [ABSTRACT FROM AUTHOR]

In 2018, Ferrari et al. wrote a paper called "Phase Transition for Infinite Systems of Spiking Neurons" in which they introduced a continuous time stochastic model of interacting neurons. This model consists in a countable number of neurons, each of them having an integer-valued membrane potential, which value determine the rate at which the neuron spikes. This model has also a parameter γ , corresponding to the rate of the leak times of the neurons, that is, the times at which the membrane potential of a given neuron is spontaneously reset to its resting value (which is 0 by convention). As its title says, it was proven in this previous article that this model presents a phase transition phenomenon with respect to γ . Here we prove that this model also exhibits a metastable behavior. By this we mean that if γ is small enough, then the re-normalized time of extinction of a finite version of this system converges toward an exponential random variable of mean 1 as the number of neurons goes to infinity. [ABSTRACT FROM AUTHOR]

In most countries, alcohol consumption distributions have been shown to possess universal features. Their unimodal right-skewed shape is usually modeled in terms of the Lognormal distribution, which is easy to fit, test, and modify. However, empirical distributions often deviate considerably from the Lognormal model, and both Gamma and Weibull distributions appear to better describe the survey data. In this paper we explain the appearance of these distributions by means of classical methods of kinetic theory of multi-agent systems. The microscopic variation of alcohol consumption of agents around a universal social accepted value of consumption, is built up introducing as main criterion for consumption a suitable value function in the spirit of the prospect theory of Kahneman and Twersky. The mathematical properties of the value function then determine the unique macroscopic equilibrium which results to be a generalized Gamma distribution. The modeling of the microscopic kinetic interaction allows to clarify the meaning of the various parameters characterizing the generalized Gamma equilibrium. [ABSTRACT FROM AUTHOR]

The paper starts by giving a motivation for this research and justifying the considered stochastic diffusion models for cosmic microwave background (CMB) radiation studies. Then it derives the exact solution in terms of a series expansion to a hyperbolic diffusion equation on the unit sphere. The Cauchy problem with random initial conditions is studied. All assumptions are stated in terms of the angular power spectrum of the initial conditions. An approximation to the solution is given and analysed by finitely truncating the series expansion. The upper bounds for the convergence rates of the approximation errors are derived. Smoothness properties of the solution and its approximation are investigated. It is demonstrated that the sample Hölder continuity of these spherical fields is related to the decay of the angular power spectrum. Numerical studies of approximations to the solution and applications to CMB data are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In the simplest model of single-file diffusion, N point particles wander on a segment of the x axis of length L, with hard core interactions, which prevent passing, and with overdamped Brownian dynamics λ x ˙ = η (t) , where η (t) has the form of Gaussian white noise with zero mean. In 1965 Harris showed that in the limit N → ∞ , L → ∞ with constant ρ = N / L , the mean square displacement of a tagged particle grows subdiffusively, as t 1 / 2 , for long times. Recently, it has been shown that the proportionality constants of the t 1 / 2 law for randomly-distributed initial positions of the particles and for equally-spaced initial positions are not the same, but have ratio 2 . In this paper we consider point particles on the x axis, which collide elastically, and which move according to (i) random-acceleration dynamics x ¨ = η (t) and (ii) Langevin dynamics x ¨ + λ x ˙ = η (t) . The mean square displacement and mean-square velocity of a tagged particle are analyzed for both types of dynamics and for random and equally-spaced initial positions and Gaussian-distributed initial velocities. We also study tagged particle statistics, for both types of dynamics, in the spreading of a compact cluster of particles, with all of the particles initially at the origin. [ABSTRACT FROM AUTHOR]

The emergence of deterministic and irreversible macroscopic behavior from deterministic and reversible microscopic dynamics is understood as a result of the law of large numbers. In this paper, we prove on the basis of the theory of algorithmic randomness that Martin-Löf random initial microstates satisfy an irreversible macroscopic law in the Kac infinite chain model. We find that the time-reversed state of a random state is not random as well as it violates the macroscopic law. [ABSTRACT FROM AUTHOR]

The reconstruction problem on an infinite tree, is to collect and analyze massive data samples at the nth level of the tree to identify whether there is non-vanishing information of the root, as n goes to infinity. This problem has wide applications in various fields such as biology, information theory, and statistical physics. Although it has been studied in numerous contexts, the existing literature with rigorous reconstruction thresholds established are very limited. In this paper, we study the noise channel in terms of a 4 × 4 -state asymmetric transition matrix with community effects. By means of refined analyses of moment recursion, in-depth concentration estimates, and thorough investigations on an asymptotic 4-dimensional nonlinear second order dynamical system, we give the first rigorous result on asymmetric noisy channels with community effects by providing the exact conditions that the reconstruction bound is not tight, which yields the hybrid-hard phase in corresponding stochastic block models. [ABSTRACT FROM AUTHOR]

We prove a theorem on the minimal specific energy for a ± 1 charged particles system, interacting through a class of pair potential v, that may be stated as follows: suppose v may be represented by a scale mixtures of d-dimensional Euclid's hat. If the number of particles n is even, then their interacting energy U n divided by n is minimized by a constant B at the configurations with total charge zero and all particles collapsed to a point; if n is odd, then the ratio U n / (n - 1) is minimized by a constant B ¯ = B at the configurations with total charge ± 1 and all particles collapsed to a point. The theorem is then used to investigate the convergence of the Mayer series for a gas of ± 1 charged particles interacting through the two-dimensional Yukawa pair potential v for inverse temperatures in the collapse interval [ 4 π , 8 π) . The convergence is proved in the present paper up to the second threshold 6 π using the decomposition of the Yukawa potential into scales of modified Bessel functions (standard) and into scale mixtures of Euclid's hat. Moreover, assuming that (i) neutral subclusters of size smaller than an odd number k > 1 do not collapse inside a cluster of size larger than k for β in the threshold interval [ 8 π (k - 2) / (k - 1) , 8 π k / (k + 1)) and (ii) they satisfy a technical condition, then the Mayer series, discarding the first even coefficients of order smaller than k, converges. [ABSTRACT FROM AUTHOR]

In this paper, we study herding phenomena of agents in financial markets arising from the combined effect of (1) non-coordinated collective interactions between agents and (2) concurrent reactions of agents to dynamic market signals. By interpreting the expected price of an asset and the favorability on the asset as the position and the velocity in phase space, respectively, we construct an agent-based particle model for explaining herding behavior in finance. We then define two types of herding functionals to this model, and show that they satisfy a Gronwall type estimate and a LaSalle type invariance property, respectively. As a result, we show the herding behavior of the agents. Various numerical tests are presented to numerically verify theoretical results. [ABSTRACT FROM AUTHOR]

The ellipsoidal statistical (ES) model of the Boltzmann equation for a polyatomic gas with constant specific heats (calorically perfect gas), proposed by Andries et al. (Eur J Mech B Fluids 19:813, 2000), is extended to a polyatomic gas with temperature-dependent specific heats (thermally perfect gas). Then, the new model equation is applied to investigate the structure of a plane shock wave with special interest in CO 2 gas, which is known to have a very large bulk viscosity, and in the case of relatively strong shock waves. A numerical analysis, as well as an asymptotic analysis for large bulk viscosity, is performed in parallel to the previous paper by two of the present authors (Kosuge and Aoki, in: Phys Rev Fluids 3:023401, 2018), where the structure of a shock wave in CO 2 gas was investigated using the ES model for a polyatomic gas with constant specific heats. From the numerical and analytical results, the effect of temperature-dependent specific heats on the structure of a shock wave is clarified. [ABSTRACT FROM AUTHOR]

We study the hydrodynamic scaling limit for the Glauber–Kawasaki dynamics. It is known that, if the Kawasaki part is speeded up in a diffusive space-time scaling, one can derive the Allen–Cahn equation which is a kind of the reaction–diffusion equation in the limit. This paper concerns the scaling that the Glauber part, which governs the creation and annihilation of particles, is also speeded up but slower than the Kawasaki part. Under such scaling, we derive directly from the particle system the motion by mean curvature for the interfaces separating sparse and dense regions of particles as a combination of the hydrodynamic and sharp interface limits. [ABSTRACT FROM AUTHOR]

In this paper, we prove a quantitative version of the semiclassical limit from the Hartree to the Vlasov equation with singular interaction, including the Coulomb potential. To reach this objective, we also prove the propagation of velocity moments and weighted Schatten norms which implies the boundedness of the space density of particles uniformly in the Planck constant. [ABSTRACT FROM AUTHOR]

The paper is concerned with the asymptotic behavior of the correlation functions of the characteristic polynomials of non-Hermitian random matrices with independent entries. It is shown that the correlation functions behave like that for the Complex Ginibre Ensemble up to a factor depending only on the fourth absolute moment of the common probability law of the matrix entries. [ABSTRACT FROM AUTHOR]