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We re-examine the combined semi-classical and mean-field limit in the N-body fermionic Schrödinger equation with pure state initial data using the Husimi measure framework. The Husimi measure equation involves three residue types: kinetic, semiclassical, and mean-field. The main result of this paper is to provide better estimates for the kinetic and mean-field residue than those in Chen et al. (J Stat Phys 182(2):1–41, , 2021). Especially, the estimate for the mean-field residue is shown to be smaller than the semiclassical residue by a mixed-norm estimate of the two-particle reduced density matrix factorization. Our analysis also updates the oscillation estimate parts in the residual term estimates appeared in Chen et al. (J Stat Phys 182(2):1–41, , 2021). [ABSTRACT FROM AUTHOR]

We examine type D ASEP, a two-species interacting particle system which generalizes the usual asymmetric simple exclusion process. For certain cases of type D ASEP, the process does not give priority for one species over another, even though there is nontrivial interaction between the two species. For those specific cases, we prove that the type D ASEP is self-dual with respect to an independent product of q-Krawtchouk polynomials. The type D ASEP was originally constructed in [15], using the type D quantum groups U q (so 6) and U q (so 8) . That paper claimed that certain states needed to be “discarded” in order to ensure non-negativity. Here, we also provide a more efficient argument for the same claim. [ABSTRACT FROM AUTHOR]

Stochastic modeling of chemical reaction systems based on master equations has been an indispensable tool in physical sciences. In the long-time limit, the properties of these systems are characterized by stationary distributions of chemical master equations. In this paper, we describe a novel method for computing stationary distributions analytically, based on a parallel formalism between stochastic chemical reaction systems and second quantization. Anderson, Craciun, and Kurtz showed that, when the rate equation for a reaction network admits a complex-balanced steady-state solution, the corresponding stochastic reaction system has a stationary distribution of a product form of Poisson distributions. In a formulation of stochastic reaction systems using the language of second quantization initiated by Doi, product-form Poisson distributions correspond to coherent states. Pursuing this analogy further, we study the counterpart of squeezed states in stochastic reaction systems. Under the action of a squeeze operator, the time-evolution operator of the chemical master equation is transformed, and the resulting system describes a different reaction network, which does not admit a complex-balanced steady state. A squeezed coherent state gives the stationary distribution of the transformed network, for which analytic expression is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider the two-species asymmetric simple exclusion process (ASEP) consisting of N - 1 first-class particles and one second-class particle. We assume that all particles are located at arbitrary positions but the second-class particle is the rightmost particle at time t = 0 . We find the exact formula of the distribution of the second-class particle’s position at time t by directly using the transition probabilities of the two-species ASEP, which is a different approach from the coupling method Tracy and Widom used in [J Phys A 42:425002, 2009]. [ABSTRACT FROM AUTHOR]

In this paper, pointwise preimage entropies for (noninvertible) continuous maps on any subset (not necessarily compact or invariant) are introduced via Carathéodory–Pesin construction. We first prove Brin–Katok local preimage entropy formula. After comparing several possible versions of metric preimage entropies, conditions to make these notions coincide are found. We also establish an inverse variational principle and obtain a variational inequality for preimage entropy of saturated sets. In particular, we prove that the preimage entropy of the set of generic points for an ergodic measure equals the metric preimage entropy of the measure, which extends Bowen’s theorem to preimage entropy. Stable entropy for partially hyperbolic endomorphisms on any subset determined by the primage structure of stable manifolds, are also defined via Carathéodory–Pesin construction. Corresponding properties and variational principles for stable entropy are also investigated. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the convergence of the perturbed diffusion process to its unperturbed limit. Our aim is to provide explicit estimates for the rates of strong and weak convergence. We also describe the exact asymptotic behavior of these convergence when the perturbation parameter tends to zero. [ABSTRACT FROM AUTHOR]

Consider the mixed p-spin models with general environments such that the covariance of Hamiltonian process is non-negative. In this paper, we prove the universality of the superconcentration phenomenon. Precisely, we show that the variance of the free energy grows sublinearly in the size of its expectation when the disordered random variable satisfies some moment matching conditions. Additionally, we also study the universality of first and second moments of the free energy of a spin glass model on general hypergraphs. [ABSTRACT FROM AUTHOR]

This paper’s major purpose is to continue the work of Zhu and Ma in [J Stat Phys 182(1):19, 2021]. To begin, the g -almost product property, more general irregular and regular sets, and some new notions of the Banach upper density recurrent points and transitive points of free semigroup actions are introduced. Furthermore, under the g -almost product property and other conditions, we coordinate the Banach upper recurrence, transitivity with (ir)regularity, and obtain lots of generalized multifractal analyses for general observable functions of free semigroup actions. Finally, statistical ω -limit sets are used to consider the upper capacity topological entropy of the sets of Banach upper recurrent points and transitive points of free semigroup actions, respectively. Our analysis generalizes the results obtained by Huang et al. in [Nonlinearity 32(7):2721–2757, 2019] and Pfister and Sullivan in [Ergodic Theory Dynam Syst 27(3):929–956, 2007]. [ABSTRACT FROM AUTHOR]

The work is devoted to some mathematical problems of dynamics of collisional plasma. The difficulty is that in plasma case we have at least three different length scales: Debye radius rD, mean free path l and macroscopic length L. This is true even for the simplest model (plasma of electrons with a neutralizing background of infinitely heavy ions), considered in the paper. We study (at the formal level of mathematical rigour) solutions of the VLPE, having the typical length of the order l>>rD, and try to clarify some mathematical questions related to corresponding limit. In particular, we study the existence of the limit for electric field and show that, generally speaking, it does not exist because of rapidly oscillating terms. An approximate asymptotic formula for the oscillating electric field near this limit is derived from VLPE. Still the limiting equations, which are used in many publications by physicists, can lead in some cases to correct results for the distribution function. Both these conclusions are confirmed by more explicit analysis of the linearized Vlasov-Poisson equation. We also study the well-posedness of limiting kinetic equations and the corresponding criterion in the class of weakly inhomogeneous initial data. It is shown that the collisional effects do not play an important role in this problem. In particular the equations are well-posed in the case of small deviations from equilibrium, as it was already known for related collisionless models. [ABSTRACT FROM AUTHOR]

The 3D Euler equations, precisely local smooth solutions of class Hs with s>5/2 are obtained as a mean field limit of finite families of interacting curves, the so called vortex filaments, described by means of the concept of 1-currents. This work is a continuation of a previous paper, where a preliminary result in this direction was obtained, with the true Euler equations replaced by a vector valued non linear PDE with a mollified Biot-Savart relation. [ABSTRACT FROM AUTHOR]

From the known statistics of fiber-fiber contacts in random fiber networks, an analytic estimate is obtained for the variance of local porosity in random fiber suspensions and evolving filtrate networks. The variance of local porosity, and hence the distribution of projected areal density, seem to depend on fiber geometry only through the cube of mean diameter. Also, the coefficient of variation of local flow rate perpendicular to the plane of the pad is, to a first approximation, independent of the mode of flow. Analytic estimates are obtained also for the effect of fiber clumping on the variance of local porosity of pads for small inspection zones. [ABSTRACT FROM AUTHOR]

In this paper we study a family of nonlinear σ -models in which the target space is the super manifold H 2 | 2 N . These models generalize Zirnbauer’s H 2 | 2 nonlinear σ -model (Zirnbauer in Commun Math Phys 141(3):503–522, 1991). The latter model has a number of special features which aid in its analysis: through a remarkable technique from symplectic geometry colloquial known as supersymmetric localization, the partition function of the H 2 | 2 model is equal to one independent of the coupling constants. Our main technical observation is to generalize this fact to H 2 | 2 N models as follows: the partition function is a multivariate polynomial of degree n = N - 1 , increasing in each variable. As an application, these facts provide estimates on the Fourier and Laplace transforms of the ’t-field’ when we specialize to Z 2 . We show that this field has fluctuations which are at least those of a massless free field. In addition we show that small fractional moments of e t v - t 0 decay at least polynomially fast in the distance of v to 0. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the impact of bond-dilution disorder on the critical behavior of the stochastic SIR model. Monte Carlo simulations were conducted using square lattices with first- and second-nearest neighbor interactions. Quenched bond-diluted lattice disorder was introduced into the systems, allowing them to evolve over time. By employing percolation theory and finite-size scaling analysis, we estimate both the critical threshold and leading critical exponent ratios of the model for different bond-dilution rates (p). An examination of the average size of the percolating cluster and the size distribution of non-percolating clusters of recovered individuals was performed to ascertain the universality class of the model. The simulation results strongly indicate that the present model belongs to a new universality class distinct from that of 2D dynamical percolation, depending on the specific p value under consideration. [ABSTRACT FROM AUTHOR]

The justification of hydrodynamic limits in non-convex domains has long been an open problem due to the singularity at the grazing set. In this paper, we investigate the unsteady neutron transport equation in a general bounded domain with the in-flow, diffuse-reflection, or specular-reflection boundary condition. Using a novel kernel estimate, we demonstrate the optimal L 2 diffusive limit in the presence of both initial and boundary layers. Previously, this result was only proved for convex domains when the time variable is involved. Our approach is highly robust, making it applicable to all basic types of physical boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global clustering coefficient (a.k.a transitivity) and clique number of graphs generated by a preferential attachment random graph model with an additional feature of allowing edge connections between existing vertices. Specifically, at each time step t, either a new vertex is added with probability f(t), or an edge is added between two existing vertices with probability 1 - f (t) . We establish concentration inequalities for the global clustering and clique number of the resulting graphs under the assumption that f(t) is a regularly varying function at infinity with index of regular variation - γ , where γ ∈ [ 0 , 1) . We also demonstrate an inverse relation between these two statistics: the clique number is essentially the reciprocal of the global clustering coefficient. [ABSTRACT FROM AUTHOR]

Unlike the single species gases, the transport coefficients such as Fick, Soret, Dufour coefficients arise in the hydrodynamic limit of multi-species gas mixtures. To the best of the authors’ knowledge, no multi-component relaxational models is reported that produces all these values correctly. In this paper, we establish the existence of unique stationary mild solutions to the BGK models for gas mixtures which produces the correct Fick coefficients in the Navier–Stokes limit for inert gases (Brull in Eur J Mech B 33:74–86, 2012), and for reactive gases (Brull and Schneider in Commun Math Sci 12(7):1199–1223, 2014) in a unified manner. [ABSTRACT FROM AUTHOR]

Local quantum Bernoulli noises (LQBNs) is a family of local annihilation operators and local creation operators acting on Bernoulli functions. In this paper, based on LQBNs, we define the local quantum joint entropy and local quantum coherence information. And we find local quantum entropy has a surprising property, which is not satisfies the additivity for tensor product state. Furthermore, we give some properties of local quantum coherent information based on LQBNs. [ABSTRACT FROM AUTHOR]

Unlike the case for classical particles, the literature on BGK type models for relativistic gas mixture is extremely limited. There are a few results in which such relativistic BGK models for gas mixture are employed to compute transport coefficients. However, to the best knowledge of authors, relativistic BGK models for gas mixtures with complete presentation of the relaxation operators are missing in the literature. In this paper, we fill this gap by suggesting a BGK model for relativistic gas mixtures for which the existence of each equilibrium coefficients in the relaxation operator is rigorously guaranteed in a way that all the essential physical properties are satisfied such as the conservation laws, the H-theorem, the capturing of the correct equilibrium state, the indifferentiability principle, and the recovery of the classical BGK model in the Newtonian limit. [ABSTRACT FROM AUTHOR]

Granular gases demand models capable of capturing their distinct characteristics. The widely employed inelastic hard-sphere model (IHSM) introduces complexities that are compounded when incorporating realistic features like surface roughness and rotational degrees of freedom, resulting in the more intricate inelastic rough hard-sphere model (IRHSM). This paper focuses on the inelastic rough Maxwell model (IRMM), presenting a more tractable alternative to the IRHSM and enabling exact solutions. Building on the foundation of the inelastic Maxwell model (IMM) applied to granular gases, the IRMM extends the mathematical representation to encompass surface roughness and rotational degrees of freedom. The primary objective is to provide exact expressions for the Navier–Stokes–Fourier transport coefficients within the IRMM, including the shear and bulk viscosities, the thermal and diffusive heat conductivities, and the cooling-rate transport coefficient. In contrast to earlier approximations in the IRHSM, our study unveils inherent couplings, such as shear viscosity to spin viscosity and heat conductivities to counterparts associated with a torque-vorticity vector. These exact findings provide valuable insights into refining the Sonine approximation applied to the IRHSM, contributing to a deeper understanding of the transport properties in granular gases with realistic features. [ABSTRACT FROM AUTHOR]

This paper establishes the large deviation principle (LDP) of certain types of nonconventional ergodic averages, namely, 1 N S N ∗ and 1 N S N # on N (defined later). The LDP for both averages are presented and such a result extends the preceding work of (Carinci et al. in Indag Math 23(3):589–602, 2012) to some specific cases of d-multiple averages for d ≥ 3 . [ABSTRACT FROM AUTHOR]

Studying independent sets of maximum size is equivalent to considering the hard-core model with the fugacity parameter λ tending to infinity. Finding the independence ratio of random d-regular graphs for some fixed degree d has received much attention both in random graph theory and in statistical physics. For d ≥ 20 the problem is conjectured to exhibit 1-step replica symmetry breaking (1-RSB). The corresponding 1-RSB formula for the independence ratio was confirmed for (very) large d in a breakthrough paper by Ding, Sly, and Sun. Furthermore, the so-called interpolation method shows that this 1-RSB formula is an upper bound for each d ≥ 3 . For d ≤ 19 this bound is not tight and full-RSB is expected. In this work we use numerical optimization to find good substituting parameters for discrete r-RSB formulas ( r = 2 , 3 , 4 , 5 ) to obtain improved rigorous upper bounds for the independence ratio for each degree 3 ≤ d ≤ 19 . As r grows, these formulas get increasingly complicated and it becomes challenging to compute their numerical values efficiently. Also, the functions to minimize have a large number of local minima, making global optimization a difficult task. [ABSTRACT FROM AUTHOR]

We consider bootstrap percolation and diffusion in sparse random graphs with fixed degrees, constructed by configuration model. Every vertex has two states: it is either active or inactive. We assume that to each vertex is assigned a nonnegative (integer) threshold. The diffusion process is initiated by a subset of vertices with threshold zero which consists of initially activated vertices, whereas every other vertex is inactive. Subsequently, in each round, if an inactive vertex with threshold θ has at least θ of its neighbours activated, then it also becomes active and remains so forever. This is repeated until no more vertices become activated. The main result of this paper provides a central limit theorem for the final size of activated vertices. Namely, under suitable assumptions on the degree and threshold distributions, we show that the final size of activated vertices has asymptotically Gaussian fluctuations. [ABSTRACT FROM AUTHOR]

In this paper, we study phase transitions of a class of time-inhomogeneous diffusion processes associated with the φ 4 model. We prove that when γ < 0 , the system has no phase transition and when γ > 0 , the system has a phase transition and we study the phase transition of the system through qualitative and quantitative methods. We further show that, as the strength of the mean field tends to 0, the solution and stationary distribution of such system converge locally uniformly in L 2 and Wasserstein distance respectively to those of corresponding system without mean field. [ABSTRACT FROM AUTHOR]

The paper is devoted to study of radially symmetric solutions to the spatially homogeneous Landau kinetic equation for Coulomb forces and related model equations. Two kinds of kinetic models for the Landau equation are introduced in order to understand a possibility of blow-up of solution. They differ by a diffusion coefficient. The model of the first kind is exactly solvable. Its global in time solution is constructed and studied. The model of the second type is more complicated. The power moments (in velocities) for this model and for the Landau equations are studied. The propagation in time of the exponential moment of the third order is proved for solutions of the Landau equation. Informally speaking, this means that a typical high-energy tail for solutions having indata with compact support looks like exp [ - b (t) | v | k ] with some k ≥ 3 . In particular, this means that a lower bound with Maxwellian tail exp [ - a (t) | v | 2 ] is impossible for solutions of the Landau equation [ABSTRACT FROM AUTHOR]

In this paper, for supersymmetric quantum integrable spin chains with rational G l (N | M) -invariant R -matrices, we construct a coupled master T -operator which represents a generating function for two-folds commuting quantum transfer matrices. We show that the functional relations for the quantum transfer matrices are equivalent to an infinite set of Hirota bilinear equations of the modified universal character hierarchy. Also the free fermion representation of the tau function of the supersymmetric quantum two-component spin chains will be given with the help of two sets of Clifford algebras. [ABSTRACT FROM AUTHOR]

In this paper we analyze the metastable behavior for the Ising model that evolves under Kawasaki dynamics on the hexagonal lattice H 2 in the limit of vanishing temperature. Let Λ ⊂ H 2 a finite set which we assume to be arbitrarily large. Particles perform simple exclusion on Λ , but when they occupy neighboring sites they feel a binding energy - U < 0 . Along each bond touching the boundary of Λ from the outside to the inside, particles are created with rate ρ = e - Δ β , while along each bond from the inside to the outside, particles are annihilated with rate 1, where β is the inverse temperature and Δ > 0 is an activity parameter. For the choice Δ ∈ (U , 3 2 U) we prove that the empty (resp. full) hexagon is the unique metastable (resp. stable) state. We determine the asymptotic properties of the transition time from the metastable to the stable state and we give a description of the critical configurations. We show how not only their size but also their shape varies depending on the thermodynamical parameters. Moreover, we emphasize the role that the specific lattice plays in the analysis of the metastable Kawasaki dynamics by comparing the different behavior of this system with the corresponding system on the square lattice. [ABSTRACT FROM AUTHOR]

The coupling through both drag force and volume fraction (of gas) of a kinetic equation of Vlasov type and a system of Euler or Navier–Stokes type (in which the volume fraction explicity appears) leads to the so-called thick sprays equations. Those equations are used to describe sprays (droplets or dust specks in a surrounding gas) in which the volume fraction of the disperse phase is non negligible. As for other multiphase flows systems, the issues related to the linear stability around homogeneous solutions is important for the applications. We show in this paper that this stability indeed holds for thick sprays equations, under physically reasonable assumptions. The analysis which is performed makes use of Lyapunov functionals for the linearized equations. [ABSTRACT FROM AUTHOR]

We derive macroscopic equations for a generalized contact process that is inspired by a neuronal integrate and fire model on the lattice Z d . The states at each lattice site can take values in 0 , … , k . These can be interpreted as neuronal membrane potential, with the state k corresponding to a firing threshold. In the terminology of the contact processes, which we shall use in this paper, the state k corresponds to the individual being infectious (all other states are noninfectious). In order to reach the firing threshold, or to become infectious, the site must progress sequentially from 0 to k. The rate at which it climbs is determined by other neurons at state k, coupled to it through a Kac-type potential, of range γ - 1 . The hydrodynamic equations are obtained in the limit γ → 0 . Extensions of the microscopic model to include excitatory and inhibitory neuron types, as well as other biophysical mechanisms, are also considered. [ABSTRACT FROM AUTHOR]

This is a comment on our recent paper (Grela et al. in J Stat Phys 183:49, 2021). In this comment we provide an easier derivation of the effective Langevin equation for vicious Brownian bridges in the flat-to-flat geometry. This derivation shows that it is not necessary to invoke the intermediate step of mapping to a Dyson Brownian bridge. The result can be directly derived using the Karlin–McGregor formula. [ABSTRACT FROM AUTHOR]

In this paper we present a numerical study of a mathematical model of spiking neurons introduced by Ferrari et al. (J Stat Phys 172(6):1564–1575, 2018). In this model we have a countable number of neurons linked together in a network, each of them having a membrane potential taking value in the integers, and each of them spiking over time at a rate which depends on the membrane potential through some rate function ϕ . Beside being affected by a spike each neuron can also be affected by leaking. At each of these leak times, which occur for a given neuron at a fixed rate γ , the membrane potential of the neuron concerned is spontaneously reset to 0. A wide variety of versions of this model can be considered by choosing different graph structures for the network and different activation functions. It was rigorously shown that when the graph structure of the network is the one-dimensional lattice with a hard threshold for the activation function, this model presents a phase transition with respect to γ , and that it also presents a metastable behavior. By the latter we mean that in the sub-critical regime the re-normalized time of extinction converges to an exponential random variable of mean 1. It has also been proven that in the super-critical regime the renormalized time of extinction converges in probability to 1. Here, we investigate numerically a richer class of graph structures and activation functions. Namely we investigate the case of the two dimensional and the three dimensional lattices, as well as the case of a linear function and a sigmoid function for the activation function. We present numerical evidence that the result of metastability in the sub-critical regime holds for these graphs and activation functions as well as the convergence in probability to 1 in the super-critical regime. [ABSTRACT FROM AUTHOR]

This paper gives an affirmative answer to the question of the global existence of Boltzmann equations without angular cutoff in the L ∞ -setting. In particular, we show that when the initial data is close to equilibrium and the perturbation is small in L 2 ∩ L ∞ with a polynomial decay tail, the Boltzmann equation has a unique global solution in the weighted L 2 ∩ L ∞ -space. In order to overcome the difficulties arising from the singular cross-section and the low regularity, a De Giorgi type argument is crafted in the kinetic context with the help of the averaging lemma. More specifically, we use a strong averaging lemma to obtain suitable L p -estimates for level-set functions. These estimates are crucial for constructing an appropriate energy functional to carry out the De Giorgi argument. Similar as in Alonso et al. (Rev Mat Iberoam, 2020), we extend local solutions to global ones by using the spectral gap of the linearised Boltzmann operator. The convergence to the equilibrium state is then obtained as a byproduct with relaxations shown in both L 2 and L ∞ -spaces. [ABSTRACT FROM AUTHOR]

We consider the frog model with geometric lifetime (parameter 1 - p ) on homogeneous trees of dimension d. In 2002, Alves et al.(Electron J Probab 7:21, 2002) proved that there exists a critical lifetime parameter p c ∈ (0 , 1) above which infinitely many frogs are activated with positive probability, and they gave lower and upper bounds for p c . Since then, the literature on this model focussed on refinements of the upper bound. In the present paper, we improve the bounds for p c on both sides. We also provide a discussion comparing the bounds of the literature and their proofs. Our proofs are based on coupling. [ABSTRACT FROM AUTHOR]

Our previous work presented explicit formulas for the generalized zeta function and the generalized Ihara zeta function corresponding to the Grover walk and the positive-support version of the Grover walk on the regular graph via the Konno–Sato theorem, respectively. This paper extends these walks to a class of walks including random walks, correlated random walks, quantum walks, and open quantum random walks on the torus by the Fourier analysis. [ABSTRACT FROM AUTHOR]

In this paper, we study periodic solutions in distribution of mean-field stochastic differential equations. We introduce the notion of upper and lower solutions of mean-field stochastic differential equations. With the help of the comparison principle, we prove the existence of periodic solutions in distribution. [ABSTRACT FROM AUTHOR]

The two-body stochastic dealer model is revisited to provide an exact solution to the average order-book profile using the kinetic approach. The dealer model is a microscopic financial model where individual traders make decisions on limit-order prices stochastically and then reach agreements on transactions. In the literature, this model was solved for several cases: an exact solution for two-body traders N = 2 and a mean-field solution for many traders N ≫ 1 . Remarkably, while kinetic theory plays a significant role in the mean-field analysis for N ≫ 1 , its role is still elusive for the case of N = 2 . In this paper, we revisit the two-body dealer model N = 2 to clarify the utility of the kinetic theory. We first derive the exact master-Liouville equations for the two-body dealer model. We next illustrate the physical picture of the master-Liouville equation from the viewpoint of the probability currents. The master-Liouville equations are then solved exactly to derive the order-book profile and the average transaction interval. Furthermore, we introduce a generalised two-body dealer model by incorporating interaction between traders via the market midprice and exactly solve the model within the kinetic framework. We finally confirm our exact solution by numerical simulations. This work provides a systematic mathematical basis for the econophysics model by developing better mathematical intuition. [ABSTRACT FROM AUTHOR]

For original Kuramoto models with nonidentical oscillators, it is impossible to realize complete phase synchronization. However, this paper reveals that complete phase synchronization can be achieved for a large class of high-dimensional Kuramoto models with nonidentical oscillators. Under the topology of strongly connected digraphs, a necessary and sufficient condition for complete phase synchronization is proposed. Under the condition, an open set in the region of synchronization attraction is obtained, and the limit set of the system trajectories is derived. Finally, some simulations are provided to validate the obtained theoretic results. [ABSTRACT FROM AUTHOR]

We study the self-similar solutions and related questions for the modified Boltzmann equation. This equation formally coincides with the classical spatially homogeneous Boltzmann equation in the presence of an artificial force term proportional to a matrix A. The modification is connected with applications to homoenergetic solutions of the spatially inhomogeneous Boltzmann equation. Our study is restricted to the case of Maxwell-type interactions. We investigate existence and uniqueness of self-similar solutions and their role as attractors for large values of time. Similar questions were studied recently under assumption of sufficient smallness of norm ‖ A ‖ without explicit estimates of that smallness. In this paper we fill this gap and prove, in particular, that all important facts related to self-similar solutions remain valid for moderately small values ‖ A ‖ = O (10 - 1) in appropriate dimensionless units. [ABSTRACT FROM AUTHOR]

In this paper, we construct the global-in-time solutions of the Cauchy problem to the Vlasov–Maxwell–Landau system with Coulomb potential near Maxwellians and strong uniform background magnetic field. Our analysis is motivated by the nonlinear energy method developed recently in Guo (Indiana Univ Math J 53(4):1081–1094, 2004), Liu et al. (Physica D 188(3):178–192, 2004), Liu and Yu (Commun Math Phys 246(1):133–179, 2004) for the Boltzmann equation and the interpolation method in negative Sobolev space developed recently in Guo and Wang (Commun Partial Differ Equ 37(12):2165–2208, 2012). [ABSTRACT FROM AUTHOR]

In this paper we continue to study small amplitude solutions of the damped cubic NLS equation, driven by a random force [the study was initiated in our previous work Dymov and Kuksin (Commun Math Phys 382:951–1014, 2021) and continued in Dymov et al. (The large-period limit for equations of discrete turbulence 2021, arXiv:2104.11967)]. We write solutions of the equation as formal series in the amplitude and discuss the behaviour of this series under the wave turbulence limit, when the amplitude goes to zero, while the space-period goes to infinity. [ABSTRACT FROM AUTHOR]

The truncated Korteweg–De Vries (TKdV) system—a shallow-water wave model with Hamiltonian structure that exhibits weakly turbulent dynamics—has been found to accurately predict the anomalous wave statistics observed in recent laboratory experiments. Majda et al. (Proc Natl Acad Sci 116(10):3982–3987, 2019) developed a TKdV statistical mechanics framework based on a mixed Gibbs measure that is supported on a surface of fixed energy (microcanonical) and takes the usual canonical form in the Hamiltonian. This paper reports two rigorous results regarding the surface-displacement distributions implied by this ensemble, both in the limit of the cutoff wavenumber Λ growing large. First, we prove that if the inverse temperature vanishes, displacement statistics converge to Gaussian as Λ → ∞ . Second, we prove that if nonlinearity is absent and the inverse-temperature satisfies a certain physically-motivated scaling law, then displacement statistics converge to Gaussian as Λ → ∞ . When the scaling law is not satisfied, simple numerical examples demonstrate symmetric, yet highly non-Gaussian, displacement statistics to emerge in the linear system, illustrating that nonlinearity is not a strict requirement for non-normality in the fixed-energy ensemble. The new results, taken together, imply necessary conditions for the anomalous wave statistics observed in previous numerical studies. In particular, non-vanishing inverse temperature and either the presence of nonlinearity or the violation of the scaling law are required for displacement statistics to deviate from Gaussian. The proof of this second theorem involves the construction of an approximating measure, which we find also elucidates the peculiar spectral decay observed in numerical studies and may open the door for improved sampling algorithms. [ABSTRACT FROM AUTHOR]

The main aim of the present paper is to prove the existence of a phase transition in quantum Markov chain (QMC) scheme for the Ising type models on a Cayley tree. Note that this kind of models do not have one-dimensional analogous, i.e. the considered model persists only on trees. In this paper, we provide a more general construction of forward QMC. In that construction, a QMC is defined as a weak limit of finite volume states with boundary conditions, i.e. QMC depends on the boundary conditions. Our main result states the existence of a phase transition for the Ising model with competing interactions on a Cayley tree of order two. By the phase transition we mean the existence of two distinct QMC which are not quasi-equivalent and their supports do not overlap. We also study some algebraic property of the disordered phase of the model, which is a new phenomena even in a classical setting. [ABSTRACT FROM AUTHOR]

In this paper we consider a stochastic system with two connected nodes, whose unidirectional connection is variable and depends on point processes associated to each node. The input node is represented by an homogeneous Poisson process, whereas the output node jumps with an intensity that depends on the jumps of the input nodes and the connection intensity. We study a scaling regime when the rate of both point processes is large compared to the dynamics of the connection. In neuroscience, this system corresponds to a neural network composed by two neurons, connected by a single synapse. The strength of this synapse depends on the past activity of both neurons, the notion of synaptic plasticity refers to the associated mechanism. A general class of such stochastic models has been introduced in Robert and Vignoud (Stochastic models of synaptic plasticity in neural networks, 2020, arxiv: 2010.08195) to describe most of the models of long-term synaptic plasticity investigated in the literature. The scaling regime corresponds to a classical assumption in computational neuroscience that cellular processes evolve much more rapidly than the synaptic strength. The central result of the paper is an averaging principle for the time evolution of the connection intensity. Mathematically, the key variable is the point process, associated to the output node, whose intensity depends on the past activity of the system. The proof of the result involves a detailed analysis of several of its unbounded additive functionals in the slow-fast limit, and technical results on interacting shot-noise processes. [ABSTRACT FROM AUTHOR]

We analyze the free energy and the overlaps in the 2-spin spherical Sherrington Kirkpatrick spin glass model with an external field for the purpose of understanding the transition between this model and the one without an external field. We compute the limiting values and fluctuations of the free energy as well as three types of overlaps in the setting where the strength of the external field goes to zero as the dimension of the spin variable grows. In particular, we consider overlaps with the external field, the ground state, and a replica. Our methods involve a contour integral representation of the partition function along with random matrix techniques. We also provide computations for the matching between different scaling regimes. Finally, we discuss the implications of our results for susceptibility and for the geometry of the Gibbs measure. Some of the findings of this paper are confirmed rigorously by Landon and Sosoe in their recent paper which came out independently and simultaneously. [ABSTRACT FROM AUTHOR]

In this paper we study coupled fast-slow ordinary differential equations (ODEs) with small time scale separation parameter ε such that, for every fixed value of the slow variable, the fast dynamics are sufficiently chaotic with ergodic invariant measure. Convergence of the slow process to the solution of a homogenized stochastic differential equation (SDE) in the limit ε to zero, with explicit formulas for drift and diffusion coefficients, has so far only been obtained for the case that the fast dynamics evolve independently. In this paper we give sufficient conditions for the convergence of the first moments of the slow variable in the coupled case. Our proof is based upon a new method of stochastic regularization and functional-analytical techniques combined via a double limit procedure involving a zero-noise limit as well as considering ε to zero. We also give exact formulas for the drift and diffusion coefficients for the limiting SDE. As a main application of our theory, we study weakly-coupled systems, where the coupling only occurs in lower time scales. [ABSTRACT FROM AUTHOR]