On quantum analogue of dynamical stabilisation of inverted harmonic oscillator by time periodical uniform field

Quantum analogue of stabilised forced oscillations around an unstable equilibrium position is explored by solving the non-stationary Schrödinger equation (NSE) of the inverted harmonic oscillator (IHO) driven periodically by spatial uniform field of frequency \(\Omega \), amplitude \(F_{0}\) and phase \(\phi \), i.e. the system with the Hamiltonian of \(\hat{{H}}=(\hat{{p}}^{2}/2m)-(m\omega ^{2}x^{2}/2)-F_0 x\sin \) \(\left( {\Omega t+\phi } \right) \). The NSE has been solved both analytically and numerically by Maple 15 in dimensionless variables \(\xi = x\sqrt{m\omega /\hbar }\hbox {, }f_0 =F_0 /\omega \sqrt{\hbar m\omega }\) and \(\tau =\omega t\). The initial condition (IC) has been specified by the wave function (w.f.) of a generalised Gaussian type which suits well the corresponding quantum IC operator. The solution obtained demonstrates the non-monotonous behaviour of the coordinate spreading \(\sigma \left( \tau \right) \hbox { =}\sqrt{\big ( {\overline{\Delta \xi ^{2}\big ( \tau \big )} } \big )}\) which decreases first from quite macroscopic values of \(\sigma _{0} =2^{12,\ldots ,25}\) to minimal one of \(\sim \!(1/\sqrt{2})\) at times \(\tau <\tau _0 =0.125\ln \!\left( {16\sigma _0^4 +1} \right) \) and then grows back unlimitedly. For certain phases \(\phi \) depending on the \(\Omega /\omega \) ratio and \(n=\log _2\!\sigma _0 \), the mass centre of the packet \(\xi _{\mathrm {av}}( \tau )= \overline{\hat{{x}}(\tau )} \cdot \sqrt{m\omega /\hbar }\) delays approximately two natural ‘periods’ \(\sim \!(4\pi /\omega )\) in the area of the stationary point and then escapes to ‘\(+\)’ or ‘?’ infinity in a bifurcating way.  For ‘resonant’ \(\Omega =\omega \), the bifurcation phases \(\phi \) fit well with the regression formula of Fermi–Dirac type of argument n with their asymptotic \(\phi ( {\Omega ,n\rightarrow \infty } )\) obeying the classical formula \(\phi _{\mathrm {cl}} ( \Omega )=-\hbox {arctg} \, \Omega \) for initial energy \(E = 0\) in the wide range of \(\Omega =2^{-4},...,2^{7}\).     

Heuristic Relative Entropy Principles with Complex Measures: Large-Degree Asymptotics of a Family of Multi-variate Normal Random Polynomials

Let \(z\in \mathbb {C}\), let \(\sigma ^2>0\) be a variance, and for \(N\in \mathbb {N}\) define the integrals

$$\begin{aligned} E_N^{}(z;\sigma ) := \left\{ \begin{array}{ll} {\frac{1}{\sigma }} \!\!\!\displaystyle \int _{\mathbb {R}}\! (x^2+z^2) \frac{e^{-\frac{1}{2\sigma ^2} x^2}}{\sqrt{2\pi }}dx&{}\quad \text{ if }\, N=1,\\ {\frac{1}{\sigma }} \!\!\!\displaystyle \int _{\mathbb {R}^N}\! \prod \prod \limits _{1\le k<l\le N}\!\! e^{-\frac{1}{2N}(1-\sigma ^{-2}) (x_k-x_l)^2} \prod _{1\le n\le N}\!\!\!\!(x_n^2+z^2) \frac{e^{-\frac{1}{2\sigma ^2} x_n^2}}{\sqrt{2\pi }}dx_n &{}\quad \text{ if }\, N>1. \end{array}\right. \!\!\! \end{aligned}$$

These are expected values of the polynomials \(P_N^{}(z)=\prod _{1\le n\le N}(X_n^2+z^2)\) whose 2N zeros \(\{\pm i X_k\}^{}_{k=1,\ldots ,N}\) are generated by N identically distributed multi-variate mean-zero normal random variables \(\{X_k\}^{N}_{k=1}\) with co-variance \(\mathrm{{Cov}}_N^{}(X_k,X_l)=(1+\frac{\sigma ^2-1}{N})\delta _{k,l}+\frac{\sigma ^2-1}{N}(1-\delta _{k,l})\). The \(E_N^{}(z;\sigma )\) are polynomials in \(z^2\), explicitly computable for arbitrary N, yet a list of the first three \(E_N^{}(z;\sigma )\) shows that the expressions become unwieldy already for moderate N—unless \(\sigma = 1\), in which case \(E_N^{}(z;1) = (1+z^2)^N\) for all \(z\in \mathbb {C}\) and \(N\in \mathbb {N}\). (Incidentally, commonly available computer algebra evaluates the integrals \(E_N^{}(z;\sigma )\) only for N up to a dozen, due to memory constraints). Asymptotic evaluations are needed for the large-N regime. For general complex z these have traditionally been limited to analytic expansion techniques; several rigorous results are proved for complex z near 0. Yet if \(z\in \mathbb {R}\) one can also compute this “infinite-degree” limit with the help of the familiar relative entropy principle for probability measures; a rigorous proof of this fact is supplied. Computer algebra-generated evidence is presented in support of a conjecture that a generalization of the relative entropy principle to signed or complex measures governs the \(N\rightarrow \infty \) asymptotics of the regime \(iz\in \mathbb {R}\). Potential generalizations, in particular to point vortex ensembles and the prescribed Gauss curvature problem, and to random matrix ensembles, are emphasized.

     

Two Phases of the Non-Commutative Quantum Mechanics with the Generalized Uncertainty Relations

We consider the quantum mechanics on the noncommutative plane with the generalized uncertainty relations \({\Delta } x_{1} {\Delta } x_{2} \ge \frac {\theta }{2}, {\Delta } p_{1} {\Delta } p_{2} \ge \frac {\bar {\theta }}{2}, {\Delta } x_{i} {\Delta } p_{i} \ge \frac {\hbar }{2}, {\Delta } x_{1} {\Delta } p_{2} \ge \frac {\eta }{2}\). We show that the model has two essentially different phases which is determined by \(\kappa = 1 + \frac {1}{\hbar ^{2} } (\eta ^{2} - \theta \bar {\theta })\). We construct a operator \(\hat {\pi }_{i}\) commuting with \(\hat {x}_{j} \) and discuss the harmonic oscillator model in two dimensional non-commutative space for three case κ > 0, κ = 0, κ < 0. Finally, we discuss the thermodynamics of a particle whose hamiltonian is related to the harmonic oscillator model in two dimensional non-commutative space.     

Fully constrained Majorana neutrino mass matrices using $$\varvec{\varSigma (72\times 3)}$$

In 2002, two neutrino mixing ansatze having trimaximally mixed middle (\(\nu _2\)) columns, namely tri-chi-maximal mixing (\(\text {T}\chi \text {M}\)) and tri-phi-maximal mixing (\(\text {T}\phi \text {M}\)), were proposed. In 2012, it was shown that \(\text {T}\chi \text {M}\) with \(\chi =\pm \,\frac{\pi }{16}\) as well as \(\text {T}\phi \text {M}\) with \(\phi = \pm \,\frac{\pi }{16}\) leads to the solution, \(\sin ^2 \theta _{13} = \frac{2}{3} \sin ^2 \frac{\pi }{16}\), consistent with the latest measurements of the reactor mixing angle, \(\theta _{13}\). To obtain \(\text {T}\chi \text {M}_{(\chi =\pm \,\frac{\pi }{16})}\) and \(\text {T}\phi \text {M}_{(\phi =\pm \,\frac{\pi }{16})}\), the type I see-saw framework with fully constrained Majorana neutrino mass matrices was utilised. These mass matrices also resulted in the neutrino mass ratios, \(m_1:m_2:m_3=\frac{\left( 2+\sqrt{2}\right) }{1+\sqrt{2(2+\sqrt{2})}}:1:\frac{\left( 2+\sqrt{2}\right) }{-1+\sqrt{2(2+\sqrt{2})}}\). In this paper we construct a flavour model based on the discrete group \(\varSigma (72\times 3)\) and obtain the aforementioned results. A Majorana neutrino mass matrix (a symmetric \(3\times 3\) matrix with six complex degrees of freedom) is conveniently mapped into a flavon field transforming as the complex six-dimensional representation of \(\varSigma (72\times 3)\). Specific vacuum alignments of the flavons are used to arrive at the desired mass matrices.     

Holography of massive M2-brane theory: non-linear extension

We investigate the gauge/gravity duality between the \(\mathcal{N} = 6\) mass-deformed ABJM theory with \(\hbox {U}_k(N)\times \hbox {U}_{-k}(N)\) gauge symmetry and the 11-dimensional supergravity on LLM geometries with SO(2,1)\(\times \)SO(4)/\({\mathbb {Z}}_k\) \(\times \)SO(4)/\({\mathbb {Z}}_k\) isometry, in terms of a KK holography, which involves quadratic order field redefinitions. We establish the quadratic order KK mappings for various gauge invariant fields in order to obtain the canonical 4-dimensional gravity equations of motion and to reduce the LLM solutions to an asymptotically AdS\(_4\) gravity solutions. The non-linearity of the KK maps indicates that we can observe the true purpose of the non-linear KK holography of the LLM solutions. We read the vacuum expectation value of conformal dimension two operator from the asymptotically AdS\(_4\) gravity solutions. For the LLM solutions which are represented by square-shaped Young diagrams, we compare the vacuum expectation value obtained from the holographic procedure with the result obtained from the field theory, which is given by \(\langle \mathcal{O}^{(\Delta =2)}\rangle =\sqrt{k}N^{\frac{3}{2}}f_{(\Delta =2)}+\mathcal{O}(N)\), where \(f_{\Delta }\) is independent of N. Based on this result, we examine the gauge/gravity duality in the large N limit and finite k. We also show that the vacuum expectation values of the massive KK graviton modes are vanishing as expected by the supersymmetry.     

Domain and Range of the Modified Wave Operator for Schrödinger Equations with a Critical Nonlinearity

We study the final problem for the nonlinear Schrödinger equation

$i{\partial }_{t}u+\frac{1}{2}\Delta u=\lambda|u|^{\frac{2}{n}}u,\quad (t,x)\in {\mathbf{R}}\times \mathbf{R}^{n},$

where\(\lambda \in{\bf R},n=1,2,3\). If the final data\(u_{+}\in {\bf H}^{0,\alpha }=\left\{ \phi \in {\bf L}^{2}:\left( 1+\left\vert x\right\vert \right) ^{\alpha }\phi \in {\bf L}^{2}\right\} \) with\(\frac{ n}{2} < \alpha < \min \left( n,2,1+\frac{2}{n}\right) \) and the norm\(\Vert \widehat{u_{+}}\Vert _{{\bf L}^{\infty }}\) is sufficiently small, then we prove the existence of the wave operator in L ². We also construct the modified scattering operator from H ^0,α to H ^0,δ with\(\frac{n}{2} < \delta < \alpha\).

     

Thermodynamic implications of the gravitationally induced particle creation scenario

A rigorous thermodynamic analysis has been done as regards the apparent horizon of a spatially flat Friedmann–Lemaitre–Robertson–Walker universe for the gravitationally induced particle creation scenario with constant specific entropy and an arbitrary particle creation rate \(\Gamma \). Assuming a perfect fluid equation of state \(p=(\gamma -1)\rho \) with \(\frac{2}{3} \le \gamma \le 2\), the first law, the generalized second law (GSL), and thermodynamic equilibrium have been studied, and an expression for the total entropy (i.e., horizon entropy plus fluid entropy) has been obtained which does not contain \(\Gamma \) explicitly. Moreover, a lower bound for the fluid temperature \(T_f\) has also been found which is given by \(T_f \ge 8\left( \frac{\frac{3\gamma }{2}-1}{\frac{2}{\gamma }-1}\right) H^2\). It has been shown that the GSL is satisfied for \(\frac{\Gamma }{3H} \le 1\). Further, when \(\Gamma \) is constant, thermodynamic equilibrium is always possible for \(\frac{1}{2}<\frac{\Gamma }{3H} < 1\), while for \(\frac{\Gamma }{3H} \le \text {min}\left\{ \frac{1}{2},\frac{2\gamma -2}{3\gamma -2} \right\} \) and \(\frac{\Gamma }{3H} \ge 1\), equilibrium can never be attained. Thermodynamic arguments also lead us to believe that during the radiation phase, \(\Gamma \le H\). When \(\Gamma \) is not a constant, thermodynamic equilibrium holds if \(\ddot{H} \ge \frac{27}{4}\gamma ^2 H^3 \left( 1-\frac{\Gamma }{3H}\right) ^2\), however, such a condition is by no means necessary for the attainment of equilibrium.     

Electrical conductivity in single crystals of GaSe<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>Te<Subscript>1 – <Emphasis Type="Italic">x</Emphasis></Subscript> solid solutions in strong electrical fields

This paper presents some results of studying the Poole–Frenkel effect with allowance for shielding in layered GaSe and GaTe single crystals and their solid solutions in strong electrical fields of up to 10⁵ V/cm at temperatures of 103–250 K. According to the relationship \(\left(\frac{\sigma}{\sigma(0)}\right)^{1/2}\) log\(\frac{\sigma}{\sigma(0)} = E\sqrt{\frac{\varepsilon}{4\pi n(0)kT}}\), there exists a linear dependence between \(\left(\frac{\sigma}{\sigma(0)}\right)^{1/2}\) log\(\frac{\sigma}{\sigma(0)}\) and the electrical field E (σ is the electrical conductivity in strong electrical fields, and σ(0) is the electrical conductivity in the ohmic region). The slopes of these lines have been determined at different temperatures (103–250 K) by estimating the concentration of current carriers n(0) = 3 × 10¹³–5 × 10¹⁵ cm^–3 in the ohmic region of the electrical conductivity of solid solutions of layered GaSe _x Te_1–x single crystals (x = 1.00, 0.95, 0.90, 0.80, 0.70, 0.30, 0.20, 0.10, 0).     

The strong decays of <Emphasis Type="Italic">X</Emphasis>(3940) and <Emphasis Type="Italic">X</Emphasis>(4160)

The new mesons X(3940) and X(4160) have been found by Belle Collaboration in the processes \(e^+e^-\rightarrow J/\psi D^{(*)}{\bar{D}}^{(*)}\). Considering X(3940) and X(4160) as \(\eta _c(3S)\) and \(\eta _c(4S)\) states, the two-body open charm OZI-allowed strong decay of \(\eta _c(3S)\) and \(\eta _c(4S)\) are studied by the improved Bethe–Salpeter method combined with the \(^3P_0\) model. The strong decay width of \(\eta _c(3S)\) is \(\Gamma _{\eta _c(3S)}=(33.5^{+18.4}_{-15.3})\) MeV, which is close to the result of X(3940); therefore, \(\eta _c(3S)\) is a good candidate of X(3940). The strong decay width of \(\eta _c(4S)\) is \(\Gamma _{\eta _c(4S)}=(69.9^{+22.4}_{-21.1})\) MeV, considering the errors of the results, it is close to the lower limit of X(4160). But the ratio of the decay width \(\frac{\Gamma (D{\bar{D}}^*)}{\Gamma (D^*{\bar{D}}^*)}\) of \(\eta _c(4S)\) is larger than the experimental data of X(4160). According to the above analysis, \(\eta _c(4S)\) is not the candidate of X(4160), and more investigations of X(4160) is needed.     

A new decay mode of higher charmonium

We calculate the \(\Lambda _c{\bar{\Lambda }}_c\) partial decay width of the excited vector charmonium states around 4.6 GeV with the quark pair creation model. We find that the partial decay width of the \(\Lambda _c{\bar{\Lambda }}_c\) mode can reach up to several MeV for \(\psi (4S,~5S,~6S)\). In contrast, the partial \(\Lambda _c{\bar{\Lambda }}_c\) decay width of the states \(\psi (3D,~4D,~5D)\) is less than one MeV. If the enhancement Y(4630) reported by the Belle Collaboration in \(\Lambda _c{\bar{\Lambda }}_c\) invariant-mass distribution is the same structure as Y(4660), the Y(4660) resonance is most likely to be a S-wave charmonium state.     

Higher Spin Dirac Operators Between Spaces of Simplicial Monogenics in Two Vector Variables

The higher spin Dirac operator \(\mathcal{Q}_{k,l}\) acting on functions taking values in an irreducible representation space for \(\mathfrak{so}(m)\) with highest weight \((k+\frac{1}{2},l+\frac{1}{2},\frac{1}{2},\ldots,\frac{1}{2})\), with k, l?∈?\(\mathbb{N}\) and \(k\geqslant l\), is constructed. The structure of the kernel space containing homogeneous polynomial solutions is then also studied.     

A New Set of Limiting Gibbs Measures for the Ising Model on a Cayley Tree

For the Ising model (with interaction constant J>0) on the Cayley tree of order k≥2 it is known that for the temperature T≥T _c,k =J/arctan?(1/k) the limiting Gibbs measure is unique, and for T<T _c,k there are uncountably many extreme Gibbs measures. In the Letter we show that if \(T\in(T_{c,\sqrt{k}}, T_{c,k_{0}})\), with \(\sqrt{k} then there is a new uncountable set \({\mathcal{G}}_{k,k_{0}}\) of Gibbs measures. Moreover \({\mathcal{G}}_{k,k_{0}}\ne {\mathcal{G}}_{k,k'_{0}}\), for k ₀≠k′₀. Therefore if \(T\in (T_{c,\sqrt{k}}, T_{c,\sqrt{k}+1})\), \(T_{c,\sqrt{k}+1} then the set of limiting Gibbs measures of the Ising model contains the set {known Gibbs measures}\(\cup(\bigcup_{k_{0}:\sqrt{k}.     

New feature of low $$p_{T}$$ charm quark hadronization in pp collisions at $$\sqrt{s}=7$$s=7 TeV

Treating the light-flavor constituent quarks and antiquarks whose momentum information is extracted from the data of soft light-flavor hadrons in pp collisions at \(\sqrt{s}=7\) TeV as the underlying source of chromatically neutralizing the charm quarks of low transverse momenta (\(p_{T}\)), we show that the experimental data of \(p_{T}\) spectra of single-charm hadrons \(D^{0,+}\), \(D^{*+}\) \(D_{s}^{+}\), \(\varLambda _{c}^{+}\) and \(\varXi _{c}^{0}\) at mid-rapidity in the low \(p_{T}\) range (\(2\lesssim p_{T}\lesssim 7\) GeV/c) in pp collisions at \(\sqrt{s}=7\) TeV can be well understood by the equal-velocity combination of perturbatively created charm quarks and those light-flavor constituent quarks and antiquarks. This suggests a possible new scenario of low \(p_{T}\) charm quark hadronization, in contrast to the traditional fragmentation mechanism, in pp collisions at LHC energies. This is also another support for the exhibition of the soft constituent quark degrees of freedom for the small parton system created in pp collisions at LHC energies.     

Phase Transition and Uniqueness of Levelset Percolation

The main purpose of this paper is to introduce and establish basic results of a natural extension of the classical Boolean percolation model (also known as the Gilbert disc model). We replace the balls of that model by a positive non-increasing attenuation function \(l:(0,\infty ) \rightarrow [0,\infty )\) to create the random field \(\Psi (y)=\sum _{x\in \eta }l(|x-y|),\) where \(\eta \) is a homogeneous Poisson process in \({\mathbb {R}}^d.\) The field \(\Psi \) is then a random potential field with infinite range dependencies whenever the support of the function l is unbounded. In particular, we study the level sets \(\Psi _{\ge h}(y)\) containing the points \(y\in {\mathbb {R}}^d\) such that \(\Psi (y)\ge h.\) In the case where l has unbounded support, we give, for any \(d\ge 2,\) a necessary and sufficient condition on l for \(\Psi _{\ge h}(y)\) to have a percolative phase transition as a function of h. We also prove that when l is continuous then so is \(\Psi \) almost surely. Moreover, in this case and for \(d=2,\) we prove uniqueness of the infinite component of \(\Psi _{\ge h}\) when such exists, and we also show that the so-called percolation function is continuous below the critical value \(h_c\).     

A comparison on absorption coefficients for secondary electron emission obtained from two different formulas

In this work, the absorption coefficients for secondary electron emißsion, α and β, that appeared respectively in the two different formulas, \(\delta (E_p ) = k\int_0^\infty {\left( {\frac{{dE}}{{dz}}} \right)E_p \exp ( - \alpha z)dz} \) and \(\delta (E_p ) = k\int_0^\infty {\left( {\frac{{dE}}{{dz}}} \right)E_p \exp ( - \alpha z)dz} \), were derived with a standard deviation rate analysis method based on a Monte Carlo simulated secondary electron yield, δ(E_p). Both the energy dissipation in depth for primary electrons, \(\left( {dE/dz} \right)E_p \), and the depth distribution for the number of secondary electrons including cascade electrons, n(z, E_p), were obtained by the same Monte Carlo method, in which the discrete inelastic scattering model was employed. The calculation results for Cu and Mg show that the n(z, E_p)-curve differs significantly from the \(\left( {dE/dz} \right)E_p \)-curve, and thus as well as a from b, for varied incidence angles (0°–80°) and low-energy primary electrons (up to 3 keV). The absorption coefficient β-values derived from a realistic depth distribution of cascade secondary electrons, n(z, E_p), then describe more accurately the nature of attenuation behavior of secondary electrons than a-values that associated with the approximate formula.     

Measurement of CP asymmetries in neutralino production at the ILC

We study the prospects to measure the CP-sensitive triple-product asymmetries in neutralino production \(e^{+} e^{-} \to\tilde{\chi}^{0}_{i}\tilde{\chi}^{0}_{1}\) and subsequent leptonic two-body decays \(\tilde{\chi}^{0}_{i} \to \tilde{\ell}_{R} \ell\), \(\tilde{\ell}_{R} \to \tilde{\chi}^{0}_{1} \ell\), for ?=e,μ, within the Minimal Supersymmetric Standard Model. We include a full detector simulation of the International Large Detector for the International Linear Collider. The simulation was performed at a center-of-mass energy of \(\sqrt{s}=500\) GeV, including the relevant Standard Model background processes, a realistic beam energy spectrum, beam backgrounds and a beam polarization of 80% and ?60% for the electron and positron beams, respectively. In order to effectively disentangle different signal samples and reduce SM and SUSY backgrounds we apply a method of kinematic reconstruction. Assuming an integrated luminosity of 500 fb^?1 collected by the experiment and the performance of the current ILD detector, we arrive at a relative measurement accuracy of 10% for the CP-sensitive asymmetry in our scenario. We demonstrate that our method of signal selection using kinematic reconstruction can be applied to a broad class of scenarios and it allows disentangling processes with similar kinematic properties.     

Phenomenology of light- and strange-quark simultaneous production at high energies

This letter presents an extension of EPL116(2017)62001 to light- and strange-quark nonequilibrium chemical phase-space occupancy factors (γ_q,s). The resulting damped trigonometric functionalities relating γ_q,s to the nucleon-nucleon center-of-mass energies (\(\sqrt {{s_{NN}}} \)) looks very similar except different coefficients. The phenomenology of the resulting γ_q,s(\(\sqrt {{s_{NN}}} \)) describes a rapid decrease at \(\sqrt {{s_{NN}}} \) ? 7GeV followed by a faster increase up to ~20 GeV. Then, both γ_q,s become nonsensitive to \(\sqrt {{s_{NN}}} \). Although these differ from γ _s (\(\sqrt {{s_{NN}}} \))obtained at γ _q (\(\sqrt {{s_{NN}}} \))=1, various particle ratios including K⁺/π⁺, K^?/π^?, Λ/π^?, Λ?/π^?, Ξ⁺/π⁺, and Ω/π^?, can well be reproduced, as well. We conclude that γ_q,s(\(\sqrt {{s_{NN}}} \)) should be instead determined from fits of various particle yields and ratios but not merely from fits to the particle ratio K⁺/π⁺.     

Summability of Connected Correlation Functions of Coupled Lattice Fields

We consider two nonindependent random fields \(\psi \) and \(\phi \) defined on a countable set Z. For instance, \(Z=\mathbb {Z}^d\) or \(Z=\mathbb {Z}^d\times I\), where I denotes a finite set of possible “internal degrees of freedom” such as spin. We prove that, if the cumulants of \(\psi \) and \(\phi \) enjoy a certain decay property, then all joint cumulants between \(\psi \) and \(\phi \) are \(\ell _2\)-summable in the precise sense described in the text. The decay assumption for the cumulants of \(\psi \) and \(\phi \) is a restricted \( \ell _1\) summability condition called \(\ell _1\)-clustering property. One immediate application of the results is given by a stochastic process \(\psi _t(x)\) whose state is \(\ell _1\)-clustering at any time t: then the above estimates can be applied with \(\psi =\psi _t\) and \(\phi =\psi _0\) and we obtain uniform in t estimates for the summability of time-correlations of the field. The above clustering assumption is obviously satisfied by any \(\ell _1\)-clustering stationary state of the process, and our original motivation for the control of the summability of time-correlations comes from a quest for a rigorous control of the Green–Kubo correlation function in such a system. A key role in the proof is played by the properties of non-Gaussian Wick polynomials and their connection to cumulants     

Equilibration in the Kac Model Using the GTW Metric $$d_2$$

We use the Fourier based Gabetta–Toscani–Wennberg metric \(d_2\) to study the rate of convergence to equilibrium for the Kac model in 1 dimension. We take the initial velocity distribution of the particles to be a Borel probability measure \(\mu \) on \(\mathbb {R}^n\) that is symmetric in all its variables, has mean \(\vec {0}\) and finite second moment. Let \(\mu _t(dv)\) denote the Kac-evolved distribution at time t, and let \(R_\mu \) be the angular average of \(\mu \). We give an upper bound to \(d_2(\mu _t, R_\mu )\) of the form \(\min \left\{ B e^{-\frac{4 \lambda _1}{n+3}t}, d_2(\mu ,R_\mu )\right\} ,\) where \(\lambda _1 = \frac{n+2}{2(n-1)}\) is the gap of the Kac model in \(L^2\) and B depends only on the second moment of \(\mu \). We also construct a family of Schwartz probability densities \(\{f_0^{(n)}: \mathbb {R}^n\rightarrow \mathbb {R}\}\) with finite second moments that shows practically no decrease in \(d_2(f_0(t), R_{f_0})\) for time at least \(\frac{1}{2\lambda }\) with \(\lambda \) the rate of the Kac operator. We also present a propagation of chaos result for the partially thermostated Kac model in Tossounian and Vaidyanathan (J Math Phys 56(8):083301, 2015).