Related papers: A Leibniz rule of distributional pairing and hyper…
A generalization of the Gibbs-von Neumann relative entropy is proposed based on the quantum BBGKY [Bogolyubov-Born-Green-Kirkwood-Yvon] hierarchy as the nonequilibrium entropy for an N-body system. By using a generalization of the…
We study quantum and classical many-body Hamiltonian systems that combine integrable contact interactions with generic long-range two-body potentials. We show that the dynamics of local observables can be cast into a generalized…
We apply the results established in arXiv:2109.15263 to prove some new fractional Leibniz rules involving $BV^{\alpha,p}$ and $S^{\alpha,p}$ functions, following the distributional approach adopted in the previous works arXiv:1809.08575,…
Starting from the BBGKY hierarchy, describing the kinetics of nonlinear particle system, we obtain the relevant entropy and stationary distribution function. Subsequently, by employing the Lorentz transformations we propose the relativistic…
As follows from the Schwartz Impossibility Theorem, multiplication of two distributions is in general impossible. Nevertheless, often one needs to multiply a distribution by a discontinuous function, not by an arbitrary distribution. In the…
A solution to the BBGKY hierarchy for nonequilibrium distribution functions is obtained within modified boundary conditions. The boundary conditions take into account explicitly both the nonequilibrium one-particle distribution function as…
A BBGKY-like hierarchy is derived from the non-equilibrium Redfield equation. Two further approximations are introduced and each can be used to truncate and solve the hierarchy. In the first approximation such a truncation is performed by…
Theoretical description and simulation of large quantum coherent systems out of equilibrium remains a daunting task. Here we are developing a new approach to it based on the Pechukas-Yukawa formalism, which is especially convenient in case…
In classical kinetic theory, the BBGKY hierarchy is an infinite chain of integro-differential equations that describes the full time-reversal-invariant (Liouville) system of interacting (quasi)-particles in terms of $N$-particle…
A generalization of the Gibbs entropy postulate is proposed based on the BBGKY hierarchy as the nonequilibrium entropy for a system of N interacting particles. This entropy satisfies the basic principles of thermodynamics in the sense that…
Let $L$ be the Dunkl Laplacian on the Euclidean space $\mathbb{R}^N$ associated with a normalized root system $R$ and a multiplicity function $k(\nu)\geq 0$, $\nu\in R$. We establish a Leibniz-type rule for the fractional powers of $L$ on…
We prove a Leibniz rule for BV functions in a complete metric space that is equipped with a doubling measure and supports a Poincar\'e inequality. Unlike in previous versions of the rule, we do not assume the functions to be locally…
A duality formula, of the Hardy and Littlewood type for multidimensional Gaussian sums, is proved in order to estimate the asymptotic long time behavior of distribution of Birkhoff sums $S_n$ of a sequence generated by a skew product…
We implement a general numerical calculation that allows for a direct comparison between nonlinear Hamiltonian dynamics and the Boltzmann-Gibbs canonical distribution in Gibbs $\Gamma$-space. Using paradigmatic first-neighbor models,…
We present a derivation of the kinetic equation describing the secular evolution of spatially inhomogeneous systems with long-range interactions, the so-called inhomogeneous Landau equation, by relying on a functional integral formalism. We…
The fractional Leibniz rule is generalized by the Coifman-Meyer estimate. It is shown that the arbitrary redistribution of fractional derivatives for higher order with the corresponding correction terms.
In this paper, a sum rule means a relationship between a functional defined on a subset of all probability measures on $\mathbb{R}$ involving the reverse Kullback-Leibler divergence with respect to a particular distribution and recursion…
We propose a general approach, named by us hyperstatistics, to treat complex systems, in which Boltzmann-Gibbs statistics breaks down in domains of the system. Hyperstatistics preserves the concavity of nonadditive $q$-entropy. We obtain…
A well-defined variational principle for gravitational actions typically requires to cancel boundary terms produced by the variation of the bulk action with a suitable set of boundary counterterms. This can be achieved by carefully…
We establish a coboundary condition for a sequence of ergodic sums (i.e.~Birkhoff partial sums) to be almost surely uniformly distributed mod $1$. Applications are given when the sequence is generated by a Gibbs-Markov map. In particular,…