arXiv subjects

Irina Navrotskaya

Publications and source records attributed to Irina Navrotskaya.

3 recordsLinked to original sources

arXiv2015

Some measure-theoretic properties of U-statistics applied in statistical physics

This paper investigates the relationship between various measure-theoretic properties of U-statistics with fixed sample size $N$ and the same properties of their kernels. Specifically, the random variables are replaced with elements in some measure space $(Λ; dx)$, the resultant real-valued functions on $Λ^N$ being called generalized $N$-means. It is shown that a.e. convergence of sequences, measurability, essential boundedness and, under certain conditions, integrability with respect to probability measures of generalized $N$-means and their kernels are equivalent. These results are crucial for the solution of the inverse problem in classical statistical mechanics in the canonical formulation.

math.CA↗

arXiv2015

Some measure-theoretic properties of generalized means

If $Λ$ is a measure space, $u:Λ^{m}\rightarrow \Bbb{R}$ is a given function and $N\geq m,$ the function $U(x_{1},...,x_{N})=\left( \begin{array}{l} N \\ m \end{array} \right) ^{-1}\sum_{1\leq i_{1}<\cdots <i_{m}\leq N}u(x_{i_{1}},...,x_{i_{m}}) $ is called the generalized $N$-mean with kernel $u,$ a terminology borrowed from $U$-statistics. Physical potentials for systems of particles are also defined by generalized means. This paper investigates whether various measure-theoretic concepts for generalized $N$-means are equivalent to the analogous concepts for their kernels: a.e. convergence of sequences, measurability, essential boundedness and integrability with respect to absolutely continuous probability measures. The answer is often, but not always, positive. This information is crucial in some problems addressing the existence of generalized means satisfying given conditions, such as the classical Inverse Problem of statistical physics (in the canonical ensemble).

math.FA↗

arXiv2015

The Classical Inverse Problem for Multi-Particle Densities in the Canonical Ensemble Formulation

We provide sufficient conditions for the solution of the classical inverse problem in the canonical distribution for multi-particle densities. Specifically, we show that there exists a unique potential in the form of a sum of m-particle (m greater then 1) interactions producing a given m-particle density. The existence and uniqueness of the solution to the multi-particle inverse problem is essential for the numerical simulations of matter using effective potentials derived from structural data. Such potentials are often employed in coarse- grained modeling. The validity of the multi-particle inverse conjecture also has implications for liquid state theory. For example, it provides the first step in proving the existence of the hierarchy of generalized Ornstein-Zernike relations. For the grand canonical distribution, the multi-particle inverse problem has been solved by Chayes and Chayes [J. Stat. Physics 36, 471-488 (1984)]. However, the setting of the canonical ensemble presents unique challenges arising from the impossibility of uncoupling interactions when the number of particles is fixed.

math-ph↗