A theory of pair connectedness is developed for fluid as well as lattice systems when the presence of physical clusters of particles in the system is explicitly taken into account. Activity and density expansions, an Ornstein-Zernicke relation and the Percus-Yevick approximation are established in analogy with the theory of the pair-correlation function. A simple application to the percolation problem is given; for a lattice the results are compared with the known solution of the Bethe lattice. For a fluid system the theory is used to investigate the relation of percolation and condensation in a Van der Waals gas, the result shows that an infinite cluster of particles is formed in the gaseous phase, along the coexistence curve, before the critical point is reached.

https://iopscience.iop.org/article/10.1088/0305-4470/10/7/011/pdf

Pair connectedness and cluster size

(2 + 1) [XTpXpH12, CTH11]. + [Zuc11b]. 0 [Fed17]. 1 [BELP15, CAS11, Cor16, Fed17, GDL10, GBL16, Hau16, JV19, KT12, KM19c, Li19, MN14b, Nak17, Pal11, Pan14, RT14, RBS16b, RY12, SS18c, Sug10, dOP18]. 1 + 1 [Sak18, CP15b]. 1/2 [MD10]. 1/f [FDR12]. 1/n [Per17]. 1/|x− y| [MSV10, MSV13]. 13 [DFL17]. 1 ≤ p ≤ ∞ [Dud13]. 1/f [HPF15]. 2 [AB19, ADS19, BF12, BNT13, DSS15, EKD12, Her13, Ily12, Lan10, Li12, Li19, LZ11, Ny13, Ost16, PSS16, ST14, Sch13b, TJ15, WPB15, dWL10]. 2 + 1 [dWL14]. 2.5 [BC15a]. 2R [WLEC17]. 2× 2 [CLTT13]. 3 [BCF19, BLS17, ESPP14, Kar18, SH16, SWKS14, dCCS19]. 3/2 [DK10]. 38 [Cam13]. 4 [BBS14, Zha14]. 4× 4 [LN19a]. 5/2 [DK10, EKD12]. 6 [EC11]. 8 [Zha14]. 90◦ [YM11]. 3 [Afz12]. 1−x [EFO11]. 13 [CDCL18]. 2 [ML15, QR13, ST11c]. 4 [HBB10]. 6 [BCL10a, BCL10b, EFO11]. x [EFO11].

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A Complete Bibliography of the Journal of Statistical Physics: 2000{2009

Large systems of interacting particles and the porous medium equation

The quantitative characterization of the arrangement of microstructural features is important for understanding the properties or function of materials or tissues. In this paper, the well‐known analysis of nearest neighbour distances is modified in order to separate clusters from a background of randomly disposed features and to characterize significant parameters of such sets. In addition, simple procedures are indicated to detect and to characterize regular arrangements, and computer simulated examples are used to demonstrate the efficiency of the suggested algorithms. Each step of the procedures can be used for three‐dimensional arrangements as well.

The characterization of the arrangement of feature centroids in planes and volumes

The determination of characteristic geometric properties of (usually 3-dimensional) objects by investigations of sections, projections, intersections with test sets, or other transformed images is a problem inherent to most of the experimental sciences. Surprisingly enough, formulas for such purposes have been established by people working in quite different fields of life, materials and earth sciences for over a hundred years until, some twenty years ago, the common theoretical (and in fact mathematical) background was realized. The new discipline dealing with problems of this type was called stereology. It was only then that mathematicians pointed out that most of the stereological results stem from classical formulas in integral geometry and that the validity of the formulas presupposes certain random structures either of the underlying sets or of the images taken of them. The development of stereological models therefore was greatly influenced by the growth of stochastic geometry. On the other hand, parts of stochastic geometry (random sets, point processes of convex bodies) have their roots in stereological questions. It seems, however, that stereology as an important field of applications of (convex) geometry is not as well-known to geometers as it should be. This was the motivation for the following survey which, although it is of an introductory nature, will present some of the recent developments with detailed, yet not exhaustive references. Since there are several geometric models which can serve as a basis of stereology, a lot of cross connections with other mathematical fields, and a variety of stereological problems of especial type, we had to restrict ourselves to a part of stereology which allows a unified treatment within a justifiable extent. The model, we present here, is based on convex geometry (methods from differential geometry and geometric measure theory are mentioned only occasionally) and the integral geometry of convex bodies (and unions thereof) is a crucial part of our considerations. The theory of random sets and point processes is investigated mainly in view of random versions of the kinematic integral formulas. We have, however, tried to be more complete with the references and give short comments to further developments and the corresponding literature at the end of each section.

Stereology: A Survey for Geometers

We consider classical systems of particles in R
 d
 interacting by a stable pair potential with finite range. We are engaged in subdividing every particle configuration into clusters of interacting particles and studying the cluster distributions corresponding to equilibrium particle distributions.

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Equilibrium distributions of physical clusters

We give the definition of Poisson point processes with exclusion by their local conditional distributions, treat the existence and uniqueness problem and their applications in percolation theory.

Poisson point processes with exclusion

We study the hydrodynamic behavior of a one-dimensional nearest neighbor gradient system with respect to a positive convex potential Φ. In the hydrodynamic limit the density distribution is shown to evolve according to the nonlinear diffusion equation δρ,(q)/δt= (δ2/dq2){F([1/ρ1(q)]), with F= −Φ.

The hydrodynamic limit of a one-dimensional nearest neighbor gradient system

We study the hydrodynamic limit of a deterministic one-dimensional particle system with nearest neighbour interaction and an additional regularizing force. Under its evolution mass and momentum are conserved. In the limit with Euler scaling their macroscopic distributions are shown to be governed by the compressible Navier–Stokes equations with a density dependent viscosity.

The Hydrodynamic Limit of a Deterministic Particle System with Conservation of Mass and Momentum

We prove that a version of the minimal entropy production principle holds rigorously for the nearest neighbor gradient system, whose hydrodynamic behavior we treated in an earlier paper, and study its relation to the macroscopic mass current and local equilibrium of higher order.

Michael Mürmann

Papers

Equilibrium distributions of physical clusters

Poisson point processes with exclusion

The hydrodynamic limit of a one-dimensional nearest neighbor gradient system

The Hydrodynamic Limit of a Deterministic Particle System with Conservation of Mass and Momentum

The nearest neighbor gradient system. A rigorous model for a version of the minimal entropy production principle