Statistical mechanics of magnetic non magnetic transitions of rare earth ions in metals

Using a simple model, and by means of a phenomenological extension of the Green's function method, it has been possible to give a more precise mathematical footing to the ideas put forward by Hirst, Maple, Wohleben and Coles regarding the demagnetization of rare earth ions in metals.     

Surface-induced phase transformations: multiple scale and mechanics effects and morphological transitions

Strong, surprising, and multifaceted effects of the width of the external surface layer Δ(ξ) and internal stresses on surface-induced pretransformation and phase transformations (PTs) are revealed. Using our further developed phase-field approach, we found that above some critical Δ(ξ)(*), a morphological transition from fully transformed layer to lack of surface pretransformation occurs for any transformation strain ε(t). It corresponds to a sharp transition to the universal (independent of ε(t)), strongly increasing the master relationship of the critical thermodynamic driving force for PT X(c) on Δ(ξ). For large ε(t), with increasing Δ(ξ), X(c) unexpectedly decreases, oscillates, and then becomes independent of ε(t). Oscillations are caused by morphological transitions of fully transformed surface nanostructure. A similar approach can be developed for internal surfaces (grain boundaries) and for various types of PTs and chemical reactions.     

A characterization of first-order phase transitions for superstable interactions in classical statistical mechanics

We give bounds on finite-volume expectations for a set of boundary conditions containing the support of any tempered Gibbs state and prove a theorem connecting the behavior of Gibbs states to the differentiability of the pressure for continuum statistical mechanical systems with long-range superstable potentials. Convergence of grand canonical Gibbs states is also studied.     

Statistical mechanics of viscoelasticity

A unified treatment of viscoelasticity is developed in the framework of non-equilibrium classical statistical mechanics. An exact correspondence between Mori's continued-fractions formalism and rheological circuits is shown.     

Statistical mechanics of interfaces

On the basis of van der Waals theory for interfaces we evaluate explicitly the small-wavevector behavior of the pair correlation function along an interface. A correction to the density profile is also found. The results obtained are in full accord with capillary wave theory when wave amplitudes are regarded as small.     

Statistical mechanics of money

In a closed economic system, money is conserved. Thus, by analogy with energy, the equilibrium probability distribution of money must follow the exponential Boltzmann-Gibbs law characterized by an effective temperature equal to the average amount of money per economic agent. We demonstrate how the Boltzmann-Gibbs distribution emerges in computer simulations of economic models. Then we consider a thermal machine, in which the difference of temperatures allows one to extract a monetary profit. We also discuss the role of debt, and models with broken time-reversal symmetry for which the Boltzmann-Gibbs law does not hold. The instantaneous distribution of money among the agents of a system should not be confused with the distribution of wealth. The latter also includes material wealth, which is not conserved, and thus may have a different (e.g. power-law) distribution. Received 22 June 2000     

Statistical mechanics of structural fluctuations

The theory of mesoscopic fluctuations is applied to inhomogeneous solids consisting of chaotically distributed regions with different crystalline structure. This approach makes it possible to describe statistical properties of such mixture by constructing a renormalized Hamiltonian. The relative volumes occupied by each of the coexisting structures define the corresponding geometric probabilities. In the case of a frozen heterophase system these probabilities should be given a priori. And in the case of a thermal heterophase mixture the structural probabilities are to be defined self-consistently by minimizing a thermodynamical potential. This permits to find the temperature behavior of the probabilities which is especially important near the points of structural phase transitions. The presence of these structural fluctuations yields a softening of a crystal and a decrease of the effective Debye temperature. These effects can be directly seen by nuclear gamma resonance since the occurrence of structural fluctuations is accompanied by a noticeable sagging of the Mössbauer factor at the point of structural phase transition. The structural fluctuations also lead to the attenuation of sound and increase of isothermic compressibility.     

Statistical mechanics of colliding beams

The observed performance of electron-positron colliding-beam storage rings is poorly understood theoretically. The problem of a storage ring's behavior is a particular instance of statistical mechanics in an external environment that varies periodically with time, in the limit of weak—and not necessarily isotropic—friction and (additive) noise. As a practical starting point for a general theory of such problems, the following ansatz is suggested: Phase space submanifolds (tori) with fixed canonical actions are manifolds of approximately equal probability density. Such an approach is especially well suited to analysis of the long-time effects of nonlinear resonance on storage ring behavior. Formal consequences of this ansatz and some associated conceptual difficulties are discussed. These issues are considered from the standpoint of “two-time” analysis. An example of a concrete application, closely related to Kramers' analysis of noise-induced barrier crossing, is provided. This paper is meant to be self-contained, so that it can be understood by readers outside the storage-ring community.     

Statistical mechanics of steiner trees

The minimum weight Steiner tree (MST) is an important combinatorial optimization problem over networks that has applications in a wide range of fields. Here we discuss a general technique to translate the imposed global connectivity constrain into many local ones that can be analyzed with cavity equation techniques. This approach leads to a new optimization algorithm for MST and allows us to analyze the statistical mechanics properties of MST on random graphs of various types.     

Statistical mechanics of relativistic plasmas

Based on the Klimontovich method of construction of a relativistic statistical mechanics the binary correlation function of a relativistic plasma is considered. Up to the order e⁴ for spatially homogeneous systems the relativistic binary correlation function can be expressed by the relativistic dielectric tensor, where higher-order correlation functions are neglected.In the case of thermodynamic equilibrium the relativistic dielectric tensor is calculated. The corresponding binary correlation function is determined up to the first relativistic order. The result is compared with those of Trubnikov/Kosachev (1968) and Krizan (1969).     

Statistical mechanics of convex bodies

The difficulties inherent in the construction of two-dimensional pressure ensembles are discussed, and are tackled by defining an energy cost depending on the convex hull of the set of particles. An energy proportional to the area of the convex hull is not able to prevent evaporation of the system, whereas an energy proportional to the area of the circumcircle of the convex hull ensures a thermodynamic behavior. In the latter model, which turns out to be exactly solvable, various characterizations are given of the geometry of a typical state.     

Statistical mechanics of elastic moduli

We consider continuous systems of particles in the framework of classical statistical mechanics and derive a general expression for the static elastic moduli tensor in terms of correlation functions. We find sufficient conditions for the vanishing of the shear modulus. Relationships between these conditions and others insuring translational or rotational invariance are discussed.     

Statistical mechanics of superfluid models

It is shown that the equilibrium thermodynamic properties of some superfluid models can be calculated by using density matrices defined in suitable irreducible representations. The approach enables one to introduce in a fundamental manner temperature-dependent order parameters and can be used to obtain general formulations of the problems of superfluids.     

Statistical mechanics of semiflexible polymers

We present the statistical-mechanical theory of semiflexible polymers based on the connection between the Kratky-Porod model and the quantum rigid rotator in an external homogeneous field, and treatment of the latter using the quantum mechanical propagator method. The expressions and relations existing for flexible polymers can be generalized to semiflexible ones, if one replaces the Fourier-Laplace transform of the end-to-end polymer distance, 1/(k ²/3 + p), through the matrix , where D and M are related to the spectrum of the quantum rigid rotator, and considers an appropriate matrix element of the expression under consideration. The present work provides also the framework to study polymers in external fields, and problems including the tangents of semiflexible polymers. We study the structure factor of the polymer, the transversal fluctuations of a free end of the polymer with fixed tangent of another end, and the localization of a semiflexible polymer onto an interface. We obtain the partition function of a semiflexible polymer in half space with Dirichlet boundary condition in terms of the end-to-end distribution function of the free semiflexible polymer, study the behaviour of a semiflexible polymer in the vicinity of a surface, and adsorption onto a surface.Received: 23 March 2004, Published online: 23 July 2004PACS: 36.20.-r Macromolecules and polymer molecules - 61.41. + e Polymers, elastomers, and plastics - 82.35.Gh Polymers on surfaces; adhesion     

Statistical mechanics of classical systems

An algebra of thermodynamic operators of the fluctuations of physical quantities in classical statistical mechanics is found and its properties studied. A method is proposed for obtaining equations that describe the equilibrium and nonequilibrium statistical ensembles of classical systems.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 6–11, May, 1980.