The inverse problem in classical statistical mechanics

We address the problem of whether there exists an external potential corresponding to a given equilibrium single particle density of a classical system. Results are established for both the canonical and grand canonical distributions. It is shown that for essentially all systems without hard core interactions, there is a unique external potential which produces any given density. The external potential is shown to be a continuous function of the density and, in certain cases, it is shown to be differentiable. As a consequence of the differentiability of the inverse map (which is established without reference to the hard core structure in the grand canonical ensemble), we prove the existence of the Ornstein-Zernike direct correlation function. A set of necessary, but not sufficient conditions for the solution of the inverse problem in systems with hard core interactions is derived.Work partially supported by NSF grant PHY-8117463Work partially supported by NSF grant PHY-8116101 A01     

Properties of the functional formalism and their application to the theory of classical fluids

A general formalism is derived relating any generating functional of a hierarchy of functions to some other functionals yieldingUrsell, Husimi, and similar expansions of the original hierarchy and vice versa. There are two expansions starting with an equation of the O.-Z. type. This formalism is applied to the grand partition function with an external potential which is a generating functional for the molecular distribution functions. When the external potential is induced by adding particles to the system we obtain several hierarchies of integral equations related to each other in a simple fashion. As the Kirkwood-Salsburg, Mayer-Montroll, Green equations, the P. Y., HNC and a HNC similar approximation with their extensions are special cases of these hierarchies the relations between them become transparent. At the same time the heuristic feature in the choice of functionals and independent functions in earlier derivations of some of these equations is removed.     

Density matrix for particle production processes in an external field

The general problem of finding the density matrices of particles and antiparticles produced by an external electromagnetic field fron an arbitrary initial state is solved. Expressions are obtained for the density matrices of particles and antiparticles produced from a state with given number of initial particles and from a state described by an initial grand canonical ensemble.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fiz., No. 6, pp. 77–81, June, 1980.     

The classical fluid of associating hard rods in an external field

Two models of one-dimensional fluids of associating hard rods in an arbitrary external field are investigated. In the first model particles can only form dimers, while in the second model, which has been solved previously by Percus, aggregates of any size coexist. In both cases the grand canonical potential and the external potential are found exactly as functionals of the density. It is shown that Wertheim's thermodynamic perturbation theory of polymerization provides a straightforward route to the exact solution by expanding the functional space to include more density parameters. This suggests that Wertheim's theory should be used also for studying the structure (and not only the thermo-dynamics) of real associating fluids.     

Pressures for a One-Component Plasma on a Pseudosphere

The classical (i.e., non-quantum) equilibrium statistical mechanics of a two-dimensional one-component plasma (a system of charged point-particles embedded in a neutralizing background) living on a pseudosphere (an infinite surface of constant negative curvature) is considered. In the case of a flat space, it is known that, for a one-component plasma, there are several reasonable definitions of the pressure, and that some of them are not equivalent to each other. In the present paper, this problem is revisited in the case of a pseudosphere. General relations between the different pressures are given. At one special temperature, the model is exactly solvable in the grand canonical ensemble. The grand potential and the one-body density are calculated in a disk, and the thermodynamic limit is investigated. The general relations between the different pressures are checked on the solvable model.     

Wall-Induced Density Profiles and Density Correlations in Confined Takahashi Lattice Gases

We propose a general formalism to study the static properties of a system composed of particles with nearest neighbor interactions that are located on the sites of a one-dimensional lattice confined by walls (confined Takahashi lattice gas). Linear recursion relations for generalized partition functions are derived, from which thermodynamic quantities, as well as density distributions and correlation functions of arbitrary order can be determined in the presence of an external potential. Explicit results for density profiles and pair correlations near a wall are presented for various situations. As a special case of the Takahashi model we consider in particular the hard rod lattice gas, for which a system of nonlinear coupled difference equations for the occupation probabilities has been presented by Robledo and Varea. A solution of these equations is given in terms of the solution of a system of independent linear equations. Moreover, for zero external potential in the hard-rod system we specify various central regions between the confining walls, where the occupation probabilities are constant and the correlation functions are translationally invariant in the canonical ensemble. In the grand canonical ensemble such regions do not exist.     

Statistical mechanics of fluids at spherical structureless walls

Statistical mechanical theories of spherical fluid interfaces are discussed in the context of fluids in contact with structureless walls. The thermodynamic route to the surface tension leads to a formula involving gradients of the external field, which is especially suited to the study of fluid-wall systems. The surface tension is found to be determined by the curvature dependence of the density in the region of the wall. For hard walls, potential distribution theory is used to obtain the exact relationship between the statistical mechanical surface tension expression and the grand potential. The accuracy of simple scaled particle theory calculations of the surface tension is estimated from predictions for the equation of state of pair potential fluids with hard core plus attractive tail interactions. Problems with the mechanical route to the curvature dependence of the surface tension are discussed. The planar wall and results for lower dimensionality are included in appendices.     

Chain formation in a monolayer of dipolar hard spheres under an external field

The phase behavior of a monolayer of dipolar hard spheres under an external field, which makes all dipoles of the monolayer orientate along its direction, is investigated. Using integral equation theory in the reference hypernetted chain (RHNC) approximation we calculate the correlation functions, which are used to obtain the response matrix of grand potential with respect to density fluctuations. The smallest eigenvalue of this response matrix determines the stability of the monolayer. When the smallest ei...     

Null surface quantization and quantum theory of massless fields in asymptotically flat space-time

The quantum theory of massless fields in an asymptotically simple space-time is developed. The Schwinger dynamical principle and the Penrose conformal technique are exploited to derive the commutation relations on proper null surfaces in a curved space-time and on null infinities. The explicit expression for theS matrix in an asymptotically simple space-time is presented. The general expression for a density matrix describing particles created in an external field is also given and its possible applications are discussed briefly.     

The thermodynamic limit for an imperfect Boson gas

We give a rigorous treatment in the infinite volume limit of a model Hamiltonian representing an imperfect Boson gas. In particular we obtain the exact expression for the mean particle density in the infinite volume limit as a function of the chemical potential, and show that the density function has a singularity at the critical density for Bose-Einstein condensation. We prove that, unlike the ideal Boson gas, the imperfect Boson gas has the same behaviour in the infinite volume limit for the grand canonical ensemble as for the canonical ensemble, and is moreover stable under small perturbations. We finally exhibit the possibility of ordinary condensation and prove that a system in an intermediate situation between two pure phases consists of a simple mixture of the two phases involved.     

Confined Coulomb Systems with Adsorbing Boundaries: The Two-Dimensional Two-Component Plasma

Using a solvable model, the two-dimensional two-component plasma, we study a Coulomb gas confined in a disk and in an annulus with boundaries that can adsorb some of the negative particles of the system. We obtain explicit analytic expressions for the grand potential, the pressure and the density profiles of the system. By studying the behavior of the disjoining pressure we find that without the adsorbing boundaries the system is naturally unstable, while with attractive boundaries the system is stable because of a positive contribution from the surface tension to the disjoining pressure. The results for the density profiles show the formation of a positive layer near the boundary that screens the adsorbed negative particles, a typical behavior in charged systems. We also compute the adsorbed charge on the boundary and show that it satisfies a certain number of relations, in particular an electro-neutrality sum rule.     

Thermodynamics of DBI Black Holes in Anti-de Sitter Spacetime

Through the gauge field theory, we obtain the solution of the DBI-AdS black hole. In the meantime, according to the relations between the action and the grand partition function, we obtain the grand partition function in the DBI-AdS black hole. The temperature and the potential of the DBI-AdS black hole are gained from differential of the grand partition function. With the thermodynamic relations, other thermodynamics are also obtained. The solution and the thermodynamics of the DBI-AdS black hole are turned out that they can reduce to the case of a charged black hole in four-dimensional spacetimes.     

Thermodynamics of particle systems in the presence of external macroscopic fields

We study the grand partition function of a system of identical particles interacting via a superstable potential in the presence of an external field depending on a scale factor. We discuss the case when the scale factor increases to infinity (macroscopic limit for the external potential) and we prove rigorously a link between the so obtained pressure and the usual one (barometric formula).Research partially supported by the Consiglio Nazionale delle Ricerche.     

Universal thermodynamics of degenerate quantum gases in the unitarity limit

We perform a systematic study of the thermodynamics of quantum gases in the unitarity limit. Our study is based on a "universality hypothesis" for the relevant energy scales which is supported by experiments and can be proven in the Boltzmann regime. It implies a universal form for the grand potential, which is specified by only a few universal numbers in the degenerate limit. This hypothesis provides a simple way to determine the density profile of a trapped fermion superfluid. It implies a superfluid bump in the density and gives the general expression of the second sound velocity of a homogeneous superfluid at unitarity.     

A microscopic theory of the lambda transition

Starting with a microscopic hamiltonian for a many-boson system with a hardcore interaction, the grand potential of the system, which contains the order-parameter of the lambda transition as one of the thermodynamical variables, is derived by making use of the finite temperature loop expansion. The divergence difficulty caused by the hardcore interaction is circumvented by the conventional field theoretic perturbational renormalization such that the chemical potential is renormalized instead of the conventional mass renormalization. The grand potential obtained consists of the superfluid part and the finite temperature elementary excitation part. The elementary excitation energy spectrum shows the Goldstone boson mode, namely, the photon, for the zero external field. A non-vanishing external field destroys such a Goldstone boson mode by causing an energy gap at zero momentum. The chemical potential and the critical temperature are also obtained for the weak coupling case. It is shown how the Bose-Einstein condensation is affected by the hardcore interaction.     

Thermodynamics of a 4‐site Hubbard model by analytical diagonalization

By use of the conservation laws a four‐site Hubbard model coupled to a particle bath within an external magnetic field in z‐direction was diagonalized. The analytical dependence of both the eigenvalues and the eigenstates on the interaction strength, the chemical potential and magnetic field was calculated. It is demonstrated that the low temperature behaviour is determined by a delicate interplay between many‐particle states differing in electron number and spin if the electron density is away from half‐filling. The grand partition sum is calculated and the specific heat, the susceptibility as well as various correlation functions and spectral functions are given in dependence of the interaction strength, the electron occupation and the applied magnetic field. For both the grand canonical and the canonical ensemble the high‐temperature crossing points of the specific heat are calculated. Whereas in the weak correlation regime the universal value calculated by second order perturbation theory for several Hubbard systems being in the thermodynamic limit is confirmed, these crossing points vanish for intermediate to strong correlation.     

Mean field and the confined single homopolymer

We develop a statistical model for a confined homopolymeric chain molecule based on a monomer grand ensemble representation. The molecule is subject to a confining external field, a backbone interaction, and an attractive interaction between any pair of monomers. An exact minimum principle for the thermodynamics of the backbone in an external field is obtained, and a controlled mean field approximation results in a modified minimum principle from which relevant physical quantities such as monomer density can be found. We explore the limit in which the chain is subject to tight confinement, and make a preliminary investigation of a prototypical system.     

Mobility in a periodic potential: A simple derivation

A simple derivation is given for the mobility of Brownian particles in a periodic potential in the overdamped regime. The method makes use of the fact that the steady state current density, in response to a uniform external force, is uniform in space and can be expressed as a product of the particle density and mean velocity field. To lowest order in the external force, the particle density is given by the equilibrium density in the absence of the external force.     

表面沸腾中接触角对液体过热度的影响

表面沸腾中接触角对液体过热度的影响刘朝，曾丹苓（重庆大学热力工程系重庆６３００４４）关键词液体过热度；接触角；涨落；统计系综１引言液体极限过热度的研究是研究沸腾热物理过程的基础。更为重要的是，液体的过热现象直接涉及到许多工程问题和技术安全问题。在实际...