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The exponential family of random graphs is one of the most promising class of network models. Dependence between the random edges is defined through certain finite subgraphs, analogous to the use of potential energy to provide dependence between particle states in a grand canonical ensemble of statistical physics. By adjusting the specific values of these subgraph densities, one can analyze the influence of various local features on the global structure of the network. Loosely put, a phase transition occurs when a singularity arises in the limiting free energy density, as it is the generating function for the limiting expectations of all thermodynamic observables. We derive the full phase diagram for a large family of 3-parameter exponential random graph models with attraction and show that they all consist of a first order surface phase transition bordered by a second order critical curve., 14 pages, 8 figures

We explore differences between canonical and grand canonical ensembles of 500–2000 free electrons confined in a spherical well with a radius from 1.2 to 2 nm. The averaged occupation numbers of the electronic energy levels and their variances are calculated. For isolated Ag and Au particles, the sum of the variances of all occupation numbers differs from the corresponding bulk-metal value by a factor of 0.005 to 2.

We revisit the expansion recently proposed by Pulvirenti and Tsagkarogiannis for a system of $N$ continuous particles in the canonical ensemble. Under the sole assumption that the particles interact via a tempered and stable pair potential and are subjected to the usual free boundary conditions, we show the analyticity of the Helmholtz free energy at low densities and, using the Penrose tree graph identity, we establish a lower bound for the convergence radius which happens to be identical to the lower bound of the convergence radius of the virial series in the grand canonical ensemble established by Lebowitz and Penrose in 1964. We also show that the (Helmholtz) free energy can be written as a series in power of the density whose $k$ order coefficient coincides, modulo terms $o(N)/N$, with the $k$-order virial coefficient divided by $k+1$, according to its expression in terms of the $m$-order (with $m\le k+1$) simply connected cluster integrals first given by Mayer in 1942. We finally give an upper bound for the $k$-order virial coefficient which slightly improves, at high temperatures, the bound obtained by Lebowitz and Penrose.

We consider a system of multicolour disordered lattice gas, following closely the (monocolour) introduced by Faggionato and Martinelli(3,4). We study the projection on the monocolour system and we derive an estimate of the closeness between grand canonical and canonical Gibbs measures.

We consider the grand canonical pressure for Coulombic matter with nuclear charges ∼Zin a magnetic field Band at nonzero temperature. We prove that its asymptotic limit as Z→∞ with B/Z3→0 can be obtained by minimizing a Thomas–Fermi type pressure functional.

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.

We study a phase transition in a non-equilibrium model first introduced in ref. 5, using the Yang–Lee description of equilibrium phase transitions in terms of both canonical and grand canonical partition function zeros. The model consists of two different classes of particles hopping in opposite directions on a ring. On the complex plane of the diffusion rate we find two regions of analyticity for the canonical partition function of this model which can be identified by two different phases. The exact expressions for both distribution of the canonical partition function zeros and their density are obtained in the thermodynamic limit. The fact that the model undergoes a second-order phase transition at the critical point is confirmed. We have also obtained the grand canonical partition function zeros of our model numerically. The similarities between the phase transition in this model and the Bose–Einstein condensation has also been studied.

When described in a grand canonical ensemble, a finite Coulomb system exhibits charge fluctuations. These fluctuations are studied in the case of a classical (i.e., non-quantum) system with no macroscopic average charge. Assuming the validity of macroscopic electrostatics gives, on a three-dimensional finite large conductor of volume V, a mean square charge 〈Q 2〉 which goes as V 1/3. More generally, in a short-circuited capacitor of capacitance C, made of two conductors, the mean square charge on one conductor is 〈Q 2〉=TC, where T is the temperature and C the capacitance of the capacitor. The case of only one conductor in a grand canonical ensemble is obtained by removing the other conductor to infinity. The general formula is checked in the weak-coupling (Debye–Huckel) limit for a spherical capacitor. For two-dimensional Coulomb systems (with logarithmic interactions), there are exactly solvable models which reveal that, in some cases, macroscopic electrostatics is not applicable even for large conductors. This is when the charge fluctuations involve only a small number of particles. The mean square charge on one two-dimensional system alone, in the grand canonical ensemble, is expected to be, at most, one squared elementary charge.

Using physical arguments and partition theoretic methods, we demonstrate under general conditions, that the eigenvalues w(m) of the grand canonical density matrix decay rapidly with their index m, like w(m)∼exp[−βB−1(ln m)1+1/α], where B and α are positive constants, O(1), which may be computed from the spectrum of the Hamiltonian. We compute values of B and α for several physical models, and confirm our theoretical predictions with numerical experiments. Our results have implications in a variety of questions, including the behaviour of fluctuations in ensembles, and the convergence of numerical density matrix renormalization group techniques.

The system under consideration is a classical homopolymeric chain under an arbitrary external field, and in the grand canonical ensemble of its monomeric units. The ideal case of only symmetric next neighbor interactions is first analyzed in the relative density format. Arbitrary monomer-monomer interactions are introduced in a graphical perturbation series, and the leading order is expressed as a relative density functional with the aid of a sequence of redundant fields under whose variation the thermodynamic potential—here the excess grand potential—is stationary. Various reduction methods are suggested to compress the set of redundant fields of this overcomplete description.

The Arak process is a solvable stochastic process which generates coloured patterns in the plane. Patterns are made up of a variable number of random non-intersecting polygons. We show that the distribution of Arak process states is the Gibbs distribution of its states in thermodynamic equilibrium in the grand canonical ensemble. The sequence of Gibbs distributions forms a new model parameterised by temperature. We prove that there is a phase transition in this model, for some non-zero temperature. We illustrate this conclusion with simulation results. We measure the critical exponents of this off-lattice model and find they are consistent with those of the Ising model in two dimensions.

In the framework of the grand-canonical ensemble of statistical mechanics, we give an exact diagrammatic representation of the density profiles in a classical multicomponent plasma near a dielectric wall. By a reorganization of Mayer diagrams for the fugacity expansions of the densities, we exhibit how the long-range of both the self-energy and pair interaction are exponentially screened at large distances from the wall. However, the self-energy due to Coulomb interaction with images still diverges in the vicinity of the dielectric wall and the variation of the density is drastically different at short or large distances from the wall. This variation is involved in the inhomogeneous Debye-Huckel equation obeyed by the screened pair potential. Then the main difficulty lies in the determination of the latter potential at every distance. We solve this problem by devising a systematic expansion with respect to the ratio of the fundamental length scales involved in the two coulombic effects at stake. (The application of this method to a plasma confined between two ideally conducting plates and to a quantum plasma will be presented elsewhere). As a result we derive the exact analytical perturbative expressions for the density profiles up to first order in the coupling between charges. The mean-field approach displayed in Paper I is then justified.

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.

The collective variables method with a reference system is developed for the case of the grand canonical ensemble for a multicomponent continuous system. The method is used to investigate phase transitions in a binary system. For a binary symmetrical system the relations between microscopic parameters determining the alternation of gas-liquid and separation phase transitions are found. The functional of the grand partition function of the symmetrical mixture is examined in the framework of parameters containing the separation point. The system is described with two sets of collective variables: ρk, a set connected with the gas-liquid critical point, andck, a set connected with the separation phenomenon. The fourfold basic density measure is constructed inck-variable phase space which contains the variablecv connected with the order parameter of the system. It is shown that the problem can be reduced to the 3D Ising model in an external field.

The liquid-vapor interface of a confined fluid at the condensation phase transition is studied in a combined hydrostatic/mean-field limit of classical statistical mechanics. Rigorous and numerical results are presented. The limit accounts for strongly repulsive short-range forces in terms of local thermodynamics. Weak attractive longer-range ones, like gravitational or van der Waals forces, contribute a self-consistent mean potential. Although the limit is fluctuationfree, the interface is not a sharp Gibbs interface, but its structure is resolved over the range of the attractive potential. For a fluid of hard balls with ∼−r−6 interactions the traditional condensation phase transition with critical point is exhibited in the grand ensemble: A vapor state coexists with a liquid state. Both states are quasiuniform well inside the container, but wall-induced inhomogeneities show up close to the boundary of the container. The condensation phase transition of the grand ensemble bridges a region of negative total compressibility in the canonical ensemble which contains canonically stable proper liquid-vapor interface solutions. Embedded in this region is a new, strictly canonical phase transition between a quasiuniform vapor state and a small droplet with extended vapor atmosphere. This canonical transition, in turn, bridges a region of negative total specific heat in the microanonical ensemble. That region contains subcooled vapor states as well as superheated very small droplets which are microcanonically stable.

We rederive the Coulomb expansion of the electron gas average energy at finite temperature, starting from scratch, i.e., using only the framework of the grand canonical ensemble and not the finite-T Green's function formalism. We recover the analytical expressions of the exchange and correlation energy in both the high-T and theT=0 limits. We explicitly show the origin of the crossover of the correlation energy leading term frome 4 lne 2 at zero temperature toe 3 at finiteT. We also discuss the relative importance of exchange and correlation in both limits.

The Hertel-Thirring cell model for unstable systems (of purely attractive particles) is solved in the canonical ensemble for arbitrary dimensions. The differences between the phase transitions found in the canonical and in the microcanonical ensemble are discussed. The cluster phase (with a complete collapse in the ground state) exhibits the nonextensive character of the cell model. The results of the cell model are compared with molecular-dynamics simulations of a one-dimensional model with a rectangular-well pair potential. The simulations support the relevance of the cell model to characterize basic properties of gravitational systems.

The grand partition function for the two-dimensional, two-component plasma at Γ=2 is evaluated exactly in a finite system for various interfaces: a charged hard wall (the so-called primitive electrode model), a second two-component plasma of different fugacity separated by an impermeable membrane (the ideally polarizable interface), and a metal wall separated by an impermeable barrier. For each of these models the surface tension is calculated directly from the asymptotic expansion of the grand partition function.

The particle and charge correlations of the two-dimensional Coulomb gas are studied in the dielectric phase. A term-by-term analysis of the low-fugacity expansions suggests that the large-distance behaviors of the particle correlations are governed by multipolar interactions, similar to what happens in a system of permanent dipoles. These behaviors are compatible with the asymptotic structure of the BGY hierarchy equations; on the other hand, a new identity for the dielectric constant ɛ is used to show that the four-particle correlations decay as the dipole-dipole potential 1/r2 when two neutral pairs are separated by a large distancer. Near the zero-density critical point of the Kosterlitz-Thouless transition, we resum the low-fugacity expansions of both 1/ɛ and the charge correlation C(r). We thus retrieve the coupling constant flow equations of the renormalization group as well as the effective interaction energy of the iterated mean-field theory by Kosterlitz and Thouless. The coupling constant at the RG fixed point is then identified with 1/ɛ. The nonanalyticity of 1/ɛ at the transition turns out to coincide with the divergence of the low-fugacity series for this quantity. The leading term in the large-distance behavior of C(r) is found to be the same as for external charges. Moreover, we exhibit the subleading terms which also contribute to 1/ɛ.

The two-dimensional, two-component plasma is considered in doubly periodic boundary conditions with the positive and negative charges confined to separate interlacing rectangular lattices. It is shown that at the special couplingΓ=2, on a lattice of 2M 1×2M 2 sites, the grand partition function can be written as a double integral over a product of determinants of dimension 2M 2×2M 2. On the basis of a conjecture regarding the zero distribution of the grand partition function, the large-M 2 behavior of the determinant is given and the pressure evaluated exactly.

This paper deals with the grand canonical entropy of a lattice gas mixture. The entropy is a function of the multisite densities corresponding to the interaction pattern of the system in question. It is first evaluated for a nearest-neighborinteraction, one-dimensional simple lattice gas to show how the structure of bulk fluid is locally maintained. Generalization requires one set of interrelations among multisite densities presented in closed form for an arbitrary lattice, and one set between Boltzmann factors and multisite densities which is written down for simply connected lattices. Application is made to two-row lattices, which turn out to have local behavior from this viewpoint, as do all single-row or Bethe lattices with complete range-p interactions. Nonlocal examples are also given, and suggestions made for approximation sequences in general lattices.

A classical gas with short-range interaction in the grand canonical ensemble is studied. Ifp(β, z) denotes the thermodynamic pressure at inverse temperatureβ and activityz, then it follows from the Mayer expansion thatp(β, z) is infinitely differentiable providedβ andβz are sufficiently small. Here it is shown that there existsβ 0>0 such thatp(β, z) is infinitely differentiable ifβ0. One can interpret this result as saying that (β 0)−1 is an upper bound on the critical temperature for the system.

For a classical system of interacting particles we prove, in the microcanonical ensemble formalism of statistical mechanics, that the thermodynamic-limit entropy density is a differentiable function of the energy density and that its derivative, the thermodynamic-limit inverse temperature, is a continuous function of the energy density. We also prove that the inverse temperature of a finite system approaches the thermodynamic-limit inverse temperature as the volume of the system increases indefinitely. Finally, we show that the probability distribution for a system of fixed size in thermal contact with a large system approaches the Gibbs canonical distribution as the size of the large system increases indefinitely, if the composite system is distributed microcanonically.

A lower bound is obtained for the grand canonical partition function (and hence for the pressure) of a charge symmetric system with positive definite interaction. For the Coulomb interaction the lower bound on the pressure is the Debye-Huckel approximation.

The grand canonical ensemble of a two-dimensional Coulomb system with +/- 1 charges is proved to have screening phenomena in its high-temperature region. The Coulomb potential in a finite region ..lambda.. is assumed to be (-..delta../sub ..lambda..)/sup -1/, where ..delta../sub ..lambda.. is the Laplacian with zero boundary conditions on ..lambda... The hard-core condition is not assumed. The model is set up by separating (-..delta../sub ..lambda../)/sup -1/ into a short-range part and a long-range part depending on a parameter lambda. The self-energies are subtracted only for the short-range part and therefore a choice of lambda is a choice of subtraction of self-energies. The method of proof is in general the same as that of Brydges-Federbush Debye screening, except that here a modification for the short-range part of the potentials is needed.

A model classical fluid is constructed by assuming that the direct correlation functionc(r − r′) is independent of any applied external field. Thermodynamic consistency requires thatc(r − r′) ⩾ 0, and permits explicit representation of the model by a many-body interaction potential. In the canonical ensemble, the model shows a phase transition to an infinite density condensed phase, but in the grand canonical ensemble only an anomalous transition to zero density vapor is found to stably exist.

For classical gases of particles interacting through nonnegative, many-body interactions of short range it is verified that the corresponding grand canonical Gibbs measures have the global Markov property for sufficiently low values of the chemical activity. This yields the existence of a (nonsymmetric in general) transfer matrix formalism for such systems.

It is shown that the basic assumptions of the classical density functional approach are rigorously correct forH-stable systems in the grand canonical ensemble. Moreover, it is established that the set of all single-particle densities is convex. These results are derived by providing necessary and sufficient conditions for the solution of the classical inverse problem for single-particle densities. Analogous results are obtained for the solution of the higher-order correlation inverse problem, and the ramifications of these results for the validity of two-body decomposition of forces are discussed.

We consider one-dimensional grand-canonical continuum Gibbs states corresponding to slowly decaying, superstable, many-body interactions. Absence of phase transitions, in the sense of uniqueness of the tempered Gibbs state, is proved for interactions with an Nth body hardcore for arbitrarily large N.

We present a new technique to simulate the contact of a molecular dynamics system with a thermal wall. A canonical ensemble is obtained, and its statistical and thermodynamic fluctuations are studied. The values of the specific heat found by simulation agree with the experimental data. By means of thermal walls at different temperatures, thermal gradients are obtained. The values of the thermal conductivity are consistent with the experimental data.

We consider a classical dipole gas in the grand canonical ensemble. We prove that in dimensions greater than or equal to three, and for all temperatures, the free energy and the charges-dipoles correlation functions have an expansion in powers ofz, the fugacity of the system, which is asymptotic to all orders. We also give some information about the decay of correlations.

This is a general and exact study of multiple Hamiltonian walks (HAW) filling the two-dimensional (2D) Manhattan lattice. We generalize the original exact solution for a single HAW by Kasteleyn to a system ofmultiple closed walks, aimed at modeling a polymer melt. In 2D, two basic nonequivalent topological situations are distinguished. (1) the Hamiltonian loops are allrooted andcontractible to a point:adjacent one to another, and, on a torus,homotopic to zero. (2) the loops can encircle one another and, on a torus, canwind around it. Forcase 1, the grand canonical partition function and multiple correlation functions are calculated exactly as those of multiple rooted spanningtrees or of a massive 2Dfree field, critical at zero mass (zero fugacity). The conformally invariant continuum limit on a Manhattantorus is studied in detail. The melt entropy is calculated exactly. We also consider the relevant effect of free boundary conditions. The number of single HAWs on Manhattan lattices with other perimeter shapes (rectangular, Kagome, triangular, and arbitrary) is studied and related to the spectral theory of the Dirichlet Laplacian. This allows the calculation of exact shape-dependent configuration exponents y. An exact surface critical exponent is obtained. Forcase 2, nested and winding Hamiltonian circuits are allowed. An exact equivalence to thecritical Q-state Potts model exists, whereQ1/2 is the walk fugacity. The Hamiltonian system is then always critical (forQ

We consider a classical system, in a ν-dimensional cube Ω, with pair potential of the formq(r) + γ v φ(γr). Dividing Ω into a network of cells ω1, ω2,..., we regard the system as in a metastable state if the mean density of particles in each cell lies in a suitable neighborhood of the overall mean densityρ, withρ and the temperature satisfying $$f_0 (\rho ) + \tfrac{1}{2}\alpha \rho ^2 > f(\rho ,0 + )$$ and $$f''_0 (\rho ) + 2\alpha > 0$$ wheref(ρ, 0+) is the Helmholz free energy density (HFED) in the limit γ→ 0; α = ∫ φ(r)d v r andf 0 (ρ) is the HFED for the caseφ = 0. It is shown rigorously that, for periodic boundary conditions, the conditional probability for a system in the grand canonical ensemble to violate the constraints at timet > 0, given that it satisfied them at time 0, is at mostλt, whereλ is a quantity going to 0 in the limit $$|\Omega | \gg \gamma ^{ - v} \gg |\omega | \gg r_0 \ln |\Omega |$$ Here,r 0 is a length characterizing the potentialq, andx ≫ y meansx/y → +∞. For rigid walls, the same result is proved under somewhat more restrictive conditions. It is argued that a system started in the metastable state will behave (over times ≪λ −1) like a uniform thermodynamic phase with HFED f0(ρ) + 1/2αρ2, but that having once left this metastable state, the system is unlikely to return.

A two-dimensional model of polymer chain folding invented by Zwanzig and Lauritzen is here studied using a grand ensemble and transfer matrix method. Due to the character of the model, there are no extensive parameters in the grand ensemble and the dispersion in system size is large, raising doubts about the validity and usefulness of the ensemble. We find it possible to define a thermodynamic limit such that it leads to near equivalence between the canonical and grand ensembles in the limit of large systems. The transfer matrix in this case is a nonlocal operator on a space of L2 functions, and the eigenvalue equation is a homogeneous Fredholm integral equation of the second kind which can be completely solved in terms of Bessel functions. The grand partition function can then be expressed as a sum of powers of the known eigenvalues. It is an easy matter to reproduce the second-order phase transition in the canonical ensemble found in the original work on the model. The investigation is extended to yield the probability densities describing the length of a segment and the correlations among segments. The concept of a local width of the folded chain is found to break down at higher temperatures, while critical correlations are characterized by infinite range, as expected. Apart from physical and methodological implications, the new solution provides striking illustrations of some basic ideas concerning phase transitions.

A constant-magnetization ensemble is introduced in order to study classical, anisotropic Heisenberg systems. Existence, uniform convergence, and convexity properties are proved for an appropriate thermodynamic potential. The thermodynamic equivalence of this ensemble with the more common canonical ensemble is also established. In a subsequent paper, this formulation is used to obtain an exact statistical mechanical solution of classical Heisenberg systems with long-range Kac interactions.