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Showing papers in "Journal of Statistical Physics in 1996"


Journal ArticleDOI
TL;DR: In this article, a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory is derived, and the first member of the hierarchy is the Euler system based on Maxwellian velocity distributions, while the second member is based on nonisotropic Gaussian velocity distributions.
Abstract: This paper presents a systematicnonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory. The first member of the hierarchy is the Euler system, which is based on Maxwellian velocity distributions, while the second member is based on nonisotropic Gaussian velocity distributions. The closure proceeds in two steps. The first ensures that every member of the hierarchy is hyperbolic, has an entropy, and formally recovers the Euler limit. The second involves modifying the collisional terms so that members of the hierarchy beyound the second also recover the correct Navier-Stokes behavior. This is achieved through the introduction of a generalization of the BGK collision operator. The simplest such system in three spatial dimensions is a “14-moment” closure, which also recovers the behavior of the Grad “13-moment” system when the velocity distributions lie near local Maxwellians. The closure procedure can be applied to a general class of kinetic theories.

903 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that there is no nontrivial, homogeneous, local, unitary, scalar cellular automata in one dimension, and that the homogeneity condition can be overcome by a quantum cellular automaton with exactly unitary partitioning.
Abstract: A natural architecture for nanoscale quantum computation is that of a quantum cellular automaton. Motivated by this observation, we begin an investigation of exactly unitary cellular automata. After proving that there can be no nontrivial, homogeneous, local, unitary, scalar cellular automaton in one dimension, we weaken the homogeneity condition and show that there are nontrivial, exactly unitary, partitioning cellular automata. We find a one-parameter family of evolution rules which are best interpreted as those for a one-particle quantum automaton. This model is naturally reformulated as a two component cellular automaton which we demonstrate to limit to the Dirac equation. We describe two generalizations of this automaton, the second, of which, to multiple interacting particles, is the correct definition of a quantum lattice gas.

730 citations


Journal ArticleDOI
TL;DR: In this article, the authors review the consistent histories formulations of quantum mechanics developed by Griffiths, Omnes and Gell-Mann and Hartle, and describe the c of consistent sets.
Abstract: We review the consistent histories formulations of quantum mechanics developed by Griffiths, Omnes and Gell-Mann and Hartle, and describe the c of consistent sets. We illustrate some general features of consistent sets by a few simple lemmas and examples. We consider various interpretations of the formalism, and examine the new problems which arise in reconstructing the past and predicting the future. It is shown that Omnes' characterisation of true statements — statements which can be deduced un- conditionally in his interpretation — is incorrect. We examine critically Gell-Mann and Hartle's interpretation of the formalism, and in particular their discussions of communi- cation, prediction and retrodiction, and conclude that their explanation of the apparent persistence of quasiclassicality relies on assumptions about an as yet unknown theory of experience. Our overall conclusion is that the consistent histories approach illustrates the

296 citations


Journal ArticleDOI
TL;DR: In this article, the authors consider self-avoiding walks on a simple cubic lattice in which neighboring pairs of vertices of the walk (not connected by an edge) have an associated pair-wise additive energy.
Abstract: We consider self-avoiding walks on the simple cubic lattice in which neighboring pairs of vertices of the walk (not connected by an edge) have an associated pair-wise additive energy. If the associated force is attractive, then the walk can collapse from a coil to a compact ball. We describe two Monte Carlo algorithms which we used to investigate this collapse process, and the properties of the walk as a function of the energy or temperature. We report results about the thermodynamic and configurational properties of the walks and estimate the location of the collapse transition.

295 citations


Journal ArticleDOI
TL;DR: In this paper, the authors analyze different mechanisms of entropy production in statistical mechanics, and propose formulas for the entropy production ratee(μ) in a state μ, when μ is steady state describing the long term behavior of a system.
Abstract: We analyze different mechanisms of entropy production in statistical mechanics, and propose formulas for the entropy production ratee(μ) in a state μ. When μ is steady state describing the long term behavior of a system we show thate(μ)≥0, and sometimes we can provee(μ)>0.

217 citations


Journal ArticleDOI
TL;DR: In this paper, the authors investigated the distribution of roots of polynomials of high degree with random coefficients which appear naturally in the context of quantum chaotic dynamics and showed that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane.
Abstract: We investigate the distribution of roots of polynomials of high degree with random coefficients which, among others, appear naturally in the context of “quantum chaotic dynamics.” It is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane. In order to further increase this tendency, we study in detail the particular case of self-inversive random polynomials and show that for them a finite portion of all roots lies exactly on the unit circle. Correlation functions of these roots are also computed analytically, and compared to the correlations of eigenvalues of random matrices. The problem of ergodicity of chaotic wavefunctions is also considered. For that purpose we introduce a family of random polynomials whose roots spread uniformly over phase space. While these results are consistent with random matrix theory predictions, they provide a new and different insight into the problem of quantum ergodicity Special attention is devoted to the role of symmetries in the distribution of roots of random polynomials.

171 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that the chaoticity hypothesis, which is analogous to Ruelle's principle for turbulence, implies Onsager reciprocity and fluctuation-dissipation theorem in reversible models for coexisting transport phenomena.
Abstract: It is shown that the chaoticity hypothesis recently introduced in statistical mechanics, which is analogous to Ruelle's principle for turbulence, implies the Onsager reciprocity and the fluctuation-dissipation theorem in various reversible models for coexisting transport phenomena.

162 citations


Journal ArticleDOI
TL;DR: In this article, the authors used boundary weights and reflection equations to obtain families of commuting double-row transfer matrices for interaction-round-a-face models with fixed boundary conditions.
Abstract: We use boundary weights and reflection equations to obtain families of commuting double-row transfer matrices for interaction-round-a-face models with fixed boundary conditions. In particular, we consider the fusion hierarchy of the Andrews-Baxter-Forrester (ABF) models, for which we obtain diagonal, elliptic solutions to the reflection equations, and find that the double-row transfer matrices satisfy functional equations with the same form as in the case of periodic boundary conditions.

159 citations


Journal ArticleDOI
TL;DR: In this paper, a new way to implement solid obstacles in lattice Boltzmann models is presented, where unknown populations at the boundary nodes are derived from the locally known populations with the help of a second-order Chapman-Enskog expansion and Dirichlet boundary conditions with a given momentum.
Abstract: A new way to implement solid obstacles in lattice Boltzmann models is presented. The unknown populations at the boundary nodes are derived from the locally known populations with the help of a second-order Chapman-Enskog expansion and Dirichlet boundary conditions with a given momentum. Steady flows near a flat wall, arbitrarily inclined with respect to the lattice links, are then obtained with a third-order error. In particular, Couette and Poiseuille flows are exactly recovered without the Knudsen layers produced for inclined walls by the bounce back condition.

156 citations


Journal ArticleDOI
TL;DR: A survey of the dynamics of polygonal billiards in polygons is given in this article, with an emphasis on the material that was not in the literature in 1985.
Abstract: In the ten or so years since the publication of ref. 19 polygonal billiards have remained an active subject of research in the mathematics and physics literature. As a result, our understanding of the subject, although still far from being complete, is much better than it was ten years ago. This survey attempts to give a broad overview of the dynamics of billiards in polygons, with an emphasis on the material that was not in the literature in 1985. This is not a survey of the publications on polygonal billiards, and we apologize to authors whose work has not been included. The selection of topics has been strongly influenced by the personal taste of the author, and by space limitations. Thus, we do not discuss quantum polygonal billiards (in particular, quantum chaos; see, e.g., refs. 41, 44, and 45). We hope that somebody will write a survey on this important subject in the near future. A fascinating aspect of the subject is the interplay between the geometric shape of the billiard table (i.e., a planar curve) and the qualitative features of the billiard dynamics. Hence it is instructive to compare polygonal billiard tables with other classes of billiard tables. In particular, the smooth, strictly convex tables and the dispersing (or Sinai) tables produce strikingly different types of billiard dynamics, We recommend refs. 46 and 15 for a general overview of the subject and a comparative study of billiard dynamics for various types of billiard tables.

153 citations


Journal ArticleDOI
TL;DR: In this paper, the authors apply large deviation theory to particle systems with a random mean-field interaction in the McKean-Vlasov limit and describe large deviations and normal fluctuations around the MCV equation.
Abstract: We apply large-deviation theory to particle systems with a random mean-field interaction in the McKean-Vlasov limit. In particular, we describe large deviations and normal fluctuations around the McKean-Vlasov equation. Due to the randomness in the interaction, the McKean-Vlasov equation is a collection of coupled PDEs indexed by the state space of the single components in the medium. As a result, the study of its solution and of the finite-size fluctuation around this solution requires some new ingredient as compared to existing techniques for nonrandom interaction.


Journal ArticleDOI
TL;DR: In this article, an energy-transport model is derived from the Boltzmann transport equation of semiconductors under the hypothesis that the energy gain or loss of the electrons by the phonon collisions is weak.
Abstract: An energy-transport model is rigorously derived from the Boltzmann transport equation of semiconductors under the hypothesis that the energy gain or loss of the electrons by the phonon collisions is weak. Retaining at leading order electron-electron collisions and elastic collisions (i.e., impurity scattering and the “elastic part” of phonon collisions), a rigorous diffusion limit of the Boltzmann equation can be carried over, which leads to a set of diffusion equations for the electron density and temperature. The derivation is given in both the degenerate and nondegenerate cases.

Journal ArticleDOI
TL;DR: In this paper, the authors derived hydrodynamic equations for systems not in local thermodynamic equilibrium, that is, where the local stationary measures are non-Gibbsian and do not satisfy detailed balance with respect to the microscopic dynamics.
Abstract: We derive hydrodynamic equations for systems not in local thermodynamic equilibrium, that is, where the local stationary measures are “non-Gibbsian” and do not satisfy detailed balance with respect to the microscopic dynamics. As a main example we consider thedriven diffusive systems (DDS), such as electrical conductors in an applied field with diffusion of charge carriers. In such systems, the hydrodynamic description is provided by a nonlinear drift-diffusion equation, which we derive by a microscopic method ofnonequilibrium distributions. The formal derivation yields a Green-Kubo formula for the bulk diffusion matrix and microscopic prescriptions for the drift velocity and “nonequilibrium entropy” as functions of charge density. Properties of the hydrodynamic equations are established, including an “H-theorem” on increase of the thermodynamic potential, or “entropy”, describing approach to the homogeneous steady state. The results are shown to be consistent with the derivation of the linearized hydrodynamics for DDS by the Kadanoff-Martin correlation-function method and with rigorous results for particular models. We discuss also the internal noise in such systems, which we show to be governed by a generalizedfluctuation-dissipation relation (FDR), whose validity is not restricted to thermal equilibrium or to time-reversible systems. In the case of DDS, the FDR yields a version of a relation proposed some time ago by Price between the covariance matrix of electrical current noise and the bulk diffusion matrix of charge density. Our derivation of the hydrodynamic laws is in a form—the so-called “Onsager force-flux form” which allows us to exploit the FDR to construct the Langevin description of the fluctuations. In particular, we show that the probability of large fluctuations in the hydrodynamic histories is governed by a version of the Onsager “principle of least dissipation,” which estimates the probability of fluctuations in terms of the Ohmic dissipation required to produce them and provides a variational characterization of the most probable behavior as that associated to least (excess) dissipation. Finally, we consider the relation of longrange spatial correlations in the steady state of the DDS and the validity of ordinary hydrodynamic laws. We also discuss briefly the application of the general methods of this paper to other cases, such as reaction-diffusion systems or magnetohydrodynamics of plasmas.

Journal ArticleDOI
TL;DR: In this article, the authors consider two-dimensional overdamped double-well systems perturbed by white noise and quantify the non-Arrhenius behavior of a system at the bifurcation point, by using the Maslov-WKB method to construct an approximation to the quasistationary probability distribution of the system that is valid in a boundary layer near the separatrix.
Abstract: We consider two-dimensional overdamped double-well systems perturbed by white noise. In the weak-noise limit the most probable fluctuational path leading from either point attractor to the separatrix (the most probable escape path, or MPEP) must terminate on the saddle between the two wells. However, as the parameters of a symmetric double-well system are varied, a unique MPEP may bifurcate into two equally likely MPEPs. At the bifurcation point in parameter space, the activation kinetics of the system become non-Arrhenius. We quantify the non-Arrhenius behavior of a system at the bifurcation point, by using the Maslov-WKB method to construct an approximation to the quasistationary probability distribution of the system that is valid in a boundary layer near the separatrix. The approximation is a formal asymptotic solution of the Smoluchowski equation. Our construction relies on a new scaling theory, which yields “critical exponents” describing weak-noise behavior at the bifurcation point, near the saddle.

Journal ArticleDOI
TL;DR: In this article, it was shown that adding a small quantum perturbation and/or increasing the temperature produce only smooth deformations of the phase diagrams of a given system, which can involve bosons or fermions and can be of infinite range but decaying exponentially with the size of the bonds.
Abstract: Starting from classical lattice systems ind≥2 dimensions with a regular zerotemperature phase diagram, involving a finite number of periodic ground states, we prove that adding a small quantum perturbation and/or increasing the temperature produce only smooth deformations of their phase diagrams. The quantum perturbations can involve bosons or fermions and can be of infinite range but decaying exponentially fast with the size of the bonds. For fermions, the interactions must be given by monomials of even degree in creation and annihilation operators. Our methods can be applied to some anyonic systems as well. Our analysis is based on an extension of Pirogov-Sinai theory to contour expansions ind+1 dimensions obtained by iteration of the Duhamel formula.

Journal ArticleDOI
TL;DR: In this article, it was shown that the probability of an interface becomes proportional to its area and the surface tension converges to the van der Waals surface tension, based on the analysis of the rate functionals for Gibbsian large deviations.
Abstract: , d¸2, and fix the temperature below its Lebowitz-Penrose critical value. We prove that when the Kac scaling parameter ° vanishes, the log of the probability of an interface becomes proportional to its area and the surface tension, related to the proportionality constant, converges to the van der Waals surface tension. The results are based on the analysis of the rate functionals for Gibbsian large deviations and on the proof that they i-converge to the perimeter functional of geometric measure theory (which extends the notion of area). Our consider- ations include non smooth interfaces proving that the Gibbsian probability of an interface depends only on its area and not on its regularity.

Journal ArticleDOI
TL;DR: In this article, the authors studied metastability and nucleation for the Blume-Capel model with spin variables taking values in {−1,0, +1}. They considered large but finite volume, small fixed magnetic fieldh, and chemical potential λ in the limit of zero temperature.
Abstract: We study metastability and nucleation for the Blume-Capel model: a ferromagnetic nearest neighbor two-dimensional lattice system with spin variables taking values in {−1,0, +1}. We consider large but finite volume, small fixed magnetic fieldh, and chemical potential λ in the limit of zero temperature; we analyze the first excursion from the metastable −1 configuration to the stable +1 configuration. We compute the asymptotic behavior of the transition time and describe the typical tube of trajectories during the transition. We show that, unexpectedly, the mechanism of transition changes abruptly when the lineh=2λ is crossed.

Journal ArticleDOI
TL;DR: For the zero-temperature Glauber dynamics of the q-state Potts model, the fractionr(q, t) of spins which never flip up to timet decays like a power law when the initial condition is random as mentioned in this paper.
Abstract: For the zero-temperature Glauber dynamics of theq-state Potts model, the fractionr(q, t) of spins which never flip up to timet decays like a power lawr(q, t)∼t−θ(q) when the initial condition is random. By mapping the problem onto an exactly soluble one-species coagulation model (A+A→A) or alternatively by transforming the problem into a free-fermion model, we obtain the exact expression of θ(q) for all values ofq. The exponent π(q) is in general irrational, θ(3)=0.53795082..., θ(4)=0.63151575..., ..., with the exception ofq=2 andq=∞, for which θ(2)=3/8 and θ(∞)=1.

Journal ArticleDOI
TL;DR: In this article, an integrable model on the simple cubic lattice was formulated and the Boltzmann weights of the model obey the vertex-type tetrahedron equation.
Abstract: In this paper we formulate an integrable model on the simple cubic lattice. TheN-valued spin variables of the model belong to edges of the lattice. The Boltzmann weights of the model obey the vertex-type tetrahedron equation. In the thermodynamic limit our model is equivalent to the Bazhanov-Baxter model. In the case whenN=2 we reproduce Korepanov's and Hietarinta's solutions of the tetrahedron equation as special cases.

Journal ArticleDOI
TL;DR: In this article, the problem of computing the one-dimensional configuration sums of the ABF model in regime III is mapped onto a problem of evaluating the grandcanonical partition function of a gas of charged particles obeying certain fermionic exclusion rules.
Abstract: The problem of computing the one-dimensional configuration sums of the ABF model in regime III is mapped onto the problem of evaluating the grandcanonical partition function of a gas of charged particles obeying certain fermionic exclusion rules. We thus obtain a newfermionic method to compute the local height probabilities of the model. Combined with the originalbosonic approach of Andrews, Baxter, and Forrester, we obtain a new proof of (some of) Melzer's polynomial identities. In the infinite limit these identities yield Rogers-Ramanujan type identities for the Virasoro characters χ l,1 (r−l,r) (q) as conjectured by the Stony Brook group. As a result of our work the corner transfer matrix and thermodynamic Bethe Ansatz approaches to solvable lattice models are unified.

Journal ArticleDOI
TL;DR: In this paper, the one-dimensional configuration sums of the ABF model were computed using the fermionic technique introduced in part I of this paper and proved polynomial identities for finitizations of the Virasoro characters conjectured by Melzer.
Abstract: We compute the one-dimensional configuration sums of the ABF model using the fermionic technique introduced in part I of this paper. Combined with the results of Andrews, Baxter, and Forrester, we prove polynomial identities for finitizations of the Virasoro characters $$\chi _{b.a}^{(r - 1.r)} (q)$$ as conjectured by Melzer. In the thermodynamic limit these identities reproduce Rogers-Ramanujan-type identities for the unitary minimal Virasoro characters conjectured by the Stony Brook group. We also present a list of additional Virasoro character identities which follow from our proof of Melzer's identities and application of Bailey's lemma.


Journal ArticleDOI
TL;DR: In this article, a new Ising spin-glass model with highly disordered coupling magnitudes was proposed, in which a greedy algorithm for producing ground states is exact, and it was shown that the procedure for determining (infinite volume) ground states for this model can be related to invasion percolation with the number of ground states identified as 2N.
Abstract: We propose a new Ising spin-glass model on Zd of Edwards-Anderson type, but with highly disordered coupling magnitudes, in which a greedy algorithm for producing ground states is exact. We find that the procedure for determining (infinite-volume) ground states for this model can be related to invasion percolation with the number of ground states identified as 2N, whereN=N(d) is the number of distinct global components in the “invasion forest.” We prove thatN(d)=∞ if the invasion connectivity function is square summable. We argue that the critical dimension separatingN=1 andN=∞ isdc=8. WhenN(d)=∞, we consider free or periodic boundary conditions on cubes of side lengthL and show that frustration leads to chaoticL dependence withall pairs of ground states occurring as subsequence limits. We briefly discuss applications of our results to random walk problems on rugged landscapes.

Journal ArticleDOI
TL;DR: In this paper, a numerical simulation algorithm that is well suited for the study of noise-induced transport processes is presented. But the algorithm is not suitable for the simulation of complex systems and it does not preserve detailed balance for systems in equilibrium.
Abstract: We present a numerical simulation algorithm that is well suited for the study of noise-induced transport processes. The algorithm has two advantages over standard techniques: (1) it preserves the property of detailed balance for systems in equilibrium and (2) it provides an efficient method for calculating nonequilibrium currents. Numerical results are compared with exact solutions from two different types of correlation ratchets, and are used to verify the results of perturbation calculations done on a three-state ratchet.

Journal ArticleDOI
TL;DR: In this paper, a unified theoretical framework for the anisotropic Kondo model and the boundary sine-Gordon model was developed, which allows to find new results for both models.
Abstract: We develop a unified theoretical framework for the anisotropic Kondo model and the boundary sine-Gordon model. They are both boundary integrable quantum field theories with a quantum-group spin at the boundary which takes values, respectively, in standard or cyclic representations of the quantum groupSU(2)q. This unification is powerful, and allows us to find new results for both models. For the anisotropic Kondo problem, we find exact expressions (in the presence of a magnetic field) for all the coefficients in the Anderson-Yuval perturbative expansion. Our expressions hold initially in the very anisotropic regime, but we show how to continue them beyond the Toulouse point all the way to the isotropic point using an analog of dimensional regularization. The analytic structure is transparent, involving only simple poles which we determine exactly, together with their residues. For the boundary sine-Gordon model, which describes an impurity in a Luttinger liquid, we find the nonequilibrium conductance for all values of the Luttinger coupling. This is an intricate computation because the voltage operator and the boundary scattering do not commute with each other.

Journal ArticleDOI
TL;DR: This paper presents the first provably polynomial-time approximation algorithms for computing the number of coverings with any specified number of monomers ind-dimensional rectangular lattice with periodic boundaries, for any fixed dimensiond, and in two-dimensional lattices with fixed boundaries.
Abstract: We study the problem of counting the number of coverings of ad-dimensional rectangular lattice by a specified number of monomers and dimers This problem arises in several models in statistical physics, and has been widely studied A classical technique due to Fisher, Kasteleyn, and Temperley solves the problem exactly in two dimensions when the number of monomers is zero (the dimer covering problem), but is not applicable in higher dimensions or in the presence of monomers This paper presents the first provably polynomial-time approximation algorithms for computing the number of coverings with any specified number of monomers ind-dimensional rectangular lattices with periodic boundaries, for any fixed dimensiond, and in two-dimensional lattices with fixed boundaries The algorithms are based on Monte Carlo simulation of a suitable Markov chain, and, in constrast to most Monte Carlo algorithms in statistical physics, have rigorously derived performance guarantees that do not rely on any assumptions The method generalizes to counting coverings of any finite vertex-transitive graph, a class which includes most natural finite lattices with periodic boundary conditions

Journal ArticleDOI
TL;DR: In this paper, the authors studied the behavior of the two-dimensional nearest neighbor ferromagnetic Ising model under an external magnetic field and showed that the boundary effect dominates in the system if the linear size of the system is of orderB/h withB small enough, while if B is large enough, then the external field dominates.
Abstract: We continue our study of the behavior of the two-dimensional nearest neighbor ferromagnetic Ising model under an external magnetic fieldh, initiated in our earlier work. We strengthen further a result previously proven by Martirosyan at low enough temperature, which roughly states that for finite systems with (−)-boundary conditions under a positive external field, the boundary effect dominates in the system if the linear size of the system is of orderB/h withB small enough, while ifB is large enough, then the external field dominates in the system. In our earlier work this result was extended to every subcritical value of the temperature. Here for every subcritical value of the temperature we show the existence of a critical valueB 0 (T) which separates the two regimes specified above. We also find the asymptotic shape of the region occupied by the (+)-phase in the second regime, which turns out to be a “squeezed Wulff shape”. The main step in our study is the solution of the variational problem of finding the curve minimizing the Wulff functional, which curve is constrained to the unit square. Other tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field, extended to all temperatures below the critical one.

Journal ArticleDOI
TL;DR: In this article, the perturbation expansion for a general class of many-fermion systems with a nonnested, nonspherical Fermi surface is renormalized to all orders.
Abstract: The perturbation expansion for a general class of many-fermion systems with a nonnested, nonspherical Fermi surface is renormalized to all orders. In the limit as the infrared cutoff is removed, the counterterms converge to a finite limit which is differentiable in the band structure. The map from the renormalized to the bare band structure is shown to be locally injective. A new classification of graphs as overlapping or nonoverlapping is given, and improved power counting bounds are derived from it. They imply that the only subgraphs that can generater factorials in ther th order of the renormalized perturbation series are indeed the ladder graphs and thus give a precise sense to the statement that “ladders are the most divergent diagrams.” Our results apply directly to the Hubbard model at any filling except for half-filling. The half-filled Hubbard model is treated in another place.

Journal ArticleDOI
TL;DR: In this paper, the Coulomb system is assumed to be confined, by walls made of an ideal conductor material; this choice corresponds to simple Dirichlet boundary conditions for the massless Gaussian field.
Abstract: When a classical Coulomb system has macroscopic conducting behavior, its grand potential has universal finite-size corrections similar to the ones which occur in the free energy of a simple critical system: the massless Gaussian field. Here, the Coulomb system is assumed to be confined, by walls made of an ideal conductor material; this choice corresponds to simple (Dirichlet) boundary conditions for the Gaussian field. For ad-dimensional (d≥2) Coulomb system confined in a slab of thicknessW, the grand potential (in units ofkBT) per unit area has the universal term Γ(d/2)ζ(d)/2dπd/2Wd−1. For a two-dimensional Coulomb system confined, in a disk of radiusR, the grand potential (in units ofkBT) has the universal term (1/6) lnR. These results, of general validity, are checked on two-dimensional solvable models.