Correlation inequalities and the thermodynamic limit for classical and quantum continuous systems II. Bose-Einstein and Fermi-Dirac statistics

We study quantum mechanical systems of particles with Bose or Fermi statistics interacting via two-body potentials of positive type in thermal equilibrium. We rewrite partition functions, reduced density matrices (RDMs), and correlation functions in terms of Wiener and Gaussian functional integrals (sine-Gordon transformation). This permits us, e.g., to apply correlation inequalities. Our main results include an analysis of stability versus instability in the grand canonical ensemble and, for charge-conjugation-invariant systems, upper and lower bounds on RDMs, the existence of the thermodynamic limit of pressure, RDMs and correlation functions, an inequality comparing correlations with Fermi statistics to ones with Bose statistics, and inequalities which are important in the study of Bose-Einstein condensation and of superconductivity.This research was done in part during the author's stay at the Department of Physics of Princeton University and was partially supported by the NSF under grant NSF PHY 76-80958.     

Temperature correlators in the two-component one-dimensional gas

The quantum non-relativistic two-component Bose and Fermi gases with infinitely strong point-like coupling between particles in one space dimension are considered. Time- and temperature-dependent correlation functions are represented in the thermodynamic limit as Fredholm determinants of integrable linear integral operators.     

A metric space of interactions and the thermodynamic limit

A metric space of interactions is formed for classical continuous systems and for quantum and classical lattice systems. It is shown that the thermodynamic limit of the grand canonical pressure exists on an extended class of potentials. In each neighborhood of each superstable lower regular, weakly tempered pair interaction and for each of a countable number of test functions there is an interaction for which the Fisher thermodynamic limit of the correlation functionals applied to the test function exists.     

Entropy inequalities

Some inequalities and relations among entropies of reduced quantum mechanical density matrices are discussed and proved. While these are not as strong as those available for classical systems they are nonetheless powerful enough to establish the existence of the limiting mean entropy for translationally invariant states of quantum continuous systems.Work supported by National Science Foundation Grant GP-9414.     

Dynamical pair correlations of classical and quantum fluids perturbed with weak long-range forces

We study time dependent correlation functions of ideal classical and quantum gases using methods of equilibrium statistical mechanics. The basis for this is the path integral formalism of quantum mechanical systems. By this approach the statistical mechanics of a quantum mechanical system becomes the equivalent of a classical polymer problem in four dimensions where imaginary time is the fourth dimension. Several non-trivial results for quantum systems have been obtained earlier by this analogy. Here we will focus upon particle dynamics. First ideal gases are considered. Then interactions, that are assumed weak and of long range, are added, and methods of classical statistical mechanics are applied to obtain the leading contribution. Comparison is performed with known results of kinetic theory. These results demonstrate how methods developed for systems in thermal equilibrium also is applicable outside equilibrium. Thus, more generally, we have reason to expect that these methods will be accurate and useful for other situations of interacting many-body systems consisting of quantized particles too. To indicate so we sketch the computation of the induced Casimir force between parallel plates filled with ions for the situation where the ions are quantized, but the interaction remains electrostatic. Further in this respect we establish expressions for a leading correction to ab initio calculations for the energies of the quantized electrons of molecules. To our knowledge these two latter applications go beyond earlier results.     

Estimates of general Mayer graphs. I: Construction of upper bounds for a given graph by means of sets of subgraphs

A large number of physical quantities (thermodynamic and correlation functions, scattering amplitudes, intermolecular potentials, etc. ...) can be expressed as sums of an infinite number of multiple integrals of the following type: $$\Gamma \left( {x_1 ,. . . , x_n } \right) = \int {\prod {f_L \left( {x_{i,} x_j } \right)dx_{n + 1} . . . dx_{n + k} } }$$ These are called Mayer graphs in statistical mechanics, Feynman graphs in quantum field theory, and multicenter integrals in quantum chemistry. We call themn-graphs here. In a preceding note [Physics Letters 62A:295 (1977)], we have proposed a new estimation method which provides upper bounds for arbitraryn-graphs. This article is devoted to developing in detail the foundations of this method. As a first application, we prove that all virial coefficients of polar systems are finite. More generally, we show on some examples that our estimation method can givefinite upper bounds forn-graphs occurring in the perturbative developments of thermodynamic functions of neutral, polar, and ionized gases and of Green's functions of Euclidean quantum field theories (thus improving Weinberg's theorem), as also in variational approximations of intermolecular potentials. Our estimation method is based on the Hölder inequality which is an improvement over the mean value estimation method, employed by Riddell, Uhlenbeck, and Groeneveld, except for the hard-sphere gas, where both methods coincide. The method is applied so far only to individual graphs and not to thermodynamic functions.     

General Decomposition of Radial Functions on Rn and Applications to N-Body Quantum Systems

We present a generalization of the Fefferman–de la Llave decomposition of the Coulomb potential to quite arbitrary radial functions V on Rⁿ going to zero at infinity. This generalized decomposition can be used to extend previous results on N-body quantum systems with Coulomb interaction to a more general class of interactions. As an example of such an application, we derive the high density asymptotics of the ground state energy of jellium with Yukawa interaction in the thermodynamic limit, using a correlation estimate by Graf and Solovej.     

Quasiclassical Statistical Thermodynamics and New Padé Approximations for the Free Charges in Strongly-Coupled Plasma

HNC equations in combination with effective quasi-classical potentials are used to calculate correlation functions and the thermodynamic properties of the free charges in semi-classical non-degenerate quantum plasmas. The interactions of the free particles are taken into account via effective potentials obtained from the Slater sum method. Analytical formulae reproducing the known limits and the HNC-results are constructed. Finally quantum effects are included as corrections by using known analytical results. This method is used to develope new Padé approximations for the subsystem of the free charges in mass-symmetrical as well as for mass-unsymmetrical hydrogen-like plasmas. The most essential result of our investigations is, that in the classical limit the scaling properties correspond to the OCP, e.g. the thermodynamic functions follow for large coupling strength Γ a Berlin-Montroll-Rosenfeld asymptotics via a Γ + b Γ^v + c ln Γ + d. Including quantum effects, the coefficients depend on the temperature, e.g. the slope a(T) increases with decreasing T converging to the classical limit. The new formulae are compared with earlier variants.     

Quantum Thermodynamic Uncertainties in Nonequilibrium Systems from Robertson-Schrödinger Relations

Thermodynamic uncertainty principles make up one of the few rare anchors in the largely uncharted waters of nonequilibrium systems, the fluctuation theorems being the more familiar. In this work we aim to trace the uncertainties of thermodynamic quantities in nonequilibrium systems to their quantum origins, namely, to the quantum uncertainty principles. Our results enable us to make this categorical statement: For Gaussian systems, thermodynamic functions are functionals of the Robertson-Schrödinger uncertainty function, which is always non-negative for quantum systems, but not necessarily so for classical systems. Here, quantum refers to noncommutativity of the canonical operator pairs. From the nonequilibrium free energy, we succeeded in deriving several inequalities between certain thermodynamic quantities. They assume the same forms as those in conventional thermodynamics, but these are nonequilibrium in nature and they hold for all times and at strong coupling. In addition we show that a fluctuation-dissipation inequality exists at all times in the nonequilibrium dynamics of the system. For nonequilibrium systems which relax to an equilibrium state at late times, this fluctuation-dissipation inequality leads to the Robertson-Schrödinger uncertainty principle with the help of the Cauchy-Schwarz inequality. This work provides the microscopic quantum basis to certain important thermodynamic properties of macroscopic nonequilibrium systems.     

Quantum Hall Phases and Plasma Analogy in Rotating Trapped Bose Gases

A bosonic analogue of the fractional quantum Hall effect occurs in rapidly rotating trapped Bose gases: There is a transition from uncorrelated Hartree states to strongly correlated states such as the Laughlin wave function. This physics may be described by effective Hamiltonians with delta interactions acting on a bosonic N-body Bargmann space of analytic functions. In a previous paper (Rougerie et al. in Phys. Rev. A 87:023618, 2013) we studied the case of a quadratic plus quartic trapping potential and derived conditions on the parameters of the model for its ground state to be asymptotically strongly correlated. This relied essentially on energy upper bounds using quantum Hall trial states, incorporating the correlations of the Bose-Laughlin state in addition to a multiply quantized vortex pinned at the origin. In this paper we investigate in more details the density of these trial states, thereby substantiating further the physical picture described in (Rougerie et al. in Phys. Rev. A 87:023618, 2013), improving our energy estimates and allowing to consider more general trapping potentials. Our analysis is based on the interpretation of the densities of quantum Hall trial states as Gibbs measures of classical 2D Coulomb gases (plasma analogy). New estimates on the mean-field limit of such systems are presented.     

Classical bounds on quantum partition functions

Explicit bounds on the quantum partition functions are given in terms of classical partition functions, incorporating effective pair potentials, which account for Fermi- and Bose-statistics, respectively. The bounds may be used for the limit 0 and eventually for showing the interchangeability of the classical with the thermodynamic limit. A simple derivation of the thermodynamic limit for free particles with general dispersions is given.Work supported in part by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich, Project No. 3569.     

Two-Dimensional Coulomb Systems on a Surface of Constant Negative Curvature

We study the equilibrium statistical mechanics of classical two-dimensional Coulomb systems living on a pseudosphere (an infinite surface of constant negative curvature). The Coulomb potential created by one point charge exists and goes to zero at infinity. The pressure can be expanded as a series in integer powers of the density (the virial expansion). The correlation functions have a thermodynamic limit, and remarkably that limit is the same one for the Coulomb interaction and some other interaction law. However, special care is needed for defining a thermodynamic limit of the free energy density. There are sum rules expressing the property of perfect screening. These generic properties can be checked on the Debye–Hückel approximation, and on two exactly solvable models, the one-component plasma and the two-component plasma, at some special temperature.     

Uniform upper bounds (and thermodynamic limit) for the correlation functions of symmetric coulomb-type systems

Uniform upper bounds are proven for the correlation functions in the strictly charge-neutral canonical and grand canonical ensembles for charge-symmetric two-component systems. For the grand canonical ensemble the increase of the correlation functions along the thermodynamic-limit sequence is shown as well, implying the existence of the states. The particles have bounded pair interactions of positive type. Both classical and quantum systems with Boltzmann statistics are considered. Coulomb systems with regularized interactions are included as a special case.     

Temperature dependence near phase transitions in classical and quant. mech. canonical statistics

The free energy of a many-body system, classical as well as quantum mechanical, is represented by a finite Laplace transform of the interaction phase volume for stable potentials for which the thermodynamic limit exists. Possible singular and regular phase transitions are discussed with respect to the temperature dependence and are classified by the behaviour of a limiting density of complex temperature zeros. The phenomenologically known types are reproduced.     

Tan relations in one dimension

We derive exact relations that connect the universal C/k⁴-decay of the momentum distribution at large k with both thermodynamic properties and correlation functions of two-component Fermi gases in one dimension with contact interactions. The relations are analogous to those obtained by Tan in the three-dimensional case and are derived from an operator product expansion of the one- and two-particle density matrix. They extend earlier results by Olshanii and Dunjko (2003) [24] for the bosonic Lieb–Liniger gas. As an application, we calculate the pair distribution function at short distances and the dimensionless contact in the limit of infinite repulsion. The ground state energy approaches a universal constant in this limit, a behavior that also holds in the three-dimensional case. In both one and three dimensions, a Stoner instability to a saturated ferromagnet for repulsive fermions with zero range interactions is ruled out at any finite coupling.     

Thermodynamics of boson and fermion systems with fractal distribution functions

Starting with the fractal-inspired distribution functions for Maxwell-Boltzmann, Bose-Einstein, and Fermi systems, as reported by Büyükkili? and Demirhan, we obtain the corresponding probability distributions and study their thermodynamic behavior. We compare our results with those corresponding to ideal gases (q=1) and Bose-Einstein and Fermi systems with quantum group symmetry. In particular, we show that the Hamiltonian that gives the Bose-Einstein generalized distribution function can be interpreted as a q deformation of the ideal gas Hamiltonian.     

Breakdown of universality for unequal-mass Fermi gases with infinite scattering length

We treat small trapped unequal-mass two-component Fermi gases at unitarity within a nonperturbative microscopic framework and investigate the system properties as functions of the mass ratio κ, and the numbers N1 and N2 of heavy and light fermions. While equal-mass Fermi gases with infinitely large interspecies s-wave scattering length a(s) are universal, we find that unequal-mass Fermi gases are, for sufficiently large κ and in the regime where Efimov physics is absent, not universal. In particular, the (N?,N?) = (2, 1) and (3, 1) systems exhibit three-body and four-body resonances at κ=12.314(2) and 10.4(2), respectively, as well as surprisingly large finite-range effects. These findings have profound implications for ongoing experimental efforts and quantum simulation proposals that utilize unequal-mass atomic Fermi gases.     

The classical field limit of scattering theory for non-relativistic many-boson systems. I

We study the classical field limit of non-relativistic many-boson theories in space dimensionn≧3. When ?→0, the correlation functions, which are the averages of products of bounded functions of field operators at different times taken in suitable states, converge to the corresponding functions of the appropriate solutions of the classical field equation, and the quantum fluctuations are described by the equation obtained by linearizing the field equation around the classical solution. These properties were proved by Hepp [6] for suitably regular potentials and in finite time intervals. Using a general theory of existence of global solutions and a general scattering theory for the classical equation, we extend these results in two directions: (1) we consider more singular potentials, (2) more important, we prove that for dispersive classical solutions, the ?→0 limit is uniform in time in an appropriate representation of the field operators. As a consequence we obtain the convergence of suitable matrix elements of the wave operators and, if asymptotic completeness holds, of theS-matrix.     

Thermalization in nature and on a quantum computer

In this work, we show how Gibbs or thermal states appear dynamically in closed quantum many-body systems, building on the program of dynamical typicality. We introduce a novel perturbation theorem for physically relevant weak system-bath couplings that is applicable even in the thermodynamic limit. We identify conditions under which thermalization happens and discuss the underlying physics. Based on these results, we also present a fully general quantum algorithm for preparing Gibbs states on a quantum computer with a certified runtime and error bound. This complements quantum Metropolis algorithms, which are expected to be efficient but have no known runtime estimates and only work for local Hamiltonians.     

Quantum nondemolition measurements on two-level atomic systems and temporal Bell inequalities

The evolution of a two-level system subjected to stimulated transitions which is undergoing a sequence of measurements of the level occupation probability is evaluated. Its time correlation function is compared to the one obtained through the pure Schr?dinger evolution. Systems of this kind have been recently proposed for testing the quantum mechanical predictions against those of macrorealistic theories, by means of temporal Bell inequalities. The classical requirement of noninvasivity, needed to define correlation functions in the realistic case, finds a quantum counterpart in the quantum nondemolition condition. The consequences on the observability of quantum mechanically predicted violations to temporal Bell inequalities are drawn and compared to the already dealt case of the rf-SQUID dynamics. Received: 28 March 1996 / Revised version: 13 August 1996