Ursell operators in statistical physics of dense systems: the role of high order operators and of exchange cycles

The purpose of this article is to discuss cluster expansions in dense quantum systems, as well as their interconnection with exchange cycles. We show in general how the Ursell operators of order l≥ 3 contribute to an exponential which corresponds to a mean-field energy involving the second operator U₂, instead of the potential itself as usual - in other words, the mean-field correction is expressed in terms of a modification of a local Boltzmann equilibrium. In a first part, we consider classical statistical mechanics and recall the relation between the reducible part of the classical cluster integrals and the mean-field; we introduce an alternative method to obtain the linear density contribution to the mean-field, which is based on the notion of tree-diagrams and provides a preview of the subsequent quantum calculations. We then proceed to study quantum particles with Boltzmann statistics (distinguishable particles) and show that each Ursell operator U_n with n≥ 3 contains a “tree-reducible part”, which groups naturally with U₂ through a linear chain of binary interactions; this part contributes to the associated mean-field experienced by particles in the fluid. The irreducible part, on the other hand, corresponds to the effects associated with three (or more) particles interacting all together at the same time. We then show that the same algebra holds in the case of Fermi or Bose particles, and discuss physically the role of the exchange cycles, combined with interactions. Bose condensed systems are not considered at this stage. The similarities and differences between Boltzmann and quantum statistics are illustrated by this approach, in contrast with field theoretical or Green's functions methods, which do not allow a separate study of the role of quantum statistics and dynamics. Received 18 October 2001     

Quantum statistics and isospin invariance

The level density of an ideal Bose or Fermi gas is written in terms of usual phase-space integrals taking isospin conservation into accoutt. This cluster decomposition includes quantum statistics corrections between equal as well as unequal charge states of the considered particles. The isospin weights are given. These results are used to formulate a simple isospin-invariant statistical bootstrap model with Bose statistics. In the framework of this model the production of neutral pions in e⁺e^? and $\text{N}\text{N}$ annihilation is investigated.     

Exact Maxwell–Boltzmann,Bose–Einstein and Fermi–Dirac statistics

The exact Maxwell–Boltzmann (MB), Bose–Einstein (BE) and Fermi–Dirac (FD) entropies and probabilistic distributions are derived by the combinatorial method of Boltzmann, without Stirling's approximation. The new entropy measures are explicit functions of the probability and degeneracy of each state, and the total number of entities, N. By analysis of the cost of a “binary decision”, exact BE and FD statistics are shown to have profound consequences for the behaviour of quantum mechanical systems.     

Infinite-volume limit of continuous n-particle quantum systems

The Feynman–Kac theorem is applied in order to establish the infinite-volume limit behaviour of the free energy per particle of continuous n-particle quantum systems with bounded separable 2-body interactions defined in the configuration space of particle positions. The mean-field character of such systems is demonstrated.A similar technique is applied to n-particle quantum systems with separable interactions defined in the space of particle momenta and spins. Three examples of systems with separable interactions are given and solved, one of which deals with an electron gas interacting with localized impurity spins in a dilute magnetic alloy (DMA) and extension of Kondo’s resistivity formula for DMA to temperatures close to 0 K.Most of the results are generalizations or more detailed presentations of those published earlier.     

Quantum liquids of particles with generalized statistics

We propose a phenomenological approach to quantum liquids of particles obeying generalized statistics of a fermionic type, in the spirit of the Landau Fermi liquid theory. The approach is developed for fractional exclusion statistics. We discuss both equilibrium (specific heat, compressibility, and Pauli spin susceptibility) and nonequilibrium (current and thermal conductivities, thermopower) properties. Low-temperature quantities have the same temperature dependences as for the Fermi liquid, with the coefficients depending on the statistics parameter. The novel quantum liquids provide an explicit realization of systems with a non-Fermi liquid Lorentz ratio in two and more dimensions. Consistency of the theory is verified by deriving the compressibility and f-sum rules.     

On the effect of fractional statistics on quantum ion acoustic waves

In this paper, I study the effect of a small deviation from the Fermi–Dirac statistics on the quantum ion acoustic waves. For this purpose, a quantum hydrodynamic model is developed based on the Polychronakos statistics, which allows for a smooth interpolation between the Fermi and Bose limits, passing through the case of classical particles. The model includes the effect of pressure as well as quantum diffraction effects through the Bohm potential. The equation of state for electrons obeying fractional statistics is obtained and the effect of fractional statistics on the kinetic energy and the coupling parameter is analyzed. Through the model, the effect of fractional statistics on the quantum ion acoustic waves is highlighted, exploring both linear and weakly nonlinear regimes. It is found that fractional statistics enhance the amplitude and diminish the width of the quantum ion acoustic waves. Furthermore, it is shown that a small deviation from the Fermi–Dirac statistics can modify the type structures, from bright to dark soliton. All known results of fully degenerate and non-degenerate cases are reproduced in the proper limits.     

Plane Waves Propagating in Gases Composed of Composite Particles

The quantum discrete kinetic equations are solved to study the propagation of plane waves in a system of composite particles with hard-sphere interactions and the filling factor (ν) being 1/2. We compare the dispersion relations thus obtained by the relevant Pauli-blocking parameter B which describes the different-statistics particles for the quantum analog of the discrete Boltzmann system when B is positive (Bose gases), zero (Boltzmann gases), and negative (Fermi Gases). We found, as the effective magnetic field being zero (ν = 1/2 using the composite fermion formulation), the electric field effect will induce anomalous dispersion relations.     

Renormalized Perturbation Expansion in Kinetic Theory

Ladder summation techniques are applied to the recently developed c-number diagram expansion for kinetic equations describing the relaxation of quantum fluids. By way of the resummation, the kinetic equations are reformulated in terms of a T-matrix which describes the scattering of two particles influenced by the remaining particles of the system via the particle statistics (Bose or Fermi). At sufficiently high temperatures (Maxwell-Boltzmann statistics), the T-matrix introduced coincides with the usual T-matrix of conventional two-body scattering theory in free space. In low-temperature Fermi systems, the T-matrix differs from the reaction matrix of the Brueckner-Goldstone theory because the “healing” property of the two-body wave function does not obtain.     

Variational principles and linked-cluster exp S expansions for static and dynamic many-body problems

The exp S formalism for the ground state of a many-body system is derived from a variational principle. An energy functional is constructed using certain n-body linked-cluster amplitudes with respect to which the functional is required to be stationary. By using two different sets of amplitudes one either recovers the normal exp S method or obtains a new scheme called the extended exp S method. The same functional can be used also to obtain the average values of any operators as well as the linear response to static perturbations. The theory is extended to treat dynamical phenomena by introducing time dependence to the cluster amplitudes. This allows the calculation of both nonlinear dynamical behaviour and of dynamical linear response and Green's functions. Practical approximation schemes are considered. In a SUB n approximation the m-body amplitudes are restricted to the order $\text{m ? n}$ and the energy functional is a finite-order multinomial in the amplitudes to be variationally determined. It is shown that the solution corresponds to summing well-defined subsets of Goldstone diagrams. These subsets are conveniently specificed in terms of tree structures, the normal or extended generalized time ordering g.t.o. trees. The extended exp S method is in the SUB n approximation able to sum, in addition to the normal SUB n diagrams, a set which contains m-body cluster amplitudes of arbitrarily high order (m > n) in the ordinary sense. The article also discusses how the SUB n truncation schemes must be modified to be able to treat a system with a strong repulsive core in the two-body interaction. The method is formulated for the general cases of Bose and Fermi systems which may or may not conserve total particle number. It is shown that the simplest approximation, SUB 1, in the extended exp S method agrees with the mean field theory, which is the coherent-state approximation in the boson case or the Hartree-Fock approximation in the fermion case. It is argued that the extended exp S method already in low-order approximations can realistically treat a great variety of diverse many-body problems, even including systems which may undergo ground-state phase transitions. A few applications are described in more detail. The Bose liquid is treated in the extended SUB 2 approximation. It is shown that the ground-state results in the uniform limit are exact and agree with the hypernetted-chain approximation. The modifications due to hard-core interactions and the non-linear equations of motion are also discussed in this case. For Fermi systems it is shown that the supercondictive phase transition of the BCS model Hamiltonian and the deformation phase transition of the Lipkin model are properly obtained by the extended exp S method in a low-order approximation.     

A unified grand canonical description of the nonextensive thermostatistics of the quantum gases: Fractal and fractional approach

In this paper, the particles of quantum gases, that is, bosons and fermions are regarded as g-ons which obey fractional exclusion statistics. With this point of departure the thermostatistical relations concerning the Bose and Fermi systems are unified under the g-on formulation where a fractal approach is adopted. The fractal inspired entropy, the partition function, distribution function, the thermodynamics potential and the total number of g-ons have been found for a grand canonical g-on system. It is shown that from the g-on formulation; by a suitable choice of the parameters of the nonextensivity q, the parameter of the fractional exclusion statistics g, nonextensive Tsallis as well as extensive (q=1) standard thermostatistical relations of the Bose and Fermi systems are recovered. Received 17 September 1999     

Equilibration in intermediate-energy heavy-ion collisions within a relativistic mean-field two-fluid model

A two-fluid model with meanσ- andω-fields is formulated for the treatment of heavy-ion collisions at incident energies around 1 GeV/u. In this energy range Fermi and Bose statistics for baryons and pions, respectively, cannot be replaced by Boltzmann statistics. The collisional coupling between the two fluids is formulated in terms of the effective nucleon-nucleon cross-sections in nuclear medium taking Pauli blocking into account. For two counterstreaming nuclear fluids the comparison of results obtained from our relativistic mean-field two-fluid model (RMF-TFM) and the relativistic Landau-Vlasov equation shows good agreement in the gross properties of the equilibration process.     

Large Deviations in Rarefied Quantum Gases

The probability of observing a large deviation (LD) in the number of particles in a region in a dilute quantum gas contained in a much larger region V is shown to decay as exp[–||F], where || is the volume of and F is the change in the appropriate free energy density, the same as in classical systems. However, in contrast with the classical case, where this formula holds at all temperatures and chemical potentials our proof is restricted to rarefied gases, both for the typical and observed density, at least for Bose or Fermi systems. The case of Boltzmann statistics with a bounded repulsive potential can be treated at all temperatures and densities. Fermions on a lattice in any dimension, or in the continuum in one dimension, can be treated at all densities and temperatures if the interaction is small enough (depending on density and temperature), provided one assumes periodic boundary conditions.     

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.     

Non-Abelian Chern-Simons particles in an external magnetic field

The quantum mechanics and thermodynamics of SU(2) non-Abelian Chern-Simons particles (non-Abelian anyons) in an external magnetic field are addressed. We derive the N-body Hamiltonian in the (anti-) holomorphic gauge when the Hilbert space is projected onto the lowest Landau level of the magnetic field. In the presence of an additional harmonic potential, the N-body spectrum depends linearly on the coupling (statistics) parameter. We calculate the second virial coefficient and find that in the strong magnetic field limit it develops a step-wise behavior as a function of the statistics parameter, in contrast to the linear dependence in the case of Abelian anyons. For small enough values of the statistics parameter we relate the N-body partition functions in the lowest Landau level to these of SU(2) bosons and find that the cluster (and virial) coefficients dependence on the statistics parameter cancels.     

Traffic of indistinguishable particles in complex networks

In this paper, we apply a simple walk mechanism to the study of the traffic of many indistinguishable particles in complex networks. The network with particles stands for a particle system, and every vertex in the network stands for a quantum state with the corresponding energy determined by the vertex degree. Although the particles are indistinguishable, the quantum states can be distinguished. When the many indistinguishable particles walk randomly in the system for a long enough time and the system reaches dynamic equilibrium, we find that under different restrictive conditions the particle distributions satisfy different forms, including the Bose--Einstein distribution, the Fermi--Dirac distribution and the non-Fermi distribution (as we temporarily call it). As for the Bose--Einstein distribution, we find that only if the particle density is larger than zero, with increasing particle density, do more and more particles condense in the lowest energy level. While the particle density is very low, the particle distribution transforms from the quantum statistical form to the classically statistical form, i.e., transforms from the Bose distribution or the Fermi distribution to the Boltzmann distribution. The numerical results fit well with the analytical predictions.     

Thermodynamics of two-parameter quantum group Bose and Fermi gases

The high and low temperature thermodynamical properties of the two-parameter deformed quantum group Bose and Fermi gases with SU _p/q (2) symmetry are studied. Starting with a SU _p/q (2)-invariant bosonic as well as fermionic Hamiltonian, several thermodynamical functions of the system such as the average number of particles, internal energy and equation of state are derived. The effects of two real independent deformation parameters p and q on the properties of the systems are discussed. Particular emphasis is given to a discussion of the Bose-Einstein condensation phenomenon for the two-parameter deformed quantum group Bose gas. The results are also compared with earlier undeformed and one-parameter deformed versions of Bose and Fermi gas models. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

Multilocal Fermionization

We present a simple isomorphism between the algebra of one real chiral Fermi field and the algebra of n real chiral Fermi fields. The isomorphism preserves the vacuum state. This is possible by a “change of localization”, and gives rise to new multilocal symmetries generated by the corresponding multilocal current and stress–energy tensor. The result gives a common underlying explanation of several remarkable recent results on the representation of the free Bose field in terms of free Fermi fields (Anguelova, arXiv:1112.3913, 2011; Anguelova, arXiv:1206.4026, 2012), and on the modular theory of the free Fermi algebra in disjoint intervals (Casini and Huerta, Class Quant Grav 26:185005, 2009; Longo et al., Rev Math Phys 22:331–354, 2010)     

On bound states in the continuum of N-body systems and the virial theorem

We prove generalized versions of the quantum mechanical virial theorem and apply them to the investigation of the spectrum of N body Hamiltonians. We show, in particular, that for N particles interacting through 2-body potentials which may have singularities but “don't wiggle too much,” no positive energy bound state can exist. We also prove results on the absence of bound states with energy bigger than some value E⁰ ? − ∞ and extend them to the case of N particles interacting through ν-body forces (ν = 1, 2,…, N) and with an external electromagnetic field. Also some remarks for the case of a Dirac electron in an external potential are given as well as for some problems with boundary conditions. A by-product of this investigation is the unitarity of the S matrix and the strong asymptotic completeness for systems of N particles interacting by 2-body forces which are not restricted to be purely repulsive.     

Distribution functions for a two-dimensional non-interacting quantum electron gas in an external magnetic field

The exact n-body distribution functions are calculated for a two-dimensional, non-interacting quantum electron gas in an external magnetic field for any temperature and density. At low tempertures and filled lowest Landau level (LLL), these functions are identical to the exact distribution functions obtained by Jancovici [1981, Phys. Rev. Lett., 46, 386] for the classical two-dimensional one-component plasma (2DOCP) at the special plasma parameter Γ = 2, thus establishing that the 2DOCP provides an exact classical Boltzmann factor which describes the ideal LLL quantum state associated with the integral quantum Hall effect.     

A Note on the Quantum Widom–Rowlison Model

Using the renormalization methods we show that the symmetry breaking in the quantum Widom–Rowlison model of particles obeying Boltzmann statistics occurs at any value of the inverse temperature >0 once the activity of the particles is sufficiently large.