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arXiv:cond-mat/9907042

Thermodynamic Limit in Chern-Simons System of Particles with MB Statistics

W. I. Skrypnik

Published Jul 2, 1999• 0 citations
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Abstract

The reduced density matrices (RDMs) are calculated in the thermodynamic limit for the Chern-Simons non-relativistic particle system and Maxwell-Boltzmann (MB) statistics. It is established that they are zero outside of a diagonal and well-behaved after a renormalization, depending on an arbitrary real number, if the condition of neutrality holds.

DOI
10.48550/arxiv.cond-mat/9907042

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The reduced density matrices (RDMs) are calculated in the thermodynamic limit for the Chern-Simons non-relativistic particle system and Maxwell-Boltzmann (MB) statistics. It is established that they are zero outside of a diagonal and well-behaved after a renormalization, depending on an arbitrary real number, if the condition of neutrality holds.
Gibbs states and dynamics are described for systems of charged particles interacting via eective electromagnetic eld, characterized by a (singular) vector potential with zero electrostatic component(magnetic interaction).The mean{ eld type limit for the Gibbs (grand canonical) correlation functions for the 2-d Chern{Simons (CS) system (the vector potential of the n-particle system is a skew derivative of the Coulomb potential energy of the n-particle system) is performed.It is found that the limit is not unique.For 1-d quantum system of particles with Maxwell{Boltzmann statistics and analog of CS interaction ( the vector potential for the n-particle system is the derivative of the 1-d Coulomb potential energy of the system) the Gibbs reduced density matrices are found in the thermodynamic limit.It is shown that in 1-d systems singular interactions may produce anyonic statistics.
Quantum systems of particles interacting via an effective electromagnetic potential with zero electrostatic component are considered (magnetic interaction). It is assumed that the j th component of the effective potential for n particles equals the partial derivative with respect to the coordinate of the jth particle of “magnetic potential energy” of n particles almost everywhere. The reduced density matrices for small values of the activity are computed in the thermodynamic limit for d-dimensional systems with short-range pair magnetic potentials and for one-dimensional systems with long-range pair magnetic interaction, which is an analog of the interaction of three-dimensional Chern-Simons electrodynamics (“magnetic potential energy” coincides with the one-dimensional Coulomb (electrostatic) potential energy).
The new approach in quantum statistical mechanics proposed by the author in previous publications is extended to systems of nonzero-spin particles. It is shown that properties of particles relevant to their spin can well be incorporated into the approach proposed. In so doing the principal ideas of the approach undergo no changes and lead to a hierarchy of equations for reduced density matrices at thermodynamic equilibrium that generalizes the hierarchy obtained earlier for spinless particles. Thermodynamics based on the hierarchy is worked out as well. The hierarchy and relevant thermodynamics enable one to calculate both thermodynamical properties of a system of particles and structural ones. With the help of perturbation expansions it is shown that the hierarchy has a solution and the solution is unique at least in the case of a weakly interacting system. Using the example of a hard-sphere system wherein triplet correlations are disregarded it is demonstrated that the hierarchy may serve as basis for deriving various results. Comparison of these results with results of other theoretical treatments evidences complete agreement within the limits allowed by the approximation used.
Spectral properties of the reduced density matrix (RDM) of permutational invariant quantum many-body systems are investigated. The RDM block diagonalization which accounts for all symmetries of the Hamiltonian is achieved. The analytical expression of the RDM spectrum is provided for arbitrary parameters and rigorously proved in the thermodynamical limit. The existence of several sum rules and recurrence relations among RDM eigenvalues is also demonstrated and the distribution function of RDM eigenvalues (including degeneracies) characterized. In particular, we prove that the distribution function approaches a two-dimensional Gaussian in the limit of large subsystem sizes n>>1. As a physical application we discuss the von Neumann entropy (VNE) of a block of size n for a system of hard-core bosons on a complete graph, as a function of n and of the temperature T. The occurrence of a crossover of VNE from purely logarithmic behavior at T=0 to a purely linear behavior in n for T≥Tc, is demonstrated.
In this work we present a classical kinetic model of intermediate statistics. In the case of Brownian particles we show that the Fermi-Dirac (FD) and Bose-Einstein (BE) distributions can be obtained, just as the Maxwell-Boltzmann (MB) distribution, as steady states of a classical kinetic equation that intrinsically takes into account an exclusion-inclusion principle. In our model the intermediate statistics are obtained as steady states of a system of coupled nonlinear kinetic equations, where the coupling constants are the transmutational potentials ${\mathrm{\ensuremath{\eta}}}_{\mathrm{\ensuremath{\kappa}}\mathrm{\ensuremath{\kappa}}\ensuremath{'}}$. We show that, besides the FD-BE intermediate statistics extensively studied from the quantum point of view, we can also study the MB-FD and MB-BE ones. Moreover, our model allows us to treat the three-state mixing FD-MB-BE intermediate statistics. For boson and fermion mixing in a D-dimensional space, we obtain a family of FD-BE intermediate statistics by varying the transmutational potential ${\mathrm{\ensuremath{\eta}}}_{\mathrm{BF}}$. This family contains, as a particular case, when ${\mathrm{\ensuremath{\eta}}}_{\mathrm{B}\mathrm{F}=0}$, the quantum statistics recently proposed by L. Wu, Z. Wu, and J. Sun [Phys. Lett. A 170, 280 (1992)]. When we consider the two-dimensional FD-BE statistics, we derive an analytic expression of the fraction of fermions. When the temperature T\ensuremath{\rightarrow}\ensuremath{\infty}, the system is composed by an equal number of bosons and fermions, regardless of the value of ${\mathrm{\ensuremath{\eta}}}_{\mathrm{BF}}$. On the contrary, when T\ensuremath{\rightarrow}0, ${\mathrm{\ensuremath{\eta}}}_{\mathrm{BF}}$ becomes important and, according to its value, the system can be completely bosonic or fermionic, or composed both by bosons and fermions.