Statistical mechanics of the isothermal lane-emden equation

For classical point particles in a box with potential energy H^(N)=N ^–1(1/2) _ij=1 ^N V(x _i,x _j) we investigate the canonical ensemble for largeN. We prove that asN the correlation functions are determined by the global minima of a certain free energy functional. Locally the distribution of particles is given by a superposition of Poisson fields. We study the particular case =[–L, L] andV(x, y)=}- cos(x–y),L}>0, }>0.References     

The effect of a constant number of particles on the pair correlation function and the density profile in a one-dimensional lattice gas

A one-dimensional lattice gas (Ising model) of lengthL and with nearest-neighbor couplingJ is considered in a canonical ensemble with fixed number of particlesN=L/2. Exact expressions and asymptotic forms for largeL are derived for the density-density correlation function, using periodic boundary conditions, and for the density (magnetization) profile, using antisymmetric boundary conditions. The density-density correlation function,g, assumes for temperaturesT> T, withT = 2J(_BlnL)^–1 and forL large, the formg(x) =g ^gc(x) +BL ^–1 +a(x)L ^–1 +O(L^–2) wherex is a distance between considered lattice sites,B is known from earlier work of Lebowitz and Percus,(1b) anda(x) decays exponentially forx . For TT, the correlation function and the density profile behave differently, the latter exhibiting a step in the middle of the interface.     

Quantum groupSU(1, 1) ⋊ Z2 and “super-tensor” products

A quantum analogue of the groupSU(1,1)Z ₂—the normalizer ofSU(1, 1) inSL ₂(C)—is introduced and studied. Although there isno correctly defined tensor product in the category of *-representations of the quantum algebraC[SU(1, 1)] _q of regular functions, some categories of *-representations ofC[SU(1, 1)Z ₂] _q turn out to be endowed with a certainZ ₂-graded structure which can be considered as a super-generalization of the monoidal category structure. This quantum effect may be considered as a step to understanding the concept of quantum topological locally compact group.In fact, there seems to be afamily of quantum groupsSU(1, 1)Z ₂ parameterized by unitary characters T ¹ of the fundamental group of the two-dimensional symplectic leaf ofSU(1, 1)/T, whereT is the subgroup of diagonal matrices.It is shown that thequasi-classical analogues of the results of the paper are connected with the decomposition of Schubert cells of the flag manifoldSL ₂(C)_R/B (whereB is the Borel subgroup of upper-triangular matrices) into symplectic leaves.Supported by the Rosenbaum Fellowship.     

Realizability of a model in infinite statistics

Following Greenberg and others, we study a space with a collection of operatorsa(k) satisfying the q-mutator relationsa(l)a (k)a(l)= _k,l (corresponding forq=±1 to classical Bose and Fermi statistics). We show that then!×n! matrixA _n (q) representing the scalar products ofn-particle states is positive definite for alln ifq lies between –1 and +1, so that the commutator relations have a Hilbert space representation in this case (this has also been proved by Fivel and by Bozejko and Speicher). We also give an explicit factorization ofA _n (q) as a product of matrices of the form(1–q ^jT)^±1 with 1jn andT a permutation matrix. In particular,A _n (q) is singular if and only ifq ^M=1 for some integerM of the formk ²–k, 2kn.     

A comment on an exact solution with a stiff equation of state

Recently, an analytical stellar model with a stiff equation of state and density distribution= _c (1-r ²/r _o ² ) was presented. We show that such a solution cannot exist.     

Branching rules for conformal embeddings

We give explicit formulas for the branching rules of the conformal embeddingssu(n(n+1)/2)₁su(n) _n+2,su(n(n–1)/2)₁su(n) _n–2,sp(n)₁so(n)₄su(2) _n , andso(m+n)₁so(m)₁ so(n)₁ withm andn odd.This research was supported in part by CONICET, CONICOR and SECYT.     

A numerical study of sparse random matrices

A numerical study is presented for the eigensolution statistics of largeN×N real and symmetric sparse random matrices as a function of the mean numberp of nonzero elements per row. The model shows classical percolation and quantum localization transitions atp _c =1 andp _q >1, respectively. In the rigid limitp=N we demonstrate that the averaged density of states follows the Wigner semicircle law and the corresponding nearest energy-level-spacing distribution functionP(S) obeys the Wigner surmise. In the very sparse matrix limitpN, withp>p _q a singularity (E))1/¦E¦ is found as¦E¦ 0 and exponential tails develop in the high-¦E¦ regions, but theP(S) distribution remains consistent with level repulsion. The localization properties of the model are examined by studying both the eigenvector amplitude and the density fluctuations. The valuep _q 1.4 is roughly estimated, in agreement with previous studies of the Anderson transition in dilute Bethe lattices.     

The Generalized Canonical Ensemble and Its Universal Equivalence with the Microcanonical Ensemble

This paper shows for a general class of statistical mechanical models that when the microcanonical and canonical ensembles are nonequivalent on a subset of values of the energy, there often exists a generalized canonical ensemble that satisfies a strong form of equivalence with the microcanonical ensemble that we call universal equivalence. The generalized canonical ensemble that we consider is obtained from the standard canonical ensemble by adding an exponential factor involving a continuous function g of the Hamiltonian. For example, if the microcanonical entropy is C², then universal equivalence of ensembles holds with g taken from a class of quadratic functions, giving rise to a generalized canonical ensemble known in the literature as the Gaussian ensemble. This use of functions g to obtain ensemble equivalence is a counterpart to the use of penalty functions and augmented Lagrangians in global optimization. linebreak Generalizing the paper by Ellis et al. [J. Stat. Phys. 101:999–1064 (2000)], we analyze the equivalence of the microcanonical and generalized canonical ensembles both at the level of equilibrium macrostates and at the thermodynamic level. A neat but not quite precise statement of one of our main results is that the microcanonical and generalized canonical ensembles are equivalent at the level of equilibrium macrostates if and only if they are equivalent at the thermodynamic level, which is the case if and only if the generalized microcanonical entropy s–g is concave. This generalizes the work of Ellis et al., who basically proved that the microcanonical and canonical ensembles are equivalent at the level of equilibrium macrostates if and only if they are equivalent at the thermodynamic level, which is the case if and only if the microcanonical entropy s is concave.     

Decay of the Two-Point Function in One-Dimensional O(N) Spin Models with Long-Range Interactions

Using Griffiths and Lieb–Simon type inequalities, it is shown that the two-point function of ferromagnetic spin models with N components in one dimension decays like the interaction J(n)n ^– provided that 1N4 and T>T _c.     

Non-linear multi-plane wave solutions of self-dual Yang-Mills theory

New solutions of self-dual Yang-Mills (SDYM) equations are constructed in Minkowski space-time for the gauge groupSL(2, ). After proposing a Lorentz covariant formulation of Yang's equations, a set of Ansätze for exact non-linear multiplane wave solutions are proposed. The gauge fields are rational functions ofe ^x·ki (K _i ² =0, 1iN) for these Ansätze. At least, three families of multisoliton type solutions are derived explicitly. Their asymptotic behaviour shows that non-linear waves scatter non-trivially in Minkowski SDYM.On leave from LPTHE Université Paris VI, 4, Place Jussieu, Tour 16, ler étage, F-75230 Paris Cedex 05, France     

Scaled Limit and Rate of Convergence for the Largest Eigenvalue from the Generalized Cauchy Random Matrix Ensemble

In this paper, we are interested in the asymptotic properties for the largest eigenvalue of the Hermitian random matrix ensemble, called the Generalized Cauchy ensemble GCyE, whose eigenvalues PDF is given by

Orthospaces and quantum logic

In this paper we construct the ortholattices arising in quantum logic starting from the phenomenologically plausible idea of a collection of ensembles subject to passing or failing various tests. A collection of ensembles forms a certain kind of preordered set with extra structure called anorthospace; we show that complete ortholattices arise as canonical completions of orthospaces in much the same way as arbitrary complete lattices arise as canonical completions of partially ordered sets. We also show that the canonical completion of an orthospace of ensembles is naturally identifiable as the complete lattice of properties of the ensembles, thereby revealing exactlywhy ortholattices arise in the analysis of tests or experimental propositions. Finally, we axiomatize the hitherto implicit concept of test and show how they may be correlated with properties of ensembles.     

The one dimensional anderson model with off diagonal disorder: band centre anomaly

We study theE-dependence of the Lyapounov exponent <(E)> of an electron with energyE in the one dimensional Anderson model with off diagonal disorder. In the neighbourhood of the band centre we find for nonzero disorder E)>log^–1 E0 forE0, but all even moments of Re(E) diverge logarithmically. As the probability of Re (E)=0 decreases to zero forE0 we conclude that the electron is always exponentially localised.     

Projective quantum spaces

Associated to the standard SU _q (n) R-matrices, we introduce quantum spheresS _q ^2n-1 , projective quantum spaces _q ^n-1 , and quantum Grassmann manifoldsG _k( _q ⁿ ). These algebras are shown to be homogeneous spaces of standard quantum groups and are also quantum principle bundles in the sense of T. Brzeziski and S. Majid.     

The Micro-Canonical Point Vortex Ensemble: Beyond Equivalence

The fluid limit N is constructed for a sequence of ensembles of N classical point vortices in a finite domain ² whose ensemble densities (w.r.t. Liouville measure) are Gaussian approximations to (E-H). Letting the variance 0 after N has been taken, one recovers the special class of nonlinear stationary Euler flows that is expected from the micro-canonical ensemble. The construction improves over previous ones which either had to regularize the logarithmic singularities of the point vortex Hamiltonian or had to assume equivalence of ensembles. In particular, nonequivalence between micro-canonical and canonical ensemble prevails for certain geometries where conditionally stable configurations with negative 'global vortex pair-specific heat' can and do exist in the micro-canonical but not in the canonical ensemble.     

New inequalities from local realism

The probabilistic formulation of local realism is shown to imply the existence of physically meaningful limits for arbitrary linear combinations of joint probabilities. The set of the so generated inequalities (setA) is wider than the previously known set of inequalities for linear combinations of correlation functions (setB). One particular inequality of the setA is shown to be violated by the probabilities of the Garg-Mermin model. The same model satisfies instead all the inequalities of the setB. As a consequence, the Garg-Mermin model is nonlocal and the setA provides physical restrictions not contained in the setB. 1. In the adopted formalism it is implicitly assumed that physical properties of the type are not created in the act of measurement. IfB(b) is measured on the systems, the setT is split into two parts,T(b _±), corresponding to the resultsB(b) = ±1, respectively. AlsoS is split intoS(b _±) from the existing correlation between and systems. If it is possible to predict that a measurement ofA(a) on the's of, say,S(b ₊) will give the results ±1 with respective probabilitiesP _±, then, on the basis of the probabilistic criterion of reality, we can attribute a physical property ₊ toS(b ₊) such that p(a ₊, ₊) is the probability ofA(a) = +1 inS(b ₊), p(a _–, ₊) is the probability ofA(a) = –1 inS(b ₊).It is natural to assume that ₊ belongs toS(b ₊) also ifA(a) isnot measured. In so doing, we exclude that future measurements create, with a retroaction in time, the physical properties of the statistical ensembles on which these measurements are performed.     

Special energies and special frequencies

Special frequencies have been asserted to be zeros of the density of frequencies corresponding to a random chain of coupled oscillators. Our investigation includes both this model and the random one-dimensional Schrödinger operator describing an alloy or its discrete analogue. Using the phase method we exactly determine a bilateral Lific asymptotic of the integrated density of statesk(E) at special energiesE _s , which is not only of the classical type exp(–c/|E–E _s |^1/2) but also exp(–c/|E–E _s |) is a typical behaviour. In addition, other asymptotics occur, e.g. |E–E _s | ^c , which show thatk(E) need not bek .     

On the interface stiffness of the random field ising model

Recent results of Grinstein, Ma, Villain and Binder on interface roughening incontinuum andlattice random field Ising models are related by introducing an effective interface stiffness function {ei247-1}. Ford3 dimensions the continuum theory is shown to be valid for non-zero random field strengthh for all temperatures and on a length scaleL>l _d (h,T) _d (h,T). Ford=2 and smallT a smeared spin-glass transition occurs at ₂(h,T)h. It is argued, that for 3<d<5 interface roughening occurs only forh larger than a critical field strengthh _R (T).     

Finite size scaling at first order phase transitions?

It is demonstrated that finite size scaling at first order phase transitions is something basically very simple: As the number of particlesN in the system goes to infinity,s _N (), the entropy per particle, rapidly approaches its limiting behaviours (). Onces () has been determined, the thermal behaviour of the infinite system is completely known and in case of a first order phase transition the specific heat exhibitis a -function singularity. If, however, the specific heatc _N (T) per particle is calculated from the canonical partition functionZ _N ()=d exp {N[s _N ()-]}, then even ifs _N () is replaced by its limiting forms (),c _N (T) only exhibits a peak with a finite maximum value proportional toN which is due to the explicit factorN in front of the angular bracket in the exponent. This is theN-dependence which has recently been called finite size scaling at first order phase transitions. The entropys _N () can very efficiently be determined in the dynamical ensemble.     

On orthogonal and symplectic matrix ensembles

The focus of this paper is on the probability,E (O;J), that a setJ consisting of a finite union of intervals contains no eigenvalues for the finiteN Gaussian Orthogonal (=1) and Gaussian Symplectic (=4) Ensembles and their respective scaling limits both in the bulk and at the edge of the spectrum. We show how these probabilities can be expressed in terms of quantities arising in the corresponding unitary (=2) ensembles. Our most explicit new results concern the distribution of the largest eigenvalue in each of these ensembles. In the edge scaling limit we show that these largest eigenvalue distributions are given in terms of a particular Painlevé II function.