On Ruelle’s Construction of the Thermodynamic Limit for the Classical Microcanonical Entropy

In 1969 Ruelle published his construction of the thermodynamic limit, in the sense of Fisher, for the quasi-microcanonical entropy density of classical Hamiltonian N-body systems with stable and tempered pair interactions. Here, “quasi-microcanonical” refers to the fact that he discussed the entropy defined with a regularized microcanonical measure as ln (N!⁻¹ ∫ χ _{{ℰ−▵ℰ<H<ℰ}}d^6N X) rather than defined with the proper microcanonical measure as ln (N!⁻¹ ∫ δ(ℰ−H) d^6N X). Replacing δ(ℰ−H) by χ _{{ℰ−▵ℰ<H<ℰ}} seems to have become the standard procedure for rigorous treatments of the microcanonical ensemble hence. In this note we make a very elementary technical observation to the effect that Ruelle’s proof (still based on regularization) does establish the thermodynamic limit also for the entropy density defined with the proper microcanonical measure. We also show that with only minor changes in the proof the regularization of δ(ℰ−H) is actually not needed at all.     

Microcanonical Entropy and Dynamical Measure of Temperature for Systems with Two First Integrals

We consider a generic classical many particle system described by an autonomous Hamiltonian H(x ¹,…,x ^N+2) which, in addition, has a conserved quantity V(x ¹,…,x ^N+2)=v, so that the Poisson bracket {H,V} vanishes. We derive in detail the microcanonical expressions for entropy and temperature. We show that both of these quantities depend on multidimensional integrals over sub-manifolds given by the intersection of the constant energy hyper-surfaces with those defined by V(x ¹,…,x ^N+2)=v. We show that temperature and higher order derivatives of entropy are microcanonical observable that, under the hypothesis of ergodicity, can be calculated as time averages of suitable functions. We derive the explicit expression of the function that gives the temperature.     

Mean square number fluctuation for a fermion source and its dependence on neutrino mass for the universal cosmic neutrino background

Using the general formulation for obtaining chemical potentialμ of an ideal Fermi gas of particles at temperature T, with particle rest mass m₀ and average density 〈N〉/V, the dependence of the mean square number fluctuation 〈ΔN ²〉/V on the particle mass m₀ has been calculated explicitly. The numerical calculations are exact in all cases whether rest mass energym ₀c² is very large (non-relativistic case), very small (ultra-relativistic case) or of the same order as the thermal energy k_BT. Application of our results to the detection of the universal very low energy cosmic neutrino background (CNB), from any of the three species of neutrinos, shows that it is possible to estimate the neutrino mass of these species if from approximate experimental measurements of their momentum distribution one can extract, someday, not only the density 〈N _v〉/V but also the mean square fluctuation 〈Δ _v ² 〉/V. If at the present epoch, the universe is expanding much faster than thermalization rate for CNB, it is shown that our analysis leads to a scaled neutrino massm _v instead of the actual massm _0v .     

Efficient Hopfield pattern recognition on a scale-free neural network

Neural networks are supposed to recognise blurred images (or patterns) of N pixels (bits) each. Application of the network to an initial blurred version of one of P pre-assigned patterns should converge to the correct pattern. In the “standard" Hopfield model, the N “neurons” are connected to each other via N² bonds which contain the information on the stored patterns. Thus computer time and memory in general grow with N². The Hebb rule assigns synaptic coupling strengths proportional to the overlap of the stored patterns at the two coupled neurons. Here we simulate the Hopfield model on the Barabási-Albert scale-free network, in which each newly added neuron is connected to only m other neurons, and at the end the number of neurons with q neighbours decays as 1/q ³. Although the quality of retrieval decreases for small m, we find good associative memory for 1 ≪ m ≪ N. Hence, these networks gain a factor N/m ≫ 1 in the computer memory and time. Received 12 January 2003 Published online 11 April 2003 RID="a" ID="a"e-mail: stauffer@thp.uni-koeln.de     

Interlaced Particle Systems and Tilings of the Aztec Diamond

Motivated by the problem of domino tilings of the Aztec diamond, a weighted particle system is defined on N lines, with line j containing j particles. The particles are restricted to lattice points from 0 to N, and particles on successive lines are subject to an interlacing constraint. It is shown that this particle system is exactly solvable, to the extent that not only can the partition function be computed exactly, but so too can the marginal distributions. These results in turn are used to give new derivations within the particle picture of a number of known fundamental properties of the tiling problem, for example that the number of distinct configurations is 2 ^N(N+1)/2, and that there is a limit to the GUE minor process, which we show at the level of the joint PDFs. It is shown too that the study of tilings of the half Aztec diamond—not known from earlier literature—also leads to an interlaced particle system, now with successive lines 2n−1 and 2n (n=1,…,N/2−1) having n particles. Its exact solution allows for an analysis of the half Aztec diamond tilings analogous to that given for the Aztec diamond tilings.     

Aerosol deposition of detonation nanodiamonds used as nucleation centers for the growth of nanocrystalline diamond films and isolated particles

The aerosol deposition of detonation nanodiamonds (DNDs) on a silicon substrate is comprehensively studied, and the possibility of subsequent growth of nanocrystalline diamond films and isolated particles on substrates coated with DNDs is demonstrated. It is shown that a change in the deposition time and the weight concentration of DNDs in a suspension in the range 0.001–1% results in a change in the shape of DND agglomerates and their number per unit substrate surface area N _s from 10⁸ to 10¹¹ cm⁻². Submicron isolated diamond particles are grown on a substrate coated with DND agglomerates at N _s ≈ 108 cm⁻² using microwave plasma-enhanced chemical vapor deposition. At N _s ≈ 10¹⁰ cm⁻², thin (∼100 nm) nanodiamond films with a root-mean-square surface roughness less than 15 nm are grown.     

Swelling of two-dimensional polymer rings by trapped particles

The mean area of a two-dimensional Gaussian ring of N monomers is known to diverge when the ring is subject to a critical pressure differential, p _c ∼ N ^-1. In a recent publication (Eur. Phys. J. E 19, 461 (2006)) we have shown that for an inextensible freely jointed ring this divergence turns into a second-order transition from a crumpled state, where the mean area scales as 〈A〉 ∼ N, to a smooth state with 〈A〉 ∼ N ². In the current work we extend these two models to the case where the swelling of the ring is caused by trapped ideal-gas particles. The Gaussian model is solved exactly, and the freely jointed one is treated using a Flory argument, mean-field theory, and Monte Carlo simulations. For a fixed number Q of trapped particles the criticality disappears in both models through an unusual mechanism, arising from the absence of an area constraint. In the Gaussian case the ring swells to such a mean area, 〈A〉 ∼ NQ, that the pressure exerted by the particles is at p _c for any Q. In the freely jointed model the mean area is such that the particle pressure is always higher than p _c, and 〈A〉 consequently follows a single scaling law, 〈A〉 ∼ N ² f (Q/N), for any Q. By contrast, when the particles are in contact with a reservoir of fixed chemical potential, the criticality is retained. Thus, the two ensembles are manifestly inequivalent in these systems. An erratum to this article is available at .     

Higher-order semiclassical energy expansions for potentials with non-integer powers

In this paper, we present a semiclassical eigenenergy expansion for the potential |x|^α when α is a positive rational number of the form2n/m (n is a positive integer and m is an odd positive integer). Remarkably, this expansion is found to be identical to the WKB expansion obtained for the potentialx ^N(N-even), if2n/m is replaced byN. Taking the limitm → 2 of the above expansion, we obtain an explicit asymptotic energy expansion of symmetric odd power potentials |x|^2j+1 (j- positive integer). We then show how to develop approximate semiclassical expansions for potentials |x|^α when α is any positive real number.     

On the stability of symmetric four-particle molecules

The stability of four-particle molecules m ₁⁺ m ₂⁺ m ₃⁻m₄⁻ with unit charges of particles is studied for different masses of the particles. It is shown that all molecules containing two like particles with masses m ₃ = m ₄ are stable against dissociation. Their stability results from the stability of symmetric molecules with m ₁ = m ₂ and m ₃ = m ₄ and from the stability of molecules with masses of particles m ₃ = m ₄ and m ₁ = ∞ containing one infinitely heavy particle.     

ωN final state interactions and ω-meson production from heavy-ion collisions

We calculate the elastic and inelastic ωN→ωN, →πN, →ρN, →ρπN, →ππN, →σN reactions within a boson exchange approximation where the ωρπ coupling constant and form factor are fixed by the reaction πN→ωN in comparison to the experimental data. We find rather large ωN cross sections at low relative momenta of the ω-meson which leads to a substantial broadening of the ω-meson width in nuclear matter. The implications of the ωN final state interactions are studied for ω production in ¹²C +¹²C, ⁴⁰Ca +⁴⁰Ca and ⁵⁸Ni +⁵⁸Ni reactions at about 2 · A GeV within the HSD transport approach; the drastic changes of the transverse mass spectra relative to a general m _T-scaling (for π⁰ and η mesons) might be controlled experimentally by the TAPS Collaboration. Received: 28 April 1999 / Revised version: 7 June 1999     

On the Mean-Field Spherical Model

Exact solutions are obtained for the mean-field spherical model, with or without an external magnetic field, for any finite or infinite number N of degrees of freedom, both in the microcanonical and in the canonical ensemble. The canonical result allows for an exact discussion of the loci/ of the Fisher zeros of the canonical partition function. The microcanonical entropy is found to be nonanalytic for arbitrary finite N. The mean-field spherical model of finite size N is shown to be equivalent to a mixed isovector/isotensor σ-model on a lattice of two sites. Partial equivalence of statistical ensembles is observed for the mean-field spherical model in the thermodynamic limit. A discussion of the topology of certain state space submanifolds yields insights into the relation of these topological quantities to the thermodynamic behavior of the system in the presence of ensemble nonequivalence.     

Zeros of Airy Function and Relaxation Process

One-dimensional system of Brownian motions called Dyson’s model is the particle system with long-range repulsive forces acting between any pair of particles, where the strength of force is β/2 times the inverse of particle distance. When β=2, it is realized as the Brownian motions in one dimension conditioned never to collide with each other. For any initial configuration, it is proved that Dyson’s model with β=2 and N particles, $\mbox {\boldmath $\mbox {\boldmath , is determinantal in the sense that any multitime correlation function is given by a determinant with a continuous kernel. The Airy function (z){\rm Ai}(z) is an entire function with zeros all located on the negative part of the real axis ℝ. We consider Dyson’s model with β=2 starting from the first N zeros of Ai(z){\rm Ai}(z) , 0>a ₁>⋅⋅⋅>a _N , N≥2. In order to properly control the effect of such initial confinement of particles in the negative region of ℝ, we put the drift term to each Brownian motion, which increases in time as a parabolic function: Y _j (t)=X _j (t)+t ²/4+{d ₁+∑ _ℓ=1 ^N (1/a _ℓ )}t,1≤j≤N, where d₁=Ai￠(0)/Ai(0)d_{1}={\rm Ai}'(0)/{\rm Ai}(0) . We show that, as the N→∞ limit of $\mbox {\boldmath $\mbox {\boldmath , we obtain an infinite particle system, which is the relaxation process from the configuration, in which every zero of (z){\rm Ai}(z) on the negative ℝ is occupied by one particle, to the stationary state m_Ai\mu_{{\rm Ai}} . The stationary state m_Ai\mu_{{\rm Ai}} is the determinantal point process with the Airy kernel, which is spatially inhomogeneous on ℝ and in which the Tracy-Widom distribution describes the rightmost particle position.     

Capacitance-voltage behaviour inp-type ɛ-GaSe single crystal ∥c at different temperatures

Summary  Capacitance-voltage measurements have been carried out onp-type ɛ-GaSe single crystal ∥c in the temperature range 300 to 360 K, with applied voltages of -1, 0 and +1 V. TheC-V measurements in this temperature range have shown a shift in capacitanceC and conductanceG to the higher values with an increase in temperature. The depletion layer widthW, the Debye length L_D and the doping densityN _α have been worked out and plots ofN _α vs. W have shown a decrease inW with an increase in temperature. The plots of L_D vs. N _α vary as 1/N_α^1/2, which gives N_αL_D ⋍ 3.3 × 10¹¹ charges/m² for doping density of 10¹⁶m⁻³. The values ofG at different temperatures have been used to obtain the activation energies, which are found to be ΔE ⋍ 0.11 eV for -1 and +1 V applied voltages, and ΔE ⋍ 0.06 eV for zero volt. The authors of this paper have agreed to not receive the proofs for correction.     

A Stochastic Model Associated with Enskog Equation and Its Kinetic Limit

 We examine a system of particles in which the particles travel deterministically in between stochastic collisions. The collisions are elastic and occur with probability ɛ ^d when two particles are at a distance σ. When the number of particles N goes to infinity and Nɛ ^d goes to a nonzero constant, we show that the particle density converges to a solution of the Enskog Equation. Received: 29 January 2002 / Accepted: 30 July 2002 Published online: 14 November 2002 RID="*" ID="*" Research supported in part by NSF Grant DMS-0072666     

Quasi-long-range order in the random anisotropy Heisenberg model

The random field and random anisotropy N-vector models are studied with the functional renormalization group in 4−ε dimensions. The random anisotropy Heisenberg (N=3) model has a phase with an infinite correlation length at low temperatures and weak disorder. The correlation function of the magnetization obeys a power law 〈m(r ₁)m(r ₂)〉∼|r ₁−r ₂|^{− 0.62ε}. The magnetic susceptibility diverges at low fields as χ∼H ^−1+0.15ε. In the random field N-vector model the correlation length is finite at arbitrarily weak disorder for any N>3. Pis’ma Zh. éksp. Teor. Fiz. 70, No. 2, 130–135 (25 July 1999) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Exact Analytical Solution of the Schrödinger Equation for an N-Identical Body-Force System

A mathematical method is presented for solving the Schr?dinger equation for a system of identical body forces. The N-body forces are more easily introduced and treated within the hyperspherical harmonics. The problem of the N-body potential has been used at the level of both classical and quantum mechanics. The hypercentral interacting potential is assumed to depend on the hyperradius x = (ξ₁² + ξ₂² + ⋯ + ξ_N−1²)^1/2 only, where ξ₁,ξ₂,…,ξ_N−1 are Jacobi relative coordinates which are functions of N-particle relative positions r₁₂,r₂₃,…,r_N1. The problem of the harmonic oscillator and the Coulomb-type potential has been widely studied in different contexts. Using the N-body potential V(x) = ax² + bx − (c/x) as an example, and assuming an ansatz for the eigenfunction, an exact analytical solution of the Schr?dinger equation for an N-body system in three dimensions is obtained. This method is also applicable to some other types of potentials for N-identical interacting particles.     

Quasi-Stationary States for Particle Systems in the Mean-Field Limit

We prove that the N particles approximation of a class of stable stationary solutions of the Vlasov equation is uniformly valid on a time scale N ^β for β>0 (explicitly given in various cases) much longer than the usual log N scale. The vortex blob method in dimension 2 is also discussed. The result applies to a class of stationary solutions more general than in a previous work.     

PHOBOS at RHIC: Some global observations

Particle production in Au+Au collisions has been measured in the PHOBOS experiment at RHIC for a range of collision energies for a large span of pseudorapidities, |η| < 5.4. Three empirical observations have emerged from this data set which require theoretical examination. First, there is clear evidence of limiting fragmentation. Namely, particle production in central Au + Au collisions, when expressed as dN/dη′ ( η′ ≡ – y_beam), becomes energy independent at high energy for a broad region of η′ around η′ = 0. This energy-independent region grows with energy, allowing only a limited region (if any) of longitudinal boost-invariance. Second, there is a striking similarity between particle production in e⁺e⁻and Au + Au collisions (scaled by the number of participating nucleon pairs). Both the total number of produced particles and the longitudinal distribution of produced particles are approximately the same in e⁺e⁻and in scaled Au + Au. This observation This presentation is based in large part on the PHOBOS summary talk by M Baker at the16th Int. Conf. on Ultrarelativistic Nucleus- Nucleus Collisions, Quark Matter 2002, Nantes, France was not predicted and has not been explained. Finally, particle production has been found to scale approximately with the number of participating nucleon pairs for (N _part ) > 65. This scaling occurs both for the total multiplicity and for highp _T particles (3 <p _T < 4.5 GeV/c). This presentation is based in large part on the PHOBOS summary talk by M Baker at the16th Int. Conf. on Ultrarelativistic Nucleus-Nucleus Collisions, Quark Matter 2002, Nantes, France     

Production of colored scalar particles in <Emphasis Type="Italic">pp</Emphasis> collisions and masses of scalar gluons from future LHC data

The production of pairs of scalar particles belonging to an arbitrary color multiplet of the SU _c (3) group in proton-proton collisions is considered, and the differential and total cross sections for the corresponding partonic processes are obtained. The total cross section for the production of octets of scalar gluons F ₁ and F ₂ at LHC is calculated versus their mass, and their dominant decays, which are necessary for their searches, are discussed. It is shown that, for mF ₁ ≲ 1000 GeV, the number of signal $ t\tilde tb\tilde b $ t\tilde tb\tilde b events fromthe decays of the scalar gluon F ₁ may exceed substantially (by not less than three standard deviations) the Standart Model background and that the cross section for the production of scalar gluons F ₁ and F ₂ having masses in the region m _F1,m _F2 ≲ 1300 GeV may be sufficient for the efficient (N _events ≲ 100–1000 at L = 10–100 fb⁻¹) production of these particles at LHC.