Realization of holonomic constraints and freezing of high frequency degrees of freedom in the light of classical perturbation theory. Part II

As in Part I of this paper, we consider the problem of the energy exchanges between two subsystems, of which one is a system of harmonic oscillators, while the other one is any dynamical system ofn degrees of freedom. Such a problem is of interest both for the realization of holonomic constraints of classical mechanics, and for the freezing of the internal degrees of freedom in molecular collisions. The results of Part I, which referred to the particular case =1, are here extended to the more difficult case >1. For the rate of energy transfer we find exponential estimates of Nekhoroshev's type, namely of the form exp (_*/)^1/a , where is a positive real number giving the size of the involved frequencies, and _* anda are constants. For the particularly relevant constanta we find in generala=1/ however, in the particular case when the frequencies are equal (collision of identical molecules), we finda=1 independently of , as conjectured by Jeans in the year 1903.     

True self-avoiding walk on critical percolation clusters and lattice animal

The statistics of true-self-avoiding walk model on two dimensional critical percolation clusters and lattice animals are studied using real-space renormalization group method. The correlation length exponents 's are found to be _TSAW ^pc 0.576 and _TSAW ^LA 0.623 respectively for the critical percolation clusters and lattice animals.     

Real-space renormalization group for directed percolation system

We use the recently proposed real-space renormalization group method to study the critical behavior of directed percolation system in two dimensions. The correlation length exponents and are found to be 1.76 and 1.15. These results are in good agreements with the best known values.     

Treatment of droplike clusters in nucleation theory using Reiss's method

The expression of the correction factor in nucleation theory is derived by extending the method Reiss used recently. is the factor appearing in the number of critical nuclei (formed as a vapor condenses into liquid drops) as a correction to the conventional theory. It is shown that=p _l / _g /kT, wherep _l is the pressure of the liquid phase inside the drop, _g , is the volume per molecule in the vapor phase,k is the Boltzmann constant, andT is the absolute temperature. The difference between this and Reiss's expression isp _l , which replaces hisp _g (the vapor pressure in equilibrium with the drop). The derived in this paper is compatible with the expression= _g / _l ( _l is the molecular volume in the liquid phase) previously proposed by the present author.     

Nonlinear Response of the Semiconductor Laser with Pump Current Modulation

The influence of tuning the lasing monochromatic radiation frequency _g within the amplification band on the nonlinear response of the semiconductor laser with harmonic modulation of pump current is investigated theoretically. It is established that the principal features of the behavior of the nonlinear amplitude-detuning characteristic (ADC) are determined by the relation between the current modulation frequency _m and the main resonance frequency of the laser _r. If _{m r}, then with increase in _g the response decreases monotonically mainly due to the decrease of its dynamic component. The exception is provided by the spectral regions where peaks on the ADC appear because of the explicitly nonlinear lasing regimes (period doubling, chaos, etc.) When _m < _r, the resonance conditions for induced oscillations are satisfied only for definite spectral intervals within the amplification band and a dip appears on the low-frequency side of the ADC. With decreasing _m, the dip boundary shifts to a more high-frequency region of the band corresponding to smaller local resonance frequencies. The peaks on the ADC corresponding to the radiation period doubling shift to the region of smaller values of _g on increase in _m.     

Stochastic linear maps and transition probability

Some aspects of the transition probability P(, ) between states , on unital ^*-algebras are discussed. It is shown that P increases under the action of any stochastic linear map T, i.e., P(T, T)P(, ). Some properties of P are derived in starting from a recently-proved characterization of the quantity in question.     

A Far IR Study of NbN in the Normal and Superconductive State

A thin film of NbN (thickness t = 300 Å), has been deposited on an MgO and a Si wafer. Both samples have been studied by transmission from 10 or 20 to 120 cm^–1, and have exhibited one maximum of transmission at a given frequency like the classical superconductors as Pb, Sn or Hg in the superconductive state. From the Far IR experimental data, the characteristic temperature _c, and the gap frequency (_gap () = 2 (), () being the energy gap) are immediately obtained (for instance for the NbN / MgO sample, _c = 15.5 K; _g (5 K) = 39.7 cm^–1), and it is seen that as expected from the BCS theory for a weak coupling. To fit the data we had to adjust only two additionnal parameters: collision and plasma frequency, _c () and _p (including all carriers). At = 5 K, thebest fit for the NbN / MgO sample is obtained with _c = 371 cm^–1 and _p = 12,600cm^–1.     

Surviving extrema for the action on the twisted SU(∞) one point lattice

We give a simplified construction of twist eating configurations, based on a theorem due to Frobenius. These configurations are defined through the equation:U U U ⁺ U ⁺ =exp(2in /N) withU SU(N), =1 tod andn an antisymmetric matrix with integer entries. In the (Twisted)-Eguchi-Kawai model they yield extrema some of which survive forN. Comparison is made with the Monte Carlo data of the internal energy in the small coupling region.     

The tortuosity of occupied crossings of a box in critical percolation

We consider the length of an occupied crossing of a box of size [0,n]×[0, 3n] ^D–1 (in the short direction) in standard (Bernoulli) bond percolation on ^D at criticality. Let ¦s _n¦ be the length of the shortest such crossing. It is believed that ¦s _n¦ ^1+c in some sense for somec>0. Here we show that if the correlation length(p) satisfies (p)p _c}–p) ^– for some <1, then with a probability tending to 1, ¦s _n¦>/C ₁ n ^1/(logn)^–(1–)/. The assumption (p)C ₃(p _c–p)^– with <1 has been rigorously established^(1,2) for largeD, but cannot hold⁽³⁾ forD=2. In the latter case, let ¦l _n¦ be the length of the lowest occupied crossing of the square [0,n]². We outline a proof ofP _pc(¦l_n¦ n ^1+c)n ^– for somec, >0. We also obtain a result about the length of optimal paths in first-passage percolation.     

Ising Models on Hyperbolic Graphs II

We consider Ising models with ferromagnetic interactions and zero external magnetic field on the hyperbolic graph (v, f), where v is the number of neighbors of each vertex and f is the number of sides of each face. Let T _c be the critical temperature and T _c =supTT _c: ^f=( ⁺+ ^–)/2, where ^f is the free boundary condition (b.c.) Gibbs state, ⁺ is the plus b.c. Gibbs state and ^– is the minus b.c. Gibbs state. We prove that if the hyperbolic graph is self-dual (i.e., v=f) or if v is sufficiently large (how large depends on f, e.g., v35 suffices for any f3 and v17 suffices for any f17) then 0<T _c <T _c, in contrast with that T _c =T _c for Ising models on the hypercubic lattice Z ^d with d2, a result due to Lebowitz.⁽²²⁾ While whenever T<T _c , ^f=( ⁺+ ^–)/2. The last result is an improvement in comparison with the analogous statement in refs. 28 and 33, in which it was only proved that ^f=( ⁺+ ^–)/2 when TT _c and it remains to show in both papers that ^f =( ⁺+ ^–)/2 whenever T<T _c . Therefore T _c and T _c divide [0, ] into three intervals: [0, T _c ), (T _c , T _c), and (T _c, ] in which ^{+ –} but ^f =( ⁺+ ^–)/2, ^{+ –} and ^f ( ⁺+ ^–)/2, and ⁺= ^–, respectively.