Classical “freezing” of plane rotations: A proof of the Boltzmann-Jeans conjecture

Using simple known methods and results of classical perturbation theory, especially those due to Nekhoroshev and Neishtadt, we study the energy exchanges between the rotational and the translational degrees of freedom in a particular model representing the planar motion of a rigid body in a bounded analytic potential. We prove that, if the angular velocity is initially large, then the energy exchanges are small,O( ^–1), for times growing exponentially with, |t|exp. We also deduce that in a scattering process from a (smooth) potential barrier, the overall change in the rotational energy of the incoming body is exponentially small in, exp(–. The results are interpreted in the light of an old conjecture by Boltzmann and Jeans on the existence of very large time scales for equilibrium in statistical systems containing high-frequency degrees of freedom (purely classical freezing of the high-frequency degrees of freedom); the rotating object is, in this interpretation, a (classical) molecule, which moves in an external field, or collides with the wall of a container. Two different limits of large are considered, namely the limit of large rotational energy, and (as is interesting for the molecular interpretation) the limit of point mass, at finite rotational energy.     

Enhancement of activated decay of metastable states by resonant pumping

We consider the effect of a high-frequency pumping cost on the escape rate of a classical underdamped Brownian particle out of a deep potential well. The energy dependence of the oscillation frequency(E) is assumed to be weak on the scale of thermal energy,_E(0)T(0)T/V₀ (0)[_E(0) is the derivative of(E) atE= 0,V ₀ is the barrier height,V ₀ T]. The quadratic-in- contribution to the decay rate is calculated in two different regimes: (1) for the case of resonance of the pumping frequency with the nth harmonic of the internal motion at an energye, when = n(e); (2) for a rollout region of the basic resonance near the bottom of the potential well, when ¦-(0)¦ and is the damping coefficient. In the latter case the absorption spectrum and the enhancement of the decay rate are calculated as functions of two reduced parameters, the anharmonicity of the potential,v _E (0)T/, and the resonance mismatch, [ —(0)]/. It is shown that the effect of the pumping increases with diminishing ¦v¦ and at small v is proportional tov ^–1. In this regime, the dependence on is stepwise: the pumping contribution is large for v > 0 and small for v < 0. In the frame of our theory, the decay rate is invariant against the simultaneous alternation of the signs of andv. The spectrum of the energy absorption has the standard Lorentzian shape in the absence of anharmonicity,v=0, and with increasing of ¦v¦ shifts and widens retaining its bell-shape form.     

Experimental survey of the sticking problem in muon catalyzed dt fusion

The sticking process dt + n, which constitutes the most severe limit to the number of fusions which a muon can catalyze, is reviewed. Many attempts were made to determine by calculations and measurements the probability for initial sticking _s ⁰ (immediately after dt fusion) and for final sticking _s (after the came to rest). Previous results based on neutron disappearance rates and on the observation of -X-rays were controversial and also in some disagreement with theory. New data are reported from PSI on direct observation of final sticking, using a setup with the St. Petersburg ionization chamber. These data mark a significant improvement in reliability and may clarify questions concerning previous discrepancies. The new results is _s(0.56±0.04)%, lower than the theory prediction _s=(0.65±0.03)%, at medium density.     

Dynamics of vector spin glasses: The Gabay-Toulouse line

The dynamics of ann-component vector spin glass with infinite range interactions are investigated near and above the Gabay-Toulouse (GT) line. The local transverse susceptibility _T for 0 varies along the whole GT lineT _c1 (H) as ^v , with a field and temperature independent critical exponentv=1/2. The longitudinal susceptibility _L () remains analytic for all (T, H)T _c1 (H), except for a cross-over fromv=1 tov=1/2 forH0 at the freezing temperatureT=T _f . The dynamic susceptibilities _T () and _L () are already coupled above the GT line via self-energy terms. BelowT _c1 (H), this coupling is strongly enhanced by other mechanisms.     

Rate equations in the hopping transport

The usual kinetic equations for the site occupation probabilities in an external field are solved exactly in a simple one-dimensional periodic model with two kinds of atoms using a) free boundary conditions and order of limitsN, 0 needed for a proper treatment of the dc conductivity here b) boundary conditions with metallic contacts and order of limitsN, 0 and c) the same boundary conditions but reversed order of limiting processes 0,N typical of e.g. numerical and percolation treatments. (N and are the number of sites and frequency.) It is demonstrated that though the bulk dc conductivity is the same in all three cases, local bulk properties of the material are strongly dependent on the régime used. The role of the order of all three limiting processes 0,N+ andn+ (N–n+) for local shifts of the chemical potential _n in the dc limit is examined (n is the number of the relevant site calculated from a boundary of the chain). It is shown especially that the rate equation treatment (régime a) on the one hand and numerical or percolation treatments (régime c) on the other hand never yield the same bulk values of _r.     

Eigenfunctions of the Koch snowflake domain

We obtain non-tangential boundary estimates for the Dirichlet eigenfunctions _n and their gradients {_n } for a class of planar domains with fractal boundaries. This class includes the quasidiscs and, in particular, snowflake-type domains. When applied to the case when is the Koch snowflake domain, one of our main results states that {₁()} tends to or 0 as approaches certain types of boundary points (where ₁ > 0 denotes the ground state eigenfunction of the Dirichlet Laplacian on ). More precisely, let Ob (resp., Ac) denote the set of boundary points which are vertices of obtuse (resp., acute) angles in an inner polygonal approximation of the snowflake curve . Then given Ob (resp., Ac), we show that {₁()} (resp., 0) as tends to in within a cone based at . Moreover, we show that blowup of {₁} also occurs at all boundary points in a Cantor-type set. These results have physical relevance to the damping of waves by fractal coastlines, as pointed out by Sapovalet al. in their experiments on the Koch drum.Research partially supported by the National Science Foundation under Grant DMS-9207098.     

An investigation of finite-size scaling for systems with long-range interaction: The spherical model

A method is suggested for the derivation of finite-size corrections in the thermodynamic functions of systems with pair interaction potential decaying at large distancesr asr ^{–d –} , whered is the space dimensionality and>0. It allows for a unified treatment of short-range (=2) and long-range (<2) interaction. The asymptotic analysis is illustrated by the mean spherical model of general geometryL ^d–d× ^d subject to periodic boundary conditions. The Fisher-Privman equation of state is generalized to arbitrary real values ofd, 0d. It is shown that the-expansion may be used to study the breakdown of standard finite-size scaling at the borderline dimensionalities.     

Plasma instability in a strong,circularly polarized electromagnetic wave

Conclusions In summary, we have studied the parametric instability of a cold electron plasma in a strong, circularly polarized electromagnetic wave, taking into account the relativistic nature of the electron motion. We have shown that an instability of longitudinal waves occurs at frequencies _p/ and _p/, while an instability of transverse waves occurs at frequencies ₀±₀ and ₀±_p/. The results of a numerical solution of the dispersion relation for large pump amplitudes, in which case the approximation of a weak coupling between waves is not applicable, show that the growth rates and the bands of the instabilities increase with increasing pump amplitude. We have found that in the case of the excitation of electromagnetic plasma waves, even at low pump amplitudes, the relativistic nature of the electron motion can be of decisive importance for the effect.T. G. Shevchenko Kiev State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 19, No. 10, pp. 1481–1488, October, 1976.     

Ultrasound in spin glasses

The change of the sound velocity v(,T) and the damping of sound waves (,T) in spin glasses are calculated in the frame-work of an Ising model with a random distribution of exchange interactions. The calculation is based on linearized equations of motion for the spins and on an improved mean field approximation which includes the Onsager reaction field. Near to the freezing temperatureT _f and at high temperatures v(,T) and (,T) turn out to be approximately proportional to the real and the imaginary parts of the dynamical susceptibility. For the special case of infinite range interactions atT=T _f one has v(, T_f) ( )^1/2 and (, T_f) (/)^1/2 where is the relaxation time of independent spins. However, already slightly aboveT _f the frequency dependence of both quantities becomes rather small for RKKY spin glasses. At high temperatures both, v(,T) and (,T) vary asT ^–1.SFB 125 Aachen-Jülich-Köln     

Entropy of Bogoliubov automorphisms of the canonical anticommutation relations

We compute the entropyh _A ( _U ) in the sense of Connes, Narnhofer and Thirring of Bogoliubov automorphisms _U of the CAR-algebra with respect to invariant quasifree states _A with 0A1 having pure point spectrum.Supported in part by a grant from the National Science Foundation     

Production of neutral mesons in the reaction12C+12C at 2 GeV/u

A systematic study of neutral meson production in heavy-ion reactions at 2 GeV/u was started with a¹²C+¹²C experiment using the photon spectrometer TAPS. Special emphasis is put on the possible observation of the-meson in a heavy-ion reaction exploiting the decay channel ⁰.Presented at the International School-Workshop Relativistic Heavy-Ion Physics, Prague (Czech Republic), 19–23 September 1994.     

Scaling Limit and Critical Exponents for Two-Dimensional Bootstrap Percolation

Consider a cellular automaton with state space {0,1} ² where the initial configuration _0 is chosen according to a Bernoulli product measure, 1s are stable, and 0s become 1s if they are surrounded by at least three neighboring 1s. In this paper we show that the configuration _n at time n converges exponentially fast to a final configuration , and that the limiting measure corresponding to is in the universality class of Bernoulli (independent) percolation. More precisely, assuming the existence of the critical exponents , , and , and of the continuum scaling limit of crossing probabilities for independent site percolation on the close-packed version of ² (i.e. for independent *-percolation on ), we prove that the bootstrapped percolation model has the same scaling limit and critical exponents.This type of bootstrap percolation can be seen as a paradigm for a class of cellular automata whose evolution is given, at each time step, by a monotonic and nonessential enhancement [Aizenman and Grimmett, J. Stat. Phys. 63: 817--835 (1991); Grimmett, Percolation, 2nd Ed. (Springer, Berlin, 1999)     

Recursion method for the density of states and spectral dimension of central-force elastic percolation network

We use the recursion method to calculate the vibrational density of states () of site diluted central-force elastic percolation network on the triangular lattice. We find that () is proportional to at long wavelengths. At shorter wavelengths, i.e. in the fracton regime, () is proportional to , where is the spectral dimension of the elastic percolation network. A log-log plot of () vs gives a straight line in the fracton regime with a negative slope. From this we estimate {ie333-3} to be 0.625.     

Theory of electric field-induced optical second harmonic generation in semiconductors

The nonlinear optical coefficientd _ij(kl) (2, 0,, ) for electric field-induced optical second harmonic generation in semiconductors is calculated by means of a perturbation treatment. The result is given directly as a Brillouin zone integral over a resonant energy denominator. A simplified energy band structure model is used to carry out the Brillouin zone integral. The analytic closed-form expression ford _ij(kl) (2, 0,, ) thus obtained permits the calculation of the absolute value of its spectrum from available energy band parameters. The dispersion ofd ₁₁₍₁₁₎ (2, 0,, ) of Ge is numerically calculated, second harmonic photon energies being close to theE ₀ andE ₁ gaps. The results show pronounced resonant behaviours.     

Excited-state absorption and third-order optical nonlinearities in symmetric π-electron organic molecules

Excited-State Absorption (ESA), Two-Photon Absorption (TPA) and the third-order polarizability (;,, – ) have been investigated for a model dichloride derivative of a symmetrically substituted benzylidene analine (SBAC), using a multielectron configuration-interaction procedure. The calculations indicate that SBAC exhibits ESA across the visible region of the spectrum, but that it is not as extensive as for molecules such as the phthalocyanines. The magnitude of the third-order polarizability is dominated by resonance enhancement from a very strongA _g B _u one-photon absorption. The calculated off-resonance value for (;,, – ) suggests that SBAC is a potential candidate for ultrafast switching applications.     

Dynamics of Ising spin glasses far below the lower critical dimension: The one-dimensional case and small clusters

The Glauber model is studied for symmetric distributions (±J and gaussian) of the nearest-neighbour interactionJ, including a magnetic field. For small clusters of spins (closed rings ofN bonds, withN7) the complex magnetic susceptibility () and the time-dependent remanent magnetizationm(t) are found exactly for given bond configurations {J _ij } by diagonalization of the Liouville operator; apart from the ±J model, the average over {J _ij } must be done numerically by simple random sampling Monte Carlo. Nevertheless our accuracy is much better than corresponding dynamic Monte Carlo simulations, even if one considers the extrapolation toN.We analyze the results along the lines of corresponding experimental work, studying the frequency-dependence of the peak in (), theT lnt-scaling ofm(t) at low temperaturesT, and the decomposition of () into a spectrum of relaxation times. Many results are strikingly similar to experimental data for systems such as the Holmium-Borate spinglass or the superparamagnet Eu_0.05Sr_0.95S, for instance. Frequency-dependent critical fieldsH _c () in theH-T plane are also identified but do not have the familiar Almeida-Thouless shape, however.Dedicated to B. Mühlschlegel on the occasion of his 60th birthday     

A Boltzmann map for quantum oscillators

We define a map on the space of quasifree states of the CCR or CAR of more than one harmonic oscillator which increases entropy except at fixed points ofx. The map is the composition of a doubly stochastic map T^* and the quasifree reductionQ. Under mixing conditions onT, iterates of take any initial state to the Gibbs state, provided that the oscillator frequencies are mutually rational. We give an example of a system with three degrees of freedom with energies ₁, ₂, and ₃ mutually irrational, but obeying a relation n₁₁+n₂₂=n₃₃,n _i . The iterated Boltzmann map converges from an initial statep to independent Gibbs states of the three oscillators at betas (inverse temperatures) ₁,₂, ₃ obeying the equation n₁₁₁+n₂₂₂=n₃₃₃. The equilibrium state can be rewritten as a grand canonical state. We show that for two, three, or four fermions we can get the usual rate equations as a special case.     

Dynamic scaling for spin glasses near the de Almeida-Thouless line

We calculate the dynamic local susceptibility () of an Ising spin glass near the de Almeida-Thouless (AT) line within the soft spin dynamics for the Sherrington-Kirkpatrick model. We find a crossover from analytic behaviour of () at =0 above the AT line to a power law behaviour ()(–i) ^v on the AT line and discuss the analytic properties of the crossover function. The frequencyscale is proportional to ^1/v , where measures the distance from the AT line. We determine the spectrum of relaxation times which diverge as ^1–1/v . The average relaxation time diverges as ^–1/v wherev1/2. In addition we determine the absolute frequency scale and prove the consistency of the ansatz of Sompolinsky and Zippelius ()–(0)(–i) ^v at and below the AT line.     

On the spectrum of Schrödinger operators with a random potential

We investigate the spectrum of Schrödinger operatorsH of the type:H =–+q _i ()f(x–x _i + _i ())(q _i () and _i () independent identically distributed random variables,i ^d ). We establish a strong connection between the spectrum ofH and the spectra of deterministic periodic Schrödinger operators. From this we derive a condition for the existence of forbidden zones in the spectrum ofH . For random one- and three-dimensional Kronig-Penney potentials the spectrum is given explicitly.     

Models of optical absorption in amorphous semiconductors at the absorption edge — A review and re-evaluation

Davis-Mott and Tauc models of optical absorption at the absorption edge in the high absorption coefficient region (10⁴cm^–1) are carefully reviewed with regard to their theoretical foundations, assumptions, mathematical derivations, and results. The full implications of these models are exploited, and it is found that the Davis-Mott model for negligible matrix elements between localised states could account for the cubic power law behaviour of with photon energy of some amorphous semiconductors such as a-Si. A fractional power law to find the optical band gapE _opt, of the form [ (–E _opt)^r; 2r3] based on Davis-Mott model is proposed in which the indexr can be a function of disorder. The Tauc model has further been extended to the case of negligible matrix elements between localised states, in which the same square power law for vs. with the same meaning of the optical gap as in the original Tauc model has resulted. A consideration of the case of unequal matrix elements for those transitions between localised states and those between extended states is also included. The meaning ofE _opt has been re-assessed and it is emphasized that it is an extrapolation of delocalised states to the zero of the density of states rather than a threshold energy for the onset of some kind of optical transitions.This paper is cordially dedicated to Professor C. A. Hogarth who taught us the Physics of amorphous materials.