Decay of correlations in spin systems

We investigate the decay of initial correlations in a spin system where each spin relaxes independently by an intramolecular mechanism. The equation of motion for the spin density matrix is assumed to be the Redfield equation, which is of the form of a quantum mechanical master equation. Our analysis of this problem is based on the techniques of Shuler, Oppenheim, and coworkers, who have studied the decay of correlations in systems which can be described by classical stochastic master equations. We find that the off-diagonal elements of the reduced spin density matrices approach their equilibrium values faster than the diagonal elements. The Ursell functions, which are a measure of the correlations in the system, decay to their zero equilibrium values faster than the spin density matrix except for the furthest off-diagonal elements. Far off-diagonal matrix elements of the spin density matrix approach equilibrium at the same rate as the Ursell functions, which is the important difference between the quantum mechanical model studied here and the classical models studied earlier.Supported in part by the National Science Foundation.     

Integral equations for N-particle scattering

Integral equations are derived for the N-particle transition operators. The equations couple together only transition operators between two-body channels. The kernel of the equations becomes connected after a single iteration. Transition operators involving channels with three or more particles can be obtained by quadratures from the solution of the equations. It is also shown that the N-particle equations can be reduced to multichannel two-body equations by the use of the quasiparticle method.     

A soluble model for quantum mechanical dissipation

In order to derive the equations for dissipation and noise in a quantum mechanical system it is necessary to include the equations of motion of a suitably chosen bath interacting with the system. In this way the standard treatment arrives at an approximate master equation for the density matrix of the system, at the expense of a number ofad hoc assumptions. These assumptions are here scrutinized on the basis of an exactly soluble model. The conclusion is: the bath must obey certain specifications; the interaction must be weak; and the temperature must be so high that the interaction energy is within the classical domain rather than occurring in quanta. Some additional comments concerning dissipation in quantum mechanics are relegated to an appendix.     

Integral Representation for the Eigenfunctions of a Quantum Periodic Toda Chain

An integral representation for the eigenfunctions of a quantum periodic Toda chain is constructed for the N-particle case. The multiple integral is calculated using the Cauchy residue formula. This gives the representation which reproduces the particular results obtained by Gutzwiller for the N=2,3 and 4-particle chain. Our method of solving the problem combines the ideas of Gutzwiller and the R-matrix approach of Sklyanin with the classical results in the theory of Whittaker functions. In particular, we calculate Sklyanin's invariant scalar product from the Plancherel formula for the Whittaker functions derived by Semenov-Tian-Shansky thus obtaining a natural interpretation of the Sklyanin measure in terms of the Harish-Chandra function.     

N-particle correlations in the McKean model

In the McKean model the BBGKY hierarchy is equivalent to a simple hierarchy of coupled equations for thep-particle correlation functions. Approximate solutions are obtained by truncating the hierarchy. The convergence of the truncation method is studied by comparison with the exact solution for the model, which can be given in closed form. In the long-time limit the exact solution is linearized around the equilibrium value, showing the decay of the correlations. It turns out thatp-particle correlations decayp times faster than the nonequilibrium one-particle distribution.     

Time relaxation of the solutions of master equations for large systems

The time relaxation behavior of the solutions of certain classes of discrete master equations is studied in the limit of an infinite number of states. Depending on the range of the transition matrix, a relaxation behavior is found reaching from at ^–1/2 law for short range, over enhanced relaxation to an exponential relaxation for the extreme long-range case. The behavior in the limit of a continuous family of states is also discussed.     

Relativistic classical mechanics and canonical formalism

The analysis of interacting relativistic many-particle systems provides a theoretical basis for further work in many diverse fields of physics. After a discussion of the nonrelativisticN-particle systems we describe two approaches for obtaining the canonical equations of the corresponding relativistic forms. A further aspect of our approach is the consideration of the constants of the motion.     

硼玻璃中Dy3+及Sm3+的辐射跃迁几率和无辐射跃迁几率

硼玻璃中掺杂Nd3+,Er3+,Tm3+等小能隙的稀土离子,由于硼玻璃声子能量大,多声子无辐射几率大,影响其发光效率。但由于硼玻璃熔点低,加工容易,对于民用上常用的Sm3+,Dy3+,Eu3+,Tb3+等大能隙物质,多声子无辐射跃迁不是主要因素的情况下,发光效率将如何是本文要研究的问题。另外,辐射跃迁性质的研究对于能量传递机理的研究也是很有意义的。     

A mixed basis approach in the SGP-limit

A perturbation method for computing quick estimates of the echo decay in pulsed spin echo gradient NMR diffusion experiments in the short gradient pulse limit is presented. The perturbation basis involves (relatively few) dipole distributions on the boundaries generating a small perturbation matrix in O(s²) time, where s denotes the number of boundary elements. Several approximate eigenvalues and eigenfunctions to the diffusion operator are retrieved. The method is applied to 1D and 2D systems with Neumann boundary conditions.     

Dynamics of quantum correlations

The density matrix describing the state of a sybsystem of a physical system whose time dependence is assumed to follow a Schrödinger equation does not itself obey a von Neumann equation. The behavior of the eigenvalues and eigenfunctions of this density matrix is studied. An expression for the rate at which initially separating systems are de-separated is derived in perturbation theory. Indications are given that coherent photon states are more stable in the presence of charged particles than photon number eigenstates. The possible dynamical origin of super selection rules is discussed. A simple model is solved analytically.     

Dynamics of fluctuations in a reactive system of low spatial dimension

We study, using master equation techniques, the time evolution of the average concentration and fluctuations in the two-speciesn-molecule reactionA+(n-1)XnX in one dimension described by a Glauber-type dynamical lattice model for the specific casesn=2 (bimolecular) andn=3 (trimolecular). The evolution is found to be quite different from that described by the Mean-Field equations even for the bimolecular case, where the steady state is meanfield. For the trimolecular process, the values of fluctuation correlations in the nonequilibrium steady state are well predicted by the fixed points of the dynamical equations obtained from the master equation. In addition, three-point fluctuation correlations are found to play an important role in both processes and are accounted for by an extended Bethe-type ansatz. The bimolecular system shows no memory effects of initial conditions, while the trimolecular system is characterized by memory effects in terms of the average concentration, fluctuations as well as the entropy. The spatial decay of fluctuation correlations is found to be short range at the steady state for the trimolecular system.     

Decay of correlations in vibrational relaxation

We investigate the decay of initial vibrational correlations in a dilute gas mixture of diatomic molecules and structureless particles. We use the techniques of Davis and Oppenheim to derive an equation for vibrational relaxation which is suitable for correlated systems. We then use the Landau-Teller transition probabilities and solve for the one-and two-molecule distribution functions and the two-molecule correlation functions. We find that the correlations decay faster than the distribution functions, which agrees with the results of Oppenheim, Shuler,et al. for other systems.     

A new class of integrable systems and its relation to solitons

We present and study a class of finite-dimensional integrable systems that may be viewed as relativistic generalizations of the Calogero-Moser systems. For special values of the coupling constants we obtain N-particle systems that are intimately related to the N-soliton solutions of the sine-Gordon and Korteweg-de Vries equations, among other ones.     

An extended master-equation approach applied to aggregation in freeway traffic

We restudy the master-equation approach applied to aggregation in a one-dimensional freeway, where the decay transition probabilities for the jump processes are reconstructed based on a car-following model. According to the reconstructed transition probabilities, the clustering behaviours and the stochastic properties of the master equation in a one-lane freeway traffic model are investigated in detail The numerical results show that the size of the clusters initially below the critical size of the unstable cluster and initially above that of the unstable cluster all enter the same stable state, which also accords with the nucleation theory and is known from the result in earlier work. Moreover, we have obtained more reasonable parameters of the master equation based on some results of cellular automata models.     

N = 4 mechanics,WDVV equations and polytopes

N = 4 superconformal n-particle quantum mechanics on the real line is governed by two prepotentials, U and F, which obey a system of partial nonlinear differential equations generalizing the Witten—Dijkgraaf—Verlinde—Verlinde (WDVV) equation for F. The solutions are encoded by the finite Coxeter systems and certain deformations thereof, which can be encoded by particular polytopes. We provide A _n and B ₃ examples in some detail. Turning on the prepotential U in a given F background is very constrained for more than three particles and nonzero central charge. The standard ansatz for U is shown to fail for all finite Coxeter systems. Three-particle models are more flexible and based on the dihedral root systems.     

New inequalities for Ising ferromagnets

It is shown that for Ising ferromagnets which obey the Lee-Yang theorem the Ursell functions or cumulants of the magnetization variable at nonzero external field satisfy series of inequalities. Several relations connecting Ursell functions with nonzero and zero field are derived.     

The elimination of fast variables in complex chemical reactions. III. Mesoscopic level (irreducible case)

The master equation for a complex chemical reaction cannot always be reduced to a simpler master equation, even if there are fast and slow individual reaction steps. Nevertheless the elimination of intermediates can be carried out with the help of the-expansion. This is illustrated with a well-known complex reaction: the dissociation of N₂O₅. It is shown that the intrinsic fluctuations in the N₂O₅ decay are larger than those implied by the master equation suggested by the macroscopic rate law.     

Critical behavior and associated conformal algebra of the Z3 potts model

The conformal algebra for operators of theZ ₃ model at the phase transition point is built. Critical exponents are found in this approach as solutions of simple algebraic equations, which are consistency conditions of the algebra. Multipoint correlation functions obey linear differential equations. Some solutions are given for the four-point correlation functions of theZ ₃ model at the phase transition point.