On the time evolution of the states of infinitely extended particles systems

We consider an infinite classical system of interacting particles in , . We study the time evolution of a particular class of nonequilibrium states. More precisely, the states we consider are Gibbs with respect to a Hamiltonian which differs from the Hamiltonian governing the motion by an external field (possibly not localized), satisfying certain conditions. It is proved that the time-evolved states satisfy superstable estimate and are described by correlation functions obeying the BBGKY hierarchy in a weak form.     

On averaging of time evolution of quantum spin lattice systems

An investigation of a procedure of obtaining irreversible dynamics for systems in contact with surroundings is given.Research supported by M. Skodowska-Curie Fund Grant No. 0IP74-01416.     

On the time evolution of large systems: A soluble model with two-body interactions

We study the time evolution of a large system consisting of two kinds of particles that can be transformed into each other by an external field. The introduction of strong enough two-body interactions leads to the emergence of two time scales. The first one is connected with (small) asymptotic oscillations, and the second characteristic time describes a monotonic decay of the initial state. We also solve exactly and study a second model in which the two-body forces are replaced by a pair-production interaction.     

On the time evolution of generator coordinates

Collective potential energy surfaces were determined in the mass region with 50<(N, Z)<82 by fitting experimental level spectra andB(E 2)-values on the basis of the generalized collective model of Gneuss and Greiner. While the nuclides withN=80 and withZ=52 are rather soft vibrators we find an abrupt transition to asymmetric rotators forN=78 and forZ=54 tending to more symmetric shapes further away from the closed shells. The results predict the position of levels not yet observed andB(E 2)-values.     

Macroscopic description of time evolution of quantum systems

The evolution of a quantum system in which a specified set of variables is subjected to measurement over the time interval [t_o, t] is considered. It is shown that if the specified set of variables is macroscopically complete (i.e., it permits continuous measurement), then the expected values of these variables satisfy a closed system of integral equations. The quantum state of the system is then described by a Gibbs density matrix.     

On the time and cell dependence of the coarse-grained entropy. I

We consider a finite, thermally isolated, classical system which passes from an equilibrium state $\text{A}$ by the removal of an internal constraint to another equilibrium state $\text{B}$ after an empirical relaxation time. In the phase space of the system, cells are introduced according to the set of measuring instruments used and their experimental inaccuracies. It is shown that the coarse-grained entropy S_cg(t) tends to its new equilibrium value in general faster than the expectation values of the macroscopic variables to their new equilibrium values. We then investigate the dependence of S_cg(t) on the size of the phase cells. For fixed t, we find a lower bound on S_cg(t) by doing to the limit of infinite accuracy of the measuring instruments. In the limit t → ∞, this lower bound on S_cg(t) also converges to the equilibrium entropy of $\text{B}$. These properties strongly support the opinion that S_cg(t) is a proper microscopic expression for the entropy for equilibrium and nonequilibrium. Finally, explicit calculations of S_cg(t) for the model of a point particle enclosed in a one-dimensional box are presented which confirm the general results.     

On the time and cell dependence of the coarse-grained entropy. II

We present calculations of the coarse-grained entropy S_cg(t) for the model of a classical point particle enclosed in a two-dimensional box with perfectly reflecting walls. We find in comparison with the one-dimensional case that the fluctuations of S_cg(t) and of the expectation values of the position have decreased, and that S_cg(t) does not take its asymptotic value at regular time intervals. The times after which S_cg(t) and the expectation values have approximately reached their equilibrium values, are about equal; recurrence times are rather clearly separated from this time. Finally, we give a general estimate on the change of an expectation value due to a refinement of the cells.     

On the time evolution of transmittivity in magnetic fluids

When a magnetic fluid is subjected to a magnetic field, a part of the magnetic particles in the fluid agglomerates to form chains. Thus, the ferrofluid becomes optically anisotropic. In this work we describe optically observed patterns in some magnetic fluid films in applied parallel magnetic fields and optical effects of these, especially the optical transmittance. The most interesting experimental observation is that concerning the time dependence of relative transmittivity . For kerosene base ferrofluids relax rapidly at coupling and decoupling magnetic field, but for a transformer-oil magnetic fluid the relaxation times can attain (5–10) minutes, depending on the intensity of applied magnetic field.     

Numerical approaches to time evolution of complex quantum systems

We examine several numerical techniques for the calculation of the dynamics of quantum systems. In particular, we single out an iterative method which is based on expanding the time evolution operator into a finite series of Chebyshev polynomials. The Chebyshev approach benefits from two advantages over the standard time-integration Crank-Nicholson scheme: speedup and efficiency. Potential competitors are semiclassical methods such as the Wigner-Moyal or quantum tomographic approaches. We outline the basic concepts of these techniques and benchmark their performance against the Chebyshev approach by monitoring the time evolution of a Gaussian wave packet in restricted one-dimensional (1D) geometries. Thereby the focus is on tunnelling processes and the motion in anharmonic potentials. Finally we apply the prominent Chebyshev technique to two highly non-trivial problems of current interest: (i) the injection of a particle in a disordered 2D graphene nanoribbon and (ii) the spatiotemporal evolution of polaron states in finite quantum systems. Here, depending on the disorder/electron-phonon coupling strength and the device dimensions, we observe transmission or localisation of the matter wave.     

Nonstationary quantum mechanics. II. On the increase of entropy with time

On the basis of the discussion in our previous work (T1) and the reexamination of the concept of entropy of a pure quantum state it is shown that a natural explanation of the process of entropy increase exists. A new definition of an entropy operator is given and it is shown that there is not any increase of entropy when one confines oneself to the usual formalism of quantum mechanics and uses this definition. Some other attempts of proving theH theorem are briefly discussed.     

Dissipative time evolution of observables in non-equilibrium statistical quantum systems

We discuss differential– versus integral–equation based methods describing out–of thermal equilibrium systems and emphasize the importance of a well defined reduction to statistical observables. Applying the projection operator approach, we investigate on the time evolution of expectation values of linear and quadratic polynomials in position and momentum for a statistical anharmonic oscillator with quartic potential. Based on the exact integro-differential equations of motion, we study the first and naive second order approximation which breaks down at secular time-scales. A method is proposed to improve the expansion by a non–perturbative resummation of all quadratic operator correlators consistent with energy conservation for all times. Motion cannot be described by an effective Hamiltonian local in time reflecting non-unitarity of the dissipative entropy generating evolution. We numerically integrate the consistently improved equations of motion for large times. We relate entropy to the uncertainty product, both being expressible in terms of the observables under consideration. Received: 21 July 1998 / Revised version: 28 September 1998 / Published online: 2 November 1998     

On the time evolution automorphisms of the CCR-algebra for quantum mechanics

In ordinary quantum mechanics for finite systems, the time evolution induced by Hamiltonians of the form is studied from the point of view of *-automorphisms of the CCRC*-algebra (see Ref. [1, 2]). It is proved that those Hamiltonians do not induce *-automorphisms of this algebra in the cases: a) and b)V L (,dx) L ¹ (,dx), except when the potential is trivial.     

非马尔科夫耗散系统长时演化下的极限环振荡现象

本文研究结构化环境中非马尔科夫耗散系统在长时演化下可能出现的极限环振荡现象. 对于欧姆型谱密度环境中的二能级系统, 由于体系只允许一个束缚态模, 给定初态系统在Bloch空间的长时演化将收敛于一个极限环. 研究揭示了极限环半径与环心位置同环境谱密度函数间的关系. 对于多带光子晶体环境中的二能级系统, 由于其可以存在多个束缚态, 研究展现了系统在长时演化下可能出现的收敛于环面或周期或准周期的振荡行为. 有关环面的特征量与环境谱密度间的量化关系同样得以刻画. 论文随后讨论了两比特系统关联量在局域非马尔科夫耗散环境中长时演化可能出现的特征行为.     

Noise and conditional entropy evolution

We study the convergence properties of the conditional (Kullback–Leibler) entropy in stochastic systems. We have proved general results showing that asymptotic stability is a necessary and sufficient condition for the monotone convergence of the conditional entropy to its maximal value of zero. Additionally we have made specific calculations of the rate of convergence of this entropy to zero in a one-dimensional situation, illustrated by Ornstein–Uhlenbeck and Rayleigh processes, higher dimensional situations, and a two-dimensional Ornstein–Uhlenbeck process with a stochastically perturbed harmonic oscillator and colored noise as examples. We also apply our general results to the problem of conditional entropy convergence in the presence of dichotomous noise. In both the one-dimensional and multidimensional cases we show that the convergence of the conditional entropy to zero is monotone and at least exponential. In the specific cases of the Ornstein–Uhlenbeck and Rayleigh processes, as well as the stochastically perturbed harmonic oscillator and colored noise examples, we obtain exact formulae for the temporal evolution of the conditional entropy starting from a concrete initial distribution. The rather surprising result in this case is that the rate of convergence of the entropy to zero is independent of the noise amplitude.     

On the violation of the holographic viscosity versus entropy KSS bound in non-relativistic systems

A computation of the quotient of shear viscosity and entropy density, or the Kovtun, Son and Starinets (KSS) number η/s, is performed in the non-relativistic and classical regime, first in chiral perturbation theory, and then in the SO(g+1)/SO(g) non-linear sigma model in the large g limit. Both are field theories stemming from a renormalizable sigma model, but, in spite of that, we explicitly calculate how one avoids the KSS bound by increasing the number of degenerate pions sufficiently. However, we argue that particle production could still keep the validity of the KSS bound in the weak sense. We also show how a large number of molecular isomers (which we estimate in terms of simple molecular properties) could avoid the bound in the strong sense. This might be possible with carbon based molecules. We finally argue that a measurement of η/s in heavy-ion collisions might be turned into an upper bound on the number of hadron resonances. PACS  11.15.Pg; 12.38.Mh; 25.75.q; 51.20.+d     

Network structure entropy and its dynamical evolution for recurrence networks from earthquake magnitude time series

Based on the theory of complex network, we construct a recurrence network for earthquakemagnitude time series from California. Network structure entropy and its dynamicalevolution of the network is studied. It is found that the network structure entropy of therecurrence network exhibits a peculiar behavior: it stays at a small value before mainshock, jumps to a great value at the main shock, and then recovers to normal valuesgradually. The network structure entropy therefore provides us an approach to characterizemain shocks quantitatively.