A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions

Combining order reduction approach and L1 discretization, a box-type scheme is presented for solving a class of fractional sub-diffusion equation with Neumann boundary conditions. A new inner product and corresponding norm with a Sobolev embedding inequality are introduced. A novel technique is applied in the proof of both stability and convergence. The global convergence order in maximum norm is O(τ^2−α + h²). The accuracy and efficiency of the scheme are checked by two numerical tests.     

Numerical method for the time evolution of the Dirac equation

We propose an iterative procedure for the numerical evolution of Dirac equation. It is a fast and stable version of the coordinate-space predictor-corrector method and can be extended to (3+1) dimensions for the relativistic mean-field calculation of heavy-ion collisions. A detailed example is worked out for the case of the Bottcher-Strayer pair production mechanism in (1+1) dimensions.     

凸形晶粒的各向异性三维von Neumann方程研究

三维晶粒长大规律是材料科学研究的核心问题之一, 本文通过考虑实际多晶组织中晶界能和晶界迁移率的不均匀性和各向异性因素对晶粒三晶棱处两面角大小的影响, 借助经典体视学中晶粒界面积分平均曲率与平均切直径的关系, 经推导得到了适合于凸形晶粒的一般性三维von Neumann方程, 结果表明实际凸形晶粒的准确长大速率可以表示为晶粒的平均切直径、三晶棱总长度和三晶棱处两面角的函数. 所得方程经过了Kelvin十四面体和5种规则多面体验证, 对于三维von Neumann方程(Nature, 2007, 446:1053)进一步推广并应用于实际金属和陶瓷材料具有重要的意义.     

Optimization of the imaginary time step evolution for the Dirac equation

Taking the single neutron levels of 12C in the Fermi sea as examples,the optimization of the imaginary time step(ITS) evolution with the box size and mesh size for the Dirac equation is investigated.For the weakly bound states,in order to reproduce the exact single-particle energies and wave functions,a relatively large box size is required.As long as the exact results can be reproduced,the ITS evolution with a smaller box size converges faster,while for both the weakly and deeply bound states,the ITS evolu...     

q-Gaussians in the porous-medium equation: stability and time evolution

The stability of q-Gaussian distributions as particular solutions of the linear diffusion equation and its generalized nonlinear form, , the porous-medium equation, is investigated through both numerical and analytical approaches. An analysis of the kurtosis of the distributions strongly suggests that an initial q-Gaussian, characterized by an index q_i, approaches asymptotically the final, analytic solution of the porous-medium equation, characterized by an index q, in such a way that the relaxation rule for the kurtosis evolves in time according to a q-exponential, with a relaxation index q_rel ≡q_rel(q). In some cases, particularly when one attempts to transform an infinite-variance distribution (q_i ≥ 5/3) into a finite-variance one (q < 5/3), the relaxation towards the asymptotic solution may occur very slowly in time. This fact might shed some light on the slow relaxation, for some long-range-interacting many-body Hamiltonian systems, from long-standing quasi-stationary states to the ultimate thermal equilibrium state.     

Scattering theory for quasi-free time automorphisms of C1-algebras and von Neumann algebras

Relations between scattering theory in L² space, for quasi-free automorphism groups of C¹-algebras and for those of their representations are established and it is shown that wave operators as time limits do not exist in these representations. Finally, the connection of scattering theory and dynamical stability is discussed.     

A New Inequality for the von Neumann Entropy

Strong subadditivity of von Neumann entropy, proved in 1973 by Lieb and Ruskai, is a cornerstone of quantum coding theory. All other known inequalities for entropies of quantum systems may be derived from it. Here we prove a new inequality for the von Neumann entropy which we prove is independent of strong subadditivity: it is an inequality which is true for any four party quantum state, provided that it satisfies three linear relations (constraints) on the entropies of certain reduced states.     

Continuity of the von Neumann Entropy

A general method for proving continuity of the von Neumann entropy on subsets of positive trace-class operators is considered. This makes it possible to re-derive the known conditions for continuity of the entropy in more general forms and to obtain several new conditions. The method is based on a particular approximation of the von Neumann entropy by an increasing sequence of concave continuous unitary invariant functions defined using decompositions into finite rank operators. The existence of this approximation is a corollary of a general property of the set of quantum states as a convex topological space called the strong stability property. This is considered in the first part of the paper.     

Inequalities for traces on von Neumann algebras

A number of useful inequalities, which are known for the trace on a separable Hilbert space, are extended to traces on von Neumann algebras. In particular, we prove the Golden rule, Hölder inequality, and some convexity statements.Battelle Fellow, 1970–1971.     

On the Embedding of von Neumann Subalgebras

For a von Neumann algebra with a cyclic and separating vector it will be shown that the von Neumann subalgebras with the same cyclic vector can uniquely be characterized by one-parametric operator-valued functions obeying a set of conditions. Since the properties contain no reference to the subalgebra these operator-valued functions will be called characteristic functions. On the set of characteristic functions there exists a natural topology under which this set is complete. Received: 3 December 1998 /Accepted: 15 February 1999     

Free Schrödinger equation analyzed in terms of the wave equation

The free time-dependent Schrödinger equation in three-dimensional space is analyzed as a special case of the wave equation in five-dimensional spacetime. This approach transforms the separation of variables in the parabolic Schrödinger equation into the separation of variables in a nonparabolic equation. Then one can solve the problem using the last theorem of V. N. Shapovalov (see Differents. Uravn., No. 10, 1864 (1980)) on the necessary and sufficient conditions for the complete separation of variables. Other advantages of this approach are also discussed.Translated from Izvestiya Vysshikh uchebnykh Zavedenii, Fizika, No. 7, pp. 59–64, July 1990.     

Structure of the Time Projection for Stopping Times in von Neumann Algebras

We give an explicit formula for the time projection in an arbitrary von Neumann algebra from which all its basic properties can be easily derived. The analysis of the situation when this time projection is a conditional expectation is also performed.     

Frobenius theory for positive maps of von Neumann algebras

Frobenius theory about the cyclic structure of eigenvalues of irreducible non negative matrices is extended to the case of positive linear maps of von Neumann algebras. Semigroups of such maps and ergodic properties are also considered.     

Inner automorphisms of von Neumann algebras

It is first shown that a *-automorphism of a factor is inner if and only if it is asymptotically equal to the identity automorphism. Then it is shown that a periodic *-automorphism of a von Neumann algebra is inner if and only if its fixed point algebra is a normal subalgebra of .     

Equivalent construction of the infinitesimal time translation operator in algebraic dynamics algorithm for partial differential evolution equation

We give an equivalent construction of the infinitesimal time translation operator for partial differential evolution equation in the algebraic dynamics algorithm proposed by Shun-Jin Wang and his students. Our construction involves only simple partial differentials and avoids the derivative terms of δ function which appear in the course of computation by means of Wang-Zhang operator. We prove Wang’s equivalent theorem which says that our construction and Wang-Zhang’s are equivalent. We use our construction to deal with several typical equations such as nonlinear advection equation, Burgers equation, nonlinear Schrodinger equation, KdV equation and sine-Gordon equation, and obtain at least second order approximate solutions to them. These equations include the cases of real and complex field variables and the cases of the first and the second order time derivatives.     

Perturbation theory of von Neumann entropy

In quantum information theory, von Neumann entropy plays an important role; it is related to quantum channel capacities. Only for a few states can one obtain their entropies. In a continuous variable system, numeric evaluation of entropy is not easy due to infinite dimensions. We develop the perturbation theory for systematically calculating von Neumann entropy of a non-degenerate system as well as a degenerate system.     

Boundary conditions for the subdiffusion equation

The boundary conditions for the subdiffusion equations are formulated using the continuous-time random walk model, as well as several versions of the random walk model on an irregular lattice. It is shown that the boundary conditions for the same equation in different models have different forms, and this difference considerably affects the solutions of this equation.     

Extension of covariant POV-measures in von Neumann algebras

We discuss the extension of POV-measures transforming covariantly with respect to automorphic group representations on von Neumann algebras.