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1.
The unsteady shock wave diffraction over a 90° sharp corner in gases of arbitrary particle statistics is simulated using an
accurate and direct algorithm for solving the semiclassical Boltzmann equation with relaxation time approximation in phase
space. The numerical method is based on the usage of discrete ordinate method for discretizing the velocity space of the distribution
function; whereas a second order accurate TVD scheme (Harten in J. Comput. Phys. 49(3):357–393, 1983) with Van Leer’s limiter (J. Comput. Phys. 32(1):101–136, 1979) is used for evolving the solution in physical space and time. The specular reflection surface boundary condition is assumed.
The complete diffraction patterns are recorded using various flow property contours. Different range of relaxation times approximately
corresponding to continuum, slip and transitional regimes are considered and the equilibrium Euler limit solution is also
computed for comparison. The effects of gas particles that obey the Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics
are examined and depicted. 相似文献
2.
Anomalous transport is usually described either by models of continuous time random walks (CTRWs) or, otherwise, by fractional Fokker-Planck equations (FFPEs). The asymptotic relation between properly scaled CTRW and fractional diffusion process has been worked out via various approaches widely discussed in literature. Here, we focus on a correspondence between CTRWs and time and space fractional diffusion equation stemming from two different methods aimed to accurately approximate anomalous diffusion processes. One of them is the Monte Carlo simulation of uncoupled CTRW with a Le?vy α-stable distribution of jumps in space and a one-parameter Mittag-Leffler distribution of waiting times. The other is based on a discretized form of a subordinated Langevin equation in which the physical time defined via the number of subsequent steps of motion is itself a random variable. Both approaches are tested for their numerical performance and verified with known analytical solutions for the Green function of a space-time fractional diffusion equation. The comparison demonstrates a trade off between precision of constructed solutions and computational costs. The method based on the subordinated Langevin equation leads to a higher accuracy of results, while the CTRW framework with a Mittag-Leffler distribution of waiting times provides efficiently an approximate fundamental solution to the FFPE and converges to the probability density function of the subordinated process in a long-time limit. 相似文献
3.
Brownian motion of the particles with repulsive interaction is investigated. When the potential condition is satisfied, the eigenvalue problem of interaction Fokker-Planck equation
under certain conditions can be transformed to that of a many-particle Schrödinger equation. Using the Green's function method,
we obtain the effective single-variable Fokker-Planck equation in the low density limit. We find that the diffusion of coupled Brownian particles in quenched disorder media is also anomalous in 2D. The Mittag-Leffler relaxation of pancake vortices is investigated by fractional Fokker-Planck equation. 相似文献
4.
Integral Representation of the Linear Boltzmann Operator for Granular Gas Dynamics with Applications
We investigate the properties of the collision operator Q associated to the linear Boltzmann equation for dissipative hard-spheres arising in granular gas dynamics. We establish that, as in the case of non-dissipative interactions, the gain
collision operator is an integral operator whose kernel is made explicit. One deduces from this result a complete picture
of the spectrum of Q in an Hilbert space setting, generalizing results from T. Carleman (Publications Scientifiques de l’Institut Mittag-Leffler,
vol. 2, 1957) to granular gases. In the same way, we obtain from this integral representation of Q
+ that the semigroup in L
1(ℝ3×ℝ3,dx
⊗
dv) associated to the linear Boltzmann equation for dissipative hard spheres is honest generalizing known results from Arlotti (Acta Appl. Math. 23:129–144, 1991). 相似文献
5.
Mark J. Everitt 《Journal of Russian Laser Research》2009,30(5):412-417
We explore the quantum–classical crossover of two coupled, identical, superconducting quantum interference device (SQUID)
rings. The motivation for this work is based on a series of recent papers. In [1] we showed that the entanglement characteristics of chaotic and periodic (entrained) solutions of the Duffing oscillator
differed significantly and that in the classical limit entanglement was preserved only in the chaotic-like solutions. However,
Duffing oscillators are a highly idealized toy system. Motivated by a wish to explore more experimentally realizable systems,
we extended our work in [2, 3] to an analysis of SQUID rings. In [3] we showed that the two systems share a common feature. That is, when the SQUID ring’s trajectories appear to follow (semi)classical
orbits, entanglement persists. Our analysis in [3] was restricted to the quantum-state diffusion unraveling of the master equation – representing unit efficiency heterodyne
detection (or ambi-quadrature homodyne detection). Here we show that very similar behavior occurs using the quantum jumps
unraveling of the master equation. Quantum jumps represents a discontinuous photon counting measurement process. Hence, the
results presented here imply that such persistent entanglement is independent of measurement process and that our results
may well be quite general in nature. 相似文献
6.
I. P. Guk 《Technical Physics》1998,43(4):353-357
Analogs of the Lagrange equation for particles evolving in a space of fractal dimension are obtained. Two cases are considered:
1) when the space is formed by a set of material points (a so-called fractal continuum), and 2) when the space is a true fractal.
In the latter case the fractional integrodifferential formalism is utilized, and a new principle for devising a fractal theory,
viz., a generalized principle of least action, is proposed and used to obtain the corresponding Lagrange equation. The Lagrangians
for a free particle and a closed system of interacting particles moving in a fractal continuum are derived.
Zh. Tekh. Fiz. 68, 7–11 (February 1990) 相似文献
7.
Motivated by a recent use of Glauber dynamics for Monte Carlo simulations of path integral representation of quantum spin
models (Krzakala et al. in Phys. Rev. B 78(13):134428, 2008), we analyse a natural Glauber dynamics for the quantum Ising model with a transverse field on a finite graph G. We establish strict monotonicity properties of the equilibrium distribution and we extend (and improve) the censoring inequality
of Peres and Winkler to the quantum setting. Then we consider the case when G is a regular b-ary tree and prove the same fast mixing results established in Martinelli et al. (Commun. Math. Phys. 250(2):301–334, 2004) for the classical Ising model. Our main tool is an inductive relation between conditional marginals (known as the “cavity
equation”) together with sharp bounds on the operator norm of the derivative at the stable fixed point. It is here that the
main difference between the quantum and the classical case appear, as the cavity equation is formulated here in an infinite
dimensional vector space, whereas in the classical case marginals belong to a one-dimensional space. 相似文献
8.
Barkai E Metzler R Klafter J 《Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics》2000,61(1):132-138
We generalize the continuous time random walk (CTRW) to include the effect of space dependent jump probabilities. When the mean waiting time diverges we derive a fractional Fokker-Planck equation (FFPE). This equation describes anomalous diffusion in an external force field and close to thermal equilibrium. We discuss the domain of validity of the fractional kinetic equation. For the force free case we compare between the CTRW solution and that of the FFPE. 相似文献
9.
We study the limit of quasilocal energy defined in Wang and Yau (Phys Rev Lett 102(2):021101, 2009; Commun Math Phys 288(3):919–942, 2009) for a family of spacelike 2-surfaces approaching null infinity of an asymptotically flat spacetime. It is shown that Lorentzian
symmetry is recovered and an energy-momentum 4-vector is obtained. In particular, the result is consistent with the Bondi–Sachs
energy-momentum at a retarded time. The quasilocal mass in Wang and Yau (Phys Rev Lett 102(2):021101, 2009; Commun Math Phys 288(3):919–942, 2009) is defined by minimizing quasilocal energy among admissible isometric embeddings and observers. The solvability of the Euler-Lagrange
equation for this variational problem is also discussed in both the asymptotically flat and asymptotically null cases. Assuming
analyticity, the equation can be solved and the solution is locally minimizing in all orders. In particular, this produces
an optimal reference hypersurface in the Minkowski space for the spatial or null exterior region of an asymptotically flat
spacetime. 相似文献
10.
Summary Diffusion of a fractal cluster of dimensiond
f
in a three-dimensional space is investigated. The diffusion process is assumed to be modelled by a standard parabolic diffusion
equation, although a more general case represented by the Fokker-Planck-Kolmogoroff equation is also introduced. The mean-square
displacement of the cluster mass centre is analysed and its anomalous behaviour is presented and critically discussed. The
results obtained can be applied to describe some effects which may occur during the diffusion-limited cluster-cluster aggregation
process, especially when the viscosity of the solvent is changed in time and/or a directed transport of clusters is present
in the system.
Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994. 相似文献
11.
Ioannis Anapolitanos 《Letters in Mathematical Physics》2011,98(1):1-31
In the mean-field regime we prove convergence, with explicit bounds, of N-particle density matrices satisfying the time-dependent von Neumann equation with factorized initial data to a product of
one particle density matrices satisfying the Hartree–von Neumann equation. To prove explicit bounds we generalize techniques
developed by Pickl (in A simple derivation of mean field limits for quantum systems. ArXiv:0907.4464, 2009) and Knowles–Pickl (in Commun. Math. Phys. 298(1):101–138, 2010). 相似文献
12.
Zhigang Wu 《Journal of statistical physics》2009,137(1):149-164
The Cauchy problem of the relativistic Enskog equation with near-vacuum data is considered in this paper. Under the same assumption
as that in Jiang (J. Stat. Phys. 127:805–812, 2007) for the relativistic Enskog equation, we obtain the uniform L
∞-stability of the solution. What’s more important, is that for two new types of the scattering cross section σ, we give the global existence and L
1(x,v)-stability for mild solution when the initial data lies in the space L
1(x,v). As a corollary, we have a BV-type estimate. It is worth mentioning that the stability results in this paper can be applied to the case in Jiang (J. Stat.
Phys. 127:805–812, 2007). 相似文献
13.
In this paper, we study the fractional number operator as an analog of the finite-dimensional fractional Laplacian. An important relation with the Ornstein-Uhlenbeck process is given. Using a semigroup approach, the solution of the Cauchy problem associated to the fractional number operator is presented. By means of the Mittag-Leffler function and the Laplace transform, we give the solution of the Caputo time fractional diffusion equation and Riemann-Liouville time fractional diffusion equation in infinite dimensions associated to the fractional number operator. 相似文献
14.
Inna Ya. Minchina Mikhail I. Bagatskii Vadim G. Manzhelii Pavel I. Muromtsev 《Czechoslovak Journal of Physics》1996,46(Z1):533-534
Quantum diffusion in solid hydrogen containing 0.02–0.25 mol.% neon has been investigated by the calorimetric method in temperature
range 1–3 K. The concentrations of orthohydrogen were 0.23; 0.5 and 1 mol. %. The parameter studied was characteristic configurational
relaxation time τ. Heat capacity is very sensitive to space distribution of orthohydrogen molecules. Therefore, the determination
of configuration relaxation rate has been performed by observing the time dependence of heat capacity. A neon impurity in
the indicated concentration is observed to accelerate quantum diffusion in hydrogen. The magnitude of the effect diminishes
as the temperature increases. 相似文献
15.
Fractional generalization of the diffusion equation includes fractional derivatives with respect to time and coordinate. It had been introduced to describe anomalous kinetics of simple dynamical systems with chaotic motion. We consider a symmetrized fractional diffusion equation with a source and find different asymptotic solutions applying a method which is similar to the method of separation of variables. The method has a clear physical interpretation presenting the solution in a form of decomposition of the process of fractal Brownian motion and Levy-type process. Fractional generalization of the Kolmogorov-Feller equation is introduced and its solutions are analyzed. (c) 1997 American Institute of Physics. 相似文献
16.
V. M. Zolotarev V. V. Uchaikin V. V. Saenko 《Journal of Experimental and Theoretical Physics》1999,88(4):780-787
The superdiffusion equation with a fractional Laplacian Δ
α/2 in N-dimensional space describes the asymptotic (t→∞) behavior of a generalized Poisson process with the range (discontinuity) distribution density ∼|x|−α−1. The solutions of this equation belong to a class of spherically symmetric stable distributions. The main properties of
these solutions are given together with their representations in the form of integrals and series and the results of numerical
calculations. It is shown that allowance for the finite velocity of free particle motion for α>1 merely amounts to a reduction in the diffusion coefficient with the form of the distribution remaining stable. For α<1 the situation changes radically: the expansion velocity of the diffusion packet exceeds the velocity of free particle motion
and the superdiffusion equation becomes physically meaningless.
Zh. éksp. Teor. Fiz. 115, 1411–1425 (April 1999) 相似文献
17.
The regularized determinant of the Paneitz operator arises in quantum gravity [see Connes in (Noncommutative geometry, 1994), IV.4.γ]. An explicit formula for the relative determinant of two conformally related metrics was computed by Branson in (Commun Math Phys 178:301–309, 1996). A similar formula holds for Cheeger’s half-torsion, which plays a role in self-dual field theory [see Juhl in (Families of conformally covariant differential operators, q-curvature and holography. Progress in Mathematics, vol 275, 2009)], and is defined in terms of regularized determinants of the Hodge laplacian on p-forms (p < n/2). In this article we show that the corresponding actions are unbounded (above and below) on any conformal four-manifold. We also show that the conformal class of the round sphere admits a second solution which is not given by the pull-back of the round metric by a conformal map, thus violating uniqueness up to gauge equivalence. These results differ from the properties of the determinant of the conformal Laplacian established in (Commun Math Phys 149:241–262, 1992), (Ann Math 142:171–212, 1995), (Commun Math Phys 189:655–665, 1997). 相似文献
18.
Francesco Mainardi 《Entropy (Basel, Switzerland)》2020,22(12)
In this survey we stress the importance of the higher transcendental Mittag-Leffler function in the framework of the Fractional Calculus. We first start with the analytical properties of the classical Mittag-Leffler function as derived from being the solution of the simplest fractional differential equation governing relaxation processes. Through the sections of the text we plan to address the reader in this pathway towards the main applications of the Mittag-Leffler function that has induced us in the past to define it as the Queen Function of the Fractional Calculus. These applications concern some noteworthy stochastic processes and the time fractional diffusion-wave equation We expect that in the next future this function will gain more credit in the science of complex systems. Finally, in an appendix we sketch some historical aspects related to the author’s acquaintance with this function. 相似文献
19.
Xicheng Zhang 《Communications in Mathematical Physics》2012,311(1):133-155
In this article we study the fractal Navier-Stokes equations by using the stochastic Lagrangian particle path approach in
Constantin and Iyer (Comm Pure Appl Math LXI:330–345, 2008). More precisely, a stochastic representation for the fractal Navier-Stokes equations is given in terms of stochastic differential
equations driven by Lévy processes. Based on this representation, a self-contained proof for the existence of a local unique
solution for the fractal Navier-Stokes equation with initial data in
\mathbb W1,p{{\mathbb W}^{1,p}} is provided, and in the case of two dimensions or large viscosity, the existence of global solutions is also obtained. In
order to obtain the global existence in any dimensions for large viscosity, the gradient estimates for Lévy processes with
time dependent and discontinuous drifts are proved. 相似文献
20.
N. S. Averkiev L. E. Golub G. E. Pikus 《Journal of Experimental and Theoretical Physics》1998,86(4):780-789
A theory of weak localization is constructed for p-type semiconductor structures with a complex Γ8 valence band. An equation for the Cooperon is obtained and solved in the case when spin relaxation cannot be treated as a
perturbation. The anomalous magnetoresistance is calculated in bulk samples as a function of the external deformation and
in quantum wells as a function of the doping level. The results of the theory are represented in a form that allows direct
comparison with experiment.
Zh. éksp. Teor. Fiz. 113, 1429–1445 (April 1998) 相似文献