The Linear Fokker-Planck Equation for the Ornstein-Uhlenbeck Process as an (Almost) Nonlinear Kinetic Equation for an Isolated N-Particle System

It is long known that the Fokker-Planck equation with prescribed constant coefficients of diffusion and linear friction describes the ensemble average of the stochastic evolutions in velocity space of a Brownian test particle immersed in a heat bath of fixed temperature. Apparently, it is not so well known that the same partial differential equation, but now with constant coefficients which are functionals of the solution itself rather than being prescribed, describes the kinetic evolution (in the N→∞ limit) of an isolated N-particle system with certain stochastic interactions. Here we discuss in detail this recently discovered interpretation. An erratum to this article can be found at     

Equilibrium Fluctuations for Interacting Ornstein-Uhlenbeck Particles

 We study the hydrodynamic density fluctuations of an infinite system of interacting particles on ℝ ^d . The particles interact between them through a two body superstable potential, and with a surrounding fluid in equilibrium through a random viscous force of Ornstein-Uhlenbeck type. The stationary initial distribution is the Gibbs measure associated with the potential and with a given temperature and fugacity. We prove that the time-dependent density fluctuation field converges in law, under diffusive scaling of space and time, to the solution of a linear stochastic partial differential equation driven by white noise. Received: 10 July 2001 / Accepted: 9 September 2002 Published online: 8 January 2003 RID="*" ID="*" We thank J. Fritz for fruitful discussions, in particular about the existence of the infinite dynamics. A special thanks to L. Bertini for help in the proof of the spectral gap estimate (cf. Appendix B). Communicated by H. Spohn     

On the Asymptotic Behavior of an Ornstein-Uhlenbeck Process with Random Forcing

We study the asymptotic behavior of an inertial tracer particle in a random force field. We show that there exists a probability measure, under which the process describing the velocity and environment seen from the vantage point of the moving particle is stationary and ergodic. This measure is equivalent to the underlying probability for the Eulerian flow. As a consequence of the above we obtain the law of large numbers for the trajectory of the tracer. Moreover, we prove also some decorrelation properties of the velocity of the particle, which lead to the existence of a non-degenerate asymptotic covariance tensor. The research of both authors was supported by the Polish Committee for Scientific Research (KBN) grant No. 2PO3A03123.     

Central Limit Theorem for Locally Interacting Fermi Gas

We consider a locally interacting Fermi gas in its natural non-equilibrium steady state and prove the Quantum Central Limit Theorem (QCLT) for a large class of observables. A special case of our results concerns finitely many free Fermi gas reservoirs coupled by local interactions. The QCLT for flux observables, together with the Green-Kubo formulas and the Onsager reciprocity relations previously established [JOP4], complete the proof of the Fluctuation-Dissipation Theorem and the development of linear response theory for this class of models. UMR 6207, CNRS, Université de la Méditerranée, Université de Toulon et Université de Provence.     

Large Detuning Limit for the Multipartite Systems Interacting with Electromagnetic Fields

We study the dynamics of the multipartite systems nonresonantly interacting with electromagnetic fields, focusing on the large detuning limit for the effective Hamiltonian. Due to the many-particle interference effects, the more rigorous large detuning condition for neglecting the rapidly oscillating terms for the effective Hamiltonian should be N 1/2 g, instead of g usually used in the literature even in the case of multipartite systems, with N the number of microparticles involved, g the coupling strength, the detuning. This result is significant since merely the satisfaction of the original condition will result in the invalidity of the effective Hamiltonian and the errors of the parameters associated with the detuning in the multipartite case.     

Integrable Deformation of Gaussian Distribution and the Ornstein-Uhlenbeck Process

An integrable deformation of one-dimensional Gaussian distribution in terms of a continuous Moser hierarchy is described explicitly. Here the hierarchy governs the level dynamics of the semi-infinite Toda molecule. The action of the second-order flow is shown to be equivalent to that of the Ornstein–Uhlenbeck diffusion process.     

Homogenization of Ornstein-Uhlenbeck Process in Random Environment

We consider a tracer particle moving in a random environment. The velocity of the tracer is modelled by an Ornstein-Uhlenbeck process which takes into account inertia and friction. The medium results in a possibly unbounded random potential. We prove an invariance principle for this kind of motion. The method used is generalized in order to obtain a central limit theorem for a large class of process, the most interesting application being a tagged particle in a medium of infinitely many Ornstein-Uhlenbeck particles.     

The Rotational Limit of the Interacting Boson Model 3

The SU(3) limit of the isospin invariant IBM-IBM3 is studied. The decomposition rules are given for N≤9. An analytical formula for the decomposition of the U(6) [N, 1] is given. Typical spectrum is discussed. Different forms of the interaction and their relation are obtained. Transition operators are also discussed.     

Weyl Semimetallic Phase in an Interacting Lattice System

By using Wilsonian renormalization group methods we rigorously establish the existence of a Weyl semimetallic phase in an interacting three dimensional fermionic lattice system, by showing that the zero temperature Schwinger functions are asymptotically close to the ones of massless Dirac fermions. This is done via an expansion which is convergent in a region of parameters, which includes the quantum critical point discriminating between the semimetallic and the insulating phase.     

Hydrodynamic Limit of Coagulation-Fragmentation Type Models of k-Nary Interacting Particles

Hydrodynamic limit of general k-nary mass exchange processes with discrete mass distribution is described by a system of kinetic equations that generalize classical Smoluchovski's coagulation equations and many other models that are intensively studied in the current mathematical and physical literature. Existence and uniqueness theorems for these equations are proved. At last, for k-nary mass exchange processes with k>2 an alternative nondeterministic measure-valued limit (diffusion approximation) is discussed.     

成像型任意反射面速度干涉仪研制

成像型任意反射面速度干涉仪是激光驱动冲击波研究的重要诊断设备,主要应用于惯性约束聚变脉冲整形实验、高压状态方程实验研究和材料特性研究.基于传统任意反射面速度干涉仪的特点,研制了一台基于神光Ⅲ原型装置的成像型任意反射面速度干涉仪,该系统可用于测量自由面速度及透明介质中的冲击波速度.给出了成像型任意反射面速度干涉仪的详细设计要点.对系统性能进行了分析及测试,结果表明时间分辨优于30 ps,物方空间分辨约为10 μm,测速范围为6～45 km/s.     

Absence of Negative Energy Spectrum for N-Particle Hamiltonians

We consider the spectrum of the quantum Hamiltonian H for a system of N one-dimensional particles. H is given by $H = \sum\nolimits_{i = 1}^n { - \frac{1}{{2m_i }}\frac{{\partial ^2 }}{{\partial x_i^2 }}} + \sum {_{1 \leqslant i < j \leqslant N} } V_{ij} \left( {x_i - x_j } \right)$ acting in L ²(R ^N ). We assume that each pair potential is a sum of a hard core for |x|≤a, a>0, and a function V _ij (x), |x|>a, with $\smallint _a^\infty \left| {x - a} \right|\left| {V_{ij} \left( x \right)} \right|dx < \infty $ . We give conditions on V ^? _ij (x), the negative part of V _ij (x), which imply that H has no negative energy spectrum for all N. For example, this is the case if V ^? _ij (x) has finite range 2a and $$2m_i \smallint _a^{2a} \left| {x - a} \right|\left| {V_{ij}^ - \left( x \right)} \right|dx < 1.$$ If V ^? _ij is not necessarily small we also obtain a thermodynamic stability bound inf?σ(H)≥?cN, where 0<c<∞, is an N-independent constant.     

On the Spectrum of the Generator of an Infinite System of Interacting Diffusions

We study the spectrum of the operator