Stochastic differential equations

Kampen, N.G. Van

doi:10.1016/0370-1573(76)90029-6

Cited by 708 publications

(303 citation statements)

References 115 publications

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“…(9) to derive an explicit expression for the evolution of the average of ξξ T . A clear exposition of this derivation, together with a very detailed discussion of its domain of validity has been given by van Kampen [16]. We just outline the basic steps: (a) Rewrite Eq.…”

Section: Theorymentioning

confidence: 99%

“…While the original problem was governed by the Hessian matrix of the potential, of size N × N , the new (reduced) one is controlled by the Laplacian of the potential, △V(t), a scalar function of time. Thereafter, △V(t) is treated as Gaussian white noise and the 2×2 system of differential equations is solved using the methods developed by van Kampen and others [16]. See Ref.…”

Section: Introductionmentioning

confidence: 99%

“…This and other examples [5,18] In this paper we present an alternative theoretical approach in which the validity domains of the successive approximations can be precisely delimited. The basic idea is to employ van Kampen's methods [16] to solve the original system of 2N differential equations for the evolution of tangent vectors. By applying this scheme to a three dimensional dilute gas, Barnett et al [7] established a link between the Lyapunov exponent and the self-diffusion coefficient (see also [19]).…”

Section: Introductionmentioning

confidence: 99%

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Theoretical estimates for the largest Lyapunov exponent of many-particle systems

Vallejos

Anteneodo

2002

Phys. Rev. E

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The largest Lyapunov exponent of an ergodic Hamiltonian system is the rate of exponential growth of the norm of a typical vector in the tangent space. For an N -particle Hamiltonian system, with a smooth Hamiltonian of the type p 2 +V(q), the evolution of tangent vectors is governed by the Hessian matrix V of the potential. Ergodicity implies that the Lyapunov exponent is independent of initial conditions on the energy shell, which can then be chosen randomly according to the microcanonical distribution. In this way a stochastic process V(t) is defined, and the evolution equation for tangent vectors can now be seen as a stochastic differential equation. An equation for the evolution of the average squared norm of a tangent vector can be obtained using the standard theory in which the average propagator is written as a cummulant expansion. We show that if cummulants higher than the second one are discarded, the Lyapunov exponent can be obtained by diagonalizing a smalldimension matrix, which, in some cases, can be as small as 3×3. In all cases the matrix elements of the propagator are expressed in terms of correlation functions of the stochastic process. We discuss the connection between our approach and an alternative theory, the so-called geometric method.

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Section: Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Theoretical estimates for the largest Lyapunov exponent of many-particle systems

Vallejos

Anteneodo

2002

Phys. Rev. E

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“…where we make use of the fact that ξ s † (t) and ξ †N (t) are uncorrelated [6] and the operator S z (t ′ ) at time t ′ is not affected by fluctuation at a latter time t. Following Bourret [16,17] and van Kampen [18] we now make decoupling approximation ( which implies that the correlation of fluctuations ξ N (t)…”

Section: Discussion Of An Experimental Scheme and Conclusionmentioning

confidence: 99%

Modified Bloch equations in the presence of a nonstationary bath

Chaudhuri¹,

Banik²,

Deb³

et al. 1999

Eur. Phys. J. D

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“…Since the time scale of the experiment is large compared to the characteristic phase fluctuation time of the lasers, we take a statistical average over the fluctuating phases [29]. We assume that both lasers are uncorrelated, so that we can apply the "decorrelation approximation" [29,32] to obtain a system of equations, which, when solved, yields the density matrix of the nD 3/2 state, from which the fluorescence intensities in each polarization are obtained.…”

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confidence: 99%

Electric-Field-Induced Symmetry Breaking of Angular Momentum Distribution in Atoms

et al. 2006

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We report the experimental observation of alignment to orientation conversion in the 7D 3/2 and 9D 3/2 states of Cs in the presence of an external dc electric field, and without the influence of magnetic fields or atomic collisions. Initial alignment of angular momentum states was created by two-step excitation with linearly polarized laser radiation. The appearance of transverse orientation of angular momentum was confirmed by the observation of circularly polarized light. We present experimentally measured signals and compare them with the results of a detailed theoretical model based on the optical Bloch equations. 32.60.+i,32.80.Qk Atomic physics experiments are able to place sensitive limits on the permanent electric dipole moment (EDM) of the electron because highly polarizable atoms can generate internal electric fields that are orders of magnitude higher than can be achieved for free particles [1,2]. Such limits on the electron EDM are of great interest to fundamental physics because a permanent EDM of a fundamental particle implies CP violation as long as the CPT theorem is assumed to hold. Such a CP violation would be a signature of new physics beyond the standard model [3]. The most sensitive upper limit to date on the electron EDM has been achieved by searching for a small precession around an external electric field of the angular momentum distribution of an ensemble of atoms [4]. Various sophisticated experimental techniques prevent angular momentum precession caused by mechanisms other than an EDM from contaminating the signal. One such phenomenon, known as alignment to orientation conversion (AOC), is known to deform the atomic and molecular angular momentum distributions under the influence of combined electric and magnetic fields or collisions. Moreover, it has been shown theoretically [5] that AOC can be induced by a purely electric field without the need for magnetic fields or collisions whenever the initial alignment is not exactly perpendicular or parallel to the external electric field. In this letter, we report the experimental observation of AOC in the presence of only an electric field. Our measured signals are described accurately by a detailed theoretical model, which could be of interest in studying potential backgrounds for electron EDM searches in atomic or molecular systems.Angular momentum alignment means that the population of atomic sublevels m J varies with |m J |, but +m J and −m J levels are populated equally. When the +m J and −m J levels are unequally populated, the ensemble is oriented. One can describe the anisotropic spatial distribution of angular momenta J in an ensemble by an atomic density matrix ρ [6]. An ensemble is described as * Electronic address: mauzins@latnet.lv aligned if the distribution of angular momenta contains a net electric quadrupole moment or oriented if there is a net magnetic dipole moment. Orientation can be classified as longitudinal or transverse, with reference to the symmetry axis. The breaking of the reflection symmetry of an initially a...

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Stochastic differential equations

Cited by 708 publications

References 115 publications

Theoretical estimates for the largest Lyapunov exponent of many-particle systems

Theoretical estimates for the largest Lyapunov exponent of many-particle systems

Modified Bloch equations in the presence of a nonstationary bath

Electric-Field-Induced Symmetry Breaking of Angular Momentum Distribution in Atoms

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