Kinetic equations from Hamiltonian dynamics: Markovian limits

Spohn, Herbert

doi:10.1103/revmodphys.52.569

Cited by 975 publications

(1,054 citation statements)

References 107 publications

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“…Therefore, to conclude the proof of (1.7), it suffices to show that: 1) every limit point of the family { ( ) , } =1 is a solution of the infinite hierarchy (2.2), and 2) the solution to (2.2) is unique. This strategy has already been used to derive the nonlinear Hartree equations for the effective dynamics of so-called mean-field systems (see [27,13,4,9]) to derive the cubic nonlinear Schrödinger equation with different (and simpler) scalings of the interaction potential (see [8,11]) and to derive the nonlinear Schrödinger equation in a one-dimensional setting (see [1,2]). We remark that the first derivation of the Hartree equation was obtained using a different method in [17,14].…”

Section: Resolution Of the Correlation Structure For Large Potentialmentioning

confidence: 99%

Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential

Erdős¹,

Schlein

Yau

2009

J. Amer. Math. Soc.

142

246

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Consider a system of N N bosons in three dimensions interacting via a repulsive short range pair potential N 2 V ( N ( x i − x j ) ) N^2V(N(x_i-x_j)) , where x = ( x 1 , … , x N ) \mathbf {x}=(x_1, \ldots , x_N) denotes the positions of the particles. Let H N H_N denote the Hamiltonian of the system and let ψ N , t \psi _{N,t} be the solution to the Schrödinger equation. Suppose that the initial data ψ N , 0 \psi _{N,0} satisfies the energy condition \[ ⟨ ψ N , 0 , H N ψ N , 0 ⟩ ≤ C N \langle \psi _{N,0}, H_N \psi _{N,0} \rangle \leq C N \] and that the one-particle density matrix converges to a projection as N → ∞ N \to \infty . Then, we prove that the k k -particle density matrices of ψ N , t \psi _{N,t} factorize in the limit N → ∞ N \to \infty . Moreover, the one particle orbital wave function solves the time-dependent Gross-Pitaevskii equation, a cubic nonlinear Schrödinger equation with the coupling constant proportional to the scattering length of the potential V V . In a recent paper, we proved the same statement under the condition that the interaction potential V V is sufficiently small. In the present work we develop a new approach that requires no restriction on the size of the potential.

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Section: Resolution Of the Correlation Structure For Large Potentialmentioning

confidence: 99%

Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential

Erdős¹,

Schlein

Yau

2009

J. Amer. Math. Soc.

142

246

View full text Add to dashboard Cite

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“…However, when the interaction is tiny (e.g., as in the case of a diluted environment) the dynamics of ρ K becomes approximately free from memory effects [16] (Markovian), and it is dictated by a Lindblad master equation [13] of the forṁ…”

Section: Propagation In Environmentmentioning

confidence: 99%

Neutral kaons in medium: decoherence effects

Andrianov¹,

Taron²,

Tarrach³

2001

Physics Letters B

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We consider departures from hamiltonian dynamics in the evolution of neutral kaons due to their interactions with environment that generate entanglement among them. We propose a phenomenological model of stochastic re-scattering and estimate the coefficients of the effective hamiltonian and decoherence terms. Finally we analyze the interplay between the weak interaction and matter effects and propose the observables suitable to measure the matter characteristics of CPT violation and decoherence.

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“…which has been studied by many authors; see [48,7,20,1,16] for some pioneer works (in these works, the convergence (11) was derived using the BBGKY hierarchy, a method that is less quantitative than our approach).…”

Section: Theorem 2 (Kinetic Estimate) Letmentioning

confidence: 99%