Quantum probability distributions in the early Universe. II. The quantum Langevin equation

1988 ◽  
Vol 38 (4) ◽  
pp. 1131-1140 ◽  
Author(s):  
F. R. Graziani
Keyword(s):  
Early Universe ◽  
Langevin Equation ◽  
Probability Distributions ◽  
Quantum Probability ◽  
Quantum Langevin Equation

scholarly journals Fokker–Planck representations of non-Markov Langevin equations: application to delayed systems

2019 ◽  
Vol 377 (2153) ◽  
pp. 20180131 ◽  
Author(s):  
Luca Giuggioli ◽  
Zohar Neu
Keyword(s):  
Langevin Equation ◽  
Complex Dynamics ◽  
Probability Distributions ◽  
Delay Systems ◽  
Langevin Equations ◽  
Delayed Systems ◽  
Temporal Features ◽  
Fokker Planck Equations ◽  
Random Fluctuations ◽  
Fokker Planck

Noise and time delays, or history-dependent processes, play an integral part in many natural and man-made systems. The resulting interplay between random fluctuations and time non-locality are essential features of the emerging complex dynamics in non-Markov systems. While stochastic differential equations in the form of Langevin equations with additive noise for such systems exist, the corresponding probabilistic formalism is yet to be developed. Here we introduce such a framework via an infinite hierarchy of coupled Fokker–Planck equations for the n -time probability distribution. When the non-Markov Langevin equation is linear, we show how the hierarchy can be truncated at n  = 2 by converting the time non-local Langevin equation to a time-local one with additive coloured noise. We compare the resulting Fokker–Planck equations to an earlier version, solve them analytically and analyse the temporal features of the probability distributions that would allow to distinguish between Markov and non-Markov features. This article is part of the theme issue ‘Nonlinear dynamics of delay systems’.

An approach towards a simple quantum Langevin equation

2011 ◽  
Vol 511 (4-6) ◽  
pp. 471-481 ◽  
Author(s):  
Joshua M. Jackson ◽  
Pietrina L. Brucia ◽  
Michael Messina
Keyword(s):  
Langevin Equation ◽  
Quantum Langevin Equation ◽  
Simple Quantum

scholarly journals Dissipative quantum tunneling: quantum Langevin equation approach

1988 ◽  
Vol 128 (1-2) ◽  
pp. 29-34 ◽  
Author(s):  
G.W. Ford ◽  
J.T. Lewis ◽  
R.F. O'Connell
Keyword(s):  
Langevin Equation ◽  
Quantum Tunneling ◽  
Equation Approach ◽  
Quantum Langevin Equation

Probabilistic aspects of the SU(2)-harmonic oscillator

2000 ◽  
Vol 37 (1) ◽  
pp. 306-314
Author(s):  
Shunlong Luo
Keyword(s):  
Harmonic Oscillator ◽  
Coherent States ◽  
Probability Distributions ◽  
Quantum Probability ◽  
Dimensional Sphere ◽  
Large Radius ◽  
Two Dimensional ◽  
Bargmann Transform ◽  
Gaussian Distributions ◽  
Binomial Distributions

In the framework of quantum probability, we present a simple geometrical mechanism which gives rise to binomial distributions, Gaussian distributions, Poisson distributions, and their interrelation. More specifically, by virtue of coherent states and a toy analogue of the Bargmann transform, we calculate the probability distributions of the position observable and the Hamiltonian arising in the representation of the classic group SU(2). This representation may be viewed as a constrained harmonic oscillator with a two-dimensional sphere as the phase space. It turns out that both the position observable and the Hamiltonian have binomial distributions, but with different asymptotic behaviours: with large radius and high spin limit, the former tends to the Gaussian while the latter tends to the Poisson.

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