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Register a new userConstructing a solution to the characteristic equation for the Langevin diffusion model with orthogonal perturbations
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Added by
Elena V. Karachanskaya
Publication date
2021
fields
Physics
and research's language is
English
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For the Langevin model of the dynamics of a Brownian particle with perturbations orthogonal to its current velocity, in a regime when the particle velocity modulus becomes constant, an equation for the characteristic function $psi (t,lambda )=Mleft[exp (lambda ,x(t))/V={rm v}(0)right]$ of the position $x(t)$ of the Brownian particle. The obtained results confirm the conclusion that the model of the dynamics of a Brownian particle, which constructed on the basis of an unconventional physical interpretation of the Langevin equations, i. e. stochastic equations with orthogonal influences, leads to the interpretation of an ensemble of Brownian particles as a system with wave properties. These results are consistent with the previously obtained conclusions that, with a certain agreement of the coefficients in the original stochastic equation, for small random influences and friction, the Langevin equations lead to a description of the probability density of the position of a particle based on wave equations. For large random influences and friction, the probability density is a solution to the diffusion equation, with a diffusion coefficient that is lower than in the classical diffusion model.
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This article is dedicated to the following class of problems. Start with an $Ntimes N$ Hermitian matrix randomly picked from a matrix ensemble - the reference matrix. Applying a rank-$t$ perturbation to it, with $t$ taking the values $1le t le N$, we study the difference between the spectra of the perturbed and the reference matrices as a function of $t$ and its dependence on the underlying universality class of the random matrix ensemble. We consider both, the weaker kind of perturbation which either permutes or randomizes $t$ diagonal elements and a stronger perturbation randomizing successively $t$ rows and columns. In the first case we derive universal expressions in the scaled parameter $tau=t/N$ for the expectation of the variance of the spectral shift functions, choosing as random-matrix ensembles Dysons three Gaussian ensembles. In the second case we find an additional dependence on the matrix size $N$.
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