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29

Non-dissipative diffusion in classical and quantum systems 

Zahid Zakir [1]

Abstract

     The theory of non-dissipative diffusion is constructed on an example of diffusion of a light particle in a dilute medium of heavy particles and it is shown that in low dissipative systems there are specific effects of non-dissipativity similar to quantum effects. In a non-dissipativity region mean energy of the light particle is conserved and processes are described by two non-linear diffusion equations with forward and backward time derivatives. Then these two diffusion equations are linearized and give one linear Schrödinger equation for complex amplitudes of probability. As a result, in the non-dissipative classical diffusion should be added probability amplitudes and there holds the superposition principle for these amplitudes. A mean square length of free passage and a mean square momentum define an elementary phase volume and a diffusion constant and they obey the uncertainty relations. It is shown that the formalism of quantum mechanics describes the classical non-dissipative diffusion with a homogeneous diffusion constant and that quantum mechanics is only a particular case when the elementary phase volume of free passage is universal and equal to the Planck constant.

PACS: 02.50.Ey, 03.65.Ta , 05.40.Jc,

Key words: stochastic processes, quantum mechanics, Brownian motion

Vol. 3, No 3,  p. 16 – 35, v1,        29 August 2008

Online: TPAC: 2798-011 v2,  28 September 2012; DOI: 10.9751/TPAC.2798-011

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[1] Centre for Theoretical Physics and Astrophyics, Tashkent, Uzbekistan

      zahidzakir@theor-phys.org