**Theory of conservative diffusion **in **classical and quantum systems**

**Theory of conservative diffusion**in**classical and quantum systems****Abstract**

In previous paper [1] conservative diffusion of light particles in a dilute medium of heavy particles was studied by analogue with Brownian motion. In the present paper the theory is formulated in more consistent “hydrodynamic” form by using a conservativity condition only, that mean energy of a light particle is conserved. As a model is taken diffusion of very cold light gas in warm heavy gas a time interval before relaxation when light gas remains cold. Unlike Lorentz’s gas, where thermal energies of light and heavy atoms are equal, here their thermal speeds are equal and this leads to the effects of conservativity similar to quantum effects. Such conservative diffusion is described by two equations – the continuity equation and the energy conservation condition, non-linear under the probability density. At introduction of a complex probability amplitude the equations linearized and turn to the Schrödinger equation. As a result, one must add not probability of alternatives, but probability amplitudes. A free pass length and corresponding momentum define an elementary phase volume and the diffusion coefficient. The predicting new quasi-quantum effects in classical systems are discussed. It is shown that the formalism of quantum mechanics describes the classical conservative diffusion with a constant diffusion coefficient and that quantum mechanics is a particular case of such diffusion in the vacuum where the elementary phase volume of free passage is equal to the Planck constant.

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

*Key words*:* quantum mechanics, diffusion, Brownian motion, kinetic theory of gases*

Vol. **9**, No 1, p. 18 –32, v1, 6 May 2014

Online: TPAC: 4874-036 v1, 6 May 2014; DOI: 10.9751/TPAC.4874-036

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

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