**Non-dissipative diffusion in classical and quantum systems**

**Non-dissipative diffusion in classical and quantum systems**

**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

**Download pdf** 932 kb

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

1. New result:

The theory of non-dissipative diffusion is constructed on an example of diffusion of a light particle in a dilute medium of heavy particles.

2. New result:

In a non-dissipativity region mean energy of a 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.

3. New result:

A mean square length of free passage and a mean square momentum define an elementary phase volume and a diffusion constant, and they are obey the uncertainty relations.

4. New result:

it is shown that in low dissipative systems there are specific effects of non-dissipativity similar to quantum effects.

5. New result:

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.