Let's consider the Brownian motion of some particles in ideal gas of particles of another sort
supposing that concentration of the former is not very big, and we can regard their motion as
a number of random transpositions. It means that while considering the motion
of the particles of the first sort we can take into account only collisions
with the second sort of particles and we can regard the distribution of the second sort of particles
as uniform. Accepting such a suppositions we can obtain as the consequence that all points
of space are equal for the particles of the first sort. It means that we can
describe the motion of all the particles with fixed initial position in terms of
conditional probabilities, taking all the possible pathways into account.
(Statistical **path integrals**)

First of all, let's find out the average square of distance between the current position of
some particle and it's initial position. (It's evident, that average radius-vector of
the particle is equal to zero on principles of symmetry):

**r**^{2} = е_{i,j}**r**_{i}**r**_{j} =
е_{i}**r**_{i}^{2} +
е _{i№ j} **r**_{i}**r**_{j}

But the latter sum is equal to zero because all transpositions are statistically independent.
Finally:

*< r^{2}> = n*<D r^{2}> = <D r^{2}>*t /<D t>*

Here

where

Finally, the integral conditions for required probability distribution of the particles
in the space are:

*p( r,t) = *т...т

where

е

One of possible solutions of this integral equation is:

*p( r) = B*Exp(C*(r - r_{0})^{2})*

Here

And exactly this solution satisfies our integral conditions! (It can be checked easily by considering only two integrals. Try to evaluate this:

т

Here is another approach to this problem of diffusion theory.
Diffusion equation for small concentration is:

*n v = D*grad n*

where

The result of two previous equations is:

Here D is Laplacian. The functions

are Green functions of this equation - it's solutions with delta-function as initial conditions. In our problem all the particles initially have the same positions, and their distribution are delta-function!

Real-time model of Brownian motion Brownian.zip

Let's consider some crystal with structural defects, atoms of some other element. Let's accept, that electric current can't pass through the defect. When concentration of defects is greater than some critical value, the whole crystal becomes non-conductor. The main task of percolation theory is to determine this critical value for different configurations of crystal structures. When the concentration of defects is equal to this critical value, the slight change in concentration leads to the great change in conductivity. There is another interesting effect: when the concentration of defects is near to this critical value, the clusters of non-defect crystal cells are the fractal structures! For simple 2D grid the critical concentration of defects is 0.41, and for simple 3D grid - 0.69.

Here are the program source to view and explore 2D and 3D crystals, and to process the results
of the experiments (defining the critical value, see Less Square Method)

Percol.zip

Let's consider a container filled with ideal gas. There is a small hole in the container,
so the gas is flowing out rather slowly. Let's find out how the temperature of the gas
depends on time. All the parameters of any statistical system can be described with
some distribution function. For example, if considered parameter is absolute value of velocity,
then the number of particles with velocity from *v* to *v + dv* is:

*dN = N*f(v)*dv*

where N is the total number of particles in the system.
It is evident that

т*f(v)dv = 1*

The function *f(v)* may change in time. Average velocity of particles in the
system is:

*<v> = *т*vf(v)dv*

Average square of velocity is:

*<v ^{2}> = *т

This value can be considered as the temperature of the gas because kinetic energy of the particle in classic mechanics is proportional to the square of its velocity, and the temperature is proportional to average energy of particles in the system. Due to the hole in the container number of particles decreases, and the rate of change of number of particles with velocities from

where

where

Here is the program, which models the gas outflow process in vacuum and evaluates the temperature of gas as the function of time Gas.zip

Entropy is the meausure of chaotic behaviour of any statistical assembly.
An universal defintion for entropy is based on distribution function of
statistical assembly, but not *dS = DQ/T*, which is most frequently
used formala for entropy. Generally:

*S = S _{i} p_{i} ln(1/p_{i})*

where

Put your computer into thermostat.
Fill the hard disk of your computer with random data and evaluate it's entropy,
using distribution function for values of bytes. Then archive all the data.
Measure the segregated heat *Q* and CPU temperature *T*.
Evaluate entropy of your hard disk again. Is *Q/T* equal to
change in HDD's entropy? ;))) I think it depends on frequency of your CPU :))))

Now let's prove such a well-known formulae for entropy:

(i) *dS = DQ/T* and

(ii) *S = ln N*

where *DQ* is infinitesimal amount of heat, *T* is thermodynamic temperature,
*N* is the number of microstates within the given energy state.
Formula (i) is special case of general formula applied to equilibrium **Gibbs distribution**.
For this distribution *p _{i} = A*Exp(-b*E_{i})*

where

After some mathematical work, such a formula can be derived from considered distribution and general formula:

Note that all

Evaluating

Comparing this, (*) and (**), we prove the initial formula:

Formula (ii) is simply proved by

Obviously

Finally, let's derive the formula for previously used **Gibbs distribution**.
The value of entropy culminates in equilibrium state (And equilibrium is the most possible state for
all the physical phenomena). All the probabilities must be varied to define
position of extremum
with additional conditions according to defintion of probability and the law of conservation of
energy:

*dS = S _{i} d(p_{i} ln(1/p_{i})) =
S_{i} dp_{i} ln(1/p_{i}) +
S_{i} p_{i} dp_{i}/p_{i} =
S_{i} dp_{i} ln(1/p_{i}) = 0
*

Using Lagrange method to define position of extremum with additional conditions, we obtain:

where

where

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