The Boltzmann Equation
INTRODUCTION
In the most general definition, a Boltzmann Equation (BE) is an
equation for the evolution of a probability density during time furnished with an initial
condition. This means that there are BEs for problems with discrete and continuous states
allowing discrete and continuous time evolution.
The following is an example of a linear BE for a continuous-time problem.
At the beginning, BEs appeared in the theory of particle transport
(more than 100 years ago) and the unknown figuring in these was the particle density in
the phase-space. These were, of course, continuous-time and continuous-state equations.
This was also the problem I was assigned in my Laurea thesis.
During my work, however, I understoood that this was a very reductive approach and
emphasized how a deterministic approach to the problem could have been easily dismounted
with theoretical arguments.
I began backing the point of "uncertainty" in these processes and finally
understood that integral operators were the most general description of stochastic
processes.
This soon brought me to generalize these equations to discrete-time as in the following
example.
I have later pulled out a lot of new applications by simply
inspecting processes different from particle transport and making generalizations.
I now employ these equations (mostly non-linear) in a wide range of stochastic problems
with good results.
SOLVING
In my thesis work toward a degree in Nuclear Engineering I have
worked out a general method for finding solutions to the INTEGRO-DIFFERENTIAL Boltzmann
equation by the application of FUNCTIONAL ANALYSIS concepts.
This method has actually been applied successfully to the solution of the long standing
problem of ELASTIC ISOTROPIC SCATTERING in a neutron flux.
The result is the exact espression for the evolution in time of the neutron velocity
ditribution.
Then I have exactly solved the equation for BGK scattering in the presence of an external
force. This is an equation in two variables solvable with these techniques.
Other minor scattering problems have been given an answer by this OPERATOR
approach and I am now working to the construction of a general scheme of solution
building for MULTI-INTEGRAL-PARTIAL-DERIVATIVE EQUATIONS which represent the most general
case of INTEGRO-DIFFERENTIAL EQUATIONS.
I am also doing some research on the NON-LINEAR case.
The technique used already covers the case of DIFFERENCE rather
than DIFFERENTIAL time dependence which applies to those problems dealing with
DISCRETE-TIME MATHEMATICS such as CONTROL THEORY and AI.
As I see it, PARTIAL-DERIVATIVE EQUATIONS are only a particular case of the Boltzmann
ones.
The use of Boltzmann equations is widening fastly in pure and applied sciences and
solutions are needed in several fields of applications ranging from civil engineering
(traffic control) to electronics and medicine; from military simulations (nuclear
explosions, battle simulations and behavior modeling etc.) to
high-end financial-market research.
In pure and applied sciences man faces problems of ever-growing difficulty which need to
be solved with increasingly powerful formalism. This affects physics as engineering,
computing science as managing. If you want to be one of those who lead in these fields you
must give yourself very powerful weapons.