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# Ludwig Boltzmann

Ludwig Boltzmann

Ludwig Boltzmann (February 20, 1844September 5, 1906) was an Austrian physicist famous for the invention of statistical mechanics.

Boltzmann was born in Vienna, Austria-Hungary (now Austria).

Biography here

Boltzmann committed suicide in 1906 by hanging while on holiday in Duino near Trieste in Italy. The motivation behind the suicide remains unclear, but it may have been related to his lingering resentment over scientific establishment's rejection of his theories. Today, his formula for entropy S is famous:

$S = k_{B} \; \ln P \; ,$

where kB = 1.380658(12) × 10-23 J K-1 is the Boltzmann constant and P is the number of possible microscopic states which give the same thermodynamical state that a system may be in. Indeed this formula, as he published it in the nomenclature of his day,

$S = k \; \log W$

is engraved on Boltzmann's tombstone at the Vienna Zentralfriedhof.

## Boltzmann's Kinetic Equation

Ludwig Boltzmann is not only famous for his work on statistical mechanics, he also did much work on kinetics; indeed he also has a famous equation named after him in the field of kinetics as well. The Boltzmann kinetic equation is given by the following

$\frac{\partial f}{\partial t}+ v \frac{\partial f}{\partial x}+ F \frac{\partial f}{\partial v} = \frac{\partial f}{\partial x}\left.{\!\!\frac{}{}}\right|_{c}.$

where f represents an arbitary distribution function (see Maxwell-Boltzmann distribution), F is a force, t is the time and v is an average velocity of particles.

The Boltzmann kinetic equation basically describes the time and spatial variation of a group of particles within a given volume. The first term on the left hand side of the equation represents the time variation of the energy distribution function of the particles. The second term give the spatial variation of the distribution function and the third term describes the effect of a force on the particles. The contribution of these three terms are then related to the effect of collisions between the particles, which is taken into account by the right hand side of the equation.

In order to obtain the macroscopic physical quantities from the Boltzmann equation, one integrates the distribution function over energy space, for example

$\int f d E = n$

Where E is the energy and n is the density of the particles. This integral equation simply states that if one integrates over all of the particles at all energies, within a given volume, then the result is the density of particles.

In principle the above equation completely describes the dynamics of an ensemble of particles, given appropriate boundary conditions. This first order differential equation is deceptively simple though, as f can represent an arbitrary distribution function. Also, the force acting on the particles depends directly on the their velocity distribution function, f. This coupling between the particle dynamics and the forces results in the Boltzmann equation being horribly difficult to integrate and solve. The collision operator on the right hand side of the equation also presents problems, since the exact form of the collision operator is in general not well known. An exact solution of the above equation would therefore require the solution of a full N body problem. Since the particle density within a gas is of the order of $10^{15} \rightarrow 10^{22}$ this is a hopelessly impossible task.

There are though various numerical techniques available for solving the boltzmann equation, such as particle in cell simulations.