Boltzmann was born in Vienna (then the capital of the Austrian Empire) in 1844. After the death of his father (a tax official) in 1859, the 18 year old Boltzmann attended the University of Vienna as a student of physics, with a focus on the mathematical aspects of these theories.
Prior to this point in time, the theory of atoms - best enunciated in their modern form by John Dalton in the first decade of the nineteenth century - lay mostly in the domain of chemistry. Chemists had come to recognize over the previous century that the chemical elements that they so fervently studied could best be understood as if composed of individual tiny "corpuscles" of matter. What was missing, however - and what prevented the theory from fully being incorporated into the models used by physicists - was any real mathematical theory which demonstrated their individual motions.
Boltzmann's Statistics
Surely Boltzmann's most important and long-lasting contribution to the realm of mathematical physics (a contribution which is still taught regularly to this day) lies in the realm of the "kinetic theory." This theory, presupposing the reality of atoms and the molecules that they form, holds that the measurement of the "heat" of a given gas is nothing more than a statistical measurement of the motions of the individual particles which make up the gas.
Boltzmann, along with another prominent physicist of the nineteenth century, James Clerk Maxwell, studied the nature of gases in relation to the prevailing kinetic theory (much of which was the result of Maxwell's work) and was able to develop a number of crucial mathematical laws which seemed to govern the behavior of these atom-based compounds.
Among the many prominent mathematical/physical tools which resulted from his work are:
- The Maxwell-Boltzmann Distribution: After recognizing that the minuscule scale of the atomic theory, combined with the utterly virulent nature of any atom-based substance meant that the measurement of the velocities of individual particles would surely prove nearly impossible (though, according to the classical physics of the day, still fundamentally possible), Boltzmann and Maxwell recognized that the velocity of the particles within a sample of gas could only be measured in a purely statistical sense, resulting as a property of the pressure and diffusion of a given sample in bulk. This use of statistical mathematics in measuring the properties of groups of atoms quickly became the prevailing method, and remains an integral part of physical theory, even within quantum mechanics.
- Boltzmann's Constant: Though the actual nature of this physical constant was not arrived at until after Boltzmann's death (through the work of the great Max Planck), it was based squarely on work performed by Boltzmann in the 1870's, specifically in regard to the nature of the energy carried by an individual particle within a gas. Boltzmann was able to recognize a fundamental relationship between the absolute temperature and the kinetic energy in particles, though he himself never did attempt to calculate the actual value of this relationship (which is now known to be somewhere very close to 1.3807 x 10^-23 joules per degree kelvin).
- The Boltzmann Equation: This is the equation which uses Boltzmann's constant in order to calculate. Although Boltzmann had not personally derived the equation which bears his name (actually, there are at least three equations named after him, but this is surely the most famous) it was nevertheless inscribed on his tomb in Vienna: S = k ln W (where S is the entropy of a given system, k is Boltzmann's constant, and W is the number of variables required to define a system).
The Boltzmann Legacy
Clearly Boltzmann's work had a great impact in physics, demonstrating just how accurately mathematics could be used to model real-world behaviors of even things so tiny that they cannot possibly be seen with the most powerful of microscopes.
By recognizing that there are certain things which are simply beyond the capacity of the human mind to fully measure (such as the motions of billions of individual, constantly colliding and changing, particles within a sample of gas), Ludwig Boltzmann was able to provide a mathematical foundation which finally allowed the world of physics in the 19th century to collide with the world of chemistry. The motions of particles could not be understood individually, as little individual units obeying the laws of Newton, but they could still be explained in a real, physical way which led to highly accurate predictions and explanations of a great many phenomena.
These statistical methods continue to thrive and grow in importance today, even though the classical mechanics of the 19th century have been so thoroughly replaced by the quantum mechanics of the 20th and 21st centuries. Boltzmann, though unknown by so many today, built with his brilliant mathematical mind a lasting legacy in the world of math and science.
References:
Boltzmann Biography. J.J. O'Conner and E.F. Robertson. www-groups.dcs.st-and.ac.uk/~history/Biographies/Boltzmann.html
The Boltzmann Equation. Eric Weisstein's World of Physics. scienceworld.wolfram.com/physics/BoltzmannEquation.html
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